The European Physical Journal C

, 74:2981

QCD and strongly coupled gauge theories: challenges and perspectives

  • N. Brambilla
  • S. Eidelman
  • P. Foka
  • S. Gardner
  • A. S. Kronfeld
  • M. G. Alford
  • R. Alkofer
  • M. Butenschoen
  • T. D. Cohen
  • J. Erdmenger
  • L. Fabbietti
  • M. Faber
  • J. L. Goity
  • B. Ketzer
  • H. W. Lin
  • F. J. Llanes-Estrada
  • H. B. Meyer
  • P. Pakhlov
  • E. Pallante
  • M. I. Polikarpov
  • H. Sazdjian
  • A. Schmitt
  • W. M. Snow
  • A. Vairo
  • R. Vogt
  • A. Vuorinen
  • H. Wittig
  • P. Arnold
  • P. Christakoglou
  • P. Di Nezza
  • Z. Fodor
  • X. Garcia i Tormo
  • R. Höllwieser
  • M. A. Janik
  • A. Kalweit
  • D. Keane
  • E. Kiritsis
  • A. Mischke
  • R. Mizuk
  • G. Odyniec
  • K. Papadodimas
  • A. Pich
  • R. Pittau
  • J.-W. Qiu
  • G. Ricciardi
  • C. A. Salgado
  • K. Schwenzer
  • N. G. Stefanis
  • G. M. von Hippel
  • V. I. Zakharov
Open Access
Review

DOI: 10.1140/epjc/s10052-014-2981-5

Cite this article as:
Brambilla, N., Eidelman, S., Foka, P. et al. Eur. Phys. J. C (2014) 74: 2981. doi:10.1140/epjc/s10052-014-2981-5
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Abstract

We highlight the progress, current status, and open challenges of QCD-driven physics, in theory and in experiment. We discuss how the strong interaction is intimately connected to a broad sweep of physical problems, in settings ranging from astrophysics and cosmology to strongly coupled, complex systems in particle and condensed-matter physics, as well as to searches for physics beyond the Standard Model. We also discuss how success in describing the strong interaction impacts other fields, and, in turn, how such subjects can impact studies of the strong interaction. In the course of the work we offer a perspective on the many research streams which flow into and out of QCD, as well as a vision for future developments.

1 Overview

1This document highlights the status and challenges of strong-interaction physics at the beginning of a new era initiated by the discovery of the Higgs particle at the Large Hadron Collider at CERN. It has been a concerted undertaking by many contributing authors, with a smaller group of conveners and editors to coordinate the effort. Together, we have sought to address a common set of questions: What are the latest achievements and highlights related to the strong interaction? What important open problems remain? What are the most promising avenues for further investigation? What do experiments need from theory? What does theory need from experiments? In addressing these questions, we aim to cast the challenges in quantum chromodynamics (QCD) and other strongly coupled physics in a way that spurs future developments.

A core portion of the scientific work discussed in this document was nurtured in the framework of the conference series on “Quark Confinement and the Hadron Spectrum,” which has served over the years as a discussion forum for people working in the field. The starting point of the current enterprise can be traced to its Xth edition (http://www.confx.de), held in Munich in October, 2012. Nearly 400 participants engaged in lively discussions spurred by its seven topical sessions. These discussions inspired the chapters that follow, and their organization is loosely connected to the topical sessions of the conference: Light Quarks; Heavy Quarks; QCD and New Physics; Deconfinement; Nuclear and Astroparticle Physics; Vacuum Structure and Confinement; and Strongly Coupled Theories. This document is an original, focused work that summarizes the current status of QCD, broadly interpreted, and provides a vision for future developments and further research. The document’s wide-angle, high-resolution picture of the field is current through March 15, 2014.

1.1 Readers’ guide

We expect that this work will attract a broad readership, ranging from practitioners in one or more subfields of QCD, to particle or nuclear physicists working in fields other than QCD and the Standard Model (SM), to students starting research in QCD or elsewhere. We should note that the scope of QCD is so vast that it is impossible to cover absolutely everything. Any omissions stem from the need to create something useful despite the numerous, and sometimes rapid, advances in QCD research. To help the reader navigate the rest of the document, let us begin with a brief guide to the contents of and rationale for each chapter.

Section 2 is aimed at all readers and explains the aims of this undertaking in more detail by focusing on properties and characteristics that render QCD a unique part of the SM. We also highlight the broad array of problems for which the study of QCD is pertinent before turning to a description of the experiments and theoretical tools that appear throughout the remaining chapters. Section 2 concludes with a status report on the determination of the fundamental parameters of QCD, namely, the gauge coupling \({\alpha _{\mathrm{s}}}\) and the quark masses.

The wish to understand the properties of the lightest hadrons with the quark model, concomitant with the observation of partons in deep-inelastic electron scattering, sparked the emergence of QCD. We thus begin in Sect. 3 with this physics, discussing not only the current status of the parton distribution functions, but also delving into many aspects of the structure and dynamics of light-quark hadrons at low energies. Section 3 also reviews the hadron spectrum, including exotic states beyond the quark model, such as glueballs, as well as chiral dynamics, probed through low-energy observables. Certain new-physics searches for which control over light-quark dynamics is essential are also described.

Heavy-quark systems have played a crucial role in the development of the SM, QCD especially. Their large mass, compared to the QCD scale, leads to clean experimental signatures and opens up a new theoretical toolkit. Section 4 surveys these theoretical tools in systems such as quarkonium, i.e., bound states of a heavy quark and a heavy antiquark, and hadrons consisting of a heavy quark bound to light degrees of freedom. Highlights of the chapter include an up-to-date presentation of the exotic states \(X\), \(Y\), \(Z\) that have been discovered in the charmonium and bottomonium regions, the state of the art of lattice-QCD calculations, and an extended discussion of the status of our theoretical understanding of quarkonium production at hadron and electron colliders. The latest results for \(B\)- and \(D\)-meson semileptonic decays, which are used to determine some SM parameters and to look for signs of new physics, are also discussed.

Control of QCD for both heavy and light quarks, and for gluons as well, is the key to many searches for physics beyond the SM. Section 5 reviews the possibilities and challenges of the searches realized through precision measurements, both at high energy through collider experiments and at low energy through accelerator, reactor, and table-top experiments. In many searches, a comparably precise theoretical calculation is required to separate SM from non-SM effects, and these are reviewed as well. This chapter has an extremely broad scope, ranging from experiments with multi-TeV \(pp\) collisions to those with ultracold neutrons and atoms; ranging from top-quark physics to the determinations of the weak-mixing angle at low energies; ranging from searches for new phenomena in quark-flavor violation to searches for permanent electric dipole moments.

In Sect. 5, QCD is a tool to aid the discovery of exotic phenomena external to QCD. The next three chapters treat a rich array of as-yet unexplored phenomena that emerge from QCD in complex, many-hadron systems. Section 6 begins this theme with a discussion of deconfinement in the context of the quark–gluon plasma and heavy-ion collisions. We first give a description of this novel kind of matter and of our present knowledge of the QCD phase diagram, based on the most recent measurements. We then turn to describing near-equilibrium properties of the quark–gluon plasma and its approach to equilibrium. We explain theorists’ present understanding, focusing on ideas and techniques that are directly connected to QCD. Hard probes such as jet quenching and quarkonium suppression as methods to scrutinize the quark–gluon plasma properties are also discussed. The chapter ends with a parallel between thermal field theory calculations in QCD and cosmology and with a note on the chiral magnetic effect.

Section 7 covers cold, dense hadronic systems, including nuclear and hypernuclear physics and also the ultra-dense hadronic matter found in neutron stars, noting also the new phases that are expected to appear at even higher densities. These topics are informed not only by theory and terrestrial experiments but also by astrophysical observations.

At this point the reader finds Sect. 8, which focuses on the biggest question in QCD: the nature of confinement. No experiment has detected a colored object in isolation, suggesting that colored objects are trapped inside color-singlet hadrons. Section 8 focuses on theoretical aspects of confinement and the related phenomenon of chiral-symmetry breaking, and how they arise in non-Abelian gauge theories.

QCD provides a loose prototype of strongly coupled theories, which are reviewed in Sect. 9. Supersymmetry, string theory, and the AdS/CFT correspondence all play a role in this chapter. These ideas modify the dynamics of gauge theories profoundly. Non-supersymmetric theories are also described here, though they are most interesting when the fermion content is such that the dynamics differ markedly from those of QCD, because they then are candidate models of electroweak symmetry breaking. Conformal symmetry is also presented here, both to help understand the phase diagram of non-Abelian gauge theories and to develop additional models of new physics. New exact results in field theories, sometimes inspired by string theory, are put forward, and their connection to computations of scattering amplitudes in QCD, with many legs or at many loops, is discussed. Section 9 further discusses techniques devised for strongly coupled particle physics and their interplay with condensed-matter physics.

Sections 39 all contain a section on future directions discussing the most important open problems and challenges, as well as the most interesting avenues for further research. The Appendix provides a list of acronyms explaining the meaning of abbreviations used throughout the review for laboratories, accelerators, and scientific collaborations. Where available, we provide links to web sites with more information.

2 The nature of QCD

2QCD is the sector of the Standard Model (SM) of particle physics that describes the strong interactions of quarks and gluons. From a modern perspective, both the SM and general relativity are thought to be effective field theories, describing the low-energy limit of a more fundamental framework emergent at high energies. To begin, we would like to focus on one specific theoretical aspect, because it shows how QCD plays a special role in the SM.

In quantum field theory, couplings are best understood as depending on an energy scale; roughly speaking, this is the scale at which the quantum field theory—understood to be an effective field theory—is defined. In some cases, such as that of the hypercharge coupling or the Higgs self-coupling in the SM, this energy dependence is such that the coupling increases with increasing energy. This behavior predicts the failure of the theory at the shortest distance scales. QCD, on the other hand, is asymptotically free, which means the following. The QCD Lagrangian in the zero-quark-mass limit is scale invariant, and the interactions of the quarks are determined by the dimensionless parameter \({\alpha _{\mathrm{s}}}\). The theory at the quantum (loop) level generates a fundamental, dimensionful scale \(\Lambda _\mathrm{QCD}\) which controls the variation of the coupling constant \({\alpha _{\mathrm{s}}}\) with energy scale. In QCD (unlike QED), the coupling decreases with increasing energy—as spectacularly confirmed in the kinematic variation of cross-section measurements from high-precision, deep-inelastic scattering data. The decrease is just fast enough that QCD retains its self-consistency in all extreme energy regimes: high center-of-mass scattering energies, of course, but also high temperatures and large baryon chemical potentials, etc. In this way, QCD is the paradigm of a complete physical theory.

Asymptotic freedom allows accurate calculations at high energy with perturbation theory. The success of the technique does not remove the challenge of understanding the non-perturbative aspects of the theory. The two aspects are deeply intertwined. The Lagrangian of QCD is written in terms of quark and gluon degrees of freedom which become apparent at large energy but remain hidden inside hadrons in the low-energy regime. This confinement property is related to the increase of \(\alpha _\mathrm{s}\) at low energy, but it has never been demonstrated analytically. We have clear indications of the confinement of quarks into hadrons from both experiments and lattice QCD. Computations of the heavy quark–antiquark potential, for example, display a linear behavior in the quark–antiquark distance, which cannot be obtained in pure perturbation theory. Indeed the two main characteristics of QCD: confinement and the appearance of nearly massless pseudoscalar mesons, emergent from the spontaneous breaking of chiral symmetry, are non-perturbative phenomena whose precise understanding continues to be a target of research. Even in the simpler case of gluodynamics in the absence of quarks, we do not have a precise understanding of how a gap in the spectrum is formed and the glueball spectrum is generated. Glueball states are predictions of QCD, and their mass spectrum can be obtained with lattice-QCD calculations. They have not, however, been unambiguously observed; their predicted mass and width can be significantly modified by \(q\bar{q}\) mixing effects.

The vacuum of QCD is also difficult to characterize. One possibility is to characterize the vacuum in terms of several non-perturbative objects. Such a parameterization has been introduced first in the sum rules approach, yielding a separation of short- and long-distance physics based on techniques derived from the existence of asymptotic freedom in QCD. These ideas have proven to be of profound importance, though the specifics have been supplanted, broadly speaking, by effective field theories in QCD, which, as discussed further in Sect. 2.3, systematically separate the high- and low-energy contributions.

Once a low-energy (non-perturbative), gauge-invariant quantity has been defined, one could use it to investigate the low-energy degrees of freedom which could characterize it and their relation to the confinement mechanism. Even in the absence of quarks, there is a fascinating and complex landscape of different possible topological objects: monopoles, vortices, calorons, or dyons, which are investigated using different methods; either lattice-QCD calculations or QCD vacuum models can be used to this end. Some of the recent research in this sector is addressed in Sect. 8.

2.1 Broader themes in QCD

Many of the most influential ideas in field theory have emerged while trying to understand QCD. The renormalization-group methods of Kenneth Wilson, where short-distance degrees of freedom are systematically removed, or “integrated out,” began with attempts to understand the scale invariance of the strong interaction. These ideas flourished in critical phenomena and statistical mechanics, before returning to particle physics after the asymptotic freedom of gauge theories was discovered. It is this view of renormalization that provides QCD the high-energy self-consistency we have discussed, and has also led to one of the two key facets of modern effective field theory. The other key lies in the work of Steven Weinberg, who argued on the grounds of unitarity and analyticity that the correct effective Lagrangian would consist of all the operators with the desired fields and symmetries. This idea is crucial to the analysis of QCD, because it allows the introduction of an effective theory whose fields differ from the original ones. For example, the chiral Lagrangian contains pions and, depending on the context, other hadron fields, but not quarks and gluons. Certainly, QCD has been at the heart of the development of most of our tools and ideas in the construction of the Standard Model.

QCD also has a distinguished pedigree as a description of experimental observations. It is a merger of two insightful ideas, the quark model and the parton model, which were introduced to explain, respectively, the discovery of the hadron “zoo” in the 1960s and then the deep-inelastic scattering events seen in the early 1970s. The acceptance of QCD was forced on us by several discoveries, such as the \(J/\psi \) and other charmonium states in 1974, the analogous \(\Upsilon \) and bottomonium states in 1977, and the first observation of three-jet events, evoking the gluon, in 1979.

Some themes in QCD recur often enough that they appear in many of the chapters to follow, so we list them here:

QCD gives rise to the visible mass of the Universe, including everyday objects—the confinement scale, \(\Lambda _\mathrm{QCD}\), sets the mass of the proton and the neutron. Similar dynamics could, conceivably, play a role in generating the mass of other forms of matter. Thus, the confinement mechanism pertains to the origin of mass.

QCD controls many parameters of the SM—QCD is needed to determine \({\alpha _{\mathrm{s}}}\), the six masses of the quarks, and the strong CP-violating parameter, as well as the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix. These tally to 12 parameters, out of the 19 of the SM (or 26–28 with neutrino masses and mixing). The quark masses and CKM parameters stem from, and the strong-CP parameter is connected to, the poorly understood Yukawa couplings of quarks to the Higgs boson; furthermore, \({\alpha _{\mathrm{s}}}\) may unify with the other gauge couplings. Thus, quark couplings play a direct role in the search for a more fundamental theory.

QCD describes the SM background to non-SM physics—in the high-energy regime, where the coupling constant is small and perturbation theory is applicable, QCD predicts the calculable background to new phenomena precisely. For example, QCD calculations of the background were instrumental to the Higgs discovery, and, indeed, QCD is ubiquitous at hadron colliders where direct contributions of new physics are most actively sought. Thus, QCD plays a fundamental role in our investigations at the high-energy frontier.

In the low-energy regime, QCD is often the limiting factor in the indirect search for non-SM physics—this is true in all searches for new physics in hadronic systems, be it in the study of CP violation in \(B\) decays, or in permanent electric dipole moment searches in hadrons and nuclei. In addition, QCD calculations of hadronic effects are also needed to understand the anomalous magnetic moment of the muon, as well as aspects of neutrino physics. Thus, QCD also plays a fundamental role in searches for new physics at the intensity frontier.

Nuclear matter has a fascinating phase diagram—at non-zero temperature and non-zero chemical potential, QCD exhibits a rich phase diagram, which we continue to explore. The QCD equation of state, the possibility of phase transitions and/or crossovers, and the experimental search for the existence of a critical point are all current topics of research. In lattice QCD one can also alter the number of fermions and the number of colors in order to study different scenarios. In addition to the hadronic phase, different states of QCD matter are predicted, such as the quark–gluon plasma, quarkyonic matter, and a color superconductor state with diquark matter. Experiments studying heavy-ion collisions have shown the quark–gluon plasma to be a surprising substance. For example, it seems to be a strongly coupled, nearly perfect liquid with a minimal ratio of shear viscosity to entropy density. Thus, QCD matter in extreme conditions exhibits rich and sometimes unexpected behavior.

QCD impacts cosmology—probing the region of the QCD phase diagrams at large temperature allows us to probe conditions which have not existed since the beginning of the universe. The new state of matter formed in heavy-ion collisions existed microseconds after the Big Bang, before hadrons emerged as the universe cooled. Thus, characterizing the quark–gluon plasma provides information about the early universe.

QCD is needed for astrophysics—the region of the QCD phase diagram at large chemical potential provides information on the system under conditions of high pressure and large density, as is the case for astrophysical objects such as compact stars. These stars could be neutron stars, quark stars, or hybrids somewhere in between these pure limits. Moreover, one can use astrometric observational data on such objects to help characterize the QCD equation of state. Thus, terrestrial accelerator experiments and astrophysical observations are deeply connected.

QCD is a prototype of strongly coupled theories—strongly coupled gauge theories have been proposed as alternatives to the SM Higgs mechanism. Strongly coupled mechanisms may also underlie new sectors of particle physics that could explain the origin of dark matter. Furthermore, the relation between gauge theories and string theories could shed light on the unification of forces. Thus, QCD provides a launching pad for new models of particle physics.

QCD inspires new computational techniques for strongly interacting systems—as the prototype of an extremely rich, strongly coupled system, the study of QCD requires a variety of analytical tools and computational techniques, without which progress would halt. These developments fertilize new work in allied fields; for example, QCD methods have helped elucidate the universal properties of ultracold atoms. Conversely, developments in other fields may shed light on QCD itself. For example, the possibility of designing arrays of cold atoms in optical lattices with the gauge symmetry and fermion content of QCD is under development. If successful, this work could yield a kind of quantum computer with QCD as its specific application. Thus, the challenge of QCD cross-fertilizes other fields of science.

2.2 Experiments addressing QCD

In this section, we offer a brief overview of the experimental tools of QCD. We discuss \(e^{+} e^{-}\) colliders, fixed-target machines, hadron colliders, and relativistic heavy-ion colliders from a QCD perspective.

From the 1960s to 1990s, \(e^+e^{-}\) colliders evolved from low center-of-mass energies \(\sqrt{s}\sim 1\) GeV with modest luminosity to the Large Electron Positron (LEP) collider with \(\sqrt{s}\) up to \(209\) GeV and a vastly greater luminosity. Along the way, the \(e^+e^{-}\) colliders PETRA (at DESY) and PEP (at SLAC) saw the first three-jet events. A further breakthrough happened at the end of 1990s with the advent of the two \(B\)-factories at KEK and SLAC and the operation of lower-energy, high-intensity colliders in Beijing, Cornell, Frascati, and Novosibirsk. Experiments at these machines are particularly good for studies of quarkonium physics and decays of open charm and bottom mesons, in a way that spurred theoretical developments. The copious production of \(\tau \) leptons at \(e^+e^{-}\) colliders led to a way to measure \({\alpha _{\mathrm{s}}}\) via their hadronic decays. Measurements of the hadronic cross section at various energy ranges play a useful role in understanding the interplay of QCD and QED.

Experiments with electron, muon, neutrino, photon, or hadron beams impinging on a fixed target have been a cornerstone of QCD. Early studies of deep inelastic scattering at SLAC led to the parton model. This technique and the complementary production of charged lepton pairs (the so-called Drell–Yan production) have remained an important tool for understanding proton structure. Later, the Hadron–Elektron Ring Anlage (HERA) continued this theme with \(e^{-}p\) and \(e^+p\) colliding beams. In addition to nucleon structure, fixed-target experiments have made significant contributions to strangeness and charm physics, as well as to the spectroscopy of light mesons, and HERA searched for non-SM particles such as leptoquarks. This line of research continues to this day at Jefferson Lab, J-PARC, Mainz, Fermilab, and CERN; future, post-HERA \(ep\) colliders are under discussion.

The history of hadron colliders started in 1971 with \(pp\) collisions at CERN’s Intersecting Storage Rings (ISR), at a center-of-mass energy of 30 GeV. The ISR ran for more than 10 years with \(pp\) and \(p\bar{p}\) collisions, as well as with ion beams: \(pd\), \(dd\), \(p\alpha \), and \(\alpha \alpha \). During this time, its luminosity increased by three orders of magnitude. This machine paved the way for the successful operation of proton–antiproton colliders: the S\(p\bar{p}\)S at CERN with \(\sqrt{s}=630\) GeV in the 1980s, and the \(p\bar{p}\) Tevatron at Fermilab with \(\sqrt{s}=1.96\) TeV, which ran until 2011. Currently, the Large Hadron Collider (LHC) collides \(pp\) beams at the highest energies in history, with a design energy of 14 TeV and luminosity four orders of magnitude higher than the ISR. Physics at these machines started from studies of jets at the ISR and moved to diverse investigations including proton structure, precise measurements of the \(W\) mass, searches for heavy fundamental particles leading to discoveries of the top quark and Higgs, production of quarkonia, and flavor physics.

At the same time, pioneering experiments with light ions (atomic number, \(A\), around 14) at relativistic energies started in the 1970s at LBNL in the United States and at JINR in Russia. The program continued in the 1980s with fixed-target programs at the CERN SPS and BNL AGS. These first experiments employed light-ion beams (\(A \sim 30\)) on heavy targets (\(A \sim 200\)). In the 1990s, the search for the quark–gluon plasma continued with truly heavy-ion beams (\(A \sim 200\)). In this era, the maximum center of mass energy per nucleon was \(\sqrt{s_{NN}} \sim 20\) GeV. With the new millennium the heavy-ion field entered the collider era, first with the Relativistic Heavy-Ion Collider (RHIC) at BNL at \(\sqrt{s_{NN}}=200\) GeV and, in 2010, the LHC at CERN, reaching the highest currently available energy, \(\sqrt{s_{NN}}=2.76\) TeV.

The goal of heavy-ion physics is to map out the nuclear-matter phase diagram, analogous to studies of phase transitions in other fields. Proton-proton collisions occur at zero temperature and baryon density, while heavy-ion collisions can quantify the state of matter of bulk macroscopic systems. The early fixed-target experiments probed moderate values of temperature and baryon density. The current collider experiments reach the zero baryon density, high-temperature regime, where the quark–gluon plasma can be studied under conditions that arose in the early universe.

While the phase structure observed in collider experiments suggests a smooth crossover from hadronic matter to the quark–gluon plasma, theoretical arguments, augmented by lattice QCD computations, suggest a first-order phase transition at non-zero baryon density. The critical point where the line of first-order transitions ends and the crossover regime begins is of great interest. To reach the needed temperature and baryon density, two new facilities—FAIR at GSI and NICA at JINR—are being built.

Work at all these facilities, from \(e^+e^{-}\) machines to heavy-ion colliders, require the development of novel trigger systems and detector technologies. The sophisticated detectors used in these experiments, coupled to farms of computers for on-line data analysis, permit the study of unprecedentedly enormous data samples, thus enabling greater sensitivity in searches for rare processes.

2.3 Theoretical tools for QCD

The theory toolkit to study QCD matter is quite diverse, as befits the rich set of phenomena it describes. It includes QCD perturbation theory in the vacuum, semiclassical gauge theory, and techniques derived from string theory. Here we provide a brief outline of some of the wider ranging techniques.

a. Effective Field Theories (EFTs): Effective field theories are important tools in modern quantum field theory. They grew out of the operator-product expansion (OPE) and the formalism of phenomenological Lagrangians and, thus, provide a standard way to analyze physical systems with many different energy scales. Such systems are very common from the high-energy domain of particle physics beyond the Standard Model to the low-energy domain of nuclear physics.

Crucial to the construction of an EFT is the notion of factorization, whereby the effects in a physical system are separated into a high-energy factor and a low-energy factor, with each factor susceptible to calculation by different techniques. The high-energy factor is typically calculated by making use of powerful analytic techniques, such as weak-coupling perturbation theory and the renormalization group, while the low-energy part may be amenable to lattice gauge theory or phenomenological methods. A key concept in factorization is the principle of universality, on the basis of which a low-energy factor can be determined from one theoretical or phenomenological calculation and can then be applied in a model-independent way to a number of different processes. Factorization appeared first in applications of the OPE to QCD, where a classification of operators revealed a leading (set of) operator(s), whose short-distance coefficients could be calculated in a power series in \({\alpha _{\mathrm{s}}}\).

Apart from their theoretical appeal, EFTs are an extremely practical tool. In many cases they allow one to obtain formally consistent and numerically reliable predictions for physical processes that are of direct relevance for experiments. The essential role of factorization was realized early on in the analysis of deep inelastic scattering data in QCD and is codified in the determination of parton distribution functions from experiment, allowing SM predictions in new energy regimes.

Several properties of EFTs are important: they have a power counting in a small parameter which permits rudimentary error assessment for each prediction; they can be more predictive if they have more symmetry; they admit an appropriate definition of physical quantities and supply a systematic calculational framework; finally, they permit the resummation of large logarithms in the ratio of physical scales. For example, an object of great interest, investigated since the inception of QCD, is the heavy quark–antiquark static energy, which can be properly defined only in an EFT and subsequently calculated with lattice gauge theory.

The oldest example is chiral EFT for light-quark systems, with roots stemming from the development of current algebra in the 1960s. Chiral EFT has supplied us with an increasingly accurate description of mesons and baryons, and it is an essential ingredient in flavor-physics studies. The EFT description of pion–pion scattering, together with the data on pionium formation, has given us a precise way to confirm the standard mechanism of spontaneous breaking of chiral symmetry in QCD. Chiral effective theory has also allowed lattice QCD to make contact with the physical region of light-quark masses from simulations with computationally less demanding quark masses. For more details, see Sects. 3 and 5.

In the case of the heavy quark–antiquark bound states known as quarkonium, the EFT known as Non-relativistic QCD (NRQCD) separates physics at the scale of the heavy-quark mass from those related to the dynamics of quarkonium binding. This separation has solved the problem of uncontrolled infrared divergences in theoretical calculations and has opened the door to a systematic improvement of theoretical predictions. It has given us the tools to understand the data on the quarkonium production cross section at high-energy colliders, such as the Tevatron, the \(B\) factories, and the LHC. It has also made it clear that a complete understanding of quarkonium production and decay involves processes in which the quark–antiquark pairs are in a color-octet state, as well as processes in which the pairs are in a color-singlet state. New, lower-energy EFTs, such as potential NRQCD (pNRQCD) have given greater control over some technical aspects of theoretical calculations and have provided a detailed description of the spectrum, decays, and transitions of heavy quarkonia. These EFTs allow the precise extraction of the Standard Model parameters, which are relevant for new-physics searches, from the data of current and future experiments. See Sects. 3 and 4 for applications of NRQCD and pNRQCD.

In the case of strong-interaction processes that involve large momentum transfers and energetic, nearly massless particles, Soft Collinear Effective Field Theory (SCET) has been developed. It has clarified issues of factorization for high-energy processes and has proved to be a powerful tool for resumming large logarithms. SCET has produced applications over a wide range of topics, including heavy-meson decays, deep-inelastic scattering, exclusive reactions, quarkonium-production processes, jet event shapes, and jet quenching. Recent developments regarding these applications can be found in Sects. 3, 4, and 5.

In  finite-temperature and  finite-density physics, EFTs such as Hard Thermal Loop (HTL), Electric QCD, Magnetic QCD, \(\mathrm{NRQCD }_\mathrm{HTL }\), or p\(\mathrm{NRQCD }_\mathrm{HTL }\) have allowed progress on problems that are not accessible to standard lattice QCD, such as the evolution of heavy quarkonia in a hot medium, thermodynamical properties of QCD at the very high temperatures, the thermalization rate of non-equilibrium configurations generated in heavy-ion collision experiments, and the regime of asymptotic density. These developments are discussed in Sect. 6.

In nuclear physics, chiral perturbation theory has been generalized to provide a QCD foundation to nuclear structure and reactions. EFTs have allowed, among other things, a model-independent description of hadronic and nuclear interactions in terms of parameters that will eventually be determined in lattice calculations, new solutions of few-nucleon systems that show universality and striking similarities to atomic systems near Feshbach resonances, derivation of consistent currents for nuclear reactions, and new approaches to understanding heavier nuclei (such as halo systems) and nuclear matter. Some recent developments are discussed in Sect. 7.

b. Lattice gauge theory: In the past decade, numerical lattice QCD has made enormous strides. Computing power, combined with new algorithms, has allowed a systematic simulation of sea quarks for the first time. The most recently generated ensembles of lattice gauge fields now have 2+1+1 flavors of sea quark, corresponding to the up and down, strange, and charm quarks. Most of this work uses chiral EFT to guide an extrapolation of the lightest two quark masses to the physical values. In some ensembles, however, the (averaged) up and down mass is now as light as in nature, obviating this step. Many quantities now have sub-percent uncertainties, so that the next step will require electromagnetism and isospin breaking (in the sea).

Some of the highlights include baryon masses with errors of 2–4 %; pion, kaon, and \(D\)-meson matrix elements with total uncertainty of 1–2 %; \(B\)-meson matrix elements to within 5–8 %. The light quark masses are now known directly from QCD (with the chiral extrapolation), with few per cent errors. Several of the best determinations of \({\alpha _{\mathrm{s}}}\) combine perturbation theory (lattice or continuum) with non-perturbatively computed quantities; these are so precise because the key input from experiment is just the scale, upon which \({\alpha _{\mathrm{s}}}\) depends logarithmically. A similar set of analyses yield the charm- and bottom-quark masses with accuracy comparable to perturbative QCD plus experiment. Lattice QCD has also yielded a wealth of thermodynamic properties, not least showing that the deconfinement transition (at small chemical potential) is a crossover, and the crossover temperature has now been found reproducibly.

Vigorous research, both theoretical and computational, is extending the reach of this tool into more demanding areas. The computer calculations take place in a finite spatial box (because computers’ memories are finite), and two-body states require special care. In the elastic case of \(K\rightarrow \pi \pi \) transitions, the required extra computing is now manageable, and long-sought calculations of direct CP violation among neutral kaons, and related decay rates, now appear on the horizon. This success has spurred theoretical work on inelastic, multi-body kinematics, which will be required before long-distance contributions to, say, \(D\)-meson mixing can be computed. Nonleptonic \(B\) and \(D\) decays will also need these advances, and possibly more. In the realm of QCD thermodynamics, the phase diagram at non-zero chemical potential suffers from a fermion sign problem, exactly as in many condensed-matter problems. This problem is difficult, and several new ideas for workarounds and algorithms are being investigated.

c. Other non-perturbative approaches: The theoretical evaluation of a non-perturbative contribution arising in QCD requires non-perturbative techniques. In addition to lattice QCD, many models and techniques have been developed to this end. Among the most used techniques are: the limit of the large number of colors, generalizations of the original Shifman–Vainshtein–Zakharov sum rules, QCD vacuum models and effective string models, the AdS/CFT conjecture, and Schwinger–Dyson equations. Every chapter reports many results obtained with these alternative techniques.

2.4 Fundamental parameters of QCD

Precise determinations of the quark masses and of \({\alpha _{\mathrm{s}}}\) are crucial for many of the problems discussed in the chapters to come. As fundamental parameters of a physical theory, they require both experimental and theoretical input. Because experiments detect hadrons, inside which quarks and gluons are confined, the parameters cannot be directly measured. Instead, they must be determined from a set of relations of the form
$$\begin{aligned} \,[M_\mathrm{HAD}(\Lambda _\mathrm{QCD}, m_q)]^\mathrm{TH}=[M_\mathrm{HAD}]^\mathrm{EXP}. \end{aligned}$$
(2.1)
One such relation is needed to determine \(\Lambda _\mathrm{QCD}\), the parameter which fixes the value of \({\alpha _{\mathrm{s}}}(Q^2)\), the running coupling constant, at a squared energy scale \(Q^2\); another six are needed for the (known) quarks—and yet another for the CP-violating angle \(\bar{\theta }\). The quark masses and \({\alpha _{\mathrm{s}}}\) depend on the renormalization scheme and scale, so that care is needed to ensure that a consistent set of definitions is used. Some technical aspects of these definitions (such as the one known as the renormalon ambiguity) are continuing objects of theoretical research and can set practical limitations on our ability to determine the fundamental parameters of the theory. In what follows, we have the running coupling and running masses in mind.

Measurements of \({\alpha _{\mathrm{s}}}\) at different energy scales provide a direct quantitative verification of asymptotic freedom in QCD. From the high-energy measurement of the hadronic width of the \(Z\) boson, one obtains \({\alpha _{\mathrm{s}}}(M_Z)=0.1197\pm 0.0028\) [1]. From the lower-energy measurement of the hadronic branching fraction of the \(\tau \) lepton, one obtains, after running to the \(Z\) mass, \({\alpha _{\mathrm{s}}}(M_Z^2)=0.1197\pm 0.0016\) [1]. At intermediate energies, several analyses of quarkonium yield values of \({\alpha _{\mathrm{s}}}\) in agreement with these two; see Sect. 4.4. The scale of the \(\tau \) mass is low enough that the error assigned to the latter value remains under discussion; see Sect. 3.5.3 for details. Whatever one makes of these issues, the agreement between these two determinations provides an undeniable experimental verification of the asymptotic freedom property of QCD.

One can combine \({\alpha _{\mathrm{s}}}\) extractions from different systems to try to obtain a precise and reliable “world-average” value. At present most (but not all) individual \({\alpha _{\mathrm{s}}}\) measurements are dominated by systematic uncertainties of theoretical origin, and, therefore, any such averaging is somewhat subjective. Several other physical systems, beyond those mentioned above, are suitable to determine \({\alpha _{\mathrm{s}}}\). Those involving heavy quarks are discussed in Sect. 4.4. Lattice QCD provides several different \({\alpha _{\mathrm{s}}}\) determinations. Recent ones include [2, 3, 4, 5], in addition to those mentioned in Sect. 4.4. Some of these determinations quote small errors, because the non-perturbative part is handled cleanly. They therefore may have quite an impact in world-averages, depending on how those are done. For example, lattice determinations dominate the error of the current PDG world average [1]. Fits of parton-distribution functions (PDFs) to collider data also provide a good way to determine \({\alpha _{\mathrm{s}}}\). Current analyses involve sets of PDFs determined in next-to-next-to-leading order (NNLO) [6, 7, 8, 9]. Effects from unknown higher-order perturbative corrections in those fits are difficult to assess, however, and have not been addressed in detail so far. They are typically estimated to be slightly larger than the assigned uncertainties of the NNLO extractions. Jet rates and event-shape observables in \(e^+e^{-}\) collisions can also provide good sensitivity to \({\alpha _{\mathrm{s}}}\). Current state-of-the-art analyses involve NNLO fixed-order predictions [10, 11, 12, 13, 14, 15, 16, 17], combined with the resummation of logarithmically enhanced terms. Resummation for the event-shape cross sections has been performed both in the traditional diagrammatic approach [18] and within the SCET framework [19, 20, 21]. One complication with those extractions is the precise treatment of hadronization effects. It is by now clear [22] that analyses that use Monte Carlo generators to estimate them [19, 20, 22, 23, 24] tend to obtain larger values of \({\alpha _{\mathrm{s}}}\) than those that incorporate power corrections analytically [25, 26, 27, 28, 29]. Moreover, it may not be appropriate to use Monte Carlo hadronization with higher-order resummed predictions [25, 26, 27]. We also mention that analyses employing jet rates may be less sensitive to hadronization corrections [30, 31, 32, 33]. The SCET-based results of Refs. [26, 28] quote remarkably small errors; one might wonder if the systematics of the procedure have been properly assessed, since the extractions are based only on thrust. In that sense, we mention analogous analyses that employ heavy-jet mass, the \(C\)-parameter, and broadening are within reach and may appear in the near future. Note that if one were to exclude the event-shape \({\alpha _{\mathrm{s}}}\) extractions that employ Monte Carlo hadronization, the impact on the PDG average could be quite significant. Related analyses employing deep-inelastic scattering data can also be performed [34].

Light-quark masses are small enough that they do not have a significant impact on typical hadronic quantities. Nevertheless, the observed masses of the light, pseudoscalar mesons, which would vanish in the zero-quark-mass limit, are sensitive to them. Moreover, various technical methods are available in which to relate the quark and hadron masses in this case. We refer to Sects. 3.4.2 and 3.4.3 for discussions of the determination of the light-quark masses from lattice QCD and from chiral perturbation theory. To determine light-quark masses, one can take advantage of chiral perturbation theory, lattice-QCD computations, and QCD sum rule methods. Current progress in the light-quark mass determinations is largely driven by improvements in lattice QCD.

Earlier lattice simulations use \(N_\mathrm{f}=2\) flavors of sea quark (recent results include Refs. [35, 36]), while present ones use \(N_\mathrm{f}=2+1\) (recent results include Refs. [37, 38, 39, 40]). The influence of charmed sea quarks will soon be studied [41, 42]. In addition, some ensembles no longer require chiral extrapolations to reach the physical mass values. The simulations are almost always performed in the isospin limit, \(m_u=m_d{=:}m_{ud}\), \(m_{ud}=(m_u+m_d)/2\), therefore what one can directly obtain from the lattice is \(m_\mathrm{s}\), the average \(m_{ud}\), and their ratio. We mention that there is a new strategy to determine the light-quark masses which consists in computing the ratio \(m_\mathrm{c}/m_\mathrm{s}\), combined with a separate calculation for \(m_\mathrm{c}\), to obtain \(m_\mathrm{s}\) [2, 43]. The advantage of this method is that the issue of lattice renormalization is traded for a continuum renormalization in the determination of \(m_\mathrm{c}\). With additional input regarding isospin-breaking effects, from the lattice results in the isospin limit one can obtain separate values for \(m_u\) and \(m_d\); see Sect. 3.4.2 for additional discussion. With the present results, one obtains that \(m_u\ne 0\), so that the strong-CP problem is not solved by having a massless \(u\) quark [1, 44, 45]; see Sect. 5.7 for further discussion of this issue.

In contrast, heavy-quark masses also affect several processes of interest; for instance, the \(b\)-quark mass enters in the Higgs decay rate for \(H\rightarrow b\bar{b}\). Many studies of Higgs physics do not, however, use the latest, more precise determinations of \(m_b\). The value of the top-quark mass is also necessary for precision electroweak fits. To study heavy-quark masses, \(m_Q\), one can exploit the hierarchy \(m_Q\gg \Lambda _\mathrm{QCD}\) to construct heavy-quark effective theories that simplify the dynamics; and additionally take advantage of high-order, perturbative calculations that are available for these systems; and of progress in lattice-QCD computations. One of the best ways to determine the \(b\) and \(c\) masses is through sum-rule analyses, that compare theoretical predictions for moments of the cross section for heavy-quark production in \(e^+e^{-}\) collisions with experimental data (some analyses that appeared in recent years include [46, 47, 48, 49]) or lattice QCD (e.g., [2]). In those analyses, for the case of \(m_\mathrm{c}\), the approach with lattice QCD gives the most precise determination, and the errors are mainly driven by perturbative uncertainties. For \(m_b\), the approach with \(e^+e^{-}\) data still gives a better determination, but expected lattice-QCD progress in the next few years may bring the lattice determination to a similar level of precision. A complementary way to obtain the \(c\)-quark mass is to exploit DIS charm production measurements in PDF fits [50]. The best measurement of the top-quark mass could be performed at a future \(e^+e^{-}\) collider, but improvements on the mass determination, with respect to the present precision, from LHC measurements are possible.

3 Light quarks

3.1 Introduction

3The study of light-quark physics is central to the understanding of QCD. Light quarks represent a particularly sensitive probe of the strong interactions, especially of non-perturbative effects.

In the two extreme regimes of QCD, namely, in the low-energy regime where the energies are (much) smaller than a typical strong interaction scale \(\sim \)\(m_\rho \), and in the high-energy regime where the energies are much higher than that scale, there are well-established theoretical methods, namely, Chiral Perturbation Theory (ChPT) and perturbative QCD, respectively, that allow for a discussion of the physics in a manner consistent with the fundamental theory, and thus permit in this way to define and quantify effects in a more or less rigorous way. The intermediate-energy regime is less developed as there are no analytic methods that need allow for a complete discussion of the physics, thus requiring the introduction of methods which that need require some degree of modeling. However, as discussed in this chapter, methods based fundamentally on QCD, such as those based on the framework of Schwinger–Dyson equations, have made great advances, and a promising future lies ahead. Advances in lattice QCD, in which the excited hadron spectrum can be analyzed, are opening new perspectives for understanding the intermediate-energy regime of QCD; and one should expect that this will result in new strategies, methods, and ideas. Progress on all of the mentioned fronts continues, and in this chapter a representative number of the most exciting developments are discussed.

Never before has the study of the strong interactions had as many sources of experimental results as today. Laboratories and experiments around the world, ranging from low- to high-energy accelerators, as well as in precision nonaccelerator physics, give unprecedented access to the different aspects of QCD, and to light-quark physics in particular. In this chapter a broad sample of experiments and results from these venues will be given.

The objective of this chapter is to present a selection of topics in light-quark physics: partonic structure of light hadrons, low-energy properties and structure, excited hadrons, the role of light-quark physics in extracting fundamental QCD parameters, such as \(\alpha _\mathrm{s}\) at the GeV scale, and also of theoretical methods, namely, ChPT, perturbative QCD, Schwinger–Dyson equations, and lattice QCD.

This chapter is organized as follows: Sect. 3.2 is devoted to hadron structure and contains the following topics: parton distributions (also including their transverse momentum dependence), hadron form factors, and generalized parton distributions (GPDs), lattice QCD calculations of form factors and moments of the parton distributions, along with a discussion of the proton radius puzzle; finally, the light pseudoscalar meson form factors, the neutral pion lifetime, and the charged pion polarizabilities complete the section. Section 3.3 deals with hadron spectroscopy and summarizes lattice QCD and continuum methods and results, along with a detailed presentation of experimental results and perspectives. Section 3.4 addresses chiral dynamics, including studies based on ChPT and/or on lattice QCD. In Sect. 3.5 the role of light quarks in precision tests of the Standard Model is discussed, with the hadronic contributions to the muon’s anomalous magnetic moment as a particular focus. The running of the electroweak mixing angle, as studied through the weak charge of the proton, and the determination of the strong coupling \({\alpha _{\mathrm{s}}}\) from \(\tau \) decay are also addressed. Finally, Sect. 3.6 presents some thoughts on future directions.

3.2 Hadron structure

3.2.1 Parton distribution functions in QCD

The description of hadrons within QCD faces severe difficulties because the strength of the color forces becomes large at low energies and the confinement properties of quarks and gluons cannot be ignored. The main concepts and techniques for treating this non-perturbative QCD regime are discussed in Sect. 8, which is devoted to infrared QCD. Here, we focus on those quantities that enter the description of hadronic processes in which a large momentum scale is involved, thus enabling the application of factorization theorems. Factorization theorems provide the possibility (under certain assumptions) to compute the cross section for high-energy hadron scattering by separating short-distance from long-distance effects in a systematic way. The hard-scattering partonic processes are described within perturbative QCD, while the distribution of partons in a particular hadron—or of hadrons arising from a particular parton in the case of final-state hadrons—is encoded in universal parton distribution functions (PDFs) or parton fragmentation functions (PFFs), respectively. These quantities contain the dynamics of long-distance scales related to non-perturbative physics and thus are taken from experiment.

To see how factorization works, consider the measured cross section in deep-inelastic scattering (DIS) for the generic process lepton + hadron \(A \rightarrow \mathrm{lepton^{\prime }}\) + anything else \(X\):
$$\begin{aligned} \mathrm{d}\sigma = \frac{\mathrm{d}^{3}\mathbf {k}'}{2s |\mathbf {k}'|} \frac{1}{(q)^2} L_{\mu \nu }(k,q) W^{\mu \nu }(p,q) \, , \end{aligned}$$
(3.1)
where \(k\) and \(k'\) are the incoming and outgoing lepton momenta, \(p\) is the momentum of the incoming nucleon (or other hadron), \(s=(p+k)^2\), and \(q\) is the momentum of the exchanged photon. The leptonic tensor \(L_{\mu \nu }(k,q)\) is known from the electroweak Lagrangian, whereas the hadronic tensor \(W^{\mu \nu }(p,q)\) may be expressed in terms of matrix elements of the electroweak currents to which the vector bosons couple, viz., [51]
$$\begin{aligned} W^{\mu \nu }= \frac{1}{4\pi } \int _{}^{}\mathrm{d}^4y e^{iq\cdot y} \sum _{X} \left\langle A|j^\mu (y)|X\right\rangle \left\langle X|j^\nu (0)|A\right\rangle \, .\nonumber \\ \end{aligned}$$
(3.2)
For \(Q^2=-q^2\) large and Bjorken \(x_B=Q^2/2p\cdot q\) fixed, \(W^{\mu \nu }\) can be written in the form of a factorization theorem to read
$$\begin{aligned} W^{\mu \nu }(p,q)&= \sum _{a} \int _{x_B}^{1} \frac{\mathrm{d}x}{x}f_{a/A}(x, \mu ) \nonumber \\&\times H_{a}^{\mu \nu }(q,xp, \mu , \alpha _\mathrm{s}(\mu )) + \text{ remainder }, \end{aligned}$$
(3.3)
where \(f_{a/A}(x, \mu )\) is the PDF for a parton \(a\) (gluon, \(u\), \(\bar{u}\), \(\ldots \)) in a hadron \(A\) carrying a fraction \(x\) of its momentum and probed at a factorization scale \(\mu \), \(H^{\mu \nu }_a\) is the short-distance contribution of partonic scattering on the parton \(a\), and the sum runs over all possible types of parton \(a\). In (3.3), the (process-dependent) remainder is suppressed by a power of \(Q\).
In DIS experiments, \(lA \rightarrow l^{\prime }X\), we learn about the longitudinal distribution of partons inside hadron \(A\), e.g., the nucleon. The PDF for a quark \(q\) in a hadron \(A\) can be defined in a gauge-invariant way (see [51] and references cited therein) in terms of the following matrix element:
$$\begin{aligned} f_{q/A}(x,\mu )&= \frac{1}{4\pi } \int _{}^{} dy^{-} e^{-i x p^{+} y^{-}} \langle p| \bar{\psi }(0^+,y^{-},\mathbf{{0}}_\mathrm{T}) \nonumber \\&\times \gamma ^{+} \mathcal {W}(0^{-},y^{-}) \psi (0^+,0^{-},\mathbf{{0}}_\mathrm{T}) |p \rangle \, , \end{aligned}$$
(3.4)
where the light-cone notation, \(v^{\pm }=(v^0\pm v^3)/\sqrt{2}\) for any vector \(v^\mu \), was used. Here, \(\mathcal {W}\) is the Wilson line operator in the fundamental representation of \(\mathrm{SU}(3)_\mathrm{c}\),
$$\begin{aligned} \mathcal {W}(0^{-},y^{-}) = \mathcal{P} \exp \left[ ig \int _{0^{-}}^{y^{-}} dz^{-} A_{a}^{+}(0^+, z^{-}, {\mathbf {0}}_\mathrm{T})t_a \right] \nonumber \\ \end{aligned}$$
(3.5)
along a lightlike contour from \(0^{-}\) to \(y^{-}\) with a gluon field \(A_a^\mu \) and the generators \(t_a\) for \(a=1,2,\dots ,8\). Here, \(g\) is the gauge coupling, such that \({\alpha _{\mathrm{s}}}=g^2/4\pi \). Analogous definitions hold for the antiquark and the gluon—the latter in the adjoint representation. These collinear PDFs (and also the fragmentation functions) represent light-cone correlators of leading twist in which gauge invariance is ensured by lightlike Wilson lines (gauge links). The factorization scale \(\mu \) dependence of PDFs is controlled by the DGLAP (Dokshitzer–Gribov–Lipatov–Altarelli–Parisi) [52, 53, 54] evolution equation [55, 56]. The PDFs represent the universal part in the factorized cross section of a collinear process such as (3.3). They are independent of any specific process in which they are measured. It is just this universality of the PDFs that ensures the predictive power of the factorization theorem. For example, the PDFs for the Drell–Yan (DY) process [57] are the same as in DIS, so that one can measure them in a DIS experiment and then use them to predict the DY cross section [51, 58].

The predictive power of the QCD factorization theorem also relies on our ability to calculate the short-distance, process-specific partonic scattering part, such as \(H_{a}^{\mu \nu }\) in (3.3), in addition to the universality of the PDFs. Since the short-distance partonic scattering part is insensitive to the long-distance hadron properties, the factorization formalism for scattering off a hadron in (3.3) should also be valid for scattering off a partonic state. Applying the factorization formalism to various partonic states \(a\), instead of the hadron \(A\), the short-distance partonic part, \(H^{\mu \nu }_a\) in (3.3), can be systematically extracted by calculating the partonic scattering cross section on the left and the PDFs of a parton on the right of (3.3), order-by-order in powers of \(\alpha _\mathrm{s}\) in perturbative QCD. The validity of the collinear factorization formalism ensures that any perturbative collinear divergence of the partonic scattering cross section on the left is completely absorbed into the PDFs of partons on the right. The Feynman rules for calculating PDFs and fragmentation functions have been derived in [55, 56] having recourse to the concept of eikonal lines and vertices. Proofs of factorization of DIS and the DY process can be found in [51] and the original works cited therein.

One of the most intriguing aspects of QCD is the relation between its fundamental degrees of freedom, quarks and gluons, and the observable hadrons, such as the proton. The PDFs are the most prominent non-perturbative quantities describing the relation between a hadron and the quarks and gluons within it. The collinear PDFs, \(f(x\!,\mu )\), give the number density of partons with longitudinal momentum fraction \(x\) in a fast-moving hadron, probed at the factorization scale \(\mu \). Although they are not direct physical observables, as the cross sections of leptons and hadrons are, they are well defined in QCD and can be systematically extracted from data of cross sections, if the factorization formulas of the cross sections with perturbatively calculated short-distance partonic parts are used. Our knowledge of PDFs has been much improved throughout the years by many surprises and discoveries from measurements at low-energy, fixed-target experiments to those at the LHC—the highest energy hadron collider in the world. The excellent agreement between the theory and data on the factorization scale \(\mu \)-dependence of the PDFs has provided one of the most stringent tests for QCD as the theory of strong interaction. Many sets of PDFs have been extracted from the QCD global analysis of existing data, and a detailed discussion of the extraction of PDFs will be given in the next subsection.

Understanding the characteristics and physics content of the extracted PDFs, such as the shape and the flavor dependence of the distributions, is a necessary step in searching for answers to the ultimate question in QCD: of how quarks and gluons are confined into hadrons. However, the extraction of PDFs depends on how well we can control the accuracy of the perturbatively calculated short-distance partonic parts. As an example, consider the pion PDF. Quite recently, Aicher, Schäfer, and Vogelsang [59] addressed the impact of threshold resummation effects on the pion’s valence distribution \(v^\pi \equiv u_v^{\pi ^+}\!=\! \bar{d}_v^{\pi ^+}\!=\!d_v^{\pi ^{-}}\!=\! \bar{u}_v^{\pi ^{-}}\) using a fit to the pion–nucleon E615 DY data [60]. They found a fall-off much softer than linear, which is compatible with a valence distribution behaving as \(xv^{\pi }=(1-x)^{2.34}\) (see Fig. 1). This softer behavior of the pion’s valence PDF is due to the resummation of large logarithmic higher-order corrections—“threshold logarithms”—that become particularly important in the kinematic regime accessed by the fixed-target DY data for which the ratio \(Q^2/s\) is large. Here \(Q\) and \(\sqrt{s}\) denote the invariant mass of the lepton pair and the overall hadronic center-of-mass energy, respectively. Because threshold logarithms enhance the cross section near threshold, the fall-off of \(v^\pi \) becomes softer relative to previous NLO analyses of the DY data. This finding is in agreement with predictions from perturbative QCD [61, 62] in the low-\(x\) regime and from Dyson–Schwinger equation approaches [63] in the whole \(x\) region. Moreover, it compares well with the CERN NA10 [64] DY data, which were not included in the fit shown in Fig. 1 (see [59] for details). Resummation effects on the PDFs in the context of DIS have been studied in [65].
Fig. 1

Valence distribution of the pion obtained in [59] from a fit to the E615 Drell–Yan data [60] at \(Q=4\) GeV, compared to the NLO parameterizations of [61] Sutton–Martin–Roberts–Stirling (SMRS) and [62] Glück–Reya–Schienbein (GRS) and to the distribution obtained from Dyson–Schwinger equations by Hecht et al. [63]. From [59]

Going beyond a purely longitudinal picture of hadron structure, one may keep the transverse (spacelike) degrees of freedom of the partons unintegrated and achieve in this way a three-dimensional image of the hadronic structure by means of transverse-momentum-(\(k_\mathrm{T}\))-dependent (TMD) distribution and fragmentation functions; see, e.g., [66] for a recent review. Such \(x\)- and \(k_{T}\)-dependent quantities provide a useful tool to study semi-inclusive deep inelastic scattering (SIDIS) \(lH^{\uparrow } \rightarrow l^{\prime } h X\) (HERMES, COMPASS, JLab at 12 GeV experiments), the Drell–Yan (DY) process \(H_{1}^{(\uparrow )} H_{2}^{\uparrow } \rightarrow l^{+} l^{-} X\) (COMPASS, PAX, GSI, RHIC experiments), or lepton-lepton annihilation to two almost back-to-back hadrons \(e^{+}e^{-} \rightarrow h_{1} h_{2} X\) (Belle, BaBar experiments), in which events naturally have two very different momentum scales: \(Q \gg q_\mathrm{T}\), where \(Q\) is the invariant mass of the exchanged vector boson, e.g., \(\gamma ^*\) or \(Z^0\), and \(q_\mathrm{T}\) is the transverse momentum of the observed hadron in SIDIS or the lepton-pair in DY, or the momentum imbalance of the two observed hadrons in \(e^+e^{-}\) collisions. It is the two-scale nature of these scattering processes and corresponding TMD factorization formalisms [58, 67, 68] that enable us to explore the three-dimensional motion of partons inside a fast moving hadron. The large scale \(Q\) localizes the hard collisions of partons, while the soft scale \(q_\mathrm{T}\) provides the needed sensitivity to access the parton \(k_\mathrm{T}\). Such a two-scale nature makes these observables most sensitive to both the soft and collinear regimes of QCD dynamics, and has led to the development of the soft-collinear effective theory approach in QCD (see Sect. 7.2.1 for more details and references).

In contrast to collinear PDFs which are related to collinear leading-twist correlators and involve only spin-spin densities, TMD PDFs (or simply, TMDs) parameterize spin-spin and momentum-spin correlations, and also single-spin and azimuthal asymmetries, such as the Sivers [69] and Collins [70, 71] effects in SIDIS. The first effect originates from the correlation of the distribution of unpolarized quarks in a nucleon with the transverse polarization vector \(S_\mathrm{T}\). The second one stems from the similar correlation between \(k_\mathrm{T}\) and \(S_\mathrm{T}\) in the fragmentation function related to the quark polarization. The important point is that the Sivers asymmetry in the DY process flips sign relative to the SIDIS situation owing to the fact that the corresponding Wilson lines point in opposite time directions as a consequence of time reversal. This directional (path) dependence breaks the universality of the distribution functions in SIDIS, DY production, \(e^{+} e^{-}\) annihilation [72], and other hadronic processes that contain more complicated Wilson lines [73], and lead to a breakdown of the TMD factorization [74, 75, 76, 77]. On the other hand, the Collins function seems to possess universal properties in SIDIS and \(e^{+} e^{-}\) processes [78]. Both asymmetries have been measured experimentally in the SIDIS experiments at HERMES, COMPASS, and JLab Hall A [79, 80, 81, 82, 83]. The experimental test of the breakdown of universality, i.e., a signal of process dependence, in terms of these asymmetries and their evolution effects is one of the top-priority tasks in present-day hadronic physics and is pursued by several collaborations.

Theoretically, the effects described above arise because the TMD field correlators have a more complicated singularity structure than PDFs, which is related to the lightlike and transverse gauge links entering their operator definition [84, 85, 86]:
$$\begin{aligned}&\Phi _{ij}^{q[C]}(x, {\mathbf {k}}_{T};n) = \int \frac{d(y\cdot p) \, \mathrm{d}^2 \varvec{y}_{T}}{(2\pi )^3} e^{-ik \cdot y} \nonumber \\&\quad \times \left\langle p| \bar{\psi }_{j}(y)\mathcal {W}(0,y|C)\psi _{i}(0) |p\right\rangle _{y\cdot n=0}, \end{aligned}$$
(3.6)
where the contour \(C\) in the Wilson line \(\mathcal {W}(0,y|C)\) has to be taken along the color flow in each particular process. For instance, in the SIDIS case (see Fig. 2 for an illustration), the correlator contains a Wilson line at \(\infty ^{-}\) that does not reduce to the unity operator by imposing the light-cone gauge \(A^{+}=0\). This arises because in order to have a closed Wilson line, one needs in addition to the two eikonal attachments pointing in the minus direction on either side of the cut in Fig. 2, an additional detour in the transverse direction. This detour is related to the boundary terms that are needed as subtractions to make higher-twist contributions gauge invariant, see [66] for a discussion and references. Hence, the sign reversal between the SIDIS situation and the DY process is due to the change of a future-pointing Wilson line into a past-pointing Wilson line as a consequence of CP invariance (noting CPT is conserved in QCD) [71]. In terms of Feynman diagrams this means that the soft gluons from the Wilson line have “cross-talk” with the quark spectator (or the target remnant) after (before) the hard scattering took place, which emphasizes the importance of the color flow through the network of the eikonal lines and vertices. The contribution of the twist-three fragmentation function to the single transverse spin asymmetry in SIDIS within the framework of the \(k_\mathrm{T}\) factorization is another open problem that deserves attention.
Fig. 2

Factorization for SIDIS of extra gluons into gauge links (double lines). Figure from [66]

The imposition of the light-cone gauge \(A^{+}=0\) in combination with different boundary conditions on the gluon propagator makes the proof of the TMD factorization difficult—already at the one-loop order—and demands the introduction of a soft renormalization factor to remove unphysical singularities [87, 88, 89]. One may classify the emerging divergences into three main categories: (i) ultraviolet (UV) poles stemming from large loop momenta that can be removed by dimensional regularization and minimal subtraction, (ii) rapidity divergences that can be resummed by means of the Collins–Soper–Sterman (CSS) [90] evolution equation in impact-parameter space, and (iii) overlapping UV and rapidity divergences that demand a generalized renormalization procedure to obtain a proper operator definition of the TMD PDFs. Rapidity divergences correspond to gluons moving with infinite rapidity in the opposite direction of their parent hadron and can persist even when infrared gluon mass regulators are included, in contrast to the collinear case in which rapidity divergences cancel in the sum of graphs. Their subtraction demands additional regularization parameters, beyond the usual renormalization scale \(\mu \) of the modified-minimal-subtraction (\(\overline{\mathrm{MS}}\)) scheme.

Different theoretical schemes have been developed to deal with these problems and derive well-defined expressions for the TMD PDFs. Starting from the factorization formula for the semi-inclusive hadronic tensor, Collins [58] recently proposed a definition of the quark TMD PDF which absorbs all soft renormalization factors into the distribution and fragmentation functions, expressing them in the impact-parameter \(b_\mathrm{T}\) space. Taking the limit \(b_\mathrm{T}\rightarrow 0\), these semi-integrated PDFs reduce to the collinear case.

However, this framework has been formulated in the covariant Feynman gauge in which the transverse gauge links vanish so that it is not clear how to treat T-odd effects in axial gauges within this framework. Moreover, the CSS \(b_\mathrm{T}\)-space approach [90] to the evolution of the TMD PDFs requires an extrapolation to the non-perturbative large-\(b_\mathrm{T}\) region in order to complete the Fourier transform in \(b_\mathrm{T}\) and derive the TMDs in \(k_\mathrm{T}\)-space. Different treatments or approximations of the non-perturbative extrapolation could lead to uncertainties in the derived TMDs [91]. For example, the TMDs based on Collins’ definition predicts [92, 93, 94] asymmetries for DY processes that are a bit too small, while a more recent analysis [95, 96], which derives from the earlier work in [67, 68, 97] employing a different treatment on the extrapolation to the large \(b_\mathrm{T}\) region, seems to describe the evolution of the TMD PDF for both the SIDIS and the DY process in the range \(2\)\(100\) GeV\(^2\) reasonably well.

An alternative approach [98, 99, 100] to eliminate the overlapping UV-rapidity divergences employs the renormalization-group properties of the TMD PDFs to derive an appropriate soft renormalization factor composed of Wilson lines venturing off the light cone in the transverse direction along cusped contours. The soft factor encodes contributions from soft gluons with nearly zero center-of-mass rapidity. The presence of the soft factor in the approach of [98, 99, 100], entailed by cusp singularities in the Wilson lines, obscures the derivation of a correct factorization because it is not clear how to split and absorb it into the definition of the TMD PDFs to resemble the collinear factorization theorem. An extension of this approach, relevant for spin observables beyond leading twist, was given in [101].

Several different schemes to study TMD PDFs and their evolution have also been proposed [102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113], which are based on soft collinear effective theory (SCET). One such framework [108, 109, 110] has been shown in [114] to yield equivalent results to those obtained by Collins in [58]. A detailed comparison of the Ji-Ma-Yuan scheme [68, 97] with that of Collins [58] was given in [96]. The universality of quark and gluon TMDs has been studied in a recent work by Mulders and collaborators [115] in which it was pointed out that the whole process (i.e., the gauge link) dependence can be isolated in gluonic pole factors that multiply the universal TMDs of definite rank in the impact-parameter space. An analysis of non-perturbative contributions at the next-to-next-to-leading-logarithmic (NNLL) level to the transverse-momentum distribution of \(Z/\gamma ^*\) bosons, produced at hadron colliders, has been presented in [116].

Last but not least, Sudakov resummation within \(k_\mathrm{T}\) factorization of single and double logarithms is an important tool not only for Higgs boson production in \(pA\) collisions, but also for heavy-quark pair production in DIS, used in the theoretical study of saturation phenomena that can be accessed experimentally at RHIC and the LHC (see, [117] for a recent comprehensive analysis). All these achievements notwithstanding, the TMD factorization formalism and the theoretical framework for calculating the evolution of TMD PDFs and radiative corrections to short-distance dynamics beyond one-loop order have not been fully developed. Complementary to these studies, exploratory calculations of TMD nucleon observables in dynamical lattice QCD have also been performed, which employ nonlocal operators with “staple-shaped,” process-dependent Wilson lines—see, for instance, [118].

3.2.2 PDFs in the DGLAP approach

The PDFs are essential objects in the phenomenology of hadronic colliders and the study of the hadron structure. In the collinear factorization framework, the PDFs are extracted from fits to experimental data for different processes—they are so-called global fits. The typical problem that a global fit solves is to find the set of parameters \(\{p_i\}\) that determine the functional form of the PDFs at a given initial scale \(Q_0^2\), \(f_i(x,Q^2_0,\{p_i\})\) so that they minimize a quality criterion in comparison with the data, normally defined by the best \(\chi ^2\). The calculation of the different observables involves i) the evolution of the PDFs to larger scales \(Q^2>Q^2_0\) by means of the DGLAP evolution equations and ii) the computation of this observable by the factorized hard cross section at a given order in QCD. Several observables are known at next-to-next-to-leading order (NNLO) at present, and this order is needed for precision analyses. This conceptually simple procedure has been tremendously improved during the last years to cope with the stringent requirements of more and more precise analyses of the data in the search of either Standard Model or Beyond the Standard Model physics. For recent reviews on the topic we refer the readers to [119, 120, 121, 122].

A standard choice of the initial parameterization, motivated by Regge theory, is
$$\begin{aligned} f_i(x,Q^2_0)=x^{\alpha _i}(1-x)^{\beta _i}g_i(x), \end{aligned}$$
(3.7)
where \(g_i(x)\) is a function whose actual form differs from group to group. Typical modern sets involve of the order of 30 free parameters and the released results include not only the best fit (the central value PDFs) but also the set of error PDFs to be used to compute uncertainty bands. These uncertainties are based on Hessian error analyses which provide eigenvectors of the covariance matrix (ideally) determined by the one-sigma confidence level criterion or \(\chi ^2=\chi ^2_\mathrm{min}+\Delta \chi ^2\), with \(\Delta \chi ^2=1\). Notice, however, that when applied to a large set of experimental data from different sources it has long been realized that a more realistic treatment of the uncertainties requires the inclusion of a tolerance factor \(T\) so that \(\Delta \chi ^2=T^2\) [123, 124].

An alternative approach which naturally includes the study of the uncertainties is based on Monte Carlo [125], usually by constructing replicas of the experimental data which encode their covariance matrix. This approach is employed by the NNPDF Collaboration [125, 126], which also makes use of neural networks for the parameterizations of (3.7). In this case, the neural networks provide an unbiased set of basis functions in the functional space of the PDFs. The Monte Carlo procedure provides a number of PDF replicas \(N_\mathrm{rep}\) and any observable is computed by averaging over these \(N_\mathrm{rep}\) sets of PDFs. The main advantage of this method is that it does not require assumptions on the form of the probability distribution in parameter space (assumed to be a multi-dimensional Gaussian in the procedure explained in the previous paragraph). As a bonus, the method also provides a natural way of including new sets of data or checking the compatibility of new sets of data, without repeating the tedious and time-consuming procedure of a whole global fit. Indeed, in this approach, including a new set of data would change the relative weights of each of the \(N_\mathrm{rep}\) sets of PDFs, so that a new observable can be computed by averaging over the \(N_\mathrm{rep}\) sets now each one with a different weight [127, 128, 129]. This Bayesian reweighing procedure has also been adapted to the Hessian errors PDFs, where a Monte Carlo representation is possible by simply generating the PDF sets through a multi-Gaussian distribution in the parameter space [130].

Modern sets of unpolarized PDFs for the proton include MSTW08 [131], CT10 [132], NNPDF2.3 [133], HERAPDF [134], ABM11 [8], and CJ12 [135]. Comparison of some of these sets can be found in Fig. 3 as well as of their corresponding impact on the computation of the Higgs cross section at NNLO [136]. Following similar procedures, nuclear PDFs are also available, that is, nCTEQ [137], DSSZ [138], EPS09 [139], and HKN07 [140], as are polarized PDFs [141, 142, 143, 144, 145].
Fig. 3

(Upper figure) Gluon–gluon luminosity to produce a resonance of mass \(M_X\) for different PDFs normalized to that of NNPDF 2.3. (Lower figure) The corresponding uncertainties in the Higgs cross section from PDFs and \(\alpha _\mathrm{s}(M_Z)\). Figures from [136]

3.2.3 PDFs and nonlinear evolution equations

Linear evolution equations such as the DGLAP or the Balitsky–Fadin–Kuraev–Lipatov (BFKL) equations assume a branching process in which each parton in the hadronic wave function splits into two lower-energy ones. The divergence of this process in the infrared makes the distributions more and more populated in the small-\(x\) region of the wave function. In this situation it was proposed long ago that a phenomenon of saturation of partonic densities should appear at small enough values of the fraction of momentum \(x\) [146], or otherwise the unitarity of the scattering amplitudes would be violated. This idea has been further developed into a complete and coherent formalism known as the Color Glass Condensate (CGC, see, e.g., [147] for a recent review).

