Physics at the \(e^+ e^-\) linear collider
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Abstract
A comprehensive review of physics at an \(e^+e^-\) linear collider in the energy range of \(\sqrt{s}=92\) GeV–3 TeV is presented in view of recent and expected LHC results, experiments from low-energy as well as astroparticle physics. The report focusses in particular on Higgs-boson, top-quark and electroweak precision physics, but also discusses several models of beyond the standard model physics such as supersymmetry, little Higgs models and extra gauge bosons. The connection to cosmology has been analysed as well.
1 Executive summary
1.1 Introduction
With the discovery of a Higgs boson with a mass of about \(m_H= 125\) GeV based on data runs at the large hadron collider in its first stage at \(\sqrt{s}=7\) and 8 TeV, the striking concept of explaining ‘mass’ as consequence of a spontaneously broken symmetry received a decisive push forward. The significance of this discovery was acknowledged by the award of the Nobel prize for physics to Higgs and Englert in 2013 [1, 2, 3, 4]. The underlying idea of the Brout–Englert–Higgs (BEH) mechanism is the existence of a self-interacting Higgs field with a specific potential. The peculiar property of this Higgs field is that it is non-zero in the vacuum. In other words the Higgs field provides the vacuum with a structure. The relevance of such a field not only for our understanding of matter but also for the history of the universe is obvious.
The discovery of a Higgs boson as the materialisation of the Higgs field was the first important step in accomplishing our present level of understanding of the fundamental interactions of nature and the structure of matter that is adequately described by the standard model (SM). In the SM the constituents of matter are fermions, leptons and quarks, classified in three families with identical quantum properties. The electroweak and strong interactions are transmitted via the gauge bosons described by gauge field theories with the fundamental symmetry group \(SU(3)_C\times SU(2)_L\times U(1)_Y\).
Is there just one Higgs?
Does the Higgs field associated to the discovered particle really cause the corresponding couplings with all particles? Does it provide the right structure of the vacuum?
Is it a SM Higgs (width, couplings, spin)? Is it a pure \({\textit{CP}}\)-even Higgs boson as predicted in the SM, or is it a Higgs boson from an extended Higgs sector, possibly with some admixture of a \({\textit{CP}}\)-odd component? To which model beyond the standard model (BSM) does it point?
The LHC has excellent prospects for the future runs^{1} 2 and 3 where proton–proton beams collide with an energy of \(\sqrt{s}=13\) TeV starting in spring 2015, continued by runs with a foreseen high luminosity upgrade in the following decade [6]. High-energy \(e^+e^-\)-colliders have already been essential instruments in the past to search for the fundamental constituents of matter and establish their interactions. The most advanced design for a future lepton collider is the International Linear Collider (ILC) that is laid out for the energy range of \(\sqrt{s}=90\) GeV–1 TeV [7, 8]. In case a drive beam accelerator technology can be applied, an energy frontier of about 3 TeV might be accessible with the Compact Linear Collider (CLIC) [9, 10].
At an \(e^+e^-\) linear collider (LC) one expects rather clean experimental conditions compared to the conditions at the LHC where one has many overlapping events due to the QCD background from concurring events. A direct consequence is that one does not need any trigger at an LC but can use all data for physics analyses. Due to the collision of point-like particles the physics processes take place at the precisely and well-defined initial energy \(\sqrt{s}\), both stable and measurable up to the per-mille level. The energy at the LC is tunable which offers to perform precise energy scans and to optimise the kinematic conditions for the different physics processes, respectively. In addition, the beams can be polarised: the electron beam up to about 90 %, the positron beam up to about 60 %. With such a high degree of polarisation, the initial state is precisely fixed and well known. Due to all these circumstances the final states are generally fully reconstructable so that numerous observables as masses, total cross sections but also differential energy and angular distributions are available for data analyses.
The quintessence of LC physics at the precision frontier is high luminosity and beam polarisation, tunable energy, precisely defined initial state and clear separation of events via excellent detectors. The experimental conditions that are necessary to fulfil the physics requirements have been defined in the LC scope documents [11].
Such clean experimental conditions for high-precision measurements at a LC are the ‘sine qua non’ for resolving the current puzzles and open questions. They allow one to analyse the physics data in a particularly model-independent approach. The compelling physics case for a LC has been described in numerous publications as, for instance [7, 8, 12, 13, 14, 15, 16], a short and compact overview is given in [17].
Although the SM has been tremendously successful and its predictions experimentally been tested with accuracies at the quantum level, i.e. significantly below the 1-per-cent level, the SM cannot be regarded as the final theory describing all aspects of nature. Astro-physical measurements [18, 19] are consistent with a universe that contains only 4 % of the total energy composed of ordinary mass but hypothesise the existence of dark matter (DM) accounting for 22 % of the total energy that is responsible for gravitational effects although no visible mass can be seen. Models accounting for DM can easily be embedded within BSM theories as, for instance, supergravity [20]. The strong belief in BSM physics is further supported by the absence of gauge coupling unification in the SM as well as its failure to explain the observed existing imbalance between baryonic and antibaryonic matter in our universe. Such facets together with the experimental data strongly support the interpretation that the SM picture is not complete but constitutes only a low-energy limit of an all-encompassing ‘theory of everything’, embedding gravity and quantum theory to describe all physical aspects of the universe. Therefore experimental hints for BSM physics are expected to manifest themselves at future colliders and model-independent strategies are crucial to determine the underlying structure of the model.
A priori there are only two approaches to reveal signals of new physics and to manifest the model of BSM at future experiments. Since the properties of the matter and gauge particles in the SM may be affected by the new energy scales, a ‘bottom-up’ approach consists in performing high precision studies of the top, Higgs and electroweak gauge bosons. Deviations from those measurements to SM predictions reveal hints to BSM physics. Under the assumption that future experiments can be performed at energies high enough to cross new thresholds, a ‘top-down’ approach becomes also feasible where the new particles or interactions can be produced and studied directly.
Obviously, the complementary search strategies at lepton and hadron colliders are predestinated for such successful dual approaches. A successful high-energy LC was already realised in the 1990s with the construction and running of the SLAC Linear Collider (SLC) that delivered up to \(5\times 10^{10}\) particles per pulse. Applying in addition highly polarised electrons enabled the SLC to provide the best single measurement of the electroweak mixing angle with \(\delta \sin ^2\theta _W \sim 0.00027\).
However, such a high precision manifests a still-existing inconsistency, namely the well-known discrepancy between the left–right polarisation asymmetry at the Z-pole measured at SLC and the forward–backward asymmetry measured at LEP [21]. Both values lead to measured values of the electroweak mixing angle \(\sin ^2\theta _\mathrm{eff}\) that differ by more than 3\(\sigma \) and point to different predictions for the Higgs mass, see Sect. 4 for more details. Clarifying the central value as well as improving the precision is essential for testing the consistence of the SM as well as BSM models.
Another example for the relevance of highest precision measurements and their interplay with most accurate theoretical predictions at the quantum level is impressively demonstrated in the interpretation of the muon anomalous moment \(g_{\mu }-2\) [22]. The foreseen run of the \(g_{\mu }-2\) experiment at Fermilab, starting in 2017 [23, 24], will further improve the current experimental precision by about a factor of 4 and will set substantial bounds to many new physics models via their high sensitivity to virtual effects of new particles.
The detectors are designed to improve the momentum resolution from tracking by a factor 10 and the jet-energy resolution by a factor 3 (in comparison with the CMS detector) and excellent \(\tau ^{\pm }\)-, b-, \(\bar{b}\)- and c, \(\bar{c}\)-tagging capabilities [8], are expected.
As mentioned before, another novelty is the availability of the polarisation of both beams, which can precisely project out the interaction vertices and can analyse its chirality directly.
The experimental conditions to achieve such an unprecedented precision frontier at high energy are high luminosity (even about three orders of magnitude more particles per pulse, \(5\times 10^{13}\) than at the SLC), polarised electron/positron beams, tunable energy, luminosity and beam-energy stability below \(0.1\,\%\) level [11]. Assuming a finite total overall running time it is a critical issue to divide up the available time between the different energies, polarisations and running options in order to maximise the physical results. Several running scenarios are thoroughly studied [27].
In the remainder of this chapter we summarise the physics highlights of this report. The corresponding details can be found in the following chapters. Starting with the three safe pillars of LC physics – Higgs-, top- and electroweak high precision physics – Sect. 2 provides a comprehensive overview about the physics of EWSB. Recent developments in LHC analyses as well as on the theory side are included, alternatives to the Higgs models are discussed. Section 3 covers QCD and in particular top-quark physics. The LC will also set a new frontier in experimental precision physics and has a striking potential for discoveries in indirect searches. In Sect. 4 the impact of electroweak precision observables (EWPO) and their interpretation within BSM physics are discussed. Supersymmetry (SUSY) is a well-defined example for physics beyond the SM with high predictive power. Therefore in Sect. 5 the potential of a LC for unravelling and determining the underlying structure in different SUSY models is discussed. Since many aspects of new physics have strong impact on astroparticle physics and cosmology, Sect. 6 provides an overview in this regard.
The above-mentioned safe physics topics can be realised at best at different energy stages at the linear collider. The possible staged energy approach for a LC is therefore ideally suited to address all the different physics topics. For some specific physics questions very high luminosity is required and in this context also a high-luminosity option at the LC is discussed, see [27] for technical details. The expected physics results of the high-luminosity LC was studied in different working group reports [28, 29], cf. Sect. 2.3.
Such an optimisation of the different running options of a LC depends on the still awaited physics demands. The possible physics outcome of different running scenarios at the LC are currently under study [27], but fixing the final running strategy is not yet advisable.
One should note, however, that such a large machine flexibility is one of the striking features of a LC.
1.2 Physics highlights
Many of the examples shown in this review are based on results of [8, 9, 10, 30, 31] and references therein.
1.2.1 Higgs physics
The need for precision studies of the new boson, compatible with a SM-like Higgs, illuminates already the clear path for taking data at different energy stages at the LC.
For a Higgs boson with a mass of 125 GeV, the first envisaged energy stage is at about \(\sqrt{s}=250\) GeV: the dominant Higgs-strahlung process peaks at \(\sqrt{s}=240\) GeV. This energy stage allows the model-independent measurement of the cross section \(\sigma (HZ)\) with an accuracy of about 2.6 %, cf. Sect. 2.3. This quantity is the crucial ingredient for all further Higgs analyses, in particular for deriving the total width via measuring the ratio of the partial width and the corresponding branching ratio. Already at this stage many couplings can be determined with high accuracy in a model-independent way: a striking example is the precision of 1.3 % that can be expected for the coupling \(g_{HZZ}\), see Sect. 2.3 for more details.
The precise determination of the mass is of interest in its own right. However, it has also high impact for probing the Higgs physics, since \(m_H\) is a crucial input parameter. For instance, the branching ratios \(H\rightarrow ZZ^*\), \(WW^*\) are very sensitive to \(m_H\): a change in \(m_H\) by 200 MeV shifts \(\mathrm{BR}(H\rightarrow ZZ^*\) by 2.5 %. Performing accurate threshold scans enables the most precise mass measurements of \(\delta m_H=40\) MeV. Furthermore and – of more fundamental relevance – such threshold scans in combination with measuring different angular distributions allow a model-independent and unique determination of the spin.
Another crucial quantity in the Higgs sector is the total width \({\varGamma }_H\) of the Higgs boson. The prediction in the SM is \({\varGamma }_H=4.07\) MeV for \(m_H=125\) GeV [32]. The direct measurement of such a small width is neither possible at the LHC nor at the LC since it is much smaller than any detector resolution. Nevertheless, at the LC a model-independent determination of \({\varGamma }_H\) can be achieved using the absolute measurement of Higgs branching ratios together with measurements of the corresponding partial widths. An essential input quantity in this context is again the precisely measured total cross section of the Higgs-strahlung process. At \(\sqrt{s}=500\) GeV, one can derive the total width \({\varGamma }_H\) with a precision of 5 % based on a combination of the \(H\rightarrow ZZ^*\) and \(WW^*\) channels. Besides this model-independent determination, which is unique to the LC, constraints on the total width can also be obtained at the LC from a combination of on- and off-shell Higgs contributions [33] in a similar way as at the LHC [34]. The latter method, however, relies on certain theoretical assumptions, and also in terms of the achievable accuracy it is not competitive with the model-independent measurement based on the production cross section \(\sigma (ZH)\) [33].
At higher energy such off-shell decays of the Higgs boson to pairs of W and Z bosons offer access to the kinematic dependence of higher-dimensional operators involving the Higgs boson. This dependence allows for example the test of unitarity in BSM models [35, 36].
Another very crucial quantity is accessible at \(\sqrt{s}=500\) GeV: the \(t\bar{t}H\)-coupling. Measuring the top-Yukawa coupling is a challenging endeavour since it is overwhelmed from \(t\bar{t}\)-background. At the LHC one expects an accuracy of 25 % on basis of 300 fb\(^{-1}\) and under optimal assumptions and neglecting the error from theory uncertainties. At the LC already at the energy stage of \(\sqrt{s}=500\) GeV, it is expected to achieve an accuracy of \({\varDelta } g_{ttH}/g_{ttH}\sim 10\) %, see Sect. 2. This energy stage is close to the threshold of ttH production, therefore the cross section for this process should be small. But thanks to QCD-induced threshold effects the cross section gets enhanced and such an accuracy should be achievable with 1 ab\(^{-1}\) at the LC. It is of great importance to measure this Yukawa coupling with high precision in order to test the Higgs mechanism and verify the measured top mass \(m_t=y_{ttH} v/\sqrt{2}\). The precise determination of the top Yukawa coupling opens a sensitive window to new physics and admixtures of non-SM contributions. For instance, in the general two-Higgs-doublet model the deviations with respect to the SM value of this coupling can typically be as large as \(\sim 20\,\%\).
Another important property of the Higgs boson that has to be determined is the \({\textit{CP}}\) quantum number. In the SM the Higgs should be a pure \({\textit{CP}}\)-even state. In BSM models, however, the observed boson state a priori can be any admixture of \({\textit{CP}}\)-even and \({\textit{CP}}\)-odd states, it is of high interest to determine limits on this admixture. The HVV couplings project out only the \({\textit{CP}}\)-even components, therefore the degree of \({\textit{CP}}\) admixture cannot be tackled via analysing these couplings. The measurements of \({\textit{CP}}\)-odd observables are mandatory to reveal the Higgs \({\textit{CP}}\)-properties: for instance, the decays of the Higgs boson into \(\tau \) leptons provides the possibility to construct unique \({\textit{CP}}\)-odd observables via the polarisation vector of the \(\tau \)s, see further details in Sect. 2.
1.2.2 Top-quark physics
1.2.3 Beyond standard model physics – “top-down”
Supersymmetry The SUSY concept is one of the most popular extensions of the SM since it can close several open questions of the SM: achieving gauge unification, providing DM candidates, stabilising the Higgs mass, embedding new sources for \({\textit{CP}}\)-violation and also potentially neutrino mixing. However, the symmetry has to be broken and the mechanism for symmetry breaking is completely unknown. Therefore the most general parametrisation allows around 100 new parameters. In order to enable phenomenological interpretations, for instance, at the LHC, strong restrictive assumptions on the SUSY mass spectrum are set. However, as long as it is not possible to describe the SUSY breaking mechanism within a full theory, data interpretations based on these assumptions should be regarded as a pragmatic approach. Therefore the rather high limits obtained at the LHC for some coloured particles exclude neither the concept of SUSY as such, nor do they exclude light electroweak particles, nor relatively light scalar quarks of the third generation.
Already the energy stage at \(\sqrt{s}=350\) GeV provides a representative open window for the direct production of light SUSY particles, for instance, light higgsino-like scenarios, leading to signatures with only soft photons. The resolution of such signatures will be extremely challenging at the LHC but is feasible at the LC via the ISR method, as discussed in Sect. 5.
Dark matter physics Weakly interacting massive particles (WIMPs) are the favourite candidates as components of the cold DM. Neutral particles that interact only weakly provide roughly the correct relic density in a natural way. Since there are no candidates for DM in the SM, the strong observational evidence for DM clearly points to physics beyond the SM. Due to precise results from cosmological observations, for instance [46, 47], bounds on the respective cross section and the mass of the DM candidates can be set in the different models. Therefore, in many models only rather light candidates are predicted, i.e. with a mass around the scale of EWSB or even lighter. That means, for instance for SUSY models with R-parity conservation, that the lightest SUSY particle, should be within the kinematical reach of the ILC. The lowest threshold for such processes is pair production of the WIMP particle. Since such a final state, however, escapes detection, the process is only visible if accompanied by radiative photons at the LC that recoil against the WIMPs, for instance, the process \(e^+e^-\rightarrow \gamma \chi \chi \) [48], where \(\chi \) denotes the WIMP particle in general with a spin \(S_{\chi }=0,\frac{1}{2},1\). Such a process can be realised in SUSY models, in universal extra dimensions, little Higgs theories etc. The dominant SM background is radiative neutrino production, which can, efficiently be suppressed via the use of beam polarisation.
