QCD and strongly coupled gauge theories: challenges and perspectives

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 3–9 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 [2–5], 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) [6–9]. 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 [10–17], 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 [19–21]. 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, 22–24] tend to obtain larger values of ${\alpha _{\mathrm{s}}}$ than those that incorporate power corrections analytically [25–29]. Moreover, it may not be appropriate to use Monte Carlo hadronization with higher-order resummed predictions [25–27]. We also mention that analyses employing jet rates may be less sensitive to hadronization corrections [30–33]. 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. [37–40]). 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 [46–49]) 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) [52–54] 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 [74–77]. 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 [79–83]. 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 [84–86]:

$$\begin{aligned}&\Phi _{ij}^{q[C]}(x, {\mathbf {k}}_{T};n) = \int \frac{d(y\cdot p) \, \mathrm{d}^2 \varvec{y}_{T}}{(2\pi )^3} e^{-ik \cdot y} \nonumber \\&\quad \times \left\langle p| \bar{\psi }_{j}(y)\mathcal {W}(0,y|C)\psi _{i}(0) |p\right\rangle _{y\cdot n=0}, \end{aligned}$$

(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 [87–89]. 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 [92–94] 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 [98–100] 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 [98–100], 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 [102–113], which are based on soft collinear effective theory (SCET). One such framework [108–110] 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 [119–122].

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 [127–129]. 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 [141–145].

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 [149–152], 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, 155–159]) 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 [180–182]. 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 [187–189] 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, 193–197]. 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 [201–204]), and more complex quantities such as GPDs have also been tackled recently [205–210], as reviewed in [211, 212]). Furthermore, several groups have reported lattice results on the strangeness content of the nucleon [213–222], as well as the strangeness contribution to the nucleon spin [223–229]. 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, 235–243]. 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 [244–247], 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 $\langle x\rangle _{u-d}$ [206–208, 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, 251–253]

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, 256–262]. 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, 273–277]. 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 [286–289] 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 [