The CGC formalism is usually formulated in terms of correlators of Wilson lines on the light cone in a color singlet state. The simplest one contains two Wilson lines and can be related to the dipole cross section; higher-order correlators can sometimes be simplified to the product of two-point correlators, especially in the large-\(N_\mathrm{c}\) limit [148]. The nonlinear evolution equation of the dipole amplitudes is known in the large-\(N_\mathrm{c}\) limit with NLO accuracy [149, 150, 151, 152], and the LO version of it is termed the Balitsky–Kovchegov equation [153, 154]. The evolution equations at finite-\(N_\mathrm{c}\) are known as the B-JIMWLK equations (using the acronyms of the authors in [153, 155, 156, 157, 158, 159]) and can be written as an infinite hierarchy of coupled nonlinear differential equations in the rapidity variable, \(Y=\log (1/x)\), of the n-point correlators of the Wilson lines. These equations are very difficult to solve numerically. However, it has been checked that in the large-\(N_\mathrm{c}\) approximation, the BK equations provide very accurate results [160]. The NLO BK equations (or rather their leading NLO contributions) provide a good description of the HERA and other small-\(x\) physics data with a reduced number of free parameters [161] (Fig. 4).
Fig. 4

Fit using the NLO BK nonlinear evolution equations of the combined H1/ZEUS HERA data. Figure from [161]

One of the main interests of the CGC formalism is that it provides a general framework in which to address some of the fundamental questions in the theory of high-energy nucleus-nucleus collisions, in particular, with respect to the initial stages in the formation of a hot and dense QCD medium and how local thermal equilibrium is reached (see, e.g., [162] and references therein). The phenomenological analyses of different sets of data in such collisions deal with the multiplicities [163]; the ridge structure in the two-particle correlations in proton-nucleus collisions, which indicate very long-range rapidity correlations [164]; or the coupling of the CGC-initial conditions with a subsequent hydrodynamical evolution [165]. These are just examples of the potentialities of the formalism to provide a complete description of such complicated systems.

3.2.4 GPDs and tomography of the nucleon

Quarks and gluons carry color charge, and it is very natural to ask how color is distributed inside a bound and color neutral hadron. Knowing the color distribution in space might shed some light on how color is confined in QCD. Unlike the distribution of electromagnetic charge, which is given by the Fourier transform of the nucleon’s electromagnetic form factors (see the next subsection), it is very unlikely, if not impossible, to measure the spatial distribution of color in terms of scattering cross sections of color-neutral leptons and hadrons. This is because the gluon carries color, so that the nucleon cannot rebound back into a nucleon after absorbing a gluon. In other words, there is no elastic nucleon color form factor. Fortunately, in the last 20 years, remarkable progress has been made in both theory and experiment to make it possible to obtain spatial distributions of quarks and gluons inside the nucleons. These distributions, which are also known as tomographic images, are encoded in generalized parton distribution functions (GPDs) [166, 167].

GPDs are defined in terms of generalized parton form factors [168], e.g., for quarks,
$$\begin{aligned}&F_{q}(x,\xi ,t) \!=\!\! \int \!\frac{dy^{-}}{2\pi } e^{-i x p^{+} y^{-}} \langle p'| \bar{\psi }({\textstyle \frac{1}{2}}y^{-}){\textstyle \frac{1}{2}}\gamma ^{+} \psi (-{\textstyle \frac{1}{2}}y^{-}) |p \rangle \nonumber \\&\quad \equiv H_q(x,\xi ,t) \left[ \overline{\mathcal{U}}(p')\gamma ^\mu \mathcal{U}(p)\right] \frac{n_{\mu }}{p\cdot n} \nonumber \\&\quad \quad + E_q(x,\xi ,t) \left[ \overline{\mathcal{U}}(p') \frac{i\sigma ^{\mu \nu }(p'-p)_{\nu }}{2M} \mathcal{U}(p) \right] \frac{n_{\mu }}{p\cdot n}, \end{aligned}$$
(3.8)
where the gauge link between two quark field operators and the factorization scale dependence are suppressed, \(\mathcal{U}\)’s are hadron spinors, \(\xi =(p'-p)\cdot n/2\) is the skewness, and \(t=(p'-p)^2\) is the squared hadron momentum transfer. In (3.8), the factors \(H_q(x,\xi ,t)\) and \(E_q(x,\xi ,t)\) are the quark GPDs. Unlike PDFs and TMDs, which are defined in terms of forward hadronic matrix elements of quark and gluon correlators, like those in (3.4) and (3.6), GPDs are defined in terms of non-forward hadronic matrix elements, \(p'\ne p\). Replacing the \(\gamma ^\mu \) by \(\gamma ^\mu \gamma _5\) in (3.8) then defines two additional quark GPDs, \(\widetilde{H}_q(x,\xi ,t)\) and \(\widetilde{E}_q(x,\xi ,t)\). Similarly, gluon GPDs are defined in terms of nonforward hadronic matrix elements of gluon correlators.

Taking the skewness \(\xi \rightarrow 0\), the squared hadron momentum transfer \(t\) becomes \(-{\overrightarrow{\Delta }_{\perp }^2}\). Performing a Fourier transform of GPDs with respect to \({\overrightarrow{\Delta }}_\perp \) gives the joint distributions of quarks and gluons in their longitudinal momentum fraction \(x\) and transverse position \(b_\perp \), \(f_a(x,b_\perp )\) with \(a=q,\bar{q},g\), which are effectively equal to the tomographic images of quarks and gluons inside the hadron. Combining the GPDs and TMDs, one could obtain a comprehensive three-dimensional view of the hadron’s quark and gluon structure.

Taking the moments of GPDs, \(\int \mathrm{d}x\, x^{n-1} H_a(x,\xi ,t)\) with \(a=q,\bar{q},g\), gives generalized form factors for a large set of local operators that can be computed with lattice QCD, as discussed in the next subsection, although they cannot be directly measured in experiments. This connects the hadron structure to lattice QCD—one of the main tools for calculations in the non-perturbative sector of QCD. For example, the first moment of the quark GPD, \(Hq(x, 0, t)\), with an appropriate sum over quark flavors, is equal to the electromagnetic Dirac form factor \(F_1(t)\), which played a major historical role in exploring the internal structure of the proton.

GPDs also play a critical role in addressing the outstanding question of how the total spin of the proton is built up from the polarization and the orbital angular momentum of quarks, antiquarks, and gluons. After decades of theoretical and experimental effort following the European Muon Collaboration’s discovery [169], it has been established that the polarization of all quarks and antiquarks taken together can only account for about 30 % of the proton’s spin, while about 15 % of proton’s spin likely stems from gluons, as indicated by RHIC spin data [170]. Thus, after all existing measurements, about one half of the proton’s spin is still not explained, which is a puzzle. Other possible additional contributions from the polarization of quarks and gluons in unmeasured kinematic regions, related to the orbital momentum of quarks and gluons, could be the major source of the missing portion of the proton’s spin. In fact, some GPDs are intimately connected with the orbital angular momentum carried by quarks and gluons [171]. Ji’s sum rule is one of the examples that quantify this connection [172],
$$\begin{aligned} J_q = \frac{1}{2} \lim _{t\rightarrow 0} \int _0^1 \mathrm{d}x\, x \left[ H_q(x,\xi ,t) + E_q(x,\xi ,t) \right] , \end{aligned}$$
(3.9)
which represents the total angular momentum \(J_q\) (including both helicity and orbital contributions) carried by quarks and antiquarks of flavor \(q\). A similar relation holds for gluons. The \(J_q\) in (3.9) is a generalized form factor at \(t=0\) and could be computed in lattice QCD [173].

GPDs have been introduced independently in connection with the partonic description of deeply virtual Compton scattering (DVCS) by Müller et al. [174], Ji [175], and Radyushkin [176]. They have also been used to describe deeply virtual meson production (DVMP) [177, 178], and more recently timelike Compton scattering (TCS) [179]. Unlike PDFs and TMDs, GPDs are defined in terms of correlators of quarks and gluons at the amplitude level. This allows one to interpret them as an overlap of light-cone wave functions [180, 181, 182]. Like PDFs and TMDs, GPDs are not direct physical observables. Their extraction from experimental data relies upon QCD factorization, which has been derived at the leading twist-two level for transversely polarized photons in DVCS [178] and for longitudinally polarized photons in DVMP [183]. The NLO corrections to the quark and gluon contributions to the coefficient functions of the DVCS amplitude were first computed by Belitsky and Müller [184]. The NLO corrections to the crossed process, namely, TCS, have been derived by Pire et al. [185].

Initial experimental efforts to measure DVCS and DVMP have been carried out in recent years by collaborations at HERA and its fixed target experiment HERMES, as well as by collaborations at JLab and the COMPASS experiment at CERN. To help extract GPDs from cross-section data for exclusive processes, such as DVCS and DVMP, various functional forms or representations of GPDs have been proposed and used for comparing with existing data. Radyushkin’s double distribution ansatz (RDDA) [176, 186] has been employed in the Goloskokov–Kroll model [187, 188, 189] to investigate the consistency between the theoretical predictions and the data from DVMP measurements. More discussions and references on various representations of GPDs can be found in a recent article by Müller [168].
Fig. 5

Connections among various partonic amplitudes in QCD. The abbreviations are explained in the text

The PDFs, TMDs, and GPDs represent various aspects of the same hadron’s quark and gluon structure probed in high-energy scattering. They are not completely independent and, actually, they are encoded in the so-called mother distributions, or the generalized TMDs (GTMDs), which are defined as TMDs with non-forward hadronic matrix elements [190, 191]. In addition to the momentum variables of the TMDs, \(x\) and \({\overrightarrow{k}}\!\!_\perp \), GTMDs also depend on variables of GPDs, the skewness \(\xi \) and the hadron momentum transfer \(\Delta ^\mu =(p'-p)^\mu \) with \(t=\Delta ^2\). The Fourier transform of GTMDs can be considered as Wigner distributions [192], the quantum-mechanical analog of classical phase-space distributions. The interrelationships between GTMDs and the PDFs, TMDs, and GPDs are illustrated in Fig. 5.

Comprehensive and dedicated reviews on the derivation and phenomenology of GPDs can be found in Refs. [168, 193, 194, 195, 196, 197]. More specific and recent reviews of the GPD phenomenology and global analysis of available data can be found in Ref. [198] for both the DVCS and DVMP processes, and in Ref. [199] for DVCS asymmetry measurements of different collaborations pertaining to the decomposition of the nucleon spin.

With its unprecedented luminosity, the updated 12 GeV program at JLab will provide good measurements of both DVCS and DVMP, which will be an excellent source of information on quark GPDs in the valence region. It is the future Electron–Ion Collider (EIC) that will provide the ultimate information on both quark and gluon GPDs, and the tomographic images of quarks and gluons inside a proton with its spin either polarized or unpolarized [200].

3.2.5 Hadron form factors

The internal structure of hadrons—most prominently of the nucleon—has been the subject of intense experimental and theoretical activities for decades. Many different experimental facilities have accumulated a wealth of data, mainly via electron–proton (\(ep\)) scattering. Electromagnetic form factors of the nucleon have been measured with high accuracy, e.g., at MAMI or MIT-Bates. These quantities encode information on the distribution of electric and magnetic charge inside the nucleon and also serve to determine the proton’s charge radius. The HERA experiments have significantly increased the kinematical range over which structure functions of the nucleon could be determined accurately. Polarized \(ep\) and \(\mu p/d\) scattering at HERMES, COMPASS, and JLab, provide the experimental basis for attempting to unravel the spin structure of the nucleon. Furthermore, a large experimental program is planned at future facilities (COMPASS-II, JLab at 12 GeV, PANDA@FAIR), designed to extract quantities such as GPDs, which provide rich information on the spatial distributions of quarks and gluons inside hadrons. This extensive experimental program requires equally intense theoretical activities, in order to gain a quantitative understanding of nucleon structure.

a. Lattice-QCD calculations Simulations of QCD on a space-time lattice are becoming increasingly important for the investigation of hadron structure. Form factors and structure functions of the nucleon have been the subject of lattice calculations for many years (see the recent reviews [201, 202, 203, 204]), and more complex quantities such as GPDs have also been tackled recently [205, 206, 207, 208, 209, 210], as reviewed in [211, 212]). Furthermore, several groups have reported lattice results on the strangeness content of the nucleon [213, 214, 215, 216, 217, 218, 219, 220, 221, 222], as well as the strangeness contribution to the nucleon spin [223, 224, 225, 226, 227, 228, 229]. Although calculations of the latter quantities have not yet reached the same level of maturity concerning the overall accuracy compared to, say, electromagnetic form factors, they help to interpret experimental data from many experiments.

Lattice-QCD calculations of baryonic observables are technically more difficult than those of the corresponding quantities in the mesonic sector. This is largely due to the increased statistical noise which is intrinsic to baryonic correlation functions, and which scales as \(\exp (m_\mathrm{N}-\frac{3}{2}m_\pi )\), where \(m_\mathrm{N}\) and \(m_\pi \) denote the nucleon and pion masses, respectively. As a consequence, statistically accurate lattice calculations are quite expensive. It is therefore more difficult to control the systematic effects related to lattice artifacts, finite-volume effects, and chiral extrapolations to the physical pion mass in these calculations. Statistical limitations may also be responsible for a systematic bias due to insufficient suppression of the contributions from higher excited states [230].

Many observables also require the evaluation of so-called “quark-disconnected” diagrams, which contain single quark propagators forming a loop. The evaluation of such diagrams in lattice QCD suffers from large statistical fluctuations, and specific methods must be employed to compute them with acceptable accuracy. In a lattice simulation, one typically considers isovector combinations of form factors and other quantities, for which the above-mentioned quark-disconnected diagrams cancel. It should be noted that hadronic matrix elements describing the \({\pi }N\) sigma term or the strangeness contribution to the nucleon are entirely based on quark-disconnected diagrams. With these complications in mind, it should not come as a surprise that lattice calculations of structural properties of baryons have often failed to reproduce some well-known experimental results.

In the following we summarize the current status of lattice investigations of structural properties of the nucleon. The Dirac and Pauli form factors, \(F_1\) and \(F_2\), are related to the hadronic matrix element of the electromagnetic current \(V_{\mu }\) via
$$\begin{aligned}&\left\langle N(p^\prime ,s^\prime )| V_{\mu }(x) | N(p,s)\right\rangle \nonumber \\&\quad = \bar{u}(p^\prime ,s^\prime ) \left( \gamma _{\mu } F_1(Q^2) - \sigma _{\mu \nu }\frac{Q_\nu }{2m_\mathrm{N}}\, F_2(Q^2) \right) u(p,s),\nonumber \\ \end{aligned}$$
(3.10)
where \(p,s\) and \(p^\prime ,s^\prime \) denote the momenta and spins of the initial- and final-state nucleons, respectively, and \(Q^2=-q^2\) is the negative squared momentum transfer. The Sachs electric and magnetic form factors, \(G_\mathrm{E}\) and \(G_\mathrm{M}\), which are related to the electron–proton scattering cross section via the Rosenbluth formula, are obtained from suitable linear combinations of \(F_1\) and \(F_2\), i.e.,
$$\begin{aligned}&G_\mathrm{E}(Q^2) = F_1(Q^2) + \frac{Q^2}{(2m_\mathrm{N})^2}F_2(Q^2),\nonumber \\&G_\mathrm{M}(Q^2)=F_1(Q^2)+F_2(Q^2). \end{aligned}$$
(3.11)
The charge radii associated with the form factors are then derived from
$$\begin{aligned} \left\langle r_i^2 \right\rangle = -6\left. \frac{d F_i(Q^2)}{d Q^2}\right| _{Q^2=0},\quad i=1,2 . \end{aligned}$$
(3.12)
Analogous relations hold for the electric and magnetic radii, \(\langle {r_\mathrm{E}^2}\rangle \) and \(\langle {r_\mathrm{M}^2}\rangle \).
Currently there is a large deviation between experimental determinations of \(\langle r_\mathrm{E}^2 \rangle \) using muonic hydrogen and electronic systems that is called the “proton radius puzzle”, see Sect. 3.2.6 for further discussion.
Fig. 6

The dependence of the nucleon’s isovector electric form factor \(G_\mathrm{E}\) on the Euclidean four-momentum transfer \(Q^2=-q^2\) for near-physical pion masses, as reported by the LHP Collaboration [231] and the Mainz group [232]. The phenomenological parameterization of experimental data is from [233]

There are many cases in which lattice QCD calculations of observables that describe structural properties of the nucleon compare poorly to experiment. For instance, the dependence of nucleon form factors on \(Q^2\) computed on the lattice is typically much flatter compared to phenomenological parameterizations of the experimental data, at least when the pion mass (i.e., the smallest mass in the pseudoscalar channel) is larger than about 250 MeV. It is then clear that the values of the associated charge radii are underestimated compared to experiment [206, 235, 236, 237, 238, 239, 240, 241, 242, 243]. The situation improved substantially after results from simulations with substantially smaller pion masses became available, combined with techniques designed to reduce or eliminate excited-state contamination. The data of [231] and [232] show a clear trend towards the \(Q^2\)-behavior seen in a fit of the experimental results as the pion mass is decreased from around 200 MeV to almost its physical value (see Fig. 6). Since different lattice actions are employed in the two calculations, the results are largely independent of the details of the fermionic discretization. A key ingredient in more recent calculations is the technique of summed operator insertions [244, 245, 246, 247], for which excited state contributions are parametrically suppressed. Alternatively one can employ multi-exponential fits including the first excited state [231, 248] and solve the generalized eigenvalue problem for a matrix correlation function [249], or study the dependence of nucleon matrix elements for a wide range of source-sink separations [250]. Results for the pion mass dependence of the Dirac radius, \(\langle r_1^2\rangle \), from [234] are shown in Fig. 7, demonstrating that good agreement with the PDG value [1] can be achieved. Similar observations also apply to the Pauli radius and the anomalous magnetic moment.

The axial charge of the nucleon, \(g_A\), and the lowest moment of the isovector parton distribution function, \(\langle x\rangle _{u-d}\) are both related to hadronic matrix elements with simple kinematics, since the initial and final nucleons are at rest. Furthermore, no quark-disconnected diagrams must be evaluated. If it can be demonstrated that lattice simulations accurately reproduce the experimental determinations of these quantities within the quoted statistical and systematic uncertainties, this would constitute a stringent test of lattice methods. In this sense \(g_A\) and \(\langle x\rangle _{u-d}\) may be considered benchmark observables for lattice QCD.
Fig. 7

The dependence of the isovector Dirac radius \(\langle r_1^2\rangle \) on the pion mass from [234]. Filled blue symbols denote results based on summed operator insertions, designed to suppress excited-state contamination

Calculations based on relatively heavy pion masses have typically overestimated \(\langle x\rangle _{u-d}\) [206, 207, 208, 239, 240, 251] by about 20 %. Moreover, it was found that \(\langle x\rangle _{u-d}\) stays largely constant as a function of the pion mass (see Fig. 8). Lower values have been observed in [252, 253], but given that the overall pion mass dependence in that calculation is quite weak, it is still difficult to make contact with the phenomenological estimate. Other systematic errors, such as lattice artifacts or insufficient knowledge of renormalization factors, may well be relevant for this quantity. Recent calculations employing physical pion masses, as well as methods to suppress excited state contamination [234, 254], have reported a strong decrease of \(\langle x\rangle _{u-d}\) near the physical value of \(m_\pi \). Although the accuracy of the most recent estimates does not match the experimental precision, there are hints that lattice results for \(\langle x\rangle _{u-d}\) can be reconciled with the phenomenological estimate.
Fig. 8

The dependence of the first moment of the isovector PDF plotted versus the pion mass. Lattice results are compiled from [207, 234, 240, 251, 252, 253]

The strategy of controlling the bias from excited states and going towards the physical pion mass has also helped to make progress on \(g_A\), which, compared to \(\langle x\rangle _{u-d}\), is a simpler quantity. It is the matrix element of the axial current, i.e., a quark bilinear without derivatives, whose normalization factor is known with very good accuracy. Lattice simulations using pion masses \(m_\pi > 250\) MeV typically underestimate \(g_A\) by \(10\)\(15~\%\) [206, 236, 237, 239, 240, 242, 256, 257, 258, 259, 260, 261, 262]. Even more worrisome is the observation that the data from these simulations show little or no tendency to approach the physical value as the pion mass is decreased. However, although some of the most recent calculations using near-physical pion masses and addressing excited state contamination [247, 248, 255] produce estimates which agree with experiment (see Fig. 9), there are notable exceptions: the authors of [234] still find a very low result, despite using summed insertions which may be attributed to a particularly strong finite-size effect in \(g_A\). The effects of finite volume have also been blamed for the low estimates reported in [263, 264].

The current status of lattice-QCD calculations of structural properties of the nucleon can be summarized by noting that various sources of systematic effects are now under much better control, which leads to a favorable comparison with experiment in many cases. Simulations employing near-physical pion masses and techniques designed to eliminate the bias from excited-state contributions have been crucial for this development. Further corroboration of these findings via additional simulations that are subject to different systematics is required. Also, the statistical accuracy in the baryonic sector must be improved.
Fig. 9

Compilation of recent published results for the axial charge in QCD with \(N_\mathrm{f}=2+1+1\) dynamical quarks [248] (upper panel), \(N_\mathrm{f}=2+1\) [234, 237] (middle panel), as well as two-flavor QCD [236, 247, 255, 256] (lower panel)

b. Poincaré-covariant Faddeev approach The nucleons’ electromagnetic [265] as well as axial and pseudoscalar [266] form factors have been calculated in the Poincaré-covariant Faddeev framework based on Landau-gauge QCD Green’s functions. The latter are determined in a self-consistent manner from functional methods and, if available, compared to lattice results. Over the last decade, especially the results for corresponding propagators and some selected vertex functions have been established to an accuracy that they can serve as precise input to phenomenological calculations, see also the discussion in Sect. 8.2.

The main idea of the Poincaré-covariant Faddeev approach is to exploit the fact that baryons will appear as poles in the six-quark correlation function. Expanding around the pole one obtains (in a similar way as for the Bethe–Salpeter equation) a fully relativistic bound-state equation. The needed inputs for the latter equation are (i) the tensor structures of the bound-state amplitudes, which rest solely on Poincaré covariance and parity invariance and provide a partial-wave decomposition in the rest frame, see, e.g., [267, 268] and references therein for details; (ii) the fully dressed quark propagators for complex arguments; and (iii) the two- and three-particle irreducible interaction kernels. In case the three-particle kernel is neglected, the bound-state equation is then named the Poincaré-covariant Faddeev equation. The two-particle-irreducible interaction kernel is usually modeled within this approach, and mesons and baryons are then both considered in the so-called rainbow-ladder truncation, which is the simplest truncation that fully respects chiral symmetry and leads to a massless pion in the chiral limit.

In [265, 266] the general expression for the baryon’s electroweak currents in terms of three interacting dressed quarks has been derived. It turns out that in the rainbow-ladder truncation the only additional input needed is the fully dressed quark-photon vertex which is then also calculated in a consistent way. It is important to note that this vertex then contains the \(\rho \)-meson pole, a property which appears essential to obtaining the correct physics.

In the actual calculations a rainbow-ladder gluon-exchange kernel for the quark-quark interaction, which successfully reproduces properties of pseudoscalar and vector mesons, is employed. Then the nucleons’ Faddeev amplitudes and form factors are computed without any further truncations or model assumptions. Nevertheless, the resulting quark-quark interaction is flavor blind,4 and by assumption it is a vector-vector interaction and thus in contradiction to our current understanding of heavy-quark scalar confinement, cf. Sect. 8.2. References [269, 270] lays out an alternative description of the phenomenology of confinement, based on the interconnections of light-front QCD, holography, and conformal invariance, with wide-ranging implications for the description of hadron structure and dynamics.
Fig. 10

The vector meson, nucleon, and \(\Delta \)/\(\Omega \) masses as a function of the pion mass squared in the Poincaré-covariant Faddeev approach (adapted from [278])

Fig. 11

The nucleons’ electromagnetic form factors in the Poincaré-covariant Faddeev approach (adapted from [265])

Therefore the challenge posed to the Poincaré-covariant Faddeev approach is to extend in a systematically controlled way beyond the rainbow-ladder and the Faddeev truncations. Given the fact that non-perturbative calculations of the full quark–gluon vertex and three-gluon vertex have been published recently and are currently improved, this will become feasible in the near future. Nevertheless, already the available results provide valuable insight, and, as can be inferred from the results presented below, in many observables the effects beyond rainbow-ladder seem to be on the one hand surprisingly small and on the other hand in its physical nature clearly identifiable.

Figure 10 shows the results for some selected hadron masses using two different interaction models, see [271] for the MT and [272] for the AFW model. (The main phenomenological difference between these two models is that the AFW model reproduces the \(\eta ^\prime \) mass via the Kogut–Susskind mechanism beyond rainbow-ladder whereas the (older) MT model does not take this issue into account.) As one can see, both model calculations compare favorably with lattice results [206, 235, 237, 238, 273, 274, 275, 276, 277]. Given the fact that the baryon masses are predictions (with parameters fixed from the meson sector) and that a rainbow-ladder model kernel has been used instead of a calculated one, the agreement is even somewhat better than expected.

In Fig. 11 the results for the electromagnetic form factors of the proton and neutron are shown. It is immediately visible that the agreement with the experimental data at large \(Q^2\) is good. In addition, there is also good agreement with lattice data at large quark masses. These two observations lead to the expectation that the difference of the calculated results with respect to the observed data is due to missing pion-cloud contributions in the region of small explicit chiral symmetry breaking. This is corroborated by the observation that the pion-loop corrections of ChPT are compatible with the discrepancies appearing in Fig. 11. This can be deduced in a qualitative way from Fig. 12. The results of the Faddeev approach are, like the lattice results, only weakly dependent on the current quark mass (viz., the pion mass squared). Whereas lattice results are not (yet) available at small masses, the Faddeev calculation can be performed also in the chiral limit. However, pion loop (or pion cloud) effects are not (yet) contained in this type of calculations. Correspondingly there are deviations at the physical pion mass. To this end it is important to note that in the isoscalar combination of the anomalous magnetic moment leading-order pion effects are vanishing. As a matter of fact, the Faddeev approach gives the correct answer within the error margin of the calculation. Details can be found in [265].
Fig. 12

Results for the nucleon’s isoscalar and isovector anomalous magnetic moments and isovector Dirac radius in the Poincaré-covariant Faddeev approach as compared to lattice QCD results and experiment (stars) (adapted from [265])

Fig. 13

\(Q^2\)-evolution of the ratio of the proton’s electric form factor to a dipole form factor in the Poincaré-covariant Faddeev approach as compared to experimental data (adapted from [265])

Last but not least, the \(Q^2\)-evolution of the proton’s electric form factor in the multi-GeV region is a topic which has attracted a lot of interest in the last decade. Contrary to some expectations (raised by experimental data relying on the Rosenbluth separation) data from polarization experiments have shown a very strong decrease of the ratio of the proton’s electric to magnetic form factor. Even the possibility that the proton’s electric form factor possesses a zero at \(Q^2 \approx 9\) GeV\(^2\) is in agreement with the data. However, more details will be known only after the 12 GeV upgrade of JLab is fully operational. In this respect it is interesting to note that the quite complex Dirac–Lorentz structure of the proton’s Faddeev amplitude quite naturally leads to a strong decrease for \(Q^2>2~\)GeV\(^2\) as shown in Fig. 13. Several authors attribute the difference between the data relying on Rosenbluth separation and polarized-target data to two-photon processes, see, e.g.,  [279]. This has initiated a study of two-photon processes in the Faddeev approach, and an extension to study Compton scattering has made first but important progress [280].

In [266] the axial and pseudoscalar form factors of the nucleon have been calculated in this approach. It is reassuring that the Goldberger–Treiman relation is fulfilled for the results of these calculations for all values of the current quark mass. On the other hand, the result for the axial charge is underestimated by approximately 20 %, yielding \(g_A\approx 1\) in the chiral limit, which is again attributed to missing pion effects. This is corroborated by the finding that the axial and pseudoscalar form factors agree with phenomenological and lattice results in the range \(Q^2>1\ldots 2\) GeV\(^2\). In any case, the weak current-quark mass dependence of \(g_A\) in the Faddeev approach deserves further investigation.

Decuplet, i.e., spin-3/2, baryons possess four electromagnetic form factors. These have been calculated in the Poincaré-covariant Faddeev approach for the \(\Delta \) and the \(\Omega \) [281], and the comments made above for the electric monopole and magnetic dipole form factors for the nucleon also apply here. The electric quadrupole (E2) form factor is in good agreement with the lattice QCD data and provides further evidence for the deformation of the electric charge contribution from sphericity. The magnetic octupole form factor measures the deviation from sphericity of the magnetic dipole distribution, and the Faddeev approach predicts nonvanishing but small values for this quantity.

Summarizing, the current status of results within the Poincaré-covariant Faddeev approach is quite promising. The main missing contributions beyond rainbow-ladder seem to be pionic effects, and it will be interesting to see whether future calculations employing only input from first-principle calculations will verify a picture of a quark core (whose rich structure is mostly determined by Poincaré and parity covariance) plus a pion cloud.

3.2.6 The proton radius puzzle

The so-called proton radius puzzle began as a disagreement at the 5\(\sigma \) level between its extraction from a precise measurement of the Lamb shift in muonic hydrogen [282] and its CODATA value [283], compiled from proton-radius determinations from measurements of the Lamb shift in ordinary hydrogen and of electron–proton scattering. A recent refinement of the muonic hydrogen Lamb shift measurement has sharpened the discrepancy with respect to the CODATA-2010 [284] value to more than 7\(\sigma \) [285]. The CODATA values are driven by the Lamb-shift measurements in ordinary hydrogen, and a snapshot of the situation is shown in Fig. 14, revealing that tensions exist between all the determinations at varying levels of significance.

The measured Lamb shift in muonic hydrogen is \(202.3706 \pm 0.0023\) meV [285], and theory [286, 287, 288, 289] yields a value of \(206.0336 \pm 0.0015 - (5.2275 \pm 0.0010)r_\mathrm{E}^2 + \Delta E_\mathrm{TPE}\) in meV [290], where \(r_\mathrm{E}\) is the proton charge radius and \(\Delta E_\mathrm{TPE}\) reflects the possibility of two-photon exchange between the electron and proton. The first number is the prediction from QED theory and experiment. The proton-radius disagreement amounts to about a 300 \(\upmu \)eV change in the prediction of the Lamb shift. Considered broadly, the topic shows explicitly how a precise, low-energy experiment interplays with highly accurate theory (QED) to reveal potentially new phenomena. We now turn to a discussion of possible resolutions, noting the review of [291].
Fig. 14

Proton radius determinations from (i) the muonic-hydrogen Lamb shift (left), (ii) electron–proton scattering (right), and (iii) the CODATA-2010 combination of the latter with ordinary hydrogen spectroscopy (center). Data taken from [290]

Since the QED calculations are believed to be well understood and indeed would have to be grossly wrong to explain the discrepancy [289] (and a recently suggested non-perturbative QED effect does not exist [292, 293]), a lot of attention has focused on the hadronic contribution arising from the proton’s structure, to which the muonic atom, given its smaller Bohr radius \(a_0(\mu )\simeq (m_e/m_{\mu }) a_0 (e)\), is much more sensitive. If the disagreement is assigned to an error in the proton-radius determination, then, as we have noted, the disagreement between the muonic-atom determination [285] (\(r_\mathrm{E}^{(\mu )}\)) and the CODATA-2010 [284] value (based on hydrogen spectroscopy as well as elastic electron–proton scattering data) (\(r_\mathrm{E}^{(e)}\)) is very large, namely,
$$\begin{aligned} r_\mathrm{E}^{(\mu )}&= 0.84087 \pm 0.00039 \,\mathrm{fm} ,\nonumber \\ r_\mathrm{E}^{(e)}&= 0.8775 \pm 0.0051 \, \mathrm{fm}. \end{aligned}$$
(3.13)
It has been argued [294] that atomic physicists measure the rest-frame proton radius, but electron-scattering data, parametrized in terms of the Rosenbluth form factors, yields the Breit-frame proton radius, and these do not coincide. A resolution by definition might be convenient, but it is not true: precisely the same definition, namely, that of (3.12), is used in both contexts [288, 292]. The value of \(r_\mathrm{E}^{(e)}\) from hydrogen spectroscopy does rely, though, on the value of the Rydberg constant \(R_\infty \) [295], and new experiments plan to improve the determination of this important quantity [285].

The precision of the experimental extraction of the vector form factor from \(ep\) scattering, from which the proton radius is extracted as per (3.12) [296], has also been questioned [297, 298]. In particular, it has been noted that the low-energy Coulomb correction from \(ep\) final-state interactions is sizeable, and this ameliorates the discrepancy between the charge radii determined from hydrogen spectroscopy and its determination in \(ep\) scattering [299].

Higher-order hadronic corrections involving two-photon processes have also been considered as a way of resolving the puzzle [300, 301]. Revised, precise dispersive reevaluations of the proton’s two-photon kernel [302] based on experimental input (photo- and electro-production of resonances off the nucleon and high-energy pomeron-dominated cross-section) yield a contribution of \(40\pm 5\ \upmu \)eV to the muonic hydrogen Lamb shift. The small uncertainty which remains is controlled with the “\(J=0\)” fixed pole of Compton scattering, i.e., the local coupling of two photons to the proton, and which is phenomenologically known only for real photons. This result is in tension with the value \(\Delta E_\mathrm{TPE}=33.2 \pm 2.0\,\upmu \mathrm{eV}\) used in [285], but it remains an order of magnitude too small to explain the discrepancy in the Lamb shift. The appearance of different energy scales in the analysis of muonic hydrogen makes it a natural candidate for the application of effective field theory techniques [301, 303]. Limitations in the ability to assess the low-energy constants would seem to make such analyses inconclusive. Nevertheless, a systematic treatment under the combined use of heavy-baryon effective theory and (potential) non-relativistic QED [303, 304] has recently been employed to determine a proton radius of \(0.8433\pm 0.0017\,\mathrm{fm}\) from the muonic hydrogen data [285], assuming that the underlying power counting determines the numerical size of the neglected terms. This result remains \(6.4\sigma \) away from the CODATA-2010 result.

To summarize, hopes that hadronic contributions to the two-photon exchange between the muon and the proton would resolve the issue quickly are starting to fade away because the correction needed to explain the discrepancy is unnaturally large [305]. Therefore, it might be useful to test ideas of physics beyond the Standard Model, i.e., a different interaction of muons and electrons, in the context of the proton radius puzzle, see Sect. 5.6 for a corresponding discussion.

3.2.7 The pion and other pseudoscalar mesons

The lightest hadron, the pion, is one of the most important strongly interacting particles and serves as a “laboratory” to test various methods within QCD, both on the perturbative and the non-perturbative side. The electromagnetic form factor at spacelike momenta has been treated by many authors over the last decades using various techniques based on collinear factorization [306, 307, 308] with calculations up to the NLO order of perturbation theory, see, e.g., [309, 310]. A novel method was recently presented in [311] which uses the Dyson–Schwinger equation framework in QCD (see [312] for a review). This analysis shows the prevalence of the leading-twist perturbative QCD result (i.e., the hard contribution) for \(Q^2F_{\pi }(Q^2)\) beyond \(Q^2 \gtrsim 8\) GeV\(^2\) in agreement with the earlier results of [310]. Furthermore, it reflects via the dressed quark propagator the scale of dynamical chiral symmetry breaking (D\(\chi \)SB) which is of paramount importance and still on the wish list of hadron physics, because a detailed microscopic understanding of this mechanism is still lacking. Moreover, our current understanding of the pion’s electromagnetic form factor in the timelike region is still marginal [313].

Nevertheless, the dual nature of the pion—being on the one hand the would-be Goldstone boson of D\(\chi \)SB and on the other hand a superposition of highly relativistic bound states of quark–antiquark pairs in quantum field theory—is basically understood and generally accepted. Furthermore, as discussed in Sect. 3.2.1, its valence parton distribution function has been recently determined with a higher precision using threshold resummation techniques [59]. Finally, the quark distribution amplitude for the pion, which universally describes its strong interactions in exclusive reactions, has been reconstructed from the world data on the pion–photon transition form factor as we will see below and is found to be wider than the asymptotic one [314].

a. Form factors of pseudoscalar mesons The two-photon processes \(\gamma ^{*}(q_{1}^2)\gamma (q_{2}^{2}) \rightarrow P\) with \(q_{1}^{2}=-Q^2\) and \(q_{2}^{2}=-q^2\sim 0\) of pseudoscalar mesons \(P=\pi ^0, \eta , \eta '\) in the high-\(Q^2\) region have been studied extensively within QCD (see [312, 315, 316] for analysis and references). This theoretical interest stems from the fact that in leading order such processes are purely electromagnetic with all strong-interaction (binding) effects factorized out into the distribution amplitude of the pseudoscalar meson in question by virtue of collinear factorization. This implies that for \(Q^2\) sufficiently large, the transition form factor for such a process can be formulated as the convolution of a hard-scattering amplitude \(T(Q^2, q^2\rightarrow 0, x) = Q^{-2}(1/x + \mathcal {O}(\alpha _\mathrm{s}))\), describing the elementary process \(\gamma ^*\gamma \longrightarrow q\bar{q}\), with the twist-two meson distribution amplitude [317]. Therefore, this process constitutes a valuable tool to test models of the distribution amplitudes of these mesons.

Several experimental collaborations have measured the cross section for \(Q^{2}F^{\gamma ^*\gamma \pi ^0}(Q^2,q^2\rightarrow 0)\) and \(Q^2F^{\gamma ^*\gamma \eta (\eta ')} (Q^2,q^2\rightarrow 0)\) in the two-photon processes \(e^+e^{-} \rightarrow e^+e^{-} \gamma ^*\gamma \rightarrow e^+e^{-} X\), where \(X=\pi ^{0}\) [318, 319, 320], \(\eta \) and \(\eta '\) [319, 321], through the so-called single-tag mode in which one of the final electrons is detected. From the measurement of the cross section the meson–photon transition form factor is extracted as a function of \(Q^2\). The spacelike \(Q^2\) range probed varies from \(0.7\)\(2.2\) GeV\(^2\) [318] (CELLO), to \(1.5\)\(9.0\) GeV\(^2\) [319] (CLEO), to \(4\)\(40\) GeV\(^2\) [320] (BaBar) and [322] (Belle). A statistical analysis and classification of all available experimental data versus various theoretical approaches can be found in [314]. The BaBar Collaboration extended substantially the range of the spacelike \(Q^2\), which had been studied before by CELLO [318] and CLEO [319] below 9 GeV\(^2\) to \(Q^2 < 40\) GeV\(^2\). While at low momentum transfers the results of BaBar agree with those of CLEO and have significantly higher accuracy, above 9 GeV\(^2\) the form factor shows rapid growth and from \(\sim 10\) GeV\(^2\) it exceeds the asymptotic limit predicted by perturbative QCD [306]. The most recent results reported by Belle [322] for the wide kinematical region \(4 \lesssim Q^2 \lesssim 40\) GeV\(^2\) have provided important evidence in favor of the collinear factorization scheme of QCD. The rise of the measured form factor \(Q^{2}F^{\gamma ^*\gamma \pi ^0}\), observed earlier by the BaBar Collaboration [320] in the high-\(Q^2\) region, has not been confirmed. This continued rise of the form factor would indicate that the asymptotic value of the form factor predicted by QCD would be approached from above and at much higher \(Q^2\) than currently accessible, casting serious doubts on the validity of the QCD factorization approach and fueling intensive theoretical investigations in order to explain it (see, for example, [323]). The results of the Belle measurement are closer to the standard theoretical expectations [306] and do not hint to a flat-like pion distribution amplitude as proposed in [323]. Further support for this comes from the data reported by the BaBar Collaboration [321] for the \(\eta (\eta ')\)-photon transition form factor that also complies with the QCD theoretical expectations of form-factor scaling at higher \(Q^2\). A new experiment by KLOE-2 at Frascati will provide information on the \(\pi \)\(\gamma \) transition form factor in the low-\(Q^2\) domain, while the BES-III experiment at Beijing will measure this form factor below \(5\) GeV\(^2\) with high statistics.

b. Neutral pion lifetime In the low-energy regime, the two-photon process \(\pi ^0\rightarrow \gamma \gamma \) is also important because one can test at once the Goldstone boson nature of the \(\pi ^0\) and the chiral Adler–Bell–Jackiw anomaly [324, 325]. While the level of accuracy achieved long ago makes existing tests satisfactory, deviations due to the nonvanishing quark masses should become observable at some point. The key quark-mass effect is due to the isospin-breaking-induced mixing: \(\pi ^0\)\(\eta \) and \(\pi ^0\)\(\eta '\), with the mixing being driven by \(m_d-m_u\). The full ChPT correction has been evaluated by several authors and an enhancement of the decay width of about \(4.5\pm 1.0\) % has been found [326, 327, 328], leading to the prediction \(\Gamma _{\pi ^0\rightarrow 2 \gamma }= 8.10\) eV.

The most recent measurement was carried out by the PRIMEX collaboration at JLab with an experiment based on the Primakoff effect [329], providing the result \(\Gamma (\pi ^0\rightarrow \gamma \gamma )=7.82\) eV with a global uncertainty of 2.8 %, which is by far the most precise result to date. Taking into account the uncertainties, it is marginally compatible with the ChPT predictions. With the aim of reducing the error down to 2 %, a second PRIMEX experiment has been completed and results of the analysis should appear soon. A measurement of \(\Gamma _{\pi ^0\rightarrow 2\gamma }\) at the per cent level is also planned in the study of two-photon collisions with the KLOE-2 detector [330]. A recent review of the subject can be found in [331].

c. Pion polarizabilities Further fundamental low-energy properties of the pion are its electric and magnetic polarizabilities \(\alpha _\pi \) and \(\beta _\pi \). While firm theoretical predictions exist based on ChPT [332, 333], the experimental determination of these quantities from pion–photon interactions using the Primakoff effect [334], radiative pion photoproduction [335], and \(\gamma \gamma \rightarrow \pi \pi \) [336], resulted in largely scattered and inconsistent results. The COMPASS experiment at CERN has performed a first measurement of the pion polarizability in pion-Compton scattering with \(190\,~{\mathrm {GeV}}/c\) pions off a Ni target via the Primakoff effect. The preliminary result, extracted from a fit to the ratio of measured cross section and the one expected for a point-like boson shown in Fig. 15, is \(\alpha _\pi =(1.9 \pm 0.7_\mathrm {stat}\pm 0.8_\mathrm {sys})\cdot 10^{-4}\,\mathrm {fm}^3\), where the relation between electric and magnetic polarizability \(\alpha _\pi =-\beta _\pi \) has been assumed [337]. This result is in tension with previous experimental results, but is in good agreement with the expectation from ChPT [332]. New data taken with the COMPASS spectrometer in 2012 are expected to decrease the statistical and systematic error, determined at COMPASS by a control measurement with muons in the same kinematic region, by a factor of about three. The data will for the first time allow an independent determination of \(\alpha _\pi \) and \(\beta _\pi \), as well as a first glimpse on the polarizability of the kaon. Studies of the charged pion polarizability have been proposed and approved at JLab, where the photon beam delivered to Hall D will be used for the Primakoff production of \(\pi ^+\pi ^{-}\) of a nuclear target. A similar study of the \(\pi ^0\) polarizability will also be possible.
Fig. 15

Determination of the pion polarizability at COMPASS through the process \(\pi ^{-} \mathrm {Ni}\rightarrow \pi ^{-}\gamma \mathrm {Ni}\) [337]

3.3 Hadron spectroscopy

In contrast to physical systems bound by electromagnetic interactions, the masses of light hadrons are not dominated by the masses of their elementary building blocks but are to a very large extent generated dynamically by the strong force. The coupling of the light quarks to the Higgs field is only responsible for \({\sim } 1~\%\) of the visible mass of our present-day universe, the rest is a consequence of the interactions between quarks and gluons. While at high energies the interactions between partons become asymptotically free, allowing systematic calculations in QCD using perturbation theory, the average energies and momenta of partons inside hadrons are below the scale where perturbative methods are justified. As a consequence, the fundamental degrees of freedom of the underlying theory of QCD do not directly manifest themselves in the physical spectrum of hadrons, which, rather, are complex, colorless, many-body systems. One of the main goals of the physics of strong interactions for many years has been the determination and the understanding of the excitation spectrum of these strongly bound states. In the past, phenomenological models have been developed, which quite successfully describe certain aspects of the properties of hadrons in terms of effective degrees of freedom, e.g., the quark model [338, 339], the bag model [340, 341], the flux-tube model [342], or QCD sum rules [343]. A full understanding of the hadron spectrum from the underlying theory of QCD, however, is still missing. Nowadays, QCD solved numerically on a discrete spacetime lattice [344] is one of the most promising routes towards this goal.

On the experimental side, significant advances in the light-quark sector have been made in the last few years. Data with unprecedented statistical accuracy have become available from experiments at both electron and hadron machines, often coupled with new observables related to polarization or precise determination of the initial and final-state properties. In the light-meson sector, the unambiguous identification and systematic study of bound states beyond the constituent quark degrees of freedom, e.g., multiquark states or states with gluonic degrees of freedom (hybrids, glueballs), allowed by QCD due to its non-Abelian structure, is within reach of present and future generations of experiments. For a recent review, see e.g. [345]. For the light baryons, photoproduction experiments shed new light on the long-standing puzzle of missing resonances. Here, the recent progress is summarized in [346].

On the theoretical side, hadron spectroscopy has received a huge boost from lattice QCD. Simulations with dynamical up, down, and strange quarks are now routinely performed, and in many cases the need for chiral extrapolations is becoming obsolete thanks to the ability to simulate at or near the physical values of the up and down quark masses [347, 348]. This concerns, in particular, lattice calculations of the masses of the lightest mesons and baryons [349, 350, 351], which show excellent agreement with experiment. Lattice-QCD calculations for the masses of higher-lying mesons, baryons, as well as possible glueball and hybrid states can provide guidance for experiments to establish a complete understanding of the hadron spectrum. Other theoretical tools, such as dispersion relations, provide a way to extract physically relevant quantities such as pole positions and residues of amplitudes.

3.3.1 Lattice QCD

The long-sought objective of studying hadron resonances with lattice QCD is finally becoming a reality. The discrete energy spectrum of hadrons can be determined by computing correlation functions between creation and annihilation of an interpolating operator \(\mathcal {O}\) at Euclidean times \(0\) and \(t\), respectively,
$$\begin{aligned} C(t) = \left\langle 0 \right| \mathcal {O}(t)\mathcal {O}^\dagger (0)\left| 0 \right\rangle \!. \end{aligned}$$
(3.14)
Inserting a complete set of eigenfunctions \(\left| n \right\rangle \) of the Hamiltonian \(\hat{H}\) which satisfy \(\hat{H}\left| n \right\rangle =E_k\left| n \right\rangle \), the correlation function can be written as a sum of contributions from all states in the spectrum with the same quantum numbers,
$$\begin{aligned} C(t) = \sum _n{\left| \left\langle 0 \right| \mathcal {O}\left| n \right\rangle \right| ^2 e^{-E_n t}}\!. \end{aligned}$$
(3.15)
For large times, the ground state dominates, while the excited states are subleading contributions. To measure the energies of excited states, it is thus important to construct operators which have a large overlap with a given state. The technique of smearing the quark-field creation operators is well established to improve operator overlap [245, 352, 353, 354]. A breakthrough for the study of excited states was the introduction of the distillation technique [355], where the smearing function is replaced by a cost-effective low-rank approximation. The interpolating operators are usually constructed from a sum of basis operators \(\mathcal {O}_i\) for a given channel,
$$\begin{aligned} \mathcal {O}=\sum _i{v_i}\mathcal {O}_i, \end{aligned}$$
(3.16)
and a variational method [356] is then employed to extract the best linear combination of operators within a finite basis for each state which maximizes \(C(t)/C(t_0)\). This requires the determination of all elements of the correlation matrix
$$\begin{aligned} C_{ij}(t) = \left\langle 0 \right| \mathcal {O}_i(t)\mathcal {O}_j^\dagger (0)\left| 0 \right\rangle \!, \end{aligned}$$
(3.17)
and the solution of the generalized eigenvalue problem [357, 358]
$$\begin{aligned} C(t)v_n = \lambda _n C (t_0)v_n. \end{aligned}$$
(3.18)
The procedure requires a good basis set of operators that resembles the states of interest.
Thanks to algorithmic and computational advances in recent years, lattice-QCD calculations of the lowest-lying mesons and baryons with given quantum numbers and quark content have been performed with full control of the systematics due to lattice artifacts (see the review in [359]). Figure 16 shows a 2012 compilation of lattice-QCD calculations of the light-hadron spectrum [360]. The pion and kaon masses have been used to fix the masses of light and strange quarks, and (in each case) another observable is used to set the overall mass scale. The experimentally observed spectrum of the baryon octet and decuplet states, as well as the masses of some light vector mesons, are well reproduced within a few percent of accuracy. Except for the isosinglet mesons, the calculations shown use several lattice spacings and a wide range of pion masses. They also all incorporate \(2+1\) flavors into the sea, but the chosen discretization of the QCD action differs. The consistency across all calculations suggests that the systematics, which are different for different calculations, are well controlled. This body of work is a major achievement for lattice QCD, and the precision will improve while the methods are applied to more challenging problems.
Fig. 16

Hadron spectrum from lattice QCD. Wide-ranging results are from MILC [361, 362], PACS-CS [349], BMW [350], and QCDSF [363]. Results for \(\eta \) and \(\eta '\) are from RBC & UKQCD [364], Hadron Spectrum [365] (also the only \(\omega \) mass), and UKQCD [366]. Symbol shape denotes the formulation used for sea quarks. Asterisks represent anisotropic lattices. Open symbols denote the masses used to fix parameters. Filled symbols (and asterisks) denote results. Red, orange, yellow, green, and blue stand for increasing numbers of ensembles (i.e., lattice spacing and sea quark mass). Horizontal bars (gray boxes) denote experimentally measured masses (widths). Adapted from [360]

Also for simulations of excited mesons and baryons huge progress has been made, although the control of the systematics is still much less advanced than in the case of the ground states [276, 365, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378]. These calculations are typically performed for relatively few fairly coarse lattice spacings, and no continuum extrapolation is attempted. A systematic study of finite-volume effects, as well as the extrapolation to physical quark masses, have not yet been performed. The goal of these calculations is to establish a general excitation pattern rather than to perform precision calculations. The focus at the moment is therefore on identifying a good operator basis, on disentangling various excitations in a given channel, and on separating resonances from multihadron states.

a. Light mesons As an example of recent progress, the work of the Hadron Spectrum Collaboration [365, 368, 371] is highlighted here, which recently performed a fully dynamical (unquenched) lattice-QCD calculation of the complete light-quark spectrum of mesons and baryons. The simulations are carried out on anisotropic lattices with lattice spacings \(a_\mathrm {s}\sim 0.12\,\mathrm {fm}\) and \(a_t^{-1}\sim 5.6\,\mathrm {GeV}\) in the spatial and temporal directions, respectively, and with spatial volumes of \(L^3 \sim (2.0\,\mathrm {fm})^3\) and \((2.5\,\mathrm {fm})^3\). They are performed with three flavors of order-\(a\) improved Wilson quarks, i.e., a mass-degenerate light-quark doublet, corresponding to a pion mass down to \(396\,~\mathrm {MeV}\), and a heavier quark whose mass is tuned to that of the strange quark. A large basis of smeared operators for single mesons was built using fermion bilinears projected onto zero meson momentum, including up to three gauge-covariant derivatives. No operators corresponding to multiparticle states, however, were used. The distillation method was used to optimize coupling to low-lying excited states. The correlators are analyzed by a variational method, which gives the best estimate for masses and overlaps. The spins of states are determined by projection of angular momentum eigenstates onto the irreducible representations of the hypercubic group.
Fig. 17

Light-quark meson spectrum resulting from lattice QCD [365], sorted by the quantum numbers \(J^{\mathrm{PC}}\). Note that these results have been obtained with an unphysical pion mass, \(m_\pi =396\,~\mathrm {MeV}\)

The resulting isoscalar and isovector meson spectrum is shown in Fig. 17 [365]. Quantum numbers and the quark–gluon structure of a meson state \(n\) with a given mass \(m_{n}\) are extracted by studying matrix elements \(\langle n|\mathcal {O}_i|0\rangle \), which encode the extent to which operator \(\mathcal {O}_i\) overlaps with state \(n\). States with high spins, up to \(4\), are resolved. The resulting spectrum, as well as the strange–nonstrange mixing of isoscalar mesons, compares well with the currently known states [1]. The calculated masses come out about \(15~\%\) too high, probably owing to the unphysical pion mass, \(m_\pi =396\,~\mathrm {MeV}\). The lattice-QCD simulations also predict a number of extra states, that are not yet well established experimentally. These include a series of exotic states with quantum numbers which cannot be produced by pairing a quark and an antiquark, like \(J^{\mathrm{PC}}=0^{+-},1^{-+},2^{+-},\ldots \), which have been previously postulated to exist also in various models. For some states, a significant overlap with operators containing the gluon field strength tensor has been found, making them candidates for hybrids. It is interesting to note that the quantum numbers and the degeneracy pattern predicted by lattice QCD for hybrid mesons are quite different from those of most models. Lattice QCD predicts four low-mass hybrid multiplets at masses around \(2\,~{\mathrm {GeV}}\) with quantum numbers \(1^{-+},0^{-+},1^{-\,\!-},2^{-+}\), in agreement with the bag model [379, 380], but at variance with the flux-tube model [342, 381], which predicts eight nearly degenerate hybrid multiplets. At masses larger than \(2.4\,~{\mathrm {GeV}}\), lattice QCD predicts a group of ten hybrid multiplets, in disagreement with bag and flux-tube model predictions. The pattern emerging from lattice QCD, i.e., of four low-mass and ten higher-mass multiplets, can be reproduced by a \(q\overline{q}'\) pair in an \(S\)- or \(P\)-wave coupled to a \(1^{+-}\) chromomagnetic gluonic excitation, which can be modeled by a quasi-gluon in a \(P\)-wave with respect to the \(q\overline{q}'\) pair [382].

The spectrum of glueballs has first been calculated on a lattice in pure SU(3) Yang–Mills theory, i.e. in the quenched approximation to QCD [383, 384, 385] at a lattice spacing of \(a\sim 0.1\)–0.2 fm. A full spectrum of states is predicted with the lightest one having scalar quantum numbers, \(0^{++}\), and a mass between \(1.5\,~{\mathrm {GeV}}\) and \(1.7\,~{\mathrm {GeV}}\). Also the next-higher glueball states have non-exotic quantum numbers, \(2^{+\,\!+}\) (mass \(2.3\)\(2.4\,~{\mathrm {GeV}}\)) and \(0^{-+}\) (mass \(2.3\)\(2.6\,~{\mathrm {GeV}}\)), and hence will be difficult to identify experimentally. In a simple constituent gluon picture, these three states correspond to two-gluon systems in relative \(S\) wave, with different combinations of helicities. Table 1 summarizes the quenched lattice results for the masses of the lightest glueballs.
Table 1

Continuum-limit glueball masses (in MeV) from quenched lattice QCD. The first parentheses contain the statistical errors, while the second, where present, include the scale uncertainty

\(J^{PC}\)

Bali [383]

Morningstar [384]

Chen [385]

\(0^{+\,\!+}\)

\(1,550(50)\)

\(1,730(50)(80)\)

\(1,710(50)(80)\)

\(2^{+\,\!+}\)

\(2,270(100)\)

\(2,400(25)(120)\)

\(2,390(30)(120)\)

\(0^{-\,\!+}\)

\(2,330(270)\)

\(2,590(40)(130)\)

\(2,560(35)(120)\)

While the glueball spectrum in pure SU(3) Yang–Mills theory is theoretically well defined, because the glueball operators do not mix with fermionic operators, unquenched lattice calculations are more difficult. The dynamical sea quarks will cause the glueball and flavor singlet fermionic \(0^{+\,\!+}\) interpolating operators to couple to the same physical states. In addition, decays of the \(0^{+\,\!+}\) states into two mesons are allowed for sufficiently light quark masses, and may thus play an important role and dynamically modify the properties of the glueball state. Hence, lattice-QCD calculations of the glueball spectrum with dynamical \(q\overline{q}\) contributions are still at a relatively early stage [386, 387, 388]. One particular problem is the unfavorable signal-to-noise ratio of the relevant correlation functions, which requires large statistics. The authors of [388], using 2+1 flavors of ASQTAD improved staggered fermions and a variational technique which includes glueball scattering states, found no evidence for large effects from including dynamical sea quarks. Their mass for the \(0^{+\,\!+}\) glueball, \(1795(60)\,~\mathrm {MeV}\), is only slightly higher compared to the quenched result of [385]. Figure 18 shows the glueball masses calculated in [388], compared to some experimental meson masses. No extrapolation to the continuum, however, was performed, and no fermionic scattering states were included. Much higher statistics will be needed for precise unquenched calculations of flavor singlet sector on the lattice, with a \(200\,~\mathrm {MeV}\) resolution needed, e.g., to distinguish the three isoscalar mesons in the \(1.5\,~{\mathrm {GeV}}\) mass range. A technique designed to overcome the problem of an exponentially increasing noise-to-signal ratio in glueball calculations has been proposed and tested in the quenched approximation [390]. However, it is not known whether it can be generalized to full QCD.
Fig. 18

Glueball masses resulting from unquenched lattice QCD [388], compared with experimental meson masses [1, 389]. From [388]

b. Light baryons In addition to the meson spectrum, also the spectrum of baryons containing the light quarks \(u, d\), and \(s\) has been calculated recently by different groups [276, 367, 369, 370, 371, 372, 374, 375, 376, 377, 391]. While the focus lies mostly on establishing the spectral pattern of baryon resonances, the possibility of the existence of hybrid baryons has also been addressed. For instance, the Hadron Spectrum Collaboration has obtained spectra for \(N\) and \(\Delta \) baryons with \(J\le \frac{7}{2}\) and masses up to \({\sim } 1.9M_\Omega \) [371]. The well-known pattern of organizing the states in multiplets of \(\mathrm{SU}(6)\times O(3)\), where the first is the spin-flavor group, clearly emerges when checking the overlap of the states with the different source/sink operators. The multiplicity of states observed is similar to that of the non-relativistic quark model. The first excited positive-parity state, however, is found to have significantly higher mass than its negative spin-parity partner, in contrast to the experimental ordering of the \(N(1440)\frac{1}{2}^+\) and the \(N(1535)\frac{1}{2}^{-}\). The chiral behavior of the observed level structure in these calculations has been analyzed in detail in [276, 370, 375, 377]. Furthermore, no obvious pattern of degenerate levels with opposite parity (parity doubling) emerges from the simulations for higher masses, in contrast to indications from experiments [392]. Lattice-QCD calculations have been extended to include excited hyperons [372, 376, 377]. In particular, the nature of the \(\Lambda (1405)\) has been the subject of the study in [372]. Moreover, lattice QCD presents the possibility of testing for the presence of excited glue in baryons (hybrids) for the first time, and it has been carried out in [374]. In contrast to the meson sector, however, all possible \(J^P\) values for baryons can be built up from states consisting of three quarks with non-vanishing orbital angular momentum between them, so that there is no spin-exotic signature of hybrid baryons. It is found that their multiplet structure is compatible with a color-octet chromomagnetic excitation with quantum numbers \(J^P=1^{+-}\), coupling to three quarks in a color-octet state and forming a color-neutral object, as in the case of hybrid mesons. Also the mass splitting between the \(qqq\) states and the hybrid states is the same as observed for the meson sector, indicating a common bound-state structure for hybrid mesons and baryons.

c. Future directions These very exciting developments still lack two aspects: First, there is the issue of controlling systematic effects such as lattice artifacts, finite-volume effects, and long chiral extrapolations owing to the use of unphysical quark masses. Second, the fact that hadron resonances have a non-zero width is largely ignored in the calculations described above, i.e., resonances are treated as stable particles. While the first issue will be dealt with once gauge ensembles with finer lattice spacing and smaller pion masses are used for spectrum calculations, the second problem requires a different conceptual approach. The position and width of a resonance are usually determined from the scattering amplitude. However, as noted in [393], the latter cannot be determined directly from correlation functions computed in Euclidean space-time. Lüscher pointed out in his seminal work [394, 395, 396, 397] that the phase shift of the scattering amplitude in the elastic region can be determined from the discrete spectrum of multi-particle states in a finite volume. When plotted as a function of \({m_\pi }L\), resonances can be identified via the typical avoided level crossing.

The Lüscher formalism, which was originally derived for the center-of-mass frame of mass-degenerate hadrons, has since been generalized to different kinematical situations [398, 399, 400, 401, 402, 403, 404]. Numerical applications of the method are computationally quite demanding, since they require precise calculations for a wide range of spatial volumes, as well as the inclusion of multi-hadron interpolating operators. Most studies have therefore focused on the simplest case, i.e., the \(\rho \)-meson [405, 406, 407, 408, 409, 410, 411]. As reviewed in [412], other mesonic channels such as \(K\pi \), \(D\pi \), and \(D^{*}\pi \), as well as the \(N\pi \) system (i.e., the \(\Delta \)-resonance) have also been considered. While the feasibility of extracting scattering phase shifts via the Lüscher method has been demonstrated, lattice calculations of resonance properties are still at an early stage. In spite of the technical challenges involved in its implementation, the Lüscher method has been extended to the phenomenologically more interesting cases of multi-channel scattering [413, 414, 415] and three-particle intermediate states [416, 417].

3.3.2 Continuum methods

Although lattice calculations will provide answers to many questions in strong QCD, the development of reliable analytical continuum methods is a necessity to develop an intuitive understanding of QCD from first principles, to construct advanced phenomenological models, and to address computationally challenging tasks like the extrapolation to physical quark masses or large hadronic systems. The tools at our disposal include effective theories such as ChPT, Dyson–Schwinger methods, fixed gauge Hamiltonian QCD approaches, and QCD sum rule methods.

A study of baryon resonances with various models has been carried out since time immemorial. In recent times, dynamical models based on meson-baryon degrees of freedom have received much attention [418, 419, 420, 421], in particular in the case of \(S\) wave resonances, such as the \(\Lambda (1405)\). These models use effective Lagrangians to couple light mesons to the ground-state baryons, and in this way generate resonances dynamically. Since baryons couple strongly to the continuum, it is known that meson-baryon dynamics plays an important role; one would like to eventually understand how to better quantify that role by using improved models. One can speculate that this can also be an interesting topic of exploration in the framework of lattice QCD, where the possibility of varying the quark masses can illuminate how excited baryon properties change with the pion mass. Models of baryons based on the Schwinger–Dyson equations have also been studied [422, 423], and are being developed into important tools to study excited baryons with a framework anchored in the principles of QCD.

In the spirit of effective theories, one approach based on the \(1/N_\mathrm{c}\) expansion has been developed [424, 425, 426, 427]. In the limit of large \(N_\mathrm{c}\), a spin-flavor dynamical symmetry emerges in the baryon sector, which is broken at subleading order in \(1/N_\mathrm{c}\) and thus provides a starting point for the description of baryon observables in a power series in \(1/N_\mathrm{c}\). As in every effective theory, it is necessary to give inputs, namely baryon observables determined phenomenologically, and the \(1/N_\mathrm{c}\) expansion serves to organize and relate them at each order in the expansion. The framework is presented as an expansion in composite operators, where quantities or observables are expanded on a basis of operators at a given order in \(1/N_\mathrm{c}\), and the coefficients of the expansion, which encode the QCD dynamics, are determined by fitting to the observables. It has been applied to baryon masses [428, 429, 430, 431, 432, 433, 434], partial decay widths [435, 436, 437], and photocouplings [438]. Through those analyses it is observed that the different effects, which are classified by their \(\mathrm{SU}(2 N_\mathrm{f})\times O(3)\) structure (\(N_\mathrm{f}\) is the number of light flavors) and by their power in \(1/N_\mathrm{c}\), seem to follow the natural order of the \(1/N_\mathrm{c}\) expansion, that is, they have natural magnitude. An interesting challenge is the implementation of the \(1/N_\mathrm{c}\) expansion constraints in models, in order to have a more detailed understanding of the dynamics. One such nice and illustrative example has been given in [439].

3.3.3 Experiments

The fundamental difficulty in studying the light-hadron spectrum is that in most cases resonances do not appear as isolated, narrow peaks. Instead, states have rather large widths of several hundred \(~\mathrm {MeV}\) and consequently overlap. Peaks observed in a spectrum may be related to thresholds opening up or interference effects rather than to genuine resonances, not to speak of kinematic reflections or experimental acceptance effects. In addition, nonresonant contributions and final-state effects may also affect the measured cross section. Partial wave or amplitude analysis (PWA) techniques are the state-of-the-art way to disentangle contributions from individual, and even small, resonances and to determine their quantum numbers. Multiparticle decays are usually modeled using the phenomenological approach of the isobar model, which describes multiparticle final states by sequential two-body decays into intermediate resonances (isobars), that eventually decay into the final state observed in the experiment. Event-based fits allow one to take into account the full correlation between final-state particles. Coupled-channel analyses are needed to reliably extract resonance parameters from different reactions or final states.

One notoriously difficult problem is the parameterization of the dynamical properties of resonances. Very often, masses and widths of resonances are determined from Breit–Wigner parameterizations, although this approach is strictly only valid for isolated, narrow states with a single decay channel. For two-body processes, e.g., the K-matrix formalism provides a way to ensure that the amplitudes fulfill the unitarity condition also in the case of overlapping resonances. The rigorous definition of a resonance is by means of a pole in the second (unphysical) Riemann sheet of the complex energy plane. For poles deep in the complex plane, however, none of the above approaches yield reliable results, although they might describe the data well. The correct analytical properties of the amplitude are essential for an extrapolation from the experimental data (real axis) into the complex plane in order to determine the pole positions. Dispersion relations provide a rigorous way to do this by relating the amplitude at any point in the complex plane to an integral over the (imaginary part of the) amplitude evaluated on the real axis (i.e., the data) making use of Cauchy’s theorem.

a. Scalar mesons and glueballs The identification and classification of scalar mesons with masses below \(2.5\,~{\mathrm {GeV}}\) is a long-standing puzzle. Some of them have large decay widths and couple strongly to the two-pseudoscalar continuum. The opening of nearby thresholds such as \(K\overline{K}\) and \(\eta \eta \) strongly distort the resonance shapes. In addition, non-\(q\overline{q}'\) scalar objects like glueballs and multi-quark states are expected in the mass range below \(2\,~{\mathrm {GeV}}\), which will mix with the states composed of \(q\overline{q}'\). The Particle Data Group (PDG) currently lists the following light scalars [1], sorted according to their isospin: (\(I=0\)) \(f_0(500)\), \(f_0(980)\), \(f_0(1370)\), \(f_0(1500)\), \(f_0(1710)\), (\(I=1/2\)) \(K_0^*(800)\) (listed as still requiring confirmation), \(K_0^*(1430)\), (\(I=1\)) \(a_0(980)\), \(a_0(1450)\). One possible interpretation is that the scalars with masses below \(1\,~{\mathrm {GeV}}\) form a new nonet with an inverted mass hierarchy, with the wide, isoscalar \(f_0(500)\) as the lightest member, the \(K_0^*(800)\) (neutral and charged), and the isospin-triplet \(a_0(980)\), which does not have any \(s\)-quark content in the quark model, and its isospin-singlet counterpart \(f_0(980)\) as the heaviest members. The high masses of the \(a_0(980)\) and the \(f_0(980)\) and their large coupling to \(K\bar{K}\) could be explained by interpreting them as tightly bound tetraquark states [440] or \(K\bar{K}\) molecule-like objects [441]. The scalar mesons above \(1\,~{\mathrm {GeV}}\) would form another nonet, with one supernumerary isoscalar state, indicating the presence of a glueball in the \(1.5\,~{\mathrm {GeV}}\) mass region mixing with the \(q\bar{q}\) states [442]. Other interpretations favor an ordinary \(q\bar{q}\) nonet consisting of \(f_0(980)\), \(a_0(980)\), \(K_0^*(1430)\), and \(f_0(1500)\) [443, 444]. The \(f_0(1370)\) is interpreted as an interference effect. The \(K_0^*(800)\) is not required in this model, and the supernumerary broad \(f_0(500)\) would then have a large admixture of a light glueball. In view of these different interpretations it is important to clarify the properties of scalar mesons. An updated review on the topic can be found, e.g., in the PDG’s “Note on Scalar Mesons below \(2\,~{\mathrm {GeV}}\)” [1].