This direct relation between neutrino physics and high-energy physics is striking. It allows one to directly test whether the measured neutrino mixing angles can be embedded within a theoretical model of high predictive power, namely a bi-linear R-parity violation model in SUSY, based on precise measurements of neutralino branching ratios [53, 54] at a future \(e^+e^-\) linear collider.
1.2.4 Beyond standard model physics – “bottom-up”
1.2.5 Synopsis
\(\sqrt{s}/\)GeV | 92,160 | 240 | 350 | 500 | 1000 | 3000 | Threshold scans required |
---|---|---|---|---|---|---|---|
Higgs | |||||||
\(m_H\) | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) |
\({\varGamma }_{\mathrm{tot}}\) | – | – | \(\times \) | \(\times \) | |||
\(g_{c,b}\) | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | ||
\(g_{ttH}\) | – | – | – | \(\times \) | \(\times \) | ||
\(g_{HHH}\) | – | – | – | \(\times \) | \(\times \) | \(\times \) | |
\(m_{H,A}^{\mathrm{SUSY}}\) | – | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) |
Top | |||||||
\(m_{t}^{\mathrm{th}}\) | – | – | \(\times \) | \(\times \) | |||
\(m_{t}^{\mathrm{cont}}\) | – | – | – | \(\times \) | (\(\times \)) | (\(\times \)) | |
\(A_{\mathrm{FB}}^t\) | – | – | \(\times \) | \(\times \) | |||
\(g_{Z,\gamma }\) | – | – | – | \(\times \) | |||
\(g_{FCNC}\) | – | – | – | \(\times \) | \(\times \) | (?) | |
Electroweak precision observables | |||||||
\(\sin ^2\theta _\mathrm{eff}\)(Z-pole) | \(\times \) | (\(\times \)) | |||||
\(m_W^{\mathrm{th}}\) | \(\times \) | \(\times \) | |||||
\(m_W^{\mathrm{cont}}\) | \(\times \) | \(\times \) | \(\times \) | (\(\times \)) | (\(\times \)) | ||
\({\varGamma }_Z\) | \(\times \) | \(\times \) | |||||
\(A_{\mathrm{LR}}\) | \(\times \) | ||||||
\(A_{\mathrm{FB}}\) | \(\times \) | ||||||
SUSY | |||||||
Indirect search | \(\times \) | \(\times \) | \(\times \) | ||||
Direct search | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | \(\times \) |
Light higgsinos | – | – | \(\times \) | \(\times \) | \(\times \) | ||
Parameter determination | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | |
Quantum numbers | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | |
Extrapolations | – | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) |
\(\nu \) mixing | |||||||
\(\theta ^2_{23}\) | – | – | \(\times \) | \(\times \) | |||
Dark matter | |||||||
Effective-field-theory | – | – | – | \(\times \) | \(\times \) | \(\times \) | |
Non-relativistic | – | – | \(\times \) | \(\times \) | \(\times \) | \(\times \) | |
Extra gauge bosons | |||||||
Indirect search \(m_{z'}\) | \(\times \) | – | – | \(\times \) | \(\times \) | \(\times \) | |
\(v'_f\), \(a'_f\) | – | – | – | \(\times \) | \(\times \) | (\(\times \)) | |
\(m_{W'}\) | \(\times \) | – | – | \(\times \) | \(\times \) | \(\times \) | |
Direct search | – | – | – | – | – | \(\times \) | \(\times \) |
2 Higgs and electroweak symmetry breaking^{3}
After a brief description of the physical basis of the Higgs mechanism, we summarise the crucial results for Higgs properties in the standard model as expected from measurements at LHC and ILC/CLIC, based on the respective reports. Extensions of the SM Higgs sector are sketched thereafter, discussed thoroughly in the detailed reports which follow: portal models requiring analyses of invisible Higgs decays, supersymmetry scenarios as generic representatives of weakly coupled Higgs sectors, and finally strong interaction elements as suggested by Little Higgs models and composite models motivated by extended space dimensions.
2.1 Résumé^{5}
The Brout–Englert–Higgs mechanism [1, 2, 3, 4, 57] is a central element of particle physics. Masses are introduced consistently in gauge theories for vector bosons, leptons and quarks, and the Higgs boson itself, by transformation of the interaction energy between the initially massless fields and the vacuum expectation value of the Higgs-field. The non-zero value of the Higgs field in the vacuum, at the minimum of the potential breaking the electroweak symmetry, is generated by self-interactions of the Higgs field. The framework of the SM [58, 59, 60] demands the physical Higgs boson as a new scalar degree of freedom, supplementing the spectrum of vectorial gauge bosons and spinorial matter particles.
This concept of mass generation has also been applied, mutatis mutandis, to extended theories into which the SM may be embedded. The new theory may remain weakly interacting up to the grand-unification scale, or even the Planck scale, as familiar in particular from supersymmetric theories, or novel strong interactions may become effective already close to the TeV regime. In such theories the Higgs sector is enlarged compared with the SM. A spectrum of several Higgs particles is generally predicted, the lightest particle often with properties close to the SM Higgs boson, and others with masses typically in the TeV regime.
A breakthrough on the path to establishing the Higgs mechanism experimentally has been achieved by observing at LHC [61, 62] a new particle with a mass of about 125 GeV and couplings to electroweak gauge bosons and matter particles compatible, cum grano salis, with expectations for the Higgs boson in the (SM) [63, 64, 65, 66].
2.1.1 Zeroing in on the Higgs particle of the SM
The mass, the lifetime (width) and the spin/\({\textit{CP}}\) quantum numbers must be measured as general characteristics of the particle;
The couplings of the Higgs particle to electroweak gauge bosons and to leptons/quarks must be proven to rise (linearly) with their masses;
The self-coupling of the Higgs particle, responsible for the potential which generates the non-zero vacuum value of the Higgs field, must be established.
Since the Higgs mechanism provides the closure of the SM, the experimental investigation of the mechanism, connected with precision measurements^{6} of the properties of the Higgs particle, is mandatory for the understanding of the microscopic laws of nature as formulated at the electroweak scale. However, even though the SM is internally consistent, the large number of parameters, notabene mass and mixing parameters induced in the Higgs sector, suggests the embedding of the SM into a more comprehensive theory (potentially passing on the way through even more complex structures). Thus observing specific patterns in the Higgs sector could hold essential clues to this underlying theory.
Cross sections in units of fb for Higgs-strahlung and W-boson fusion of Higgs bosons in the SM for a set of typical ILC/CLIC energies with beam polarisations: \(P(e^-,e^+)=(-0.8,+0.3)\) for ILC at 250 and 500 GeV, \((-0.8,+0.2)\) for ILC at 1 TeV, and \((-0.8,0)\) for CLIC at 3 TeV
250 GeV | 500 GeV | 1 TeV | 3 TeV | |
---|---|---|---|---|
\(\sigma [e^+e^- \rightarrow ZH]\) | 318 | 95.5 | 22.3 | 2.37 |
\(\sigma [e^+e^- \rightarrow \bar{\nu }_e \nu _e H]\) | 36.6 | 163 | 425 | 862 |
(a) Higgs particle: mass and \(J^{{\textit{CP}}}\)
Already for quite some time, precision analyses of the electroweak parameters, like the \(\rho \)-parameter, suggested an SM Higgs mass of less than 161 GeV in the intermediate range [21], above the lower LEP2 limit of 114.4 GeV [72] (for a review see [73]). The mass of the new particle observed close to 125 GeV at LHC, agrees nicely with this expectation.
(b) Higgs couplings to SM particles
Expected accuracy with which fundamental and derived Higgs couplings can be measured; the deviations are defind as \(\kappa :=g/g_{\mathrm{SM}}=1\pm {\varDelta }\) compared to the SM at the LHC/HL-LHC, LC and in combined analyses of the HL-LHC and LC [29]. The fit assumes generation universality: \(\kappa _u\equiv \kappa _c\equiv \kappa _t\), \(\kappa _d\equiv \kappa _s\equiv \kappa _b\), and \(\kappa _\mu \equiv \kappa _\tau \). The 95 % CL upper limit of potential couplings to invisible channels is also given
Coupling | LHC (%) | HL-LHC (%) | LC (%) | HL-LHC \(+\) LC (%) |
---|---|---|---|---|
HWW | 4–6 | 2–5 | 0.3 | 0.1 |
HZZ | 4–6 | 2–4 | 0.5 | 0.3 |
Htt | 14–15 | 7–10 | 1.3 | 1.3 |
Hbb | 10–13 | 4–7 | 0.6 | 0.6 |
\(H\tau \tau \) | 6–8 | 2–5 | 1.3 | 1.2 |
\(H\gamma \gamma \) | 5–7 | 2–5 | 3.8 | 3.0 |
Hgg | 6–8 | 3–5 | 1.2 | 1.1 |
\(H \mathrm{invis}\) | – | – | 0.9 | 0.9 |
A special role is played by the loop-induced \(\gamma \gamma \) width which can most accurately be measured by Higgs fusion-formation in a photon collider.
Potential deviations of the couplings from the SM values can be attributed to the impact of physics beyond the SM. Parameterizing these effects, as naturally expected in dimensional operator expansions, by \(g_H = g_H^{\mathrm{SM}} [1 + v^2 / {{\Lambda }}^2_*]\), the BSM scale is estimated to \({\Lambda }_*>\) 550 GeV for an accuracy of 20 % in the measurement of the coupling, and 2.5 TeV for 1 %, see also [78]. The shift in the coupling can be induced either by mixing effects or by loop corrections to the Higgs vertex. Such mixing effects are well known in the supersymmetric Higgs sector where in the decoupling limit the mixing parameters in the Yukawa vertices approach unity as \(\sim v^2/m^2_A\). Other mixing effects are induced in Higgs-portal models and strong interaction Higgs models with either universal or non-universal shifts of the couplings at an amount \(\xi = (v/f)^2\), which is determined by the Goldstone scale f of global symmetry breaking in the strong-interaction sector; with \(f \sim 1\) TeV, vertices may be modified up to the level of 10 %. Less promising is the second class comprising loop corrections of Higgs vertices. Loops, generated for example by the exchange of new \(Z'\)-bosons, are suppressed by the numerical coefficient \(4\pi ^2\) (reduced in addition by potentially weak couplings). Thus the accessible mass range, \(M < {\Lambda }_*/ 2\pi \sim \) 250 GeV, can in general be covered easily by direct LHC searches.
(c) Higgs self-couplings
(d) Invisible Higgs decays
The observation of cold DM suggests the existence of a hidden sector with a priori unknown, potentially high complexity. The Higgs field of the SM can be coupled to a corresponding Higgs field in the hidden sector, \(\tilde{\mathscr {V}} = \eta |\phi _{\mathrm{SM}}|^2 |\phi _{hid}|^2\), in a form compatible with all standard symmetries. Thus a portal could be opened from the SM to the hidden sector [80, 81]. Analogous mixing with radions is predicted in theories incorporating extra-space dimensions. The mixing of the Higgs fields in the two sectors induces potentially small universal changes in the observed Higgs couplings to the SM particles and, moreover, Higgs decays to invisible hidden states (while this channel is opened in the canonical SM only indirectly by neutrino decays of Z pairs). Both signatures are a central target for experimentation at LC, potentially allowing the first sighting of a new world of matter in the Higgs sector.
In summary, essential elements of the Higgs mechanism in the SM can be determined at \(e^+ e^-\) linear colliders in the 250 to 500 GeV and 1 to 3 TeV modes at high precision. Improvements on the fundamental parameters by nearly an order of magnitude can be achieved in such a faciliy. Thus a fine-grained picture of the Higgs sector as third component of the SM can be drawn at a linear collider, completing the theory of matter and forces at the electroweak scale. First glimpses of a sector beyond the SM are possible by observing deviations from the SM picture at scales far beyond those accessible at colliders directly.
2.1.2 Supersymmetry scenarios
The hypothetical extension of the SM to a supersymmetric theory [82, 83] is intimately connected with the Higgs sector. If the SM is embedded in a grand unified scenario, excessive fine tuning in radiative corrections would be needed to keep the Higgs mass near the electroweak scale, i.e. 14 orders of magnitude below the grand-unification scale. A stable bridge can be constructed, however, in a natural way if matter and force fields are assigned to fermion–boson symmetric multiplets with masses not spread more than order TeV. In addition, by switching the mass (squared) of a scalar field from positive to negative value when evolved from high to low scales, supersymmetry offers an attractive physical explication of the Higgs mechanism. It should be noted that supersymmetrisation of the SM is not the only solution of the hierarchy problem, however, it joins in nicely with arguments of highly precise unification of couplings, the approach to gravity in local supersymmetry, and the realisation of cold DM. Even though not yet backed at present by the direct experimental observation of supersymmetric particles, supersymmetry remains an attractive extension of the SM, offering solutions to a variety of fundamental physical problems.
To describe the Higgs interaction with matter fields by a superpotential, and to keep the theory anomaly-free, at least two independent Higgs iso-doublets must be introduced, coupling separately to up- and down-type matter fields. They are extended eventually by additional scalar superfields, etc.
(a) Minimal supersymmetric model MSSM
Extending the SM fields to super-fields and adding a second Higgs doublet defines the minimal supersymmetric standard model (MSSM). After gauge symmetry breaking, three Goldstone components out of the eight scalar fields are aborbed to provide masses to the electroweak gauge bosons while five degrees of freedom are realised as new physical fields, corresponding to two neutral \({\textit{CP}}\)-even scalar particles \(h^0,H^0\); one neutral \({\textit{CP}}\)-odd scalar particle \(A^0\); and a pair of charged \(H^\pm \) scalar particles [84, 85, 86, 87].
Since the quadri-linear Higgs couplings are pre-determined by the (small) gauge couplings, the mass of the lightest Higgs particle is small. The bound, \(M_{h^0} < M_Z | \cos 2\beta |\) at lowest order, with \(\tan \beta \) accounting for Goldstone–Higgs mixing, is significantly increased, however, to \(\sim \)130 GeV by radiative corrections, adding a contribution of order \(3 M^4_t/2 \pi ^2 v^2\, \log M^2_{\tilde{t}}/M^2_t + mix\) for large top and stop masses. To reach a value of 125 GeV, large stop masses and/or large tri-linear couplings are required in the mixings.
Predictions for production and decay amplitudes deviate, in general, from the SM not only because of modified tree couplings but also due to additional loop contributions, as \(\tilde{\tau }\) loops in the \(\gamma \gamma \) decay mode of the lightest Higgs boson.
To accommodate a 125-GeV Higgs boson in minimal supergravity the quartet of heavy Higgs particles \(H^0,A^0,H^\pm \) is shifted to the decoupling regime with order TeV masses. The properties of the lightest Higgs boson \(h^0\) are very close in this regime to the properties of the SM Higgs boson.
Additional channels open in single Higgs production \(\gamma \gamma \rightarrow A^0,H^0\), completely exhausting the multi-TeV energy potential \({\sqrt{s}}_{\gamma \gamma }\) of a photon collider.
(b) Extended supersymmetry scenarios
The minimal supersymmetry model is quite restrictive by connecting the quadri-linear couplings with the gauge couplings, leading naturally to a small Higgs mass, and grouping the heavy Higgs masses close to each other. The simplest extension of the system introduces an additional iso-scalar Higgs field [88, 89], the next-to-minimal model (NMSSM). This extension augments the Higgs spectrum by two additional physical states, \({\textit{CP}}\)-even and \({\textit{CP}}\)-odd, which mix with the corresponding MSSM-type states.
The bound on the mass of the lightest MSSM Higgs particle is alleviated by contributions from the tri-linear Higgs couplings in the superpotential (reducing the amount of ‘little fine tuning’ in this theory). Loop contributions to accommodate a 125-GeV Higgs boson are reduced so that the bound on stop masses is lowered to about 100 GeV as a result.
The additional parameters in the NMSSM render the predictions for production cross sections and decay branching ratios more flexible, so that an increased rate of \(pp \rightarrow \mathrm{Higgs} \rightarrow \gamma \gamma \), for instance, can be accomodated more easily than within the MSSM.
Motivations for many other extensions of the Higgs sector have been presented in the literature. Supersymmetry provides an attractive general framework in this context. The new structures could be so rich that the clear experimental environment of \(e^+ e^-\) collisions is needed to map out this Higgs sector and to unravel its underlying physical basis.
2.1.3 Composite Higgs bosons
Not long after pointlike Higgs theories had been introduced to generate the breaking of the electroweak symmetries, alternatives have been developed based on novel strong interactions [90, 91]. The breaking of global symmetries in such theories gives rise to massless Goldstone bosons which can be absorbed by gauge bosons to generate their masses. This concept had been expanded later to incorporate also light Higgs bosons with mass in the intermediate range. Generic examples for such theories are Little Higgs Models and theories formulated in higher dimensions, which should be addressed briefly as generic examples.
(a) Little Higgs models
If new strong interactions are introduced at a scale of a few 10 TeV, the breaking of global symmetries generates a Goldstone scale f typically reduced by one order of magnitude, i.e. at a few TeV. The spontaneous breaking of large global groups leads to an extended scalar sector with Higgs masses generated radiatively at the Goldstone scale. The lightest Higgs mass is delayed, by contrast, acquiring mass at the electroweak scale only through collective symmetry breaking at higher oder.
Such a scenario [92] can be realised, for instance, in minimal form as a non-linear sigma model with a global SU(5) symmetry broken down to SO(5). After separating the Goldstone modes which provide masses to gauge bosons, ten Higgs bosons emerge in this scenario which split into an isotriplet \(\Phi \), including a pair of doubly charged \(\Phi ^{\pm \pm }\) states with TeV-scale masses, and the light standard doublet h. The properties of h are affected at the few per-cent level by the extended spectrum of the fermion and gauge sectors. The new TeV triplet Higgs bosons with doubly charged scalars can be searched for very effectively in pair production at LC in the TeV energy range.
(b) Relating to higher dimensions
An alternative approach emerges out of gauge theories formulated in five-dimensional anti-de-Sitter space. The AdS/CFT correspondence relates this theory to a four-dimensional strongly coupled theory, the fifth components of the gauge fields interpreted as Goldstone modes in the strongly coupled four-dimensional sector. In this picture the light Higgs boson appears as a composite state with properties deviating to order \((v/f)^2\) from the standard values [93], either universally or non-universally with alternating signs for vector bosons and fermions.
2.2 The SM Higgs at the LHC: status and prospects^{7}
In July 2012 the ATLAS and CMS experiments at the LHC announced the discovery of a new particle with a mass of about 125 GeV that provided a compelling candidate for the Higgs boson in the framework of the standard model of particle physics (SM). Both experiments found consistent evidence from a combination of searches for three decay modes, \(H\rightarrow \gamma \gamma \), \(H\rightarrow ZZ\rightarrow 4l\) and \(H\rightarrow WW\rightarrow 2 l2\nu \) (\(l=e,\mu \)), with event rates and properties in agreement with SM predictions for Higgs-boson production and decay. These findings, which were based on proton–proton collision data recorded at centre-of-mass energies of 7 and 8 TeV and corresponding to an integrated luminosity of about 10 fb\(^{-1}\) per experiment, received a lot of attention both within and outside the particle physics community and were eventually published in [62, 94, 95, 96].
Since then, the LHC experiments have concluded their first phase of data taking (“Run1”) and significantly larger datasets corresponding to about 25 fb\(^{-1}\) per experiment have been used to perform further improved analyses enhancing the signals in previously observed decay channels, establishing evidence of other decays and specific production modes as well as providing more precise measurements of the mass and studies of other properties of the new particle. Corresponding results, some of them still preliminary, form the basis of the first part of this section, which summarises the status of the ATLAS and CMS analyses of the Higgs boson candidate within the SM.
The second part gives an outlook on Higgs-boson studies during the second phase (“Run2”) of the LHC operation scheduled to start later this year and the long-term potential for an upgraded high-luminosity LHC.
2.2.1 Current status
The initial SM Higgs-boson searches at the LHC were designed for a fairly large Higgs mass window between 100 and 600 GeV, most of which was excluded by the ATLAS and CMS results based on the data sets recorded in 2011 [99, 100]. In the following we focus on the analyses including the full 2012 data and restrict the discussion to decay channels relevant to the discovery and subsequent study of the 125 GeV Higgs boson.
\(H\rightarrow \gamma \gamma \): the branching fraction is very small but the two high-energy photons provide a clear experimental signature and a good mass resolution. Relevant background processes are diphoton continuum production as well as photon-jet and dijet events. The most recent ATLAS [101] and CMS [104] analyses yield signals with significances of \(5.2\sigma \) and \(5.7\sigma \), respectively, where \(4.6\sigma \) and \(5.2\sigma \) are expected.
\(H\rightarrow ZZ\rightarrow 4\ell \): also this decay combines a small branching fraction with a clear experimental signature and a good mass resolution. The selection of events with two pairs of isolated, same-flavour, opposite-charge electrons or muons results in the largest signal-to-background ratio of all currently considered Higgs-boson decay channels. The remaining background originates mainly from continuum ZZ, Z+jets and \(t\bar{t}\) production processes. ATLAS [105] and CMS [102] report observed (expected) signal significances of \(8.1\sigma \) (\(6.2\sigma \)) and \(6.8\sigma \) (\(6.7\sigma \)).
\(H\rightarrow WW\rightarrow 2\ell 2\nu \): the main advantage of this decay is its large rate, and the two oppositely charged leptons from the W decays provide a good experimental handle. However, due to the two undetectable final-state neutrinos it is not possible to reconstruct a narrow mass peak. The dominant background processes are WW, Wt, and \(t\bar{t}\) production. The observed (expected) ATLAS [103] and CMS [106] signals have significances of \(6.1\sigma \) (\(5.8\sigma \)) and \(4.3\sigma \) (\(5.8\sigma \)).
\(H\rightarrow bb\): for a Higgs-boson mass of 125 GeV this is the dominant Higgs-boson decay mode. The experimental signature of b quark jets alone is difficult to exploit at the LHC, though, so that current analyses focus on the Higgs production associated with a vector boson Z or W. Here, diboson, vector boson+jets and top production processes constitute the relevant backgrounds.
\(H\rightarrow \tau \tau \): all combinations of hadronic and leptonic \(\tau \)-lepton decays are used to search for a broad excess in the \(\tau \tau \) invariant mass spectrum. The dominant and irreducible background is coming from \(Z\rightarrow \tau \tau \) decays; further background contributions arise from processes with a vector boson and jets, top and diboson production.
In the following, we summarise the status of SM Higgs boson analyses of the full 2011/2012 datasets with ATLAS and CMS. The discussion is based on preliminary combinations of ATLAS and published CMS results collected in [112, 113], respectively; an ATLAS publication of Higgs-boson mass measurements [114]; ATLAS [115] and CMS [116] constraints on the Higgs boson width; studies of the Higgs boson spin and parity by CMS [117] and ATLAS [65, 118, 119]; and other results on specific aspects or channels referenced later in this section.
ATLAS and CMS have also studied the relative contributions from production mechanisms mediated by vector bosons (VBF and VH processes) and gluons (ggF and ttH processes), respectively. For example, Fig. 19 shows ATLAS results constituting a 4.3\(\sigma \) evidence that part of the Higgs-boson production proceeds via VBF processes [112].
Couplings to other particles The Higgs-boson couplings to other particles enter the observed signal strengths via both the Higgs production and decay. Leaving other SM characteristics unchanged, in particular assuming the observed Higgs-boson candidate to be a single, narrow, \({\textit{CP}}\)-even scalar state, its couplings are tested by introducing free parameters \(\kappa _X\) for each particle X, such that the SM predictions for production cross sections and decay widths are modified by a multiplicative factor \(\kappa ^2_X\). This includes effective coupling modifiers \(\kappa _{g}\), \(\kappa _\gamma \) for the loop-mediated interaction with gluons and photons. An additional scale factor modifies the total Higgs boson width by \(\kappa ^2_H\).
Summaries of CMS results [113] from such coupling studies are presented in Fig. 23. Within each of the specific sets of assumptions, consistency with the SM expectation is found. Corresponding studies by CMS [113] yield the same conclusions. It should be noted, however, that this does not yet constitute a complete, unconstrained analysis of the Higgs-boson couplings.
Mass Current measurements of the Higgs-boson mass are based on the two high-resolution decay channels \(H\rightarrow \gamma \gamma \) and \(H\rightarrow ZZ\rightarrow 4\ell \). Based on fits to the invariant diphoton and four-lepton mass spectra, ATLAS measures [114] \(m_H=125.98\pm 0.42{\mathrm {(stat)}}\pm 0.28{\mathrm {(sys)}}\) and \(m_H=124.51\pm 0.52{\mathrm {(stat)}}\pm 0.06{\mathrm {(sys)}}\), respectively. A combination of the two results, which are consistent within 2.0 standard deviations, yields \(m_H=125.36\pm 0.37{\mathrm {(stat)}}\pm 0.18{\mathrm {(sys)}}.\) An analysis [113] of the same decays by CMS finds consistency between the two channels at 1.6\(\sigma \); see Fig. 25. The combined result \(m_H=125.02^{+0.26}_{-0.27}{\mathrm {(stat)}}^{+0.14}_{-0.15}{\mathrm {(sys)}}\) agrees well with the corresponding ATLAS measurement.
Other decay channels currently do not provide any significant contributions to the overall mass precision but they can still be used for consistency tests. For example, CMS obtains \(m_H=128^{+7}_{-5}\) and \(m_H=122\pm 7\) GeV from the analysis of WW [106] and \(\tau \tau \) [109] final states, respectively.
Width Information on the decay width of the Higgs boson obtained from the above mass measurements is limited by the experimental resolution to about 2 GeV, whereas the SM prediction for \({\varGamma }_H\) is about 4 MeV.
Spin and parity Within the SM, the Higgs boson is a spin-0, \({\textit{CP}}\)-even particle. Since the decay kinematics depend on these quantum numbers, the \(J^P=0^+\) nature of the SM Higgs boson can be used as constraint to increase the sensitivity of the SM analyses. After dropping such assumptions, however, these analyses can also be used to test against alternative spin–parity hypotheses. These studies are currently based on one or several of the bosonic decays modes discussed above: \(H\rightarrow \gamma \gamma \), \(H\rightarrow ZZ\rightarrow 4\ell \), and \(H\rightarrow WW\rightarrow 2\ell 2\nu \).
Including the spin-1 hypotheses in the analyses of the decays into vector bosons provides a test independent of the \(H\rightarrow \gamma \gamma \) channel, where \(J = 1\) is excluded by the Landau–Yang theorem, and implies the assumptions that the signals observed in the two-photon and VV final states are not originating from a single resonance. A representative sample of spin-2 alternatives to SM hypothesis is considered, also including different assumptions concerning the dominant production mechanisms.
For example, Fig. 29 shows the results obtained from CMS analyses of the \(H\rightarrow ZZ\rightarrow 4\ell \) and \(H\rightarrow WW\rightarrow 2\ell 2\nu \) channels [117]. Agreement with the SM (\(J^P=0^+\)) within \(1.5\sigma \) and inconsistency with alternative hypotheses at a level of at least \(3\sigma \) is found. Corresponding ATLAS studies [65, 118, 119] yield similar conclusions.
Other analyses In addition to the results discussed above, a number of other analyses have been performed, making use of the increase in the available data since the first Higgs boson discovery in different ways. These include, for example, measurements of differential distributions in \(H\rightarrow \gamma \gamma \) [123] and \(H\rightarrow ZZ\) [124] events and searches for rarer decays, such as \(H\rightarrow \mu \mu \) [125, 126], \(H\rightarrow ee\) [126], \(H\rightarrow Z\gamma \) [127, 128], decays to heavy quarkonia states and a photon [129], and invisible modes [130, 131]. These searches are not expected to be sensitive to a SM Higgs boson signal based on the currently available data and thus are as of now mainly relevant for the preparation for the larger datasets expected from LHC Run2 and/or for using Higgs boson events as a probe for effects beyond the SM.
Additional production modes are searched for as well. Here, top-associated production is of particular interest because it would provide direct access to the top-Higgs Yukawa coupling. While the results from recent analyses [132, 133, 134, 135] of these complex final states do not quite establish a significant signal yet, they demonstrate a lot of promise for LHC Run2, where, in addition to larger datasets, an improved signal-to-background ratio is expected due to the increased collision energy.
2.2.2 Future projections
Studies of longer-term Higgs physics prospects currently focus on the scenario of an LHC upgraded during a shutdown starting in 2022 to run at a levelled luminosity of \(5\times 10^{34}\) cm\(^{-2}\)s\(^{-1}\), resulting in a typical average of 140 pile-up events per bunch crossing. This so-called HL-LHC is expected to deliver a total integrated luminosity of 3000 fb\(^{-1}\) to be compared to a total of 300 fb\(^{-1}\) expected by the year 2022.
The following summary of SM Higgs boson analysis prospects for such large datasets is based on preliminary results by the ATLAS and CMS Collaborations documented in [136, 137], respectively. While the prospects for measurements of other Higgs boson properties are being studied as well, the discussion below focusses on projections concerning signal strength measurements and coupling analyses.
ATLAS investigates the physics prospects for 14 TeV datasets corresponding to the same integrated luminosities as CMS but here the expected detector performance is parameterised based on efficiency and resolution modifications at the detector object level. These are obtained from full simulations corresponding to current and/or upgraded ATLAS detector components assuming values for the number of pile-up events per bunch crossing ranging from 40 to 200. The theoretical uncertainties are assumed to be similar to those used in recent analysis of the Run1 data but some of the experimental systematic uncertainties are re-evaluated taking into account, e.g., the expected improved background estimates due to an increased number of events in data control regions.
Relative uncertainty on the determination of the signal strength expected for the CMS experiment for integrated luminosities of 300 fb\(^{-1}\) and 3000 fb\(^{-1}\) [137] and the two uncertainty scenarios described in the text
\(\mathscr {L}\) | 300 fb\(^{-1}\) | 3000 fb\(^{-1}\) | ||
---|---|---|---|---|
Scenario | 2 (%) | 1 (%) | 2 (%) | 1 (%) |
\(\gamma \gamma \) | 6 | 12 | 4 | 8 |
WW | 6 | 11 | 4 | 7 |
ZZ | 7 | 11 | 4 | 7 |
bb | 11 | 14 | 5 | 7 |
\(\tau \tau \) | 8 | 14 | 5 | 8 |
\(Z\gamma \) | 62 | 62 | 20 | 24 |
\(\mu \mu \) | 40 | 42 | 14 | 20 |
Relative uncertainty on the signal strength projected by ATLAS for different production modes using the combination of Higgs final states based on integrated luminosities of 300 fb\(^{-1}\) and 3000 fb\(^{-1}\) [136], assuming a SM Higgs boson with a mass of 125 GeV and branching ratios as in the SM
\(\mathscr {L}\) | 300 fb\(^{-1}\) | 3000 fb\(^{-1}\) | ||
---|---|---|---|---|
Uncertainties | All (%) | No theory (%) | All (%) | No theory (%) |
\(gg\rightarrow H\) | 12 | 6 | 11 | 4 |
VBF | 18 | 15 | 15 | 9 |
WH | 41 | 41 | 18 | 18 |
qqZH | 80 | 79 | 28 | 27 |
ggZH | 371 | 362 | 147 | 138 |
ttH | 32 | 30 | 16 | 10 |
Relative uncertainty on the determination of the coupling scale factor ratios expected for the CMS experiment for integrated luminosities of 300 fb\(^{-1}\) and 3000 fb\(^{-1}\) [137] and the two uncertainty scenarios described in the text
\(\mathscr {L}\) | 300 fb\(^{-1}\) | 3000 fb\(^{-1}\) | ||
---|---|---|---|---|
Scenario | 2 (%) | 1 (%) | 2 (%) | 1 (%) |
\(\kappa _\gamma \cdot \kappa _Z/\kappa _H\) | 4 | 6 | 2 | 5 |
\(\kappa _W/\kappa _Z\) | 4 | 7 | 2 | 3 |
\(\lambda _{tg}=\kappa _t/\kappa _g\) | 13 | 14 | 6 | 8 |
\(\lambda _{bZ}=\kappa _b/\kappa _Z\) | 8 | 11 | 3 | 5 |
\(\lambda _{\tau Z}=\kappa _\tau /\kappa _Z\) | 6 | 9 | 2 | 4 |
\(\lambda _{\mu Z}=\kappa _\mu /\kappa _Z\) | 22 | 23 | 7 | 8 |
\(\lambda _{Zg}=\kappa _Z/\kappa _g\) | 6 | 9 | 3 | 5 |
\(\lambda _{\gamma Z}=\kappa _\gamma /\kappa _Z\) | 5 | 8 | 2 | 5 |
\(\lambda _{(Z\gamma )Z }=\kappa _{Z\gamma }/\kappa _Z\) | 40 | 42 | 12 | 12 |
The \(\kappa _X\) extraction requires assumptions on the total width of the Higgs boson. Without total width information, only ratios of couplings can be studied. As for the current Run1 analyses, results are obtained for several different sets of assumptions. An overview of the expected CMS precision for the most generic of these scenarios, still with a single, narrow, \({\textit{CP}}\)-even scalar Higgs boson but without further assumptions, e.g. on new-particle contributions through loops, is given in Table 6. Results from corresponding ATLAS analyses are shown in Fig. 34, where, for an integrated luminosity of 3000 fb\(^{-1}\), the experimental uncertainties range from about 2 % for the coupling scale factors between the electroweak bosons to 5–8 % for the ratios involving gluons and fermions outside the first generation.