When it comes to the lightest scalar mesons, the \(f_0(500)\), huge progress has been made in recent years towards a confirmation of its resonant nature and the determination of its pole position. Although omitted from the PDG’s compilation for many years, its existence has been verified in several phenomenological analyses of \(\pi \)\(\pi \) scattering data. As for other scalar particles, the \(f_0(500)\), also known as \(\sigma \), is produced in, e.g., \(\pi \)\(N\)-scattering or \(\bar{p}p\)-annihilation, and data is, in particular, obtained from \(\pi \)\(\pi \), \(K\)\(\bar{K}\), \(\eta \)\(\eta \), and 4\(\pi \) systems in the \(S\)-wave channel. The analyses of several processes require four poles, the \(f_0(500)\) and three other scalars, in the region from the \(\pi \)\(\pi \) threshold to \(1600\,~\mathrm {MeV}\). Hereby the missing distinct resonance structure below \(900\,~\mathrm {MeV}\) in \(\bar{p} p\)-annihilation was somehow controversial. However, by now it is accepted that also these data are described well with the standard solution requiring the existence of the broad \(f_0(500)\).

The pole position, i.e., the pole mass and related width, is also accurately determined. The combined analysis with ChPT and dispersion theory of \(\pi \)\(\pi \) scattering [445] has led to a particularly accurate determination of those parameters. The PDG quotes a pole position of \(M-i\Gamma /2\simeq \sqrt{s_\sigma } = (400\)\(550)-i(200\)\(350)\,~\mathrm {MeV}\), whereas averaging over the most advanced dispersive analyses gives a much more restricted value of \(\sqrt{s_\sigma } = (446\pm 6)-i(276\pm 5)\,~\mathrm {MeV}\). Especially relevant for the precise determination of the \(f_0(500)\) pole were recent data from the NA48/2 experiment at CERN’s Super Proton Synchrotron (SPS) on \(K^\pm \rightarrow \pi ^+\pi ^{-} e^\pm \nu \) (K\(_{e4}\)) decays [446], which have a much smaller systematic uncertainty than the older data from \(\pi N\rightarrow \pi \pi N\) scattering due to the absence of other hadrons in the final state. NA48/2 has collected \(1.13\) million K\(_{e4}\) events using simultaneous \(K^+\) and \(K^{-}\) beams with a momentum of \(60\,~{\mathrm {GeV}}\).

As mentioned above, however, there exist many, partly mutually excluding interpretations of the \(f_0(500)\): a quark–antiquark bound state, \(\pi \)\(\pi \) molecule, tetraquark, QCD dilaton, to name the most prominent ones. In addition, it will certainly also mix with the lightest glueball. From the phenomenological side it is evident that the large \(\pi \)\(\pi \) decay width is the largest obstacle in gaining more accurate information. However, it is exactly the pattern of D\(\chi \)SB which makes this width quite naturally so large. An \(f_0(500)\) with a small width could only occur if there is substantial explicit breaking of chiral symmetry (because, e.g., a large current mass would lead to \(m_\sigma < 2 m_\pi \)) or if by some other mechanism the scalar mass would be reduced.

Here a look to the electroweak sector of the Standard Model is quite enlightening. The scalar particle claimed last year by CMS and ATLAS with mass \(125\)\(126\,~{\mathrm {GeV}}\) is consistent with the Standard Model (SM) Higgs boson, cf. Sect. 5.2.3. It appears to be very narrow as its width-to-mass ratio is small. Though the mass is a free parameter of the SM, one natural explanation of its lightness relative to its “natural” mass of about \(300\)\(400\,~{\mathrm {GeV}}\) is fermion-loop mass renormalization, strongest by the top quark loop. This is one clear contribution that makes the Higgs light.5 In any case, the accident \(m_\mathrm{H} < 2 m_W\) prevents the decay \(h\rightarrow WW\). Since the longitudinal \(W\) components are the Goldstone bosons of electroweak symmetry breaking, the analogy to \(\sigma \pi \pi \) in QCD is evident. If the top quark were much lighter, or if it would be less strongly coupled (such as the nucleon to the sigma), the Higgs mass could naturally be higher by some hundreds of GeV, the decay channel to \(WW\) would open, and the Higgs would have a width comparable in magnitude to its mass. This comparison makes it plain that the \(f_0(500)\), for which no fermion that strongly couples to it is similar in mass, is naturally so broad because of the existence of pions as light would-be Goldstone bosons and its strong coupling to the two-pion channel. Unfortunately, this also implies that the nature of the \(f_0(500)\) can be only revealed by yet unknown non-perturbative methods. It has to be emphasized that the lack of understanding of the ground state in the scalar meson channel is an unresolved but important question of hadron physics.

In recent dispersive analyses [447, 448] of \(\pi \pi \) scattering data and the very recent \(K_{\ell 4}\) experimental results [446], the pole positions of the \(f_0(500)\) and \(f_0(980)\) were determined simultaneously, and the results, summarized in Table 2, are in excellent agreement with each other.
Table 2

Positions of the complex poles of the \(f_0(500)\) and \(f_0(980)\), determined in dispersive analyses [447, 448] of \(\pi \pi \) scattering data and \(K_{\ell 4}\) decays

Ref.

\(\sqrt{s_0}\) (\(~\mathrm {MeV}\))

 

\(f_0(500)\)

\(f_0(980)\)

 [447]

\(\left( 457^{+14}_{-13}\right) -i\left( 279^{+11}_{-7}\right) \)

\(\left( 996\pm 7\right) -i\left( 25^{+10}_{-6}\right) \)

 [448]

\(\left( 442^{+5}_{-8}\right) -i\left( 274^{+6}_{-5}\right) \)

\(\left( 996^{+4}_{-14}\right) -i\left( 24^{+11}_{-3}\right) \)

The situation with the lightest strange scalar, \(K_0^*(800)\) or \(\kappa \), is more complicated. A dispersive analysis of \(\pi K\rightarrow \pi K\) scattering data gives a pole position of the \(K_0^*(800)\) of \(\left( 658\pm 13\right) -i/2\left( 557 \pm 24\right) \,~\mathrm {MeV}\) [449], while recent measurements by BESII in \(J/\psi \rightarrow K_\mathrm{S} K_\mathrm{S} \pi ^+\pi ^{-}\) decays [450] give a slightly higher value for the pole position of \(\left( 764\pm 63^{+71}_{-54}\right) -i\left( 306\pm 149^{+143}_{-85}\right) \,~\mathrm {MeV}\). Similar results from dispersive analyses are expected for the \(a_0(980)\). A broad scalar with mass close to that above is also needed for the interpretation of the \(K\pi \) invariant mass spectrum observed by Belle in \(\tau ^{-} \rightarrow K^0_\mathrm{S}\pi ^{-}\nu _\tau \) decay [451]. Numerous measurements of invariant mass spectra in hadronic decays of \(D\) and \(B\) mesons are hardly conclusive because of the large number of interfering resonances involved in parameterizations and different models used in the analyses.

New data are being collected by BES III at the recently upgraded BEPCII \(e^+e^{-}\) collider in Beijing in the \(\tau \)-charm mass region at a luminosity of \(10^{33}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\) (at a center-of-mass (CM) energy of \(2\times 1.89\,~{\mathrm {GeV}}\)), with a maximum CM energy of \(4.6\,~{\mathrm {GeV}}\) [452]. In the last 3 years, the experiment has collected the world’s largest data samples of \(J/\psi \), \(\psi (2S)\), and \(\psi (3770)\) decays. These data are also being used to make a variety of studies in light-hadron spectroscopy, especially in the scalar meson sector. Recently, BES III reported the first observation of the isospin-violating decay \(\eta (1405) \rightarrow \pi ^0 f_0(980)\) in \(J/\psi \rightarrow \gamma 3\pi \) [453], together with an anomalous lineshape of the \(f_0(980)\) in the \(2\pi \) invariant mass spectra, as shown in Fig. 19.
Fig. 19

Invariant mass of \(\pi ^+\pi ^{-}\) and \(\pi ^0\pi ^0\) with the \(\pi ^+\pi ^{-}\pi ^0\) (\(3\pi ^0\)) mass in the \(\eta (1405)\) mass region, measured at BES III [453]

The \(f_0\) mass, deduced from a Breit–Wigner fit to the mass spectra, is slightly shifted compared to its nominal value, with a width of \(<11.8\,~\mathrm {MeV}\) (\(90~\%\) C.L.), much smaller than its nominal value. The observed isospin violation is \((17.9\pm 4.2)~\%\), too large to be explained by \(f_0(980)\)\(a_0(980)\) mixing, also observed recently by BES III at the \(3.4\sigma \) level [454]. Wu et al. [455] suggest that a \(K\) triangle anomaly could be large enough to account for the data.

BES III has recently performed a full PWA of \(5460\) radiative \(J/\psi \) decays to two pseudoscalar mesons, \(J/\psi \rightarrow \gamma \eta \eta \), commonly regarded as an ideal system to look for scalar and tensor glueballs. In its baseline solution, the fit contains six scalar and tensor resonances [456], \(f_0(1500)\), \(f_0(1710)\), \(f_0(2100)\), \(f_2'(1525)\), \(f_2(1810)\), and \(f_2(2340)\), as well as \(0^{+\,\!+}\) phase space and \(J/\psi \rightarrow \phi \eta \). The scalars \(f_0(1710)\), \(f_0(2100)\), and \(f_0(1500)\) are found to be the dominant contributions, with the production rate for the latter being about one order of magnitude smaller than for the first two. No evident contributions from \(f_0(1370)\) or other scalar mesons are seen. The well-known tensor resonance \(f_2'(1525)\) is clearly observed, but several \(2^{+\,\!+}\) tensor components are also needed in the mass range between \(1.8\) and \(2.5\,~{\mathrm {GeV}}\). The statistical precision of the data, however, is not yet sufficient to distinguish the contributions. Figure 20 shows the resulting PWA fit result of the \(\eta \eta \) invariant mass spectrum.

In conclusion, the situation in the scalar meson sector is still unresolved. The lightest glueball is predicted to have scalar quantum numbers and is therefore expected to mix with nearby isoscalar scalar \(q\bar{q}\)\(P\)-wave states. For recent reviews on glueballs, see [444, 457, 458]. On the experimental side, further new results from BES III, from Belle on two-photon production of meson pairs [459, 460], and from COMPASS on central production [461] may help to resolve some of the questions in the scalar sector in the future.

b. Hybrid mesons Experimental evidence for the existence of hybrid mesons can come from two sources. The observation of an overpopulation of states with \(q\overline{q}'\) quantum numbers may indicate the existence of states beyond the quark model, i.e., hybrids, glueballs, or multi-quark states. The densely populated spectrum of light mesons in the mass region between \(1\) and \(2\,~{\mathrm {GeV}}/c^2\), and the broad nature of the states involved, however, makes this approach difficult. It requires the unambiguous identification of all quark-model states of a given \(J^{\mathrm{PC}}\) nonet, a task which has been achieved only for the ground-state nonets so far. The identification of a resonant state with exotic, i.e., non-\(q\overline{q}'\) states, however, is considered a “smoking gun” for the existence of such states. Table 3 lists experimental candidates for hybrid mesons and their main properties.6

Models, as well as lattice QCD, consistently predict a light hybrid multiplet with spin-exotic quantum numbers \(J^{\mathrm{PC}}=1^{-+}\). Currently, there are three experimental candidates for a light \(1^{-+}\) hybrid [1] (for recent reviews, see [487, 488]): the \(\pi _1(1400)\) and the \(\pi _1(1600)\), observed in diffractive reactions and \(\overline{p}N\) annihilation, and the \(\pi _1(2015)\), seen only in diffraction. The \(\pi _1(1400)\) has only been observed in the \(\pi \eta \) final state, and is generally considered too light to be a hybrid meson. In addition, a hybrid should not decay into a \(P\)-wave \(\eta \pi \) system from SU(3) symmetry arguments [489]. There are a number of studies that suggest it is a nonresonant effect, possibly related to cusp effects due to two-meson thresholds. The \(\pi _1(1600)\) has been seen decaying into \(\rho \pi \), \(\eta '\pi \), \(f_1(1285)\pi \), and \(b_1(1235)\pi \). New data on the \(1^{-+}\) wave have recently been provided by COMPASS, CLEO-c, and CLAS and will be reviewed in the following.
Fig. 20

Invariant mass distribution of \(\eta \eta \) from \(J/\psi \rightarrow \gamma \eta \eta \), and the projection of the PWA fit from BES III [456]

The COMPASS experiment [490] at CERN’s Super Proton Synchrotron (SPS) is investigating diffractive and Coulomb production reactions of hadronic beam particles into final states containing charged and neutral particles. In a first analysis of the \(\pi ^{-}\pi ^{-}\pi ^+\) final state from scattering of \(190\,~{\mathrm {GeV}}\)\(\pi ^{-}\) on a Pb target, a clear signal in intensity and phase motion in the \(1^{-+}1^+\,\rho \pi \,P\) partial wave has been observed [467], as shown in Fig. 21.
Table 3

Experimental properties of low-mass hybrid candidate states with quantum numbers \(J^{PC}=1^{-\,\!+}\), \(0^{-\,\!+}\), \(2^{-\,\!+}\), \(1^{-\,\!-}\)

State

\(J^{PC}\)

Final state

Decay mode(s)

Mass \((~\mathrm {MeV})\)

Width \((~\mathrm {MeV})\)

Events

Reference

\(\pi _1(1400)\)

\(1^{-\,\!+}\)

\(\pi ^{+}\pi ^{-}\pi ^{0} \pi ^{0}\)

\(\eta \pi ^0\)

\(1257\pm 20\pm 25\)

\(354\pm 64\pm 60\)

\(24\)k

E852 [462]

\(2\pi ^{+} 2\pi ^{-}\)

\(\rho \pi \)

\(1384\pm 20\pm 35\)

\(378\pm 58\)

\(90\)k

OBELIX [463]

\(\pi ^{0} \pi ^{0}\eta (2\gamma )\)

\(\eta \pi ^0\)

\(1360\pm 25\)

\(220\pm 90\)

\(270\)k

CB [464]

\(\pi ^{-}\pi ^0\eta (2\gamma )\)

\(\eta \pi \)

\(1400\pm 20\pm 20\)

\(310\pm 50^{+50}_{-30}\)

\(53\)k

CB [465]

\(\pi ^{-}\eta (2\gamma )\)

\(\eta \pi ^{-}\)

\(1370\pm 16^{+50}_{-30}\)

\(385\pm 40^{+65}_{-105}\)

\(47\)k

E852 [466]

\(\pi _1(1600)\)

\(1^{-\,\!+}\)

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\(\rho \pi ^{-}\)

\(1660\pm 10^{+0}_{-64}\)

\(269\pm 21^{+42}_{-64}\)

\(420\)k

COMPASS [467]

\(\pi ^{-}\pi ^0\omega (\pi ^{+}\pi ^{-}\pi ^0)\)

\(b_1(1235)\pi ^{-}\)

\(1664\pm 8\pm 10\)

\(185\pm 25\pm 28\)

\(145\)k

E852 [468]

\(\pi ^{-}\pi ^{-}\pi ^{+}\eta (\gamma \gamma )\)

\(f_1(1285)\pi ^{-}\)

\(1709\pm 24\pm 41\)

\(403\pm 80\pm 115\)

\(69\)k

E852 [469]

\(\pi ^{-}\pi ^{-}\pi ^{+}\eta (\gamma \gamma )\)

\(\eta ^{\prime }\pi ^{-}\)

\(1597\pm 10^{+45}_{-10}\)

\(340\pm 40\pm 50\)

\(6\)k

E852 [470]

\(\pi _1(2015)\)

\(1^{-\,\!+}\)

\(\pi ^{-}\pi ^0\omega (\pi ^{+}\pi ^{-}\pi ^0)\)

\(b_1(1235)\pi ^{-}\)

\(2014\pm 20\pm 16\)

\(230\pm 32\pm 73\)

\(145\)k

E852 [468]

\(\pi ^{-}\pi ^{-}\pi ^{+}\eta (\gamma \gamma )\)

\(f_1(1285)\pi ^{-}\)

\(2001\pm 30\pm 92\)

\(333\pm 52\pm 49\)

\(69\)k

E852 [469]

\(\pi (1800)\)

\(0^{-\,\!+}\)

\(3\pi ^{-}2\pi ^{+}\)

\(f_0(1500)\pi ^{-}\)

\(1781\pm 5^{+1}_{-6}\)

\(168\pm 9^{+5}_{-14}\)

\(200\)k

COMPASS [471]

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\(f_0(980)\pi ^{-}\)

\(1785\pm 9^{+12}_{-6}\)

\(208\pm 22^{+21}_{-37}\)

\(420\)k

COMPASS [467]

\(\eta (\gamma \gamma )\eta (\pi ^{+}\pi ^{-}\pi ^0)\pi ^{-}\)

\(a_0(980)\eta \), \(f_0(1500)\pi ^{-}\)

\(1876\pm 18\pm 16\)

\(221\pm 26\pm 38\)

\(4\)k

E852 [472]

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\(f_0(980)\pi ^{-}\)

\(1774\pm 18\pm 20\)

\(223\pm 48\pm 50\)

\(250\)k

E852 [473]

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\((\pi \pi )_S\pi ^{-}\)

\(1863\pm 9\pm 10\)

\(191\pm 21\pm 20\)

\(250\)k

E852 [473]

\(\eta (\pi ^{+}\pi ^{-}\pi ^0)\eta (\gamma \gamma )\pi ^{-}\)

\(a_0(980)\eta \)

\(1840\pm 10\pm 10\)

\(210\pm 30\pm 30\)

\(1\)k

VES [474]

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\(f_0(980)\pi ^{-}\), \((\pi \pi )_S\pi ^{-}\)

\(1775\pm 7\pm 10\)

\(190\pm 15\pm 15\)

\(2000\)k

VES [475]

\(K^{+}K^{-}\pi ^{-}\)

\(f_0(980)\pi ^{-}\), \(K_0^*(800) K^{-}\)

\(1790\pm 14\)

\(210\pm 70\)

\(145\)k

VES [476]

\(\eta ^{\prime }(\pi ^{+}\pi ^{-}\eta (\gamma \gamma ),\rho ^0\gamma )\eta (\gamma \gamma )\pi ^{-}\)

\(\eta ^{\prime }\eta \pi ^{-}\)

\(1873\pm 33\pm 20\)

\(225\pm 35\pm 20\)

\(1.9\)k

VES [477]

\(\eta (\pi ^{+}\pi ^{-}\pi ^0)\eta (\gamma \gamma )\pi ^{-}\)

\(\eta \eta \pi ^{-}\)

\(1814\pm 10\pm 23\)

\(205\pm 18\pm 32\)

\(0.4\)k

VES [478]

\(\pi ^{+}\pi ^{-}\pi ^{-}\)

\((\pi \pi )_S\pi ^{-}\)

\(1770\pm 30\)

\(310\pm 50\)

\(120\)k

SERP [479]

\(\pi _2(1880)\)

\(2^{-\,\!+}\)

\(3\pi ^{-}2\pi ^{+}\)

\(f_2(1270)\pi ^{-}\), \(a_1(1260)\rho \),

    
 

   \(a_2(1320)\rho \)

\(1854\pm 6^{+6}_{-4}\)

\(259\pm 13^{+7}_{-17}\)

\(200\)k

COMPASS [471]

\(\eta (\gamma \gamma )\eta (\pi ^{+}\pi ^{-}\pi ^0)\pi ^{-}\)

\(a_2(1320)\eta \)

\(1929\pm 24\pm 18\)

\(323\pm 87\pm 43\)

\(4\)k

E852 [472]

\(\pi ^{-}\pi ^0\omega (\pi ^{+}\pi ^{-}\pi ^0)\)

\(\omega \rho ^{-}\)

\(1876\pm 11\pm 67\)

\(146\pm 17\pm 62\)

\(145\)k

E852 [468]

\(\pi ^{-}\pi ^{-}\pi ^{+}\eta (\gamma \gamma )\)

\(f_1(1285)\pi ^{-}\), \(a_2(1320)\eta \)

\(2003\pm 88\pm 148\)

\(306\pm 132\pm 121\)

\(69\)k

E852 [469]

\(\pi ^{0} \pi ^{0}\eta (\gamma \gamma )\eta (\gamma \gamma )\)

\(a_2(1320)\eta \)

\(1880\pm 20\)

\(255\pm 45\)

\(15\)k

CB [480]

\(\eta _2(1870)\)

\(2^{-\,\!+}\)

\(\eta (\gamma \gamma ,\pi ^{+}\pi ^{-}\pi ^0)\pi ^{+}\pi ^{-}\)

\(a_2(1320)\pi \), \(a_0(980)\pi \)

\(1835\pm 12\)

\(235\pm 23\)

 

WA102 [481]

\(2\pi ^{+}2\pi ^{-}\), \(\pi ^{+}\pi ^{-}\pi ^{0} \pi ^{0}\)

\(a_2(1320)\pi \)

\(1844\pm 13\)

\(228\pm 23\)

\(1500\)k

WA102 [482]

\(2\pi ^{+}2\pi ^{-}\)

\(a_2(1320)\pi \)

\(1840\pm 25\)

\(200\pm 40\)

\(1200\)k

WA102 [483]

\(\eta (\gamma \gamma )3\pi ^0\)

\(f_2(1270)\eta \)

\(1875\pm 20\pm 35\)

\(200\pm 25\pm 45\)

\(5\)k

CB [484]

\(\eta (\gamma \gamma )\pi ^{0} \pi ^{0}\)

\(a_2(1320)\pi \), \(a_0(980)\pi \)

\(1881\pm 32\pm 40\)

\(221\pm 92\pm 44\)

\(1.2\)k

CBall [485]

\(\rho (1450)\)

\(1^{-\,\!-}\)

\(\pi \pi \), \(4\pi \), \(e^{+}e^{-}\)

 

\(1465\pm 25\)

\(400\pm 60\)

 

PDG est. [1]

\(\rho (1570)\)

\(1^{-\,\!-}\)

\(K^{+}K^{-}\pi ^0\)

\(\phi \pi ^0\)

\(1570\pm 36\pm 62\)

\(144\pm 75 \pm 43\)

\(54\)

BABAR [486]

Fig. 21

Exotic \(1^{-+}1^+\,\rho \pi \,P\) wave observed at the COMPASS experiment [467] for 4-momentum transfer between \(0.1\) and \(1.0\,~{\mathrm {GeV}}^2\) on a Pb target and \(\pi ^{-}\pi ^{-}\pi ^+\) final state. Left intensity, right phase difference from the \(1^{+\,\!+}0^+\,\rho \pi \,S\) wave as a function of the \(3\pi \) invariant mass. The data points represent the result of the fit in mass bins, the lines are the result of the mass-dependent fit

A much bigger data set was taken by the same experiment with a liquid hydrogen target, surpassing the existing world data set by about one order of magnitude [491, 492]. For both the Pb and the H targets a large broad nonresonant contribution at lower masses is needed to describe the mass dependence of the spin-density matrix. First studies suggest that the background can be reasonably well described by Deck-like processes [493] which proceed through 1-pion exchange. A more refined analysis in bins of \(3\pi \) mass and \(t\) is being performed on the larger data set and is expected to shed more light on the relative contribution of resonant and nonresonant processes in this and other waves.

COMPASS has also presented data for \(\eta \pi \) (\(\eta \rightarrow \pi ^+\pi ^{-}\pi ^0\)) and \(\eta '\pi \) (\(\eta '\rightarrow \pi ^+\pi ^{-}\eta \), \(\eta \rightarrow \gamma \gamma \)) final states from diffractive scattering of \(\pi ^{-}\) off the H target [494], which exceed the statistics of previous experiments by more than a factor of \(5\). Figure 22 shows the intensities in the (top panel) \(2^{+\,\!+}1^+\) and (bottom panel) \(1^{-+}1^+\) waves for the \(\eta '\pi \) (black data points) and the \(\eta \pi \) final state (red data points), respectively, where the data points for the latter final state have been scaled by a phase-space factor. While the intensities in the \(D\) wave are remarkably similar in intensity and shape in both final states after normalization, the \(P\) wave intensities appear to be very different. For \(\eta \pi \), the \(P\) wave is strongly suppressed, while for \(\eta '\pi \) it is the dominant wave. The phase differences between the \(2^{+\,\!+}1^+\) and the \(1^{-+}1^+\) waves agree for the two final states for masses below \(1.4\,~{\mathrm {GeV}}\), showing a rising behavior due to the resonating \(D\) wave, while they evolve quite differently at masses larger than \(1.4\,~{\mathrm {GeV}}\), suggesting a different resonant contribution in the two final states. As for the \(3\pi \) final states, resonant, as well as nonresonant, contributions to the exotic wave have to be included in a fit to the spin-density matrix in order to describe both intensities and phase shifts [494]. Regardless of this, the spin-exotic contribution to the total intensity is found to be much larger for the \(\eta '\pi \) final state than for the \(\eta \pi \) final state, as expected for a hybrid candidate.
Fig. 22

Comparison of waves for \(\eta \pi \) (red data points) and \(\eta '\pi \) (black data points) final states. Top Intensity of the \(J^\mathrm{PC}=2^{+\,\!+}\)\(D\) wave, bottom intensity of the spin-exotic \(1^{-+}\)\(P\) wave from COMPASS [494]

The CLEO-c detector [495] at the Cornell Electron Storage Ring studied charmed mesons at high luminosities until 2008. The advantage of using charmonium states as a source for light-quark states is a clearly defined initial state, which allows one to limit the available decay modes and to select the quantum numbers through which the final state is reached. Using the full CLEO-c data sample of \(25.9\times 10^{6}\)\(\psi (2S)\) decays, an amplitude analysis of the decay chains \(\psi (2S)\rightarrow \gamma \chi _{c1}\), with \(\chi _{c1}\rightarrow \eta \pi ^+\pi ^{-}\) or \(\chi _{c1}\rightarrow \eta '\pi ^+\pi ^{-}\) has been performed [496]. For these final states, the only allowed \(S\)-wave decay of the \(\chi _{c1}\) goes through the spin-exotic \(1^{-+}\) wave, which then decays to \(\eta (')\pi \). There was no need to include a spin-exotic wave for the \(\eta \pi ^+\pi ^{-}\) final state, for which 2498 events had been observed. In the \(\eta '\pi ^+\pi ^{-}\) channel with 698 events, a significant contribution of an exotic \(\pi _1\) state decaying to \(\eta '\pi \) is required in order to describe the data, as can be seen from Fig. 23. This is consistent with the COMPASS observation of a strong exotic \(1^{-+}\) wave in the same final state in diffractive production, and is the first evidence of a light-quark meson with exotic quantum numbers in charmonium decays.
Fig. 23

Invariant mass projections from the analyses of a, b\(\chi _{c1}\rightarrow \eta \pi ^+\pi ^{-}\), and c, d\(\chi _{c1}\rightarrow \eta '\pi ^+\pi ^{-}\) measured by CLEO-c [496]. The contributions of the individual fitted decay modes are indicated by lines, the data points with full points

The CEBAF Large Acceptance Spectrometer (CLAS) [497] at Hall B of JLab is studying photo- and electro-induced hadronic reactions by detecting final states containing charged and neutral particles. Since the coverage for photon detection is limited in CLAS, undetected neutral particles are inferred mostly via energy-momentum conservation from the precisely measured 4-momenta of the charged particles. CLAS investigated the reaction \(\gamma p\rightarrow \Delta ^{+\,\!+}\eta \pi ^{-}\) in order to search for an exotic \(\pi _1\) meson decaying to the \(\eta \pi \) final state [498]. They found the \(J^{\mathrm{PC}}=2^{+\,\!+}\) wave to be dominant, with Breit–Wigner parameters consistent with the \(a_2(1320)\). No structure or clear phase motion was observed for the \(1^{-+}\) wave. Two CLAS experimental campaigns in 2001 and 2008 were dedicated to a search for exotic mesons photoproduced in the charge exchange reaction \(\gamma p\rightarrow \pi ^+\pi ^+\pi ^{-} (n)\). The intensity of the exotic \(1^{-+}1^\pm \,\rho \pi \,P\) wave, shown in Fig. 24 (left) as a function of the \(3\pi \) invariant mass, does not exhibit any evidence for structures around \(1.7\,~{\mathrm {GeV}}\). Also its phase difference relative to the \(2^{-+}1^\pm \,f_2\pi \,S\) wave does not suggest any resonant behavior of the \(1^{-+}\) wave in this mass region.
Fig. 24

Intensity of the \(1^{-+}1^\pm \,\rho \pi \,P\) waves from photoproduction at (left) CLAS [499] and (right) COMPASS [488] as a function of \(3\pi \) invariant mass

The conclusion from the CLAS experiments is that there is no evidence for an exotic \(1^{-+}\) wave in photoproduction. This is in contradiction to some models [500, 501, 502], according to which photoproduction of mesons with exotic quantum numbers was expected to occur with a strength comparable to \(a_2(1320)\) production.

The COMPASS experiment studied pion-induced reactions on a Pb target at very low values of 4-momentum transfer, which proceed via the exchange of quasi-real photons from the Coulomb field of the heavy nucleus. A partial wave analysis of this data set does not show any sign of a resonance in the exotic \(1^{-+}1^\pm \,\rho \pi \,P\) wave at a mass of \(1.7\,~{\mathrm {GeV}}\) (see Fig. 24), consistent with the CLAS observation.

While there is some evidence for an isovector member of a light \(1^{-+}\) exotic nonet, as detailed in the previous paragraphs, members of non-exotic hybrid multiplets will be more difficult to identify. Most of the light meson resonances observed until now are in fact compatible with a \(q\overline{q}'\) interpretation. Taking the lattice-QCD predictions as guidance, the lowest isovector hybrids with ordinary quantum numbers should have \(J^\mathrm{PC}=0^{-+}\), \(1^{-\,\!-}\), and \(2^{-+}\) (see Sect. 3.3.1a). In the following paragraphs, recent experimental results for states with these quantum numbers are summarized.

There is clear experimental evidence for the \(\pi (1800)\) [1]. The latest measurements of this state come from the COMPASS experiment which observes it in the \(3\pi \) and \(5\pi \) final states, using a \(190\,~{\mathrm {GeV}}\)\(\pi ^{-}\) beam impinging on a Pb target. Table 3 includes the masses and widths obtained by fitting Breit–Wigner functions to the spin density matrix. More statistics and advanced coupled-channel analyses are certainly needed to clarify the decay pattern and thus the hybrid or \(3S\)\(q\overline{q}'\) interpretation of this state.

There is growing experimental evidence for the existence of the \(\pi _2(1880)\). The latest high-statistics measurements of this state again come from COMPASS. For both Pb and H targets a clear peak is observed in the intensity of the \(2^{-+}0^+\,f_2\pi \,D\) wave of the \(3\pi \) final state [491], which is shifted in mass with respect to the \(\pi _2(1670)\), and also exhibits a phase motion relative to the latter in the \(f_2\pi \,S\) wave. This observation, however, was also explained differently, including, e.g., the interference of the \(f_2\pi \,S\) wave with a Deck-like amplitude, which shifts the true \(\pi _2\) peak to lower masses [503]. For \(5\pi \) final states [471], a total of three resonances are needed to describe the \(2^{-+}\) sector, the \(\pi _2(1670)\), the \(\pi _2(1880)\), and a high-mass \(\pi _2(2200)\). The resulting mass and width deduced from this fit for the \(\pi _2(1880)\) are also included in Table 3. A possible isoscalar partner of the \(\pi _2(1880)\), the \(\eta _2(1870)\) has also been reported [1], but needs confirmation.

The PDG lists two \(\rho \)-like excited states, the \(\rho (1450)\) and the \(\rho (1700)\), observed in \(e^+e^{-}\) annihilation, photoproduction, antiproton annihilation and \(\tau \) decays [1]. Their masses are consistent with the \(2^3S_1\) and \(1^3D_1\)\(q\overline{q}'\) states, respectively, but their decay patterns do not follow the \(^3P_0\) rule [504]. The existence of a light vector hybrid state, mixing with the \(q\overline{q}'\) states, was proposed to solve these discrepancies [505]. Recently, BaBar has reported the observation of a \(1^{-\,\!-}\) state decaying to \(\phi \pi ^0\) [486], the \(\rho (1570)\), which might be identical to an earlier observation in Serpukhov [506]. Interpretations of this signal include a new state, a threshold effect, and an OZI-suppressed decay of the \(\rho (1700)\). A very broad vector state with pole position \(M=(1576^{+49+98}_{-55-91}+\frac{i}{2} 818^{+22+64}_{-23-133})\,~\mathrm {MeV}\) has been reported by BES [507] and is listed as \(X(1575)\) by the PDG [1]. It has been interpreted to be due to interference effects in final-state interactions, and in tetraquark scenarios. In conclusion, there is no clear evidence for a hybrid state with vector quantum numbers. A clarification of the nature of the \(\rho \)-like states, especially above \(1.6\,~{\mathrm {GeV}}\), requires more data than those obtained in previous ISR measurements at BaBar and Belle, which will hopefully be reached in current \(e^+e^{-}\) experiments (CMD-3 and SND at the VEPP-2000 collider, BES III at BEPCII) as well as with ISR at the future Belle II detector.

The final test for the hybrid hypothesis of these candidate states will, of course, be the identification of the isoscalar and strange members of a multiplet. Identification of some reasonable subset of these states is needed to experimentally confirm what we now expect from lattice QCD. New experiments with higher statistical significance and better acceptance, allowing for more elaborate analysis techniques, are needed in order to shed new light on these questions.

c. Light baryons Light baryon resonances represent one of the key areas for studying the strong QCD dynamics. Despite large efforts, the fundamental degrees of freedom underlying the baryon spectrum are not yet fully understood. The determinations of baryon resonance parameters, namely quantum numbers, masses and partial widths and their structure such as electromagnetic (EM) helicity amplitudes are currently among the most active areas in hadron physics, with a convergence of experimental programs, and analysis and theoretical activities. An appraisal of the present status of the field can be found in [346]. Many important questions and open problems motivate those concerted efforts. Most important among them is the problem of missing resonances: in quark models based on approximate flavor SU(3) symmetry it is expected that resonances form multiplets; many excited states are predicted which have not been observed (for a review see [508]), with certain configurations seemingly not realized in nature at all [509]. More recently lattice-QCD calculations (at relatively large quark masses) [371] also predict a similar proliferation of states. Do (some of) those predicted states exist, and if so, is it possible to identify them in the experimental data? In addition to \(N\) and \(\varDelta \) baryons made of \(u\) and \(d\) quarks, the search for hyperon resonances remains an important challenge. Efforts in that direction are ongoing at current facilities, in particular at JLab (CLAS), where studies of \(S=-1\) excited hyperons, e.g., in photoproduction of \(\Lambda (1405)\) [510, 511], have been completed. A program to study hyperons with \(S=-1\), \(-2\), and even \(-3\) is part of the CLAS12 upgrade.

Another important task is quantifying and understanding the structure of resonances, which still is in its early stages. Experimentally, one important access to structure is provided by measurements at resonance electro-production, as exemplified by recent work [512, 513] where the EM helicity amplitudes \(A_{1/2}(Q^2)\) (electro-couplings) of the Roper and \(N\)(1520) resonances have been determined from measurements at CLAS, an effort that will continue with the CLAS12 program. An additional tool is provided by meson transition couplings which can be obtained from single meson EM production. Both experimental and theoretical studies of resonance structure are key to further progress.

Since most of the information on light-quark baryon resonances listed in [1] comes from partial wave analyses of \(\pi N\) scattering, one possible reason why many predicted resonances were not observed may be due to small couplings to \(\pi N\). Additional information may come from the observation of other final states like \(\eta N\), \(\eta ' N\), \(KY\), \(\omega N\), or \(2\pi N\). A significant number of the current and future experimental efforts are in electro- and photoproduction experiments, namely JLab (CLAS and CLAS12), Mainz (MAMI-C), Bonn (ELSA) and Osaka (SPring8-LEPS). Experiments with proton beams are being carried out at CERN (COMPASS), J-PARC (Japan; also K beam), COSY and GSI (Germany), and at the proton synchrotron at ITEP (Russia). Resonance production in charmonium decays (BES III and CLEO-c) is also an important source of new excited baryon data.
Fig. 25

Double-polarization observable \(G\) measured at CBELSA [515], (left) as a function of \(\cos {\theta _\pi }\) for four different photon energies, (right) as a function of photon energy for two different pion polar angles \(\theta _\pi \), compared to predictions by different PWA formalisms, (blue) SAID, (red) BnGa, (black) MAID

As in the light-meson sector, the broad and overlapping nature of baryon resonances in the mass region below \(2.5\,~{\mathrm {GeV}}\) requires the application of sophisticated amplitude or partial-wave analyses in order to disentangle the properties of the contributing states. Partial wave analysis is currently a very active area, with several important groups employing different methods and models. Among the groups are SAID (George Washington Univ.), MAID (Mainz), EBAC (Jefferson Lab), Bonn-Gatchina (BnGa), Bonn-Jülich, Valencia, Gießen, and others. While at present the analyses are based to the largest extent on \(\pi N\) and \(K N\) data, the large data sets already accumulated and to be acquired in the near future in photo- and electroproduction are expected to have a big impact in future analyses.

The extraction of amplitudes from the measured differential cross sections suffers from ambiguities, as the latter are bilinear products of amplitudes. These ambiguities can be resolved or at least minimized by imposing physical constraints on the amplitudes, or by measuring a well-chosen set of single and double polarization observables which further constrain the problem. For photoproduction experiments, a “complete experiment” to extract the full scattering amplitude unambiguously [514] requires a combination of linearly and circularly polarized photon beams, longitudinally and transversely polarized targets (protons and neutrons), or the polarization of the recoil nucleon, measured for each energy. These amplitudes are then expanded in terms of partial waves, which are usually truncated at some values of angular momentum. Such measurements are one of the main objectives for the near future, which will give unprecedented detailed access to established baryon resonances, means to confirm or reject less established ones and also possibly lead to the discovery of new resonances.

Even for the simplest photoproduction reaction, \(\gamma p\rightarrow \pi ^0 p\), recently investigated in a double-polarization experiment at CBELSA/TAPS (Bonn) using linearly polarized photons hitting longitudinally polarized protons [515], discrepancies between the latest PWA predictions and the data were found at rather low energies in the region of the four-star resonances \(N(1440)\), \(N(1535)\), and \(N(1520)\). Figure 25 (left) shows the observable \(G\) as a function of \(\cos {\theta _\pi }\) for four different photon energies, where \(\theta _\pi \) is the polar angle of the outgoing pion, compared to predictions by several PWA formalisms. \(G\) is the amplitude of a \(\sin {2\phi _\pi }\) modulation of the cross section in a double-polarization experiment, where \(\phi _\pi \) is the azimuthal angle of the produced pion.Figure 25 (right) shows \(G\) as a function of the photon energy \(E_\gamma \) for two selected bins in pion polar angle \(\theta _\pi \). The differences in the theory predictions arise from different descriptions of two multipoles, \(E_{0^+}\) and \(E_{2^{-}}\), in the three analyses, which are related to the properties of the \(N(1520)\)\(J^P=\frac{3}{2}^{-}\) and \(N(1535)\frac{1}{2}^{-}\) resonances, respectively.

Photoproduction of strangeness, where a hyperon is produced in association with a strange meson, e.g., \(\gamma p\rightarrow K Y\) (\(Y=\Lambda ,\Sigma \)), provides complementary access to nonstrange baryon resonances that may couple only weakly to single-pion final states. In addition, the self-analyzing weak decay of hyperons offers a convenient way to access double polarization observables, as has been recently exploited at CLAS and GRAAL. Using a beam of circularly polarized photons, the polarization transfer to the recoiling hyperon along orthogonal axes in the production plane is characterized by \(C_x\) and \(C_z\). The CLAS collaboration [516] reported that for the case of \(\Lambda \) photoproduction the polarization transfer along the photon momentum axis \(C_z\sim +1\) over a wide kinematic range (see Fig. 26), and the corresponding transverse polarization transfer \(C_x\sim C_z-1\). The magnitude of the total \(\Lambda \) polarization vector \(\sqrt{P^2+C_x^2+C_z^2}\), including the induced polarization \(P\), is consistent with unity at all measured energies and angles for a fully polarized photon beam, an observation which still lacks a proper understanding. Consistent results were obtained by GRAAL [517] for the double polarization observables \(O_{x,z}\) using linearly polarized photons.
Fig. 26

Beam-recoil observable \(C_z\) for circularly polarized photons in the reaction \(\gamma p\rightarrow K^+\Lambda \) as a function of \(\gamma \)\(p\) CM energy for different kaon polar angles \(\theta _K^{\mathrm {CM}}\) measured by CLAS [516]. The data points are compared to different models (see [516] for details)

Decays to vector mesons provide additional polarization information by a measurement of the spin-density matrix, which constrains the PWA of the reaction. Additionally, the photoproduction of \(\omega \) mesons, like that of \(\eta \), serves as an isospin filter for \(N^*\) resonances. A PWA based on a recent high-statistics CLAS measurement of the unpolarized cross section of the reaction \(\gamma p\rightarrow \omega p\) at CM energies up to \(2.4\,~{\mathrm {GeV}}\) [518] required contributions from at least two \(\frac{5}{2}^+\) resonances, identified as the \(N(1680)\frac{5}{2}^+\) and \(N(2000)\frac{5}{2}^+\), and a heavier \(N(2190)\frac{7}{2}^{-}\) resonance. The latter had previously only been observed in \(\pi N\) scattering, and was confirmed more recently by CBELSA/TAPS in \(\pi ^0\) photoproduction [519].

As a consequence of the recent high-statistics data sets from photoproduction, in particular for the reaction \(\gamma p\rightarrow K^+\Lambda \), several baryon resonances, some of which had previously been only weakly observed in \(\pi N\) scattering, have been newly proposed in a recent multichannel analysis of the Bonn-Gatchina PWA group [520] and are now listed in the 2012 PDG review [1]. Table 4 shows the new states in bold letters.
Table 4

Summary of new light-quark baryon resonances (in bold) proposed in [520] and listed in the 2012 review of particle physics [1]

\(J^P\)

Resonance region

\(1/2^{+}\)

\(N(1440)\)****

\(N(1710)\)***

\(\varvec{N(1880)}\)**

\(\mathbf {N(2100)}\)*

\(1/2^{-}\)

\(N(1535)\)****

\(N(1650)\)****

\(\varvec{N(1895)}\)**

 

\(3/2^{+}\)

 

\(N(1720)\)****

\(N(1900)\)***

\(\varvec{N(2040)}\)*

\(3/2^{-}\)

\(N(1520)\)****

\(N(1700)\)***

\(\varvec{N(1875)}\)***

\(\varvec{N(2120)}\)**

\(5/2^{+}\)

 

\(N(1680)\)****

\(\varvec{N(1860)}\)**

\(N(2000)\)**

\(5/2^{-}\)

 

\(N(1675)\)****

 

\(\varvec{N(2060)}\)**

\(3/2^{-}\)

 

\(\Delta (1700)\)***

\(\Delta (1940)\)**

 

Some solutions of the partial wave analyses of the world data seem to indicate the existence of parity doublets at higher masses [392, 509], i.e., two approximately degenerate states with the same spin but opposite parity (see also Table 4). This is consistent with predictions based on the effective restoration of chiral symmetry at high baryon masses [521, 522]. Similar patterns, however, are also predicted in models which do not make explicit reference to chiral symmetry [523, 524]. In contrast, the most recent lattice-QCD calculations of excited, higher-spin baryon masses [371] uncover no evidence for the existence of parity doublets. Thus, the question of whether or not chiral doublets exist in the upper reaches of the baryon spectrum remains unanswered.

d. Future directions Spectroscopy of light hadrons will remain an active field of research in the future. In order to arrive at a full understanding of the excitation spectrum of QCD, a departure from simplistic Breit–Wigner resonance descriptions towards a full specification of the pole positions of the amplitude in the complex plane, including dynamical effects, thresholds, cusps, is required. As masses increase, multiparticle channels open up, leading to broad and overlapping resonances. Partial-wave analysis models have to be extended to fully respect unitarity, analyticity, and crossing symmetry, in order to extract fundamental, process-independent quantities. The rigorous way of determining the poles and residues of the amplitude from experiment, which has been performed at physical values of \(s\) and \(t\), is by means of dispersion relations, which provide the correct analytic extension of the amplitudes to the complex plane. If and how these can be incorporated into fit models for multiparticle final states remains an open question. A clear separation of resonant and nonresonant contributions, a recurring question for many of the observed signals in the light meson sector, e.g., requires coupled-channel analyses of different final states, but also studies in different reactions and kinematics in order to clarify the underlying production mechanisms.

New results from running experiments are to be expected in the near future. The extraction of polarization observables for baryon resonances in electromagnetically induced reactions will continue at ELSA and MAMI, which in turn will provide input to multichannel PWA. COMPASS, whose data set with hadron beams (\(\pi \), \(p\), \(K\)) is currently being analyzed, will continue to take data for a couple of years with muon and pion beams  [525]. New experiments are on the horizon or have already started to take data, which are expected to considerably advance our understanding of the excitation spectrum of QCD. Key features of these experiments will be large data sets requiring highest possible luminosities and sensitivity to production cross sections in the sub-nanobarn region. This can only be achieved by hermetic detectors with excellent resolution and particle identification capabilities, providing a very high acceptance for charged and neutral particles.

Although not their primary goal, \(e^{+} e^{-}\) machines, operating at charmonium or bottomonium center-of-mass energies, have initiated a renaissance of hadron spectroscopy in the past few years by discovering many new and yet unexplained states containing charm and bottom quarks. In \(e^{+} e^{-}\) collisions states with photon quantum numbers are directly formed. Other states including exotics can be accessed via hadronic or radiative decays of heavy mesons, or are produced recoiling against other particles. Hadronic decays of heavy-quark states may serve as a source for light-quark states, with a clearly defined initial state facilitating the partial wave analysis. BES III at the BEPCII collider in Beijing has already started to take data in the \(\tau \)-charm region with a luminosity of \(10^{33}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\) at a CM energy of \(2\times 1.89\,~{\mathrm {GeV}}\), and will continue to do so over the next years. The Belle II experiment at SuperKEKB [526], aiming at a 40-fold luminosity increase to values of \(8\times 10^{35}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\), is expected to increase the sensitivity for new states in the charm and bottom sector dramatically, but will also feed the light-quark sector. Experiments at the LHC, especially LHCb with its excellent resolution, are also expected to deliver high-statistics data on the meson spectrum.

GlueX [527] is a new experiment which will study photoproduction of mesons with masses below \(3\,~{\mathrm {GeV}}\) at the \(12\,~{\mathrm {GeV}}\) upgrade of CEBAF at JLab. An important advantage of the experiment will be the use of polarized photons, which narrows down the possible initial states and gives direct information on the production process. Hadron spectroscopy in Hall B of JLab will be extended to a new domain of higher mass resonances and the range of higher transferred momentum using electron beams up to \(11\,~{\mathrm {GeV}}\) and the upgraded CLAS12 detector [528]. In addition to studying GPDs, CLAS12 will perform hadron spectroscopy using photoproduction of high-mass baryon and meson resonances, either by electron scattering via quasi-real photons or by high-energy real photon beams. The detector will consist of a forward detector, making use of partly existing equipment with new superconducting torus coils, and a central detector with a new \(5\,\mathrm {T}\) solenoid magnet and a barrel tracker, providing nearly \(4\pi \) solid angle coverage for hadronic final states.

PANDA, a new experiment at the FAIR antiproton storage ring HESR, is designed for high-precision studies of the hadron spectrum in the charmonium mass range [529]. In \(\overline{p}p\) annihilations, all states with non-exotic quantum numbers can be formed directly. Consequently, the mass resolution for these states is only limited by the beam momentum resolution. Spin-exotic states can be obtained in production experiments. PANDA is expected to run at center-of-mass energies between \(2.3\) and \(5.5\,~{\mathrm {GeV}}\) with a maximum luminosity of \(2\times 10^{32}\,\mathrm {cm}^{-2}\,\mathrm {s}^{-1}\). As for the \(e^+e^{-}\) machines, hadronic decays of heavy hadrons may also serve as a well-defined source for light mesons. The study of multistrange hyperons in proton–antiproton annihilation is also foreseen in the PANDA experiment.

3.4 Chiral dynamics

The low-energy regime of light hadron physics plays a key role in tests of the non-perturbative phenomena of QCD. In particular, the approximate chiral \(\mathrm{SU}_L(3)\times \mathrm{SU}_R(3)\) symmetry and its spontaneous breaking sets the stage for low-energy QCD. The rigorous description of low-energy QCD in terms of effective theories, namely Chiral Perturbation Theory (ChPT) in its various versions, the availability of fundamental experiments, and most recently the advent of lattice-QCD calculations with small quark masses, are signs of progress that continues unabated, leading to very accurate tests of QCD’s chiral dynamics.

ChPT is a low-energy effective field theory of QCD, in which the degrees of freedom are the eight Goldstone bosons of the hadronic world, corresponding to the \(\pi \), \(K\), and \(\eta \) mesons, and resulting from the spontaneous breakdown of the chiral \(\mathrm{SU}(3)_L\times \mathrm{SU}(3)_R\) symmetry in the limit of massless \(u\), \(d\), \(s\) quarks [530, 531]. ChPT can be readily extended to include the low-energy physics of ground-state light baryons, as well as that of heavy mesons and baryons.

We review here the most salient experimental and theoretical developments that have been accomplished recently in the areas of meson–meson and meson–nucleon dynamics, along with an outlook for the future.

3.4.1 \(\pi \pi \) and \(\pi K\) scattering lengths

Measurements of the \(S\)-wave \(\pi \pi \) scattering lengths represent one of the most precise tests of the \(\mathrm{SU}(2)_L^{}\times \mathrm{SU}(2)_R^{}\) sector of chiral dynamics. The NA48/2 experiment at the CERN SPS [446] has analyzed, on the basis of more than one million events, the \(K_{e4}\) decay \(K^{\pm }\rightarrow \pi ^+\pi ^{-}e^{\pm }\nu \). The analysis of the corresponding form factors, and through them of the \(\pi \pi \) final-state interactions, has led to the currently most accurate determination of the \(S\)-wave isospin-0 and isospin-2 scattering lengths \(a_0^0\) and \(a_0^2\), where \(a_\ell ^I\) denotes the channel with orbital angular momentum \(\ell \) and isospin \(I\). In this analysis, a crucial role is played by isospin breaking effects [532]. An additional improvement has been attained by combining the latter results with those of the experiment NA48/2 on the nonleptonic decay \(K^{\pm }\rightarrow \pi ^{\pm }\pi ^0\pi ^0\), with more than 60 million events, and the impact of the cusp properties at \(\pi ^0\pi ^0\) threshold, due to the mass difference between charged and neutral pions. The current results are summarized by:
$$\begin{aligned}&m_{\pi }a_0^0=0.2210\pm 0.0047_{\mathrm {stat}} \pm 0.0040_{\mathrm {syst}}, \nonumber \\&m_{\pi }a_0^2=-0.0429\pm 0.0044_{\mathrm {stat}} \pm 0.0018_{\mathrm {syst}}, \nonumber \\&m_{\pi }(a_0^0-a_0^2)=0.2639\pm 0.0020_{\mathrm {stat}} \pm 0.0015_{\mathrm {syst}}, \end{aligned}$$
(3.19)
where \(m_{\pi }\) is the charged pion mass. The agreement with the ChPT result at two-loop order [533] is striking:
$$\begin{aligned}&m_{\pi }a_0^0=0.220\pm 0.005, \nonumber \\&m_{\pi }a_0^2=-0.0444\pm 0.0010, \nonumber \\&m_{\pi }(a_0^0-a_0^2)=0.265\pm 0.004. \end{aligned}$$
(3.20)
The \(\pi \pi \) scattering amplitude is usually analyzed with the aid of the so-called Roy equations [534], which are fixed-\(t\) dispersion relations based on analyticity, crossing symmetry and unitarity. The corresponding representation has been used in [533] to check the consistency of the chiral representation and of the corresponding values of the scattering lengths and to restrict as much as possible the resulting uncertainties. Dispersion relations and Roy equations have also been used in [535], without the input of ChPT, to analyze the \(\pi \pi \) scattering amplitude; using high-energy data and the \(K_{e4}\) decay measurements, results in agreement with those of [533] have been found.

Recently, the NA48/2 collaboration also measured the branching ratio of \(K_{e4}\) decay [536], which permits the determination of the normalization of the corresponding form factors. This in turn can be used for additional tests of ChPT predictions.

On the other hand, the measurement of the \(K_{\mu 4}\) decay [537] will give access to the \(R\) form factor, which is not detectable in \(K_{e4}\) decay, since it contributes to the differential decay rate with a multiplicative factor proportional to the lepton mass squared. \(R\) is one of the three form factors associated with the matrix element of the axial vector current; it is mostly sensitive to the matrix element of the divergence of the axial vector current and hence brings information about the chiral symmetry breaking parameters.

Distinct access to the \(\pi \pi \) scattering lengths is provided through the DIRAC experiment at CERN, which measures the lifetime of the pionium atom. The atom, because of the mass difference between the charged and neutral pions, decays mainly into two \(\pi ^0\)’s. The decay width is proportional, at leading non-relativistic order, to \((a_0^0-a_0^2)^2\) [538]. Corrections coming from relativistic effects, photon radiative corrections, and isospin breaking must be taken into account to render the connection between the lifetime and the strong interaction scattering lengths more accurate: these amount to a 6 % effect [539] (and references therein). The DIRAC experiment, which started almost 10 years ago, reached last year the objective of measuring the pionium lifetime with an error smaller than 10 %. From a sample of \(21000\) pionium atoms a 4 % measurement of the difference of the \(\pi \pi \) scattering lengths has been obtained [540]:
$$\begin{aligned} m_{\pi }|a_0^0-a_0^2|=0.2533^{+0.0080}_{-0.0078}|_ {\mathrm {stat}}{}^{+0.0078}_{-0.0073}|_{\mathrm {syst}}, \end{aligned}$$
(3.21)
a result which is in agreement with those of (3.20) and (3.19), taking into account the relatively large uncertainty.

In the future, the DIRAC Collaboration also aims to measure the \(2s-2p\) energy splitting, which would allow for the separate measurements of the two \(S\)-wave scattering lengths. Another project of the collaboration is the study of the properties of the \(\pi K\) atom, in analogy with the pionium case, thus providing the \(S\)-wave \(\pi K\) scattering lengths [541, 542]. Preliminary tests of the experiment at CERN have already begun [543].

A review of the status of several scattering processes which are sensitive to the spontaneous and explicit chiral symmetry breaking of QCD can be found in [544].

The analysis of the \(\pi K\) scattering process is a particularly representative computation in ChPT in the presence of a strange quark. Calculations, similar to those of the \(\pi \pi \) scattering amplitude, have been carried out. The elastic scattering amplitude has been evaluated in one- and two-loop order [545, 546]. One finds a slow but reasonable convergence of the results at each step of the evaluation. The \(S\)-wave isospin 1/2 and 3/2 scattering lengths are found at the two-loop order:
$$\begin{aligned} m_{\pi }a_0^{1/2}=+0.220,\ \ \ \ \ \ m_{\pi }a_0^{3/2}=-0.047. \end{aligned}$$
(3.22)
The uncertainties, not quoted explicitly, depend on the variations of the parameters that enter in the modeling of the \(O(p^6)\) low energy constants.
The experimental values of the scattering lengths are obtained by using Roy–Steiner equations [534, 547], which generalize the Roy equations to the \(\pi K\) system, and high-energy data for \(\pi K\) scattering [548], leading to:
$$\begin{aligned} m_{\pi }a_0^{1/2}&= +0.224\pm 0.022,\nonumber \\ m_{\pi }a_0^{3/2}&= -0.0448\pm 0.0077. \end{aligned}$$
(3.23)
The agreement between the ChPT evaluation and the experimental output seems satisfactory, with, however, larger uncertainties than in the \(\pi \pi \) case.

Efforts are also being made to extract the \(\pi K\) phase shifts from the nonleptonic decays of \(D\) and \(B\) mesons [549, 550, 551, 552]. The results are not yet sufficiently precise to allow for quantitative comparisons with previous work.

In recent years, the lattice-QCD determination of the \(\pi \pi \) and \(\pi K\) scattering lengths is providing increasingly accurate results in full QCD [553, 554, 555, 556, 557, 558, 559]. This work is still maturing, as can be seen in the wide range of both central values and error estimates (some of which are not yet complete). A comparative summary of lattice-QCD results can be found in [559]. Once all sources of uncertainty are controlled, however, one can foresee the time when lattice QCD will compete with and even supersede the experimental extraction of scattering lengths.

3.4.2 Lattice QCD calculations: quark masses and effective couplings

While the determination of scattering lengths in lattice QCD is still at an early stage, other quantities, such as quark masses or low-energy constants (LECs) of mesonic ChPT, have been obtained with high overall precision and controlled systematic uncertainties. The “Flavour Averaging Group” (FLAG) has set itself the task of collecting and compiling the available lattice results for phenomenologically relevant quantities [44, 45]. Furthermore, FLAG provides a critical assessment of individual calculations regarding control over systematic effects. Results which satisfy a set of quality criteria are then combined to form global estimates. Here we briefly summarize the results and discussions in [45], related to determinations of the light quark masses and LECs. We focus on QCD with \(2+1\) dynamical quarks, which corresponds to a degenerate doublet of \(u,d\) quarks, supplemented by the heavier strange quark.

The FLAG estimates for the strange quark mass, \(m_\mathrm{s}\), and the average light quark mass, \(\hat{m}\equiv \frac{1}{2}(m_u+m_d)\), were obtained by combining the results of [37, 39, 40], with [2] as an important cross check. In the \(\mathrm \overline{MS}\) scheme at a scale 2 GeV one finds
$$\begin{aligned}&\hat{m} = 3.42 \pm 0.06_\mathrm{stat}\pm 0.07_\mathrm{sys}\,\mathrm{MeV}, \end{aligned}$$
(3.24)
$$\begin{aligned}&{m_\mathrm{s}} = 93.8 \pm 1.5_\mathrm{stat}\pm 1.9_\mathrm{sys}\,\mathrm{MeV}. \end{aligned}$$
(3.25)
The FLAG estimate for  the scheme- and scale-independent ratio \(m_\mathrm{s}/\hat{m}\), in which some systematic effects cancel, reads
$$\begin{aligned} m_\mathrm{s}/\hat{m} = 27.46\pm 0.15_\mathrm{stat}\pm 0.41_\mathrm{sys}. \end{aligned}$$
(3.26)
In order to provide separate estimates for the masses of the up and down quarks, one has to account for isospin breaking effects, stemming from both the strong and electromagnetic interactions. Current lattice estimates of \(m_u\) and \(m_d\) are mostly based on additional input from phenomenology [39, 40, 560]. In some cases, electromagnetic effects (i.e., corrections to Dashen’s theorem [561]) have been determined via the inclusion of a quenched electromagnetic field [562, 563]. The FLAG results for \(m_u, m_d\) are obtained by combining the global lattice estimate for \(\hat{m}\) with the ChPT estimate for the ratio \(m_u/m_d\) and phenomenological estimates of electromagnetic self-energies. In the \(\overline{\mathrm{MS}}\) scheme at 2 GeV this yields
$$\begin{aligned}&{m_u} = 2.16 \pm 0.09_\mathrm{stat+sys}\pm 0.07_\mathrm{e.m.}\,\mathrm{MeV}, \end{aligned}$$
(3.27)
$$\begin{aligned}&{m_d} = 4.68 \pm 0.14_\mathrm{stat+sys}\pm 0.07_\mathrm{e.m.}\,\mathrm{MeV}, \end{aligned}$$
(3.28)
$$\begin{aligned}&m_u/m_d = 0.46\pm 0.02_\mathrm{stat+sys}\pm 0.02_\mathrm{e.m.}. \end{aligned}$$
(3.29)
For a detailed discussion we refer the reader to the FLAG report [45]. The quark mass ratio \(Q\), defined by
$$\begin{aligned} Q^2=(m_\mathrm{s}^2-\hat{m}^2)/(m_\mathrm{d}^2-m_u^2), \end{aligned}$$
(3.30)
is a measure of isospin-breaking effects. By combining (3.24), (3.25), (3.27), and (3.28) one arrives at the lattice estimate
$$\begin{aligned} Q=22.6\pm 0.7_\mathrm{stat+sys}\pm 0.6_\mathrm{e.m.}. \end{aligned}$$
(3.31)
In addition to providing accurate values for the light quark masses, lattice QCD has also made significant progress in determining the effective couplings of ChPT. This concerns not only the LECs that arise at order \(p^2\) in the chiral expansion, i.e., the chiral condensate \(\Sigma \) and the pion decay constant in the chiral limit, \(F\), but also the LECs that appear at \(O(p^4)\). Moreover, lattice QCD can be used to test the convergence properties of ChPT, since the bare quark masses are freely tunable parameters, except for the technical limitation that simulations become less affordable near the physical pion mass.
The recent FLAG averages for the leading-order LECs for QCD with \(2+1\) flavors read [45]
$$\begin{aligned} \Sigma =(265\pm 17)^3\,\mathrm{MeV}^3, \quad F_\pi /F=1.0620\pm 0.0034, \end{aligned}$$
(3.32)
where \(F_\pi /F\) denotes the ratio of the physical pion decay constant over its value in the chiral limit. As discussed in detail in Sect. 5 of [45], there are many different quantities and methods which allow for the determination of the LECs of either SU(2) or SU(3) ChPT. The overall picture that emerges is quite coherent, as one observes broad consistency among the results, independent of the details of their extraction. For specific estimates of the \(O(p^4)\) LECs we again refer to the FLAG report. Despite the fact that the LECs can be determined consistently using a variety of methods, some collaborations [349, 564] have reported difficulties in fitting their data to SU(3) ChPT for pion masses above 400 MeV. Whether this is due to the employed “partially quenched” setting, in which the sea and valence quark masses are allowed to differ, remains to be clarified.

3.4.3 \(\mathrm{SU}(3)_L\times \mathrm{SU}(3)_R\) global fits

Due to the relatively large value of the strange quark mass with respect to the masses of the nonstrange quarks, the matter of the convergence and accuracy of \(\mathrm{SU}(3)_L^{}\times \mathrm{SU}(3)_R^{}\) ChPT becomes of great importance. In the meson sector, this has been investigated over a long period of time by Bijnens and collaborators [565] at NNLO in the chiral expansion. Taking into account new experimental data, mainly on the \(K_{e4}\) and \(K_{\ell 3}\) form factors, a new global analysis has been carried out up to \(O(p^6)\) effects [566]. The difficulty of the task comes from the fact that the number of LECs at two-loop order, \(C_i^r\), is huge and no unambiguous predictions of them are possible. One is left here with educated guesses based on naive dimensional analysis or model calculations. Several methods of estimate have been used and compared with each other. It turns out that the most consistent estimate of the LECs \(C_i^r\) comes from their evaluation with the resonance saturation scheme. One is then able to extract from the various experimental data the values of the \(O(p^4)\) LECs \(L_i^r\). \(\mathrm{SU}(3)_L\times \mathrm{SU}(3)_R\) ChPT seems now to satisfy improved convergence properties concerning the expressions of the meson masses and decay couplings, a feature which was not evident in the past evaluations. Nevertheless, the new global fit still suffers from several drawbacks, mainly related to a bad verification of the expected large-\(N_\mathrm{c}\) properties of some OZI-rule violating quantities. Another drawback is related to the difficulty of reproducing the curvature of one of the form factors of the \(K_{e4}\) decay. Incorporation of latest lattice-QCD results is expected to improve the precision of the analysis.

The question of the convergence of \(\mathrm{SU}(3)_L\times \mathrm{SU}(3)_R\) ChPT has also led some authors to adopt a different line of approach. It has been noticed that, because of the proximity of the strange quark mass value to the QCD scale parameter \(\Lambda _{\mathrm{QCD}}\), vacuum fluctuations of strange quark loops may be enhanced in OZI-rule violating scalar sectors and hence may cause instabilities invalidating the conventional counting rules of ChPT [567] in that context. To cure that difficulty, it has been proposed to treat the quantities that may be impacted by such instabilities with resummation techniques. Analyses, supported by some lattice-QCD calculations [349, 564], seem to provide a consistent picture of three-flavor ChPT [568], at least for pion masses below about 400 MeV.

The problem of including strangeness in Baryon Chiral Perturbation Theory (BChPT) is, on the other hand, still a wide open question. It is particularly striking in the quark mass expansion of the baryon masses, where very large nonanalytic terms proportional to \(M_K^3\) indicate a failure of the chiral expansion, and this happens in every known version of BChPT. In other observables, such as the axial couplings, magnetic moments, and meson-baryon scattering, certain versions of BChPT, namely, those including the decuplet baryons as explicit degrees of freedom, lead to important improvements in its convergence. As discussed later, these latter versions are motivated by the \(1/N_\mathrm{c}\) expansion, and they provide several such improvements which lend a strong support to their use.

3.4.4 \(\eta \rightarrow 3\pi \) and the nonstrange quark masses

A process of particular interest in ChPT is \(\eta \rightarrow 3\pi \) decay. This process is due to the breaking of isospin symmetry and therefore should allow for measurements of the nonstrange quark-mass difference. Nevertheless, attempts to evaluate the decay through the Dalitz plot analysis, at one-loop order [569], as well as at two-loop order [570], do not seem successful. One of the difficulties is related to the fit to the neutral-channel Dalitz plot slope parameter \(\alpha \), whose experimental value is negative, while ChPT calculations yield a positive value. To remedy difficulties inherent to higher-order effects, a dispersive approach has been suggested, in which \(\pi \pi \) rescattering effects are taken into account in a more systematic way [571].

Including new experimental data accumulated during recent years (Crystal Barrel [572], Crystal Ball [573, 574, 575], WASA [576, 577], KLOE [578]), several groups have reanalyzed the \(\eta \rightarrow 3\pi \) problem. Reference [579], using a modified non-relativistic effective field theory approach, shows that the failure to reproduce \(\alpha \) in ChPT can be traced back to the neglect of \(\pi \pi \) rescattering effects. References [580] and [581] tackle this problem using the dispersive method, which takes into account higher-order rescattering effects. The two groups use similar methods of approach and the same data, but differ in the imposed normalization conditions. The sign of the parameter \(\alpha \) is found to be negative in both works, but it is slightly greater in magnitude than the experimental value. The parameter that measures the isospin-breaking effect is \(Q\), defined in terms of quark masses; see (3.30). The value found for \(Q\) in [580] is \(Q=23.1\pm 0.7\), to be compared with the lattice-QCD evaluation \(Q=22.6\pm 0.9\) [45]. (Results of [581] are still preliminary and will not be quoted.)