Higgs self-coupling One of the most important long-term goals of the SM Higgs physics programme is the measurement of the tri-linear self-coupling \(\lambda _{HHH}\), which requires the study of Higgs boson pair production. At the LHC the dominant production mechanism is gluon–gluon fusion with a cross section of about 40 fb at \(\sqrt{s}=14\) TeV. Several combinations of Higgs decays can be considered. For example, assuming 3000 fb\(^{-1}\) of 14 TeV data [139] presents the ATLAS prospects for the search for Higgs pair production in the channel \(H(\rightarrow \gamma \gamma )H(\rightarrow bb)\), which combines the large \(H\rightarrow bb\) branching ratio with the good mass resolution of the two-photon final state. The projected diphoton mass distribution for simulated ggF-produced signal and background processes after signal selection requirements is shown in Fig. 36; the statistical analysis gives a signal yield of about eight events and signal significance of 1.3\(\sigma \). Although additional observables, the application of more sophisticated analysis techniques and the inclusion of other production modes can be expected to improve on this result, a combination with other decay channels will likely be needed to find evidence for SM Higgs pair production (or to exclude that the Higgs self-coupling strength is close to its SM expectation) with an integrated luminosity of 3000 fb\(^{-1}\).
2.3 Higgs at ILC: prospects^{9}
2.3.1 Introduction
The properties to measure are the mass, width, and \(J^{PC}\), its gauge, Yukawa, and self-couplings. The key is to confirm the mass–coupling relation. If the 125 GeV boson is the one to give masses to all the SM particles, coupling should be proportional to mass as shown in Fig. 39. Any deviation from the straight line signals physics beyond the standard model (BSM). The Higgs serves therefore as a window to BSM physics.
The expected deviation pattern for various Higgs couplings, assuming small deviations for \(\cos (\beta -\alpha ) < 0\). The arrows for Yukawa interactions are reversed for 2HDMs with \(\cos (\beta -\alpha ) > 0\)
Model | \(\mu \) | \(\tau \) | b | c | t | \(g_V\) |
---|---|---|---|---|---|---|
Singlet mixing | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) |
2HDM-I | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) |
2HDM-II (SUSY) | \(\uparrow \) | \(\uparrow \) | \(\uparrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) |
2HDM-X (Lepton-specific) | \(\uparrow \) | \(\uparrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) |
2HDM-Y (Flipped) | \(\downarrow \) | \(\downarrow \) | \(\uparrow \) | \(\downarrow \) | \(\downarrow \) | \(\downarrow \) |
Why 250–500 GeV? The ILC is an \(e^+e^-\) collider designed primarily to cover the energy range from \(\sqrt{s}=250\) to 500 GeV. This is because of the following three very well-known thresholds (Fig. 41). The first threshold is at around \(\sqrt{s}=250\) GeV, where the \(e^+e^- \rightarrow Zh\) process will reach its cross section maximum. This process is a powerful tool to measure the Higgs mass, width, and \(J^{PC}\). As we will see below, this process allows us to measure the hZZ coupling in a completely model-independent manner through the recoil mass measurement. This is a key to perform model-independent extraction of branching ratios for various decay modes such as \(h \rightarrow b\bar{b}\), \(c\bar{c}\), \(\tau \bar{\tau }\), gg, \(WW^*\), \(ZZ^*\), \(\gamma \gamma \), as well as invisible decays.
The second threshold is at around \(\sqrt{s}=350\) GeV, which is the well-known \(t\bar{t}\) threshold. The threshold scan here provides a theoretically very clean measurement of the top-quark mass, which can be translated into \(m_t(\overline{{\mathrm{MS}}})\) to an accuracy of 100 MeV. The precise value of the top mass obtained this way can be combined with the precision Higgs mass measurement to test the stability of the SM vacuum [148, 149]. The \(t\bar{t}\) threshold also enables us to indirectly access the top Yukawa coupling through the Higgs exchange diagram. It is also worth noting that with the \(\gamma \gamma \) collider option at this energy the double Higgs production: \(\gamma \gamma \rightarrow hh\) is possible, which can be used to study the Higgs self-coupling [150]. Notice also that at \(\sqrt{s}=350\,\)GeV and above, the WW-fusion Higgs production process, \(e^+e^- \rightarrow \nu \bar{\nu }h\), becomes sizeable with which we can measure the hWW coupling and accurately determine the total width.
The third threshold is at around \(\sqrt{s}=500\) GeV, where the double Higgs-strahlung process, \(e^+e^- \rightarrow Zhh\) attains its cross section maximum, which can be used to access the Higgs self-coupling. At \(\sqrt{s}=500\) GeV, another important process, \(e^+e^- \rightarrow t\bar{t}h\), will also open, though the product cross section is much smaller than its maximum that is reached at around \(\sqrt{s}=800\) GeV. Nevertheless, as we will see, QCD threshold correction enhances the cross section and allows us a reasonable measurement of the top Yukawa coupling concurrently with the self-coupling measurement.
By covering \(\sqrt{s}=250\)–500 GeV, we will hence be able complete the mass–coupling plot. This is why the first phase of the ILC project is designed to cover the energy up to \(\sqrt{s}=500\) GeV.
2.3.2 ILC at 250 GeV
Expected relative errors for the \(\sigma \times \mathrm{BR}\) measurements at \(\sqrt{s}=250\,\)GeV with \(250\,\)fb\(^{-1}\) for \(m_h=125\,\)GeV
Process | Decay mode | \({\varDelta } (\sigma \mathrm{BR})/(\sigma \mathrm{BR})\) (%) | \({\varDelta } \mathrm{BR}/\mathrm{BR}\) (%) |
---|---|---|---|
Zh | \(h \rightarrow b\bar{b}\) | 1.2 | 2.9 |
\(h \rightarrow c\bar{c}\) | 8.3 | 8.7 | |
\(h \rightarrow gg\) | 7.0 | 7.5 | |
\(h \rightarrow WW^*\) | 6.4 | 6.9 | |
\(h \rightarrow \tau \bar{\tau }\) | 4.2 | 4.9 | |
\(h \rightarrow ZZ^*\) | 19 | 19 | |
\(h \rightarrow \gamma \gamma \) | 34 | 34 |
Notice that the cross section error, \({\varDelta } \sigma _{Zh}/\sigma _{Zh}=2.5\,\%\), eventually limits the precision of the BR measurements. We hence need more data at \(\sqrt{s}=250\) GeV so as to improve the situation. We will return to the possible luminosity upgrade scenario later.
2.3.3 ILC at 500 GeV
Expected relative errors for the \(\sigma \times \mathrm{BR}\) measurements at \(\sqrt{s}=250\) GeV with \(250\,\)fb\(^{-1}\) and at \(\sqrt{s}=500\,\)GeV with 500 fb\(^{-1}\) for \(m_h=125\) GeV and \((e^{-}, e^{+})=(-0.8, +0.3)\) beam polarisation. The last column of the table shows the relative errors on the branching ratios. Then the numbers in the parentheses are for 250 fb\(^{-1}\) at \(\sqrt{s}=250\) GeV alone
Energy (GeV) Mode | \({\varDelta } (\sigma \cdot \mathrm{BR}) / (\sigma \cdot \mathrm{BR})\) | \({\varDelta } \mathrm{BR}/\mathrm{BR}\) | ||
---|---|---|---|---|
250 | 500 | \(250+500\) | ||
Zh (%) | Zh (%) | \(\nu \bar{\nu }h\) (%) | Combined (%) | |
\(h \rightarrow b\bar{b}\) | 1.2 | 1.8 | 0.66 | 2.2 (2.9) |
\(h \rightarrow c\bar{c}\) | 8.3 | 13 | 6.2 | 5.1 (8.7) |
\(h \rightarrow gg\) | 7.0 | 11 | 4.1 | 4.0 (7.5) |
\(h \rightarrow WW^*\) | 6.4 | 9.2 | 2.4 | 3.1 (6.9) |
\(h \rightarrow \tau ^+\tau ^-\) | 4.2 | 5.4 | 9.0 | 3.7 (4.9) |
\(h \rightarrow ZZ^*\) | 19 | 25 | 8.2 | 7.5 (19) |
\(h \rightarrow \gamma \gamma \) | 29–38 | 29–38 | 20–26 | 17 (34) |
The number of remaining events for the three event selection modes: \(Zhh \rightarrow (\ell \bar{\ell })(b\bar{b})(b\bar{b})\), \((\nu \bar{\nu })(b\bar{b})(b\bar{b})\), and \((q\bar{q})(b\bar{b})(b\bar{b})\) and corresponding excess and measurement sensitivities for \(m_h=120\) GeV at \(\sqrt{s}=500\) GeV with \(2\,\)ab\(^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.3)\) beam polarisation
Mode | Signal | BG | Significance | |
---|---|---|---|---|
Excess | Meas. | |||
\(Zhh \rightarrow (\ell \bar{\ell })(b\bar{b})(b\bar{b})\) | 3.7 | 4.3 | 1.5\(\sigma \) | 1.1\(\sigma \) |
4.5 | 6.0 | 1.5\(\sigma \) | 1.2\(\sigma \) | |
\(Zhh \rightarrow (\nu \bar{\nu })(b\bar{b})(b\bar{b})\) | 8.5 | 7.9 | 2.5\(\sigma \) | 2.1\(\sigma \) |
\(Zhh \rightarrow (q\bar{q})(b\bar{b})(b\bar{b})\) | 13.6 | 30.7 | 2.2\(\sigma \) | 2.0\(\sigma \) |
18.8 | 90.6 | 1.9\(\sigma \) | 1.8\(\sigma \) |
2.3.4 ILC at 1000 GeV
The numbers of signal and background events before and after selection cuts and measurement significance for \(m_h=120\,\)GeV at \(\sqrt{s}=1\,\)TeV with 2 ab\(^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.2)\) beam polarisation
Mode | No cut | After cuts |
---|---|---|
\(\nu \bar{\nu }hh\) (WW-fusion) | 272 | 35.7 |
\(\nu \bar{\nu }hh\) (Zhh) | 74.0 | 3.88 |
BG (\(t\bar{t}/\nu \bar{\nu }Zh\)) | \(7.86 \times 10^{5}\) | 33.7 |
Meas. significance | 0.30 | 4.29 |
With 2 ab\(^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.2)\) beam polarisation at \(\sqrt{s}=\) TeV, we would be able to determine the cross section for the \(e^+e^- \rightarrow \nu \bar{\nu }hh\) process to \({\varDelta } \sigma / \sigma = 23~\%\), corresponding to the self-coupling precision of \(\varDelta \lambda / \lambda = 18 (20)~\%\) with (without) the event weighting to enhance the contribution from the signal diagram for \(m_h=120\) GeV [186]. According to preliminary results from a on-going full simulation study [192], adding \(hh \rightarrow WW^*b\bar{b}\) would improve the self-coupling measurement precision by about 20 % relatively, which means \(\varDelta \lambda / \lambda = 21\,\%\) for \(m_h=125\) GeV with the baseline integrated luminosity of \(1 ab^{-1}\) at 1 TeV.
At \(\sqrt{s}=1\,\)TeV, the \(e^+e^- \rightarrow t\bar{t}h\) process is also near its cross section maximum, making concurrent measurements of the self-coupling and top Yukawa coupling possible. We will be able to observe the \(e^+e^- \rightarrow t\bar{t}h\) events with \(12\sigma \) significance in 8-jet mode and \(8.7\sigma \) significance in lepton-plus-6-jet mode, corresponding to the relative error on the top Yukawa coupling of \(\varDelta g_Y(t) / g_Y(t) = 3.1~\%\) with \(1ab^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.2)\) beam polarisation at \(\sqrt{s}=1\,\)TeV for \(m_h=125\,\)GeV [193].
Independent Higgs measurements using the Higgs-strahlung (Zh) and the WW-fusion (\(\nu \bar{\nu }h\)) processes for \(m_h=125\,\)GeV at three energies: \(\sqrt{s}=250\,\)GeV with \(250\,\)fb\(^{-1}\), \(500\,\)GeV with \(500\,\)fb\(^{-1}\) both with \((e^{-}, e^{+})=(-0.8, +0.3)\) beam polarisation, \(\sqrt{s}=1\,\)TeV with \(1ab^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.2)\) beam polarisation
\(\sqrt{s}\) | 250 GeV | 500 GeV | 1 TeV | ||
Lumi. | 250 fb\(^{-1}\) | 500 fb\(^{-1}\) | 1 ab\(^{-1}\) | ||
Process | Zh | \(\nu \bar{\nu }h\) | Zh | \(\nu \bar{\nu }h\) | \(\nu \bar{\nu }h\) |
\(\varDelta \sigma / \sigma \) | |||||
2.6% | – | 3.0% | – | – | |
Mode | \(\varDelta (\sigma \cdot \mathrm{BR}) / (\sigma \cdot \mathrm{BR})\) | ||||
\(h \rightarrow b\bar{b}\) (%) | 1.2 | 10.5 | 1.8 | 0.66 | 0.5 |
\(h \rightarrow c\bar{c}\) (%) | 8.3 | 13 | 6.2 | 3.1 | |
\(h \rightarrow gg\) (%) | 7.0 | 11 | 4.1 | 2.3 | |
\(h \rightarrow WW^*\) (%) | 6.4 | 9.2 | 2.4 | 1.6 | |
\(h \rightarrow \tau ^+\tau ^-\) (%) | 4.2 | 5.4 | 9.0 | 3.1 | |
\(h \rightarrow ZZ^*\) (%) | 18 | 25 | 8.2 | 4.1 | |
\(h \rightarrow \gamma \gamma \) (%) | 34 | 34 | 23 | 8.5 | |
\(h \rightarrow \mu ^+\mu ^-\) (%) | 100 | – | – | – | 31 |
2.3.5 ILC 250 + 500 + 1000: global fit for couplings
Expected precisions for various couplings of the Higgs boson with \(m_h=125\,\)GeV from a model-independent fit to observables listed in Table 12 at three energies: \(\sqrt{s}=250\) GeV with 250 fb\(^{-1}\), 500 GeV with 500 fb\(^{-1}\) both with \((e^{-}, e^{+})=(-0.8, +0.3)\) beam polarisation, \(\sqrt{s}=1\) TeV with \(2ab^{-1}\) and \((e^{-}, e^{+})=(-0.8, +0.2)\) beam polarisation, cf. [29] and Scen. ’Snow’ in [27]. \(^\mathrm{a}\)Values assume inclusion of \(hh\rightarrow WW^*b\bar{b}\) decays
Coupling | \(\sqrt{s}\) (GeV) | ||
---|---|---|---|
250 | 250 + 500 | 250 + 500 + 1000 | |
hZZ (%) | 1.3 | 1.0 | 1.0 |
hWW (%) | 4.8 | 1.1 | 1.1 |
hbb (%) | 5.3 | 1.6 | 1.3 |
hcc (%) | 6.8 | 2.8 | 1.8 |
hgg (%) | 6.4 | 2.3 | 1.6 |
\(h\tau \tau \) (%) | 5.7 | 2.3 | 1.6 |
\(h\gamma \gamma \) (%) | 18 | 8.4 | 4.0 |
\(h\mu \mu \) (%) | 91 | 91 | 16 |
\({\varGamma }_0\) (%) | 12 | 4.9 | 4.5 |
htt (%) | – | 14 | 3.1 |
hhh (%) | – | 83\(^\mathrm{a}\) | 21\(^\mathrm{a}\) |
2.3.6 Synergy: LHC + ILC
Maximum possible deviations when nothing but the 125 GeV boson would be found at the LHC [199]
\(\varDelta hVV\) (%) | \(\varDelta h\bar{t}t\) | \(\varDelta h\bar{b}b\) | |
---|---|---|---|
Mixed-in singlet | 6 | 6 % | 6 % |
Composite Higgs | 8 | tens of % | tens of % |
Minimal SUSY | \(<\)1 | 3 % | 10 %\(^\mathrm{a}\), 100 %\(^\mathrm{b}\) |
LHC 14 TeV, \(3ab^{-1}\) | 8 | 10 % | 15 % |
Expected Higgs precisions on normalised Higgs couplings (\(\kappa _i := g_i / g_i (\mathrm{SM})\)) for \(m_h=125\,\)GeV from model-dependent 7-parameter fits for the LHC and the ILC, where \(\kappa _c = \kappa _t =: \kappa _u\), \(\kappa _s = \kappa _b =: \kappa _d\), \(\kappa _\mu = \kappa _\tau =: \kappa _\ell \), and \({\varGamma }_\mathrm{tot} = \sum {\varGamma }_i^\mathrm{SM} \, \kappa _i^2\) are assumed
Facility | LHC | HL-LHC | ILC500 | ILC1000 |
---|---|---|---|---|
\(\sqrt{s}\,\)(GeV) | 1,400 | 14,000 | 250/500 | 250/500/1000 |
\(\int {\mathscr {L}} \mathrm{d}t\,\)(fb\(^{-1}\)) | 300/exp (%) | 3000/exp (%) | 250 + 500 (%) | 250 + 500 + 1000 (%) |
\(\kappa _\gamma \) | 5–7 | 2–5 | 8.3 | 3.8 |
\(\kappa _g\) | 6–8 | 3–5 | 2.0 | 1.1 |
\(\kappa _W\) | 4–6 | 2–5 | 0.39 | 0.21 |
\(\kappa _Z\) | 4–6 | 2–4 | 0.49 | 0.50 |
\(\kappa _\ell \) | 6–8 | 2–5 | 1.9 | 1.3 |
\(\kappa _d\) | 10–13 | 4–7 | 0.93 | 0.51 |
\(\kappa _u\) | 14–15 | 7–10 | 2.5 | 1.3 |
The different models predict different deviation patterns. The ILC together with the LHC will be able to fingerprint these models or set the lower limit on the energy scale for BSM physics.