It is possible to obtain the values of the nonstrange quark masses \(m_u\) and \(m_d\) from the value of \(Q\), provided one has additional information about the strange quark mass \(m_\mathrm{s}\) and about \(\hat{m}\). Using the lattice-QCD results \(m_\mathrm{s}=(93.8\pm 2.4)\) MeV and \(\hat{m}=(3.42\pm 0.09)\) MeV [45], calculated in the \(\overline{\mathrm{MS}}\) scheme at the running scale \(\mu =2\) GeV, one finds [580]
$$\begin{aligned} m_u=(2.23\pm 0.14)\, \mathrm {MeV},\, \, \, \, m_d=(4.63\pm 0.14)\, \mathrm {MeV}, \end{aligned}$$
(3.33)
which are in good agreement with the lattice-QCD results [45] quoted in (3.27) and (3.28).

Some qualitative differences exist between [580] and [581]. The key point concerns the Adler zeros [582] for the \(\eta \rightarrow 3\pi \) decay amplitude, whose existence is derived here as a consequence of a \(\mathrm{SU}(2)_L^{}\times \mathrm{SU}(2)_R^{}\) low-energy theorem [583], therefore not using the expansion in terms of the strange quark mass. While the two solutions are close to each other in the physical region, they differ in the unphysical region where the Adler zeros exist. The solution obtained in [580] does not seem to display, for small non-zero values of the nonstrange quark masses, any nearby Adler zeros. However, the authors of [580] point out that the quadratic slopes of the amplitude are not protected by the above mentioned symmetry and might find larger corrections than expected.

If the difference between the results of the above two approaches persists in the future, it might be an indication that the detailed properties of the \(\eta \rightarrow 3\pi \) decays are not yet fully under control. A continuous effort seems still to be needed to reach a final satisfactory answer. For the most recent appraisal of the theoretical status, see [584].

On the experimental side, the \(\eta \rightarrow 3\pi \) width is determined through the branching ratio from the measurement of the \(\eta \rightarrow \gamma \gamma \) width. For a long time, measurements of the latter using the reaction \(e^+e^{-}\rightarrow e^+e^{-}\eta \) consistently gave a significantly higher value [1] than that of the old determination via the Primakoff effect [585]. However, a reanalysis of this result based on a new calculation of the inelastic background, due to incoherent \(\eta \) photoproduction, brought the Primakoff measurement in line with those at \(e^+e^{-}\) colliders [586]. A new Primakoff measurement has been proposed by the PRIMEX Collaboration at JLab, using the 11 GeV tagged photon beam to be delivered to Hall D, with the aim of a width measurement with an error of 3 % or less. Also, the large data base collected by Hall B at JLab contains a large sample of \(\eta \rightarrow \pi ^+\pi ^{-}\pi ^0\), of the order of \(2\times 10^6\) events, which will significantly improve the knowledge of its Dalitz distribution. A recent precise measurement of \(\Gamma _{\eta \rightarrow \gamma \gamma }\) by KLOE [587] shows a high promise of the new measurement planned with KLOE-2.

Isospin-breaking effects are also being investigated with lattice-QCD calculations, as recently reviewed in [588]. The effects of the quark-mass difference \(m_d-m_u\) on kaon masses, as well as on nucleon masses, have recently been studied in [560], in which earlier references can also be found. In addition, QED effects have also been included [589, 590, 591, 592]. These concern mainly the evaluation of the corrections to Dashen’s theorem [561], which establishes, in the \(\mathrm{SU}(3)_L\times \mathrm{SU}(3)_R\) chiral limit, relationships between the electromagnetic mass differences of hadrons belonging to the same \(\mathrm{SU}(3)_V\) multiplet. A summary of results regarding the latter subject, as well as about the ratio of the nonstrange quark masses, can be found in [593]. The issue of the ChPT LECs in the presence of electromagnetism and isospin breaking through lattice-QCD calculations is also considered in [594].

3.4.5 Other tests with electromagnetic probes

One of the classic low-energy processes is \(\pi ^0\rightarrow \gamma \gamma \) decay, which tests the Goldstone boson nature of the \(\pi ^0\) and the chiral Adler–Bell–Jackiw anomaly [324, 325]. This subject is considered at the end of Sect. 3.2.7b to which the reader is referred.

One important test remaining to be improved is that of the process \(\gamma \pi \rightarrow 2\pi \), whose amplitude \(F_{3\pi }\) is fixed in the chiral and low-energy limit by the chiral box anomaly. The two existing results for \(F_{3\pi }\), namely the Primakoff one [595] from Serpukhov and the recent analysis [596] of the \(e^{-}\pi ^{-}\rightarrow e^{-}\pi ^{-}\pi ^0\) data [597], disagree with each other and with leading order ChPT. Currently, data from COMPASS using the Primakoff effect for measuring \(\gamma \pi \rightarrow 2\pi \) are under analysis (for early results on the \(2\pi \) invariant mass spectrum see the COMPASS-II proposal [598]), and this result is expected to have a significant impact for experimentally establishing this important process. Recently, and motivated by the COMPASS measurement, a new theoretical analysis of \(\gamma \pi \rightarrow 2\pi \) has been carried out [599], in which the whole kinematic domain of this process is taken into account using ChPT supplemented with dispersion relations. In particular, this analysis gives also information that can be used to describe more accurately the amplitude \(\pi ^0\gamma \gamma ^*\), important in the analysis of light-by-light scattering and the muon’s \(g-2\).Theoretical works on the corrections to the contributions of the anomaly to \(\gamma \pi \rightarrow 2\pi \) have been addressed in ChPT to higher orders in [600, 601, 602], and in the vector meson dominance model [603]. \(F_{3\pi }\) has also been calculated from the pion’s Bethe–Salpeter amplitude, see [604] and references therein. In these and related studies three key constraints are met: the quark propagator and the pion amplitude respect the axial-vector Ward identity, the full quark–photon vertex fulfills an electromagnetic Ward identity, and a complete set of ladder diagrams beyond the impulse approximation are taken into account. The three conditions are necessary to reproduce the low-energy theorem for the anomalous form factor. Results at large momentum and nonvanishing pion mass agree with the limited data and exhibit the same resonance behavior as the phenomenological vector meson dominance model. The latter property signals that a dynamically calculated quark–photon vertex contains the \(\rho \) meson pole, and that in the relevant kinematical regions the vector mesons are the key physical ingredient in this QCD-based calculation. It seems that the time for an accurate test of \(\gamma \pi \rightarrow 2\pi \) has arrived. A recent additional test of the box anomaly contributions is the decay \(\eta \rightarrow \pi ^+\pi ^{-}\gamma \), which is currently being investigated in measurements at COSY (WASA) [605].

Another test of ChPT is provided by the measurement at COMPASS of \(\pi ^{-} \gamma \rightarrow \pi ^{-}\pi ^{-}\pi ^+\) at \(\sqrt{s}\le 5 M_\pi \) with an uncertainty in the cross section of 20 %. The results have been published in [606], along with a discussion of the good agreement with the leading-order ChPT result [607].

3.4.6 Hard pion ChPT

ChPT also describes situations in which pions are emitted by heavy mesons (\(K\), \(D\), \(B\), etc.). In such decays, there are regions of phase space where the pion is hard and where chiral counting rules can no longer be applied. It has been, however, argued that chiral logarithms, calculated in regions with soft pions, might still survive in hard pion regimes and therefore might enlarge, under certain conditions, the domain of validity of the ChPT analysis [608, 609]. This line of approach has been called “Hard pion ChPT” and assumes that the chiral logarithms factorize with respect to the energy dependence in the chiral limit. Such factorization properties have been carefully analyzed in [610] using dispersion relations and explicitly shown to be violated for pion form factors by the inelastic contributions, starting at three loops. The study in [610] is presently being extended to heavy-light form factors. This will help clarify to what degree of approximation and in what regions of phase space hard pion ChPT might have practical applications in the analysis of heavy meson decays.

3.4.7 Baryon chiral dynamics

Baryon chiral dynamics still represents a challenge, but very exciting progress is being made on three fronts: experimental, theoretical, and lattice QCD. Here we highlight some of them.

Combining data from pionic hydrogen and deuterium [611, 612], the \(\pi N\) scattering lengths have been extracted with the so-called Deser formula [538, 539], leading to [613]: \(m_{\pi }a_0^{-}=(86.1\pm 0.1)\times 10^{-3}\) and \(m_{\pi }a_0^+=(7.6\pm 3.1)\times 10^{-3}\), to be compared with the leading-order predictions [614]: \(m_{\pi }a_0^{-}\simeq 80\times 10^{-3}\) and \(m_{\pi }a_0^+=0\). (\(a_0^+\) and \(a_0^{-}\) are the \(S\)-wave isospin even and isospin odd scattering lengths, respectively. They are related to the isospin \(1/2\) and \(3/2\) scattering lengths through the formulas \(a_0^+=(a_0^{1/2}+2a_0^{3/2})/3\) and \(a_0^{-}=(a_0^{1/2}-a_0^{3/2})/3\).) In the same spirit, kaon–nucleon scattering lengths have been extracted from the combined data coming from kaonic hydrogen X-ray emissions [615] and kaon deuterium scattering [616]. The latter analysis uses data coming from the recent SIDDHARTA experiment at the DA\(\Phi \)NE electron–positron collider [617].

In spite of existing huge data sets on pion–nucleon scattering, the low-energy scattering amplitudes are still not known with great precision. And yet this is the region in which low-energy theorems and ChPT predictions exist. To remedy this deficiency, a systematic construction of \(\pi N\) scattering amplitudes has been undertaken in [618] using the Roy–Steiner equations, based on a partial wave decomposition, crossing symmetry, analyticity, and dispersion relations. This approach parallels the one undertaken for \(\pi K\) scattering [548], although in the present case the spin degrees of freedom of the nucleon considerably increase the number of Lorentz invariant amplitudes. It is hoped that a self-consistent iterative procedure between solutions obtained in different channels will yield a precise description of the low-energy \(\pi N\) scattering amplitude.

Another long-standing problem in \(\pi N\) physics is the evaluation of the pion–nucleon sigma term. In general, sigma terms are defined as forward matrix elements of quark mass operators between single hadronic states and are denoted, with appropriate indices, by \(\sigma \). More generally, the sigma terms are related to the scalar form factors of the hadrons, denoted by \(\sigma (t)\), where \(t\) is the momentum transfer squared, with \(\sigma (0)\) corresponding to the conventional sigma term. The interest in the sigma terms resides in their property of being related to the mass spectrum of the hadrons and to the scattering amplitudes through Ward identities. Concerning the pion–nucleon sigma term, in spite of an existing low-energy theorem [619], its full evaluation necessitates an extrapolation of the low-energy \(\pi N\) scattering amplitude to an unphysical region [620]. The result depends crucially on the way the data are analyzed. Several contradictory results have been obtained in the past, and this has given rise to much debate. Recent evaluations of the sigma term continue to raise the same questions. In [621], a relatively large value of the sigma term is found, \(\sigma =(59\pm 7)\) MeV, while in [622], the relatively low value of [620] is confirmed, \(\sigma =(43.1\pm 12.0)\) MeV; the two evaluations remain, however, marginally compatible. One application of the equations of [618] concerns a dispersive analysis of the scalar form factor of the nucleon [623]. This has allowed the evaluation of the correction \(\Delta _{\sigma }=\sigma (2m_{\pi }^2)-\sigma (0)\) of the scalar form factor of the nucleon, needed for the extraction of the \(\pi N\) sigma term from \(\pi N\) scattering. Using updated phase shift inputs, the value \(\Delta _{\sigma }=(15.2\pm 0.4)\) MeV has been found, confirming the earlier estimate of [624].

A complementary access to the sigma term is becoming possible thanks to lattice-QCD calculations of the nucleon mass at varying values of the quark masses [625]. The current limitations reside in the relatively large quark masses used, and also in the still significant error bars from calculations which employ the lowest possible quark masses. It is however feasible that in the near future results competitive in accuracy to the ones obtained from \(\pi N\) analyses will be available from lattice QCD.

One issue that has been open for a long time is the precise value of the \(g_{\pi NN}\) coupling. A new extraction by an analysis in [626] based on the Gell-Mann–Oakes–Renner (GMO) sum rule gives \(g_{\pi NN}^2/(4\pi )=13.69(12)(15)\). This value agrees with those of analyses favoring smaller values of the coupling. It, in particular, supports the argument based on the naturalness of the Goldberger–Treiman discrepancy when extended to \(\mathrm{SU}(3)\) [627].

A theoretical development in BChPT which has been taking place over many years is the development of effective theories with explicit spin 3/2 baryons degrees of freedom. It has been known for a long time [628] that the inclusion of the spin 3/2 decuplet improved the convergence of the chiral expansion for certain key quantities. The theoretical foundation for it is found in the \(1/N_\mathrm{c}\) expansion [629, 630], the key player being the (contracted) spin-flavor symmetry of baryons in large \(N_\mathrm{c}\). This has led to formulating BChPT in conjunction with the \(1/N_\mathrm{c}\) expansion [631, 632, 633, 634], a framework which has yet to be applied extensively but which has already shown its advantages. Evidence of this is provided by several works on baryon semileptonic decays [632, 633, 635], and in particular in the analysis of the nucleon’s axial coupling [634], where the cancellations between the contributions from the nucleon and \(\varDelta \) to one-loop chiral corrections are crucial for describing the near independence of \(g_A\) with respect to the quark masses as obtained from lattice-QCD calculations [206, 236, 237, 256, 259, 261]. We expect that many further applications of the BChPT\(\otimes 1/N_\mathrm{c}\) framework will take place in the near future, and it will be interesting to see what its impact becomes in some of the most difficult problems such as baryon polarizabilities, spin-polarizabilities, \(\pi N\) scattering, etc. Further afield, and addressed elsewhere in this review, are the applications to few-nucleon effective theories, of which the effective theory in the one-nucleon sector is a part. An interesting recent development in baryon lattice QCD is the calculation of masses at varying \(N_\mathrm{c}\) [636]. Although at this point the calculations are limited to quenched QCD, they represent a new tool for understanding the validity of \(N_\mathrm{c}\) counting arguments in the real world, which will be further improved by calculations in full QCD and at lower quark masses. For an analysis of the results in [636] in the light of BChPT\(\otimes 1/N_\mathrm{c}\) framework, see [637].

A new direction worth mentioning is the application of BChPT to the study of the nucleon partonic structure at large transverse distances [638], which offers an example of the possible applications of effective theories to the soft structures accompanying hard processes in QCD.

3.4.8 Other topics

Many other subjects are in the domains of interest and expertise of ChPT and are being studied actively. We merely quote some of them: pion and eta photoproduction off protons [639, 640, 641, 642, 643, 644], pion polarizabilities [332, 333] (see also Sect. 3.2.7c) and two-pion production in \(\gamma \gamma \) collisions [645], the decay \(\eta '\rightarrow \eta \pi \pi \) [646], the electromagnetic rare decays \(\eta '\rightarrow \pi ^0\gamma \gamma \) and \(\eta '\rightarrow \eta \gamma \gamma \) [647], \(K\) meson rare decays [648, 649], hadronic light-by-light scattering [650], etc. The incorporation of the \(\eta '\) meson into the ChPT calculations is usually done in association with the \(1/N_\mathrm{c}\) expansion [651], since for finite \(N_\mathrm{c}\), the \(\eta '\) is not a Goldstone boson in the chiral limit.

The above processes enlarge the field of investigation of ChPT, by allowing for the determination of new LECs and tests of nontrivial predictions. Some of the amplitudes of these processes do not receive contributions at tree level and have as leading terms \(O(p^4)\) or \(O(p^6)\) loop contributions. Therefore they offer more sensitive tests of higher-order terms of the chiral expansion.

An important area of applications of ChPT is to weak decays, which unfortunately cannot be covered in this succinct review. Of particular current interest are the inputs to nonleptonic kaon decays, where lattice-QCD calculations have been steadily progressing and are making headway in understanding old, difficult problems such as the \(|\Delta I|=1/2\) rule [652]. We refer to [44] for a review of the current status of kaon nonleptonic decays vis-à-vis lattice QCD. Many topics in baryon physics have also not been touched upon, among them the study of low-energy aspects of the EM properties of baryons such as the study of polarizabilities, in particular, the spin polarizabilities and generalized polarizabilities as studied with electron scattering [653, 654].

3.4.9 Outlook and remarks

As a low-energy effective field theory of QCD, ChPT offers a solid and reliable framework for a systematic evaluation of various dynamical contributions, where the unknown parts are encoded within a certain number of low-energy constants (LECs). Two-flavor ChPT is well established, founded on a firm ground. The main challenge now concerns the convergence properties of three-flavor ChPT, where a definite progress in our understanding of the role of the strange quark is still missing. Efforts are being continued in this domain, and it is hoped that new results coming from lattice-QCD calculations will help clarify the situation. Another specific challenge concerns the understanding of isospin breaking, including the evaluation of electromagnetic effects, in the decay \(\eta \rightarrow 3\pi \). New domains of interest, such as the probe of hard-pion regions in heavy-particle decays, \(\eta '\) physics, and rare kaon decays, are being explored. This, together with data provided by high-precision experimental projects, gives confidence in the progress that should be accomplished in the near future.

In baryons, the present progress in lattice QCD is leading to an important understanding of the behavior of the chiral expansion thanks to the possibility of studying the quark-mass dependence of key observables. Although issues remain, such as the problem in confronting with the empirical value of \(g_A\), it is clear that lattice QCD will have a fundamental impact in our understanding of the chiral expansion in baryons. Further, the union of BChPT and the \(1/N_\mathrm{c}\) expansion represents a very promising framework for further advances in the low-energy effective theory for baryons.

3.5 Low-energy precision observables and tests of the Standard Model

3.5.1 The muon’s anomalous magnetic moment

The muon’s anomalous magnetic moment, \(a_{\mu }=(g-2)_{\mu }/2\), is one of the most precisely measured quantities in particle physics, reaching a precision of 0.54 ppm. The most recent experimental measurement, BNL 821 [655], is
$$\begin{aligned} 10^{10}a_{\mu } = 11\,659\,208.9(6.3). \end{aligned}$$
(3.34)
This result should be compared with the theoretical calculation within the Standard Model (a topic worthy of a review in itself):using the compilation in [657]. Here the leading-order (LO) hadronic vacuum polarization is taken from measurements of \(R(e^+e^{-}\rightarrow \text {hadrons})\), and the electroweak (EW) corrections have been adjusted slightly to account for the (since measured) Higgs mass \(M_\mathrm{H}=125~\text {GeV}\). While QED and electroweak contributions account for more than 99.9999 % of the value \(a_{\mu }\), the dominant errors in (3.35) stem from the hadronic vacuum polarization (HVP) and hadronic light-by-light (HLbL) scattering—they stem from QCD.

The difference between the values in (3.34) and (3.35) is \(28.5\pm 6.3_\text {expt}\pm 4.9_\text {SM}\), which is both large—larger than the EW contributions \(19.5-3.9=15.6\)—and significant—around \(3.5\sigma \). This deviation has persisted for many years and, if corroborated, would provide a strong hint for physics beyond the Standard Model. This situation has motivated two new experiments with a target precision of 0.14 ppm, FNAL 989 [658], and J-PARC P34 [659]. The new experiments have, in turn, triggered novel theoretical efforts with the objective to obtain a substantial improvement of the theoretical values of the QCD corrections to the muon anomaly. In this section, we address HVP and HLbL in turn, discussing approaches (such as \(R(e^+e^{-})\)) involving other experiments, lattice QCD, and for HLbL also models of QCD.

The principal phenomenological approach to computing the HVP contribution \(a_{\mu }^\mathrm{had;VP}\) is based on the optical theorem and proceeds by evaluating a dispersion integral, using the experimentally measured cross section for \(e^{+}e^{-}\rightarrow \mathrm hadrons\). Evaluations of various authors use the same data sets and basically agree, differing slightly in the computational methods and final uncertainties deslightly pending on the conservatism of the authors [660, 661, 662]. Note that recent measurements of \(\sigma (e^+e^{-} \rightarrow \pi ^+\pi ^{-})\), the process dominating the LO HVP contribution, performed using initial-state radiation at BaBar [663] and KLOE [664, 665] do not show complete agreement with each other and with the previous measurements based on direct scans [666, 667]. Determination of the cross section in all these experiments, in particular those using initial-state radiation, crucially depends on the rather complicated radiative corrections.

An alternative phenomenological approach is to use \(\tau \) decay to hadrons to estimate the HVP. This approach is very sensitive to the way isospin-breaking corrections are evaluated. While a model-dependent method trying to take into account various effects due to \(m_d \ne m_u\) still shows notable deviation from the \(e^+e^{-}\) based estimate [668], the authors of [661] claim that after correcting the \(\tau \) data for the missing \(\rho \)\(\omega \) mixing contribution, in addition to the other known isospin-symmetry-violating corrections, \(e^+e^{-}\) and \(\tau \)-based calculations give fully compatible results.

To complement the phenomenological approach, it is desirable to determine the contributions due to HVP from first principles. Lattice QCD is usually restricted to space-like momenta, and in [669, 670] it was shown that \(a_{\mu }^\mathrm{had;VP}\) can be expressed in terms of a convolution integral, i.e.,
$$\begin{aligned} a_{\mu }^\mathrm{VP;had} = 4\pi ^2\left( \frac{\alpha }{\pi }\right) ^2 \int _0^\infty \mathrm{d}Q^2\, f(Q^2)\left\{ \Pi (Q^2)-\Pi (0) \right\} , \end{aligned}$$
(3.36)
where the vacuum polarization amplitude, \(\Pi (Q^2)\), is determined by computing the correlation function of the vector current. Recent calculations based on this approach appeared in [671, 672, 673, 674, 675, 676, 677], and a compilation of published results is shown in Fig. 27.
Fig. 27

Compilation of recently published lattice QCD results for the leading hadronic vacuum polarization contribution to the muon’s anomalous magnetic moment. Displayed is \(10^{10}a_{\mu }^\mathrm{VP;had}\), from ETM [672, 676], CLS/Mainz [674], RBC/UKQCD [673] and Aubin et al. [671]. The position and width of the red vertical line denote the phenomenological result from dispersion theory and its uncertainty, respectively

The evaluation of the correlation function of the electromagnetic current involves quark-disconnected diagrams, which are also encountered in isoscalar form factors of the nucleon discussed earlier in this section. Given that a statistically precise evaluation is very costly, such contributions have been largely neglected so far. Another major difficulty arises from the fact that the known convolution function \(f(Q^2)\) in (3.36) is peaked at momenta around the muon mass, which is a lot smaller than the typical nonzero momentum that can be achieved on current lattices. Therefore, it appears that lattice estimates of \(a_{\mu }^\mathrm{VP;had}\) are afflicted with considerable systematic uncertainties related to the low-\(Q^2\) region. In [674] it was therefore proposed to apply partially twisted boundary conditions [678, 679] to compute the quark-connected part of the correlator. In this way, it is possible to obtain a very high density of data points, which penetrate the region where the convolution integral receives its dominant contribution.

Recently, there have been proposals which are designed to overcome this problem. In [680, 681] the subtracted vacuum polarization amplitude, \(\Pi (Q^2)-\Pi (0)\), is expressed as an integral of a partially summed vector-vector correlator, which is easily evaluated on the lattice for any given value of the \(Q^2\). Furthermore, a method designed to compute the additive renormalization \(\Pi (0)\) directly, i.e., without the need for an extrapolation to vanishing \(Q^2\), has been proposed [682].

The compilation of recent lattice results for the leading hadronic vacuum polarization contributions and their comparison to the standard dispersive approach in Fig. 27 shows that the accuracy of current lattice estimates of \(a_{\mu }^\mathrm{{VP;had}}\) is not yet competitive. In particular, statistical uncertainties will have to be considerably reduced before lattice results can challenge the accuracy of dispersion theory. One step in this direction has been taken in [683], which advocates the use of efficient noise reduction techniques, dubbed “all-mode-averaging”. Other recent activities include the study of the systematic effects related to the use of twisted boundary conditions [684] and the Ansatz used to extrapolate \(\Pi (Q^2)\) to vanishing \(Q^2\) [685].

For HLbL, a direct experimental determination analogous to those discussed for HVP is not directly available. HLbL enters in \(\mathcal{O}(\alpha ^3)\), just as the NLO HVP does. The latter, however, is assessed in a dispersion relation framework [686], similar to that of the LO piece—the piece associated with the electromagnetic dressing of the HVP is part of the final-state radiative correction to the LO HVP term [687]. As for the HLbL term, it must be calculated; we refer to [656, 688] for reviews. The diagrammatic contributions to it can be organized in a simultaneous expansion in momentum \(p\) and number of colors \(N_\mathrm{c}\) [689]; the leading contribution in \(N_\mathrm{c}\) is a \(\pi ^0\) exchange graph. The computation of HLbL requires integration over three of the four photon momenta. Detailed analysis reveals that the bulk of the integral does not come from small, virtual momenta, making ChPT of little use. Consequently heavier meson exchanges should be included as well; this makes the uncertainties in the HLbL computation more challenging to assess. We have reported the HLbL result determined by the consensus of different groups [656]. Recently there has been discussion of the charged-pion loop graph (which enters as a subleading effect) in chiral perturbation theory, arguing that existing model calculations of HLbL are inconsistent with the low-energy structure of QCD [650]. Including the omitted low-energy constants in the usual framework does modify the HLbL prediction at the 10 % level [690]. The upshot is that the uncertainties can be better controlled through measurement of the pion polarizability (or generally of processes involving a \(\pi ^+\pi ^{-} \gamma ^*\gamma ^*\) vertex such as \(e^{+} e^{-} \rightarrow e^{+} e^{-} \pi ^+\pi ^0\)), which is possible at JLab [691]. As long recognized, data on \(\pi ^0\rightarrow \gamma \gamma ^*\), \(\pi ^0\rightarrow \gamma ^*\gamma ^*\), as well as \(\pi ^0 \rightarrow e^{+} e^{-} (\gamma )\), should also help in constraining the primary \(\pi ^0\) exchange contribution. Recently, a dispersive framework for the analysis of the \(\pi \) and \(2\pi \) intermediate states (and generalizable to other mesons) to HLbL has been developed [692]; we are hopeful in regards to its future prospects.

Unfortunately, lattice-QCD calculations of HLbL are still at a very early stage. A survey of recent ideas with a status report is given in [693]. Here, we comment briefly on only two approaches: the extended Nambu–Jona–Lasinio (ENJL) model (see, e.g., [694]) and a functional approach based on calculations of Landau-gauge-QCD Green’s functions (see, e.g., [695, 696]). The latter is based on a model interaction (cf. the remarks on the Faddeev approach to nucleon observables in Sect. 3.2.5b). However, such a calculation based on input determined from first-principle calculations would be highly desirable.

In the ENJL model one has a nonrenormalizable contact interaction, and consequently a momentum-independent quark mass and no quark wave-function renormalization. The quark–photon vertex is modeled as a sum of the tree-level term and a purely transverse term containing the vector meson pole. On the other hand, the Green function approach is based on an interaction according to the ultraviolet behavior of QCD and is therefore renormalizable. The resulting quark propagator is characterized by a momentum-dependent quark mass and a momentum-dependent quark wave-function renormalization. The quark–photon vertex is consistently calculated and contains a dynamical vector meson pole. Although the different momentum dependencies cancel each other partly (which is understandable when considering the related Ward identities), remarkable differences in these calculations remain. Based on a detailed comparison the authors of [695] argue that the suppression of the quark-loop reported in the ENJL model is an artifact of the momentum-independent quark mass and the momentum restriction within the quark–photon vertex, which, in turn, are natural consequences of the contact interaction employed there. Regardless of whether one concludes from these arguments that the standard value for the hadronic light-by-light scattering contribution may be too small, one almost inevitably needs to conclude that the given comparison provides evidence that the systematic error attributed to the ENJL calculation is largely underestimated.

As it is obvious that an increased theoretical error leads to a different conclusion on the size of the discrepancy of the value for \(a_{\mu }\) between the theoretical and experimental values, an increased effort on the QCD theory side is needed. One important aspect of future lattice calculations of the hadronic light-by-light contribution is to employ them in a complementary way together with other methods. For instance, an identification of the relevant kinematics of the hadronic contribution to the photon four-point function through the cross-fertilization of different approaches might already pave the way for much more accurate computations. The forthcoming direct measurement of \(a_{\mu }\) at FNAL is expected to reduce the overall error by a factor of five. Therefore, a significant improvement of the theoretical uncertainty for the hadronic light-by-light scattering contribution down to the level of 10 % is required. Hereby the systematic comparison of different approaches such as effective models, functional methods, and lattice gauge theory may be needed to achieve this goal.

3.5.2 The electroweak mixing angle

The observed deviation between direct measurements and theoretical predictions of the muon anomalous magnetic moment—if corroborated in the future—may be taken as a strong hint for physics beyond the Standard Model. Another quantity which provides a stringent test of the Standard Model is the electroweak mixing angle, \(\sin ^2\theta _W\). There is, however, a three-sigma difference between the most precise experimental determinations of \(\sin ^2\theta _W(M_Z)_\mathrm{\overline{MS}}\) at SLD [697], measuring the left-right asymmetry in polarized \(e^{+}e^{-}\) annihilation, and LEP [698], which is based on the forward-backward asymmetry in \(Z\rightarrow b\bar{b}\). The origin of the tension between these two results has never been resolved. While an existing measurement at the Tevatron [699, 700] is not accurate enough to decide the issue, it will be interesting to see whether the LHC experiments can improve the situation.

The value of \(\sin ^2\theta _W\) can be translated into a value of the Higgs mass, given several other SM parameters as input, including the strong coupling constant \(\alpha _\mathrm{s}\), the running of the fine-structure constant \(\Delta \alpha \), and the mass of the top quark, \(m_t\). The two conflicting measurements at the \(Z\)-pole lead to very different predictions for the Higgs mass \(m_\mathrm{H}\) [701], which can be confronted with the direct Higgs mass measurement at the LHC. In order to decide whether any observed discrepancy could be a signal for physics beyond the Standard Model, further experimental efforts to pin down the value of \(\sin ^2\theta _W(M_Z)_\mathrm{\overline{MS}}\) are required.
Fig. 28

The scale dependence of the electroweak mixing angle in the \(\mathrm \overline{MS}\) scheme. The blue band is the theoretical prediction, while its width denotes the theoretical uncertainty from strong interaction effects. From [1]

In addition to the activities at high-energy colliders, there are also new (QWEAK [702]) and planned experiments (MOLLER [703], P2@MESA), designed to measure the electroweak mixing angle with high precision at low energies, by measuring the weak charge of the proton. These efforts extend earlier measurements of the parity-violating asymmetry in Møller scattering [704] and complement other low-energy determinations, based on atomic parity violation (APV) and neutrino-DIS (NuTeV). The collection of measurements across the entire accessible energy range can be used to test whether the running of \(\sin ^2\theta _W\) is correctly predicted by the SM, i.e., by checking that the different determinations can be consistently translated into a common value at the \(Z\)-pole. The current status is depicted in Fig. 28.

We will now discuss the particular importance of low-energy hadronic determinations of the electroweak mixing angle, and the role of new experiments (for an in-depth treatment, see [705]). These are based on measuring the weak charge of the proton, \(Q_W^p\), which is accessible by measuring the helicity-dependent cross section in polarized \(ep\) scattering. For a precise determination of the electroweak mixing angle, one must augment the tree-level relation \(Q_W^p=1-4\sin ^2\theta _W\) by radiative corrections [706]. It then turns out that the dominant theoretical uncertainty is associated with hadronic effects, whose evaluation involves some degree of modeling [707]. Radiative corrections arising from \({\gamma }Z\) box graphs play a particularly important role, and their contributions have been evaluated in [708, 709, 710, 711, 712]. An important feature is that they are strongly suppressed at low energies. It is therefore advantageous to measure the weak charge in low-energy \(ep\) scattering, since the dominant theoretical uncertainties in the relation between \(Q_W^p\) and \(\sin ^2\theta _W\) are suppressed.

3.5.3 \(\alpha _\mathrm{s}\) from inclusive hadronic \(\tau \) decay

As remarked several times in this review, the precise determination of \(\alpha _\mathrm{s}\) at different scales, and hereby especially the impressive agreement between experimental determinations and theoretical predictions, provides an important test of asymptotic freedom and plays a significant role in establishing QCD as the correct fundamental theory of the Strong Interaction.

Hadronic \(\tau \) decays allow for a determination of \(\alpha _\mathrm{s}\) at quite low momentum scales [713]. The decisive experimental observable is the inclusive ratio of \(\tau \) decay widths,
$$\begin{aligned} R_\tau \equiv \frac{\Gamma [\tau ^{-} \rightarrow \nu _\tau {\mathrm {hadrons} \, (\gamma )]}}{\Gamma [\tau ^{-} \rightarrow \nu _\tau e^{-}\bar{\nu }_e (\gamma )]} , \end{aligned}$$
(3.37)
which can be rigorously analyzed with the short-distance operator product expansion.

Since non-perturbative corrections are heavily suppressed by six powers of the \(\tau \) mass, the theoretical prediction is dominated by the perturbative contribution, which is already known to \(O(\alpha _\mathrm{s}^4)\) and amounts to a 20 % increase of the naive parton-model result \(R_\tau = N_C = 3\). Thus, \(R_\tau \) turns out to be very sensitive to the value of the strong coupling at the \(\tau \) mass scale; see, e.g., [714] and references therein.

From the current \(\tau \) decay data, one obtains [714]
$$\begin{aligned} \alpha _\mathrm{s}(m_\tau ^2) = 0.331\pm 0.013. \end{aligned}$$
(3.38)
The recent Belle measurement of the \(\tau \) lifetime [715] has slightly increased the central value by \(+0.002\), with respect to the previous result [716]. After evolution to the scale \(M_Z\), the strong coupling decreases to
$$\begin{aligned} \alpha _\mathrm{s}(M_Z^2) = 0.1200\pm 0.0015, \end{aligned}$$
(3.39)
in excellent agreement with the direct measurement at the \(Z\) peak, \(\alpha _\mathrm{s}(M_Z^2) = 0.1197\pm 0.0028\) [1]. Owing to the QCD running, the error on \(\alpha _\mathrm{s}\) decreases by one order of magnitude from \(\mu = m_\tau \) to \(\mu =M_Z\).

The largest source of uncertainty has a purely perturbative origin. The \(R_\tau \) calculation involves a closed contour integration in the complex \(s\)-plane, along the circle \(|s| = m_\tau ^2\). The long running of \(\alpha _\mathrm{s}(-s)\) generates powers of large logarithms, \(\log {(-s/m_\tau ^2)}=i \phi \), \(\phi \in [-\pi ,\pi ]\), which need to be resummed using the renormalization group. One gets in this way an improved perturbative series, known as contour-improved perturbation theory (CIPT) [717], which shows quite good convergence properties and a mild dependence on the renormalization scale. A naive expansion in powers of \(\alpha _\mathrm{s}(m_\tau ^2)\) (fixed-order perturbation theory, FOPT), without resumming those large logarithms, gives instead a badly-behaved series which suffers from a large renormalization-scale dependence. A careful study of the contour integral shows that, even at \(O(\alpha _\mathrm{s}^4)\), FOPT overestimates the total perturbative correction by about 11 %; therefore, it leads to a smaller fitted value for \(\alpha _\mathrm{s}\). Using CIPT one obtains \(\alpha _\mathrm{s}(m_\tau ^2) = 0.341\pm 0.013\), while FOPT results in \(\alpha _\mathrm{s}(m_\tau ^2) = 0.319\pm 0.014\) [714].

The asymptotic nature of the perturbative QCD series has been argued to play an important role even at low orders in the coupling expansion. Assuming that the fourth-order series is already governed by the lowest ultraviolet and infrared renormalons, and fitting the known expansion coefficients to ad-hoc renormalon models, one predicts a positive correction from the unknown higher orders, which results in a total perturbative contribution to \(R_\tau \) close to the naive FOPT result [718]. This conclusion is however model dependent [714]. In the absence of a better understanding of higher-order corrections, the CIPT and FOPT determinations have been averaged in (3.38), but keeping the larger error.

A precise extraction of \(\alpha _\mathrm{s}\) at such low scale necessitates also a thorough understanding of the small non-perturbative condensate contributions to \(R_\tau \). Fortunately, the numerical size of non-perturbative effects can be determined from the measured invariant-mass distribution of the final hadrons in \(\tau \) decay [719]. With good data, one could also analyze the possible role of corrections beyond the operator product expansion. The latter are called duality violations (because they signal the breakdown of quark-hadron duality underlying the operator product expansion), and there is (as yet) no first-principle theoretical description available. These effects are negligible for \(R_\tau \), because the operator-product-expansion uncertainties near the real axis are kinematically suppressed in the relevant contour integral; however, they could be more relevant for other moments of the hadronic distribution.

The presently most complete and precise experimental analysis, performed with the ALEPH data, obtains a total non-perturbative correction to \(R_\tau \), \(\delta _{\mathrm {NP}} = -(0.59\pm 0.14)~\% \) [720], in good agreement with the theoretical expectations and previous experimental determinations by ALEPH, CLEO, and OPAL [714]. This correction has been taken into account in the \(\alpha _\mathrm{s}\) determination in (3.38). A more recent fit to rescaled OPAL data (adjusted to reflect current values of exclusive hadronic \(\tau \)-decay branching ratios), with moments chosen to maximize duality violations, finds \(\delta _{\mathrm {NP}} = -(0.3\pm 1.2)~\% \) [721], in agreement with the ALEPH result but less precise because of the much larger errors of the OPAL data.

A substantial improvement of the \(\alpha _\mathrm{s}(m_\tau ^2)\) determination requires more accurate \(\tau \) spectral-function data, which should be available in the near future, and a better theoretical control of higher-order perturbative contributions, i.e., an improved understanding of the asymptotic nature of the QCD perturbative series.

Experimental knowledge on \(\alpha _\mathrm{s}\) at even lower scales (\(s\!<\!m^2_\tau \)), at the borderline of the perturbative to non-perturbative regime of QCD, could profit from lattice simulations of appropriately chosen observables. Last but not least, it should be noted that in the non-perturbative domain, i.e., at scales below 1 GeV, an unambiguous definition of the strong coupling is missing; for a corresponding discussion see, e.g., [722].

3.6 Future directions

In a broad sense, the physics of light quarks remains a key for understanding strong QCD dynamics, from its more fundamental non-perturbative effects to the varied dynamical effects which manifest themselves in the different properties of hadrons. Recent progress in the theoretical and experimental fronts has been remarkable.

Numerous experimental results keep flowing from different facilities employing hadron (J-PARC, COSY, COMPASS, VES) or electron beams (CLAS, MAMI-C, ELSA, SPring-8, CLEO-c, BESIII, KLOE-2, and CMD-3 and SND at VEPP-2000). The experiments aim at investigating the full hadron spectrum, searching, e.g., for exotic and hybrid mesons or missing baryon resonances, as well as at determining dynamical properties of those excited states such as helicity amplitudes and form factors. New facilities are planned to come into operation in the next few years, which are expected to deliver data with extremely high statistical accuracy. The upcoming 12 GeV upgrade of JLab with the new Hall D is one of the key new additions to that line of research. Also the upgraded CLAS12 detector at JLab is expected to contribute to hadron spectroscopy. Hadronic decays of heavy-quark states produced at future \(e^+e^{-}\) (Belle II) or \(p\bar{p}\) machines (PANDA) will serve as abundant source of light-quark states with clearly defined initial states. In addition, the particularly clean access to light hadron states via direct production in \(e^+e^{-}\) annihilation with initial-state radiation, as well as via \(\gamma \gamma \) fusion, is possible at Belle II. The anticipated data from these next-generation experiments should, in principle, allow us to clarify the existence and nature of hadronic resonances beyond the quark model and to determine resonance parameters reliably for states where this has not been possible in the past because of pole positions far in the complex plane, overlapping resonances, or weak couplings to experimentally accessible channels. A model- and reaction-independent characterization of resonance parameters in terms of pole positions and residues, however, also requires advances on the analysis side to develop models which respect the theoretical constraints of unitarity and analyticity.

Experiments on the ground-state mesons and baryons will continue at the intermediate- and high-energy facilities, which can have an impact in and beyond QCD. Examples include the elucidation of the spin structure of the nucleon at the partonic level, which is one of the motivations for the work currently underway on the design of an Electron Ion Collider, precision photo-production on the nucleon and of light mesons, and experiments that impact the Standard Model, such as those necessary for improving the calculation of the hadronic contributions to the muon’s anomalous magnetic moment and the measurements of the weak charge of the nucleon, which impacts the knowledge of the EW angle at lower energies. Naturally, most of the topics discussed in this review are part of the broad experimental programs in place today and planned for the near future.

On the theoretical front, LQCD is opening new perspectives. Full QCD calculations with light quark masses nearing the physical limit are becoming standard. This is allowing for unprecedented insights into the quark mass dependencies of meson and baryon observables, which especially influence the determination of numerous LECs in EFT which are poorly known from phenomenology, and also in the knowledge of form factors and moments of structure functions. The study of excited light hadrons in LQCD is one of the most important developments in recent years, with the promise of illuminating the present rather sparse knowledge of those excited states, as well as possibly leading to the “discovery” of new states which are of difficult experimental access. It is clear that the progress in LQCD will continue, turning it into a key tool for exploration and discovery, as well as a precision tool for light quark physics.

Progress also continues with analytic methods, in particular with methods rooted in QCD, such as Schwinger-Dyson equations, ChPT, dispersion theory combined with ChPT, SCET, various approaches in perturbative QCD, \(1/N_\mathrm{c}\) expansion, AdS/QCD, etc. Most analytic methods rely on experimental and/or lattice QCD information, which is currently fueling theoretical progress thanks to the abundance and quality of that information.

4 Heavy quarks

7Heavy quarks have played a crucial role in the establishing and development of QCD in particular, and the Standard Model of particle physics in general. Experimentally this is related to a clean signature of many observables even in the presence of only few rare events, which allows the study of both new emergent phenomena in the realm of QCD and new physics beyond the Standard Model. Theoretically, the clean signature may be traced back to the fact that
$$\begin{aligned} m_Q \gg \Lambda _{\mathrm{QCD}}, \end{aligned}$$
(4.1)
which implies that processes happening at the scale of the heavy-quark mass \(m_Q\) can be described by perturbative QCD and that non-perturbative effects, including the formation of background low-energy light hadrons, are suppressed by powers of \(\Lambda _{\mathrm{QCD}}/m_Q\). The hierarchy (4.1) gets complicated by lower energy scales if more than one heavy quark is involved in the physical process, but the basic fact that high-energy physics at the scale \(m_Q\) can be factorized from low-energy non-perturbative physics at the hadronic scale \(\Lambda _{\mathrm{QCD}}\) is at the core of the dynamics of any system involving a heavy quark.

The hierarchy (4.1) is usually exploited to replace QCD with equivalent Effective Field Theories (EFTs) that make manifest at the Lagrangian level the factorization of the high-energy modes from the low-energy ones. Examples are the Heavy Quark Effective Theory (HQET) [723, 724, 725, 726] suitable to describe systems made of one heavy quark, and EFTs like Non-relativistic QCD (NRQCD) [727, 728] or potential Non-relativistic QCD (pNRQCD) [729, 730], suitable to describe systems made of two or more heavy quarks. Non-relativistic EFTs [731] have been systematically used both in analytical and in numerical (lattice) calculations involving heavy quarks. Concerning lattice studies, nowadays the standard approach is to resort to EFTs when bottom quarks are involved, and to rely on full lattice QCD calculations when studying systems made of charm quarks.

The section aims at highlighting some of the most relevant progress made in the last few years in the heavy-quark sector of QCD both from the methodological and phenomenological point of view. There is no aim of completeness. It is organized in the following way. In Sect. 4.1 we discuss methodological novelties in the formulation of non-relativistic EFTs and in lattice QCD, whereas the following sections are devoted to more phenomenological aspects. These are divided in phenomenology of heavy-light mesons, discussed in Sect. 4.2 and in phenomenology of heavy quarkonia. In Sect. 4.3 we present recent progress in quarkonium spectroscopy with particular emphasis on the quarkonium-like states at and above the open flavor threshold. Section 4.4 provides an updated list of \({\alpha _{\mathrm{s}}}\) extractions from quarkonium observables. Section 4.5 summarizes our current understanding of quarkonium production. Finally, Sect. 4.6 outlines future directions.

4.1 Methods

4.1.1 Non-relativistic effective field theories

The non-relativistic EFT of QCD suited to describe a heavy quark bound into a heavy-light meson is HQET [725, 732] (see [726] for an early review). Heavy-light mesons are characterized by only two energy scales: the heavy quark mass \(m_Q\) and the hadronic scale \(\Lambda _{\mathrm{QCD}}\). Hence the HQET Lagrangian is organized as an expansion in \(1/m_Q\) and physical observables as an expansion in \(\Lambda _{\mathrm{QCD}}/m_Q\) (and \({\alpha _{\mathrm{s}}}\) encoded in the Wilson coefficients). In the limit where \(1/m_Q\) corrections are neglected, the HQET Lagrangian is independent of the flavor and spin of the heavy quark. This symmetry is called the heavy quark symmetry. Some of its phenomenological consequences for \(B\) and \(D\) decays will be discussed in Sect. 4.2.

In the case of two or more heavy quarks, the system is characterized by more energy scales. We will focus on systems made of a quark and an antiquark, i.e. quarkonia, although EFTs have been also developed for baryons made of three quarks [733, 734, 735]. For quarkonia, one has to consider at least the scale of the typical momentum transfer between the quarks, which is also proportional to the inverse of the typical distance, and the scale of the binding energy. In a non-relativistic bound state, the first goes parametrically like \(m_Qv\) and the second like \(m_Qv^2\), where \(v\) is the velocity of the heavy quark in the center-of-mass frame. An EFT suited to describe heavy quarkonia at a scale lower than \(m_Q\) but larger than \(m_Qv\) and \(\Lambda _{\mathrm{QCD}}\) is NRQCD [727, 728] (whose lattice version was formulated in [736, 737]). Also the NRQCD Lagrangian is organized as an expansion in \(1/m_Q\) and physical observables as an expansion in \(v\) (and \({\alpha _{\mathrm{s}}}\) encoded in the Wilson coefficients). In the heavy-quark bilinear sector the Lagrangian coincides with the one of HQET (see also [738]), but the Lagrangian contains also four-quark operators. These are necessary to describe heavy-quarkonium annihilation and production, which are processes happening at the scale \(m_Q\). The NRQCD factorization for heavy quarkonium annihilation processes has long been rigorously proved [728], while this is not the case for heavy quarkonium production. Due to its relevance, we devote the entire Sect. 4.1.2 to the most recent progress towards a proof of factorization for heavy quarkonium production. The state of the art of our understanding of heavy quarkonium production in the framework of NRQCD is presented in Sect. 4.5.

The power counting of NRQCD is not unique because the low-energy matrix elements depend on more than one residual energy scale. These residual scales are \(m_Qv\), \(m_Q v^2\), \(\Lambda _{\mathrm{QCD}}\) and possibly other lower energy scales. The ambiguity in the power counting is reduced and in some dynamical regimes solved by integrating out modes associated to the scale \(m_Qv\) and by replacing NRQCD by pNRQCD, an EFT suited to describe quarkonium physics at the scale \(m_Qv^2\) [729, 730]. The pNRQCD Lagrangian is organized as an expansion in \(1/m_Q\), inherited from NRQCD, and an expansion in powers of the distance between the heavy quarks. This second expansion reflects the expansion in the scale \(m_Qv^2\) relative to the scale \(m_Qv\) specific to pNRQCD. Like in NRQCD, contributions to physical observables are counted in powers of \(v\) (and \({\alpha _{\mathrm{s}}}\) encoded in the high-energy Wilson coefficients). The degrees of freedom of pNRQCD depend on the specific hierarchy between \(m_Qv^2\) and \(\Lambda _{\mathrm{QCD}}\) for the system under examination.

The charmonium ground state and the lowest bottomonium states may have a sufficiently small radius to satisfy the condition \(m_Qv^2 \gtrsim \Lambda _{\mathrm{QCD}}\). If this is the case, the degrees of freedom of pNRQCD are quark–antiquark states and gluons. The system can be studied in perturbative QCD, non-perturbative contributions are small and in general one may expect precise theoretical determinations once potentially large logarithms have been resummed by solving renormalization group equations and renormalon-like singularities have been suitably subtracted. For early applications we refer to [739, 740, 741, 742, 743, 744], for a dedicated review see [745]. As an example of the quality of these determinations, we mention the determination of the \(\eta _b\) mass in [743]. This was precise and solid enough to challenge early experimental measurements, while being closer to the most recent ones. We will come back to this and other determinations in Sect. 4.3.

Excited bottomonium and charmonium states are likely strongly bound, which implies that \(\Lambda _{\mathrm{QCD}}\gtrsim m_Qv^2\). The degrees of freedom of pNRQCD are colorless and made of color-singlet quark–antiquark and light quark states [746, 747, 748, 749, 750]. The potentials binding the quark and antiquark have a rigorous expression in terms of Wilson loops and can be determined by lattice QCD [751, 752, 753, 754]. It is important to mention that lattice determinations of the potentials have been performed so far in the quenched approximation. Moreover, at order \(1/m_Q^2\) not all the necessary potentials have been computed (the set is complete only for the spin-dependent potentials). This implies that the quarkonium dynamics in the strongly coupled regime is not yet exactly known beyond leading \(1/m_Q\) corrections.

For states at or above the open flavor threshold, new degrees of freedom may become important (heavy-light mesons, tetraquarks, molecules, hadro-quarkonia, hybrids, glueballs,\(\ldots \)). These states can in principle be described in a very similar framework to the one discussed above for states below threshold [755, 756, 757, 758]. However, a general theory does not exist so far and specific EFTs have been built to describe specific states (an example is the well-known \(X(3872)\) [759, 760, 761, 762, 763, 764]). This is the reason why many of our expectations for these states still rely on potential models.

In Sect. 4.3 we will discuss new results concerning the charmonium and bottomonium spectroscopy below, at and above threshold, the distinction being dictated by our different understanding of these systems. For instance, we will see that there has been noteworthy progress in describing radiative decays of quarkonium below threshold and that the theory is now in the position to provide for many of the transitions competitive and model-independent results.

Finally, on a more theoretical side, since the inception of non-relativistic EFTs there has been an ongoing investigation on how they realize Lorentz invariance. It has been shown in [738, 765] that HQET is reparameterization invariant. Reparameterization invariance constrains the form of the Wilson coefficients of the theory. In [766, 767] it was shown that the same constraints follow from imposing the Poincaré algebra on the generators of the Poincaré group in the EFT. Hence reparameterization invariance appears as the way in which Lorentz invariance, which is manifestly broken by a non-relativistic EFT, is retained order by order in \(1/m_Q\) by the EFT. This understanding has recently been further substantiated in [768], where the consequences of reparameterization and Poincaré invariance have been studied to an unprecedented level of accuracy.

4.1.2 The progress on NRQCD factorization

The NRQCD factorization approach to heavy quarkonium production, introduced as a conjecture [728], is phenomenologically successful in describing existing data, although there remain challenges particularly in connection with polarization observations [757]. In the NRQCD factorization approach, the inclusive cross section for the direct production of a quarkonium state \(H\) at large momentum transfer (\(p_\mathrm{T}\)) is written as a sum of “short-distance” coefficients times NRQCD long-distance matrix elements (LDMEs),
$$\begin{aligned} \sigma ^H(p_\mathrm{T},m_Q) = \sum _n \sigma _n(p_\mathrm{T},m_Q,\Lambda ) \langle 0| \mathcal{O}_n^H(\Lambda )|0\rangle . \end{aligned}$$
(4.2)
Here \(\Lambda \) is the ultraviolet cut-off of the NRQCD effective theory. The short-distance coefficients \(\sigma _n\) are essentially the process-dependent partonic cross sections to produce a \(Q\bar{Q}\) pair in various color, spin, and orbital angular momentum states \(n\) (convolved with the parton distributions of incoming hadrons for hadronic collisions), and perturbatively calculated in powers of \(\alpha _\mathrm{s}\). The LDMEs are non-perturbative, but universal, representing the probability for a \(Q\bar{Q}\) pair in a particular state \(n\) to evolve into a heavy quarkonium. The sum over the \(Q\bar{Q}\) states \(n\) is organized in terms of powers of the pair’s relative velocity \(v\), an intrinsic scale of the LDMEs. For charmonia, \(v^2\approx 0.3\), and for bottomonia, \(v^2\approx 0.1\). The current successful phenomenology of quarkonium production mainly uses only NRQCD LDMEs through relative order \(v^4\), as summarized in Table 5. The traditional color singlet model is recovered as the \(v\rightarrow 0\) limit. In case of \(P\) wave quarkonia and relativistic corrections to \(S\) state quarkonia [769], the color singlet model is incomplete, due to uncanceled infrared singularities.
Table 5

NRQCD velocity scaling of the LDMEs contributing to \({^3}S_1\) quarkonium production up to the order \(O(v^4)\) relative to the leading \({^3}S_1\) color singlet contribution [728]. Upper indices \(^{[1]}\) refer to color singlet states and upper indices \(^{[8]}\) to color octet states. The \(\langle \mathcal{O}^{J/\psi }({^3}S_1^{[1]}) \rangle \), \(\langle \mathcal{P}^{J/\psi }({^3}S_1^{[1]}) \rangle \), and \(\langle \mathcal{Q}^{J/\psi }({^3}S_1^{[1]}) \rangle \) LDMEs correspond to the leading order, and the \(O(v^{2})\) and \(O(v^4)\) relativistic correction contributions to the color singlet model. The contributions involving the \(\langle \mathcal{O}^{J/\psi }({^1}S_0^{[8]}) \rangle \), \(\langle \mathcal{O}^{J/\psi }({^3}S_1^{[8]}) \rangle \) and \(\langle \mathcal{O}^{J/\psi }({^3}P_J^{[8]}) \rangle \) LDMEs are often referred to as the Color Octet states

Relative scaling

Contributing LDMEs

1

\(\langle \mathcal{O}^{H}({^3}S_1^{[1]}) \rangle \)

\(v^{2}\)

\(\langle \mathcal{P}^{H}({^3}S_1^{[1]}) \rangle \)

\(v^3\)

\(\langle \mathcal{O}^{H}({^1}S_0^{[8]}) \rangle \)

\(v^4\)

\(\langle \mathcal{Q}^{H}({^3}S_1^{[1]}) \rangle \), \(\langle \mathcal{O}^{H}({^3}S_1^{[8]}) \rangle \), \(\langle \mathcal{O}^{H}({^3}P_J^{[8]}) \rangle \)

Despite the well-documented phenomenological successes, there remain two major challenges for the NRQCD factorization approach to heavy quarkonium production. One is the validity of the factorization itself, which has not been proved, and the other is the difficulty in explaining the polarization of produced quarkonia in high-energy scattering, as will be reviewed in Sect. 4.5. These two major challenges could well be closely connected to each other, and could also be connected to the observed tension in extracting LDMEs from global analyses of all data from different scattering processes [770]. A proof of the factorization to all orders in \(\alpha _\mathrm{s}\) is complicated because gluons can dress the basic factorized production process in ways that apparently violate factorization. Although there is a clear scale hierarchy for heavy quarkonium, \(m_Q \gg m_Q v \gg m_Q v^2\), which is necessary for using an effective field theory approach, a full proof of NRQCD factorization would require a demonstration that all partonic diagrams at each order in \(\alpha _\mathrm{s}\) can be reorganized such that (1) all soft singularities cancel or can be absorbed into NRQCD LDMEs, and (2) all collinear singularities and spectator interactions can be either canceled or absorbed into incoming hadrons’ parton distributions. So far, this has been established at all orders only for exclusive production in helicity-non-flip processes in \(e^+e^{-}\) annihilation and \(B\)-meson decay [771, 772, 773].

For heavy quarkonium production at collider energies, there is sufficient phase space to produce more than one pair of heavy quarks, and additional observed momentum scales, such as \(p_\mathrm{T}\). The NRQCD factorization in (4.2) breaks when there are co-moving heavy quarks [774, 775]. The short-distance coefficient \(\sigma _n(p_\mathrm{T},m_Q,\Lambda )\) in (4.2) for a \(Q\bar{Q}(n)\) state can have different power behavior in \(p_\mathrm{T}\) at different orders in \(\alpha _\mathrm{s}\). For example, for \(Q\bar{Q}(^3S_1^{[1]})\)-channel, the Leading Order (LO) coefficient in \(\alpha _\mathrm{s}\) is dominated by \(1/p_\mathrm{T}^8\), and the Next-to-Leading Order (NLO) dominated by \(1/p_\mathrm{T}^6\), while the Next-to-Next-to-Leading Order (NNLO) coefficient has terms proportional to \(1/p_\mathrm{T}^4\). When \(p_\mathrm{T}\) increases, the logarithmic dependence of \(\alpha _\mathrm{s}\) on the hard scale cannot compensate the power enhancement in \(p_\mathrm{T}\) at higher orders, which leads to an unwanted phenomenon that the NLO correction to a given channel could be an order of magnitude larger than the LO contribution [776, 777]. Besides the power enhancement at higher orders, the perturbative coefficients at higher orders have higher powers of large \(\ln (p_\mathrm{T}^2/m_Q^2)\)-type logarithms, which should be systematically resummed. That is, when \(p_\mathrm{T}\gg m_Q\), a new organization of the short-distance coefficients in (4.2) or a new factorization formalism is necessary. Very significant progress has been made in recent years.

Two new factorization formalisms were derived for heavy quarkonium production at large \(p_\mathrm{T}\). One is based on perturbative QCD (pQCD) collinear factorization [778, 779, 780, 781, 782, 783, 784], and the other based on soft collinear effective theory (SCET) [785, 786]. Both approaches focus on quarkonium production when \(p_\mathrm{T}\gg m_Q\), and explore potential connections to the NRQCD factorization.

The pQCD collinear factorization approach, also referred to as the fragmentation function approach [757], organizes the contributions to the quarkonium production cross section in an expansion in powers of \(1/p_\mathrm{T}\), and then factorizes the leading power (and the next-to-leading power) contribution in terms of “short-distance” production of a single-parton of flavor \(f\) (and a heavy quark pair \([Q\bar{Q}(\kappa )]\) with \(\kappa \) labeling the pair’s spin and color) convolved with a universal fragmentation function for this parton (and the pair) to evolve into a heavy quarkonium,
$$\begin{aligned} \mathrm{d}\sigma _\mathrm{H}(p_\mathrm{T},m_Q)&\approx \sum _{f} d\hat{\sigma }_\mathrm{f}(p_\mathrm{T},z)\otimes D_{f\rightarrow H}(z,m_Q)\, \nonumber \\&+ \sum _{[Q\bar{Q}(\kappa )]} d\hat{\sigma }_{[Q\bar{Q}(\kappa )]}(p_\mathrm{T},z,u,v)\nonumber \\&\&\otimes \mathcal{D}_{[Q\bar{Q}(\kappa )] \rightarrow H}(z,u,v,m_Q) , \end{aligned}$$
(4.3)
where factorization scale dependence was suppressed, \(z,u,v\) are momentum fractions, and \(\otimes \) represents the convolution of these momentum fractions. Both the single parton and heavy quark pair fragmentation functions, \(D_\mathrm{f}\) and \(\mathcal{D}_{[Q\bar{Q}(\kappa )]}\), are universal, and we can resum large logarithms by solving the corresponding evolution equations [780, 782, 783]. The factorization formalism in (4.3) holds to all orders in \(\alpha _\mathrm{s}\) in pQCD up to corrections of \(\mathcal{O}(1/p_\mathrm{T}^4)\) (\(\mathcal{O}(1/p_\mathrm{T}^2)\)) with (without) a heavy quark pair, \([Q\bar{Q}(\kappa )]\), being produced [778, 780, 783].
Including the \(1/p_\mathrm{T}\)-type power correction into the factorized production cross section in (4.3) necessarily requires modifying the evolution equation of a single parton fragmentation function as [780, 783],
$$\begin{aligned}&\frac{\partial }{\partial \ln \mu ^2}D_{f\rightarrow H}(z,\mu ^2;m_Q) = \sum _{f'} \gamma _{f\rightarrow f'} \otimes D_{f'\rightarrow H} \nonumber \\&\quad +\, \frac{1}{\mu ^2} \sum _{[Q\bar{Q}(\kappa ')]} \gamma _{f\rightarrow [Q\bar{Q}(\kappa ')]} \otimes \mathcal{D}_{[Q\bar{Q}(\kappa ')]\rightarrow H}, \end{aligned}$$
(4.4)
where \(\otimes \) represents the convolution of momentum fractions as those in (4.3), and the dependence of momentum fractions in the right-hand-side is suppressed. The first line in (4.3) is effectively equal to the well-known DGLAP evolution equation. The second term on the right of (4.3) is new, and is needed for the single-parton fragmentation functions to absorb the power collinear divergence of partonic cross sections producing a “massless” (\(m_Q/p_\mathrm{T} \sim 0\)) heavy quark pair to ensure that the short-distance hard part, \(\hat{\sigma }_{[Q\bar{Q}(\kappa )]}(p_\mathrm{T},z,u,v)\) in (4.3), is infrared and collinear safe [784]. The modified single-parton evolution equation in (4.4), together with the evolution equation of heavy quark-pair fragmentation functions [780, 783, 785],
$$\begin{aligned}&\frac{\partial }{\partial \ln \mu ^2}\mathcal{D}_{[Q\bar{Q}(\kappa )]\rightarrow H}(z,u,v,\mu ^2;m_Q) \nonumber \\&\quad = \sum _{[Q\bar{Q}(\kappa ')]} \Gamma _{[Q\bar{Q}(\kappa )]\rightarrow [Q\bar{Q}(\kappa ')]} \otimes \mathcal{D}_{[Q\bar{Q}(\kappa ')]\rightarrow H}, \end{aligned}$$
(4.5)
forms a closed set of evolution equations of all fragmentation functions in (4.3). The \(\mathcal{O}(\alpha _\mathrm{s}^2)\) evolution kernels for mixing the single-parton and heavy quark-pair fragmentation functions, \(\gamma _{f\rightarrow [Q\bar{Q}(\kappa ')]}\) in (4.4), are available [783], and the \(\mathcal{O}(\alpha _\mathrm{s})\) evolution kernels of heavy quark-pair fragmentation functions, \(\Gamma _{[Q\bar{Q}(\kappa )]\rightarrow [Q\bar{Q}(\kappa ')]}\) in (4.5), were derived by two groups [783, 786].

For production of heavy quarkonium, it is necessary to produce a heavy quark pair. The combination of the QCD factorization formula in (4.3) and the evolution equation in (4.4) presents a clear picture of how QCD organizes the contributions to the production of heavy quark pairs in terms of distance scale (or time) where (or when) the pair was produced. The first (the second) term in (4.3) describes the production of the heavy quark pairs after (at) the initial hard partonic collision. The first term in (4.4) describes the evolution of a single active parton before the creation of the heavy quark pair, and the power-suppressed second term summarizes the leading contribution to the production of a heavy quark pair at any stage during the evolution. Without the power-suppressed term in (4.4), the evolved single-parton fragmentation function is restricted to the situation when the heavy quark pair is only produced after the time corresponding to the input scale of the evolution \(\mu _0\gtrsim 2m_Q\). With perturbatively calculated short-distance hard parts [784] and evolution kernels [783, 786], the predictive power of the pQCD factorization formalism in (4.3) relies on the experimental extraction of the universal fragmentation functions at the input scale \(\mu _0\), at which the \(\ln (\mu _0^2/(2m_Q)^2)\)-type contribution is comparable to \((2m_Q/\mu _0)^2\)-type power corrections. It is these input fragmentation functions at \(\mu _0\) that are responsible for the characteristics of producing different heavy quarkonium states, such as their spin and polarization, since perturbatively calculated short-distance partonic hard parts and evolution kernels of these fragmentation functions are universal for all heavy quarkonium states.

The input fragmentation functions are universal and have a clear scale hierarchy \(\mu _0 \gtrsim 2m_Q \gg m_Q v\). It is natural to apply the NRQCD factorization in (4.2) to these input fragmentation functions as [781, 784]
$$\begin{aligned}&D_{f\rightarrow H}(z,m_Q,\mu _0) = \sum _n d_{f\rightarrow n}(z,m_Q,\mu _0)\langle 0| \mathcal{O}_n^H |0\rangle \nonumber \\&\mathcal{D}_{[Q\bar{Q}(\kappa )]\rightarrow H}(z,u,v,m_Q,\mu _0) \\&\quad = \sum _n d_{[Q\bar{Q}(\kappa )]\rightarrow n}(z,u,v,m_Q,\mu _0)\langle 0| \mathcal{O}_n^H | 0 \rangle \, . \nonumber \end{aligned}$$
(4.6)
The above NRQCD factorization for single-parton fragmentation functions was verified to NNLO [778], and was also found to be valid for heavy-quark pair fragmentation functions at NLO [787, 788]. But a proof to all orders in NRQCD is still lacking. If the factorization in (4.6) would be proved to be valid, the pQCD factorization in (4.3) is effectively a reorganization of the NRQCD factorization in (4.2) when \(p_\mathrm{T}\gg m_Q\), which resums the large logarithmic contributions to make the perturbative calculations much more reliable [781, 784]. In this case, the experimental extraction of the input fragmentation functions at \(\mu _0\) is reduced to the extraction of a few universal NRQCD LDMEs.

When \(p_\mathrm{T}\gg m_Q\), the effective theory, NRQCD, does not contain all the relevant degrees of freedom. In addition to the soft modes absorbed into LDMEs, there are also dangerous collinear modes when \(m_Q/p_\mathrm{T} \sim 0\). On the other hand, SCET [789, 790] is an effective field theory coupling soft and collinear degrees of freedom and should be more suited for studying heavy quarkonium production when \(p_\mathrm{T}\gg m_Q\). The SCET approach matches QCD onto massive SCET at \(\mu \sim p_\mathrm{T}\) and expands perturbatively in powers of \(\alpha _\mathrm{s}(p_\mathrm{T})\) with a power counting parameter \(\lambda \sim (2m_Q)/p_\mathrm{T}\). The approach derives effectively the same factorization formalism for heavy quarkonium production as that in (4.3) for the first two powers in \(\lambda \). However, the derivation in SCET, due to the way the effective theory was set up, does not address the cancellation of Glauber gluon interactions between spectators, and may face further difficulties having to do with infinite hierarchies of gluon energy scales, and therefore may be not as complete as in the pQCD approach. As expected, the new fragmentation functions for a heavy quark pair to fragment into a heavy quarkonium obey the same evolution equations derived in the pQCD collinear factorization approach. Recently, it was verified that the first-order evolution kernels for heavy-quark pair fragmentation functions calculated in both pQCD and SCET approaches are indeed consistent [782, 783, 786]. However, it is not clear how to derive the evolution kernels for mixing the single-parton and heavy quark-pair fragmentation functions, like \(\gamma _{f\rightarrow [Q\bar{Q}(\kappa ')]}\) in (4.4), in SCET [791].