2.3.7 Model-dependent global fit: example of fingerprinting
2.3.8 High luminosity ILC?
In order to improve the recoil mass measurement significantly a new luminosity upgrade option (doubling of the number of bunches plus 10 Hz collisions instead of nominal 5 Hz) was proposed for the 250 GeV running in the Snowmass 2013 process [141] (see the red box in Fig. 55). It should be noted that the number of bunches was 2625 in the original ILC design given in the reference design report [202], which was reduced to 1312 in the TDR so as to reduce the construction cost. The 10 Hz operation is practical at 250 GeV, since the needed wall plug power is lower at the lower energy. The upgrade would hence allow a factor of 4 luminosity upgrade at \(\sqrt{s}=250\) GeV. Let us now assume that after the baseline programme at \(\sqrt{s}=250\), 500, and 1000 GeV we will run at the same three energies with the luminosity upgrade, thereby achieving \(1150\,\)fb\(^{-1}\) at 250 GeV, \(1600\,\)fb\(^{-1}\) at 500 GeV, and \(2500\,\)fb\(^{-1}\) at 1000 GeV.
Similar table to Table 12 but with the luminosity upgrade described in the text: 1150 fb\(^{-1}\) at 250 GeV, 1600 fb\(^{-1}\) at 500 GeV, and 2500 fb\(^{-1}\) at 1 TeV
\(\sqrt{s}\) | 250 GeV | 500 GeV | 1 TeV | ||
Lumi. | 1150 fb\(^{-1}\) | 1600 fb\(^{-1}\) | 2.5 ab\(^{-1}\) | ||
Process | Zh | \(\nu \bar{\nu }h\) | Zh | \(\nu \bar{\nu }h\) | \(\nu \bar{\nu }h\) |
\(\varDelta \sigma / \sigma \) | |||||
1.2 % | – | 1.7 % | – | – | |
Mode | \(\varDelta (\sigma \cdot \mathrm{BR}) / (\sigma \cdot \mathrm{BR})\) | ||||
\(h \rightarrow b\bar{b}\) (%) | 0.56 | 4.9 | 1.0 | 0.37 | 0.3 |
\(h \rightarrow c\bar{c}\) (%) | 3.9 | 7.2 | 3.5 | 2.0 | |
\(h \rightarrow gg\) (%) | 3.3 | 6.0 | 2.3 | 1.4 | |
\(h \rightarrow WW^*\) (%) | 3.0 | 5.1 | 1.3 | 1.0 | |
\(h \rightarrow \tau ^+\tau ^-\) (%) | 2.0 | 3.0 | 5.0 | 2.0 | |
\(h \rightarrow ZZ^*\) (%) | 8.4 | 14 | 4.6 | 2.6 | |
\(h \rightarrow \gamma \gamma \) (%) | 16 | 19 | 13 | 5.4 | |
\(h \rightarrow \mu ^+\mu ^-\) (%) | 46.6 | – | – | – | 20 |
2.3.9 Conclusions
Coupling | \(\sqrt{s}\) (GeV) | ||
---|---|---|---|
250 | 250 + 500 | 250 + 500 + 1000 | |
hZZ (%) | 0.6 | 0.5 | 0.5 |
hWW (%) | 2.3 | 0.6 | 0.6 |
hbb (%) | 2.5 | 0.8 | 0.7 |
hcc (%) | 3.2 | 1.5 | 1.0 |
hgg (%) | 3.0 | 1.2 | 0.93 |
\(h\tau \tau \) (%) | 2.7 | 1.2 | 0.9 |
\(h\gamma \gamma \) (%) | 8.2 | 4.5 | 2.4 |
\(h\mu \mu \) (%) | 42 | 42 | 10 |
\({\varGamma }_0\) (%) | 5.4 | 2.5 | 2.3 |
htt (%) | – | 7.8 | 1.9 |
hhh (%) | – | 46\(^\mathrm{a}\) | 13\(^\mathrm{a}\) |
2.4 Higgs at CLIC: prospects^{10}
2.4.1 Introduction
The CLIC accelerator [203] offers the possibility to study \(e^+e^-\) collisions at centre-of-mass energies from 350 GeV up to 3 TeV. The novel CLIC acceleration schemes uses a two-beam acceleration scheme and normal conducting cavities, which operate at room temperature. A high-intensity drive beam generates the necessary RF power at 12 GHz, which is then used to accelerate the main beam. Compared to the ILC [204], the pulse length is significantly shorter (150 ns) with a bunch spacing of just 0.5 ns and a repetition rate of 50 Hz.
Running in the multi-TeV regime and with small intense bunches means that the CLIC detectors experience much higher backgrounds from beamstrahlung. This also leads to a long tail of the luminosity spectrum. To cope with these harsh backgrounds, the CLIC detectors plan to use highly granular detectors with time-stamping on the 10 ns level in for the tracking detectors and 1 ns level for the calorimeters in order to suppress these backgrounds [9, 10].
This section focusses on the production of heavy Higgs bosons (\(H, A, H^\pm \)), which are predicted in extended models like the 2HDM or supersymmetric models. The CLIC capabilities for studying light, SM-like Higgs bosons are summarised elsewhere [9, 10, 208].
2.4.2 Searches for heavy Higgs Bosons
The pair production processes \(e^+e^-\rightarrow H^+H^-\) and \(e^+e^-\rightarrow HA\) will give access to these heavy Higgs bosons almost up to the kinematic limit [212, 213]. Two separate scenarios have recently been studied [9, 10], with a mass of the pseudoscalar Higgs boson A of \(m_A\)=902 GeV (Model I) or \(m_A\)=742 GeV (Model II). In both scenarios, the dominant decay modes are \(HA\rightarrow b\bar{b}b\bar{b}\) and \(H^{+}H^{-}\rightarrow t\bar{b}\bar{t}b\). As already mentioned above, the analyses use the anti-\(k_T\) algorithm that has been developed for the LHC in order to suppress the background originating from \(\gamma \gamma \rightarrow \mathrm{hadrons}\).
2.5 Prospects for MSSM Higgs bosons^{11}
We will briefly review the MSSM Higgs sector, the relevance of higher-order corrections and the implications of the recent discovery of a Higgs-like state at the LHC at \(\sim 125\ \mathrm {GeV}\,\). Finally we look at the prospects in view of this discovery for MSSM Higgs physics at the LC. We will concentrate on the MSSM with real parameters.^{12} The NMSSM will be covered in Sect. 2.9.
2.5.1 The Higgs sector of the MSSM at tree level
2.5.2 The relevance of higher-order corrections
Higher-order corrections also affect the various couplings of the Higgs bosons and thus the production cross sections and branching ratios. Focusing on the light \({\mathscr {CP}}\)-even Higgs boson, the couplings to down-type fermions are modified with respect to the SM coupling by an additional factor \(-\sin \alpha /\cos \beta \), and higher-order corrections can be absorbed into the \({\mathscr {CP}}\)-even mixing angle, \(\alpha \rightarrow \alpha _\mathrm{eff}\) [228]. For large higher-order corrections which drive \(\alpha _\mathrm{eff}\rightarrow 0\) the decay widths \({\varGamma }(h \rightarrow b \bar{b})\) and \({\varGamma }(h \rightarrow \tau ^+\tau ^-)\) could be substantially smaller than in the SM [229], altering the available search modes for such a Higgs boson.
Deviations from the SM predictions can also be induced by the appearance of light virtual SUSY particles in loop-induced processes. Most promiently a light scalar top can have a strong impact on the prediction of \(gg \rightarrow h\). The additional contributions can interfere negatively with the top loop contribution, leading to a strong suppression of the production cross section [229, 237, 238]. Similarly, it was shown that light scalar taus can lead to an enhancement of up to \(\sim \)50 % of the decay width of the light \({\mathscr {CP}}\)-even Higgs to photons, \({\varGamma }(h \rightarrow \gamma \gamma )\) [239, 240].
2.5.3 Implicatios of the discovery at \(\sim \)125 GeV
Looking into pre-defined benchmark scenarios it was shown that the light \({\mathscr {CP}}\)-even Higgs boson can be interpreted as the new boson around \(125\ \mathrm {GeV}\,\). On the other hand, also the heavy \({\mathscr {CP}}\)-even Higgs boson can in principle be interpreted as the newly discovered state [248]. The latter option, however, is challenged by the latest ATLAS results on charged Higgs-boson searches [249].
Here we briefly discuss the results in two of the new benchmark scenarios [238], devised for the search for heavy MSSM Higgs bosons. In the upper plot of Fig. 61 the \(m_h^{\max }\) scenario is shown. The red area is excluded by LHC searches for the heavy MSSM Higgs bosons, the blue area is excluded by LEP Higgs searches, and the light shaded red area is excluded by LHC searches for a SM-like Higgs boson. The bounds have been obtained with HiggsBounds [250, 251, 252] (where an extensive list of original references can be found). The green area yields \(M_h= 125 \pm 3 \) GeV, i.e. the region allowed by the experimental data, taking into account the theoretical uncertainty in the \(M_h\) calculation as discussed above. The left plot also allows one to extract new lower limits on \(M_A\) and \(\tan \beta \). From this analysis it can be concluded that if the light \({\mathscr {CP}}\)-even Higgs is interpreted as the newly discovered state at \(\sim \)125 GeV, then \(\tan \beta \gtrsim 4\), \(M_A\gtrsim 200 \, \mathrm {GeV}\,\) and \(M_{H^\pm }\gtrsim 220 \, \mathrm {GeV}\,\) [238].In the lower plot of Fig. 61 we show the \(m_h^\mathrm{mod+}\) scenario that differs from the \(m_h^{\max }\) scenario in the choice of \(X_t\). While in the \(m_h^{\max }\) scenario \(X_t/M_\mathrm{SUSY} = +2\) had been chosen to maximise \(M_h\), in the \(m_h^\mathrm{mod+}\) scenario \(X_t/M_\mathrm{SUSY} = +1.5\) is used to yield a “good” \(M_h\) value over the nearly the entire \(M_A\)–\(\tan \beta \) plane, which is visible as the extended green region.In GUT based scenarios such as the CMSSM and the NUHM1^{13} it was shown that a light \({\mathscr {CP}}\)-even Higgs boson around or slightly below \(125\ \mathrm {GeV}\,\) is a natural prediction of these models [254]. These predictions take into account the current SUSY search limits (but no direct light Higgs search limits), as well as the relevant EWPO, B-physics observables and the relic Dark Matter density. In Fig. 62 we show the predictions in the CMSSM (upper) and the NUHM1 (lower plot). The red bands indicate a theory uncertainty of \(\sim \)1.5 GeV on the evaluation of \(M_h\). The green columns indicate the range of the newly discovered particle mass.
- Parameter scans in the MSSM with 19 free parameters (pMSSM–19 [253]) are naturally compatible with a light Higgs boson around \(M_h\sim 125\ \mathrm {GeV}\,\), as has been analysed in Refs. [255, 256] (see also Ref. [257] for a more recent analysis in the pMSSM–15 and Ref. [258] for an analysis in the pMSSM–19). Taking into account the available constraints from SUSY searches, Higgs searches, low-energy observables, B-physics observables and the relic abundance of Dark Matter viable scenarios can be identified that can be analysed in the upcoming LHC runs. Also the effects on the various production cross sections and branching ratios were analysed, where it was confirmed that light particles can modify in particular the decay rate to photons [239, 240].
Parameter scans in the MSSM with seven free parameters (pMSSM–7) in comparison to the pMSSM–19 have the advantage of a full sampling of the parameter space with \({\mathscr {O}}(10^7)\) points; but they have the disadvantage of potentially not including all relevant phenomenogy of the MSSM. The pMSSM–7 fits to the full set of Higgs data (and several low-energy observables) [259] allow one to show an enhancement of the \(\mathrm{BR}(h \rightarrow \gamma \gamma )\), correlated to a suppression of the decays to \(b \bar{b}\) and \(\tau ^+\tau ^-\) via the mechanisms outlined in Sect. 2.5.2 (see also Ref. [260]). In particular, these scans (while not incorporating the latest data) demonstrate that light scalar top masses are compatible with \(M_h\sim 125\ \mathrm {GeV}\,\) (see also Ref. [248]). In Fig. 63 we show \(X_t/m_{\tilde{q}_3}\) vs. the light stop mass (left plot, where \(X_t= A_t- \mu /\tan \beta \) denotes the off-diagonal entry in the scalar top mass matrix, \(A_t\) is the tri-linear Higgs-stop coupling, and \(m_{\tilde{q}_3}\) denotes the (common) diagonal soft SUSY-breaking parameter in the scalar top and bottom sector) and the light vs. the heavy stop mass (right plot) in the case that the light \({\mathscr {CP}}\)-even Higgs boson corresponds to the new state at \(\sim \)125 GeV. The coloured points passed the Higgs exclusion bounds (obtained using HiggsBounds [250, 251, 252]). The red (yellow) points correspond to the best-fit points with a \({\varDelta }\chi ^2 < 2.3 (5.99)\), see Ref. [259] for details. In the left plot one can see that the case of zero stop mixing in the MSSM is excluded by the observation of a light Higgs at \(M_h\sim 125\ \mathrm {GeV}\,\) (unless \(m_{\tilde{q}_3}\) is extremely large, see, e.g., Ref. [261]), and that values of \(|X_t/m_{\tilde{q}_3}|\) between \(\sim \)1 and \(\sim \)2.5 must be realised. For the most favoured region we find \(X_t/m_{\tilde{q}_3}= 2 \)– 2.5. Concerning the value of the lightest scalar top mass, the overall smallest values are found at \(m_{\tilde{t}_1}\sim 200 \, \mathrm {GeV}\,\), where also the regions favoured by the fit to the Higgs rates start, in the case of \(X_t\) positive. Such a light \(\tilde{t}_{1}\) is accompanied by a somewhat heavier \(\tilde{t}_{2}\), as can be seen in the right plot of Fig. 63. Values of \(m_{\tilde{t}_1}\sim 200 \mathrm {GeV}\,\) are realised for \(m_{\tilde{t}_2}\sim 600 \mathrm {GeV}\,\), which would mean that both stop masses are rather light, offering interesting possibilities for the LHC. The highest favoured \(m_{\tilde{t}_1}\) values we find are \(\sim \)1.4 TeV. These are the maximal values reached in the scan in Ref. [259], but from Fig. 63 it is obvious that the favoured region extends to larger values of both stop masses. Such a scenario would be extremely difficult to access at the LHC.
2.5.4 Prospects for the MSSM Higgs bosons at the LHC
The prime task now is to study the properties of the discovered new particle and in particular to test whether the new particle is compatible with the Higgs boson of the SM or whether there are significant deviations from the SM predictions, which would point towards physics beyond the SM. A large part of the current and future LHC physics programme is devoted to this kind of analyses.
The Higgs-boson mass can be determined down to a level of \({\mathscr {O}}(200\, \mathrm {MeV}\,)\).
For the coupling determination the following has to be kept in mind. Since it is not possible to measure the Higgs production cross sections independently from the Higgs decay (or, equivalently, the Higgs boson width^{14}), a determination of couplings is only possible if certain (theory) assumptions on the Higgs width are made, see, e.g. Ref. [196, 266]. For instance, it can be assumed that no new particles contribute to the decay width. Under this kind of assumption, going to the HL-LHC, precisions on couplings at the \(\sim \)10 % level can be achieved. Without any assumptions only ratios of couplings can be determined (see also Ref. [78] for a recent review).