In the SCET approach to heavy quarkonium production, the heavy-quark pair fragmentation functions defined in SCET are matched onto NRQCD after running the fragmentation scale down to the order of \(2m_Q\). It was argued [785] that the matching works and NRQCD results can be recovered under the assumption that the LDMEs are universal. However, the NRQCD factorization in (4.6) has not been proved to all orders in pQCD because of the potential for the input fragmentation functions to have light-parton jet(s) of order of \(m_Q\). It is encouraging that major progress has been achieved in understanding heavy quarkonium production and its factorization recently, but more work is still needed.

4.1.3 Lattice gauge theory

With ensembles at very fine lattice spacings becoming increasingly available due to the continuous growth of available computer power, simulations employing relativistic valence charm quarks are now becoming more and more common. Indeed, the first ensembles incorporating dynamical (sea) charm quarks [42, 792, 793] are beginning to become available.
Fig. 29

The charmonium spectrum from lattice simulations of the Hadron Spectrum Collaboration using \(N_\mathrm{f}=2+1\) flavors of dynamical light quarks and a relativistic valence charm quark on anisotropic lattices. The shaded boxes indicate the \(1\sigma \) confidence interval from the lattice for the masses relative to the simulated \(\eta _\mathrm{c}\) mass, while the corresponding experimental mass differences are shown as black lines. The \(D\overline{D}\) and \(D_\mathrm{s}\overline{D}_\mathrm{s}\) thresholds from lattice simulation and experiment are shown as green and grey dashed lines, respectively. From [812]

The heavy mass of the charm quark means that (since \(m_ca\!\not \ll \! 1\)) discretization effects cannot be completely neglected and have to be accounted for properly. This is possible using the Symanzik effective theory formalism [794, 795]. For any given lattice action, it is possible to formulate an effective theory (the Symanzik effective theory) defined in the continuum, which has the lattice spacing \(a\) as its dimensionful expansion parameter and incorporates all operators compatible with the symmetries of the lattice action (including Lorentz-violating term with hypercubic symmetry), and the short-distance coefficients of which are fixed by determining that it should reproduce the on-shell matrix elements of the lattice theory up to some given order in \(a\). The use of this effective theory in lattice QCD is twofold [796]: firstly, it provides a means to parameterize the discretization artifacts as a function of the lattice spacing, thus allowing an extrapolation to the \(a\rightarrow 0\) continuum limit from a fit to results obtained at a range of (sufficiently small) lattice spacings. Secondly, one can take different lattice actions discretizing the same continuum theory and consider a lattice action formed from their weighted sum with the weights chosen so as to ensure that the leading short-distance coefficients of the Symanzik effective action become zero for the resulting (improved) action. Examples of improved actions in current use are the Sheikholeslami-Wohlert (clover) action [797, 798], which removes the O(\(a\)) artifacts of the Wilson quark action, and the asqtad (\(a^2\) tadpole-improved) [799] and HISQ (Highly Improved Staggered Quark) [800] actions for staggered quarks. Likewise, it is possible to improve the lattice action for NRQCD [737, 801] so as to remove O(\(a^2\)) artifacts. The operators used to measure correlation functions may be improved in a similar fashion; cf. e.g. [802, 803, 804] for the O(\(a\)) improvement of the static-light axial and vector currents used in HQET. Finally, one can use HQET instead of the Symanzik theory to understand the cutoff effects with heavy quarks [805, 806, 807], which when applied to the Wilson or clover action is known as the Fermilab method [808].

Since the experimental discovery of the \(X(3872)\) resonance by the Belle collaboration [809], and the subsequent emergence of more and more puzzling charmonium-like states, the spectroscopy of charmonium has gained increased interest. Lattice studies of states containing charm quarks are thus of great importance, as they provide an a priori approach to charm spectroscopy. The use of relativistic charm quarks eliminates systematic uncertainties arising from the use of effective theories, leaving discretization errors as the leading source of systematic errors, which can in principle be controlled using improved actions.

A variety of lattice studies with different actions are now available, with both the HISQ [800] action [810], and O(\(a\))-improved Wilson fermions [811] having been used for a fully relativistic treatment of the charm quark. In addition, anisotropic lattices have been employed to improve the time resolution of the correlation functions to allow for better control of excited states [812] (Fig. 29). An important ingredient in all spectroscopy studies is the use of the variational method [356, 357, 358] to resolve excited states.

As flavor singlets, charmonium states also receive contributions from quark-disconnected diagrams representing quark–antiquark annihilation and mixing with glueball and light-quark states [813]. Using improved stochastic estimators, Bali et al. [814] have studied disconnected contributions, finding no resulting energy shift within the still sizeable statistical errors. The use of the new “distillation” method [355, 815] to estimate all-to-all propagators has been found to be helpful in resolving disconnected diagrams, whose contributions have been found to be small [816].

Besides the mixing with non-\(c\bar{c}\) states arising from the disconnected diagram contributions, quarkonium states above or near the open-charm threshold may also mix with molecular \(D\overline{D}\) and tetraquark states. Studies incorporating these mixings [814, 817] have found evidence for a tightly bound molecular \(D\overline{D}^*\) state. Recently, a study of \(DD^*\) scattering on the lattice [818] using Lüscher’s method [396] found the first evidence of an \(X(3872)\) candidate. It was found that the observed spectrum of states near the threshold depends strongly on the basis of operators used; in particular, the \(X(3872)\) candidate was not observed if only \(\bar{c}c\) operators, but no dimeson operators, were included in the basis, nor if the basis contained only dimeson, but no \(\bar{c}c\) operators. This was interpreted as evidence that the \(X(3872)\) might be the consequence of an accidental interference between \(\bar{c}c\) and scattering states. On the other hand, it could also be seen as rendering the results of this and similar studies doubtful in so far as it cannot be easily excluded that the inclusion of further operators might not change the near-threshold spectrum again. A significant challenge in this area is thus to clarify which operators are needed to obtain reliable physical results. Recently, it has been suggested [819] based on large-\(N\) arguments that for tetraquark operators the singly disconnected contraction is of leading order in \(1/N\) whenever it contributes. This would appear to apply also to the tetraquark operators relevant near the open-charm threshold, making use of all-to-all methods such as distillation [355, 815] (which was used in [818]) mandatory for near-threshold studies.

The spectra of the open-charm \(D\) and \(D_\mathrm{s}\) mesons have been studied by Mohler and Woloshyn [820] using the Fermilab formalism for the charm quark. It was found that while the ground state \(D\), \(D^*\), \(D_\mathrm{s}\) and \(D_\mathrm{s}^*\) masses were reasonably well reproduced, the masses of the \(D_J\) and \(D_{sJ}\) states from their simulation strongly disagreed with experiment; possible reasons include neglected contributions from mixing with multihadron states.

As for \(b\) quarks, the currently achievable lattice spacings do not allow the direct use of relativistic actions. An interesting development in this direction is the use of highly improved actions (such as HISQ [800]) to simulate at a range of quark masses around and above the physical charm quark mass, but below the physical \(b\) quark mass, in order to extrapolate to the physical \(b\) quark mass using Bayesian fits [821] incorporating the functional form of the expected discretization artifacts and \(1/m_Q\) corrections [822, 823]. This method relies on the convergence of the Symanzik expansion up to values of \(m_Qa\sim 1\), and of the heavy-quark expansion in the vicinity of the charm quark mass; neither assumption can be proven with present methods, but empirical evidence [824] suggests that at least for the heavy-quark expansion convergence is much better than might naively be expected. The removal of as many sources of discretization errors as possible, including using the \(N_\mathrm{f}=2+1+1\) HISQ MILC ensembles [825] with reduced sea quark discretization effects [826] might be helpful in addressing the question of the convergence of the expansion in \(a\).

Otherwise, simulations of \(b\) quarks need to rely on effective field theories, specifically non-perturbatively matched HQET [827, 828, 829] for heavy–light systems, and NRQCD or m(oving)NRQCD [737, 801, 830] for heavy–heavy, as well as heavy–light, systems. An important point to note in this context is that each discretization choice (such as the use of HYP1 versus HYP2 links [831, 832, 833] in the static action of HQET, or the use of different values of the stability parameter in the lattice NRQCD action [736, 834, 835]) within either approach constitutes a separate theory with its own set of renormalization constants which must be matched to continuum QCD separately.

The non-perturbative matching of HQET to quenched QCD at order \(1/m_b\) has been accomplished in [804], and subsequent applications to the spectroscopy [836] and leptonic decays [824] of the \(B_\mathrm{s}\) system have showcased the power of this approach. The extension to \(N_\mathrm{f}=2\) is well under way [837, 838], and future studies at \(N_\mathrm{f}=2+1\) are to be expected. Beyond the standard observables such as masses and decay constants, observables featuring in effective descriptions of strong hadronic interactions, such as the \(B^*B\pi \) coupling in Heavy Meson Chiral Perturbation Theory [246, 839, 840] and the \(B^{*'}\rightarrow B\) matrix element [841] have been studied successfully in this approach.

In NRQCD, until recently only tree-level actions were available. In [842, 843], the one-loop corrections to the coefficients \(c_1\), \(c_5\) and \(c_6\) of the kinetic terms in an \(\mathrm {O}(v^4)\) NRQCD lattice action have been calculated, and in [835, 844], the background field method has been used to calculate also the one-loop corrections to the coefficients \(c_2\) and \(c_4\) of the chromomagnetic \(\sigma \cdot \mathbf{B}\) and chromoelectric Darwin terms for a number of lattice NRQCD actions. Simulations incorporating these perturbative improvements [843, 845] have shown a reduced lattice-spacing dependence and improved agreement with experiment.

Matching the NRQCD action to QCD beyond tree-level has a significant beneficial effect on lattice determinations of bottomonium spectra, in particular for the case of the bottomonium \(1S\) hyperfine splitting, which moves from \(\Delta M_\mathrm{HF}(1S) = 61(14)\) MeV without the perturbative improvements [834] to \(\Delta M_\mathrm{HF}(1S) =70(9)\) MeV with the perturbative improvements [843].

The most recent determination based on lattice NRQCD, including \(O(v^6)\) corrections, radiative one-loop corrections to \(c_4\), non-perturbative four-quark interactions and the effect of \(u\), \(d\), \(s\) and \(c\) sea quarks, gives \(\Delta M_\mathrm{HF}(1S) = (62.8 \pm 6.7)\,~\mathrm {MeV}\) [846], which is to be compared to the PDG value of \(\Delta M_\mathrm{HF}(1S) = 69.3(2.9)\) MeV [1] excluding the most recent Belle data [847], or \(\Delta M_\mathrm{HF}(1S) = 64.5(3.0)\) MeV [1] when including them.

The resulting prediction for the bottomonium 2S hyperfine splitting of \(\Delta M_\mathrm{HF}(2S) = 35(3)(1)\) MeV [843] is in reasonable agreement with the Belle result \(\Delta M_\mathrm{HF}(2S) =24.3^{+4.0}_{-4.5}\) MeV [848], but disagrees with the CLEO result of Dobbs et al., \(\Delta M_\mathrm{HF}(2S) = 48.7(2.3)(2.1)\) MeV [849]; see discussion in Sect. 4.3.

Another factor with a potentially significant influence on the bottomonium hyperfine splitting is the lack of, or the inclusion of, spin-dependent interactions at higher orders in the non-relativistic expansion. In [850], it was shown that including the \(\mathrm {O}(v^6)\) spin-dependent terms in the NRQCD action leads to an increase in the 1S hyperfine splitting, moving it away from the experimental value. The results of [835] suggest that this effect will at least partially be compensated by the inclusion of perturbative corrections to the coefficients of the spin-dependent operators. The \(2S\) hyperfine splitting is not similarly affected, and the prediction \(\Delta M_\mathrm{HF}(2S) = 23.5(4.1)(2.1)(0.8)\) MeV of [850] is in excellent agreement with the Belle value [848].

The \(B_\mathrm{c}\) system combines the challenges of both the \(b\) and charm sectors, while also allowing for one of the relatively few predictions from QCD that is not to some extent a “postdiction” in that it precedes experiment, viz. the mass of the as yet undiscovered \(B_\mathrm{c}^*\) meson, which has been predicted by the HPQCD collaboration to be \(M_{B_\mathrm{c}^*}=6330(7)(2)(6)\) MeV [851] using NRQCD for the \(b\) and the HISQ action for the charm quarks. Reproducing this prediction using another combination of lattice actions might be worthwhile. For the time being, we note that the lattice prediction compares very well with the perturbative calculation of [852], which gives \(M_{B_\mathrm{c}^*}=6327(17)^{+15}_{-12}(6)\) MeV to next-to-leading logarithmic accuracy.

4.2 Heavy semileptonic decays

Semileptonic decays of \(B\) and \(D\) mesons have been extensively studied in the last years. They provide information about the CKM matrix elements \( |V_{cb}| \), \(|V_{ub}|\), \(|V_{cd}|\) and \(|V_{cs}|\) through exclusive and inclusive processes driven by \( b \rightarrow c(u)\) and \(c \rightarrow s(d)\) decays, respectively (for recent reviews see, e.g., Refs. [853, 854, 855, 856, 857]).

The leptonic decays \( B^{+} \rightarrow l^{+} \nu \) and \( D^+_{(s)} \rightarrow l^{+} \nu \) can also be used for the determination of CKM matrix elements. The advantages of semileptonic decays are that they are not helicity suppressed and new physics is not expected to play a relevant role; so, it is generally, but not always, disregarded.

In deep inelastic neutrino (or antineutrino)–nucleon scattering, single charm particles can be produced through \(dc\) and \(sc\) electroweak currents. Analyses based on neutrino and antineutrino interactions give a determination of \(|V_{cd}|\) with comparable, and often better, precision than the ones obtained from semileptonic charm decays. Not so for the determination of \(|V_{cs}|\), which suffers from the uncertainty of the s-quark sea content [1]. On-shell \(W^\pm \) decays sensitive to \(|V_{cs}|\) have also been used [858], but semileptonic \(D\) or leptonic \(D_\mathrm{s}\) decays provide direct and more precise determinations.

4.2.1 Exclusive and inclusive \(D\) decays

The hadronic matrix element for a generic semileptonic decay \(H \rightarrow P l \nu \), where \(H\) and \(P\) denote a heavy and a light pseudoscalar meson, respectively, is usually written in terms of two form factors \(f_+(q^2)\) and \(f_0(q^2)\)
$$\begin{aligned} \langle P(p_P)| J^\mu | H(p_\mathrm{H}) \rangle&= f_+(q^2) \left( p_\mathrm{H}^\mu +p_P^\mu \!-\! \frac{m^2_\mathrm{H}-m_P^2}{q^2} q^\mu \right) \nonumber \\&\!\! + f_0(q^2) \frac{m^2_\mathrm{H}-m_P^2}{q^2} q^\mu , \end{aligned}$$
(4.7)
where \( q \equiv p_\mathrm{H} - p_P\) is the momentum transferred to the lepton pair, and \(J^\mu \) denotes the heavy-to-light vector current. In the case of massless leptons, the form factor \(f_0(q^2)\) is absent and the differential decay rate depends on \(f_+(q^2) \) only.

The main theoretical challenge is the non-perturbative evaluation of the form factors. In this section, we consider \(H\) to be a \(D_{(q)}\) meson. For simplicity’s sake, one can split the non-perturbative evaluation of the form factors into two steps, the evaluation of their normalization at \(q^2=0\) and the determination of their \(q^2\) dependence.

The form factors are expected to decrease at low values of \(q^2\), that is at high values of spectator quark recoil. Indeed, in the leading spectator diagram, the probability of forming a hadron in the final state decreases as the recoil momentum of the spectator quark increases. Moreover, the form factors are expected to be analytic functions everywhere in the complex \(q^2\) plane outside a cut extending along the positive \(q^2\) axis from the mass of the lowest-lying \(c \bar{q}\) resonance. That implies they can be described by dispersion relations, whose exact form is not known a priori, but can be reasonably assumed to be dominated, at high \(q^2\), by the nearest poles to \(q^2_{\mathrm {max}}= (m_{D_{(q)}}-m_P)^2\). Pole dominance implies current conservation at large \(q^2\). We expect the form factors to have a singular behavior as \(q^2\) approaches the lowest lying poles, without reaching them, since they are beyond the kinematic cutoff. The simplest parameterization of the \(q^2\) dependence motivated by this behavior is the simple pole model, where a single pole dominance is assumed. By restricting to the form factor \(f_+(q^2)\), we have
$$\begin{aligned} f_+(q^2)= \frac{f_+(0)}{1-\frac{q^2}{m_{\mathrm {pole}}^2}}. \end{aligned}$$
(4.8)
In \(D \rightarrow \pi l \nu \) decays, the pole for \(f_+(q^2)\) corresponds to the \(c \bar{d}\) vector meson of lowest mass \(D^\star \). In \(D \rightarrow K l \nu \) and \(D_\mathrm{s} \rightarrow \eta ^{(\prime )} l \nu \) decays, the poles correspond to the \(c \bar{s}\) vector mesons and the lowest resonance compatible with \(J^P=1^{-}\) is \(D_\mathrm{s}^{*\pm }\), with mass \(M_ {D_\mathrm{s}^{*}}= 2112.3 \pm 0.5\) MeV. Form factor fits have been performed for \(D \rightarrow K(\pi ) l \nu \) by the CLEO [859] and BESIII Collaborations [860], where several models for the \(q^2\) shape have been considered. In the simple pole model, agreement with data is only reached when the value of \(m_{\mathrm {pole}}\) is not fixed at the \(D^\star _{(s)}\) mass, but is a free parameter. In order to take into account higher poles, while keeping the number of free parameters low, a modified pole model has been proposed [861], where
$$\begin{aligned} f_+(q^2)= \frac{f_+(0)}{ \left( 1-\frac{q^2}{m_{\mathrm {pole}}^2}\right) \left( 1- \alpha \frac{q^2}{m_{\mathrm {pole}}^2}\right) }. \end{aligned}$$
(4.9)
Another parameterization, known as the series or \(z\)-expansion, is based on a transformation that maps the cut in the \(q^2\) plane onto a unit circle in another variable, \(z\), and fits the form factor as a power series (in \(z\)) with improved properties of convergence [862, 863, 864]. More in detail, the first step is to remove poles by the form factors, that is, for the \(B \rightarrow K\) decays
$$\begin{aligned} \tilde{f}_0^{D \rightarrow K } (q^2)&= \left( 1 - \frac{q^2}{ m_{D^*_{s 0}}^2 }\right) f_0^{D \rightarrow K } (q^2) \nonumber \\ \tilde{f}_+^{D \rightarrow K } (q^2)&= \left( 1 - \frac{q^2}{ m_{D^*_{s}}^2 }\right) f_+^{D \rightarrow K } (q^2) \end{aligned}$$
(4.10)
The variable \(z\) is defined as
$$\begin{aligned} z(q^2) = \frac{ \sqrt{t_+ - q^2}- \sqrt{t_+ - t_0} }{ \sqrt{t_+ - q^2} + \sqrt{t_+ - t_0}} \qquad t_+ = (m_D+m_K)^2\nonumber \\ \end{aligned}$$
(4.11)
The final step consists in fitting \(\tilde{f}\) as a power series in \(z\),
$$\begin{aligned} \tilde{f}_0^{D \rightarrow K } (q^2)&= \sum _{n \ge 0} c_n z^n \nonumber \\ \tilde{f}_+^{D \rightarrow K } (q^2)&= \sum _{n \ge 0} b_n z^n \qquad c_0=b_0 \end{aligned}$$
(4.12)
Employing this parameterization, the shapes of \( f_{0,+}^{D \rightarrow K }\) form factors have been very recently estimated by the HPQCD Collaboration [865].

To evaluate the normalization of the form factors, lattice and QCD sum rules are generally employed. Lately, high statistics studies on the lattice have become available and preliminary results for \( f_{0,+}^{D \rightarrow K/\pi }\) have been presented by ETMC [866, 867], HPQCD [868] and Fermilab/MILC [869].

The most recent published \(|V_{cd}|\) estimates are from HPQCD [870], where the value of \(|V_{cd}|\) has been evaluated using the Highly Improved Staggered Quark (HISQ) action for valence charm and light quarks on MILC \(N_\mathrm{f}=2+1\) lattices with experimental inputs from CLEO [871] and BESIII [872]. The value \( |V_{cd}| = 0.223 \pm 0.010_{\mathrm {exp}} \pm 0.004_{\mathrm {lat}}\) [870], with the first error coming from experiment and the second from the lattice computation, is in agreement with the value of \(|V_{cd}|\) the same collaboration has recently extracted from leptonic decays. It also agrees, with a competitive error, with the value \( |V_{cd}| = 0.230 \pm 0.011\) [1] from neutrino scattering.

The same HPQCD collaboration gives also the most recent \(|V_{cs}| \) estimate by analyzing \(D \rightarrow K/\pi \, l \, \nu \), \(D_\mathrm{s} \rightarrow \phi /\eta _\mathrm{s} \, l \, \nu \) and using experimental inputs from CLEO [859], BaBar [873, 874], Belle [875]. and BESIII (preliminary) [860]. Their best value \( |V_{cs}| = 0.963 \pm 0.005_{\mathrm {exp}} \pm 0.014_{\mathrm {lat}}\) is in agreement with values from indirect fits [1]. The big increase in accuracy with respect to their older determinations, is due to the larger amount of data employed. Specifically they have used all experimental \(q^2\) bins, rather than just the \(q^2 \rightarrow 0\) limit or the total rate. The FLAG \(N_\mathrm{f}=2+1\) average value from semileptonic decays gives \(|V_{cs}|= 0.9746 \pm 0.0248 \pm 0.0067\) [876]. Experiments at BESIII, together with experiments at present and future flavor factories, all have the potential to reduce the errors on the measured decay branching fractions of \(D^+_{(s)}\) and \(D^0\) leptonic and semileptonic decays, in order to allow more precise comparison of these CKM matrix elements. In particular, BESIII is actively working on semileptonic charm decays; new preliminary results on the branching fractions and form factors in the parameterizations mentioned above, for the \(D \rightarrow K/\pi e \nu \) channels, have been recently reported [877].

Lattice determinations of the decay constant \(f_{D_\mathrm{s}}\) governing the leptonic decays \(D_\mathrm{s}^+\rightarrow \mu ^+\nu \) and \(D_\mathrm{s}^+\rightarrow \tau ^+\nu \) have for several years exhibited the “\(f_{D_\mathrm{s}}\) puzzle”, an apparent \((3-4)\sigma \) discrepancy between lattice determinations of \(f_{D_\mathrm{s}}\) [878, 879, 880, 881] and the value of \(f_{D_\mathrm{s}}\) inferred from experimental measurements of the branching ratios \(B(D_\mathrm{s}^+\rightarrow \mu ^+\nu )\) and \(B(D_\mathrm{s}^+\rightarrow \tau ^+\nu )\) [882, 883, 884, 885, 886]. When this discrepancy first appeared, it was immediately discussed as a signal for new physics [887]; in the meantime, however, careful investigation of all sources of systematic error, combined with increased statistics, has led to the lattice values shifting up slightly [888, 889, 890] and the experimental values shifting down noticeably [891, 892, 893, 894], thus more or less eliminating the “puzzle” [895]. However, the most recent determinations still show some tension versus the FLAG \(N_\mathrm{f}=2+1\) average value from leptonic decays \(|V_{cs}|= 1.018 \pm 0.011 \pm 0.021\) [876].

It is interesting to observe that, according to lattice determinations in [868], the form factors are insensitive to the spectator quark: The \(D_\mathrm{s} \rightarrow \eta _\mathrm{s} l \nu \) and \(D \rightarrow K l \nu \) form factors are equal within 3 %, and the same holds for \(D_\mathrm{s} \rightarrow K l \nu \) and \(D \rightarrow \pi l \nu \) within 5 %. This result, which can be tested experimentally, is expected by heavy quark symmetry to hold also for \(B\) meson decays so that the \(B_\mathrm{s} \rightarrow D_\mathrm{s}\) and \(B \rightarrow D\) form factors would be equal.

QCD light-cone sum rules have also been employed to extract \(|V_{cs}|\) and \(|V_{cd}|\) [896], giving substantial agreement on the averages and higher theoretical error with respect to the previously-quoted lattice results. By using the same data and a revised version of QCD sum rules, errors on \(|V_{cd}|\) have been reduced, but a higher average value has been obtained: \(|V_{cd}|= 0.244 \pm 0.005 \pm 0.003 \pm 0.008 \). The first and second errors are of an experimental origin and the third is due to the theoretical uncertainty [897].

Form factors for semileptonic transitions to a vector or a pseudoscalar meson have also been investigated within a model which combines heavy quark symmetry and properties of the chiral Lagrangian [898, 899, 900].

Exclusive semileptonic \(D\) decays play also a role in better understanding the composition of the \(\eta \) and \(\eta ^{\prime }\) wave functions, a long-standing problem. The transitions \(D_\mathrm{s}^{+} \rightarrow \eta ^{(\prime )} l^{+} \nu \) and \(D^+\rightarrow \eta ^{(\prime )} l^{+} \nu \) are driven by weak interactions at the Cabibbo-allowed and Cabibbo-suppressed levels, and provide us with complementary information since they produce the \(\eta ^{(\prime )}\) via their \(s \bar{s}\) and \(d \bar{d}\) components, respectively. In addition, \(\eta ^{(\prime )}\) could be excited via a \(gg\) component. That is important since it would validate, for the first time, an independent role of gluons in hadronic spectroscopy, outside their traditional domain of mediating strong interactions. Also \(B\) decays, semileptonic or hadronic, have been similarly employed (see e.g., Refs. [853, 901, 902]). Experimental evidence of glueballs is searched for in a variety of processes at several experiments, e.g., BESIII and PANDA. In 2009 the first absolute measurement of \(\mathcal{{B}} ( D_\mathrm{s}^{+} \rightarrow \eta ^{(\prime )} e^{+} \nu _e)\) [903] and the first observation of the \( D^{+} \rightarrow \eta \, e^{+} \nu _e\) decay [904] were reported by CLEO. Improved branching fraction measurements, together with the first observation of the decay mode \( D^{+} \rightarrow \eta ^\prime e^{+} \nu _e \) and the first form factor determination for \( D^{+} \rightarrow \eta \, e^{+} \nu _e\), followed in 2011 [905]. On the theoretical side, recent lattice results have become available for the values of mixing angles [364, 365], quoting values of the mixing angle \(\phi \) between \( 40^\circ \) and \(50^\circ \). The latest analysis, by ETM, leads to a value of \(\phi = (44 \pm 5)^\circ \) [906], with a statistical error only. Systematic uncertainties, difficult to estimate on the lattice, are likely to affect this result. Preliminary results by the QCDSF Collaboration [907, 908] give a mixing angle \(\theta \sim -(7^\circ , 8^\circ )\) in the octet-singlet basis, that is, in the quark-flavor basis, \(\phi = \theta +\arctan \sqrt{2} \sim 47^\circ \). Out of chorus is the lower value favored by the recent UKQCD staggered investigation [366], \(\phi = (34 \pm 3)^\circ \). All lattice analyses do not include a gluonic operator, discussing only the relative quark content. The agreement with other determinations from semileptonic decays based on different phenomenological approaches and older data is remarkable (see, e.g., Refs. [901, 909, 910, 911]). Recent experimental and theoretical progress has increased the role of semileptonic \(D\) decays with respect to traditional, low-energy analyses [912].

In the vector sector, the \(\phi \)\(\omega \) mixing is not expected as large as in the pseudoscalar one, because there is no additional mixing induced by the axial \(U(1)\) anomaly. In the absence of mixing, the state \(\omega \) has no strange valence quark and corresponds to \(|u \bar{u} + d \bar{d}\rangle /\sqrt{2} \). Cabibbo-favored semileptonic decays of \(D_\mathrm{s}\) are expected to lead to final states that can couple to \(|\bar{s} s \rangle \), in the quark flavor basis. The decay \( D^+_\mathrm{s} \rightarrow \omega e^{+} \nu _e \) occurs through \(\phi \)\(\omega \) mixing and/or Weak Annihilation (WA) diagrams, where the lepton pair couples weakly to the \(c \bar{s}\) vertex. Experimentally, only an upper limit is available on the branching fraction \(\mathcal{{B}}( D^+_\mathrm{s} \rightarrow \omega e^{+} \nu _e) <0.20~\%\), at 90 % C.L. [913].

Exclusive semileptonic \(D\) decays also offer the chance to explore possible exotic states. An interesting channel is the \(D^+_\mathrm{s} \rightarrow f_0(980) \, l^{+} \nu \) decay. The nontrivial nature of the experimentally well-established \(f_0(980)\) state has been discussed for decades and there are still different interpretations, from the conventional quark–antiquark picture, to a multiquark or molecular bound state. The channels \(D_{(s)}^{+} \rightarrow f_0(980) \, l^{+} \nu \) can be used as a probe of the hadronic structure of the light scalar resonance; more recent experimental investigation has been made available by CLEO [914]. A further handle is given by the possibility to correlate observables related to the charm semileptonic branching ratios with theoretical and experimental analyses of the hadronic \(B_\mathrm{s} \rightarrow J/\psi f_0\) decay [914, 915, 916].

The most recent experimental results on inclusive \(D^0\) and \(D_{(s)}^+\) semileptonic branching fractions have been derived using the complete CLEO-c data sets [917]. Besides being important in their own right, these measurements, due to similarities between the \(D\) and \(B\) sectors, can be helpful to improve understanding of \(B\) semileptonic decays, with the hope to reduce the theoretical uncertainty in the determination of the still-debated weak mixing parameter \(|V_{ub}|\). In [917], knowledge about exclusive semileptonic modes and form factor models is used to extrapolate the spectra below the 200 MeV momentum cutoff. The ratios of the semileptonic decay widths are determined to be \(\Gamma _{D^+}^{\mathrm {SL}}/\Gamma _{D^0}^{\mathrm {SL}} = 0.985 \pm 0.015 \pm 0.024 \) and \(\Gamma _{D^+_\mathrm{s}}^{\mathrm {SL}}/\Gamma _{D^0_\mathrm{s}}^{\mathrm {SL}} = 0.828 \pm 0.051 \pm 0.025 \). The former agrees with isospin symmetry, while the latter ratio shows an indication of difference. Significant improvements of the branching ratio measurement \(\mathcal{B} (D \rightarrow X \mu ^{+} \nu _{\mu })\) can be expected at BESIII, because of advantages provided by the capabilities of the BESIII \(\mu \) detection system [918]. The \(D^{0,\pm }\) and \(D_\mathrm{s}\) inclusive decays are differently affected by the WA diagrams, since they are Cabibbo-suppressed in the \(D^\pm \) case, Cabibbo-favored in \(D_\mathrm{s}\) decays, and completely absent in \(D^0\) decays. The semileptonic decays of \(D\) and \(D_\mathrm{s}\) can be helpful in constraining the WA matrix elements that enter the \(B \rightarrow X_u \, l \bar{\nu }\) decay, via heavy quark symmetry. By comparison of measured total semileptonic rates or moments in these channels, we can hope to extract information on the WA contributions. The “theoretical background” to take into account is the fact that such contributions compete with additional ones arising from \(\mathrm{SU}(3)\) breaking in the matrix elements, and/or from weak annihilation. However, no relevance or clear evidence of WA effects has been found considering the semileptonic widths [919] or the widths and the lepton–energy moments [920].

4.2.2 Exclusive \(B\) decays

Most theoretical approaches exploit the fact that the mass \(m_b\) of the \(b\) quark is large compared to the QCD scale that determines low-energy hadronic physics in order to build differential ratios. Neglecting the charged lepton and neutrino masses, we can recast the differential ratios as
$$\begin{aligned}&\frac{d\Gamma }{d \omega } (\bar{B}\rightarrow D\,l \bar{\nu }) = \frac{G_\mathrm{F}^2}{48 \pi ^3}\, K_1\, (\omega ^2-1)^{\frac{3}{2}}\, |V_{cb}|^2 \mathcal{G}^2(\omega )\nonumber \\&\quad \frac{d\Gamma }{d \omega }(\bar{B}\rightarrow D^*\,l \bar{\nu }) = \frac{G_\mathrm{F}^2}{48 \pi ^3} K_2 (\omega ^2-1)^{\frac{1}{2}} |V_{cb}|^2 \mathcal{F}^2(\omega )\nonumber \\ \end{aligned}$$
(4.13)
where \(K_1= (m_B+m_D)^2 m_D^3 \), \(K_2 = (m_B-m_{D^*})^2 m_{D^*}^3 \chi (\omega ) \) and \(\chi (\omega )\) is a known phase space. The semileptonic decays \( \bar{B}\rightarrow D \, l \, \bar{\nu }\) and \( \bar{B}\rightarrow D^*\, l \, \bar{\nu }\) depend on the form factors \(\mathcal{G}(\omega )\) and \(\mathcal{F}(\omega )\), respectively, where \(\omega \) is the product of the heavy quark velocities \(v_B= p_B/m_B\) and \(v_{D^{(*)}}= p_{D^{(*)}}/m_{D^{(*)}}\). The form factors, in the heavy-quark limit, are both normalized to unity at the zero recoil point \(\omega =1\). Corrections to this limit have been calculated in the lattice unquenched approximation, giving \( \mathcal{G}(1) = 1.074 \pm 0.024 \) [921] and \( \mathcal{F}(1) =0.906 \pm 0.004 \pm 0.012 \) [922], including the enhancement factor 1.007, due to the electroweak corrections to the four-fermion operator mediating the semileptonic decay.

The lattice calculations have been compared with non-lattice ones (see, e.g., Ref. [923]). By combining the heavy-quark expansion with a “BPS” expansion [924], in which \(\mu _\pi ^2=\mu ^2_G\), the following value is quoted \( \mathcal{G}(1) =1.04 \pm 0.02 \). Recently, the value \( \mathcal{F}(1) = 0.86 \pm 0.02 \) [925, 926] has been calculated, using zero recoil sum rules, including full \(\alpha _\mathrm{s}\) and estimated effects up to \(1/m_Q^5\).

Since the zero recoil point is not accessible experimentally, due to the kinematical suppression of the differential decay rates, the \(|V_{cb}|\) estimates rely on the extrapolation from \(\omega \ne 0\) to the zero recoil point. In Table 6 we list the results of the \(|V_{cb}|\) determinations obtained from the comparison of the previous form factors at zero recoil with experimental data. The errors are experimental and theoretical, respectively. The first three averages are taken by HFAG [927], the fourth one by PDG [1]. The slightly smaller values for the form factors in non-lattice determinations imply slightly higher values of \(|V_{cb}|\). In the last line, we quote the result due to an alternative lattice determination, currently available only in the quenched approximation, which consists of calculating the form factor normalization directly at values \(\omega >1\), avoiding the large extrapolation to \(\omega =1\) and thus reducing the model dependence [928]. This approach, by using 2009 BaBar data [929], gives a slightly higher value than the unquenched lattice result. The errors are statistical, systematic and due to the theoretical uncertainty in the form factor \( \mathcal{G}\), respectively. Calculations of form factors at non-zero recoil have been recently completed for \(B \rightarrow D\) semileptonic decays, giving the value \(|V_{cb}|=(38.5 \pm 1.9_\mathrm{exp+lat} \pm 0.2_\mathrm{QED}) \times 10^{-3}\) [930].
Table 6

Comparison of some exclusive determinations of \(|V_{cb}|\)

Theory

\(|V_{cb}| \times 10^{3}\)

\( \bar{B}\rightarrow D^*\, l \, \bar{\nu }\)

 

HFAG (Lattice) [922, 927, 931]

\( 39.04 \pm 0.49_{\mathrm {exp}} \pm 0.53_{\mathrm {QCD}}\)

 

                     \(\pm 0.19_{\mathrm {QED}}\)

HFAG (SR) [925, 926, 927]

\( 41.6\pm 0.6_{\mathrm {exp}}\pm 1.9_{\mathrm {th}} \)

\( \bar{B}\rightarrow D \, l \, \bar{\nu }\)

 

HFAG (Lattice) [921, 927]

\(39.70 \pm 1.42_{\mathrm {exp}} \pm 0.89_{\mathrm {th}} \)

PDG (BPS) [1, 924]

\( 40.7 \pm 1.5_{\mathrm {exp}} \pm 0.8_{\mathrm {th}} \)

BaBar (Lattice \(\omega \ne 1\)) [928, 929]

\( 41.6 \pm 1.8 \pm 1.4 \pm 0.7_{\mathrm{FF}} \)

Until a few years ago, only exclusive decays where the final lepton was an electron or a muon had been observed, since decays into a \(\tau \) lepton are suppressed because of the large \(\tau \) mass. Moreover, these modes are very difficult to measure because of the multiple neutrinos in the final state, the low lepton momenta, and the large associated background contamination. Results of semileptonic decays with a \(\tau \) in the final state were limited to inclusive and semi-inclusive measurements in LEP experiments. The first observation of an exclusive semileptonic \(B\) decay was reported by the Belle Collaboration in 2007. They measured the branching fraction \( \mathcal{{B}} (\bar{B}^0 \rightarrow D^{*+} \tau ^{-} \bar{\nu }_\tau )\) [932]. Recently the BaBar Collaboration has published results of their measurements of \(B \rightarrow D^{(*)} \tau \nu \) branching fractions normalized to the corresponding \(B \rightarrow D^{(*)} l \nu \) modes (with \(l=e , \mu \)) by using the full BaBar data sample [933]. Their results are in agreement with measurements by Belle using \(657 \times 10^6\)\(B \bar{B}\) events [934], and indicate an enhancement of order \((2 \sim 3) \sigma \) above theoretical results within the SM. It will be interesting to compare with the final Belle results on these modes using the full data sample of \(772 \times 10^6\)\(B \bar{B}\) pairs together with improved hadronic tagging. Indeed, a similar deviation from the SM has been previously observed also in leptonic decays \(B^{-} \rightarrow \tau ^{-} \bar{\nu }_\tau \), but Belle finds now a much lower value, in agreement with the SM, by using the full data set of \(B \bar{B}\) events [935]. By using Belle data and the FLAG \(N_\mathrm{f}=2+1\) determination of \(f_B\), one obtains the value \(|V_{ub}| = (3.35 \pm 0.65 \pm 0.07) \times 10^{-3}\) [876]. The accuracy is not sufficient to make this channel competitive for \( |V_{ub}|\) extraction, but the intriguing experimental situation has led to a reconsideration of SM predictions as well as exploring the possibility of new physics contributions, traditionally not expected in processes driven by the tree level semileptonic \(b\) decay. (For more details see, e.g., Refs. [854, 936].)

The analysis of exclusive charmless semileptonic decays, in particular the \(\bar{B} \rightarrow \pi l \bar{\nu }_l\) decay, is currently employed to determine the CKM parameter \(|V_{ub}|\), which plays a crucial role in the study of the unitarity constraints. Also here, information about hadronic matrix elements is required via form factors. Recent \(|V_{ub}|\) determinations have been reported by the BaBar collaboration; see Table VII of Ref. [937] (see also [853]), all in agreement with each other and with the value \( |V_{ub}| = (3.25 \pm 0.31) \times 10^{-3} \), determined from the simultaneous fit to the experimental data and the lattice theoretical predictions [937]. They are also in agreement with the Belle results for \( |V_{ub}| = (3.43 \pm 0.33) \times 10^{-3} \) extracted from the \(\bar{B} \rightarrow \pi l \bar{\nu }_l\) decay channel [938] and for \( |V_{ub}| \) from the \(\bar{B} \rightarrow \rho l \bar{\nu }_l\) decay channel, with precision of twice as good as the world average [939].

Finally, we just mention that exclusive \(B_\mathrm{s}\) decays are attracting a lot of attention due to the avalanche of recent data and to the expectation of new data. \(B_\mathrm{s}\) physics has been, and is, the domain of Tevatron and LHCb, but also present and future \(e^{+} e^{-}\) colliders can give their contribution, since the \(\Upsilon \mathrm {(5S)}\) decays in about 20 % of the cases to \(B_\mathrm{s}^{(\star )}\) meson-antimeson pairs. The measurement of the semileptonic asymmetry and its analysis are particularly interesting, since CP violation is expected to be tiny in the SM and any significant enhancement would be evidence for NP (see also [853, 940]).

4.2.3 Inclusive \(B\) decays

In most of the phase space for inclusive \( B \rightarrow X_q l \nu \) decays, long and short distance dynamics are factorized by means of the heavy quark expansion. However, the phase space region includes a region of singularity, also called endpoint or threshold region, plagued by the presence of large double (Sudakov-like) perturbative logarithms at all orders in the strong coupling.8 For \(b \rightarrow c\) semileptonic decays, the effect of the small region of singularity is not very important; in addition, corrections are not expected as singular as in the \( b \rightarrow u\) case, being cut off by the charm mass.

Recently, a global fit [927] has been performed to the width and all available measurements of moments in \( B \rightarrow X_\mathrm{c} l \nu \) decays, yielding, in the kinetic scheme \(|V_{cb}| = (41.88 \pm 0.73) \times 10^{-3}\) and in the 1S scheme \(|V_{cb}| = (41.96 \pm 0.45) \times 10^{-3}\). Each scheme has its own non-perturbative parameters that have been estimated together with the charm and bottom masses. The inclusive averages are in good agreement with the values extracted from exclusive decays in Table 6, within the errors.

In principle, the method of extraction of \(|V_{ub}|\) from inclusive \( \bar{B} \rightarrow X_u l \bar{\nu }_l\) decays follows in the footsteps of the \(|V_{cb}|\) determination from \( \bar{B} \rightarrow X_\mathrm{c} l \bar{\nu }_l\), but the copious background from the \( \bar{B} \rightarrow X_\mathrm{c} l \bar{\nu }_l\) process, which has a rate about 50 times higher, limits the experimental sensitivity to restricted regions of phase space, where the background is kinematically suppressed. The relative weight of the threshold region, where the previous approach fails, increases and new theoretical issues need to be addressed. Latest results by Belle [947] and BaBar [948] access about \( 90\) % of the \( \bar{B} \rightarrow X_u l \bar{\nu }_l\) phase space. On the theoretical side, several approaches have been devised to analyze data in the threshold region, with differences in treatment of perturbative corrections and the parameterization of non-perturbative effects.
Table 7

Comparison of inclusive determinations of \(|V_{ub}|\) [927]

Theory

\(|V_{ub}| \times 10^{3}\)

BLNP

\( 4.40 \pm 0.15^{+0.19}_{-0.21} \)

DGE

\(4.45 \pm 0.15^{+ 0.15}_{- 0.16}\)

ADFR

\(4.03 \pm 0.13^{+ 0.18}_{- 0.12}\)

GGOU

\(4.39 \pm 0.15^{ + 0.12}_ { -0.20} \)

The average values for \(|V_{ub}|\) have been extracted by HFAG [927] from the partial branching fractions, adopting a specific theoretical framework and taking into account correlations among the various measurements and theoretical uncertainties. In Table 7 we list some determinations, specifically the QCD theoretical calculations taking into account the whole set of experimental results, or most of it, starting from 2002 CLEO data [949]. They refer to the BLNP approach by Bosch, Lange, Neubert, and Paz [950], the GGOU one by Gambino, Giordano, Ossola and Uraltsev [951], the DGE one, the dressed gluon exponentiation, by Andersen and Gardi [952, 953] and the ADFR approach, by Aglietti, Di Lodovico, Ferrara, and Ricciardi, [954, 955, 956]. The results listed in Table 7 are consistent within the errors, but the theoretical uncertainty among determinations can reach 10 %. Other theoretical approaches have also been proposed in [957, 958, 959]. Notwithstanding all the experimental and theoretical efforts, the values of \(|V_{ub}|\) extracted from inclusive decays remain about two \(\sigma \) above the values given by exclusive determinations.

4.2.4 Rare charm decays

The decays driven by \( c \rightarrow u l^{+} l^{-}\) are forbidden at tree level in the standard model (SM) and proceed via one-loop diagrams (box and penguin) at leading order in the electroweak interactions. Virtual quarks in the loops are of the down type, and no breaking due to the large top mass occurs. The GIM mechanism works more effectively in suppressing flavor (charm) changing neutral currents than their strangeness and beauty analogues, leading to tiny decay rates, dominated by long-distance effects. On the other side, we expect possible enhancements due to new physics to stand out, once we exclude potentially large long-distance SM contributions.

In the SM, a very low branching ratio has been estimated for inclusive decays, largely dominated by long-distance contributions \( { \mathcal B} (D \rightarrow X_u l^{+} l^{-}) = { \mathcal B}_\mathrm {LD} (D \rightarrow X_u l^{+} l^{-}) \sim O(10^{-6})\) [960]. Long-distance contributions are assumed to proceed from intermediate vector resonances such as \( D \rightarrow X_u V\), \(V \rightarrow l^+l^{-}\), where \(V = \phi \), \(\rho \) or \(\omega \), which set the scale with branching fractions of order \(10^{-6}\). Short-distance contributions lay far behind [960, 961, 962]; the latest estimate gives \( { \mathcal B}_\mathrm {SD} (D \rightarrow X_u e^{+} e^{-}) \sim 4 \,\times \, 10^{-9}\) [962]. Handling long-distance dynamics in these processes becomes equivalent to handling several intermediate charmless resonances, in a larger number than in the case of \(B\) meson analogs. Their effect can be separated from short-distance contributions by applying selection criteria on the invariant mass of the leptonic pair.

To consider exclusive decays, let us start from \(D_{(s)}^\pm \rightarrow h^\pm l^{+} l^{-}\), with \(h \in (\pi , \rho , K, K^\star )\) and \(l \in (e, \mu )\), none of which has been observed up to now. The best experimental limits on branching fractions are \(O(10^{-6})\) or higher, at 90 % confidence level (CL), coming all from BaBar [963, 964], with a few exceptions: very old limits on \(D^{+} \rightarrow \rho ^{+} \mu ^{+} \mu ^{-}\) and \(D_\mathrm{s}^{+} \rightarrow K^{*+}(892) \mu ^{+} \mu ^{-}\) decays, given by E653 [965], and the recent limits on \(D^{+}_{(s)} \rightarrow \pi ^{\pm } \mu ^\mp \mu ^+\) decays, given by LHCb with an integrated luminosity of 1.0 \({\mathrm {fb}}^{-1}\) [966]. The BESIII collaboration will be able to reach a sensitivity of \(O(10^{-7})\) for \(D^{+} \rightarrow K^+/\pi ^{+} \, l^{+} l^{-}\) at 90 % CL with a 20 fb\(^{-1}\) data sample taken at the \(\psi (3770)\) peak [918]. The LHCb collaboration can also search for \(D_{(s)}^\pm \rightarrow h^\pm l^{+} l^{-}\) decays. The very recent update on the \( D_{(s)}^{+} \rightarrow \pi ^{+} \mu ^{+} \mu ^{-}\) channel with a 3 \({\mathrm {fb}}^{-1}\) full data sample is still orders of magnitudes above the SM prediction; new searches for the \( D_{(s)}^{+} \rightarrow K^{+} \mu ^{+} \mu ^{-}\) decays are ongoing [967]. Also decays \(D^0 \rightarrow h^0 l^{+} l^{-}\) have not been observed yet; the best experimental limits at 90 % CL are of order \(O(10^{-5})\) or higher, and are given by older analyses of CLEO [968], E653 [965] and E791 [969]. Future Super B factories are expected to reach a sensitivity of \(O(10^{-8})\) on a 90 % CL on various rare decays, including \(D^{+} \rightarrow \pi ^{+} l^{+} l^{-}\) and \(D^0 \rightarrow \pi ^0 l^{+} l^{-}\) [970].

A way to disentangle possible new physics is to choose appropriate observables containing mainly short distance contributions. Last year, hints of possible new physics (NP) have been advocated in the charm sector to explain the nonvanishing direct CP asymmetry in \(D^0 \rightarrow K^{+} K^{-} \) and \(D^0 \rightarrow \pi ^{+} \pi ^{-} \), measured by LHCb [971], confirmed by CDF [972] and supported by recent data from Belle [973]. Encouraged by these results, effects of the same kind of possible NP have been looked for in other processes, including rare charm decays. CP asymmetries can be generated by imaginary parts of Wilson coefficients in the effective Hamiltonian for \( c \rightarrow u l^{+} l^{-}\) driven decays. They have been investigated in \( D^{+} \rightarrow \pi ^{+} \mu ^{+} \mu ^{-}\) and \(D_\mathrm{s}^{+} \rightarrow K^{+} \mu ^{+} \mu ^{-}\) decays, around the \(\phi \) resonance peak in the spectrum of dilepton invariant mass, concluding that in favorable conditions their value can be as high as 10 % [974]. Older studies report investigations of semileptonic decays in the framework of other NP models, such as R-parity violating supersymmetric models, extra heavy up vector-like quark models [975], Little Higgs [962], or leptoquark models [976]. The parameter space discussed in older analyses cannot take into account the constraints given by recent LHC data, most notably the discovery of the 125 GeV resonance. In several cases, a reassessment in the updated framework could be used advantageously.

4.3 Spectroscopy

The year 2013 marks the 10th anniversary of the observation of the \(X(3872)\) charmonium-like state [809] that put an end to the era when heavy quarkonium was considered as a relatively well established bound system of a heavy quark and antiquark. Since 2003 every year has been bringing discoveries of new particles with unexpected properties, not fitting a simple \(q\bar{q}\) classification scheme. The wealth of new results is mainly from B- and c-factories, Belle, BaBar and BES III, where data samples with unprecedented statistics became available.

In this section we first describe experiments that contribute to the subject, discuss recent developments for low-lying states, then we move to the open flavor thresholds and beyond. We consider the charmonium- and bottomonium-(like) states in parallel to stress similarities between the observed phenomena in the two quarkonium sectors.

4.3.1 Experimental tools

Over the last decade the main suppliers of new information about quarkonium states have been the \(B\)-factories, the experiments working at asymmetric-energy \(e^{+}e^{-}\) colliders operated at center-of-mass energies in the \(\Upsilon \)-resonance region. Both Belle and BaBar detectors are general-purpose 4\(\pi \) spectrometers with excellent momentum resolution, vertex positioning and particle identification for charged tracks, as well as with high-resolution electromagnetic calorimeters. Although the main purpose of the \(B\)-factories is to study CP asymmetries in \(B\)-decays, these experiments allow for many other searches apart from the major goal. Charmonium states at \(B\)-factories are copiously produced in \(B\)-decays, two-photon fusion, charm quark fragmentation in \(e^{+}e^{-}\rightarrow c\bar{c}\) annihilation (mostly via double \(c\bar{c}\) production) and via initial-state radiation, when the energy of \(e^{+}e^{-}\) annihilation is dumped by emission of photons in the initial state. Both \(B\)-factories intensively studied also bottomonium states, taking data at different \(\Upsilon \) states that allow to access lower mass bottomonia via hadronic and radiative transitions. Although both \(B\)-factories completed their data taking already long ago (BaBar in 2008 and Belle in 2010), the analysis of the collected data is still ongoing, and many interesting results have been obtained recently. The data samples of the two experiments are summarized in Table 8.
Table 8

Integrated luminosities (in fb\(^{-1}\)) collected by the BaBar and Belle experiments at different \(e^{+}e^{-}\) energies

 

BaBar

Belle

\(\Upsilon ({1}{S})\)

5.7

\(\Upsilon ({2}{S})\)

14

24.1

\(\Upsilon ({3}{S})\)

30

3.0

\(\Upsilon ({4}{S})\)

433

711

Off-resonance

54

87

\(\Upsilon ({5}{S})\)

121

\(\Upsilon ({5}{S})\)- \(\Upsilon ({6}{S})\) scan

5

27

Another class of experiments where charmonium states are extensively studied are the charm-\(\tau \) factories. For the last decade BES II, CLEOc, and finally BES III have successively covered measurements of \(e^{+}e^{-}\) annihilation around the charmonium region. The BES III experiment started data taking in 2009 after a major upgrade of the BEPC \(e^{+}e^{-}\) collider and the BES II spectrometer. The BEPC II accelerator operates in the c.m. energy range of \(\sqrt{s} = (2 - 4.6)~{\mathrm {GeV}}\) and has already reached a peak luminosity close to the designed one. Starting late 2012 BES III has collected data at high energies to study \(Y(4260)\) and other highly excited charmonium-like states.
Table 9

Quarkonium states below the corresponding open flavor thresholds

State

\(M,\,~\mathrm {MeV}\)

\(\Gamma ,\,~\mathrm {MeV}\)

\(J^{PC}\)

Process (mode)

Experiment (#\(\sigma \))

Year

Status

\(\psi _2(1D)\)

\(3823.1\pm 1.9\)

\(<24\)

\(2^{-\,\!-}\)

\(B\rightarrow K(\gamma \,\chi _{c1})\)

Belle [977] (3.8)

2013

NC!

\(\eta _b(1S)\)

\(9398.0\pm 3.2\)

\(11^{+6}_{-4}\)

\(0^{-\,\!+}\)

\(\Upsilon (3S)\rightarrow \gamma \,(...)\)

BaBar [978] (10), CLEO [979] (4.0)

2008

OK

    

\(\Upsilon (2S)\rightarrow \gamma \,(...)\)

BaBar [980] (3.0)

2009

NC!

    

\(h_b(1P,2P)\rightarrow \gamma \,(...)\)

Belle [848] (14)

2012

NC!

\(h_b(1P)\)

\(9899.3\pm 1.0\)

?

\(1^{+\,\!-}\)

\(\Upsilon (10860)\rightarrow \pi ^{+}\pi ^{-}\,(...)\)

Belle [848, 981] (5.5)

2011

NC!

    

\(\Upsilon (3S)\rightarrow \pi ^0\,(...)\)

BaBar [982] (3.0)

2011

NC!

\(\eta _b(2S)\)

\(9999\pm 4\)

\(<24\)

\(0^{-\,\!+}\)

\(h_b(2P)\rightarrow \gamma \,(...)\)

Belle [848] (4.2)

2012

NC!

\(\Upsilon (1D)\)

\(10163.7\pm 1.4\)

?

\(2^{-\,\!-}\)

\(\Upsilon (3S)\rightarrow \gamma \gamma \,(\gamma \gamma \,\Upsilon (1S))\)

CLEO [983] (10.2)

2004

NC!

    

\(\Upsilon (3S)\rightarrow \gamma \gamma \,(\pi ^{+}\pi ^{-}\Upsilon (1S))\)

BaBar [984] (5.8)

2010

NC!

    

\(\Upsilon (10860)\rightarrow \pi ^{+}\pi ^{-}(\gamma \gamma \,\Upsilon (1S))\)

Belle [985] (9)

2012

NC!

\(h_b(2P)\)

\(10259.8\pm 1.2\)

?

\(1^{+\,\!-}\)

\(\Upsilon (10860)\rightarrow \pi ^{+}\pi ^{-}\,(...)\)

Belle [848, 981] (11.2)

2011

NC!

\(\chi _{bJ}(3P)\)

\(10534\pm 9\)

?

\((1,2)^{+\,\!+}\)

\(pp,p\bar{p}\rightarrow (\gamma \Upsilon (1S,2S))\,...\)

ATLAS [986] (\(>\)6), D0 [987] (5.6)

2011

OK

Experiments at hadron machines (Tevatron and LHC) can investigate quarkonium produced promptly in high-energy hadronic collisions in addition to charmonium produced in \(B\)-decays. The Tevatron experiments CDF and D0 completed their experimental program in 2010, after CERN started operating the LHC. Four LHC experiments are complementary in tasks and design. While LHCb has been optimized for mainly heavy flavor physics, ATLAS and CMS are contributing to the field by investigating certain signatures in the central rapidity range with high statistics. The LHC accelerator performance has fulfilled and even exceeded expectations. The integrated luminosity delivered to the general-purpose experiments (ATLAS and CMS) in 2011 was about 6 fb\(^{-1}\), and more than 20 fb\(^{-1}\) in 2012. The instantaneous luminosity delivered to LHCb is leveled to a constant rate due to limitations in the LHCb trigger and readout, and to collect data under relatively clean conditions. The integrated luminosity delivered to LHCb was 1 fb\(^{-1}\) and 2 fb\(^{-1}\) in 2011 and 2012, respectively.

The new \(B\) factory at KEK, SuperKEKB, will be commissioned in 2015 according to the current planning schedule. It is expected that the target integrated luminosity, 50 ab\(^{-1}\) , will be collected by 2022.

4.3.2 Heavy quarkonia below open flavor thresholds

Recently, significant progress has been achieved in the studies of the spin-singlet bottomonium states. In addition, last year two more states have been found below their corresponding open flavor thresholds, the \(\psi _2(1D)\) charmonium and the \(\chi _b(3P)\) bottomonium (in the latter case the levels with different \(J\) are not resolved), see Table 9. All these new data provide important tests of the theory, which, due to lattice and effective field theories, is rather solid and predictive below the open flavor threshold. The theory verification in this particular region becomes even more important given the difficulties of the theory for states near or above the open flavor threshold.

Spin-singlet bottomonium states do not have production or decay channels convenient for experimental studies. Therefore their discovery became possible only with the high statistics of the \(B\)-factories. An unexpected source of the spin-singlet states turned out to be the di-pion transitions from the \(\Upsilon ({5}{S})\). The states are reconstructed inclusively using the missing mass of the accompanying particles. Belle observed the \(h_{b}(1P)\) and \(h_{b}(2P)\) states in the transitions \(\Upsilon ({5}{S})\rightarrow {{\pi ^{+}\pi ^{-}}}h_{b}(nP)\) [981]. The hyperfine splittings were measured to be \((+0.8\pm 1.1)\,~\mathrm {MeV}\) for \(n=1\) and \((+0.5\pm 1.2)\,~\mathrm {MeV}\) for \(n=2\) [848]. The results are consistent with perturbative QCD expectations [988, 989, 990, 991]. This shows in particular that the spin–spin potential does not have a sizeable long-range contribution [992], an observation supported by direct lattice computations [752]. For comparison, in the charmonium sector the measured \(1P\) hyperfine splitting of \((-0.11\pm 0.17)\,~\mathrm {MeV}\) [1] is also consistent with zero with even higher accuracy.

The \(\eta _b(1S)\) is found in M1 radiative transitions from \(\Upsilon ({3}{S})\) [978, 979] and \(\Upsilon ({2}{S})\) [980]. The measured averaged hyperfine splitting \(\Delta M_\mathrm{HF}(1S)=M_{\Upsilon ({1}{S})}-M_{\eta _{b}(1S)}= (69.3\pm 2.8)~\mathrm {MeV}\) [1] was larger than perturbative pNRQCD \((41 \pm 14)\,~\mathrm {MeV}\) [743] and lattice \((60 \pm 8)\,~\mathrm {MeV}\) [850] estimates. In 2012, using a large sample of \(h_{b}(mP)\) from \(\Upsilon ({5}{S})\) Belle observed the \(h_{b}(1P)\rightarrow {\eta _{b}(1S)}\gamma \) and \(h_{b}(2P)\rightarrow {\eta _{b}(1S)}\gamma \) transitions [848]. The Belle \({\eta _{b}(1S)}\) mass measurement is more precise than the PDG2012 average and is \((11.4 \pm 3.6)\,~\mathrm {MeV}\) above the central value, which is in better agreement with the perturbative pNRQCD determination. The residual difference of about \(17\,~\mathrm {MeV}\) is consistent with the uncertainty of the theoretical determination. Also lattice determinations have improved their analyses (see Sect. 4.1.3). The latest determination based on lattice NRQCD, which includes spin-dependent relativistic corrections through \(O(v^6)\), radiative corrections to the leading spin-magnetic coupling, non-perturbative four-quark interactions and the effect of \(u\), \(d\), \(s\) and \(c\) quark vacuum polarization, gives \(\Delta M_\mathrm{HF}(1S) = (62.8 \pm 6.7)\,~\mathrm {MeV}\) [846]. Belle measured for the first time also the \({\eta _{b}(1S)}\) width, \(\Gamma _{{\eta _{b}(1S)}} = (10.8\,^{+4.0}_{-3.7}\,^{+4.5}_{-2.0})\,~\mathrm {MeV}\), which is consistent with expectations.

Belle found the first strong evidence for the \({\eta _{b}(2S)}\) with a significance of \(4.4\,\sigma \) using the \(h_{b}(2P)\rightarrow \gamma {\eta _{b}(2S)}\) transition. The hyperfine splitting was measured to be \(\Delta M_\mathrm{HF}(2S)=(24.3^{+4.0}_{-4.5})\,~\mathrm {MeV}\). The ratio \(\Delta M_\mathrm{HF}(2S)/ \Delta M_\mathrm{HF}(1S)=0.420^{+0.071}_{-0.079}\) is in agreement with NRQCD lattice calculations [843, 846, 850], the most recent of which gives \(\Delta M_\mathrm{HF}(2S)/\Delta M_\mathrm{HF}(1S)=0.425\pm 0.025\) [846] (see also Sect. 4.1.3). The measured branching fractions \( \mathcal {B}(h_{b}(1P)\rightarrow \gamma {\eta _{b}(1S)})=(49.2\pm 5.7\,^{+5.6}_{-3.3})~\%\), \( \mathcal {B}(h_{b}(2P)\rightarrow \gamma {\eta _{b}(1S)})=(22.3\pm 3.8\,^{+3.1}_{-3.3})~\%\), and \( \mathcal {B}(h_{b}(2P)\rightarrow \gamma {\eta _{b}(2S)})=(47.5\pm 10.5\,^{+6.8}_{-7.7})~\%\) are somewhat higher than the model predictions [993].

There is another claim of the \({\eta _{b}(2S)}\) signal by the group of K. Seth from Northwestern University, that used CLEO data [849]. The \(\Upsilon ({2}{S})\rightarrow {\eta _{b}(2S)}\gamma \) production channel is considered and the \({\eta _{b}(2S)}\) is reconstructed in 26 exclusive channels with up to 10 charged tracks in the final state. The measured hyperfine splitting \(\Delta M_\mathrm{HF}(2S)=(48.7\pm 3.1)\,~\mathrm {MeV}\) is \(5\,\sigma \) away from the Belle value and is in strong disagreement with theoretical expectations [994]. In [849] the contribution of final-state radiation is not considered, therefore the background model is incomplete and the claimed significance of \(4.6\,\sigma \) is overestimated. Belle repeated the same analysis with 17 times higher statistics and found no signal [995]. The Belle upper limit is an order of magnitude lower than the central value in [849]. We conclude that the evidence for the \({\eta _{b}(2S)}\) with the anomalous mass reported in [849] is refuted.

The \(n=3\) radial excitation of the \(\chi _{bJ}\) system was recently observed by ATLAS [986] and confirmed by D0 [987]. The \(\chi _{bJ}(3P)\) states are produced inclusively in the \(pp\) and \(p\overline{p}\) collisions and are reconstructed in the \(\gamma \Upsilon (1S,2S)\) channels with \(\Upsilon \rightarrow \mu ^+\mu ^{-}\). Converted photons and photons reconstructed from energy deposits in the electromagnetic calorimeter are used. The mass resolution does not allow to discern individual \(\chi _{bJ}(3P)\) states with \(J=0\), 1 and 2. A measured barycenter of the triplet \(10534\pm 9\,~\mathrm {MeV}\) is close to the quark model expectations of typically \(10525\,~\mathrm {MeV}\) [996, 997].

Potential models predict that \(D\)-wave charmonium levels are situated between the \(D \bar{D} \) and \(D \bar{D}^{*}\) thresholds [998]. Among them the states \(\eta _{c2}\) (\(J^{\mathrm{PC}}=2^{-+}\)) and \(\psi _2\) (\(J^{\mathrm{PC}}=2^{-\,\!-}\)) cannot decay to \(D \bar{D} \) because of unnatural spin–parity, and they are the only undiscovered charmonium levels that are expected to be narrow. Recently Belle reported the first evidence for the \(\psi _2(1D)\) using the \(B^+\rightarrow K^+\psi _2(1D)[\rightarrow \gamma \chi _{c1}]\) decays [977], with a mass of \(M=(3823.1\pm 1.9)\,~\mathrm {MeV}\) and width consistent with zero, \(\Gamma <24\,~\mathrm {MeV}\). The full width is likely to be very small, since the state is observed in the radiative decay and the typical charmonium radiative decay widths are at the \(O(100)\,~{\mathrm {keV}}\) level. The odd \(C\)-parity (fixed by decay products) discriminates between the \(\eta _{c2}\) and \(\psi _2\) hypotheses. No signal is found in the \(\gamma \chi _{c2}\) channel, in agreement with expectations for the \(\psi _2\) [998]. Belle measured \( \mathcal {B}(B^+\rightarrow K^+\psi _2)\times \mathcal {B}(\psi _2\rightarrow \gamma \chi _{c1})= (9.7{^{+2.8}_{-2.5}}{^{+1.1}_{-1.0}})\times 10^{-6}\). Given that one expects \( \mathcal {B}(\psi _2\rightarrow \gamma \chi _{c1})\sim 2/3\) [998], \( \mathcal {B}(B^+\rightarrow K^+\psi _2)\) is a factor of 50 smaller than the corresponding branching fractions for the \(J/\psi \), \(\psi (2S)\) and \(\chi _{c1}\) due to the factorization suppression [999, 1000].

Many of the above studies and, in particular, many discovery channels involve radiative decays. For states below threshold, theory has made in the last few years remarkable progress in the study of these decay channels. From the EFT side, pNRQCD provides now an (almost) complete description of E1 and M1 transitions [1001, 1002], which means that we have expressions for all these decay channels up to and including corrections of relative order \(v^2\). The only exception are M1 transitions for strongly bound quarkonia that depend at order \(v^2\) on a not-yet-calculated Wilson coefficient. The kind of insight in the QCD dynamics of quarkonia that one may get from having analytical expressions for these decay rates can be understood by looking at the transition \(J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma \). The PDG average for the width \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma )\) is \((1.58 \pm 0.37) \,~{\mathrm {keV}}\), which is clearly lower than the leading order estimate \(2.83\,~{\mathrm {keV}}\). Corrections of relative order \(v^2\) are positive in the case of a confining potential, whereas they are negative in the case of a Coulomb potential [1001]. Therefore the current PDG average favors an interpretation of the \(J/\psi \) as a Coulombic bound state. This interpretation may be challenged by the most recent KEDR analysis that finds \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma ) = (2.98 \pm 0.18^{+0.15}_{-0.33})\,\mathrm{keV}\) [1003]. The KEDR result has a better accuracy than the current world average and is \(3.0\,\sigma \) above its central value.

In [1004], a determination of \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma )\) based on lattice QCD in the continuum limit with two dynamical quarks, the authors find \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma ) = (2.64\pm 0.11\pm 0.03)\,~{\mathrm {keV}}\). Earlier lattice determinations of the charmonium radiative transitions in quenched lattice QCD can be found in [1005, 1006]. In [1007], a determination of \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma )\) in perturbative pNRQCD, the authors find \(\Gamma (J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma ) = (2.12\pm 0.40) \,\mathrm{keV}\). Both theoretical determinations are consistent with each other and fall in between the PDG average and the latest KEDR determination with the lattice determination favoring a somewhat larger value and the perturbative QCD determination a somewhat smaller value of the transition width. Part of the tension between data, and between data and theoretical determinations may be due to the fact that the extraction of the \(J/\psi \rightarrow \eta _\mathrm{c}(1S)\gamma \) branching fraction from the photon energy line shape in \(J/\psi \rightarrow X\gamma \) is not free from uncontrolled uncertainties [1008].
Table 10

Quarkonium-like states at the open flavor thresholds. For charged states, the \(C\)-parity is given for the neutral members of the corresponding isotriplets

State

\(M,\,~\mathrm {MeV}\)

\(\Gamma ,\,~\mathrm {MeV}\)

\(J^{PC}\)

Process (mode)

Experiment (#\(\sigma \))

Year

Status

\(X(3872)\)

\(3871.68\pm 0.17\)

\(<\!1.2\)

\(1^{+\,\!+}\)

\(B\rightarrow K(\pi ^{+}\pi ^{-}J/\psi )\)

Belle [809, 1029] (\(>\)10), BaBar [1030] (8.6)

2003

OK

    

\(p\bar{p}\rightarrow (\pi ^{+}\pi ^{-}J/\psi )\,...\)

CDF [1031, 1032] (11.6), D0 [1033] (5.2)

2003

OK

    

\(pp\rightarrow (\pi ^{+}\pi ^{-}J/\psi )\,...\)

LHCb [1034, 1035] (np)

2012

OK

    

\(B\rightarrow K(\pi ^{+}\pi ^{-}\pi ^0J/\psi )\)

Belle [1036] (4.3), BaBar [1037] (4.0)

2005

OK

    

\(B\rightarrow K(\gamma \, J/\psi )\)

Belle [1038] (5.5), BaBar [1039] (3.5)

2005

OK

     

LHCb [1040] (\(>10\))

  
    

\(B\rightarrow K(\gamma \, \psi (2S))\)

BaBar [1039] (3.6), Belle [1038] (0.2)

2008

NC!

     

LHCb [1040] (4.4)

  
    

\(B\rightarrow K(D\bar{D}^{*})\)

Belle [1041] (6.4), BaBar [1042] (4.9)

2006

OK

\(Z_{c}(3885)^{+}\)

\(3883.9\pm 4.5\)

\(25\pm 12\)

\(1^{+\,\!-}\)

\(Y(4260)\rightarrow \pi ^{-}(D\bar{D}^{*})^{+}\)

BES III [1043] (np)

2013

NC!

\(Z_{c}(3900)^{+}\)

\(3891.2\pm 3.3\)

\(40\pm 8\)

\(?^{?-}\)

\(Y(4260)\rightarrow \pi ^{-}(\pi ^{+}J/\psi )\)

BES III [1044] (8), Belle [1045] (5.2)

2013

OK

     

T. Xiao et al. [CLEO data] [1046] (\(>\)5)

  

\(Z_{c}(4020)^{+}\)

\(4022.9\pm 2.8\)

\(7.9\pm 3.7\)

\(?^{?-}\)

\(Y(4260,4360)\rightarrow \pi ^{-}(\pi ^{+}h_{c})\)

BES III [1047] (8.9)

2013

NC!

\(Z_{c}(4025)^{+}\)

\(4026.3\pm 4.5\)

\(24.8\pm 9.5\)

\(?^{?-}\)

\(Y(4260)\rightarrow \pi ^{-}(D^{*}\bar{D}^{*})^{+}\)

BES III [1048] (10)

2013

NC!

\(Z_b(10610)^{+}\)

\(10607.2\pm 2.0\)

\(18.4\pm 2.4\)

\(1^{+\,\!-}\)

\(\Upsilon (10860)\rightarrow \pi (\pi \Upsilon (1S,2S,3S))\)

Belle [1049, 1050, 1051] (\(>\)10)

2011

OK

    

\(\Upsilon (10860)\rightarrow \pi ^{-}(\pi ^{+}h_b(1P,2P))\)

Belle [1050] (16)

2011

OK

    

\(\Upsilon (10860)\rightarrow \pi ^{-}(B\bar{B}^{*})^{+}\)

Belle [1052] (8)

2012

NC!

\(Z_b(10650)^{+}\)

\(10652.2\pm 1.5\)

\(11.5\pm 2.2\)

\(1^{+\,\!-}\)

\(\Upsilon (10860)\rightarrow \pi ^{-}(\pi ^{+}\Upsilon (1S,2S,3S))\)

Belle [1049, 1050] (\(>\)10)

2011

OK

    

\(\Upsilon (10860)\rightarrow \pi ^{-}(\pi ^{+}h_b(1P,2P))\)

Belle [1050] (16)

2011

OK

    

\(\Upsilon (10860)\rightarrow \pi ^{-}(B^{*}\bar{B}^{*})^{+}\)

Belle [1052] (6.8)

2012

NC!

Bottomonium M1 transitions have been studied in perturbative pNRQCD in [1001] and [1007]. In particular, in [1007] a class of large perturbative contributions associated with the static potential has been resummed providing an improved determination of several M1 transitions: \(\Gamma ({\Upsilon (1S) \rightarrow \eta _b(1S)\gamma }) = (15.18 \pm 0.51) \,\mathrm{eV}\), \(\Gamma (h_b(1P) \rightarrow \chi _{b0}(1P)\gamma ) = (0.962\pm 0.035) \,\mathrm{eV}\), \(\Gamma ({h_b(1P) \rightarrow \chi _{b1}(1P)\gamma }) = (8.99\pm 0.55) \times 10^{-3}\,\mathrm{eV}\), \(\Gamma ({\chi _{b2}(1P) \rightarrow h_b(1P)\gamma }) = (0.118\pm 0.006) \,\mathrm{eV}\) and \(\Gamma ({\Upsilon (2S) \rightarrow \eta _b(1S)\gamma }) =6^{+26}_{-6} \, \mathrm{eV}\). The improved determination of \(\Gamma ({\Upsilon (2S) \rightarrow \eta _b(1S)\gamma })\) is particularly noteworthy because it is consistent with the most recent data, \((12.5\pm 4.9)\,\mathrm{eV}\) from BaBar [980], while the leading-order determination is off by at least one order of magnitude. Bottomonium transitions in lattice NRQCD with \(2+1\) dynamical quarks have been computed in [1009, 1010].

E1 transitions are more difficult to study both on the lattice and with analytical methods. The reason is that even at leading order they involve a non-perturbative matrix element. A complete theoretical formulation in the framework of pNRQCD can be found in [1002] with a preliminary but promising phenomenological analysis in [1011].

The theoretical status of quarkonium hadronic transitions, inclusive and exclusive hadronic and electromagnetic decays has been summarized in [757, 1012, 1013]. There has been a limited use of the pNRQCD factorization for these processes and only restricted to inclusive hadronic and electromagnetic decays [744, 745, 748, 749, 1014, 1015], while most of the recent work has concentrated on improving the expansion in the NRQCD factorization framework to higher orders in \(v\) and \(\alpha _\mathrm{s}\) [1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 1025, 1026, 1027, 1028].

4.3.3 Quarkonium-like states at open flavor thresholds

There are several states in both the charmonium and bottomonium sectors lying very close to the threshold of their decay to a pair of open flavor mesons; see Table 10. This proximity suggests a molecular structure for these states.

The \(X(3872)\) is a state very close to the \( D^{*0} \bar{ D^{0}}\)threshold, \(\delta m_{X(3872)}=m_{X(3872)}-m_{D^{*0}}-m_{D^0}=-0.11\pm 0.22\,~\mathrm {MeV}\) [1, 1053, 1054]. The decays \(X(3872)\rightarrow \rho J/\psi \) and \(X(3872)\rightarrow \omega J/\psi \) have similar branching fractions, \( \mathcal {B}_{\omega }/ \mathcal {B}_{\rho } =0.8\pm 0.3\) [1036, 1037]; this corresponds to a strong isospin violation. The favorite \(X(3872)\) interpretation is a mixture of a charmonium state \(\chi _{c1}(2P)\) and an \(S\)-wave \( D^{*0} \bar{ D^{0}}\)molecule [759, 760, 761, 762, 763, 764, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071, 1072, 1073, 1074], with the molecular component responsible for the isospin violation and the charmonium component accounting for the production in \(B\) decays and at hadron collisions. For alternative interpretations we refer to [757] and references therein. The molecular hypothesis is valid only for the spin–parity assignment \(J^{\mathrm{PC}}=1^{+\,\!+}\). Experimentally \(1^{+\,\!+}\) was favored, but \(2^{-+}\) was not excluded for a long time [1029, 1031, 1075]. This intrigue has been settled recently by LHCb with a clear preference of \(1^{+\,\!+}\) and exclusion of \(2^{-+}\) hypothesis [1035]. Progress towards a lattice understanding of the \(X(3872)\) has been discussed in Sect. 4.1.3.

The contributions to the \(\delta m_{X(3872)}\) uncertainty are \(0.17\,~\mathrm {MeV}\) from the \(X(3872)\) mass, \(0.13\,~\mathrm {MeV}\) from the \(D^0\) mass and \(0.07\,~\mathrm {MeV}\) from the \(D^{*0}-D^0\) mass difference. LHCb can improve the accuracy in \(m_{X(3872)}\) and \(m_{D^0}\), BES III and KEDR can contribute to the \(m_{D^0}\) measurement. The \(D^{*0}-D^0\) mass difference was measured 20 years ago by ARGUS and CLEO and also can be improved.

The radiative \(X(3872)\rightarrow \gamma J/\psi \) decay is established, while there is an experimental controversy regarding \(X(3872) \rightarrow \gamma \psi (2S)\) [1036, 1038, 1039], with recent LHCb evidence pointing towards existence of this channel [1040]. The dominant decay mode of the \(X(3872)\) is \( D^{*0} \bar{ D^{0}}\)[1041, 1042, 1076], as expected for the molecule, however, the absolute branching fraction is not yet determined. These questions will have to wait for Belle II data.

Charged bottomonium-like states \(Z_{b}(10610)\) and \(Z_{b}(10650)\) are observed by Belle as intermediate \(\Upsilon ({n}{S})\pi ^{\pm }\) and \(h_b(mP)\pi ^{\pm }\) resonances in the \(\Upsilon ({5}{S})\rightarrow \pi ^+\pi ^{-}\Upsilon ({n}{S})\) and \(\Upsilon ({5}{S})\rightarrow \pi ^+\pi ^{-} h_b(mP)\) decays [1050]. The nonresonant contribution is sizable for the \(\Upsilon ({5}{S})\rightarrow \pi ^+\pi ^{-}\Upsilon ({n}{S})\) decays and is consistent with zero for the \(\Upsilon ({5}{S})\rightarrow \pi ^+\pi ^{-}h_b(mP)\) decays. The mass and width of the \(Z_{b}\) states were measured from the amplitude analysis, assuming a Breit–Wigner form of the \(Z_{b}\) amplitude. The parameters are in agreement among all five decay channels, with the average values \(M_1=(10607.4\pm 2.0)\,~\mathrm {MeV}\), \(\Gamma _1=(18.4\pm 2.4)\,~\mathrm {MeV}\) and \(M_2=(10652.2\pm 1.5)\,~\mathrm {MeV}\), \(\Gamma _2=(11.5\pm 2.2)\,~\mathrm {MeV}\). The measured \(Z_{b}(10610)\) and \(Z_{b}(10650)\) masses coincide within uncertainties with the \(B\bar{B}^*\) and \(B^*\bar{B}^*\) thresholds, respectively.

Belle observed the \(Z_{b}(10610)\rightarrow B\bar{B}^*\) and \(Z_{b}(10650)\rightarrow B^*\bar{B}^*\) decays with the significances of \(8\,\sigma \) and \(6.8\,\sigma \), respectively, using the partially reconstructed \(\Upsilon ({5}{S})\rightarrow (B^{(*)}\bar{B}^*)^{\pm }\pi ^{\mp }\) transitions [1052]. Despite much larger phase space, no significant signal of the \(Z_{b}(10650)\rightarrow B\bar{B}^*\) decay was found. Assuming that the decays observed so far saturate the \(Z_b\) decay table, Belle calculated the relative branching fractions of \(Z_{b}(10610)\) and \(Z_{b}(10650)\) (Table 11).
Table 11

Branching fractions (\( \mathcal {B}\)) of \(Z_{b}\)s in per cent

Channel

\( \mathcal {B}\) of \(Z_{b}(10610)\)

\( \mathcal {B}\) of \(Z_{b}(10650)\)

\(\pi ^{+}\Upsilon ({1}{S})\)

\(0.32\pm 0.09\)

\(0.24\pm 0.07\)

\(\pi ^{+}\Upsilon ({2}{S})\)

\(4.38\pm 1.21\)

\(2.40\pm 0.63\)

\(\pi ^{+}\Upsilon ({3}{S})\)

\(2.15\pm 0.56\)

\(1.64\pm 0.40\)

\(\pi ^{+}h_{b}(1P)\)

\(2.81\pm 1.10\)

\(7.43\pm 2.70\)

\(\pi ^{+}h_{b}(2P)\)

\(4.34\pm 2.07\)

\(14.8\pm 6.22\)

\(B^{+}\bar{B}^{*0}+\bar{B}^0B^{*+}\)

\(86.0\pm 3.6\)

\(B^{*+}\bar{B}^{*0}\)

\(73.4\pm 7.0\)

The \(Z_{b}(10610)\rightarrow B\bar{B}^*\) and \(Z_{b}(10650)\rightarrow B^*\bar{B}^*\) decays are dominant with a branching fraction of about 80 %. If the \(Z_{b}(10650)\rightarrow B\bar{B}^*\) channel is included in the decay table, its branching fraction is \( \mathcal {B}(Z_{b}(10650)\rightarrow B\bar{B}^*)=(25\pm 10)\,~\%\) and all other \(Z_{b}(10650)\) branching fractions are reduced by a factor of 1.33.

Belle observed the neutral member of the \(Z_b(10610)\) isotriplet by performing a Dalitz analysis of the \(\Upsilon ({5}{S})\rightarrow \pi ^0\pi ^0\Upsilon ({n}{S})\) (\(n=1,2,3\)) decays [1051]. The \(Z_b(10610)^0\) significance combined over the \(\pi ^0\Upsilon ({2}{S})\) and \(\pi ^0\Upsilon ({3}{S})\) channels is \(6.5\,\sigma \). The measured mass value \(M_{Z_b(10610)^0}=(10609\pm 6)\,~\mathrm {MeV}\) is in agreement with the mass of the charged \(Z_{b}(10610)^{\pm }\). No significant signal of the \(Z_b(10650)^0\) is found; the data are consistent with the existence of the \(Z_b(10650)^0\) state, but the available statistics are insufficient to observe it.

To determine the spin and parity of the \(Z_b\) states, Belle performed a full six-dimensional amplitude analysis of the \(\Upsilon (5S)\rightarrow \pi ^+\pi ^{-}\Upsilon (nS)\)\((n=1,2,3)\) decays [1077]. The \(Z_b(10610)\) and \(Z_b(10650)\) are found to have the same spin–parity \(J^P=1^+\), while all other hypotheses with \(J\le 2\) are rejected at more than \(10\,\sigma \) level. The highest discriminating power is provided by the interference pattern between the \(Z_b\) signals and the nonresonant contribution.

The proximity to the \(B\bar{B}^*\) and \(B^*\bar{B}^*\) thresholds suggests that the \(Z_{b}\) states have molecular structure, i.e., their wave function at large distances is the same as that of an S-wave meson pair in the \(I^G(J^P)=1^+(1^+)\) state [1078].

The assumption of the molecular structure can naturally explain all the properties of the \(Z_{b}\) states without specifying their dynamical model [1078]. The decays into constituents [i.e. \(Z_{b}(10610)\rightarrow B\bar{B}^*\) and \(Z_{b}(10650)\rightarrow B^*\bar{B}^*\)], if kinematically allowed, should dominate. The molecular spin function, once decomposed into \(b\bar{b}\) spin eigenstates, appears to be a mixture of the ortho- and para-bottomonium components. The weights of the components are equal, therefore the decays into \(\pi \Upsilon \) and \(\pi h_b\) have comparable widths. The \(B\bar{B}^*\) and \(B^*\bar{B}^*\) states differ by a sign between the ortho- and para-bottomonium components. This sign difference is observed in the interference pattern between the \(Z_{b}(10610)\) and \(Z_{b}(10650)\) signals in the \(\pi \Upsilon \) and \(\pi h_b\) final states [1050].

The question of the dynamical model of the molecules remains open. Among different possibilities are nonresonant rescattering [1079], multiple rescatterings that result in a pole in the amplitude, known as a coupled channel resonance [1080], and deuteron-like molecule bound by meson exchanges [1081]. All these mechanisms are closely related and a successful phenomenological model should probably account for all of them. Predictions for the \(Z_b\) lineshapes that can be used to fit data would be useful to discriminate between different mechanisms.

Alternatively, the \(Z_{b}\) states are proposed to have diquark–antidiquark structure [1082]. In this model the \(B^{(*)}\bar{B}^*\) channels are not dominant and the lighter (heavier) state couples predominantly to \(B^*\bar{B}^*\) (\(B\bar{B}^*\)). The data on the decay pattern of the \(Z_{b}\) states strongly disfavor the diquark–antidiquark hypothesis.

Observation of the charged \(Z_b\) states motivated a search for their partners in the charm sector. Since late 2012 BES III has been collecting data at different energies above \(4\,~{\mathrm {GeV}}\) to study charmonium-like states.

In the course of 2013 the states \(Z_\mathrm{c}(3885)^{\pm }\rightarrow (D\bar{D}^*)^{\pm }\), \(Z_\mathrm{c}(3900)^{\pm }\rightarrow \pi ^{\pm }J/\psi \), \(Z_\mathrm{c}(4020)\rightarrow \pi ^{\pm }h_\mathrm{c}\), \(Z_\mathrm{c}(4025)\rightarrow (D^*\bar{D}^*)^{\pm }\) were observed (see Table 10). The masses and widths of the \(Z_\mathrm{c}(3885)\)/\(Z_\mathrm{c}(3900)\) and \(Z_\mathrm{c}(4020)\)/\(Z_\mathrm{c}(4025)\) pairs agree at about \(2\,\sigma \) level. All current measurements disregard the interference between the \(Z_\mathrm{c}\) signal and the nonresonant contribution, which is found to be significant in all channels (including \(\pi h_\mathrm{c}\), in contrast to the \(\pi h_b\) case). Interference effects could shift the peak position by as much as half the resonance width. A more accurate measurement of masses and widths as well as spins and parities using the amplitude analyses will help to clarify whether the above \(Z_\mathrm{c}\) pairs could be merged.

The \(Z_\mathrm{c}(3885)\) and \(Z_\mathrm{c}(3900)\) [\(Z_\mathrm{c}(4020)\) and \(Z_\mathrm{c}(4025)\)] states are close to the \(D\bar{D}^*\) [\(D^*\bar{D}^*\)] threshold. In fact, all the measured masses are about \(10\,~\mathrm {MeV}\)above the thresholds. This is a challenge for a molecular model, but could be an experimental artifact caused by neglecting the interference.

If the \(Z_\mathrm{c}(3885)\) and \(Z_\mathrm{c}(3900)\) states are merged, the properties of the resulting state agree with the expectations for the \(D\bar{D}^*\) molecular structure. The \(D\bar{D}^*\) channel is dominant [1043],
$$\begin{aligned} \frac{\Gamma [Z_\mathrm{c}(3885)^{\pm }\rightarrow (D\bar{D}^*)^{\pm }]}{\Gamma [Z_\mathrm{c}(3900)^{\pm }\rightarrow \pi ^{\pm }J/\psi ]}=6.2\pm 2.9. \end{aligned}$$
(4.14)
A \(2.1\,\sigma \) hint for the \(Z_\mathrm{c}(3900)\rightarrow \pi ^{\pm }h_\mathrm{c}\) transition [1047] implies that the state couples to both ortho- and para-charmonium, with a weaker \(\pi h_\mathrm{c}\) signal due to phase-space suppression. Finally, the spin–parity measured for the \(Z_\mathrm{c}(3885)\)\(J^P=1^+\) corresponds to \(D\bar{D}^*\) in S-wave.

Identification of the \(Z_\mathrm{c}(4020)\) or \(Z_\mathrm{c}(4025)\) as a \(D^*\bar{D}^*\) molecule is difficult. If the \(D\bar{D}^*\) molecule decays to \(\pi ^{\pm }J/\psi \), then according to heavy-quark spin symmetry the \(D^*\bar{D}^*\) molecule should also decay to \(\pi ^{\pm }J/\psi \). However, no hint of \(Z_\mathrm{c}(4020)\) or \(Z_\mathrm{c}(4025)\) is seen in the \(\pi ^{\pm }J/\psi \) final state.

It could be that the \(D^*\bar{D}^*\) molecule is not produced in the \(Y(4260)\) decays, as would be the case if the \(Y(4260)\) is a \(D_1(2420)\bar{D}\) molecule (see next section).
Table 12

Quarkonium-like states above the corresponding open flavor thresholds. For charged states, the \(C\)-parity is given for the neutral members of the corresponding isotriplets

State

\(M,\,~\mathrm {MeV}\)

\(\Gamma ,\,~\mathrm {MeV}\)

\(J^{PC}\)

Process (mode)

Experiment (#\(\sigma \))

Year

Status

\(Y(3915)\)

\(3918.4\pm 1.9\)

\(20\pm 5\)

\(0/2^{?+}\)

\(B\rightarrow K(\omega J/\psi )\)

Belle [1087] (8), BaBar [1037, 1088] (19)

2004

OK

    

\(e^{+}e^{-}\rightarrow e^{+}e^{-}(\omega J/\psi )\)

Belle [1089] (7.7), BaBar [1090] (7.6)

2009

OK

\(\chi _{c2}(2P)\)

\(3927.2\pm 2.6\)

\(24\pm 6\)

\(2^{+\,\!+}\)

\(e^{+}e^{-}\rightarrow e^{+}e^{-}(D\bar{D})\)

Belle [1091] (5.3), BaBar [1092] (5.8)

2005

OK

\(X(3940)\)

\(3942^{+9}_{-8}\)

\(37^{+27}_{-17}\)

\(?^{?+}\)

\(e^{+}e^{-}\rightarrow J/\psi \,(D\bar{D}^{*})\)

Belle [1085, 1086] (6)

2005

NC!

\(Y(4008)\)

\(3891\pm 42\)

\(255\pm 42\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\pi ^{+}\pi ^{-}J/\psi )\)

Belle [1045, 1093] (7.4)

2007

NC!

\(\psi (4040)\)

\(4039\pm 1\)

\(80\pm 10\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (D^{(*)}\bar{D}^{(*)}(\pi ))\)

PDG [1]

1978

OK

    

\(e^{+}e^{-}\rightarrow (\eta J/\psi )\)

Belle [1094] (6.0)

2013

NC!

\(Z(4050)^{+}\)

\(4051^{+24}_{-43}\)

\(82^{+51}_{-55}\)

\(?^{?+}\)

\(\bar{B}^0\rightarrow K^{-}(\pi ^{+}\chi _{c1})\)

Belle [1095] (5.0), BaBar [1096] (1.1)

2008

NC!

\(Y(4140)\)

\(4145.8\pm 2.6\)

\(18\pm 8\)

\(?^{?+}\)

\(B^{+}\rightarrow K^{+}(\phi J/\psi )\)

CDF [1097] (5.0), Belle [1098] (1.9),

2009

NC!

     

LHCb [1099] (1.4), CMS [1100] (\(>\)5)

  
     

D0 [1101] (3.1)

  

\(\psi (4160)\)

\(4153\pm 3\)

\(103\pm 8\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (D^{(*)}\bar{D}^{(*)})\)

PDG [1]

1978

OK

    

\(e^{+}e^{-}\rightarrow (\eta J/\psi )\)

Belle [1094] (6.5)

2013

NC!

\(X(4160)\)

\(4156^{+29}_{-25}\)

\(139^{+113}_{-65}\)

\(?^{?+}\)

\(e^{+}e^{-}\rightarrow J/\psi \,(D^{*}\bar{D}^{*})\)

Belle [1086] (5.5)

2007

NC!

\(Z(4200)^{+}\)

\(4196^{+35}_{-30}\)

\(370^{+99}_{-110}\)

\(1^{+\,\!-}\)

\(\bar{B}^0\rightarrow K^{-}(\pi ^{+}J/\psi )\)

Belle [1102] (7.2)

2014

NC!

\(Z(4250)^{+}\)

\(4248^{+185}_{-45}\)

\(177^{+321}_{-72}\)

\(?^{?+}\)

\(\bar{B}^0\rightarrow K^{-}(\pi ^{+}\chi _{c1})\)

Belle [1095] (5.0), BaBar [1096] (2.0)

2008

NC!

\(Y(4260)\)

\(4250\pm 9\)

\(108\pm 12\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\pi \pi J/\psi )\)

BaBar [1103, 1104] (8), CLEO [1105, 1106] (11)

2005

OK

     

Belle [1045, 1093] (15), BES III [1044] (np)

  
    

\(e^{+}e^{-}\rightarrow (f_0(980)J/\psi )\)

BaBar [1104] (np), Belle [1045] (np)

2012

OK

    

\(e^{+}e^{-}\rightarrow (\pi ^{-}Z_{c}(3900)^{+})\)

BES III [1044] (8), Belle [1045] (5.2)

2013

OK

    

\(e^{+}e^{-}\rightarrow (\gamma \,X(3872))\)

BES III [1107] (5.3)

2013

NC!

\(Y(4274)\)

\(4293\pm 20\)

\(35\pm 16\)

\(?^{?+}\)

\(B^{+}\rightarrow K^{+}(\phi J/\psi )\)

CDF [1097] (3.1), LHCb [1099] (1.0),

2011

NC!

     

CMS [1100] (\(>\)3), D0 [1101] (np)

  

\(X(4350)\)

\(4350.6^{+4.6}_{-5.1}\)

\(13^{+18}_{-10}\)

\(0/2^{?+}\)

\(e^{+}e^{-}\rightarrow e^{+}e^{-}(\phi J/\psi )\)

Belle [1108] (3.2)

2009

NC!

\(Y(4360)\)

\(4354\pm 11\)

\(78\pm 16\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\pi ^{+}\pi ^{-}\psi (2S))\)

Belle [1109] (8), BaBar [1110] (np)

2007

OK

\(Z(4430)^{+}\)

\(4458\pm 15\)

\(166^{+37}_{-32}\)

\(1^{+\,\!-}\)

\(\bar{B}^0\rightarrow K^{-}(\pi ^{+}\psi (2S))\)

Belle [1111, 1112] (6.4), BaBar [1113] (2.4)

2007

OK

     

LHCb [1114] (13.9)

  
    

\(\bar{B}^0\rightarrow K^{-}(\pi ^{+}J/\psi )\)

Belle [1102] (4.0)

2014

NC!

\(X(4630)\)

\(4634^{+9}_{-11}\)

\(92^{+41}_{-32}\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\Lambda _{c}^{+}\bar{\Lambda }_{c}^{-})\)

Belle [1115] (8.2)

2007

NC!

\(Y(4660)\)

\(4665\pm 10\)

\(53\pm 14\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\pi ^{+}\pi ^{-}\psi (2S))\)

Belle [1109] (5.8), BaBar [1110] (5)

2007

OK

\(\Upsilon (10860)\)

\(10876\pm 11\)

\(55\pm 28\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (B_{(s)}^{(*)}\bar{B}_{(s)}^{(*)}(\pi ))\)

PDG [1]

1985

OK

    

\(e^{+}e^{-}\rightarrow (\pi \pi \Upsilon (1S,2S,3S))\)

Belle [1050, 1051, 1116] (\(>\)10)

2007

OK

    

\(e^{+}e^{-}\rightarrow (f_0(980)\Upsilon (1S))\)

Belle [1050, 1051] (\(>\)5)

2011

OK

    

\(e^{+}e^{-}\rightarrow (\pi Z_b(10610,10650))\)

Belle [1050, 1051] (\(>\)10)

2011

OK

    

\(e^{+}e^{-}\rightarrow (\eta \Upsilon (1S,2S))\)

Belle [985] (10)

2012

OK

    

\(e^{+}e^{-}\rightarrow (\pi ^{+}\pi ^{-}\Upsilon (1D))\)

Belle [985] (9)

2012

OK

\(Y_b(10888)\)

\(10888.4\pm 3.0\)

\(30.7^{+8.9}_{-7.7}\)

\(1^{-\,\!-}\)

\(e^{+}e^{-}\rightarrow (\pi ^{+}\pi ^{-}\Upsilon (nS))\)

Belle [1117] (2.3)

2008

NC!

The \(Z_\mathrm{c}(4020)\) could be a candidate for hadrocharmonium, a color-neutral quarkonium state in a cloud of light meson(s) [1083]. The decay into constituent charmonium and light meson should dominate, while the decay to another charmonium is suppressed. The available experimental information on \(Z_\mathrm{c}(4020)\) agrees with this picture. The \(Z_\mathrm{c}(4025)\) is not a suitable hadrocharmonium candidate since the \(D^*\bar{D}^*\) channel dominates. Hadrocharmonium was proposed to explain the affinity of many charmonium-like states [\(Y(4260)\), \(Y(4360)\), \(Z(4050\), \(Z(4250)\), \(Z(4430)\),...] to some particular channels with charmonium and light hadrons, as discussed below [1084].

Another configuration proposed for the \(Z_\mathrm{c}\) states is a Born–Oppenheimer tetraquark [758]. In such a state a colored \(c\bar{c}\) pair is moving in the adiabatic potential created by the light degrees of freedom. This approach aims at providing a general framework for the description of all \(XYZ\) states.

To summarize, the properties of the \(Z_b(10610)\) and \(Z_b(10650)\) states are in good agreement with the assumption that they have molecular structure. The \(Z_\mathrm{c}(3885/3900)\) state is a candidate for the \(D\bar{D}^*\) molecule, while the absence of the \(Z_\mathrm{c}(4020,4025)\rightarrow \pi J/\psi \) signal disfavors the interpretation of \(Z_\mathrm{c}(4020)\) or \(Z_\mathrm{c}(4025)\) as a \(D^*\bar{D}^*\) molecule. The peak positions of the \(Z_\mathrm{c}\) signals are about \(10\,~\mathrm {MeV}\) above the \(D\bar{D}^*\) or \(D^*\bar{D}^*\) thresholds. Unless future amplitude analyses find values that are closer to the thresholds, this could be a challenge for the molecular model. Upcoming BES III results on the \(Z_\mathrm{c}\) masses, widths, branching fractions and spin-parities from the amplitude analyses, and on the search for other decay channels (\(\pi \psi (2S)\) and \(\rho \eta _\mathrm{c}\)), are crucial for interpreting the \(Z_\mathrm{c}\) states.

The \(Z_\mathrm{c}\) and \(Z_b\) states provide a very rich testing ground for phenomenological models and, given intensive experimental and theoretical efforts, one can expect progress in understanding of the hadronic systems near the open flavor thresholds.

4.3.4 Quarkonium and quarkonium-like states above open flavor thresholds

More than 10 new charmonium and charmonium-like states well above the \(D \bar{D} \) threshold have been observed in the last decade by the \(B\)-factories and other experiments; see Table 12. We discuss first the states that can be assigned to vacant charmonium levels. In 2008 Belle observed the \(\chi _{c2}(2P)\) state in \(\gamma \gamma \) collisions, later confirmed by BaBar. Almost all of the \(\chi _{c2}(2P)\) properties (diphoton width, full width, decay mode) are in nice agreement with the theory expectations, only the mass of the state is \({\sim } 50~\mathrm {MeV}\) below potential model predictions. Another two charmonium candidates (for the third and fourth radial excitations of \({\eta _{c}(1S)}\)) are observed by Belle [1085, 1086] in the double charmonium production process \(e^{+}e^{-}\rightarrow J/\psi X(3940/4160)\), that decay to \(D \bar{D}^{*}\) and \( D^{*} \bar{D}^{*} \) channels, respectively. BaBar has not reported any studies of these processes yet. While production processes and decay modes are typical of conventional charmonium, the masses of these states are significantly lower than potential model expectations (e.g., \(\eta _\mathrm{c}(4S)\) is expected to be \({\sim } 300~\mathrm {MeV}\) heavier than the observed \(X(4160)\)). The assignment can be tested by studying the angular distribution of the final state at Belle II.

For the majority of the other new particles, the assignments to the ordinary charmonium states are not well recognized. Contrary to expectations, most of the new states above the open charm threshold, the so-called “\(XYZ\) states”, decay into final states containing charmonium, but do not decay into open charm pairs with a detectable rate. This is the main reason why they are discussed as candidates for exotic states. An extended discussion on the different interpretations of these states can be found in [757] and references therein. In the following we discuss recent results on the states above open heavy flavor thresholds.

BaBar confirmed the observation of the process \(\gamma \gamma \rightarrow Y(3915)\rightarrow \omega J/\psi \) that was observed by Belle in 2009 [1090]. From angular analyses BaBar determined the \(Y(3915)\) spin–parity to be \(J^P=0^+\) [1090]. In this analysis it is assumed that in the alternative hypothesis of \(J=2\) it is produced in the helicity-2 state, analogous to the production of \(\chi _{c2}(1P)\). Given the unknown nature of \(Y(3915)\), this assumption could be unjustified. The \(J^P=0^+\) state can decay to \(D\bar{D}\) in S-wave. Since \(Y(3915)\) is \(200\,~\mathrm {MeV}\) above the \(D\bar{D}\) threshold, its width of \(20\,~\mathrm {MeV}\) looks extremely narrow and points to its exotic nature. In addition, the mass difference relative to \(\chi _{c2}(2P)\) of \(9\,~\mathrm {MeV}\) is too small [1118] to interpret the \(Y(3915)\) as \(\chi _{c0}(2P)\).

CMS and D0 studied the \(B^+\rightarrow K^+\phi J/\psi \) decays [1100, 1101] and confirmed the \(Y(4140)\) state near the \(\phi J/\psi \) threshold that was observed in 2008 by CDF [1097]. The experiments also see a second structure, the \(Y(4274)\), though the mass measurements agree only at about \(3\,\sigma \) level. The background under the \(Y(4274)\) can be distorted by reflections from the \(K^{*+}\rightarrow \phi K^+\) decays, which makes an estimate of the \(Y(4274)\) significance difficult [1100]. The \(Y(4140)\) and \(Y(4274)\) states were not seen in \(B\) decays by Belle [1098] and LHCb [1099] and in \(\gamma \gamma \) collisions by Belle [1108]. Amplitude analyses with increased statistics at the LHC could settle the controversy.

BaBar updated the \(e^+e^{-}\rightarrow \pi ^+\pi ^{-}\psi (2S)\) study using ISR photons and confirmed the \(Y(4660)\) that was earlier observed by Belle [1110]. Both BaBar and Belle updated the \(e^+e^{-}\rightarrow \pi ^+\pi ^{-} J/\psi \) analyses [1045, 1104]. Belle confirms the \(Y(4008)\) using an increased data sample. However, the mass becomes smaller, \(M=3891\pm 42\,~\mathrm {MeV}\). BaBar sees events in the same mass region, but they are attributed to a contribution with an exponential shape. BES III data taken in this region will help to clarify the existence of the \(Y(4008)\) resonance.

BES III measured the \(e^+e^{-}\rightarrow \pi ^+\pi ^{-}h_\mathrm{c}\) cross section at several energies above \(4\,~{\mathrm {GeV}}\) [1047]. Unlike the \(e^+e^{-}\rightarrow \pi ^+\pi ^{-}h_b\) reaction, the final three-body state is mainly nonresonant. The shape of the cross section looks different from that of the \(\pi ^+\pi ^{-}J/\psi \) final state and possibly exhibits structures distinct from known \(Y\) states [1119]. Since hybrids contain a \(c\bar{c}\) pair in the spin-singlet state, such structures could be candidates for hybrids. A more detailed scan by BES III is underway.

Belle performed the full amplitude analysis of the \(B^0\rightarrow K^+\pi ^{-}\psi (2S)\) decays to determine the spin–parity of the \(Z(4430)^{\pm }\) [1112], which is the first charged quarkonium-like state observed by Belle in 2007 [1095, 1120]. The \(J^P=1^+\) hypothesis is favored over the \(0^{-}\), \(1^{-}\) and \(2^{-}\) and \(2^+\) hypotheses at the levels of \(3.4\,\sigma \), \(3.7\,\sigma \), \(4.7\,\sigma \) and \(5.1\,\sigma \), respectively. The width of the \(Z(4430)^{\pm }\) became broader, \(\Gamma =200^{+49}_{-58}\,~\mathrm {MeV}\). This state and two more states, \(Z(4050)^{\pm }\) and \(Z(4250)^{\pm }\), in the \(\pi ^{\pm }\chi _{c1}\) channel are not confirmed by BaBar [1096, 1113]. The long-standing question of the \(Z(4430)^\pm \)’s existence has finally been settled by the LHCb experiment, which confirmed both the state itself and its spin–parity assignment of \(1^+\) [1114]. For the first time, LHCb has demonstrated resonant behavior of the \(Z(4430)^\pm \) amplitude. Belle has performed a full amplitude analysis of the \(\bar{B}^0\rightarrow K^{-}\pi ^+J/\psi \) decays and observed a new charged charmonium-like state \(Z(4200)^+\) and evidence for the \(Z(4430)^+\) decay to \(\pi ^+J/\psi \) [1102]. This decay is within the reach of LHCb. Further studies of \(Z(4050)^{\pm }\) and \(Z(4250)^{\pm }\) could be more difficult at LHCb because of soft photons in the final state and might have to wait for Belle II to run.

Given that the signals of \(Y(4260)\), \(Y(4360)\) and \(Y(4660)\) are not seen in the \(e^+e^{-}\rightarrow \mathrm {hadrons}\) cross section (\(R_\mathrm{c}\) scan), one can set the limit \(\Gamma [Y\rightarrow \pi ^+\pi ^{-}\psi ]\gtrsim 1\,~\mathrm {MeV}\) [1121]. This is at least one order of magnitude higher than that of \(\psi (2S)\) and \(\psi (3770)\). Recently Belle found that \(\psi \) states seen as prominent peaks in the \(R_\mathrm{c}\) scan, can also have anomalous hadronic transitions to lower charmonia. Belle observed \(\psi (4040)\) and \(\psi (4160)\) signals in the \(e^{+}e^{-}\rightarrow \eta J/\psi \) cross section measured using ISR [1122]. The partial widths are measured to be \(\Gamma [\psi (4040,4160)\rightarrow \eta J/\psi ]\sim 1\,~\mathrm {MeV}\). Thus it seems to be a general feature of all charmonium(-like) states above the open charm thresholds to have intense hadronic transitions to lower charmonia. A similar phenomenon is found in the bottomonium sector: In 2008 Belle observed anomalously large rates of the \(\Upsilon ({5}{S})\rightarrow {{\pi ^{+}\pi ^{-}}}\Upsilon ({n}{S})\) (\(n=1,~2,~3\)) transitions with partial widths of \(300-400\,~{\mathrm {keV}}\) [1116]. Recently Belle reported preliminary results on the observation of \(\Upsilon ({5}{S})\rightarrow \eta \Upsilon (1S,2S)\) and \(\Upsilon ({5}{S}) \rightarrow {{\pi ^{+}\pi ^{-}}}\Upsilon ({1}{D})\) with anomalously large rates [985]. It is proposed that these anomalies are due to rescatterings [1123, 1124]. The large branching fraction of the \(\Upsilon ({4}{S})\rightarrow \Upsilon ({1}{S})\eta \) decay observed in 2010 by BaBar could have a similar origin [1125].

The mechanism can be considered either as a rescattering of the \(D \bar{D} \) or \(B\bar{B}\) mesons, or as a contribution of the molecular component to the quarkonium wave function.

The model in which \(Y(4260)\) is a \(D_1(2420)\bar{D}\) molecule naturally explains the high probability of the intermediate molecular resonance in the \(Y(4260)\rightarrow \pi ^+\pi ^{-}J/\psi \) transitions [1126, 1127] and predicts the \(Y(4260)\rightarrow \gamma X(3872)\) transitions with high rates [1128]. Such transitions have recently been observed by BES III, with [1107]
$$\begin{aligned} \frac{\sigma [e^+e^{-}\rightarrow \gamma X(3872)]}{\sigma [e^+e^{-}\rightarrow \pi ^+\pi ^{-}J/\psi ]}\sim 11~\%. \end{aligned}$$
(4.15)
Despite striking similarities between the observations in the charmonium and bottomonium sectors, there are also clear differences. In the charmonium sector, each of the \(Y(3915)\), \(\psi (4040)\), \(\psi (4160)\), \(Y(4260)\), \(Y(4360)\) and \(Y(4660)\) decays to only one particular final state with charmonium [\(\omega J/\psi \), \(\eta J/\psi \), \({{\pi ^{+}\pi ^{-}}}J/\psi \) or \({{\pi ^{+}\pi ^{-}}}\psi (2S)\)]. In the bottomonium sector, there is one state with anomalous properties, the \(\Upsilon ({5}{S})\), and it decays to different channels with similar rates [\({{\pi ^{+}\pi ^{-}}}\Upsilon ({n}{S})\), \({{\pi ^{+}\pi ^{-}}}h_{b}(mP)\), \({{\pi ^{+}\pi ^{-}}}\Upsilon ({1}{D})\), \(\eta \Upsilon ({n}{S})\)]. There is no general model describing these peculiarities. To explain the affinity of the charmonium-like states to some particular channels, the notion of “hadrocharmonium” was proposed in [1084]. It is a heavy quarkonium embedded into a cloud of light hadron(s), thus the fall-apart decay is dominant. Hadrocharmonium could also provide an explanation for the charged charmonium-like states \(Z(4430)^+\), \(Z(4050)^+\) and \(Z(4250)^+\).

4.3.5 Summary

Quarkonium spectroscopy enjoys an intensive flood of new results. The number of spin-singlet bottomonium states has increased from one to four over the last 2 years, including a more precise measurement of the \({\eta _{b}(1S)}\) mass, \(11\,~\mathrm {MeV}\) away from the PDG2012 average. There is evidence for one of the two still missing narrow charmonium states expected in the region between the \(D \bar{D} \) and \(D \bar{D}^{*}\) thresholds. Observations and detailed studies of the charged bottomonium-like states \(Z_{b}(10610)\) and \(Z_{b}(10650)\) and first results on the charged charmonium-like states \(Z_\mathrm{c}\) open a rich phenomenological field to study exotic states near open flavor thresholds. There is also significant progress and a more clear experimental situation for the highly excited heavy quarkonium states above open flavor thresholds. Recent highlights include confirmation of the \(Y(4140)\) state by CMS and D0, observation of the decays \(\psi (4040,4160)\rightarrow \eta J/\psi \) by Belle, measurement of the energy dependence of the \(e^+e^{-}\rightarrow \pi ^+\pi ^{-}h_\mathrm{c}\) cross section by BES III, observation of the \(Y(4260)\rightarrow \gamma X(3872)\) by BES III and determination of the \(Z(4430)\) spin–parity from full amplitude analysis by Belle. A general feature of highly excited states is their large decay rate to lower quarkonia with the emission of light hadrons. Rescattering is important for understanding their properties, however, there is no general model explaining their decay patterns. The remaining experimental open questions or controversies are within the reach of the LHC or will have to wait for the next generation \(B\)-factory.

From the theoretical point of view, low quarkonium excitations are in agreement with lattice QCD and effective field theories calculations, which are quite accurate and able to challenge the accuracy of the data. Higher quarkonium excitations show some unexpected properties. Specific effective field theories have been developed for some of these excitations. Lattice studies provide a qualitative guide, but in most cases theoretical expectations still rely on models and a quantitative general theory is still missing.

4.4 Strong coupling \(\alpha _\mathrm{s}\)

There are several heavy-quark systems that are suitable for a precise determination of \({\alpha _{\mathrm{s}}}\), mainly involving quarkonium, or quarkonium-like, configurations, which are basically governed by the strong interactions. One can typically take advantage of non-relativistic effective theories, high-order perturbative calculations that are available for these systems, and of progress in lattice computations.

Using moments of heavy-quark correlators calculated on the lattice, and the continuum perturbation theory results for them [1129], the HPQCD collaboration has obtained \({\alpha _{\mathrm{s}}}(M_Z)=0.1183\pm 0.0007\) [2]. This result is very close, both in the central value and error, to the one obtained from measuring several quantities related to short-distance Wilson loops by the same collaboration [2]. The energy between two static sources in the fundamental representation, as a function of its separation, is also suitable for a precise \({\alpha _{\mathrm{s}}}\) extraction. The perturbative computation has now reached a three-loop level [1130, 1131, 1132, 1133, 1134, 1135], and lattice-QCD results with \({N}_\mathrm{f}=2+1\) sea quarks are available [1136]. A comparison of the two gives \({\alpha _{\mathrm{s}}}(M_Z)=0.1156^{+0.0021}_{-0.0022}\) [1137]. New lattice data for the static energy, including points at shorter distances, will be available in the near future, and an update of the result for \({\alpha _{\mathrm{s}}}\) can be expected, in principle with reduced errors.

Quarkonium decays, or more precisely ratios of their widths (used to reduce the sensitivity to long-distance effects), were readily identified as a good place for \({\alpha _{\mathrm{s}}}\) extractions. One complication is the dependence on color-octet configurations. The best ratio for \({\alpha _{\mathrm{s}}}\) extractions, in the sense that the sensitivity to color-octet matrix elements and relativistic effects is most reduced, turns out to be \(R_{\gamma }:=\Gamma (\Upsilon \rightarrow \gamma X)/\Gamma (\Upsilon \rightarrow X)\), from which one obtains \({\alpha _{\mathrm{s}}}(M_Z)=0.119^{+0.006}_{-0.005}\) [1138]. The main uncertainty in this result comes from the systematic errors of the experimental measurement of \(R_{\gamma }\) [1139]. Belle could be able to produce an improved measurement of \(R_{\gamma }\), which may translate into a better \({\alpha _{\mathrm{s}}}\) determination.

Very recently the CMS collaboration has presented a determination of \({\alpha _{\mathrm{s}}}\) from the measurement of the inclusive cross section for \(t\bar{t}\) production, by comparing it with the NNLO QCD prediction. The analysis is performed with different NNLO PDF sets, and the result from the NNPDF set is used as the main result. Employing \(m_t=173.2\pm 1.4\) GeV, \({\alpha _{\mathrm{s}}}(M_Z)=0.1151^{+0.0033}_{-0.0032}\) is obtained [1140], the first \({\alpha _{\mathrm{s}}}\) determination from top-quark production.

4.5 Heavy quarkonium production

Forty years after the discovery of the \(J/\psi \), the mechanism underlying quarkonium production has still not been clarified. Until the mid-90s mostly the traditional color singlet model was used in perturbative cross section calculations. The dramatic failure to describe \(J/\psi \) production at the Tevatron led, however, to a search for alternative explanations. The NRQCD factorization conjecture has by now received most acceptance, although not yet being fully established.

4.5.1 Summary of recent experimental progress

The past couple of years have seen incredible progress in measurements of quarkonium production observables, which was mainly, but not solely, due to the operation of the different LHC experiments. Here we will give an overview of the most remarkable results of the past years.

The production rates of a heavy quarkonium \(H\) are split into direct, prompt, and nonprompt contributions. Direct production refers to the production of \(H\) directly at the interaction point of the initial particles, while prompt production also includes production via radiative decays of higher quarkonium states, called feed-down contributions. Nonprompt production refers to all other production mechanisms, mainly the production of charmonia from decaying \(B\) mesons, which can be identified by a second decay vertex displaced from the interaction point.

a. \(J/\psi \)production in\(pp\)collisions The 2004 CDF transverse momentum \(p_\mathrm{T}\) distribution measurement of the \(J/\psi \) production cross section [1141] is still among the most precise heavy quarkonium production measurements. But since theory errors in all models for heavy quarkonium production are still much larger than today’s experimental errors, it is in general not higher precision which is needed from the theory side, but rather new and more diverse production observables. And this is where the LHC experiments have provided very important input. As for the \(J/\psi \) hadroproduction cross section, they have extended the CDF measurement [1141] into new kinematic regions: Obviously, the measurements have been performed at much higher center-of-mass energies than before, namely at \(\sqrt{s}=2.76\), 7, and 8 TeV. But more important for testing quarkonium production models is the fact that there are measurements which exceed the previously measured \(p_\mathrm{T}\) range both at high \(p_\mathrm{T}\), as by ATLAS [1142] and CMS [1143], and at low \(p_\mathrm{T}\), as in the earlier CMS measurement [1144], but also in the recent measurement by the PHENIX collaboration at RHIC [1145]. We note that this list is not complete, and that there have been many more \(J/\psi \) hadroproduction measurements recently than those cited here.

b.\(\psi (2S)\)and \(\chi _\mathrm{c}\)production in\(pp\)collisions\(J/\psi \) is the quarkonium which is easiest to be measured due to the large branching ratio of its leptonic decay modes, but in recent years, high precision measurements have been also performed for the \(\psi (2S)\), namely by the CDF [1146], the CMS [1143], and the LHCb [1147] collaborations. Also the \(\chi _\mathrm{c}\) production cross sections were measured via their decays into \(J/\psi \) by LHCb [1148], the first time since the CDF measurement [1149] in 2001. The \(\chi _{c2}\) to \(\chi _{c1}\) production ratio was measured at LHCb [1150], CMS [1151] and previously by CDF [1152]. These measurements are of great importance for the theory side since they allow fits of NRQCD LDMEs for these charmonia and determine direct \(J/\psi \) production data, which can in turn be compared to direct production theory predictions.

c.\(\Upsilon \)production in\(pp\)collisions \(\Upsilon (1S)\), \(\Upsilon (2S)\), and \(\Upsilon (3S)\) production cross sections were measured at the LHC by ATLAS [1153] and LHCb [1154, 1155]. Previously, \(\Upsilon \) was produced only at the Tevatron [1156, 1157].
Fig. 30

The \(J/\psi \) polarization parameter \(\alpha \equiv \lambda _\theta \) in the helicity frame as measured by CDF in Tevatron run I [1158] (a), run II [1159] (b), and by ALICE [1160] and LHCb [1161] at the LHC (right). Adapted from [1158, 1159, 1161], respectively

Fig. 31

The polarization parameter \(\lambda _\theta \) in the helicity frame for \(J/\psi \) (left) and \(\psi (2S)\) (right) production as measured by CMS [1162]. Adapted from [1162]

Fig. 32

The \(\Upsilon (1S)\) polarization parameter \(\lambda _\theta \) in the helicity frame as measured by CDF [1156, 1163], D0 [1164] and CMS [1165]. Adapted from [1163, 1165], respectively

d.Polarization measurements in\(pp\)collisions The measurements of the angular distributions of the quarkonium decay leptons are among the most challenging experimental tasks in quarkonium physics, because much more statistics and a much better understanding of the detector acceptances than in cross section measurements is needed. These angular distributions \(W(\theta ,\phi )\) are directly described by the polarization parameters \(\lambda _\theta \), \(\lambda _\phi \), and \(\lambda _{\theta \phi }\) via
$$\begin{aligned} W(\theta ,\phi )&\propto 1+\lambda _\theta \cos ^2\theta +\lambda _\phi \sin ^2\theta \cos (2\phi ) \nonumber \\&{}+\lambda _{\theta \phi }\sin (2\theta )\cos \phi , \end{aligned}$$
(4.16)
where \(\theta \) and \(\phi \) are, respectively, the polar and azimuthal angles of the positively charged decay lepton in the quarkonium rest frame. These polarization measurements pose highly nontrivial tests for quarkonium production models, and have therefore probably been the most anticipated LHC results on quarkonium. Previous Tevatron measurements tended to give ambiguous results: The CDF measurements of \(J/\psi \) polarization in Tevatron run I [1158] and II [1159] have been in partial disagreement; see Fig. 30, similar to the \(\Upsilon (1S)\) polarization measured by D0 [1164] and by CDF in Tevatron run I [1156] and II [1163]; see Fig. 32. At RHIC, \(J/\psi \) polarization has been measured by PHENIX [1166] and STAR [1167]. At the LHC, \(J/\psi \) polarization has so far been measured by ALICE [1160], LHCb [1161], and CMS [1162]; see Figs. 30 and 31. Furthermore, CMS has measured the \(\psi (2S)\) [1162] and \(\Upsilon \) [1165] polarization; see Figs. 31 and 32. None of the CDF Tevatron run II and the LHC measurements have found a strong and significant transverse or longitudinal polarization for any quarkonium. CDF at Tevatron run II and LHCb do, however, seem to prefer slight longitudinal polarizations in their helicity frame quarkonium polarization measurements, whereas in the CMS measurements there seems to be a tendency for slight transverse polarizations in the helicity frame, see Figs. 3031, and 32.

e. Recent\(ep\)collision results For testing theory predictions, in particular the universality of NRQCD long distance matrix elements, we need to consider experimental data from a variety of different production mechanisms. Very important charmonium production data have thereby in the past come from inelastic photoproduction at the \(ep\) collider HERA, which came in distributions in the transverse charmonium momentum \(p_\mathrm{T}\), the photon-proton invariant mass \(W\) and the inelasticity variable \(z\). The latest update on inclusive \(J/\psi \) production cross sections was in 2012 by the ZEUS collaboration [1168]. This publication also presented values for \(\sigma (\psi (2S))/\sigma (J/\psi )\) with error bars reduced by about two thirds relative to the previous ZEUS measurement [1169] at HERA 1. The \(J/\psi \) polarization measurements by the ZEUS [1170] and H1 [1171] collaborations were, however, still associated with such large errors that no unambiguous picture of the \(J/\psi \) polarization in photoproduction emerged. Furthermore, no \(\Upsilon \) photoproduction could be observed at HERA. Therefore, from the theory side, a new \(ep\) collider at much higher energies and luminosities than HERA, like possibly an LHeC, would be highly desired. On the other hand, there is still no NLO calculation for \(J/\psi \) production in deep inelastic scattering available, as, for example, measured most recently by H1 [1171].

f. Further production observables The LHCb experiment with its especially rich quarkonium program has also measured completely new observables which still need to be exploited fully in theory tests: For the first time in \(pp\) collisions the double \(J/\psi \) production cross section was measured [1172], as well as the production of \(J/\psi \) in association with charmed mesons [1173]. Like double charmonium production, \(J/\psi +c\overline{c}\) was previously only measured at the \(B\) factories, latest in the Belle analysis [1174], which was crucial for testing \(J/\psi \) production mechanisms in \(e^+e^{-}\) production. \(J/\psi \) production in association with \(W\) bosons has for the first time been measured by the ATLAS collaboration [1175]. Exclusive charmonium hadroproduction has been observed recently by CDF [1176] and LHCb [1177, 1178]. Exclusive production had previously been a domain of \(ep\) experiments; see [1179] for a recent update by the H1 collaboration. Another observable for which theory predictions exist is the \(J/\psi \) production rate in \(\gamma \gamma \) scattering. This observable has previously been measured at LEP by DELPHI [1180] with very large uncertainties and could possibly be remeasured at an ILC.

4.5.2 NLO tests of NRQCD LDME universality

Table 13

Overview of different NLO fits of the CO LDMEs. Analysis [770] is a global fit to inclusive \(J/\psi \) yield data from 10 different \(pp\), \(\gamma p\), \(ee\), and \(\gamma \gamma \) experiments. In [1181], fits to \(pp\) yields from CDF [1141, 1146] and LHCb [1147, 1148, 1182] were made. In [1183], three values for their combined fit to CDF \(J/\psi \) yield and polarization [1158, 1159] data are given: A default set, and two alternative sets. Analysis [1184] is a fit to the \(\chi _{c2}/\chi _{c1}\) production ratio measured by CDF [1152]. The analyses [770] and [1183] refer only to direct \(J/\psi \) production, and in the analyses [1181] and [1183] \(p_T<7\) GeV data was not considered. The color singlet LDMEs for the \({^3}S_1^{[1]}\) and \({^3}P_0^{[1]}\) states were not fitted. The values of the LDMEs given in the second through sixth column (referring to [770, 1181], and [1183]) were used for the plots of Fig. 33

 

Butenschoen, Kniehl [770]:

Gong, Wan, Wang, Zhang [1181]:

Chao, Ma, Shao, Wang, Zhang [1183]:

Ma, Wang, Chao [1184]:

   

(default set)

(set 2)

(set 3)

 

\(\langle \mathcal{O}^{J/\psi }({^3}S_1^{[1]}) \rangle /\text{ GeV }^3 \)

\(1.32\)

\(1.16\)

\(1.16\)

\(1.16\)

\(1.16\)

 

\(\langle \mathcal{O}^{J/\psi }({^1}S_0^{[8]}) \rangle /\text{ GeV }^3\)

\(0.0497\pm 0.0044\)

\(0.097\pm 0.009\)

\(0.089\pm 0.0098\)

\(0\)

\(0.11\)

 

\(\langle \mathcal{O}^{J/\psi }({^3}S_1^{[8]}) \rangle /\text{ GeV }^3\)

\(0.0022\pm 0.0006\)

\(-0.0046\pm 0.0013\)

\(0.0030\pm 0.012\)

\(0.014\)

\(0\)

 

\(\langle \mathcal{O}^{J/\psi }({^3}P_0^{[8]}) \rangle /\text{ GeV }^5\)

\(-0.0161\pm 0.0020\)

\(-0.0214\pm 0.0056\)

\(0.0126\pm 0.0047\)

\(0.054\)

\(0\)

 

\(\langle \mathcal{O}^{\psi (2S)}({^3}S_1^{[1]}) \rangle /\text{ GeV }^3\)

 

\(0.758\)

    

\(\langle \mathcal{O}^{\psi (2S)}({^1}S_0^{[8]}) \rangle /\text{ GeV }^3\)

 

\(-0.0001\pm 0.0087\)

    

\(\langle \mathcal{O}^{\psi (2S)}({^3}S_1^{[8]}) \rangle /\text{ GeV }^3\)

 

\(0.0034\pm 0.0012\)

    

\(\langle \mathcal{O}^{\psi (2S)}({^3}P_0^{[8]}) \rangle /\text{ GeV }^5\)

 

\(0.0095\pm 0.0054\)

    

\(\langle \mathcal{O}^{\chi _0}({^3}P_0^{[1]}) \rangle /\text{ GeV }^5\)

 

\(0.107\)

   

\(0.107\)

\(\langle \mathcal{O}^{\chi _0}({^3}S_1^{[8]}) \rangle /\text{ GeV }^3\)

 

\(0.0022\pm 0.0005\)

   

\(0.0021\pm 0.0005\)

Fig. 33

The predictions of the \(J/\psi \) total \(e^+e^{-}\) cross section measured by Belle [1174], the transverse momentum distributions in photoproduction measured by H1 at HERA [1171, 1185], and in hadroproduction measured by CDF [1141] and ATLAS [1142], and the polarization parameter \(\lambda _\theta \) measured by CDF in Tevatron run II [1159]. The predictions are plotted using the values of the CO LDMEs given in [770], [1181] and [1183] and listed in Table 13. The error bars of graphs ag refer to scale variations, of graph d also fit errors, errors of graph h according to [1181]. As for graphs il, the central lines are evaluated with the default set, and the error bars evaluated with the alternative sets of the CO LDMEs used in [1183] and listed in Table 13. From [1186]

The phenomenological relevance of the NRQCD factorization conjecture is closely tied to the question of whether or not the LDMEs can be shown to be universal. In this section recent works will be reviewed which aim at examining this universality at Next-to-Leading Order (NLO) in \(\alpha _\mathrm{s}\). In the case of \(\chi _{cJ}\), these tests include just the leading order of the NRQCD \(v\) expansion, formed by the \(n={^3}P_J^{[1]}\) and \(n={^3S}_1^{[8]}\) states. In the case of \({^3}S_1\) quarkonia, these tests include the terms up to relative order \(O(v^4)\) in the \(v\) expansion, namely the \(n={^3}S_1^{[1]}\) color singlet state, as well as the \(n={^1}S_0^{[8]}\), \({^3}S_1^{[8]}\), and \({^3}P_J^{[8]}\) Color Octet (CO) states; see Table 5. The relativistic corrections involving the \(\langle \mathcal{P}^{H}({^3}S_1^{[1]}) \rangle \) and \(\langle \mathcal{Q}^{H}({^3}S_1^{[1]}) \rangle \) LDMEs are, however, not part of these analyses, although they are of order \(O(v^2)\) and \(O(v^4)\) in the \(v\) expansion. There are two reasons for that: First, the corresponding NLO calculations are far beyond the reach of current techniques, and secondly, they are expected to give significant contributions to hadroproduction only at \(p_\mathrm{T}\ll m_\mathrm{c}\) and for photoproduction only at \(z\approx 1\). This behavior is inferred from the behavior at LO in \(\alpha _\mathrm{s}\) [1187, 1188] and can be understood by noting that new topologies of Feynman diagrams open up when doing the transition from the \({^3}S_1^{[1]}\) state to the CO states, but not when calculating relativistic corrections: For example, at leading order in \(\alpha _\mathrm{s}\) the slope of the transverse momentum distribution in hadroproduction is \(\mathrm{d}\sigma /dp_\mathrm{T} \approx p_\mathrm{T}^{-8}\) for the \({^3}S_1^{[1]}\) state, compared to \(\mathrm{d}\sigma /dp_\mathrm{T} \approx p_\mathrm{T}^{-6}\) for the \({^1}S_0^{[8]}\) and \({^3}P_J^{[8]}\) states and \(\mathrm{d}\sigma /dp_\mathrm{T} \approx p_\mathrm{T}^{-4}\) for the \({^3}S_1^{[8]}\) state.

The \(O(\alpha _\mathrm{s})\) corrections to the necessary unpolarized short-distance cross sections of the \(n={^1}S_0^{[8]}\), \({^3}S_1^{[1/8]}\), and \({^3}P_J^{[1/8]}\) intermediate states have been calculated for most of the phenomenologically relevant inclusive quarkonium production processes: For two-photon scattering [770, 1189], \(e^+e^{-}\) scattering [1190], photoproduction [770, 1191] and hadroproduction [1184, 1192, 1193, 1194, 1195]. The polarized cross sections have been calculated for photoproduction [1196] and hadroproduction [1181, 1183, 1197, 1198].

In [770], a global fit of the \(J/\psi \) CO LDMEs to 26 sets of inclusive \(J/\psi \) production yield data from 10 different \(pp\), \(\gamma \)p, \(\gamma \gamma \), and \(e^+e^{-}\) experiments was done; see the second column of Table 13 for the fit results. This fit describes all data, except perhaps the two-photon scattering at LEP [1180], reasonably well. This fit is overconstrained, and practically independent of possible low-\(p_\mathrm{T}\) cuts (unless such high \(p_\mathrm{T}\) cuts are chosen that all data except hadroproduction drop out of the fit [1199]). Furthermore, the resulting LDMEs are in accordance with the velocity scaling rules predicted by NRQCD; see Table 5. Thus the fit is in itself already a nontrivial test of the NRQCD factorization conjecture, especially since the high-\(z\) photoproduction region can now also be well described, which had been plagued by divergent behavior in the earlier Born analyses [1200, 1201]. However, in [1197] it was shown that these CO LDME values lead to predictions of a strong transverse \(J/\psi \) polarization in the hadroproduction helicity frame, which is in contrast to the precise CDF Tevatron run II measurement [1159]; see Fig. 33d. On the other hand, in [1183] it was shown that both the measured \(J/\psi \) hadroproduction cross sections and the CDF run II polarization measurement [1159] can, even at the highest measured \(p_\mathrm{T}\) values, be well described when choosing one of the three CO LDME sets listed in columns four through six of Table 13. These LDMEs, however, result in predictions for \(e^+e^{-}\) annihilation and photoproduction which are factors four to six above the data; see Fig. 13e–f. Third, the calculation [1181] is the first NLO polarization analysis to include feed-down contributions. To this end, the CO LDMEs of \(J/\psi \), \(\psi (2S)\) and \(\chi _{cJ}\) were fitted to CDF [1141, 1146] and LHCb [1147, 1148, 1182] unpolarized production data with \(p_\mathrm{T}>7\) GeV; see column three of Table 13. These fit results were then used for the predictions of Fig. 33e–h, taking the \(\psi (2S)\) and \(\chi _{cJ}\) feed-down contributions consistently into account. A similar analysis has recently also been performed for \(\Upsilon (1S,2S,3S)\) production [1198].

The shape of high-\(p_\mathrm{T}\)\(J/\psi \) hadroproduction yield can be nicely described by the \({^1}S_0^{[8]}\) component alone, which automatically yields unpolarized hadroproduction. Since at \(p_\mathrm{T}>10\) GeV this is already all data available, there is no tension between NRQCD predictions and current data if the validity of the NRQCD factorization conjecture is restricted to high enough \(p_\mathrm{T}\) values and the \({^3}S_1^{[1/8]}\) and \({^3}P_J^{[1/8]}\) LDMEs are very small or even put to zero, as for example in the sixth column of Fig. 33 (set 3). This is also the spirit of [1202], and of the analysis [1203], in which the NLO short distance cross sections used in [1183] are combined with \(c\overline{c}\) production via single parton fragmentation using fragmentation functions at order \(\alpha _\mathrm{s}^2\) including a leading log resummation.

To summarize, none of the proposed CO LDME sets is able to describe all of the studied \(J/\psi \) production data sets, which poses a challenge to the LDME universality. Possible resolutions include the following:
  1. 1.

    The perturbative \(v\) expansion might converge too slowly.

     
  2. 2.

    NRQCD factorization might hold for exclusive, but not inclusive, production.

     
  3. 3.

    NRQCD factorization might hold only in the region \(p_\mathrm{T}\gg M_\mathrm{onium }\). Currently, photoproduction cross sections are measured only up to \(p_\mathrm{T}=10\) GeV.

     
  4. 4.

    NRQCD factorization might not hold for polarized production.

     

4.5.3 Recent calculations of relativistic corrections

Table 14

Color singlet model predictions for \(\sigma (e^{+}e^{-}\rightarrow J/\psi +\eta _{c})\) compared to \(B\)-factory data [1204, 1205, 1206]. As for the theoretical predictions for the leading-order cross section as well as the corrections of order \(O(\alpha _{s})\), \(O(v^{2})\), and \(O(\alpha _{s} v^{2})\), we compare the results obtained in [1207, 1208, 1209]. These calculations mainly differ by different methods of color singlet LDME determinations. As for the values of [1208], the leading-order results include pure QED contributions, the \(O(\alpha _{s})\) results include interference terms with the QED contributions, and the \(O(v^{2})\) results include in part a resummation of relativistic corrections, the \(O(\alpha _{s} v^{2})\) results do, however, include the interference terms of the \(O(\alpha _{s})\) and \(O(v^{2})\) amplitudes only. The short-distance coefficients of the \(O(\alpha _{s})\) contribution used in [1207] and [1208] were taken over from [1210]. The experimental cross sections refer to data samples in which at least 2, respectively 4, charged tracks were identified

 

He, Fan,

Bodwin,

Li, Wang [1209]

 

Chao [1207]

Lee,Yu [1208]

  
 

\(\alpha _{s}(2m_{c})\)

\(\alpha _{s}(\sqrt{s}/2)\)

\(\alpha _{s}(\sqrt{s}/2)\)

\(\alpha _{s}(2m_{c})\)

\(\sigma _{LO}\)

9.0 fb

6.4 fb

4.381 fb

7.0145 fb

\(\sigma (\alpha _{s})\)

8.8 fb

6.9 fb

5.196 fb

7.367 fb

\(\sigma (v^{2})\)

2.2 fb

2.9 fb

1.714 fb

2.745 fb

\(\sigma (\alpha _{s} v^{2})\)

 

1.4 fb

0.731 fb

0.245 fb

sum

20.0 fb

\(17.6^{+8.1}_{-6.7}\) fb

12.022 fb

17.372 fb

Belle [1204]

\(33^{+7}_{-6}\pm 9\) fb (\(\ge \)4 charged tracks)

Belle [1205]

\(25.6\pm 2.8 \pm 3.4\) fb (\(\ge \)2 charged tracks)

BaBar [1206]

\(17.6\pm 2.8^{+1.5}_{-2.1}\) fb (\(\ge \)2 charged tracks)

As explained in the last section, the relativistic corrections of order \(O(v^2)\) in the NRQCD \(v\) expansion have at leading order in \(\alpha _\mathrm{s}\) in inclusive hadro- [1188] and photoproduction [1187] been shown to be less significant than the CO contributions of order \(O(v^4)\) in the NRQCD \(v\) expansion. Similarly, the \(O(v^2)\) [1211] and the technically challenging \(O(v^4)\) [769] relativistic corrections to gluon fragmentation into \({^3}S_1\) quarkonia have turned out to be small. The relativistic \(O(v^2)\) corrections to the process \(e^+e^{-}\rightarrow J/\psi +gg\) have, however, turned out to be between 20 % and 30 % [1212, 1213] relative to the leading order CS cross section, an enhancement comparable in size to the \(O(\alpha _\mathrm{s})\) CS correction [1214, 1215]. These corrections helped bring the color singlet model prediction for inclusive \(J/\psi \) production in \(e^+e^{-}\) collisions in rough agreement with experimental data [1174].