Studies in the context of the HL-LHC indicate that there might be some sensitivity on the tri-linear Higgs self-coupling; however, this will require a careful estimate of background contributions. Further studies to clarify these issues are currently in progress, see Ref. [267] for a discussion.
It can be expected that the spin 2 hypothesis can be rejected using LHC data.
A pure \({\mathscr {CP}}\)-even state can be discarded at the \(2\,\sigma \) level already from current data (assuming that the coupling strength to gauge bosons is the same one as in the \({\mathscr {CP}}\)-even case). However, the prospects for the LHC to determine a certain level of \({\mathscr {CP}}\)-odd admixture to the Higgs state are less clear [268].
Only in the lower allowed range for \(M_A\) in this scenario larger deviations from the phenomenology of the light \({\mathscr {CP}}\)-even MSSM Higgs with respect to the SM Higgs can be expected. Depending on the level of decoupling, the LHC might be able to detect this kind of deviations, e.g. in enhanced rates involving the decay to two photons or in suppressed rates in the decay to \(\tau \) leptons or b quarks.
2.5.5 Prospects for the MSSM Higgs bosons at the LC
The mass of a SM-like Higgs boson at \(\sim \)125 GeV can be determined at the level of \(50 \mathrm {MeV}\,\).
Using the Z recoil method the production cross section of a SM-like Higgs can be determined independently of the decay products, see Sect. 2.3. This allows for a model-independent measurement of the Higgs couplings at the per-cent level; see Table 18. In particular, a determination of the tri-linear Higgs self-coupling at the level of 15 % can be expected.
The spin can be determined unambiguously from a production cross section threshold scan.
The \({\mathscr {CP}}\) decomposition can be determined, in particular, using the channel \(e^+e^- \rightarrow t \bar{t} H\) [270, 271].
The reach for the heavy Higgs bosons can be extended to higher masses in particular for lower and intermediate values of \(\tan \beta \) up to \(M_A\lesssim \sqrt{s}/2\) (and possibly beyond, depending on the SUSY parameters [272]).
An indirect determination of \(M_A\) can be performed via a precise measurement of the Higgs couplings, where a sensitivity up to \(800 \mathrm {GeV}\,\) was found [273].
In the \(\gamma \gamma \) option of the LC the Higgs bosons can be produced in the s-channel, and a reach up to \(M_A\lesssim 0.8 \sqrt{s}\) can be realised [274] (see also Refs. [275, 276]).
2.6 General multi-Higgs structures^{15}
2.6.1 Introduction
Examples of the precision of SM-like Higgs observables at a \(\sqrt{s}=500 \mathrm {GeV}\,\) LC assuming a Higgs-boson mass of \(125 \, \mathrm {GeV}\,\). The results are based on the ILC set-up. For the direct measurements, an integrated luminosity of \({\mathscr {L}}^\mathrm{int} = 500~\mathrm {fb}^{-1}\) is assumed. For the indirect measurements at GigaZ, a running time of approximately one year is assumed, corresponding to \({\mathscr {L}}= \) \({\mathscr {O}}(10~\mathrm {fb}^{-1})\). Taken from Ref. [269]
Observable | Expected precision (%) |
---|---|
\(M_H\) (GeV) | 0.03 |
\(g_{HWW} \) | 1.4 |
\(g_{HZZ} \) | 1.4 |
\(g_{Hbb} \) | 1.4 |
\(g_{Hcc} \) | 2.0 |
\(g_{H\tau \tau }\) | 2.5 |
\(g_{Htt} \) | 10 |
\(g_{HHH} \) | 40 |
\(\mathrm{BR}\, (H \rightarrow \gamma \gamma )\) | 25 |
\(\mathrm{BR}\, (H \rightarrow gg)\) | 5 |
\(\mathrm{BR}\, (H \rightarrow \mathrm {invisible})\) | 0.5 |
By putting an assumption of \(\mu ^2 < 0\) (and \(\lambda > 0\)), the shape of the potential becomes like a Mexican hat, and the electroweak symmetry is broken spontaneously at the vacuum \(\langle \Phi \rangle = (0, v/\sqrt{2})^T\). Consequently, weak gauge bosons, quarks and charged leptons obtain their masses from the unique vacuum expectation value (VEV) v (\(=(\sqrt{2}G_F)^{-1/2} \simeq 246\) GeV). However, there is no theoretical principle for the SM Higgs sector, and there are many possibilities for non-minimal Higgs sectors. While the current LHC data do not contradict the predictions of the SM, most of the extended Higgs sectors can also satisfy current data. These extended Higgs sectors are often introduced to provide physics sources to solve problems beyond the SM, such as baryogenesis, DM and tiny neutrino masses. Each scenario can predict a specific Higgs sector with additional scalars.
It is also known that the introduction of the elementary scalar field is problematic from the theoretical viewpoint, predicting the quadratic divergence in the radiative correction to the mass of the Higgs boson. Such a quadratic divergence causes the hierarchy problem. There are many scenarios proposed to solve the hierarchy problem such as supersymmetry, dynamical symmetry breaking, Extra dimensions and so on. Many models based on these new paradigms predict specific Higgs sectors in their low-energy effective theories.
Therefore, experimental determination of the structure of the Higgs sector is essentially important to deeply understand EWSB and also to find direction to new physics beyond the SM. The discovery of the 125-GeV Higgs boson at the LHC in 2012 is a big step to experimentally investigate the structure of the Higgs sector. From the detailed study of the Higgs sector, we can determine the model of new physics.
What kind of extended Higgs sectors can we consider? As the SM Higgs sector does not contradict the current data within the errors, there should be at least one isospin doublet field which looks like the SM Higgs boson. An extended Higgs sector can then contain additional isospin multiplets. There can be infinite kinds of extended Higgs sectors. These extended Higgs sectors are subject to constraints from the current data of many experiments including those of the electroweak \(\rho \)-parameter and for flavour changing neutral currents (FCNCs).
Extended Higgs sectors with a multi-doublet structure, in general, receive a severe constraint from the results of FCNC experiments. The data show that FCNC processes such as \(K^0 \rightarrow \mu ^+\mu ^-\), \(B^0-\bar{B}^0\) and so on are highly suppressed [277]. In the SM with a doublet Higgs field, the suppression of FCNC processes is perfectly explained by the so-called Glashow–Illiopoulos–Miani mechanism [279]. On the other hand, in general multi Higgs-doublet models where multiple Higgs doublets couple to a quark or a charged lepton, Higgs boson-mediated FCNC processes can easily occur at the tree level. In these models, in order to avoid such dangerous FCNC processes, it is required that these Higgs-doublet fields have different quantum numbers [280].
In Sect. 2.6.2, we discuss properties of the two Higgs-doublet model (2HDM), and its phenomenology at the LHC and the ILC. The physics of the model with the Higgs sector with a triplet is discussed in Sect. 2.6.3. The possibilities of more exotic extended Higgs sectors are briefly discussed in Sect. 2.6.4.
2.6.2 Two Higgs-doublet models
The 2HDM is one of the simplest extensions of the standard Higgs sector with one scalar doublet field. The model has many typical characteristics of general extended Higgs sectors, such as the existence of additional neutral Higgs states, charged scalar states, and the source of CP violation. In fact, the 2HDM often appears in the low-energy effective theory of various new physics models which try to solve problems in the SM such as the minimal supersymmetric SM (MSSM), to some models of neutrino masses, DM, and electrowak baryogenesis. Therefore, it is useful to study properties of 2HDMs with their collider phenomenology.
As already mentioned, in general 2HDMs, FCNCs can appear via tree-level Higgs-mediated diagrams, which are not phenomenologically acceptable. The simple way to avoid such dangerous FCNCs is to impose a discrete \(Z_2\) symmetry, under which the two doublets are transformed as \(\Phi _1\rightarrow +\Phi _1\) and \(\Phi _2\rightarrow -\Phi _2\) [280, 281, 282, 283]. Then each quark or lepton can couple with only one of the two doublets, so that the Higgs-mediated FCNC processes are forbidden at the tree level.
Four possible \(Z_2\) charge assignments of scalar and fermion fields to forbid tree-level Higgs-mediated FCNCs [146]
\(\Phi _1\) | \(\Phi _2\) | \(u_R\) | \(d_R\) | \(\ell _R\) | \(Q_L\) | \(L_L\) | |
---|---|---|---|---|---|---|---|
Type-I | \(+\) | \(-\) | \(-\) | \(-\) | \(-\) | \(+\) | \(+\) |
Type-II | \(+ \) | \(-\) | \(-\) | \(+\) | \(+\) | \(+\) | + |
Type-X | \(+\) | \(-\) | \(-\) | \(-\) | \(+\) | \(+\) | \(+\) |
Type-Y | \(+\) | \(-\) | \(-\) | \(+\) | \(-\) | \(+\) | \(+\) |
The coefficients for different type of Yukawa interactions [146]. \(c_\theta =\cos \theta ,~\mathrm{and }~s_\theta =\sin \theta \) for \(\theta = \alpha ,~\beta \)
\(\xi _h^u\) | \(\xi _h^d\) | \(\xi _h^\ell \) | \(\xi _H^u\) | \(\xi _H^d\) | \(\xi _H^\ell \) | \(\xi _A^u\) | \(\xi _A^d\) | \(\xi _A^\ell \) | |
---|---|---|---|---|---|---|---|---|---|
Type-I | \(c_\alpha /s_\beta \) | \(c_\alpha /s_\beta \) | \(c_\alpha /s_\beta \) | \(s_\alpha /s_\beta \) | \(s_\alpha /s_\beta \) | \(s_\alpha /s_\beta \) | \(\cot \beta \) | \(-\cot \beta \) | \(-\cot \beta \) |
Type-II | \(c_\alpha /s_\beta \) | \(-s_\alpha /c_\beta \) | \(-s_\alpha /c_\beta \) | \(s_\alpha /s_\beta \) | \(c_\alpha /c_\beta \) | \(c_\alpha /c_\beta \) | \(\cot \beta \) | \(\tan \beta \) | \(\tan \beta \) |
Type-X | \(c_\alpha /s_\beta \) | \(c_\alpha /s_\beta \) | \(-s_\alpha /c_\beta \) | \(s_\alpha /s_\beta \) | \(s_\alpha /s_\beta \) | \(c_\alpha /c_\beta \) | \(\cot \beta \) | \(-\cot \beta \) | \(\tan \beta \) |
Type-Y | \(c_\alpha /s_\beta \) | \(-s_\alpha /c_\beta \) | \(c_\alpha /s_\beta \) | \(s_\alpha /s_\beta \) | \(c_\alpha /c_\beta \) | \(s_\alpha /s_\beta \) | \(\cot \beta \) | \(\tan \beta \) | \(-\cot \beta \) |
In Fig. 64, decay branching ratios of additional Higgs bosons H, A, and \(H^\pm \) are plotted in each type of Yukawa interaction as a function of \(\tan \beta \) for the masses of 250 GeV. For simplicity, the SM-like limit \(\sin (\beta -\alpha )=1\) is taken. In this limit, the decay modes of \(H\rightarrow W^+W^-\), ZZ, hh as well as \(A\rightarrow Zh\) are absent. In this limit, decay branching ratios of the SM-like Higgs boson are completely the same as those in the SM at the tree level, so that we cannot distinguish models by precision measurements of the couplings of the SM-like Higgs boson h.^{16}
Constraints on the Higgs potential from perturbative unitarity and vacuum stability
Constraints on the Higgs potential from electroweak precision observables
Further constraints on the Higgs sector of the 2HDM are from the electroweak precision measurements. The S, T and U parameters [304] are sensitive to the loop effects of Higgs bosons [305, 306]. The T parameter corresponds to the electroweak \(\rho \) parameter, which is severely constrained by experimental observations as has been discussed. The mass splitting between the additional Higgs bosons are strongly bounded [307, 308]. This implies that the Higgs potential has to respect the custodial SU(2) symmetry approximately.
Flavour constraints on\(m_{H^\pm }\)and \(\tan \beta \)
Flavour experiments provide strong constraints on the 2HDMs through the \(H^\pm \) contribution to the flavour mixing observables at the tree level or at the loop level [146, 309, 310]. Because the amplitudes of these processes necessarily contain the Yukawa interaction, constraints on the 2HDM strongly depends on the type of Yukawa interaction. In Ref. [311], the limits on the general couplings from flavour physics are translated into those on the (\(m_{H^\pm },\tan \beta \)) plane for all four types of Yukawa interaction in the 2HDM, see Fig. 65, where Type III and Type IV correspond to Type Y and Type X, respectively. See also the more recent studies [312, 313, 314].
A strong exclusion limit is given from the result for the branching ratio of the \(B\rightarrow X_s\gamma \) process [315]. For Type-II and Type-Y, a \(\tan \beta \)-independent lower limit of \(m_{H^\pm }\gtrsim 380\) GeV is obtained [316] by comparing with the NNLO calculation [317]. For Type-I and Type-X, on the other hand, \(\tan \beta \lesssim 1\) is excluded for \(m_{H^\pm }\lesssim 800\) GeV, while no lower bound on \(m_{H^\pm }\) is obtained.
By the results for the \(B_{d}^0\)–\(\bar{B}^0_{d}\) mixing, lower \(\tan \beta \) regions (\(\tan \beta \le 1\)) are excluded for \(m_{H^\pm }\lesssim 500\) GeV for all types of Yukawa interaction [315].
Constraints in larger \(\tan \beta \) regions are obtained only for Type-II, which come from the results for leptonic meson decay processes [315], \(B\rightarrow \tau \nu \) [318] and \(D_s\rightarrow \tau \nu \) [319]. Upper bounds on \(\tan \beta \) are obtained at around 30 for \(m_{H^\pm }\simeq 350\) GeV and around 60 for \(m_{H^\pm }\simeq 700\) GeV [311]. On the other hand, the other types do not receive any strong constraint for large \(\tan \beta \) values, because the relevant couplings behave \(\xi _A^d\xi _A^\ell =\tan ^2\beta \) for Type-II while \(\xi _A^d\xi _A^\ell =-1\) (\(\cot ^2\beta \)) for Type-X and Type-Y (Type-I).
Constraint from the data at LEP/SLC, Tevatron and also from the current LHC data
At the LEP direct search experiments, lower mass bounds on H and A have been obtained as \(m_H>92.8\) GeV and \(m_A>93.4\) GeV in the \({\textit{CP}}\)-conservation scenario [320, 321]. Combined searches for \(H^\pm \) give the lower mass bound \(m_{H^\pm }>80\) GeV, by assuming \({\mathscr {B}}(H^+\rightarrow \tau ^+\nu )+{\mathscr {B}}(H^+\rightarrow c\bar{s})=1\) [322, 323, 324].
At the Fermilab Tevatron, CDF and D0 Collaborations have studied the processes of \(p\bar{p}\rightarrow b\bar{b}H/A\), followed by \(H/A\rightarrow b\bar{b}\) or \(H/A\rightarrow \tau ^+\tau ^-\) [325, 326, 327]. By using the \(\tau ^+\tau ^-\) (\(b\bar{b}\)) decay mode, which can be sensitive for the cases of Type-II (Type-II and Type-Y), upper bounds on \(\tan \beta \) have been obtained to be from about 25 to 80 (40 to 90) for \(m_A\) from 100 to 300 GeV, respectively. For the direct search of \(H^\pm \), the decay modes of \(H^\pm \rightarrow \tau \nu \) and \(H^\pm \rightarrow cs\) have been investigated by using the production from the top quark decay \(t\rightarrow bH^\pm \) [328, 329, 330]. Upper bounds on \({\mathscr {B}}(t\rightarrow bH^\pm )\) have been obtained, which can be translated into the bound on \(\tan \beta \) in various scenarios. For Type-I with \(H^\pm \) heavier than the top quark, upper bounds on \(\tan \beta \) have been obtained to be from around 20 to 70 for \(m_{H^\pm }\) from 180 to 190 GeV, respectively [328].
Prospect of extra Higgs-boson searches at the LHC (13–14 TeV)
\(H/A (+ b\bar{b})\) inclusive and associated production followed by the \(H/A\rightarrow \tau ^+\tau ^-\) decay [342].
\(H/A+b\bar{b}\) associated production followed by the \(H/A\rightarrow b\bar{b}\) decay [342, 343, 344, 345].
\(gb\rightarrow tH^\pm \) production followed by the \(H^\pm \rightarrow tb\) decay [346, 347].
\(q\bar{q}\rightarrow HA\rightarrow 4\tau \) process [341, 348].