Similarly, in the exclusive process \(e^+e^{-}\rightarrow J/\psi +\eta _\mathrm{c}\), \(O(\alpha _\mathrm{s})\) corrections as well as relativistic corrections of \(O(v^2)\) were necessary to bring the color singlet model prediction in agreement with data; see Table 14. Recently, even \(O(\alpha _\mathrm{s} v^2)\) corrections to this process have been calculated [1209, 1216]. For a review of the history of the measurements and calculations of this process, as well as for a description of different methods to determine the LDMEs of relative order \(O(v^2)\), we refer to section 4.5.1 of [757].

As a final point of this section, we mention the interesting work [1217] in which relativistic corrections to the process \(gg\rightarrow J/\psi +g\) via color octet states formally of order \(O(v^6)\) were estimated. According to this analysis, at leading order in \(\alpha _\mathrm{s}\), they might reduce the \(O(v^4)\) CO contributions by up to 20–40 % in size.

4.5.4 Calculations using \(k_\mathrm{T}\) factorization

Color singlet model predictions for \(J/\psi \) production face many phenomenological problems: Except for \(e^+e^{-}\) annihilation, NLO color singlet model predictions are shown to lie significantly below inclusive \(J/\psi \) production data, 1–2 orders of magnitude for hadroproduction and \(\gamma \gamma \) scattering, and a factor 3–5 for photoproduction at HERA; see, for example, [770]. As in photoproduction [1218, 1219], \(J/\psi \) polarization in hadroproduction [1220] is at NLO predicted to be highly longitudinal in the helicity frame, in contrast to the CDF measurement at Tevatron run II [1159].

According to [1221, 1222], these shortcomings can be overcome when the transverse momenta \(k_\mathrm{T}\) of the initial gluons are retained. The off-shell matrix elements are then folded with unintegrated, \(k_\mathrm{T}\) dependent, Parton Distribution Functions (uPDFs). The weakest point of this approach is certainly the derivation of the uPDFs from the usual gluon PDFs using varying prescriptions. The latest analyses [1221, 1222] show very good agreement with \(J/\psi \) photoproduction data at HERA [1169, 1170, 1171, 1185] and hadroproduction at the LHC [1142, 1144, 1182]. On top of that, the \(J/\psi \) is predicted to be largely unpolarized, in line with all recent polarization measurements; see paragraph \(d\) in Sect. 4.5.1. As for hadroproduction, the conclusions are however contrary to the author’s earlier findings [1223], which show longitudinal \(J/\psi \) polarization and cross sections an order of magnitude below the CDF production data. They also disagree with the recent work [1224], where \(J/\psi \) hadroproduction at the LHC was studied in the same way, comparing to the same data [1142, 1144, 1182], even when the same uPDFs [1225, 1226] were used. Here, the color singlet predictions lie again clearly below the data, and the difference was even used to fit the CO LDMEs of NRQCD in a \(k_\mathrm{T}\) factorization approach.

We note that calculations in the \(k_\mathrm{T}\) factorization scheme can be performed for any intermediate Fock state of the NRQCD \(v\) expansion. On the other hand, even a fully worked out framework of \(k_\mathrm{T}\) factorization at NLO in \(\alpha _\mathrm{s}\) could not cure the problem of uncanceled infrared singularities in color singlet model calculations for \(P\) wave quarkonia.

4.5.5 Current trends in theory

The most prominent candidate theory for heavy quarkonium production is NRQCD, and lots of effort is going on to prove its factorization theorem on the one hand, and to show the universality of the LDMEs by comparison to data on the other. Since at the moment there are hints that at least to the orders currently considered in perturbation theory, not all data might be simultaneously described by single LDME sets, more effort will be going on to refine NRQCD calculations for specific observables or specific kinematic regimes, such as the low and high \(p_\mathrm{T}\) limits of the hadroproduction cross sections. For low \(p_\mathrm{T}\) resummation of large logarithms, the recent work [1227] followed the idea of [1228] to apply the Collins–Soper–Sterman impact parameter resummation formalism [90]. For high \(p_\mathrm{T}\) resummation, the factorization theorem of [781, 783] in terms of single and double parton fragmentation functions, and the soft-collinear effective theory approach [785, 786] can be applied. Other paths may be to apply transverse momentum-dependent PDFs in quarkonium production calculations, but the uncertainties inherent to these calculations will still need to be thoroughly investigated, as can be seen from contradicting \(k_\mathrm{T}\) factorization results. But also in more phenomenologically based models, like the color evaporation model, new predictions are still calculated [1229].

4.6 Future directions

Our understanding of heavy quark hadronic systems improves with the progress made on experimental measurements of masses, production and decay rates, the development of suitable effective field theories, perturbative calculations within these frameworks, and the progress on lattice gauge theory calculations.

Lattice simulations are obtaining a more and more prominent role in heavy quark physics. They may compute low-energy matrix elements, factorized by effective field theories, appearing in the study of quarkonia below threshold, improving our understanding of the dynamics of these systems and providing, among others, precision determinations of the strong coupling constant at low energies and the heavy quark masses. For states at and above threshold, they may eventually be able to determine the nature of the \(XYZ\) exotic states, including in particular the role that mixing between tetraquark and multihadron states plays. A possible way to address these problems that relies on lattice simulations has been very recently proposed in [1230, 1231]. Lattice simulations are also required for determining non-perturbative form factors needed in extracting the CKM matrix elements \(|V_{cb}|\), \(|V_{ub}|\), \(|V_{cs}|\) and \(|V_{cd}|\) from \(B \rightarrow D^\star /\pi l \nu \) and \(D \rightarrow K/\pi l \nu \) decays, respectively. Current gaps between lattice determinations and experimental fits of these form factors are expected to be removed by further progress in lattice simulations. The emergence of ensembles incorporating the effects of dynamical charm quarks in lattice calculations will help to establish whether charm sea contributions to charmonium spectra and to flavor observables are relevant. At the same time, the trend to finer lattice spacings (even if currently somewhat displaced by a trend to perform simulations at the physical pion mass) is likely to continue in the long run and will eventually enable the use of fully relativistic b-quarks, which will provide an important cross-check on effective field theories, and eventually for some observables replace them.

Rapid progress on the side of effective field theories is currently happening for any system involving heavy quarks. Many quantities, like spectra, decays, transitions and production cross sections, are computed in this framework with unprecedented precision in the velocity and \(\alpha _\mathrm{s}\) expansions. Noteworthy progress is happening, in particular, in the field of quarkonium production. Here, the recent Snowmass White Paper on “Quarkonium at the Frontiers of High Energy Physics” [1013] provides an excellent summary. Future work will be likely centered around the effort to search for a rigorous theoretical framework (factorization with a rigorous proof) for inclusive as well as exclusive production of quarkonia at various momentum scales. While a proof of NRQCD factorization is still lacking, performing global analyses of all available data in terms of the NRQCD factorization formalism is equally important, so that the universality of the NRQCD LDMEs can be systematically tested, which is a necessary condition for the factorization conjecture. To better test the conjecture, a resummation of various large logarithms in perturbative calculations in different production environments are critically needed.

The currently running experiments, in particular BESIII and the LHC experiments, will at this stage primarily help refine previous measurements. The LHC will in particular continue to provide measurements on heavy quarkonium production rates at unprecedented values of transverse momentum, provide better measurements on quarkonium polarization, but might also provide more diverse observables, such as associated production of a heavy quarkonium with gauge bosons, jets or other particles. The LHC will also continue to contribute to the studies of \(XYZ\) states, and determine the \(XYZ\) quantum numbers from amplitude analyses. Studies of \(Z_\mathrm{c}\) states at BESIII will continue and provide precise measurements of spin-parities and resonance parameters from multiple decay channels and amplitude analyses.

In the farther future, however, Belle II is expected to produce more and better data that will be particularly useful to reduce the uncertainties on the CKM matrix elements \(|V_{cb}|\) and \(|V_{ub}|\). Data from a larger phase space can provide more precise information to solve the long-standing discrepancy between the inclusive and exclusive measurements of \(|V_{ub}|\). Having about 100 fb\(^{-1}\) integrated luminosity from the first Belle II run at the \(\Upsilon (6S)\) resonance or at a nearby energy will be very exciting for bottomonium studies. \(\Upsilon (6S)\) deserves further studies, in particular, to clarify if \(Z_b\) states are also produced in its decays, to search for \(\Upsilon (6S) \rightarrow h_b \pi ^+\pi ^{-}\) transitions, and to measure the \(e^+e^{-} \rightarrow h_b \pi ^+\pi ^{-}\) cross section as a function of energy, which should provide important information that is needed to answer whether \(\Upsilon (6S)\) is more similar to \(\Upsilon (5S)\) or to \(Y(4260)\) in its properties. With a possible upgrade of the injection system to increase its energy from current \(11.2\) GeV, Belle II could access also more molecular states close to \(B^{(*)} \overline{B}^{(*)}\), predicted from heavy quark spin symmetry. Belle II and the LHC upgrade, as well as future higher energy/luminosity \(ep\) (electron-ion) and \(e^+e^{-}\) (Higgs factory) colliders, will provide precision measurements of heavy quarkonium production with more diverse observables in various environments, and might thereby challenge our understanding of how heavy quarkonia emerge from high-energy collisions.

5 Searching for new physics with precision measurements and computations

5.1 Introduction

9The scope of the current chapter extends beyond that of QCD. Therefore, we begin with a brief overview of the standard model (SM) in order to provide a context for the new physics searches we describe throughout.

The current SM of particle physics is a renormalizable quantum field theory based on an exact SU(3)\(_{c}\times \)SU(2)\(_{L} \times \)U(1) gauge symmetry. As a result of these features and its specific particle content, it contains additional, accidental global symmetries, of which the combination B–L is anomaly free. It also preserves the discrete spacetime symmetry CPT, but C and P and T are not separately guaranteed—and indeed P and C are violated by its explicit construction. It describes all the observed interactions of known matter, save for those involving gravity, with a minimum of 25 parameters. These parameters can be taken as the six quark masses, the six lepton masses, the four parameters each (three mixing angles and a CP-violating phase) in the CKM and Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrices which describe the mixing of quarks and neutrinos,10 respectively, under the weak interactions, and the five parameters describing the gauge and Higgs sectors. The SM encodes CP violation in the quark sector not only through a phase in the CKM matrix but also through a “would-be” parameter \(\bar{\theta }\), which the nonobservation of a permanent electric dipole moment of the neutron [1232] limits to \(\bar{\theta }< 10^{-10}\) if no other sources of CP violation operate.

The SM, successful though it is, is incomplete in that it leaves many questions unanswered. Setting aside the question of gravity, which is excluded from the onset, the SM cannot explain, e.g., why the \(W\) and \(Z\) bosons have the masses that they do, the observed pattern of masses and mixings of the fermions, nor why there are three generations. It cannot explain why \(\bar{\theta }\) is so small, nor why the baryon asymmetry of the Universe has its observed value. It does not address the nature or even the existence of dark matter and dark energy. It has long been thought that the answers to some of these questions could be linked and, moreover, would find their resolution in new physics at the weak scale. The LHC is engaged in just such a search for those distinct and new phenomena that cannot be described within the SM framework. In Sect. 5.2 we review current collider efforts and how QCD studies advance and support them.

Direct searches for new particles and interactions at colliders certainly involve precision measurements and computations, but discoveries of new physics can also be made at low energies through such efforts. There are two paths: one can discover new physics through (i) the observed failure of the symmetries of the SM, or (ii) the failure of a precision computation to confront a precision measurement. Examples from the first path include searches for permanent electric dipole moments (EDMs) and for charged-lepton flavor violation, at current levels of sensitivity, as well as searches for neutrinoless double beta decay and \(n\)\(\bar{n}\) oscillations. Prominent examples from the second path are the determination of the lepton anomalous magnetic moments, the \(g-2\) of the muon and of the electron. Taken more broadly, the second path is also realized by overconstraining the SM parameters with multiple experiments and trying to find an inconsistency among them. Updated elsewhere in this review are determinations of the weak mixing angle \(\theta _W\) (Sect. 3.5) and the strong coupling constant \(\alpha _\mathrm{s}\) (Sect. 3.5.3), which are under intense scrutiny by the QCD community. We refer to Sect. 3.5 for a discussion of the muon \(g-2\). In this chapter we review such results from quark flavor physics.

QCD plays various roles in these efforts. In the first case, the discovery of whether a SM symmetry is actually broken is essentially an experimental question, though QCD effects play a key role not only in assessing the relative sensitivity of different experiments but also in the interpretation of an experimental result in terms of the parameters of an underlying new physics model. In the second case, the importance of QCD and confinement physics is clear. QCD effects are naturally dominant in all experiments searching for new physics that involve hadrons. We emphasize that experiments in the lepton sector are not immune to such issues, because hadronic effects are simply suppressed by power(s) of the fine-structure constant \(\alpha \)—they enter virtually through loop corrections. Their ultimate importance is predicated by the precision required to discover new physics in a particular process. Generally, for fixed experimental precision, a lack of commensurate control over QCD corrections, be it in experiments at high-energy colliders or at low energies, can jeopardize our search for physics beyond the SM.

In this document, we consider the broad ramifications of the physics of confinement, with a particular focus on our ability to assess its impact in the context of QCD. This interest drives the selection of the topics which follow. We begin with a brief overview of the role of QCD in collider physics. This part particularly concerns factorization theorems and resummation, which is illustrated with a few select examples. Our discussion, however, is not comprehensive, so that we do not review here the recent and impressive progress on next-to-leading-order (NLO) predictions for multi-parton production processes; see Ref. [1233] for a recent example, or the associated development of on-shell methods, which are reviewed in [1234, 1235]. We refer to Sect. 9.2 of this document for a terse summary of these developments. Next, we move to the primary focus of the chapter, which is the role of QCD in the search for new physics in low-energy processes. There is a large array of possible observables to consider; we refer the reader to a brief, recent overview [1236], as well as to a dedicated suite of reviews [1237, 1238, 1239, 1240, 1241, 1242, 1243, 1244]. In this chapter we describe the theoretical framework in which such experiments can be analyzed before delving more deeply into examples which illustrate the themes we have described. We consider, particularly, searches for permanent electric dipole moments of the neutron and proton and precision determinations of \(\beta \)-decay correlation coefficients. We refer the reader to Sect. 3.5 for a detailed discussion of the magnetic moment of the muon. We proceed to consider the need for and the computation of particular nucleon matrix elements rather broadly before turning to a summary of recent results in flavor-changing processes and an assessment of future directions.

5.2 QCD for collider-based BSM searches

5.2.1 Theoretical overview: factorization

A general cross section for a collider process involving hadrons is not directly calculable in perturbative QCD. Any such process will involve, at least, the energy scale of the collision and scales associated with masses of the hadrons, apart from other possible scales related to the definition of the jets involved in the process or to necessary experimental cuts. There is therefore an unavoidable dependence on long-distance, non-perturbative scales, and one cannot invoke asymptotic freedom to cope with it. Factorization theorems in QCD allow us to separate, in a systematic way, short-distance and, thus, perturbatively calculable effects from long-distance non-perturbative physics, which are encoded in process-independent objects, such as the parton distribution functions (PDFs). We refer to Sect. 3.2.1 for the theoretical definition of a PDF in the Wilson line formalism and a discussion of its empirical extraction. (A summary of pertinent lattice-QCD results, notably of the lowest moment of the isovector PDF, can be found in Sect. 3.2.5a.) Factorization theorems are, therefore, essential to QCD calculations of hadronic hard-scattering processes. The simpler structure of emissions in the soft and collinear limits, which can generate low-virtuality states, are at the basis of factorization proofs. Factorized forms for the cross sections (see the next section and Sect.  3.2.1 for some examples) can be obtained via diagrammatic methods in perturbative QCD [51] or, alternatively, by employing effective field theories (EFTs) to deal with the different scales present in the process. Soft collinear effective theory (SCET) [790, 1245, 1246, 1247] is the effective theory that implements the structure of soft and collinear interactions at the Lagrangian level, and it has been extensively used in the last years for many different processes, along with traditional diagrammatic approaches. Establishing a factorized form for the cross section is also the first necessary step for performing resummations of logarithmically enhanced terms, which are key for numerical accuracy in certain portions of phase space. In the following, we discuss a few illustrative examples, which allow us not only to glimpse state-of-the-art techniques but also to gain an impression of the current challenges.

5.2.2 Outcomes for a few sample processes

We begin with single vector-boson (\(W/Z/\gamma \)) production in hadron-hadron collisions in order to illustrate an application of the procedure known as threshold resummation. The transverse momentum, \(p_\mathrm{T}\), spectrum for these processes is known at NLO [1248, 1249, 1250], and there is ongoing work to obtain the NNLO corrections. This is an extremely challenging calculation, but even without it one can improve the fixed-order results by including the resummation of higher-order terms that are enhanced in certain limits. In some cases, such resummations of the fixed-order results are necessary in order to obtain a reasonable cross section. In particular, we focus now on the large-\(p_\mathrm{T}\) region of the spectrum, where enhancements related to partonic thresholds can appear. By a partonic threshold we mean configurations in which the colliding partons have just enough energy to produce the desired final state. In these cases, the invariant mass of the jet recoiling against the vector boson is small, and the perturbative corrections are enhanced by logarithms of the jet mass over \(p_\mathrm{T}\). The idea is that one can expand around the threshold limit and resum such terms. For single-particle production this was first achieved at next-to-leading-logarithmic (NLL) accuracy in [1251]. In general the cross section also receives contributions from regions away from the partonic threshold, but due to the rapid fall-off of the PDFs at large \(x\) the threshold region often gives the bulk of the perturbative correction. SCET offers a convenient, well-developed framework in which to perform such resummations, and allows one to push them to higher orders. A typical factorized form for the partonic cross section \(d\hat{\sigma }\), for example in the \(q\bar{q}\rightarrow gZ\) channel, looks schematically as follows
$$\begin{aligned} d\hat{\sigma } \propto H \, \int \! \mathrm {d} k \, J_g(m_X^2-(2E_J)k) S_{q\bar{q}}(k), \end{aligned}$$
(5.1)
with \(m_X\) and \(E_J\) the invariant mass and energy of the radiation recoiling against the vector boson, respectively. The jet function \(J_g\) describes collinear radiation initiated (in this case) by the gluon \(g\) present at Born level, the soft function \(S_{q\bar{q}}\) encodes soft radiation, and \(H\) is the hard function which contains short-distance virtual corrections. The argument of \(J_g\) in (5.1) can be understood by recalling that the recoiling radiation \(p_X^{\mu }\) is almost massless, i.e., it consists of collinear radiation \(p_J^{\mu }\) and additional soft radiation \(p_\mathrm{S}^{\mu }\). We can then write \(m^2_X=p_X^2=(p_J^{\mu }+p_\mathrm{S}^{\mu })^2=p_J^2+2p_J\cdot p_\mathrm{S}\), up to terms of order \(p_\mathrm{S}^2\ll p_J^2\); the collinear radiation can be approximated at leading order as \(p_J^{\mu }\sim E_Jn_J^{\mu }\), with \(n_J^{\mu }\) a light-like vector, and we obtain \(p_J^2=m_X^2-(2E_J)k\), where \(k\equiv n_J\cdot p_\mathrm{S}\) is the only component of the soft radiation that is relevant in the threshold limit. The hadronic cross section \(\mathrm{d}\sigma \) is given by a further convolution with the PDFs \(f_a\) as
$$\begin{aligned} \mathrm{d}\sigma \propto \sum _{ab=q,\bar{q},g}\int \mathrm{d}x_1\mathrm{d}x_2f_a(x_1)f_b(x_2)\left[ d\hat{\sigma }_{ab}\right] , \end{aligned}$$
(5.2)
where we include a sum over all allowed partonic channels \(ab\). Resummation has now been achieved at NNLL accuracy in Refs. [1252, 1253, 1254] using SCET techniques, which are based on the renormalization group (RG) evolution of the hard, jet, and soft functions. Some NNLL results obtained using diagrammatic methods have also been presented in [1255]. All ingredients required to achieve N\(^3\)LL accuracy within the SCET framework are essentially known [1256, 1257, 1258, 1259, 1260, 1261, 1262, 1263]. A phenomenological analysis at that unprecedented level of accuracy, combined with the inclusion of electroweak corrections which are enhanced by logarithms of the \(Z\) or \(W\) mass over \(p_\mathrm{T}\) [1264], can be expected to appear in the near future. These predictions can then be used, for instance, to constrain the \(u/d\) ratio of PDFs at large \(x\) (to which we return again in Sect. 5.2.4 from a lower-energy point of view), and to help estimate the \(Z(\rightarrow \nu \bar{\nu })+\)jets background to new heavy-particle searches [1265] at the LHC.

The same vector-boson production process but in the opposite limit, i.e., at low \(p_\mathrm{T}\), is a classic example in which resummation is essential to obtain reasonable predictions, since the perturbative fixed-order calculation diverges. An all-orders resummation formula for this cross section at small \(p_\mathrm{T}\) was first obtained in [90]. All ingredients necessary for NNLL accuracy have been computed. Predictions for the cross section at this level of accuracy are discussed in Refs. [105, 106, 1266]. The factorized formulas for this process are more involved than the corresponding ones for threshold resummation in the previous paragraph. In the SCET language, they involve what is sometimes called a “collinear factorization anomaly.” This means that the treatment of singularities present in SCET diagrams requires the introduction of additional regulators, in addition to the usual dimensional regularization, such as an analytic phase-space regularization [112], or, alternatively, one can also use the so-called “rapidity renormalization group” formalism [111], which is based on the regularization of Wilson lines. In any case, this generates some additional dependence (the aforementioned collinear anomaly) on the large scale of the process, \(Q\), with respect to what one might otherwise expect. There are, by now, well-understood consistency conditions [105, 1267] that restrict the form of this \(Q\) dependence to all orders, and the factorization formula remains predictive and useful. This nuance is directly related to the definition and regularization of the TMD PDFs, which appear in the factorization formula; see Sect. 3.2.1 for further discussion of TMD PDFs. Similar issues also appear when studying the evolution of double parton distribution functions in double-parton scattering (DPS) processes [1268]; further discussion on DPS is given in Sect. 5.2.4.

As we discuss in the next section, much of the current effort is, of course, devoted to the study of the Higgs and its properties. Let us just highlight here one example where good control over QCD effects is necessary, and for which recent progress has been significant.

To optimize the sensitivity of the analyses, Higgs-search data are often separated into bins with a specific number of jets in the final state. In particular, for the Higgs coupling measurements and spin studies, the \(H\rightarrow W^+W^{-}\) decay channel is quite relevant; but in this channel there is a large background coming from \(t\bar{t}\) production, which after the tops decay can produce a \(W^+W^{-}\) pair together with two \(b\)-quark jets. To reduce this background, events containing jets with transverse momentum above a certain threshold are rejected, i.e., one focuses on the 0-jet bin, which is also known as the jet-veto cross section. This restriction on the cross section enhances the higher-order QCD corrections to the process, by terms that contain logarithms of the transverse-momentum veto scale (typically around 25–30 GeV) over the Higgs mass. One should be careful when estimating the perturbative uncertainty of fixed-order predictions for the jet-veto cross section, since the cancellation of different effects can lead to artificially small estimations. A reliable procedure to estimate it was presented in [1269], and the outcome is that the perturbative uncertainty for the jet-veto cross section is around \(20~\%\), which is comparable to the current statistical experimental uncertainty and larger than the systematic one. It is therefore desirable to improve these theoretical predictions. There has been a lot of progress, starting with [1270], which showed that the resummation could be performed at NLL accuracy, and its authors also computed the NNLL terms associated with the jet radius dependence. Subsequently, resummation of these logarithms was performed at NNLL precision [1271, 1272, 1273, 1274, 1275]. An all-orders factorization formula was also put forward in Refs. [1272, 1275] within the SCET framework; its adequacy, though, has been questioned in Refs. [1271, 1274]. In any case, the accuracy for this jet-veto cross section has significantly improved, and there is room to continue improving the understanding of jet-veto cross sections and their uncertainty.

Related to the discussion of the previous paragraph, one would also like to have resummed predictions for \(N\)-jet processes, by which we mean any process with \(N\) hard jets. Although there has been important recent progress [1276, 1277] regarding the structure of infrared singularities in gauge theories, connecting them to \(N\)-jet operators and its evolution in SCET, many multi-jet processes involve so-called nonglobal logarithms [1278]. These are logarithms that arise in observables that are sensitive to radiation in only a part of the phase space. In general they appear at the NLL level, and although several explicit computations of these kinds of terms have been performed, it is not known how to resum them in general. Their presence, therefore, hinders the way to resummation for general \(N\)-jet cross sections. One might be forced to switch to simpler observables; see, e.g., Ref. [1279], to be able to produce predictions at higher-logarithmic accuracy.

Giving their present significance, jet studies command a great part of the current focus of attention. In particular, driven in part by the new possibilities that the LHC offers, the study of jet substructure, and jet properties in general, is a growing field. Jet substructure analysis can allow one, for instance, to distinguish QCD jets from jets coming from hadronic decays of boosted heavy objects; see, e.g., Refs. [1280, 1281, 1282]. Many other new results have appeared recently, and one can certainly expect more progress regarding jet studies in the near future. This will hopefully allow for improved identification techniques in searches for new heavy particles.

5.2.3 LHC results: Higgs and top physics

The announcement in mid-2012 of the discovery of a boson of mass near 125 GeV while searching for the SM Higgs electrified the world and represents a landmark achievement in experimental particle physics. It is decidedly a new physics result and one which we hope will open a new world to us. The discovery raises several key questions: What is its spin? Its parity? Is it pointlike or composite? One particle or the beginning of a multiplet? Does it couple like the SM Higgs to quarks, leptons, and gauge bosons? No other significant deviations from SM expectations have as yet been observed, falsifying many new-physics models. Nevertheless, plenty of room remains for new possibilities, both within and beyond the Higgs sector, and we anticipate that resolving whether the new particle is “just” the SM Higgs will require years of effort, possibly extending beyond the LHC. The ability to control QCD uncertainties will be essential to the success of the effort, as we can already illustrate.

a. Higgs production and decay The observation of the Higgs candidate by the ATLAS (http://atlas.ch [1283]) and CMS (http://cern.ch/cms [1284]) collaborations was based on the study of the \(H\rightarrow \gamma \gamma \) and \(H\rightarrow ZZ \rightarrow 4{\ell }\), with \({\ell } \in e,\mu \), channels, due to the excellent mass resolution possible in these final states [1285, 1286]. The finding was supported by reasonably good statistics, exceeding 4\(\sigma \) significance, in the four-lepton channel, for which the background is small, whereas the background in \(H\rightarrow \gamma \gamma \) is rather larger. Further work has led to an observed significance of \(6.7\sigma \) (CMS) in the \(H\rightarrow ZZ\) channel alone, and to studies of the \(H\rightarrow WW\), \(H\rightarrow bb\), \(H\rightarrow \tau \tau \) decay modes as well [1286]. It is worth noting that the Bose symmetry of the observed two-photon final state precludes a \(J=1\) spin assignment to the new particle; this conclusion is also supported by further study of \(H\rightarrow ZZ\rightarrow 4 {\ell }\) [1287]. Moreover, the finding is compatible with indirect evidence for the existence of a light Higgs boson [1288]. Figure 34 shows a comparison of recent direct and indirect determinations of the \(t\) quark and \(W\) gauge boson masses; this tests the consistency of the SM. The horizontal and vertical bands result from using the observed \(W\) (LEP+Tevatron) and \(t\) (Tevatron) masses at 68 % C.L., and global fits to precision electroweak data, once the \(t\) and \(W\) direct measurements are excluded, are shown as well [1289]. The smaller set of ellipses include determinations of the Higgs mass determinations from the LHC.
Fig. 34

Direct and indirect determinations of the \(W\)-boson and \(t\)-quark masses within the SM from measurements at LEP [698] and the Tevatron [1288], and from Higgs mass \(M_\mathrm{H}\) measurements at the LHC [1283, 1284]. The nearly elliptical contours indicate constraints from global fits to electroweak data, note http://cern.ch/gfitter [1290], exclusive of direct measurements of \(M_W\) and \(m_t\) from LEP and the Tevatron [1289, 1291]. The smaller (larger) contours include (exclude) the Higgs mass determinations from the LHC. We show a September, 2013 update from a similar figure in [1289] and refer to it for all details

Fig. 35

Values of \(\sigma /\sigma _\mathrm{SM}\) for particular decay modes, or of subcombinations therein which target particular production mechanisms. The horizontal bars indicate \(\pm 1\,\sigma \) errors including both statistical and systematic uncertainties; the vertical band shows the overall uncertainty. The quantity \(\sigma /\sigma _\mathrm{SM}\) (denoted \(\mu (x,y)\) in text) is the production cross section times the branching fraction, relative to the SM expectation [1286]. (Figure reproduced from [1286], courtesy of the CMS collaboration.)

We now summarize ongoing studies of the Higgs couplings, as well as of its spin and parity, highlighting the essential role of QCD in these efforts. It is evident that the Higgs discovery opens a new experimental approach to the search for new physics, through the determination of its properties and couplings that are poorly constrained beyond the SM [1292]. The theoretical control over the requisite SM cross sections and backgrounds needed to expose new physics becomes more stringent as the constraints sharpen without observation of departures from the SM. Figure 35 shows the value of \(\sigma /\sigma _\mathrm{SM}\), namely, of the production cross section times the branching fraction, relative to the SM expectation [1286], with decay mode and targeted production mechanism, where the latter includes \(gg\), VBF, VH (WH and ZH), and \(t{\bar{t}}H\) processes. This quantity is usually called \(\mu \), and we can define, for production mode \(X\) and decay channel \(Y\),
$$\begin{aligned} \mu (X,Y) \equiv \frac{\sigma (X)\mathcal{B}(H\rightarrow Y)}{\sigma _\mathrm{SM}(X)\mathcal{B}_\mathrm{SM}(H\rightarrow Y)} , \end{aligned}$$
(5.3)
noting a global average of \(\mu =0.80\pm 0.14\) for a Higgs boson mass of 125.7 GeV [1286]. See Ref. [1293] for further results and discussion and Ref. [1294] for a succinct review. We note that \(pp\rightarrow H\) via gluon–gluon fusion is computed to NNLO \(+\) NNLL precision in QCD, with an estimated uncertainty of about \(\pm 10~\%\) by varying the renormalization and factorization scales [1294, 1295]. In contrast, the error in the computed partial width of \(H\rightarrow b\bar{b}\) is about 6 % [1296]. The Higgs partial widths are typically accessed through channels in which the Higgs appears in an intermediate state, as in (5.3). Consequently, the ratio of the Higgs coupling to a final state \(Y\) with respect to its SM value, defined as \(\kappa _Y^2 = \Gamma (H\rightarrow Y{\bar{Y}})/\Gamma _\mathrm{SM}(H\rightarrow Y{\bar{Y}})\), is determined through a multi-channel fit. The ability of the LHC to probe \(\kappa _Y\) has been forecast to be some 10–30 % [1294, 1297, 1298]. Estimates instigated by the U.S.-based Community Planning Study (Snowmass 2013) support these assessments [1292], comparing the sensitivity of the current stage of the LHC (data samples at 7–8 TeV with an integrated luminosity of \(20\,\mathrm{fb}^{-1}\)) to staged improvements at the LHC and to possible new accelerators, such as differing realizations of a linear \(e^+e^{-}\) collider. New backgrounds can appear at the LHC which were not known at LEP; e.g., a previously unappreciated background to the Higgs signal in \(H\rightarrow ZZ\) and \(H\rightarrow WW\), arising from asymmetric internal Dalitz conversion to a lepton pair, has been discovered [1299]. Nevertheless, even with conservative assessments of the eventual (albeit known) systematic errors, tests of the Higgs coupling to \(W\)’s or \(b\)-quarks of sub-10 % precision are within reach of the LHC’s high luminosity upgrade, with tests of sub-1 % precision possible at an \(e^+e^{-}\) collider [1292]. These prospects demand further refinements of the existing SM predictions, with concomitant improvements in the theoretical inputs such as \(\alpha _\mathrm{s}\), \(m_b\), and \(m_\mathrm{c}\) [1292].

Current constraints on the quantum numbers of the new boson support a \(0^+\) assignment but operate under the assumption that it is exclusively of a particular spin and parity. Of course admixtures are possible, and they can reflect the existence of CP-violating couplings; such possibilities are more challenging to constrain. Near-degenerate states are also possible and are potentially discoverable [1300]. ATLAS has studied various, possible spin and parity assignments, namely of \(J^P = 0^{-},1^+, 1^{-}, 2^+\), as alternative hypotheses to the \(0^+\) assignment associated with a SM Higgs, and excludes these at a C.L. in excess of 97.8 % [1287]. In the case of the \(2^+\), however, a specific graviton-inspired model is chosen to reduce the possible couplings to SM particles. It is worth noting that QCD effects play a role in these studies as well. In the particular example of the \(H\rightarrow \gamma \gamma \) mode, the \(J^P\) assignments of \(0^+\) and \(2^+\) are compared vis-a-vis the angular distribution of the photons with respect to the \(z\)-axis in the Collins-Soper frame [1287]. The expected angular distribution of the signal yields in the \(0^+\) case is corrected for interference effects with the nonresonant diphoton background \(gg\rightarrow \gamma \gamma \) mediated through quark loops [1301].

EFT methods familiar from the study of processes at lower energies also play an important role, and can work to disparate ends. They can be used, e.g., to describe a generalized Higgs sector [1302], providing not only a theoretical framework for the simultaneous possibility of various SM extensions therein [1303, 1304] but also a description of its CP-violating aspects [1305]. In addition, such methods can be used to capture the effect of higher-loop computations within the Standard Model. For example, the effective vertex (\(v\) is the Higgs vacuum expectation value) [1306]
$$\begin{aligned} \mathcal {L}_\mathrm{eff}= \alpha _\mathrm{s} \frac{C_1}{4v} H F^a_{\mu \nu } F^{a\ \mu \nu } \end{aligned}$$
(5.4)
couples the Higgs to the two gluons in a SU(3)\(_\mathrm{c}\)-gauge-invariant manner. It can capture this coupling in a very efficient way, yielding a difference of less than 1 % between the exact and approximate NLO cross sections for a Higgs mass of less than 200 GeV [1294]. This speeds up Monte Carlo programs, for example. All short-distance information (at the scales of \(M_\mathrm{H}\), \(m_t\), or new physics) is encoded in the Wilson coefficient \(C_1\), which is separately computed in perturbation theory.
Fig. 36

Inclusive cross section for top pair production with center-of-mass energy in \(pp\) and \(p\bar{p}\) collisions [1307], compared with experimental cross sections from CDF, D0, ATLAS, and CMS [1314]. (Figure reproduced from [1314], courtesy of the CMS collaboration.)

b. Top quark studies From the Tevatron to the LHC, the cross section for top-quark pair production \(\sigma (t\bar{t})\), in Fig. 36, grows by a factor of roughly \(30\) due to the larger phase space; from 7 pb at the 1.96 TeV center-of-mass (CM) energy of the Tevatron to some 160 pb at 7 TeV and to some 220 pb at 8 TeV. We refer to [1307] cross-section predictions at 14 TeV and to [1308] for recent cross-section results from CMS and ATLAS.

A good part of \(t\bar{t}\) production is near threshold, with a small relative velocity between the two heavy quarks. A non-relativistic, fixed-order organization of the perturbative series is appropriate. Supplementing such a NNLO calculation with a resummation of soft and Coulomb corrections at NNLL accuracy, a computation of \(\sigma (t\bar{t})\) at the LHC (7 TeV) of \(10\) pb precision has been reported [1309, 1310, 1311]. More generally, the predictions show a residual theoretical uncertainty of some \(3\)\(4~\%\), with an additional \(4\)\(4.5~\%\) uncertainty from the PDFs and the determination of \(\alpha _\mathrm{s}\) [1310, 1311]. Measurements of the \(t\bar{t}\) inclusive cross section can thus be used to extract the top-quark mass, yielding a result of \(m_t=171.4 \mathop {{}_{-5.7}}\limits ^{+5.4}\,\mathrm{GeV}\) [1310], in good agreement with the direct mass determination from the Tevatron, \(m_t=173.18 \pm 0.56\,(\mathrm{stat.})\,\pm 0.75\,(\mathrm{syst.})\, \mathrm{GeV}\) [1288], but less precise. The measurement of near-threshold \(t\bar{t}\) production at an \(e^+e^{-}\) collider, in contrast, can reduce the precision with which \(m_t\) is known by a factor of a few, spurring further theoretical refinements [1312, 1313]. Moreover, in this case, the connection to a particular top mass definition is also crisp.

c. Collider searches for new particles ATLAS and CMS continue to search for the new physics effects expected in various extensions of the SM. All searches, thus far, yield results compatible with the SM. Certain efforts concern searches for high mass \(t\bar{t}\) resonances, such as could be generated through a high mass (leptophobic) \(Z^\prime \) or Kaluza-Klein gluon, or searches for top \(+\) jet resonances, such as could be generated through a high mass \(W^\prime \) [1315, 1316, 1317, 1318, 1319]. Experimental collaborations face a new problem in collecting large top samples at the higher LHC energies: often the \(t\) and \(\bar{t}\) fly away together in a boosted frame, so that the SM decay with visible particles
$$\begin{aligned} t\bar{t}\rightarrow Wb W\bar{b}\ \ (\rightarrow 6\ \mathrm{jets\ or \ \rightarrow 2 \ jets} + 2 \mathrm{leptons} ) \end{aligned}$$
(5.5)
contains several jets that may overlap yielding “fat jets,” for which new algorithms are being developed [1320].

The constraints are sharpest for \(t\bar{t}\) resonances, which decay into lepton pairs, with exclusion limits of 2.79 TeV at 95 % C.L. for a \(Z^\prime \) (with SM-like couplings) decaying into \(e^+e^{-}\). In contrast, the 95 % C.L. exclusion limit on a leptophobic \(Z^\prime \) decaying into \(t\bar{t}\) is greater than 1.5 TeV [1319]. The parity programs at JLab (note, e.g., HAPPEX, http://hallaweb.jlab.org/experiment/HAPPEX and Q-weak, http://www.jlab.org/qweak) and MESA (http://www.prisma.uni-mainz.de/mesa.php) at Mainz are geared towards searches for similar objects, in complementary regions of parameter space, through the precision measurement of parity-violating asymmetries at low momentum transfers [705, 1321]. Moreover, a unique window on the possibility of a leptophobic \(Z^\prime \) can come from the study of parity-violating deep inelastic scattering of polarized electrons from deuterium [1322].

Significant indirect constraints exist on the possibility of an extra chiral generation of quarks from the observation of \(H\rightarrow \gamma \gamma \) [1323], as well as through the apparent production of the Higgs through \(gg\) fusion. Direct searches are mounted, however, for certain “exotic” variants of the extra generation hypothesis, be they vector-like quarks, or quarks with unusual electric charge assignments [1316, 1318]. All searches thus far are null, and \((5/3)e\)-charged up quarks, e.g., are excluded for masses below 700 GeV at 95 % C.L. [1319].

Because no new particle (beyond the Higgs-like particle) has yet appeared in the mass region below 1 TeV, direct searches for a new resonance \(R\) will likely extend to higher mass scales. This will push the QCD inputs needed for PDF fits to the limits of currently available phase space, and it is worth exploring the prospects for better control of such quantities. Precision determinations of the particle properties and couplings of the particles we know also drive a desire to understand the PDFs as accurately as possible. We also refer to Sect. 3.2.1 for a discussion of PDFs and their uncertainties.

5.2.4 Uncertainties from nucleon structure and PDFs

In order to produce a previously unknown particle \(R\), the colliding partons in the initial state, as in for instance \(g(x_1) g(x_2)\rightarrow R +X\), must each carry a significant fraction of the proton’s momentum. This makes constraining parton distribution functions at large Bjorken \(x\), particularly for \(x>0.5\), ever more important as the mass of \(R\) increases. As we have seen, the PDF and scale uncertainties are the largest uncertainties in the predicted inclusive \(t\bar{t}\) cross section. Such uncertainties are also important to the interpretation of ultra-high–energy neutrino events observed at Ice Cube [1324], whose rate may exceed that of the SM. There is currently an effort [135, 1325, 1326] to investigate this issue by combining the traditional CTEQ fits in the large-\(x\) (\(x\rightarrow 1\)) region with JLab data at lower energies. These efforts will likely wax with importance in time because, the 12 GeV upgrade at JLab will allow greatly expanded access to the large-\(x\) region [1327]. Various complications emerge as \(x\rightarrow 1\), and it is challenging to separate the additional contributions that arise. In particular, large logarithms, the so-called Sudakov double logarithms, appear in the \(x\rightarrow 1\) region, and they need to be resummed in order to get an accurate assessment of the cross section. To this end the \(x\rightarrow 1\) region has been subject to extensive theoretical investigation, both in traditional approaches based on factorization theorems [1328, 1329] and in effective field theory [1330, 1331, 1332, 1333, 1334]. Moreover, studies of deep inelastic scattering in nuclei require the assessment of Fermi-motion effects as well. The former issue is skirted in traditional global fits, based on structure functions in leading-twist, collinear factorization, by making the cut on the hadronic invariant mass \(W\) large, such as in [131] for which \(W^2 \ge 15\,\mathrm{GeV}^2\). Here, \(W^2 =M^2 + Q^2 (1-x)/x\). The global-fit approach in [135, 1325, 1326] includes both large-\(x\) and nuclear corrections and allows the \(W\) cut to be relaxed to \(W\sim 1.7\,\mathrm{GeV}\) [1335].

To obtain the \(d\) quark distribution, for example, one uses the data on the unpolarized structure function \(F_2\), e.g., from deep inelastic scattering on the proton and neutron, to find
$$\begin{aligned} \frac{d(x)}{u(x)} = \frac{4 F_{2n}(x) - F_{2p}(x)}{4F_{2p}(x) -F_{2n}(x)} , \end{aligned}$$
(5.6)
where, for brevity, we suppress the \(Q^2\) dependence. Since there are no free neutron targets, the experiments are performed with few-body nuclei, either the deuteron or \(^3\)He. For \(x\) above \(x\simeq 0.5\), the nuclear corrections become large. The CTEQ-JLab fits employ the collinear factorization formula
$$\begin{aligned} F_{2d}(x,Q^2)&= \sum _{N=p,n} \int dy S_{N/A}(y,\gamma ) F_2(x/y,Q^2) \nonumber \\&+ \Delta ^\mathrm{off}(x,Q^2) , \end{aligned}$$
(5.7)
where the deuteron structure function is computed from the parametrized nucleon \(F_2\), the modeled off-shell correction \(\Delta ^\mathrm{off}\), and the nuclear smearing function \(S_{N/A}\), computed from traditional nuclear potential theory based on the Paris, Argonne, or CD-Bonn interactions. There is clearly room for QCD-based progress in these computations. The notion of [135, 1336] is that data on the \(W^\pm \) charge asymmetry from the Tevatron [1337, 1338] can be used to fix the \(d(x)/u(x)\) ratio at large \(x\), and then precision nuclear experiments can be used to fix the nuclear corrections. Future JLab experiments, which are less sensitive to nuclear effects, can then be used to test the procedure [1336].

Of course, the higher energy run of the LHC at 14 TeV, scheduled for 2015, should also lower the \(x\) needed for a given energy reach. Taking 2 TeV as the reference CM energy for a gluon–gluon collision, doubling the LHC energy from 7 to 14 TeV increases the parton luminosity by a factor of 50 [1339], making the new physics reach at \(\mathcal{O}(1\,\text {TeV})\) less sensitive to the large \(x\) behavior of the PDFs. At 14 TeV the parton luminosity (taking this as a crude proxy for \(x\)) of the 2 TeV gluon–gluon subprocess in the 7 TeV collision is found at a CM energy of 3.3 TeV [1339]. Sorting out the PDFs in the large-\(x\) region may prove essential to establishing new physics.

Another issue for new physics searches and Higgs physics is double-parton scattering [1268, 1340]. Two hard partons collide if they coincide within a transverse area of size \(1/Q^2\) out of the total \(1/\Lambda _\mathrm{QCD}^2\). The flux factor being \(1/\Lambda _\mathrm{QCD}^2\), the probability of one hard collision scales as \(\hat{\sigma }_1 \propto ({1}/{\Lambda _\mathrm{QCD}^2}) ( {\Lambda _\mathrm{QCD}^2}/{Q^2})\). The probability of a double collision in the same \(pp\) event (this is not the same as pile up, which is the aftermath of multiple, nearly simultaneous \(pp\) events) is thus power-suppressed, \(\hat{\sigma }_2 \propto ({1}/{\Lambda _\mathrm{QCD}^2}) ( {\Lambda _\mathrm{QCD}^2}/{Q^2})^2\). The rate is small but still leads to a background about three times the signal in Higgs processes such as \(pp\rightarrow WH\rightarrow l\bar{\nu }b\bar{b}\) [1341]. It also entails power corrections to double Drell–Yan processes, an important background to four-lepton Higgs decays. Like-sign \(W^+W^+\) production has long been recognized as a viable way to identify double-parton scattering [1342, 1343] because this final state is not possible in single-parton scattering unless two additional jets are emitted (due to charge and quark-number conservation). It comes to be dominated by double scattering when the particle pairs come out almost back-to-back (typically \(|{\mathbf {p}}_{1\mathrm{T}} + {\mathbf {p}}_{2\mathrm{T}}| \sim \Lambda _\mathrm{QCD}\)).

One might suppose the differential cross section for double-parton scattering could be described as [1344]
$$\begin{aligned}&\frac{\mathrm{d}\sigma ^{DPS}}{\mathrm{d}x_1\mathrm{d}x_2\mathrm{d}x_3\mathrm{d}x_4} \propto \nonumber \\&\quad \times \int \mathrm{d}^2z_\perp F_{ij}(x_1,x_2,z_\perp ) F_{kl}(x_3,x_4,z_\perp ) \hat{\sigma }_{ik} \hat{\sigma }_{jl} , \end{aligned}$$
(5.8)
employing a distribution-like function \(F\) to describe the probability of finding the two partons in the proton at \(z_\perp \) from each other in the plane perpendicular to the momentum, with given momentum fractions \(x_i\). Quantum interference is intrinsic to this process, however, so that some knowledge of the proton at the wave function or amplitude level is needed, as a purely probabilistic description is insufficient. We refer to [1268] for a detailed analysis.

5.2.5 Complementarity with low-energy probes

Searches for unambiguous signs of new physics at high-energy colliders have so far proved null; it may be that new physics appears at yet higher energy scales or that it is more weakly coupled than has been usually assumed. In the former case, a common theoretical framework, which is model-independent and contains few assumptions, can be used to connect the constraints from collider observables to those from low-energy precision measurements; we provide an overview thereof in the next section. In the latter case, an explicit BSM model is required to connect experimental studies at high and low energy scales, and the minimal supersymmetric standard model (MSSM) is a particularly popular example. The impact of permanent electric dipole moment (EDM) searches at low energies, for example, on the appearance of CP-violating terms in the softly broken supersymmetric sector of the MSSM and its broader implications have been studied for decades [1345, 1346, 1347, 1348, 1349, 1350]. Computations of the various QCD matrix elements which appear are important to assessing the loci of points in parameter space which survive these constraints; we discuss the state of the art, albeit in simpler cases, in Sect. 5.4.4.

In the event that new physics is beyond the reach of current colliders, the connection between experimental probes at the highest and lowest energies mentioned is particularly transparent and certainly two-way. Although collider experiments limit new-physics possibilities at low energies, it is also the case that low-energy experiments limit the scope of new-physics at colliders. Before closing this section, we consider an example of how a model-independent approach employing effective Lagrangian techniques can be used in the top-quark sector as well [1351]. Usually such techniques are employed assuming the accessible energy to be no larger than the \(W\) mass [1352, 1353]. In particular, we consider the possibility that the top quark itself could have a permanent (chromo)electric or (chromo)magnetic dipole moment. This is particularly natural if the top quark is a composite particle [1354], and the large top-quark mass suggests that the effects could well be large [1355]. Although such effects could potentially be probed directly through spin observables [1356], constraints from the neutron EDM also operate [1357, 1358], to yield a severe constraint on the chromoelectric top-quark operator through its effect on the coefficient \(w\) of the Weinberg three-gluon operator
$$\begin{aligned} {\mathcal L}_{W3g}= -\frac{w}{6} f^{abc} \varepsilon ^{\mu \nu \lambda \rho } (F^a)_{\mu \sigma } (F^b)^\sigma _\nu (F^\mathrm{c})_{\lambda \rho } \end{aligned}$$
(5.9)
at low energies [1357], where \(f^{abc}\) are SU(3) structure constants. Turning to the specific numerical details, the QCD matrix element of the Weinberg operator in the neutron is needed, and the QCD sum rule calculation of [1359] has been employed to obtain the limits noted [1357]. (See Sects. 5.4.3 and 5.4.4 for further discussion of matrix elements for EDMs.) Stronger limits on the color-blind dipole moments, however, come from \(b\rightarrow s \gamma \) and \(b\rightarrow s {\ell }^+{\ell }^{-}\) decays [1357, 1360]. In the face of such constraints, the space of new-physics models to be explored at the LHC is significantly reduced [1357, 1358], and presumably can be sharpened further, even in the absence of additional experimental data, if the non-perturbative matrix element can be more accurately calculated. In the sections to follow we will find further examples of low-to-high-energy complementarity.

5.3 Low-energy framework for the analysis of BSM effects

The SM leaves many questions unanswered, and the best-motivated models of new physics are those which are able to address them. Commonly, this is realized so that the more fundamental theory has the SM as its low-energy limit. It is thus natural to analyze the possibility of physics beyond the SM within an effective field theory framework. To do this we need only assume that we work at some energy \(E\) below the scale \(\Lambda \) at which new particles appear. Consequently for \(E< \Lambda \) any new degrees of freedom are “integrated out,” and the SM is amended by higher-dimension operators written in terms of fields associated with SM particles [1361]. Specifically,
$$\begin{aligned} \mathcal{L}_\mathrm{SM} \rightarrow \mathcal{L}_\mathrm{SM} + \sum _i \frac{c_i}{\Lambda ^{D-4}} {\mathcal{O}^D_i} , \end{aligned}$$
(5.10)
where the new operators \(\mathcal{O}_i^D\) have dimension \(D>4\). We emphasize that \(\mathcal{L}_\mathrm{SM}\) contains a dimension-four operator, controlled by \(\bar{\theta }\), that can also engender CP-violating effects, though they have not yet been observed. The experimental limit on the neutron EDM implies \(\bar{\theta }<10^{-10}\) [1232], though the underlying reason for its small value is unclear. This limitation is known as the “strong CP problem”. If its resolution is in a new continuous symmetry [1362] that is spontaneously and mechanically broken at low energy, then there is a new particle, the axion [1363, 1364], which we may yet discover [1365, 1366]. The higher-dimension operators include terms which manifestly break SM symmetries and others which do not.

Since flavor-physics observables constrain the appearance of operators that are not SM invariant to energies far beyond the weak scale [1367, 1368, 1369], it is more efficient to organize the higher-dimension terms so that only those invariant under SM electroweak gauge symmetry are included. Under these conditions, and setting aside B- and L-violating operators, the leading-order (dimension-six) terms in our SM extension can be found in [1352, 1353]. Nevertheless, this description does not capture all the possibilities usually considered in dimension six because of the existence of neutrino mass. The latter has been established beyond all doubt [1], though the need for the inclusion of dynamics beyond that in the SM to explain it has, as yet, not been established. To be specific, we can use the Higgs mechanism to generate their mass.11 Since the neutrinos are all light in mass, to explore the consequences of this possibility, we must include three right-handed neutrinos explicitly in our description at low energies [1371]. Finally, if we evolve our description (valid for \(E<\Lambda \)) to the low energies (\(E \ll M_W, \Lambda \)) appropriate to the study of the weak decays of neutrons and nuclei, we recover precisely 10 independent terms, just as argued long ago by Lee and Yang starting from the assumption of Lorentz invariance and the possibility of parity nonconservation [1372]. The latter continues to be the framework in which new physics searches in \(\beta \)-decay are analyzed, as discussed, e.g., in [1240, 1373, 1374, 1375].

In order to employ the low-energy quark and gluon operator framework we have discussed in a chiral effective theory in nucleon degrees of freedom, nucleon, rather than meson, matrix elements need to be computed. Nucleon matrix elements are generally more computationally demanding than meson matrix elements in lattice QCD, since the statistical noise grows with Euclidean time \(t\) as \(\exp [(M_\mathrm{N}-3 M_\pi /2)t]\) for each nucleon in the system. Thus, results with high precision in the nucleon sector lag those in the meson sector. Furthermore, extrapolating to the physical light-quark masses is more challenging for baryons, since chiral perturbation theory converges more slowly. The latter issue is likely to be brought under control in the near future, as ensembles of lattices begin to be generated with physical \(u\) and \(d\) (and \(s\) and \(c\)) quark masses. This should greatly reduce the systematic uncertainties. Other systematics, such as finite-volume effects, renormalization, and excited-state contamination can be systematically reduced by improved algorithms and by increasing the computational resources devoted to the calculations. We refer to Sect. 3.2.5a for additional discussion.

One interesting idea from experimental physics is to perform “blind” analyses, so that the true result is hidden while the analysis is performed. Concretely what this means is that the result should only be revealed after all the systematics have been estimated. This technique has begun to be employed in lattice-QCD calculations, notably in the computation of the exclusive semileptonic decay matrix elements needed to determine the CKM matrix elements \(|V_{cb}|\) and \(|V_{ub}|\) [931, 1376]. It would be advantageous to implement this approach in lattice-QCD calculations of nucleon matrix elements as well, so that an analysis of systematic effects could be concluded on grounds independent of the specific result found. Blind analysis would help in ensuring an extremely careful analysis of systematics, and we hope the lattice community will choose to follow this approach in the next few years.

We now turn to the analysis of particular low-energy experiments to the end of discovering physics BSM and the manner in which theoretical control over confinement physics can support or limit them.

5.4 Permanent EDMs

5.4.1 Overview

The (permanent) EDM of the neutron is a measure of the distribution of positive and negative charge inside the neutron; it is nonzero if a slight offset in the arrangement of the positive and negative charges exists. This is possible if interactions are present which break the discrete symmetries of parity P and time reversal T. In the context of the CPT theorem, it also reflects the existence of CP violation, i.e., of the product of charge conjugation C and parity P, as well. Consequently, permanent EDM searches probe the possibility of new sources of CP violation at the Lagrangian level. The EDM \(\mathbf {d}\) of a nondegenerate system is proportional to its spin \(\mathbf {S}\), and it is nonzero if the energy of the system shifts in an external electric field \(\mathbf {E}\), with an interaction energy proportional to \(\mathbf {S}\cdot \mathbf {E}\).

As we have already noted, the SM nominally possesses two sources of CP violation, the single phase \(\delta \) in the Cabibbo–Kobayashi–Maskawa (CKM) matrix and the coefficient \(\bar{\theta }\) which controls the T-odd, P-odd product of the gluon field-strength tensor and its dual, namely \(\bar{\theta } ({\alpha _\mathrm{s}}/{8\pi }) F^a \tilde{F}^a\). Experimental studies of CP violation in the B system have shown that \(\delta \sim \mathcal{O}(1)\) [1368, 1369], whereas neutron EDM limits have shown that the second source of CP violation does not appear to operate. Even if a physical mechanism exists to remove the appearance of \(\bar{\theta }\), higher-dimension operators from physics BSM may still induce it, so that we use experiment to constrain this second source, as well as CP-violating effects arising from other BSM operators.

The CKM mechanism of CP violation does give rise to nonzero permanent EDMs; however, the first nontrivial contributions to the quark and charged lepton EDMs come in three- and four-loop order (for massless neutrinos), respectively, so that for the down quark \(|d_d| \sim 10^{-34}\, e\hbox {-}{\mathrm {cm}}\) [1377, 1378]. Nevertheless, there exists a well-known, long-distance chiral enhancement of the neutron EDM (arising from a pion loop and controlled by \(\log (m_\pi /M_\mathrm{N})\)), and estimates yield \(|d_n| \sim 10^{-31}\)\(10^{-33}\, e\hbox {-}{\mathrm {cm}}\) [1379, 1380, 1381], making it relatively larger but still several orders of magnitude below the current experimental sensitivity. It is worth noting that the nucleon’s intrinsic flavor content can also modify an EDM estimate [1382, 1383, 1384]. Finally, if neutrinos are massive Majorana particles, then the electron EDM induced by the CKM matrix can be greatly enhanced, though not sufficiently to make it experimentally observable [1385]. (Neutrino mixing and Majorana-mass dynamics can also augment the muon EDM in the MSSM in a manner which evades \(e\)\(\mu \) universality, motivating a dedicated search for \(d_{\mu }\) [1386].) A compilation of the results from various systems is shown in Table 15.
Table 15

Upper limits on EDMs (\(|d|\)) from different experiments. For the “Nucleus” category, the EDM values are of the \(^{199}\)Hg atom that contains the nucleus. No direct limit yet exists on the proton EDM, though such could be realized through a storage-ring experiment. Here we report the best inferred limit in brackets, which is determined by asserting that the \(^{199}\)Hg limit is saturated by \(d_p\) exclusively. Table adapted from [1387]

Category

EDM Limit (\(e-{\mathrm {cm}}\))

SM Value (\(e-{\mathrm {cm}}\))

Electron

\(1.0\times 10^{-27}\,(90\%\, \mathrm{C.L.})\) [1388]

10\(^{-38}\)

Muon

\(1.9\times 10^{-19}\,(95\%\, \mathrm{C.L.})\) [1389]

10\(^{-35}\)

Neutron

\(2.9\times 10^{-26}\,(90\%\, \mathrm{C.L.})\) [1232]

10\(^{-31}\)

Proton

\([7.9\times 10^{-25}]\quad \quad \quad \quad \) [1390]

10\(^{-31}\)

Nucleus

\(3.1\times 10^{-29}\,(95\%\, \mathrm{C.L.})\) [1390]

10\(^{-33}\)

5.4.2 Experiments, and their interpretation and implications

The last few years have seen an explosion of interest in experimental approaches to searches for electric dipole moments of particles composed of light quarks and leptons. This increased scientific interest has developed for many reasons. First, the power of the existing and achievable constraints from EDM searches on sources of CP violation BSM has become more and more widely recognized. Moreover, other sensitive experimental tests of “T” invariance come from particle decays and reactions in which the observables are only motion-reversal odd and thus do not reflect true tests of time-reversal invariance [1391]. Such can be mimicked by various forms of final-state effects which eventually limit their sensitivity. In contrast, the matrix element associated with an intrinsic particle EDM has definite transformation properties under time reversal because the initial state and the final state are the same. The consequence is that an EDM search is one of the few true null tests for time-reversal invariance. Consequently an upper bound on an EDM constitutes a crisp, non-negotiable limit, and a positive observation of an EDM at foreseeable levels of sensitivity would constitute incontrovertible evidence for T violation. Moreover, since the SM prediction is inaccessibly small, as shown in Table 15, it would also speak directly to the existence of new physics. Popular models of new physics at the weak scale generate EDMs greatly in excess of SM expectations, and the parameter space of these models is already strongly constrained by current limits. Consequently, even null results from the next generation of EDM experiments would be interesting, for these would give hints as to the energy scale at which new physics could be.

Such null results could also damage beyond repair certain theoretical explanations for generating the baryon asymmetry of the universe through the physics of the electroweak phase transition. Two of the famous Sakharov conditions for the generation of the baryon asymmetry (namely, B violation and a departure from thermal equilibrium) are already present in the SM, in principle. For a Higgs mass of some 125 GeV, however, SU(2) lattice gauge–Higgs theory simulations, as in [1392], e.g., reveal that the electroweak phase transition is not of first order. The lack of a sufficiently robust first-order phase transition can also be problematic in BSM models. Nevertheless, new mechanisms, or sources, of CP (or T) and C violation in the quark sector could make baryon production much more effective. Existing EDM constraints curtail possible electroweak baryogenesis scenarios in the MSSM severely [1393, 1394, 1395], and an improvement in the experimental bound on \(d_n\) by a factor of \(\sim \)100 could rule out the MSSM as a model of electroweak baryogenesis [1350, 1396, 1397]. This outcome would thereby favor supersymmetric models beyond the MSSM, such as in [1398, 1399, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408], or possibly mechanisms based on the two-Higgs doublet model (2HDM) [1409], or mechanisms which are not tied to the weak scale, such as leptogenesis, or dark-matter mediated scenarios. Consequently, people have come to recognize that a measurement of an EDM in any system, regardless of its complexity, is of fundamental interest. Since there are many different possibilities for generating an EDM at a microscopic level, many experiments are likely to be needed to localize the fundamental source of any EDM once observed. New ideas for EDM measurements abound and have come from scientific communities in atomic, molecular, nuclear, particle, and condensed-matter physics.

Compact overviews of this field can be found in [1387, 1410], whereas a recent theoretical review can be found in [1238]. The most stringent limits on particle EDMs come from atomic physics measurements in \(^{199}\)Hg [1390]. However, it is known that, in the pointlike, non-relativistic limit, the electron cloud of an atom shields any EDM which might be present in the nucleus—making the atomic EDM zero even if the nuclear EDM were not. This “no-go” result is known as Schiff’s theorem [1411]. As a consequence, the fantastic upper bound on the EDM in this atom places a much weaker constraint on the EDM of its nucleons.

Atomic and molecular physicists have long sought systems in which the EDM could be amplified rather than shielded by electron effects; such an amplification can indeed occur in certain polar molecules [1388, 1412]. Gross enhancements also exist in certain heavy atoms whose relativistic motion evades Schiff’s theorem, yielding an EDM which scales as \(Z^3\alpha ^2\) [1413, 1414]. More recently it has been recognized that atoms whose nuclei possess octupole deformations [1415] can have particularly enhanced atomic EDMs, by orders of magnitude over \(^{199}\)Hg [1416], in part through the resulting mixing of certain nearly-degenerate atomic energy levels. Even such enhancements do not defeat Schiff’s theorem completely, though they can come close. To be suitable for an EDM experiment, it is also necessary to be able to polarize sufficiently large ensembles of nuclei in order to perform the delicate NMR frequency-difference measurements typically needed to detect EDMs. Such needs, in concert with the desired enhancements, lead one to consider certain heavy radioactive nuclei such as radon and radium. Recently, the first direct evidence of octupole deformation in \(^{224}\)Ra has been established through measurements of Coulomb excitation of 2.85 MeV/amu rare-isotope beams at REX-ISOLDE (CERN) [1417], strengthening the confidence in the size of the Schiff moment in like systems, whose computation is dominated by many-body calculations in nuclear and atomic physics. Generally, in the presence of rigid octupole deformation, as observed in \(^{224}\)Ra, the computation of the Schiff moment is expected to be more robust [1238]. This underscores the discovery potential of an EDM measurement in such systems. Progress towards an EDM measurement in \(^{225}\)Ra, e.g., is ongoing [1418], and the sensitivity of an eventual EDM limit could be greatly increased through the enhanced isotope production capability of a megawatt-class 1 GeV proton linac [1387].

EDM searches on simpler objects such as the neutron, proton, or deuteron, e.g., are of course more directly interpretable in terms of the fundamental sources of CP violation at the quark level. The theoretical interpretation of these systems in chiral effective theory has been under intense development [1419, 1420, 1421, 1422, 1423]. Many experiments to search for a neutron EDM are in progress [1424, 1425, 1426, 1427], of which the nEDM-SNS experiment under development at ORNL is the most ambitious [1424]. Its ultimate goal is to improve the sensitivity by more than two orders of magnitude beyond the present 90 % CL bound of some \(3\times 10^{-26}\, e\hbox {-}{\mathrm {cm}}\) [1232]. This limit already constrains, e.g., the CP-violating phases in minimal supersymmetric models to assume unnaturally small values, or to make the masses of the supersymmetric partner particles larger than previously anticipated, or to make the spectrum of partner particles possess unexpected degeneracies. These experiments are broadly similar in experimental strategy to atomic physics approaches.

Over the last few years a qualitatively new approach to the measurement of particle EDMs using charged particles in storage rings [1428], exploiting the large electric fields present in such environments, has come under active development. Such an experiment would have the advantage of enlarging the spectrum of available species to include charged particles, and the ability to allow a coherent effect to accumulate over many revolutions around a ring. A variety of operators can generate an EDM, so that stringent EDM measurements on the proton, neutron, and other light nuclei are complementary and can help unravel the underlying CP-violating mechanism if a signal is seen. The theoretical insights to be gained have been studied carefully [1421, 1422, 1429]. An experimental difficulty of this approach is that one loses the clean electric-field flip used in previous experiments on electrically neutral objects to reduce systematic errors. Instead one must typically measure a rotation of the plane of polarization of a transversely polarized particle in the ring and to develop other methods to deal with systematic errors, as discussed in [1387]. Measurements in existing storage rings to quantify these instrumental issues are in progress.

Finally one can consider constraints on the EDMs of leptons. The muon EDM can be limited in part as a byproduct of the muon \(g-2\) measurements [1389], and the heavier mass of the muon amplifies its sensitivity to certain new-physics possibilities. Nevertheless, at anticipated levels of sensitivity, such experiments constrain CP-violating sources which do not simply scale with the mass of the muon. Such flavor-blind CP-violating contributions to the muon EDM are already severely constrained by electron EDM limits; rather, direct limits probe the possibility of lepton-flavor violation here as well [1386, 1430, 1431]. The electron EDM possesses stringent limits from atomic and molecular physics measurements, and in addition there are many promising approaches under development, which could achieve even higher levels of sensitivity. These range from solid-state systems at low temperature [1432, 1433] to new experiments with cold molecules [1434]. Indeed, the ACME collaboration, using ThO, has just announced a limit on \(d_e\) an order of magnitude smaller than any ever achieved before [1435]. These constraints are important in themselves and are also needed to interpret the source of an EDM if observed in an atomic physics experiment.

5.4.3 EFTs for EDMs: the neutron case

We now consider how sources of CP violation beyond the SM can generate a permanent EDM at low energies. Noting [1348], we organize the expected contributions in terms of the mass dimension of the possible CP-violating operators, in quark and gluon degrees of freedom, appearing in an effective field theory with a cutoff of \({\sim }1\) GeV:
$$\begin{aligned} \mathcal{L}&= \frac{\alpha _\mathrm{s} \bar{\theta }}{8\pi } \epsilon ^{\alpha \beta \mu \nu }F_{\alpha \beta }^a F_{\mu \nu }^a \nonumber \\&- \frac{i}{2} \sum _{i\in u,d,s}\!\left( d_i \bar{\psi }_i F_{\mu \nu }\sigma ^{\mu \nu } \gamma _5 \psi _i + {\tilde{d}}_i \bar{\psi }_i F_{\mu \nu }^a T^a\sigma ^{\mu \nu }\gamma _5 \psi _i \right) \nonumber \\&+ \frac{w}{3} f^{abc} \epsilon ^{\nu \beta \rho \delta } (F^a)_{\mu \nu } (F^b)_{\rho \delta } (F^\mathrm{c})_\beta ^{\mu } \nonumber \\&+ \sum _{i,j} C_{ij} (\bar{\psi }_i \psi _i)(\bar{\psi }_j i\gamma _5\psi _j) + \cdots \end{aligned}$$
(5.11)
with \(i,j\in u,d,s\) unless otherwise noted—all heavier degrees of freedom have been integrated out. The leading term is the dimension-four strong CP term already discussed, proportional to the parameter \(\bar{\theta }\), though it can also be induced by higher-dimension operators even in the presence of axion dynamics [1348, 1436] so that we retain it explicitly. The terms in the second line of (5.11) appear to be of dimension five, but their chirality-changing nature implies that a Higgs insertion, of form \(H/v\), say, is needed to make the operator invariant under SU(2)\(_{L}\times \)U(1) symmetry. (See (3.1) in [1437] for an explicit expression.) Therefore these operators, which determine the fermion EDMs \(d_i\) and quark chromo-EDMs (CEDM) \(\tilde{d}_i\), are suppressed by an additional factor containing a large mass scale and should be regarded as dimension-six operators in numerical effect. The remaining terms in (5.11) are the dimension-six Weinberg three-gluon operator from (5.9) with coefficient \(w\), and CP-violating four-fermion operators, characterized by coefficients \(C_{ij}\). Turning to [1353], we note that after electroweak symmetry breaking there are also chirality-changing four-fermion operators which, analogously, are of dimension eight numerically once SU(2)\(_{L}\times \)U(1) symmetry is imposed. Various extensions of the SM can generate the low-energy constants which appear, so that, in turn, EDM limits thereby constrain the new sources of CP violation which appear in such models. In connecting the Wilson coefficients of these operators and hence models of new physics to the low-energy constants of a chiral effective theory in meson and nucleon degrees of freedom requires the evaluation of non-perturbative hadron matrix elements. Parametrically, we have [1348]
$$\begin{aligned} d_n&= d_n({\bar{\theta }},d_i, {\tilde{d}}_i,w,C_{ij}) \nonumber \\ {\bar{g}}_{\pi NN}^{(i)} ,&= {\bar{g}}_{\pi NN}^{(i)}({\bar{\theta }},d_i, {\tilde{d}}_i,w,C_{ij}). \end{aligned}$$
(5.12)
Several computational aspects must be considered in connecting a model of new physics at the TeV scale to the low-energy constants of (5.11). After matching to an effective theory in SM degrees of freedom, there are QCD evolution and operator-mixing effects, as well as flavor thresholds, involved in realizing the Wilson coefficients at a scale of \({\sim }\)1 GeV. Beyond this, the hadronic matrix elements must be computed. A detailed review of all these issues can be found in [1238]. Typically QCD sum rule methods, or a SU(6) quark model, have been employed in the computation of the matrix elements [1348]; for the neutron, we refer to [1438] for a comparative review of different methods. Lattice gauge theory can also be used to compute the needed proton and neutron matrix elements, and the current status and prospects for lattice-QCD calculations are presented in the next section. We note in passing that \(d_n\) and \(d_p\) have also been analyzed in chiral perturbation theory [1439, 1440, 1441], as well as in light-cone QCD [1442]. We refer to Sect. 3.4.7 for a general discussion of chiral perturbation theory in the baryon sector.

We turn now to the evaluation of the requisite hadron matrix elements with lattice QCD.

5.4.4 Lattice-QCD matrix elements

To generate a nonzero neutron EDM, one needs interactions that violate CP symmetry, and the CP-odd \(\bar{\theta }\)-term in the SM is one possible example. The most common type of lattice-QCD EDM calculation is that of the neutron matrix element of the operator associated with the leading \(\bar{\theta }\) term. A recent combined analysis gives \(O(30~\%)\) in the statistical error alone, noting Fig. 37 for a summary, so that the precision of lattice-QCD calculations needs to be greatly improved. All-mode averaging (AMA) has been proposed to improve the current statistics even at near-physical pion mass [683].

There are currently three main approaches to computing these matrix elements using lattice QCD. One is a direct computation, studying the electromagnetic form factor \(F_3\) under the QCD Lagrangian including the CP-odd \(\theta \) term (as adopted by RBC, J/E, and CP-PACS (2005) [1443, 1445, 1446, 1447, 1448])where \(J^\mathrm{EM}_{\mu }\) is the electromagnetic current, \(\bar{u}\) and \(u\) are appropriate spinors for the neutron, and \(q\) is the transferred momentum. This requires an extrapolation of the form factors to \(q^2=0\), which can introduce systematic error and exacerbate the statistical error. Another method is introducing an external static and uniform electric field and looking at the energy difference induced between the two spin states of the nucleon at zero momentum (by CP-PACS [1445]), one can infer \(d_n\). Or, finally, one can compute the product of the anomalous magnetic moment of neutron \(\kappa _\mathrm{N}\) and \(\tan (2\alpha )\) (by QCDSF [1448]), where \(\alpha \) is the \(\gamma _5\) rotation of the nucleon spin induced by the CP-odd source. A summary of \(d_n\) calculations from dynamical lattice QCD is shown in Fig. 37, where the results are given as a function of the pion mass used in the calculation. Combining all data and extrapolating to the physical pion mass yields \(d_n^\mathrm{lat}=(0.015 \pm 0.005) \bar{\theta }\)  \(e\hbox {-}{\mathrm {fm}}\) [1449], which is the starred point in the figure. Further and more precise calculations from various groups are currently in progress, using improved techniques to reduce the statistical error, such as the aforementioned AMA [683].
Fig. 37

Summary of the latest dynamical calculations of the neutron EDM [1443, 1444, 1445, 1446, 1447, 1448] \(d_n\) as a function of \(m_\pi ^2\) from a nonzero \(\bar{\theta }\) term in QCD. The band is a global extrapolation at 68 % CL combining all the lattice points (except for [1448]) each weighted by its error bar. The leftmost star indicates the value at the physical pion mass. Figure taken from [1449]

It should also be possible to compute the nucleon matrix elements of higher-dimension operators, such as the quark electric dipole moment (qEDM) and the chromoelectric dipole moment (CEDM). This will require us to extend lattice-QCD calculations to such cases [1437], and we now discuss the prospects.

a. Quark Electric Dipole Moment In this case, the neutron EDM is induced by nonzero quark electric dipole moments, which are related to the following matrix elements of the hadronic part of the first of the effectively dimension-six operators in (5.11):The nucleon matrix elements can be accessed through direct lattice-QCD calculations with isoscalar and isovector tensor charges. There are several existing lattice-QCD calculations of the isovector tensor charge; see Fig. 38.
b. Chromoelectric Dipole Moment In this case a direct calculation of the chromoelectric dipole moments would be more challenging on the lattice, since it requires the calculation of a four-point Green function. Only a few such calculations have previously been attempted. One way to avoid this problem would be to apply the Feynman–Hellmann theorem by introducing an external electric field \(E\) to extract the matrix elements:where \(A_{\mu }(E)\) refers to the corresponding vector potential and \(G^{\nu \kappa }\) is shorthand for \((F^a)^{\nu \kappa } T^a\). Similar techniques have been widely implemented in lattice QCD to determine the strangeness contribution to the nucleon mass; one only needs to combine the idea with a nucleon matrix element calculation. Although as of the time of this review, no lattice calculation of the chromoelectric dipole moment has been attempted, we are optimistic that it will be explored within the next few years.
Fig. 38

Figures adapted from [203]. (Upper figure) Global analysis of all \(N_\mathrm{f}=2+1(+1)\) lattice calculations of \(g_\mathrm{T}\) (upper) and \(g_\mathrm{S}\) (lower) [206, 234, 251, 261, 1458] with \(m_\pi L >4\) and \(m_\pi T > 7\) cuts to avoid systematics due to small spatial or temporal extent. The leftmost points are the extrapolated values at the physical pion mass. The two bands show extrapolations with different upper pion-mass cuts: \(m_\pi ^2 <0.4\) and \(m_\pi ^2 <0.2\). The \(m_\pi L<4\) data points are marked faded within each calculation; the lattice spacings for each point are denoted by a solid line for \(a\le 0.06\) fm, dashed\(0.06 < a \le 0.09\) fm, dot–dashed\(0.09 < a \le 0.12\) fm, and dotted\(a > 0.12\) fm. (Lower figure) The allowed \(\epsilon _{S}\)\(\epsilon _{T}\) parameter region using different experimental and theoretical inputs. The outermost (green), middle (purple), and innermost (magenta) dashed lines are the constraints from the first LHC run [1463], along with near-term expectations, running to a scale of 2 GeV to compare with the low-energy experiments. The inputs for the low-energy experiments assume that limits (at 68 % CL) of \(|b|< 10^{-3}\) and \(|B_\mathrm{BSM} -b| < 10^{-3}\) from neutron \(\beta \) decay and a limit of \(g_\mathrm{T} \epsilon _\mathrm{T} < 2\times 10^{-4}\) from \(^6\)He \(\beta \) decay [1464], which is a purely Gamow–Teller transition. These low-energy experiments probe \(S,T\) interactions through possible interference terms and yield constraints on \(\mathrm{Re}(\epsilon _\mathrm{T})\) and \(\mathrm{Re}(\epsilon _\mathrm{S})\) only

Currently, lattice-QCD calculations on \(d_n\) due to the leading \(\theta \) term have statistical errors at the level of 30 % after a chiral extrapolation combining all existing dynamical data. More updates and precise calculations from various groups are currently in progress, including improved numerical techniques that will significantly reduce the errors. Within the next 5 years, lattice QCD should be able to make predictions of better than 10 % precision, and one can hope that percent-level computations will be available on a ten-year timescale.

Outside the leading-order \(\theta \) term, there are plans for calculating the dimension-six operator matrix elements by the PNDME (http://www.phys.washington.edu/users/hwlin/pndme/index.xhtml) collaboration. The matrix elements relevant to the quark electric dipole moments are rather straightforward, involving isovector and isoscalar nucleon tensor matrix elements. The latter one requires disconnected diagrams with extra explicit quark loops. They will require techniques similar to those already used to determine the strangeness contribution to the nucleon mass and the strange spin contribution to nucleon. However, the chromoelectric dipole moment is more difficult still, since it requires a four-point Green function. One alternative method we have considered would be to take a numerical derivative with the magnitude of the external electric field [1437]. We should see some preliminary results soon.

5.5 Probing non-\((V-A)\) interactions in beta decay

The measurement of non-SM contributions to precision neutron (nuclear) beta-decay measurements would hint to the existence of BSM particles at the TeV scale; if new particles exist, their fundamental high-scale interactions would appear at low energy in the neutron beta-decay Hamiltonian as new terms, where we recall (5.10) and the opening discussion of Sect. 5.3. In this case the new terms are most readily revealed by their symmetry; they can violate the so-called \(V-A\) law of the weak interactions. Specifically, in dimension six, the effective Hamiltonian takes the form
$$\begin{aligned} H_\mathrm{eff} = G_\mathrm{F} \Bigg ( J_{V-A}^\mathrm{lept} \times J_{V-A}^\mathrm{quark} + \sum _i \epsilon _i \hat{O}_i^\mathrm{lept} \times \hat{O}_i^\mathrm{quark} \Bigg ), \nonumber \\ \end{aligned}$$
(5.16)
where \(G_\mathrm{F}\) is the Fermi constant, \(J_{V-A}\) is the left-handed current of the indicated particle, and the sum includes operators of non-\((V-A)\) form which represent physics BSM. As we have noted, the new operators will enter with coefficients controlled by the mass scale of new physics; this is similar to how the dimensionful Fermi constant gave hints to the masses of the \(W\) and \(Z\) bosons of the electroweak theory prior to their discovery. Matching this to an effective theory at the nucleon level, the ten terms of the effective Hamiltonian are independent, linear combinations of the coefficients of the Lee–Yang Hamiltonian [1371, 1372]. Since scalar and tensor structures (controlled by \(\epsilon _\mathrm{S}\) and \(\epsilon _\mathrm{T}\) in \(\beta \) decay) do not appear in the SM Lagrangian, signals in these channels at current levels of sensitivity would be clear signs of BSM physics. In neutron decay, the new operators of (5.16) yield, in particular, the following low-energy coupling constants \(g_\mathrm{T}\) and \(g_\mathrm{S}\) (here, multiplied by proton and neutron spin wave functions):
$$\begin{aligned} g_\mathrm{T} \bar{u}_n \sigma _{\mu \nu } u_p&= \langle n | \overline{u}\sigma _{\mu \nu } d | p \rangle \end{aligned}$$
(5.17)
$$\begin{aligned} g_\mathrm{S} \bar{u}_n u_p&= \langle n | \overline{u} d | p \rangle . \end{aligned}$$
(5.18)
Lattice QCD is a perfect theoretical tool to determine these constants precisely.

The search for BSM physics proceeds experimentally by either measuring the Fierz interference term \(b\) (i.e., \(b m_e/E_e\)) or the neutrino asymmetry parameter \(B\) (i.e., \(B(E_e){\mathbf {S}}_n \cdot {\mathbf {p}}_\nu \)) of the neutron differential decay rate [1240]. The Fierz term can either be measured directly or indirectly, the latter through either its impact on the electron-neutrino correlation \(a {\mathbf {p}}_e\cdot {\mathbf {p}}_\nu \) or on the electron-momentum correlation with neutron-spin, \(A{\mathbf {S}}_n\cdot {\mathbf {p}}_e\). Here, \({\mathbf {S}}_n\) and \({\mathbf {p}}_{\ell }\) denote the neutron spin and a lepton momentum (\({\ell } \in (e,\nu )\)), respectively. We note, neglecting Coulomb corrections, \(b=(2/(1+3\lambda ^2))(g_\mathrm{S} \mathrm{Re(\epsilon _\mathrm{S})} - 12 \lambda g_\mathrm{T} \mathrm{Re}(\epsilon _\mathrm{T}))\) [1450], where \(\lambda =g_A/g_V\approx 1.27\). Assessing \(b\) through \(a\) or \(A\) employs an asymmetry measurement, reducing the impact of possible systematic errors. The Fierz term is nonzero only if scalar or tensor currents appear, whereas the latter contribute to the magnitude of \(B\).

There are several upcoming and planned experiments worldwide to measure the correlation coefficients in neutron decay, with plans to probe \(b\) and \(B_\mathrm{BSM}\) up to the \({\sim } 10^{-3}\) level or better, and they include PERC [1451] at the FRM-II, PERKEOIII at the ILL [1452], UCNB [1453] and UCNb [1454] at LANL, Nab at ORNL [1455, 1456], and ACORN [1457] at NIST—indeed the PERC experiment [1451] has proposed attaining \(10^{-4}\) precision. Models of QCD give rather loose bounds on \(g_\mathrm{S}\) and \(g_\mathrm{T}\); for example, \(g_\mathrm{S}\) is estimated to range between 0.25 and 1 [1458]. Consequently, lattice-QCD calculations of these quantities need not be terribly precise to have a dramatic positive impact. Indeed, determining \(g_{S,T}\) to 10–20 % (after summing all systematic uncertainties) [1458] is a useful and feasible goal. The obvious improvement in the ability to limit the coefficients of the underlying non-\((V-A)\) interactions speaks to its importance.

The PNDME collaboration reported the first lattice-QCD results for both \(g_\mathrm{S}\) and \(g_\mathrm{T}\) [1458] and gave the first estimate of the allowed region of \(\epsilon _{S}\)\(\epsilon _{T}\) parameter-space when combined with the expected experimental precision; we will return to this point in a moment. The latest review, from [203], contains a summary of these charges; to avoid the unknown systematics coming from finite-size artifacts, data with \(M_\pi L \le 4\) and \(M_\pi T \le 8\) are omitted, as shown on the lower part of Fig. 38. This figure includes updated calculations of \(g_{S,T}\) after [1458] from the PNDME and LHP collaborations. Reference [203] uses the chiral formulation given in [1459] and [1460] for the tensor and scalar charges, respectively, to extrapolate to the physical pion mass. We see that the PNDME points greatly constrain the uncertainty due to chiral extrapolation in both cases and obtain \(g_\mathrm{T}^\mathrm{lat}= 0.978 \pm 0.035\) and \(g_\mathrm{S}^\mathrm{lat}= 0.796 \pm 0.079\), where only statistical errors have been reported.

More recently, the PNDME collaboration has computed the axial, scalar, and tensor charges on two HISQ ensembles with 2+1+1 dynamical flavors at a lattice spacing of 0.12 fm and with light-quark masses corresponding to pions with masses of 310 and 220 MeV [248]. These ensembles have been generated by the MILC Collaboration [825]. Including systematic errors, the continuum and chiral extrapolation yields the estimates \(g_S = 0.72 \pm 0.32\) and \(g_T = 1.047 \pm 0.061\). In comparison, the recent LHPC results are \(g_S =1.08 \pm 0.28 \pm 0.16\) and \(g_T = 1.038 \pm 0.011 \pm 0.012\), with \(M_{\pi } \approx 150\) MeV at a single lattice spacing of \(a \approx 0.116\) fm [1460]. A different, promising path to \(g_S\) has been realized in [1461], exploiting lattice-QCD calculations of the neutron–proton mass difference in pure QCD to yield a value of \(g_S\).

As previously mentioned, the tensor and scalar charges can be combined with experimental data to determine the allowed region of parameter space for scalar and tensor BSM couplings. Using the \(g_{S,T}\) from the model estimations and combining with the existing nuclear experimental data,12 we get the constraints shown as the outermost band of the lower part of Fig. 38. Combining anticipated (in the shorter term) results from \(\beta \)-decay and existing measurements, and again, using the model inputs of \(g_{S,T}\), we see the uncertainties in \(\epsilon _{S,T}\) are significantly improved. (A limit on \(g_\mathrm{T} \epsilon _\mathrm{T}\) also comes from radiative pion decay, but it can be evaded by cancellation and has been omitted [1240, 1462].) Finally, using our present lattice-QCD values of the scalar and tensor charges, combined with the anticipated precision of the experimental bounds on the deviation of low-energy decay parameters from their SM values, we find the constraints on \(\epsilon _{S,T}\) are further improved, shown as the innermost region. These upper bounds on the effective couplings \(\epsilon _{S,T}\) would correspond to lower bounds for the scales \(\Lambda _{S,T}\) at 5.6 and 10 TeV, respectively, determined using naive dimensional analysis (\(\epsilon _i \sim (v/\Lambda _i)^2\) with \(v \sim 174~\mathrm{GeV}\)), for new physics in these channels.

There is a complication, however, that should be noted. The analysis of neutron \(\beta \) decay requires a value of the neutron axial vector coupling \(g_A\) as well (similar considerations operate for Gamow–Teller nuclear transitions); presently, this important quantity is taken from experiment because theory cannot determine it well enough, as illustrated in Fig. 39. This topic is also addressed in Sect. 3.2.5a; here we revisit possible resolutions. A crucial direction for lattice QCD is to reexamine the systematics in the nucleon matrix elements, a task that was somewhat neglected in the past when we struggled to get enough computing power to address merely statistical errors. Resources have improved, and many groups have investigated the excited-state contamination, and this seems to be under control in recent years. However, the results remain inconsistent with experiment, and more extensive studies of finite-volume corrections with high statistics will be carried out in the future. In addition, the uncertainty associated with extrapolating to a physical pion mass should be greatly improved within the next year or two. Overall, we believe \(g_A\) will be calculated to the percent-level or better (systematically and statistically) in the next few years. It is worth noting that a blind analysis should be easy to carry out for \(g_A\) since it is an overall constant in the lattice three-point correlators. Nevertheless, it is currently the case that poorly understood systematics can affect the lattice-QCD computations of the nucleon matrix elements, and those of \(g_A\) serve as an explicit example. However, those uncertainties are not so large that they undermine the usefulness of the \(g_\mathrm{S}\) and \(g_\mathrm{T}\) results. As we have shown, the lattice computations of these quantities need not be very precise to be useful.
Fig. 39

Compilation of \(g_A\) determined from experiment (top) and lattice QCD (bottom) adapted from Ref. [1437]. The lower panel shows \(g_A\) values after extrapolating to the physical pion mass collected from dynamical 2+1-flavor and 2-flavor lattice calculations using \(O(a)\)-improved fermions [209, 236, 240, 242, 247, 249, 250, 251, 255, 256, 259, 260, 261, 1460, 1469, 1470, 1471, 1472]. A small discrepancy persists: while calculations continue to tend towards values around 1.22 with a sizeable error, the experimental values are converging towards \(1.275 \pm 0.005\). A significant lattice effort will be necessary to reduce the systematics and achieve total error at the percent level

There are also other \(\beta \)-decay nucleon matrix elements induced by strong-interaction effects which enter as recoil corrections at \(\mathcal{O}(E/M)\), where \(E\) is the electron energy scale and \(M\) is the nucleon mass. The weak magnetic coupling \(f_2\) can be determined using the conserved-vector-current (CVC) hypothesis (though a fact in the SM) and the isovector nucleon magnetic moment, though this prediction, as well as that of the other matrix elements to this order, is modified by isospin-breaking effects. This makes it useful to include errors in the assessment of such matrix elements, when optimizing the parameters to be determined from experiment. Such a scheme has been developed, after that in [1369], in [1465], and the impact of such theory errors on the ability to resolve non-\((V-A)\) interactions has been studied, suggesting that it is important to study the induced tensor term \(g_2\) and other recoil-order matrix elements using lattice QCD as well. The study of [1465] shows that it is also crucial to measure the neutron lifetime extremely well, ideally to \(\mathcal{O}(0.1\,\mathrm{sec})\) precision, in order to falsify the \(V-A\) law and establish the existence of physics BSM in these processes. We refer the reader to Sect. 5.5.1 for a perspective on the neutron lifetime and its measurement.

Second-class currents gain in importance in neutron decay precisely because it is a mixed transition—and because BSM effects are already known to be so small. No direct lattice-QCD study of these isospin-breaking couplings has yet been done, but a few previous works have tried to estimate their size in hyperon decay [1466, 1467, 1468]. Perhaps particularly interesting is the analysis of the process \(\Xi ^0 \rightarrow \Sigma ^{+} {\ell }\bar{\nu }\), in which the second-class current terms emerge as SU(3)\(_\mathrm{f}\) breaking effects. In this case, [1467] \(|f_3(0)/f_1(0)| = 0.14 \pm 0.09\) and \(|g_2(0)/f_1(0)| = 0.68 \pm 0.18\); this exploratory calculation is made in the quenched approximation with a relatively heavy pion mass of 539–656 MeV. Nevertheless, this decay is a strict analog of the neutron decay process, with the \(d\) valence quark replaced by \(s\), so that one can estimate the size of \(g_2/g_A\) in neutron decay by scaling the earlier results by a factor of \(m_d/m_\mathrm{s} \sim 1/20\) [1465]. Ultimately, one can foresee results with reduced uncertainties from direct calculations on physical pion mass ensembles, using the variation of the up and down quark masses to resolve the second-class contributions in neutron decay.

High-energy colliders can constrain \(\epsilon _\mathrm{S}\) and \(\epsilon _\mathrm{T}\) in the manner shown in Fig. 38. Unfortunately, as shown in [203, 1458], the CDF and D0 results do not provide useful constraints in this context. The limits shown follow from estimating the \(\epsilon _{S,T}\) constraints from LHC current bounds and near-term expectations through an effective Lagrangian
$$\begin{aligned} \mathcal{L} = -\frac{\eta _\mathrm{S}}{\Lambda _\mathrm{S}^2}V_{ud}(\overline{u}d)(\overline{e}P_L\nu _e) -\frac{\eta _\mathrm{T}}{\Lambda _\mathrm{T}^2}V_{ud}(\overline{u}\sigma ^{\mu \nu }d) (\overline{e}\sigma _{\mu \nu }P_L\nu _e), \end{aligned}$$
(5.19)
where \(\eta _{S,T}=\pm 1\) to account for the possible sign of the couplings at low-energy. The high-energy bounds are scaled down to a scale of 2 GeV to compare with low-energy predictions. By looking at events with high transverse mass from the LHC in the \(e\nu +X\) channel and comparing with the SM \(W\) background, the authors of [203, 1458] estimated 90 %-C.L. constraints on \(\epsilon _{S,T}\) based on existing data [1463], \(\sqrt{s}=7\) TeV \(L=10 \text{ fb }^{-1}\) (the outermost (green) line) and the anticipated (null result) data sets at \(\sqrt{s}=8\) TeV \(L=25 \text{ fb }^{-1}\) (the middle (purple) line) and \(\sqrt{s}=14\) TeV \(L=300 \text{ fb }^{-1}\) (the innermost (magenta) line) of the lower panel in Fig. 38. The low-energy experiments can potentially yield much sharper constraints.

There is plenty of room for further improvements of lattice-QCD calculations of \(g_{S,T}\). Currently, there are fewer direct lattice calculations of \(g_\mathrm{T}\) and \(g_\mathrm{S}\), and the errors are roughly 10 % and 30 %, respectively. Ongoing calculations are improving control over the systematics due to chiral extrapolation and finite-volume effects. In addition, we expect more collaborations will compute these quantities, and near-future work will substantially reduce the errors. In particular, there is presently no chiral perturbation theory formula for the extrapolation to a physical pion mass, and operator matching is done either at tree- or one-loop level. Work is under way to reduce these errors, and we expect results with 5 % errors (including all systematics) on a five-year timescale.

5.5.1 The role of the neutron lifetime

The neutron lifetime value provides important input to test weak-interaction theory in the charged-current sector [1473]. It is also important for Big-Bang nucleosynthesis (BBN), which is becoming more and more important for constraining many BSM physics scenarios which produce new contributions to the relativistic particle energy density [1243]. BBN predicts the primordial abundances of the light elements (H, He, D, Li) in terms of the baryon-to-photon ratio \(\eta \), together with nuclear physics input that includes 11 key nuclear reaction cross sections, along with the neutron lifetime [1474]. As primordial neutrons are protected against \(\beta \)-decay by fusing with protons into deuterons and then into \(^4\)He, a shorter neutron lifetime results in a smaller \(^4\)He abundance (\(Y_{p}\)). The dependence of the helium abundance on changes in the neutron lifetime, the “effective” number of light neutrinos \(N_\mathrm{eff}\), and the baryon-to-photon ratio are: \(\delta Y_{p}/Y_{p}=+0.72 \delta \tau _{n}/\tau _{n}\), \(\delta Y_{p}/Y_{p}=+0.17 \delta N_\mathrm{eff}/N_\mathrm{eff}\), and \(\delta Y_{p}/Y_{p}=+0.039 \delta \eta / \eta \)  [1475, 1476]. With the precise determination of \(\eta \) from WMAP [1477] and now PLANCK [1478], the 0.2–0.3 % error on the BBN prediction for Y\(_p\) is now dominated by the uncertainty in the neutron lifetime. At the same time astrophysical measurements of the helium abundance (\(Y_{p}=0.252 \pm 0.003\) [1476, 1479]) from direct observations of the H and He emission lines from low-metallicity regions are poised for significant improvement. Astronomers are now in a position to re-observe many of the lowest abundance objects used for nebular \(^{4}\)He abundance determinations over the next 3–5 years and will continue to find additional ultralow abundance objects [1480]. Sharpening this test of BBN will constrain many aspects of nonstandard physics scenarios.

Measurements of the neutron lifetime had been thought to be approaching the 0.1 % level of precision (corresponding to a \(\sim \) 1 s uncertainty) by 2005, with the Particle Data Group [1481] reporting \(885.70\pm 0.85\) s. However, several neutron lifetime results since 2005 using ultracold neutron measurements in traps [1482, 1483, 1484, 1485] reported significantly different results from the earlier PDG average: the latest PDG value (\(880.1 \pm 1.1\) s) [1] includes all these measurements, with the uncertainty scaled up by a factor of \(2.7\). The cause of this many-sigma shift has not yet been resolved. The large discrepancies between the latest lifetime measurements using ultracold neutrons in material bottles make it clear that systematic errors in at least some previous measurements have been seriously underestimated, and precision measurements using alternative techniques are badly needed [1486]. The latest update [1487] from a Penning trap neutron lifetime experiment in a cold neutron beam [1488, 1489] gives \(\tau _{n}= 887.7 \pm 1.2\,(\mathrm{stat.})\,\pm 1.9\,(\mathrm{sys})\,\mathrm{s}\). In addition to continued measurements using the Penning trap technique, neutron lifetime measurements with ultracold neutrons now concentrate on trapping the neutrons using magnetic field gradients in an attempt to avoid what people suspect to be uncontrolled systematic errors from surface effects in material traps. A recent experiment at Los Alamos using a magneto-gravitational trap that employs an asymmetric Halbach permanent magnet array [1490] has observed encouraging results [1491].

5.6 Broader applications of QCD

Nucleon matrix elements and lattice-QCD methods are key to a broad sweep of low-energy observables which probe how precisely we understand the nature of things. We now consider a range of examples, to illustrate the breadth of the possibilities.

5.6.1 Determination of the proton radius

The charge radius of the proton \(r_p\) has not yet been precisely calculated in lattice QCD because the computation of disconnected diagrams with explicit quark loops is required. (In the case of the isovector charge radius (\(r_p - r_n\)) the disconnected diagrams cancel, so that this quantity could be more precisely calculated than \(r_p\).) Rather, it is currently determined from the theoretical analysis of experimental results. There has been great interest in \(r_p\) because the determination of this quantity from the study of the Lamb shift in muonic hydrogen [282, 285], yielding [285]
$$\begin{aligned} r_p^{(\mu H)} = 0.84087 \pm 0.00039 \,\mathrm{fm} , \end{aligned}$$
(5.20)
is some \(7\sigma \) different from the value in the CODATA-2010 compilation [284], determined from measurements of hydrogen spectroscopy (\(r_p^{(e H)}\)) and electron–proton (\(r_p^{(e p)}\)) scattering. The incompatibility of the various extractions offers a challenge to both theory and experiment.

We note that \(r_p^{(ep)}\) is by no means a directly determined quantity, because two-photon exchange effects do play a numerical role as well. Such corrections also appear in the context of the muonic-hydrogen analysis, though the effects turn out to be too small to explain the discrepancies. For example, a dispersive re-evaluation [302] of such hadronic effects based on experimental input (photo- and electroproduction of resonances off the nucleon and high-energy pomeron-dominated cross section) yields a contribution of \(40\pm 5\) \(\upmu \)eV to the muonic hydrogen Lamb shift. Even if the error were underestimated for some unknown reason, its order of magnitude is insufficient to resolve the 300 \(\upmu \)eV discrepancy between direct measurement of the muonic Lamb shift [282, 285] and its expectation determined from QED theory and conventional spectroscopy. Such difficulties have prompted much discussion [289], and we refer to Sect. 3.2.6 for further details. It is still too speculative to state that we are confronting a violation of universality in the couplings of the electron and the muon, but hope that hadron contributions to the two-photon exchange between the muon and the proton would resolve the issue seems misplaced. Nevertheless, a viable BSM model does exist which would permit the discrepancy to stand [1492]. It predicts the existence of new parity-violating muonic forces which potentially can be probed through experiments using low-energy muon beams, notably through the measurement of a parity-violating asymmetry in elastic scattering from a nuclear target. Unfortunately, this picture cannot easily explain the existing muon \(g-2\) discrepancy [1492]. Disagreement between theory and experiment lurks there also, but the precision of the discrepancy is two orders of magnitude smaller than in the muonic Lamb shift case. Indeed the muon \(g-2\) result constrains new, muon-specific forces [1493]. Planned studies of \(\mu p\) and \(e p\) scattering at PSI should offer a useful direct test on the universality of lepton–proton interactions [1494].

5.6.2 Dark-matter searches

Various threads of astronomical evidence reveal that we live in a Universe dominated by dark matter and dark energy [1]. It is commonly thought that dark matter could be comprised of an as yet unidentified weakly interacting massive particle (WIMP). Such particles in the local solar neighborhood of our own Milky Way galaxy can be constrained or discovered through low-background experiments which search for anomalous recoil events involving the scattering of WIMPs from nuclei [1495, 1496]. Supersymmetric models offer a suitable candidate particle, the neutralino, which can be made compatible with all known astrophysical constraints [1497, 1498]. The neutralino is made stable by introducing a discrete symmetry, \(R\) parity, that forbids its decay. An analogous discrete symmetry can be introduced in other, nonsupersymmetric new-physics contexts, such as in “little Higgs” models [1499], to yield an identical effect—generally, one can introduce a dark-matter parity that renders the dark-matter candidate stable.

WIMP–nuclear interactions mediated by \(Z^0\) exchange were long-ago ruled out [1498, 1500], so that the WIMP of supersymmetric models is commonly regarded as a neutralino. Current experiments probe the possibility of mediation by Higgs exchange. Consequently, the spin-independent neutralino–nucleon cross section is particularly sensitive to the strange scalar density, namely, the value of \(y=2 \langle N | \bar{s} s | N \rangle / \langle N | \bar{u} u + \bar{d} d | N \rangle \), noting [1501] and references therein, because the Higgs coupling increases with quark mass. The value of this quantity impacts the mapping of the loci of supersymmetric parameter space to the exclusion plot of WIMP mass versus the WIMP–nucleon cross section. Earlier studies relate \(y\) to the \(\pi N\) sigma term \(\Sigma _{\pi N}\) via \(y=1- \sigma _0/\Sigma _{\pi N}\) for fixed \(\sigma _0 \equiv m_l \langle N | \bar{u} u + \bar{d} d - 2 \bar{s} s | N \rangle \) [1501], where \(m_l\equiv (m_u+m_d)/2\), suggesting that the predicted neutralino–nucleon cross section depends strongly on the value of this phenomenological quantity [1502]. Its impact can be remediated, however, without recourse to assumptions in regards to SU(3)-flavor breaking; e.g., as shown in [1503], the couplings to the \(u\)- and \(d\)-quarks can be analyzed directly within the framework of \(\mathrm{SU}(2)\) chiral perturbation theory (ChPT), permitting, in addition, control over isospin-breaking effects.

The matrix elements \(m_\mathrm{s} \langle N | \bar{s} s | N \rangle \) and \(\Sigma _{\pi N} \equiv m_l\langle N | \bar{u} u + \bar{d} d | N \rangle \) can also be computed directly in lattice-QCD, via different techniques, and the sensitivity to \(\Sigma _{\pi N}\) is lessened [1502]. Several lattice-QCD groups have addressed this problem, and new results continue to emerge [1504, 1505, 1506]. The spin-independent WIMP–nucleon cross section can be predicted to much better precision than previously thought, though the cross section tends to be smaller than that previously assumed [1502], diminishing the new physics reach of a particular WIMP direct detection experiment. Heavier quark flavors can also play a significant role in mediating the gluon coupling to the Higgs, and hence to the neutralino, and the leading contribution in the heavy-quark limit is well-known [1498, 1507]—and this treatment should describe elastic scattering sufficiently well. Nevertheless, the non-perturbative scalar charm matrix element should also be considered, and it has also been recently evaluated [1508]. We note, moreover, in the case of heavy WIMP–nucleon scattering, that the renormalization-group evolution from the weak to typical hadronic scales also plays a numerically important role [1509].

The effects of the nuclear medium in mediating effects beyond the impulse approximation (for scalar-mediated interactions) have also been argued to be important [1510]. This possibility has been recently investigated on the lattice, and the effects actually appear rather modest [1511]. Nevertheless, two-body exchange currents, which appear in chiral effective theory, can be important in regions of WIMP parameter space for which the usual WIMP–nucleon interaction is suppressed [1512]. For a study in spin-dependent WIMP–nuclear scattering see [1513].

5.6.3 Neutrino physics

The physics of QCD also plays a crucial role in the analysis of neutrino experiments, particularly through the axial-vector form factor of the nucleon (and of nuclei). The value of the axial coupling of the nucleon \(g_A\), which is precisely measured in neutron \(\beta \)-decay, is key to the crisp interpretation of low-energy neutrino experiments such as SNO [1514]. In higher-energy experiments, however, the \(q^2\) dependence of the axial form factor becomes important. In particular, elucidating the axial mass \(M_A\), which reflects the rate at which the form factor changes with \(q^2\), is crucial to the interpretation of neutrino oscillation experiments at \(\mathcal{O}(1~\mathrm{GeV})\), an energy scale typical of accelerator-based studies. Commonly the value of \(M_A\) is assessed experimentally by assuming the form factor can be described by a dipole approximation,
$$\begin{aligned} G_A^\mathrm{dipole}(q^2) = \frac{g_A}{\left[ 1- q^2/M_A^2\right] ^2} , \end{aligned}$$
(5.21)
and the nuclear effects, at least for neutrino quasi-elastic scattering, are assessed within a relativistic Fermi gas model, though final-state interactions of the produced hadrons in the nucleus can also be included. A consistent description of the neutrino–nuclear cross sections with beam energy and nuclear target is essential for future investigations of the neutrino mass hierarchy and CP violation in long-baseline experiments (LBNE, T2K, NO\(\nu \)A, CNGS). Within this framework, tension exists in the empirically determined values of \(M_A\) [1462]. Moreover, recent studies at MiniBoone (http://www-boone.fnal.gov) have illustrated that the framework itself-appears to be wanting [1515, 1516]. Current and future studies at MINER\(\nu \)A (http://minerva.fnal.gov) can address these deficiencies by measuring the neutrino (and antineutrino) reaction cross sections with various nuclei [1517, 1518]. Model-independent analyses of experimental data have also been developed [1462] and have explored ways in which to relax the usual dipole parameterization of the axial form factor of the nucleon, as it is only an approximation. Nevertheless, a computation of the \(q^2\) dependence of the nucleon axial form factor within QCD is greatly desired.

The value of \(M_A\) can be estimated from the nucleon isovector axial form factor by a fit of its \(q^2\) dependence to a dipole form. Alternatively, \(r_A\), the axial radius, is calculated by taking the derivative of the form factors near \(Q^2=0\), and they are linked through \(r_A^2={12}/{M_A^2}\) (in the dipole approximation). The quantity \(r_A\) is ultimately of greater interest as it is not tied to a dipole form. Lattice-QCD calculations of axial form factors, as well as of vector form factors, tend to yield smaller slopes and, thus, prefer a larger value of \(M_A\) [206, 236, 1519, 1520]. This tendency may stem from a heavy pion mass or finite volume effects.

5.6.4 Cold nuclear medium effects

Many precision searches for new physics are undertaken within nuclear environments, be they dark-matter searches or studies of neutrino properties, and so far there is no universal understanding nor theoretical control over nuclear corrections. A common assumption is that the WIMP, or neutrino, interactions in the nucleus are determined by the sum of the individual interactions with the nucleons in the nucleus, as, e.g., in [1521, 1522]. This impulse-approximation picture treats nuclear-structure effects independently of the particle-physics interaction with a single nucleon. Nevertheless, single-particle properties can be modified in the nuclear medium, and evidence for such effects range from low-to-high energy scales. For example, at the lowest energy scales, the possibility of quenching of the Gamow–Teller strength in nuclei (with respect to its free-nucleon value) has been discussed for some time [1523, 1524, 1525, 1526], though its source is unclear. It may be an artifact of the limitations of nuclear shell-model calculations13 [1528] or a genuine effect, possibly arising from meson-exchange currents in nuclei [1529]. At larger energies, in deep-inelastic lepton scattering from nuclei, medium effects are long established, most famously through the so-called EMC effect noted in \(F_2\) structure function data [1530]. At \(\mathcal{O}(1\,\mathrm{GeV})\) energy scales important for accelerator-based, long-baseline neutrino experiments, medium effects have also been observed in the studies of \(3.5~\mathrm{GeV}\) neutrino–nuclear interactions in the MINER\(\nu \)A experiment [1517, 1518]. The inclusion of two-nucleon knock-out in addition to quasi-elastic scattering appears to be needed to explain the observed neutrino–nuclear cross sections at these energies [1531]. This is a challenging energy regime from a QCD viewpoint; the interactions of \(\mathcal{O}(1\,\mathrm{GeV})\) nucleons are not suitable for treatment in chiral effective theory or perturbative QCD.

In-medium effects may also help explain older puzzles. For example, the NuTeV (http://www-e815.fnal.gov) experiment [1532] yields a value of \(\sin ^2\theta _W\) in neutrino-nucleus scattering \({\sim } 3\sigma \) away from the SM expectation. This anomalous result can be explained, at least in part, by QCD effects, through corrections arising from modifications of the nuclear environment [1533].

Theoretical insight into these problems may prove essential to the discovery of new physics. Unfortunately, multibaryon systems are complicated to calculate in lattice QCD due to a rapid increase in statistical noise. An analogous, albeit simpler, system using many pions has been the subject of an exploratory study. This first lattice-QCD attempt to measure many-hadron modifications of the hadronic structure in a pion (\(\pi ^+\)) medium uses pion masses ranging 290–490 MeV at 2 lattice spacings [1534]. The preliminary result indicates strong medium corrections to the first moment of the pion quark-momentum fraction. With recent improvements to the efficiency of making quark contractions, which was one of the bottlenecks preventing lattice QCD from accessing even just 12-quark systems, we expect to see development toward structure calculations for light nuclei albeit at heavier pion masses within the next few years.

5.6.5 Gluonic structure

In the current global fit of the unpolarized parton distribution functions (PDFs) the gluonic contribution plays an enormously important role—roughly half of the nucleon’s momentum is carried by glue. However, gluonic structure has been notoriously difficult to calculate with reasonable statistical signals in lattice QCD, even for just the first moment. Despite these difficulties, gluonic structure has been re-examined recently, with new work providing approaches and successful demonstrations that give some hope that the problem can be addressed. Both \(\chi \)QCD [226, 1535] and QCDSF [1536] (note also [1537]) have made breakthroughs with updated studies of gluonic moments in quenched ensembles with lightest pion masses of 480 MeV. The two groups attack the problem using different techniques and show promising results, with around 15 % uncertainty when extrapolated to the physical pion mass. These methods are now being applied to gauge ensembles with dynamical sea quarks, and we expect to see updated results within a few years. Similar methods are also now used to probe the role of glue in the angular momentum of the proton [226, 1535].

Let us conclude this section more broadly and note that, in addition to these known effects, lattice-QCD matrix elements are also important to experiments which have not yet observed any events, such as \(n\)\(\bar{n}\) oscillations [1538] or proton decay [1539]. Lattice-QCD calculations can provide low-energy constants to constrain the experimental search ranges. The potential to search for new physics using these precision nucleon matrix elements during the LHC era and in anticipation of future experiments at Fermilab make lattice-QCD calculations of nucleon structure particularly timely and important.

5.7 Quark flavor physics

The majority of the SM parameters have their origin in the flavor sector. The quark and lepton masses vary widely, which is an enduring puzzle. In this section we review studies of flavor and CP violation in the quark sector, usually probed through the weak decays of hadrons. In the SM the pattern of observed quark flavor and CP violation is captured by the CKM matrix, and the pattern is sufficiently distinctive that by overconstraining its parameters with multiple experiments and by employing accurate calculations, there is hope that an inconsistency between them (and therefore new physics) will ultimately emerge. At a minimum, this effort would allow the extraction of the CKM parameters with ever increasing precision. Extensive reviews of this issue already exist; we note the massive efforts of the Heavy Flavor Averaging Group [927] and the PDG [1] for experimental matters, as well as similar reviews of lattice-QCD results [44, 45, 1540]. Thus, we concentrate here on a few highlights suggested by very recent progress or promise of principle. We turn first, however, to two topics in non-CKM flavor physics which link to searches for BSM physics at low energies.

5.7.1 Quark masses and charges

a. Light quark masses The pattern of fermion masses has no explanation in the SM, but if the lightest quark mass were to vanish, the strong CP problem would disappear. Thus in light of our discussion of permanent EDMs and the new sources of CP violation that those experimental studies may reveal, it is pertinent to summarize the latest lattice-QCD results for the light quark masses. Current computations work in the isospin limit (\(m_u=m_d\)), treating electromagnetism perturbatively. Turning to the compilation of the second phase of the Flavour Lattice Averaging Group (FLAG2) [45], we note with \(N_\mathrm{f}=2+1\) flavors (implying that a strange sea quark has been included) in the \({\overline{\mathrm{MS}}}\) scheme at a renormalization scale of \(\mu =2\) GeV that
$$\begin{aligned} {m}_\mathrm{s}&= (93.8\pm 1.5 \pm 1.9)~\mathrm{MeV} ,\, \nonumber \\ {m}_{ud}&= (3.42\pm 0.06\pm 0.07)~\mathrm{MeV} , \end{aligned}$$
(5.22)
with \(m_{ud} \equiv (m_{u} + m_{d})/2\), where the first error comes from averaging the lattice results and the second comes from the neglect of charm (and more massive) sea quarks. The \(m_\mathrm{s}\) average value employs the results of [37, 39, 40, 348], whereas the \(m_{ud}\) average value employs the results of [37, 39, 348, 1541]. To determine \(m_u\) and \(m_d\) individually additional input is needed. A study of isospin-breaking effects in chiral perturbation theory yields an estimate of \(m_u/m_d\); this with the lattice results of (5.22) yields [45]
$$\begin{aligned} {m}_u&= (2.16\pm 0.09 \pm 0.07)~\mathrm{MeV} ,\, \nonumber \\ {m}_{d}&= (4.68\pm 0.14 \pm 0.07)~\mathrm{MeV} , \end{aligned}$$
(5.23)
where the first error represents the lattice statistical and systematic errors, taken in quadrature, and the second comes from uncertainties in the electromagnetic corrections.14 The electromagnetic effects could well deserve closer scrutiny.
Nevertheless, it is apparent the determined up quark mass is definitely nonzero. This conclusion is not a new one, even if the computations themselves reflect the latest technical advances, and it is worthwhile to remark on the (\(m_u=0\)) proposal’s enduring appeal. Ambiguities in the determination of \(m_u\) have long been noted [1542, 1543]; particularly, Banks et al. [1543] have argued that both the real and imaginary parts of \(m_u\) could be set to zero if there were an accidental U(1) symmetry predicated by some new, spontaneously broken, horizontal symmetry. This would allow \(\delta \) of the CKM matrix to remain large at the TeV scale, while making \(\bar{\theta }\) small. In this picture, a nonzero \(m_u\) still exists at low scales, but it is driven by non-perturbative QCD effects (and the strong CP problem can still be solved if its impact on the EDM is sufficiently small). That is, in this picture \(m_u\) is zero at high scales but is made nonzero at low scales through additive renormalization [1543]. Namely, its evolution from its low-scale value \(\mu _u\) to a high-scale value \(m_u\) (\(m_u=0\)) would be determined by
$$\begin{aligned} \mu _u = \beta _1 m_u + \beta _2 \frac{m_d m_\mathrm{s}}{\Lambda _\mathrm{QCD}} +\cdots , \end{aligned}$$
(5.24)
where \(\beta _1\) and \(\beta _2\) are dimensionless, scheme-dependent constants. This solution has been argued to be untenable because \(m_d\) and \(m_\mathrm{s}\) are guaranteed not to vanish (independently of detailed dynamical calculations) by simple spectroscopy and no symmetry distinguishes the \(m_u=0\) point [1544, 1545]. This makes the notion of a zero up-quark mass ill-posed [1545] within strict QCD, though this does not contradict the proposal in [1543], precisely because their analysis takes the second term of (5.24) into account.

In [1543], \(\mu _u\) on the left-hand side of (5.24) was argued to hold for the mass parameter of the chiral Lagrangian. The pertinent question is whether it applies to the masses of the QCD Lagrangian, obtained from lattice gauge theory. Because \(\beta _2\) is scheme dependent, the answer depends on details of the lattice determination. Still, there is no evidence that the additive renormalization term is large enough to make the \(m_u=0\) proposal phenomenologically viable [1546]. The proposal of [1543] could be independently falsified if the residual \(\mathrm{Im}(m_u)\) effects at low energies could be shown at odds with the existing neutron EDM bounds.

b. Quark charges In the SM with a single generation, electric-charge quantization (i.e., unique \(U(1)_Y\) quantum number assignments) is predicated by the requirement that the gauge anomalies cancel, and ensures that both the neutron and atomic hydrogen carry identically zero electric charge. There has been much discussion of the fate of electric charge quantization upon the inclusion of new physics degrees of freedom, prompted by the experimental discovery that neutrinos have mass—we refer to [1547] for a review. For example, enlarging the SM with a gauge-singlet fermion, or right-handed neutrino, breaks the uniqueness of the hypercharge assignments, so that electric charge is no longer quantized unless the new neutrino is a Majorana particle [1548, 1549]. This outcome can be understood in terms a hidden \(B-L\) symmetry which is broken if the added particle is Majorana [1548, 1549, 1550, 1551]. More generally, electric charge quantization is not guaranteed in theories for which the Lagrangian contains anomaly-free global symmetries which are independent of the SM hypercharge Y [1547].15 If neutrinoless double \(\beta \) decay or neutron–antineutron oscillations are observed to occur, then the puzzle of electric-charge quantization is solved. Alternatively, if the charge neutrality of the neutron or atomic hydrogen were found to be experimentally violated, then it would suggest neither neutrinos nor neutrons are Majorana particles. Another pathway to charge quantization could lead from making the SM the low-energy limit of a grand unified theory, though this is not guaranteed even if such a larger theory occurs in nature. Ultimately, then, searches for the violation of charge neutrality, both of the neutron and of atoms with equal numbers of protons and electrons, probe for the presence of new physics at very high scales [1552]. Such a violation could also connect to the existence of new sources of CP violation [1553]. Novel, highly sensitive, experiments [1552, 1554] are under development, and plan to better existing limits by orders of magnitude.
Fig. 40

Constraints on the (neutral-current) weak couplings of the \(u\) and \(d\) quarks plotted in the \(C_{1u}\)\(C_{1d}\) plane. The band refers to the limits from APV, whereas the more vertical ellipse represents a global fit to the existing PVES data with \(Q^2< 0.63~(\mathrm{GeV})^2\). The small, more horizontal ellipse refers to the constraint determined from combining the APV and PVES data. The SM prediction as a function of \(\sin ^2\theta _W\) in the \({\overline{\mathrm{MS}}}\) scheme appears as a diagonal line; the SM best fit value is \(\sin ^2\theta _W=0.23116\) [1]. Figure taken from [1321], and we refer to it for all details

Experimental measurements of parity-violating electron scattering (PVES) observables yield significant constraints on the weak charges of the quarks and leptons, probed through the neutral current. Recently, the weak charge of the proton \(Q_W^p\) has been measured in polarized \(\varvec{e}\)\(p\) elastic scattering at \(Q^2=0.025~(\mathrm{GeV})^2\) [1321]. This result, when combined with measurements of atomic parity violation (APV), yields the weak charge of the neutron \(Q_W^n\) as well. The associated limits on the weak couplings of the quarks are shown in Fig. 40. For reference, we note that \(Q_W^p=-2(2C_{1u} + C_{1d})\), where \(C_{1i} = 2 g_A^e g_V^i\) and \(g_A^e\) and \(g_V^i\) denote the axial electron and vector quark couplings, respectively. The plot depicts an alternate way of illustrating the constraints on the \(Q^2\) evolution of \(\sin ^2\theta _W\) in the \({\overline{\mathrm{MS}}}\) scheme discussed in Sect. 3.5.

Non-perturbative QCD effects enter in this context as well, and we pause briefly to consider the extent to which they could limit the sensitivity of BSM tests in PVES. One notable effect is the energy-dependent radiative correction which arises from the \(\gamma \)\(Z\) box diagram. Dispersion techniques can be used to evaluate it [708, 709, 710, 711, 712, 1555, 1556], and the correction is demonstrably large, contributing to some 8 % of \(Q_W^p\) in the SM [1]. Currently the dispersion in its assessed error is greater than that in the predicted central value, though the expected error can be refined through the use of additional PDF data [1556]. Charge-symmetry-breaking effects in the nucleon form factors may eventually prove significant as well but are presently negligible as they should represent a \(\lesssim 1~\%\) correction [1557, 1558, 1559, 1560, 1561]. The implications of such theoretical errors, which appear manageable at current levels of sensitivity, could eventually be assessed in a framework analogous to that recently developed for neutron decay observables [1465].

5.7.2 Testing the CKM paradigm

We begin by presenting the moduli of the elements \(V_{ij}\) of the CKM matrix determined in particular charged-current processes, using the compilation of [1]:
$$\begin{aligned}&\begin{array}{|ccc|} |V_{ud}| &{} |V_{us}| &{} |V_{ub}| \\ |V_{cd}| &{} |V_{cs}| &{} |V_{cb}| \\ |V_{td}| &{} |V_{ts}| &{} |V_{tb}| \end{array} \propto \quad \quad \quad \quad \quad \quad \\ \nonumber&\begin{array}{|ccc|} 0.97425\!\pm \! 0.00022 &{} 0.2252\! \pm \! 0.0009 &{} 0.00415\! \pm \! 0.00049 \\ 0.230 \!\pm \! 0.011 &{} 1.006\! \pm \! 0.023 &{} 0.0409 \!\pm \! 0.0011 \\ 0.0084 \!\pm \! 0.0006 &{} 0.0429 \!\pm \! 0.0026 &{} 0.89 \!\pm \! 0.07 \end{array} .\quad \end{aligned}$$
(5.25)
Quark intergenerational mixing is characterized by the parameter \(\lambda \approx |V_{us}|\), with mixing of the first (second) and third generations scaling as \(\mathcal{O}(\lambda ^3)\) (\(\mathcal{O}(\lambda ^2)\)) [1562]. The CKM matrix \(V_\mathrm{CKM}\) can be written in terms of the parameters \(\lambda , A, {\bar{\rho }}, {\bar{\eta }}\); thus parametrized it is unitary to all orders in \(\lambda \) [1369, 1563]. We test the SM of CP violation by determining whether all CP-violating phenomena are compatible with a universal value of \(({\bar{\rho }}, {\bar{\eta }})\) (note [1] for the explicit connection to \(\delta \)).

Current constraints are illustrated in Fig. 41. The so-called unitarity triangle in the \({\bar{\rho }}\)\({\bar{\eta }}\) plane has vertices located at \((0,0)\), \((1,0)\), and \((\bar{\rho }_\mathrm{SM},\bar{\eta }_\mathrm{SM})\). The associated interior angles at each vertex are \(\gamma (\phi _3)\), \(\beta (\phi _1)\), and \(\alpha (\phi _2)\), respectively. The CP asymmetry \(S_{\psi K}\) is realized through the interference of \(B^0\)\(\bar{B}^0\) mixing and direct decay into \(\psi K\) and related modes. It is \(\sin 2\beta \) in the SM up to hadronic uncertainties which appear in \(\mathcal{O}(\lambda ^2)\). The other observables require hadronic input of some kind to determine the parameters of interest; lattice-QCD calculations are essential to realize the precision of the tests shown in Fig. 41.

The constraints thus far are consistent with the SM of CP violation; the upper \(S_{\psi K}\) band in the \(\bar{\rho }-\bar{\eta }\) plane arising from a discrete ambiguity has been ruled out by the determination that \(\cos 2\beta > 0\) at 95 % C.L. [1564]. Experimental studies of CP violation in the \(B\) system continue, and we note an improved constraint on \(\gamma \) of \(\gamma =67^\circ \pm 12^\circ \) from LHCb [1565], which is consistent with the SM and with earlier B-factory determinations [1566]. Certain, early anomalies in B-physics observables can be explained by a possible fourth SM-like generation [1567, 1568], and it remains an intriguing idea. Its existence, however, is becoming less and less consistent with experimental data. Direct searches have yielded nothing so far [1569], and a fourth SM-like generation is disfavored by the observation of the Higgs, and most notably of \(H\rightarrow \gamma \gamma \), as well [1323, 1570]. Flavor and CP violation are well-described by the CKM matrix [1367], so that it has become popular to build BSM models of the electroweak scale which embed this feature. That is, flavor symmetry is broken only by the standard Yukawa couplings of the SM; this paradigm is called Minimal Flavor Violation (MFV) [1571, 1572, 1573].
Fig. 41

Precision test of the SM mechanism of CP violation in charged-current processes realized through the comparison of the parameters \(\bar{\rho }\) and \(\bar{\eta }\) determined through various experimental observables and theory inputs from lattice QCD. The experimental inputs are as of September 2013, and the lattice inputs are derived from published results through April 30, 2013; the figure is an update of those in Ref. [1540]

The structure of the CKM matrix can also be tested by determining whether the empirically determined elements are compatible with unitarity. Figure 41 illustrates that unitarity is maintained if probed through the angles determined from CP-violating observables, that is, e.g., \(\alpha +\beta +\gamma =(178^{+11}_{-12})^\circ \) [1]. The most precise unitarity test comes from the first row [1], namely, of whether \(\Delta _u\equiv |V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 -1\) is nonzero. The contribution of \(|V_{ub}|^2\) is negligibly small at current levels of sensitivity, and for the last, several years the uncertainty has been dominated by that in \(|V_{us}|\) [1]. This situation changed, however, in 2013 with new, precise calculations of kaon decay parameters in lattice QCD becoming available [1574, 1575, 1576]. The quantity \(|V_{ij}|^2 (\delta |V_{ij}|)^2\) determines the impact of a CKM matrix element on the unitarity test, and by this measure that of \(|V_{us}|\) and \(|V_{ud}|\) [1577] are now comparable [1575, 1576]. Consequently, the earlier result [1]
$$\begin{aligned} \Delta _u = -0.0001\pm 0.0006 , \end{aligned}$$
(5.26)
becomes, using the average value of \(f_{K^\pm }/f_{\pi ^\pi }\) in QCD with broken isospin from [45], \(\Delta _u =0 \pm 0.0006\). By averaging over computations in \(N_\mathrm{f}=2+1+1\), \(N_\mathrm{f}=2+1\), and \(N_\mathrm{f}=2\) ensembles the improvements associated with the included (published at that time) \(N_\mathrm{f}=2+1+1\) computation [1574] is muted, begging the question of whether it is appropriate to average calculations which differ in their quenching of heavier sea quarks.16 The use of the most precise kaon results yields a tension with CKM unitarity [1575, 1576]. The value of \(|V_{us}|\) can also be determined from \(\tau \) decay, and the situation there is quite different. The inclusive \(\tau \) decay data yield a value of \(|V_{us}|\) which is less precisely determined but still different from the one assuming 3-flavor CKM unitarity by three sigma [45, 927]; more theoretical [718, 1578, 1579, 1580] and experimental work will likely be needed to determine the origin of the discrepancy. We refer to Sect. 3.5.3 for a discussion of the determination \(\alpha _\mathrm{s}\) in hadronic \(\tau \) decays, needed for a determination of \(|V_{us}|\).

The most precise determination of \(|V_{ud}|\), \(|V_{ud}|=0.97425 \pm 0.00022\) [1577], comes from the study of superallowed (\(0^+\rightarrow 0^+\)) transitions in nuclei. Its error is dominated by theoretical uncertainties, particularly from Coulomb corrections in the nuclear matrix elements and other nuclear-structure-dependent effects [1577] and from the evaluation of the \(\gamma \)\(W\) box diagram [1581, 1582, 1583]. The assessment of nuclear Coulomb corrections [1577] has been criticized as incomplete [1584, 1585], though it has been experimentally validated in a superallowed decay for which the corrections are particularly large [1586]. Another unitarity test comes from the second row; this can either be accessed directly through determinations of the \(V_{ij}\) or indirectly though the leptonic width of the \(W\), for which the hadronic uncertainties are trivially small. The former procedure yields \(\Delta _\mathrm{c} \equiv |V_{cd}|^2 + |V_{cs}|^2 + |V_{cb}|^2 -1 = 0.04 \pm 0.06\) [45], whereas the latter yields \(\Delta _\mathrm{c} = 0.002 \pm 0.027\) [1], making the indirect method more precise.

Theory plays a key and indeed expanding role in making all these tests more precise, so that increasingly the comparison between theory and experiment becomes a test field for QCD. We now consider some of the theory inputs in greater detail.

a. Theory inputs for\(V_{us}\) Until very recently, the error in \(V_{us}\) dominated that of the first-row CKM unitarity test. Here, we consider different pathways to \(V_{us}\) through meson decays; as we have noted, such efforts parallel the extraction of \(V_{us}\) from \(\tau \) decays [714, 1580, 1587, 1588].

Typically, \(V_{us}\) has been determined through \(K\rightarrow \pi {\ell } \nu \) (\(K_{{\ell }3}\)) decays and for which the following formula for the decay width applies [1589]:
$$\begin{aligned} \Gamma (K_{{\ell } 3})&= \frac{G_\mathrm{F}^2 m_K^5}{128\pi ^3} C_K^2 S_{\mathrm{EW}} (1+\delta _\mathrm{SU(2)}^{K\pi } + \delta _\mathrm{EM}^{K\ell }) \nonumber \\&\times |V_{us}|^2 |f_+^{K^0\pi ^{-}}(0)|^2 I_{K\ell } , \end{aligned}$$
(5.27)
which includes various electroweak, electromagnetic, and isospin-breaking corrections, in addition to the phase space integral \(I_{K\ell }\) and other known factors. We have separated in the second line two of the most interesting ones. The first is the wanted CKM matrix element, and the second is a hadronic form factor to be evaluated at zero-momentum transfer. The form factors \(f_{\pm }^{K\pi }(t)\) are determined by the QCD matrix elements
$$\begin{aligned}&\langle \pi (p_\pi )|{\bar{s}}\gamma _{\mu } u| K(p_K)\rangle \nonumber \\&\quad = (p_\pi + p_K)_{\mu } f_+^{K\pi }(t) + (p_\pi - p_K)_{\mu } f_-^{K\pi }(t), \end{aligned}$$
(5.28)
where \(t=(p_K- p_\pi )^2\), and we note
$$\begin{aligned} \delta _\mathrm{SU(2)}^{K \pi } = (f_+^{K\pi }(0)/f_+^{K^0\pi ^{-}}(0))^2 -1 . \end{aligned}$$
(5.29)
There are, in principle, five different widths to be determined, in \(K^\pm _{e3}, K^\pm _{\mu 3}, K_{Le3}, K_{L\mu 3}\), and \(K_{S\mu 3}\) decay, and the corrections in each can differ. Moreover, real-photon radiation also distinguishes the various processes, and it must be treated carefully to determine the experimental decay widths [1589]. Great strides have been made in the analysis of the various corrections [1589, 1590, 1591], which are effected in the context of chiral perturbation theory, and it is reasonable to make a global average of the determinations of \(V_{us} f_+(0)\) in the various modes [1]. Progress also continues to be made on the experimental front, there being new measurements of \(K^\pm \rightarrow \pi ^0 l^\pm \nu \) by the NA48/2 experiment at CERN [1592]. The updated five-channel average is \(f_+(0)|V_{us}|=0.2163 \pm 0.0005\) [1593]. The \(t\) dependence of the form factor is embedded in the evaluation of \(I_K^{\ell }\) in (5.27). NA48/2 has selected events with one charged lepton and two photons that reconstruct the \(\pi ^0\) meson and extract form factors that they fit with either a quadratic polynomial in \(t\), or a simple pole ansatz (be it scalar or vector),
$$\begin{aligned} f_{+,0}(t) = \frac{m_{v,s}^2}{m_{v,s}^2-t}, \end{aligned}$$
(5.30)
where \(f_0(t)=f_+(t) + (t/(m_K^2 - m_\pi ^2))f_-(t)\). A good fit is obtained with \(m_v= 877 \pm 6\) MeV and \(m_\mathrm{s}=1176 \pm 31\) MeV; these quantities do not precisely correspond to known particles but are of a reasonable magnitude. We detour, briefly, to note that the systematic error in the precise choice of fitting form can be mitigated through considerations of analyticity and crossing symmetry [1594, 1595]; the latter permits the use of experimental data in \(\tau \rightarrow K \pi \nu \) decays [1595] to constrain the fitting function. Finally we turn to the determination of \(f_+(0)\), for which increasingly sophisticated lattice-QCD calculations have become available. Noting [45], we report \([N_\mathrm{f} =2]\) [1596] and \([N_\mathrm{f} =2 +1]\) [1597] results:
$$\begin{aligned}&f_+(0) = 0.9560 \pm 0.0057 \pm 0.0062 \quad [N_\mathrm{f} =2] \nonumber \\&f_+(0) = 0.9667 \pm 0.0023 \pm 0.0033 \quad [N_\mathrm{f} =2 +1] \end{aligned}$$
(5.31)
as well as [1576]
$$\begin{aligned} f_+(0) = 0.9704 \pm 0.0024 \pm 0.0022 \quad [N_\mathrm{f} =2 +1 +1]. \end{aligned}$$
(5.32)
Using the last value for \(f_+(0)\), which attains a physical value of the pion mass, and those of \(|V_{us}| f_+(0)\) and \(V_{ud}\) we have reported, yields \(\Delta _u=-0.00115\pm 0.00040 \pm 0.0043\), where the first (second) error is associated with \(V_{us}\) (\(V_{ud}\)), and roughly a \(2\sigma \) tension with unitarity [1576].
As a final topic we consider the possibility of determining \(V_{us}/V_{ud}\) from the ratio of \(K_{\ell 2 (\gamma )}\) and \(\pi _{\ell 2 (\gamma )}\) decay widths with the use of the decay constant ratio \(f_K/f_\pi \) computed in lattice QCD [1598]. This method competes with the \(K_{\ell 3}\) decays in precision. In a recent development, the isospin-breaking effects which enter can now be computed using lattice-QCD methods as well; the method is based on the expansion of the Euclidean functional integral in the terms of the up-down mass difference [560, 590]. Generally, the separation of isospin-breaking effects in terms of up-down quark mass and electromagnetic contributions is one of convention, because the quark masses themselves accrue electromagnetic corrections which diverge in the ultraviolet [560, 1599]. Technically, however, the pseudoscalar meson decay constants are only defined within pure QCD, so that
$$\begin{aligned} \frac{f_{K^+}}{f_{\pi ^+}} = \frac{f_{K}}{f_{\pi }} \left( 1 + \delta _\mathrm{SU(2)}\right) , \end{aligned}$$
(5.33)
where \(f_K/f_\pi \) are evaluated in the isospin-symmetric (\(m_u=m_d\)) limit. Thus we can crisply compare the ChPT determination of \(\delta _\mathrm{SU(2)}\) [1600] with a completely different non-perturbative method. Namely, noting [1599], we have \(\delta _\mathrm{SU(2)}^\mathrm{ChPT}=-0.0021\pm 0.0006\) [1600], whereas \(\delta _\mathrm{SU(2)}^\mathrm{lattice}=-0.0040\pm 0.0003 \pm 0.0002\) [590] and \(\delta _\mathrm{SU(2)}^\mathrm{lattice}=-0.0027\pm 0.006\) [1575]. Thus tension exists in the various assessments of SU(2)-breaking effects, and it will be interesting to follow future developments.
b.\(B\)and \(D\)form factors Lattice-QCD methods can also be used for the computation of the \(B\) and \(D\) meson form factors in exclusive semileptonic decays, yielding ultimately additional CKM matrix elements once the appropriate partial widths are experimentally determined. Generally, CKM information can be gleaned from both exclusive and inclusive (to a final state characterized by a quark flavor \(q\), as in \(B\rightarrow X_q {\ell } \nu \) decay) \(B\) meson decays, and different theoretical methods figure in each. In the inclusive case, the factorization of soft and hard degrees of freedom is realized using heavy quark effective theory, and the needed non-perturbative ingredients are determined through fits to data. As we have noted, lattice-QCD methods can be employed in the exclusive channels, and the leptonic process \(B\rightarrow \tau \nu \), along with a lattice-QCD computation of the decay constant \(f_B\), also