In Fig. 66, the contour plots of the expected exclusion regions [\(2\sigma \) confidence level (CL)] in the \((m_\phi ,\tan \beta )\) plane are shown at the LHC \(\sqrt{s}=14\) TeV with the integrated luminosity of 300 fb\(^{-1}\) (thick solid lines) and 3000 fb\(^{-1}\) (thin dashed lines), where \(m_\phi \) represents common masses of additional Higgs bosons. From the left panel to the right panel, the results for Type-I, Type-II, Type-X and Type-Y are shown separately. Following the analysis in Ref. [338], the reference values of the expected numbers of signal and background events are changed at the several values of \(m_\phi \) [295, 340], which makes sharp artificial edges of the curves in Fig. 66.
For Type-I, H / A production followed by the decay into \(\tau ^+\tau ^-\) can be probed for \(\tan \beta \lesssim 3\) and \(m_{H,A}\le 350\) GeV, where the inclusive production cross section is enhanced by the relatively large top Yukawa coupling with the sizeable \(\tau ^+\tau ^-\) branching ratio. The \(tH^\pm \) production decaying into \(H^\pm \rightarrow tb\) can be used to search \(H^\pm \) in relatively smaller \(\tan \beta \) regions. \(H^\pm \) can be discovered for \(m_{H^\pm } < 800\) GeV and \(\tan \beta \lesssim 1\) (2) for the integrated luminosity of 300 fb\(^{-1}\) (3000 fb\(^{-1}\)).
For Type-II, the inclusive and the bottom-quark-associated production processes of H / A with the decay into \(\tau ^+\tau ^-\) or the \(b \overline{b}\) can be used to search H and A for relatively large \(\tan \beta \). They can also be used in relatively small \(\tan \beta \) regions for \(m_{H,A}\lesssim 350\) GeV. \(H^\pm \) can be searched by the \(tH^\pm \) production with \(H^\pm \rightarrow tb\) decay for \(m_{H^\pm }\gtrsim 180\) GeV for relatively small and large \(\tan \beta \) values. The region of \(m_{H^\pm }\gtrsim 350\) GeV (500 GeV) could be excluded with the 300 fb\(^{-1}\) (3000 fb\(^{-1}\)) data.
For Type-X, H and A can be searched via the inclusive production and HA pair production by using the \(\tau ^+\tau ^-\) decay mode, which is dominant. The inclusive production could exclude the region of \(\tan \beta \lesssim 10\) with \(m_{H,A}\lesssim 350\) GeV. Regions up to \(m_{H,A}\simeq 500\) GeV (700 GeV) with \(\tan \beta \gtrsim 10\) could be excluded by using the pair production with the 300 fb\(^{-1}\) (3000 fb\(^{-1}\)) data. The search for \(H^\pm \) is similar to that for Type-I.
Finally, for Type-Y, the inclusive production of H and A f ollowed by \(H/A \rightarrow \tau ^+\tau ^-\) can be searched for the regions of \(\tan \beta \lesssim 2\) and \(m_{H,A}\le 350\) GeV. The bottom-quark associated production of H and A with \(H/A\rightarrow b\bar{b}\) can be searched for the regions of \(\tan \beta \gtrsim 30\) up to \(m_{H,A}\simeq 800\) GeV. The search of \(H^\pm \) is similar to that for Type-II.
If \(H^\pm \) is discovered at the LHC, its mass could be determined immediately [338, 352]. Then the determination of the type of the Yukawa interaction becomes important. At the LHC, however, we would not completely distinguish the types of Yukawa interaction, because the Type-I and Type-X, or Type-II and Type-Y have a common structure for the \(tbH^\pm \) interaction. In addition, as seen in Fig. 66, there can be no complementary process for the neutral Higgs-boson searches in some parameter regions; e.g., \(m_{H,A}\gtrsim 350\) GeV with relatively small \(\tan \beta \), depending on the type of the Yukawa interaction. At the ILC, on the other hand, as long as \(m_{H,A}\lesssim 500\) GeV, the neutral Higgs bosons can be produced and investigated almost independent of \(\tan \beta \). Therefore, it is quite important to search for the additional Higgs bosons with the mass of 350–500 GeV, and to determine the models and parameters at the ILC, even after the LHC.
Notice that the above results are obtained in the SM-like limit, \(\sin (\beta -\alpha )=1\). A deviation from the SM-like limit causes appearance of additional decay modes such as \(H\rightarrow W^+W^-\), ZZ, hh as well as \(A\rightarrow Zh\) [86, 353, 354, 355]. Especially, for Type-I with a large value of \(\tan \beta \), branching ratios of these decay modes can be dominant even with a small deviation from the SM-like limit [146, 354]. Therefore, searches for additional Higgs bosons in these decay modes can give significant constraints on the deviation of \(\sin (\beta -\alpha )\) from the SM-like limit [336, 337], which is independent of coupling constants of hVV.
Prospect for the searches for the additional Higgs bosons at the ILC
At LCs the main production mechanisms of additional Higgs bosons in the 2HDM are \(e^+ e^- \rightarrow H A\) and \(e^+ e^- \rightarrow H^+ H^-\), where a pair of additional Higgs bosons is produced via gauge interactions as long as kinematically allowed. For energies below the threshold, the single production processes, \(e^+e^- \rightarrow H(A) f \bar{f}\) and \(e^+ e^- \rightarrow H^\pm f \bar{f}'\) are the leading contributions [356]. They are enhanced when the relevant Yukawa couplings \(\phi f \bar{f}^{(')}\) are large. The cross sections of these processes have been studied extensively [206, 356, 357, 358], mainly for the MSSM or for the Type-II 2HDM.
Expected signatures to be observed at the LHC and ILC for the benchmark scenarios with \(m_\phi =220\) GeV [295]. Observable final states are listed as the signatures of additional Higgs bosons, H, A and \(H^{\pm }\). LHC300, LHC3000, ILC500 represent the LHC run of 300, 3000 fb\(^{-1}\) luminosity, ILC run of 500 GeV, respectively
\((m_\phi , \tan \beta )\) | Type-I | Type-II | Type-X | Type-Y | ||||
---|---|---|---|---|---|---|---|---|
H, A | \(H^\pm \) | H, A | \(H^\pm \) | H, A | \(H^\pm \) | H, A | \(H^\pm \) | |
(220 GeV, 20) | ||||||||
LHC300 | \(-\) | \(-\) | \(\tau \tau \), bb | tb | \(4\tau \) | \(-\) | bb | tb |
LHC3000 | \(-\) | \(-\) | \(\tau \tau \), bb | tb | \(4\tau \) | \(-\) | bb | tb |
ILC500 | \(4b,2b2\tau ,4g\)\(,2b2g,2\tau 2g\) | tbtb | \(4b,2b2\tau \),\(4\tau \) | \(tbtb,tb\tau \nu ,\)\(\tau \nu \tau \nu ^{}\) | \(4\tau \) | \(tb\tau \nu \),\(\tau \nu \tau \nu \) | 4b | tbtb, tbcb |
(220 GeV, 7) | ||||||||
LHC300 | – | – | \(\tau \tau \) | tb | \(4\tau \) | – | – | tb |
LHC3000 | – | tb | \(\tau \tau \) | tb | \(\tau \tau ,4\tau \) | – | – | tb |
ILC500 | \(4b,2b2\tau ,4g, \,2b2g,2\tau 2g\) | tbtb | \(4b,2b2\tau ,4\tau \) | \(tbtb,tb\tau \nu \), \(\tau \nu \tau \nu \) | \(2b2\tau ,4\tau \) | \(tbtb,tb\tau \nu \), \(\tau \nu \tau \nu \) | 4b | tbtb, tbcb |
(220 GeV, 2) | ||||||||
LHC300 | \(-\) | tb | \(\tau \tau \) | tb | \(\tau \tau ,4\tau \) | tb | \(-\) | tb |
LHC3000 | \(\tau \tau \) | tb | \(\tau \tau \) | tb | \(\tau \tau ,4\tau \) | tb | \(-\) | tb |
ILC500 | \(4b,2b2\tau ,4g, 2b2g,2\tau 2g\) | tbtb | \(4b,2b2\tau \), \(4\tau ,2b2g\) | tbtb,\(tb\tau \nu \) | \(4b,2b2\tau \), \(4\tau \) | tbtb,\(tb\tau \nu \) | \(4b,2b2\tau ,\,2b2g\) | tbtb |
Both the pair and the single production processes of additional Higgs bosons mostly result in four-particle final states (including neutrinos). In Ref. [295], the cross sections of various four-particle final states are studied for given masses of additional Higgs bosons and \(\tan \beta \) with setting \(\sin (\beta -\alpha )=1\), and draw contour curves where the cross sections are 0.1 fb. This value is chosen commonly for all processes as it could be regarded as a typical order of magnitude of the cross section of the additional Higgs boson production [358]. In addition, this value can also be considered as a criterion for observation with the expected integrated luminosity at the ILC [56, 206]. Certainly, the detection efficiencies are different for different four-particle final states, and the decay of unstable particles such as tau leptons and top quarks have to be considered if they are involved. We here restrict ourselves to simply compare the various four-particle production processes in four types of Yukawa interaction in the 2HDMs with taking the criterion of 0.1 fb as a magnitude of the cross sections. Expected background processes and a brief strategy of observing the signatures are discussed in Ref. [295].
In Fig. 67, contour plots of the cross sections of four-particle production processes through H and/or A are shown in the \((m_{H/A},\tan \beta )\) plane (upper figures), and those through \(H^\pm \) are shown in the\((m_{H^\pm },\tan \beta )\) plane (lower figures) for the collision energy to be \(\sqrt{s}=500\) GeV. From left to right, the figures correspond to the results in Type 1, Type II, Type X and Type Y. We restrict ourselves to consider the degenerated mass case, \(m_H=m_A\).
In Fig. 68, contour plots of the cross sections of four-particle production processes through H and/or A are shown in the \((m_{H/A},\tan \beta )\) plane (upper figures), and those through \(H^\pm \) are shown in the\((m_{H^\pm },\tan \beta )\) plane (lower figures) for the collision energy to be \(\sqrt{s}=1\) TeV. From left to right, the figures correspond to the results in Type 1, Type II, Type X and Type Y. We restrict ourselves to consider the degenerated mass case, \(m_H=m_A\).
We here give a comment on the SM background processes and their cross sections [295]. In general, for the four-particle production processes, the SM background cross sections are larger for \(\sqrt{s}=250\) GeV, but decrease with the collision energy. The typical orders of cross sections are of the order of 1–10 fb for the \(Z/\gamma \)-mediated processes, and of the order of 10–100 fb for the processes which are also mediated by \(W^\pm \). For the four-quark production processes, gluon exchange diagrams also contribute. In order to reduce the background events, efficient kinematical cuts are required.
The cross section of the 4t production is very small in the SM. Therefore, a clean signature can be expected to be detected in this mode. Detailed studies on the signal and background processes for tbtb production can be found in Ref. [357], and the signal-to-background analysis for the \(4\tau \) production can be found in Ref. [359] with the reconstruction method of the masses of additional Higgs bosons.
Finally, we discuss some concrete scenarios to show the complementarity of direct searches for the additional Higgs bosons in the 2HDMs at the LHC and the ILC. As benchmark scenarios, three cases \(\tan \beta =2\), 7 and 20 are considered for \(m_\phi =220\) GeV and \(\sin (\beta -\alpha )=1\), where \(m_\phi \) represents the common mass of H, A and \(H^\pm \). In Table 21, the expected signatures of H / A and \(H^\pm \) are summarised to be observed at the LHC with 300, 3000 fb\(^{-1}\) and at the ILC with \(\sqrt{s}=500\) GeV.
First, for the case of \((m_\phi , \tan \beta ) = (220~\mathrm{GeV}, 20)\). no signature is predicted for Type-I, while different signatures are predicted for Type-II, Type-X and Type-Y at the LHC with 300 and 3000 fb\(^{-1}\). Therefore those three types could be discriminated at the LHC. On the other hand, at the ILC with \(\sqrt{s}=500\) GeV, all the four types of the Yukawa interaction including Type-I predict signatures which are different from each other. Therefore, complete discrimination of the type of Yukawa interaction could be performed at the ILC.
Next, we turn to the second case with \((m_\phi , \tan \beta ) = (220~\mathrm{GeV}, 7)\). At the LHC with 300 fb\(^{-1}\), Type-I cannot be observed, while Type-II, Type-X and Type-Y are expected to be observed with different signatures. At the LHC with 3000 fb\(^{-1}\), the signature of Type-I can also be observed with the same final state as Type-Y. Type-I and Type-Y can be basically separated, because for Type-Y the signals can be observed already with 300 fb\(^{-1}\), while for Type-I that can be observed only with 3000 fb\(^{-1}\). Therefore, at the LHC with 3000 fb\(^{-1}\), the complete discrimination can be achieved. At the ILC, the four types of Yukawa interaction can also be separated by a more variety of the signatures for both channels with the neutral and charged Higgs bosons.
Finally, for the case of \((m_\phi , \tan \beta ) = (220~\mathrm{GeV}, 2)\), signals for all the four types of Yukawa interaction can be observed at the LHC with 300 fb\(^{-1}\). However, the signatures of Type-I and Type-Y are identical, so that the two types cannot be discriminated. With 3000 fb\(^{-1}\), the difference between the Type-I and Type-Y emerges in the H / A signature. Therefore the two types can be discriminated at this stage. Again, at the ILC, the four types can also be separated with a more variety of the signatures for both channels with the neutral and charged Higgs bosons.
Fingerprinting the type of the 2HDM by precision measurement of the Higgs couplings at the ILC
The analysis including radiative corrections has been done recently [293, 294]. We show the one-loop results for the Yukawa couplings in the planes of fermion scale factors. In Fig. 70, predictions of various scale factors are shown on the \(\kappa _\tau \) vs. \(\kappa _b\) (upper panels), and \(\kappa _\tau \) vs. \(\kappa _c\) (bottom panels) planes. When we consider the case with \(\sin (\beta -\alpha )\ne 1\), the sign dependence of \(\cos (\beta -\alpha )\) to \(\kappa _f\) is also important. We here show the both cases with \(\cos (\beta -\alpha )<0\). The value of \(\tan \beta \) is discretely taken as \(\tan \beta \)=1, 2, 3 and 4. The tree-level predictions are indicated by the black dots, while the one-loop corrected results are shown by the red for \(\sin ^2(\beta -\alpha )=0.99\) and blue for \(\sin ^2(\beta -\alpha )=0.95\) regions where the values of \(m_\Phi \) and M are scanned over from 100 GeV to 1 TeV and 0 to \(m_\Phi \), respectively. All the plots are allowed by the unitarity and vacuum stability bounds.
Even when we take into account the one-loop corrections to the Yukawa couplings, this behaviour; i.e., predictions are well separated among the four types of THDMs, does not so change as we see the red and blue coloured regions. Therefore, we conclude that all the 2HDMs can be distinguished from each other by measuring the charm, bottom and tau Yukawa couplings precisely when the gauge couplings hVV are deviated from the SM prediction with \(\mathscr {O}(1)\) %.^{17}
2.6.3 Higgs triplet models
The potential respects additional global symmetries in some limits. First, there is the global U(1) symmetry in the potential in the limit of \(\mu = 0\), which conserves the lepton number. As long as we assume that the lepton number is not spontaneously broken, the triplet field does not carry the VEV; i.e., \(v_{\varDelta }=0\). Next, an additional global SU(2) symmetry appears in the limit where \(\mu = \lambda _5 =0\). Under this SU(2) symmetry, \(\Phi \) and \({\varDelta }\) can be transformed with the different SU(2) phases. All the physical triplet-like Higgs bosons are then degenerate in mass.
The condition for the vacuum stability bound has been derived in Ref. [395], where we require that the Higgs potential is bounded from below in any direction of the large scalar fields region. The unitarity bound in the HTM has been discussed in Ref. [395]. In Fig. 72, the excluded regions by the unitarity bound and the vacuum stability condition are shown for \(\lambda _1 = m_h^2/(2v^2)\simeq 0.13\) in the \(\lambda _4\)–\(\lambda _5\) plane [375]. We take \(\lambda _{\varDelta }=1.5\) (3) in the left (right) panel. Excluded regions by the unitarity and vacuum stability bounds are shown.
The fraction of the VEVs \(\tan \beta \) and the scaling factors \(\kappa _f\) and \(\kappa _V\) in the extended Higgs sectors with universal Yukawa couplings [340]
\(\tan \beta \) | \(\kappa _f\) | \(\kappa _V^{}\) | |
---|---|---|---|
Doublet-singlet model | – | \(\cos \alpha \) | \(\cos \alpha \) |
Type-I THDM | \(v_0/v_\text {ext}^{}\) | \(\cos \alpha /\sin \beta =\sin (\beta -\alpha )+\cot \beta \cos (\beta -\alpha )\) | \(\sin (\beta -\alpha )\) |
GM model | \(v_0/(2\sqrt{2}v_\text {ext}^{})\) | \(\cos \alpha /\sin \beta \) | \(\sin \beta \cos \alpha -\tfrac{2\sqrt{6}}{3} \cos \beta \sin \alpha \) |
Doublet-septet model | \(v_0/(4v_\text {ext}^{})\) | \(\cos \alpha /\sin \beta \) | \(\sin \beta \cos \alpha -4 \cos \beta \sin \alpha \) |
2.6.4 Other exotic models
In Fig. 78, the predictions for the scale factors of the universal Yukawa coupling \(\kappa _F\) and the gauge coupling \(\kappa _V\) are plotted in exotic Higgs sectors for each set of mixing angles. The current LHC bounds, expected LHC and ILC sensitivities for \(\kappa _F\) and \(\kappa _V\) are also shown at the 68.27 % CL. Therefore, exotic Higgs sectors can be discriminated by measuring \(\kappa _V\) and \(\kappa _F\) precisely. For details, see Refs. [339, 340].
2.6.5 Summary
Although the Higgs boson with the mass 125 GeV was found at the LHC, knowledge about the structure of the Higgs sector is very limited. Since there are no theoretical principles for the minimal Higgs sector with one Higgs doublet, there are many possibilities of non-minimal Higgs sectors. Such extended Higgs sectors appear in many new physics models beyond the SM. Therefore, the Higgs sector is a window to new physics, and we can explore new physics from clarifying the structure of the Higgs sector by coming collider experiments. At the LHC, direct discovery of additional Higgs bosons can be expected as long as they are not too heavy. On the other hand, the Higgs sector can also be explored by precisely measuring the properties of the discovered Higgs boson h accurately. The precision measurements will be performed partially at the high luminosiity LHC with 3000 fb\(^{-1}\). Using the high ability of the ILC for measuring the Higgs-boson couplings, we can further test extended Higgs sectors, and consequently narrow down the new physics models.
2.7 Higgs physics in strong-interaction scenarios^{20}
In composite Higgs models, the deviations from the SM point \(a=b=1\) are controlled by the ratio of the weak scale over the compositeness scale f. In these models the Higgs boson is a composite bound state which emerges from a strongly interacting sector [421, 422, 423, 424, 425, 426]. The good agreement with the electroweak precision data is achieved by a mass gap that separates the Higgs scalar from the other resonances of the strong sector. This mass gap arises dynamically in a natural way if the strongly interacting sector has a global symmetry G, which is spontaneously broken at a scale f to a subgroup H so that the coset G / H contains a fourth Nambu–Goldstone boson which is identified with the Higgs boson. Composite Higgs models can be viewed as a continuous interpolation between the SM and technicolour type models. With the compositeness scale of the Higgs boson given by the dynamical scale f, the limit \(\xi \equiv v^2/f^2 \rightarrow 0\) corresponds to the SM where the Higgs boson appears as an elementary light particle and the other resonances of the strong sector decouple. In the limit \(\xi \rightarrow 1\) the Higgs boson does not couple to the \(V_L\) any longer and other (heavy) resonances are necessary to ensure unitarity in the gauge boson scattering. The \(\xi \rightarrow 1\) limit corresponds to the technicolour paradigm [90, 91] where the strong dynamics directly breaks the electroweak symmetry down to the electromagnetism subgroup.
2.7.1 Effective Lagrangian and Higgs couplings
Parameters | SILH | MCHM4 | MCHM5 |
---|---|---|---|
a | \(1-c_H\xi /2\) | \(\sqrt{1-\xi }\) | \(\sqrt{1-\xi }\) |
b | \(1-2 c_H \xi \) | \(1-2\xi \) | \(1-2\xi \) |
\(b_3\) | \(-\frac{4}{3}\xi \) | \(-\frac{4}{3} \xi \sqrt{1-\xi }\) | \(-\frac{4}{3} \xi \sqrt{1-\xi }\) |
c | \(1-(c_H/2+c_y) \xi \) | \(\sqrt{1-\xi }\) | \(\frac{1-2\xi }{\sqrt{1-\xi }}\) |
\(c_2\) | \(-(c_H+3c_y)\xi /2\) | \(-\xi /2\) | \(-2\xi \) |
\(d_3\) | \(1+(c_6 - 3 c_H/2) \xi \) | \(\sqrt{1-\xi }\) | \(\frac{1-2\xi }{\sqrt{1-\xi }}\) |
\(d_4\) | \(1+(6 c_6 - 25 c_H/3) \xi \) | \(1-7 \xi /3\) | \(\frac{1-28\xi (1-\xi )/3}{1-\xi }\) |
The Higgs anomalous couplings affect both the Higgs production and decay processes. The Higgs boson branching ratios of a 125 GeV Higgs boson are shown in Fig. 79 for MCHM5. For \(\xi =0.5\) the Higgs boson becomes fermiophobic and the branching ratios into fermions and gluons vanish, while the ones into gauge bosons become enhanced. As explained above, in MCHM4 the branching ratios are unchanged. The modified production cross sections can easily be obtained from the corresponding SM results by rescaling with the appropriate coupling modification factors squared. As the QCD couplings are not affected the higher order QCD corrections can be taken over from the SM, while the EW corrections would change and have to be omitted as they are not available so far.
The anomalous couplings can be tested by a measurement of the Higgs interaction strengths. In case of a universal coupling modification as, e.g., in MCHM4 the production rates and the total width have to be tested. At an \(e^+e^-\) linear collider an accuracy of a few per-cent can be achieved in the measurement of the SM Higgs couplings to gauge bosons and fermions [56]. For an investigation of the prospects for the determination of \(\xi \) at the LHC, see Ref. [440]. In Ref. [367] a study of Higgs couplings performed in the context of genuine dimension-six operators showed that a sensitivity of up to \(4\pi f \sim 40\) TeV can be reached for a 120 GeV Higgs boson already at 500 GeV with \(1ab^{-1}\) integrated luminosity. At the high-energy phase of the CLIC project, i.e., at 3 TeV with \(2ab^{-1}\) integrated luminosity, the compositeness scale of the Higgs boson will be probed up to 60–90 TeV [441]. Also the total width of a 125 GeV Higgs boson can be measured at a few per-cent precisely already at the low-energy phase of the ILC programme.
2.7.2 Strong processes
If no new particles are discovered at the LHC, deviations from the SM predictions for production and decay rates can point towards models with strong dynamics. It is, however, only the characteristic signals of a composite Higgs boson in the high-energy region which unambiguously imply the existence of new strong interactions. Since in the composite Higgs scenario the \(V_L V_L\) scattering amplitude is not fully unitarised the related interaction necessarily becomes strong and eventually fails tree-level unitarity at the cutoff scale. The VV scattering therefore becomes strong at high energies. As the transversely polarised vector boson scattering is numerically large in the SM, the test of the energy growth in longitudinal gauge boson scattering is difficult at the LHC [418]. Another probe of the strong dynamics at the origin of EWSB is provided by longitudinal vector boson fusion in Higgs pairs which also grows with the energy. For the test of strong double Higgs production the high-luminosity upgrade of the LHC would be needed, however [418]. Besides testing the high-energy behaviour in strong double Higgs production, new resocances unitarising the scattering amplitudes can be searched for. The ILC has been shown to be able to test anomalous strong gauge couplings up to a scale \(\sim \)3 TeV and exclude \(\rho \)-like resonances below 2.5 TeV [56].
2.7.3 Non-linear Higgs couplings
The double Higgs-strahlung process dominates at low energies, and in the MCHM4 and MCHM5 it is always smaller than in the SM, which is due to the suppressed Higgs-gauge couplings. On the other hand, the WW fusion process, which becomes important for higher c.m. energies, is enhanced compared to the SM for non-vanishing values of \(\xi \) [442, 443]. This are due to interference effects related to the anomalous Higgs couplings. Furthermore, the amplitude grows like the c.m. energy squared contrary to the SM where it remains constant. The sensitivity of double Higgs-strahlung and gauge boson fusion processes to the tri-linear Higgs self-coupling of the corresponding model can be studied by varying the Higgs tri-linear coupling in terms of the respective self-interaction of the model in consideration, hence \(\lambda _{HHH} (\kappa )= \kappa \, \lambda _{HHH}^{{\mathrm{MCHM4,5}}}\). This gives an estimate of how accurately the Higgs pair production process has to be measured in order to extract \(\lambda _{HHH}\) within in the investigated model with a certain precision. Note, however, that this does not represent a test of models beyond the actually investigated theory. Figure 82 shows for the SM and for the MCHM5 with three representative \(\xi \) values (\(\xi =0.2,0.5,0.8\)) the normalised double Higgs production cross sections for Higgs-strahlung and gauge boson fusion, respectively, at two c.m. energies, \(\sqrt{s}=500\) GeV and 1 TeV, as a function of the modification factor \(\kappa \). The cross sections are normalised with respect to the double Higgs production cross sections at \(\kappa =1\) of the respective model. As can be inferred from the figure, both Higgs-strahlung and double Higgs production are more sensitive to \(\lambda _{HHH}\) at lower c.m. energies. This is due to the suppression of the propagator in the diagrams which contain the tri-linear Higgs self-coupling with higher energies. In addition in WW fusion the t- and u-channel diagrams, insensitive to this coupling, become more important with rising energy. A high-energy \(e^+e^-\) collider can exploit the WW fusion process to study the deviations in the coupling between two Higgs bosons and two gauge bosons by looking at the large \(m_{HH}\) invariant mass distribution [441]. The sensivity obtained on \(\xi \) via this process is almost an order of magnitude better than the one obtained from the study of double Higgs-strahlung [441].
The parton level analysis in Refs. [442, 443] showed that both double Higgs-strahlung and WW fusion have, in the 4b final state from the decay of the two 125 GeV Higgs bosons, sensitivity to a non-vanishing \(\lambda _{HHH}\) at the 5\(\sigma \) level in almost the whole \(\xi \) range, with the exception of \(\xi =0.5\) in MCHM5, where the tri-linear Higgs coupling vanishes, cf. Table 23.
2.7.4 Top sector
The fermionic sector of composite Higgs models, in particular the top sector, also shows an interesting phenomenology. With the fermion coupling strengths being proportional to their masses the top quark has the strongest coupling to the new sector and is most sensitive to new physics. It is hence natural to consider one of the two top helicities to be partially composite. The top-quark mass then arises through linear couplings to the strong sector. ATLAS and CMS already constrained the top partners to be heavier than 600–700 GeV at 95 % confidence level [444]. The associated new heavy top quark resonances have been shown to influence double Higgs production through gluon fusion [445, 446]. At \(e^+e^-\) colliders these new resonances can be searched for either in single or in pair production [447].
2.7.5 Summary
2.8 The Higgs portal^{21}
A large fraction of matter in the universe is dark and not incorporated in the SM. Nevertheless, this new kind of invisible matter is expected to interact with the SM fields, naturally by gravitational interaction. However, another path could be opened by a Higgs portal which connects the SM Higgs field with potential Higgs fields in the dark sector, respecting all symmetry principles and well-founded theoretical SM concepts like renormalisability.
Even though the particles of the novel sector are invisible, the portal nevertheless induces observable signals in the SM, in the Higgs sector in particular. Mixings among Higgs bosons of the SM and of the dark sector modify Higgs couplings to the SM particles and give rise to invisible Higgs decays (beyond the cascades to neutrinos).
From Eq. (66), we need to interpret the strong Higgs exclusion for heavy Higgs masses as a sign of a highly suppressed production cross section for heavier Higgs-like resonances. That heavy Higgs copies need to be weakly coupled in simple model-building realisations is already known from the investigation of electroweak precision measurements performed during the LEP era. This complements the requirement to include unitarising degrees of freedom for longitudinal gauge boson scattering \(V_LV_L\rightarrow V_L V_L\)\((V=W^\pm , Z)\), and, constraining to less extent, massive quark annihilation to longitudinal gauge bosons \(q\bar{q}\rightarrow V_LV_L\). Saturating all three of these requirements fixes key characteristics of the phenomenological realisation of the Higgs mechanism, and does not allow dramatic modifications of the couplings \(\{g_i\}\) in Eq. (66) away from the SM expectation of a light Higgs – the common predicament of electroweak-scale model building. In this sense gaining additional sensitivity to invisible Higgs decays (or the Higgs total width in general) beyond the limitations of the LHC hadronic environment is crucial to the understanding of electroweak physics at the desired level, before the picture will be clarified to the maximum extent possible at a LC.
The aforementioned Higgs-portal model [80, 81, 455] provides a theoretically well-defined, renormalisable, and yet minimal framework to explore both effects in a consistent way [460]: the influence of \({\varGamma }_{\text {inv}}\) on the Higgs phenomenology is captured, while heavier Higgs boson-like particles with suppressed couplings are naturally incorporated. Therefore, the Higgs-portal model not only provides a well-motivated SM Higgs sector extension in the context of DM searches^{22} and current data, but it represents an ideal model to generalise the SM in its phenomenologically unknown parameters to facilitate the SM’s validation by constraining the additional portal parameters beyond introducing biases (e.g. \({\varGamma }_H^{\text {tot}}\equiv {\varGamma }_H^{\text {SM}}\)).
We have also included cascade decays \({\varGamma }^{HH}_2\) (if they are kinematically allowed for \(M_2\ge 2M_1\)) and the possibility for a hidden partial decay width in Eq. (72b). The latter naturally arise if the hidden sector has matter content with \(2m\le m_{H_1}\), i.e. in models with light DM candidates. Weak coupling of the heavier Higgs-like state is made explicit when correlating the Higgs-portal model with electroweak precision constraints [460].
Generically, the branching ratio of the heavier Higgs boson to two light Higgs states is small (Fig. 84) and kinematically suppressed, so that a direct measurement of the cascade decay at the LHC is challenging. Measurement strategies targeting invisible Higgs-boson decays at the LHC [466] are based on measurements in weak boson fusion [467] and associated production [468, 469]. Recent re-analysis of the monojet+Higgs production [452, 470], however, suggest that additional sensitivity can be gained in these channels, at least for the 7 and 8 TeV data samples.
In Fig. 86 we show a hypothetical situation, where \(H_2\) is discovered at the LHC with \({\mathscr {R}}_2=0.4\); the error is given by a more precise measurement at a 350 GeV LC, see Fig. 84. The measurement of \({\mathscr {J}}_2=0.4\) allows one to reconstruct \(\sin ^2\chi \), which can be seeded to a reconstruction algorithm [460] that yields the full Higgs-portal potential Eq. (68).
To summarise, the Higgs portal can open the path to the dark sector of matter and can allow crucial observations on this novel kind of matter in a global way. While first hints may be expected from LHC experiments, high-precision analyses of Higgs properties and the observation of invisible decays at LC can give rise to a first transparent picture of a new world of matter.
2.9 The NMSSM^{23}
In view of the mass of 125–126 GeV of the at least approximately Standard Model-like Higgs boson \(H_{\mathrm {SM}}\) measured at the LHC, the NMSSM has received considerable attention: In contrast to the MSSM, no large radiative corrections to the Higgs mass (implying fine tuning in parameter space) are required in order to obtain \(M_{H_{\mathrm {SM}}}\) well above \(M_Z\), the upper bound on \(M_{H_{\mathrm {SM}}}\) at tree level in that model. In the NMSSM, additional tree-level contributions to \(M_{H_{\mathrm {SM}}}\) originate from the superpotential Eq. (77) [89]. Also a mixing with a lighter mostly singlet-like Higgs boson can increase the mass of the mostly Standard-Model-like Higgs boson [475], in which case one has to identify \(H_{\mathrm {SM}}\) with \(H_2\). Both effects allow one to obtain \(M_{H_{\mathrm {SM}}}\sim \) 125–126 GeV without fine tuning and, moreover, such a mixing could easily explain an enhanced branching fraction of this Higgs boson (from now on denoted as \(H_{125}\)) into \(\gamma \gamma \) [260, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485].
Depending on the mixing angles, on the masses of the additional Higgs bosons and on their branching fractions, the LHC can be blind to the extended Higgs sector of the NMSSM beyond the mostly standard model-like state. Then the detection of the additional states will be possible only at a LC. Also if hints for such an extended Higgs sector are observed at the LHC, only a LC will be able to study its properties in more detail. Earlier studies of the detection of NMSSM Higgs bosons at \(e^+ e^-\) colliders can be found in [486, 487, 488, 489, 490, 491].
For \(M_{H_1} < 114\) GeV, the upper bounds on \(R_1^{bb}\) in Fig. 87 follow from the LEP II constraints in [321]. Still, even for \(M_{H_1} < 110\) GeV, a detection of \(H_1\) at a LC is possible (but difficult at the LHC within the semiconstrained NMSSM). From Fig. 88 one finds that, if \(M_{H_1} > 114\) GeV, \(R_2^{bb}\) can assume all possible values from 0 to 1. Note that \(R_1^{bb}\) and \(R_2^{bb}\) satisfy approximately \(R_1^{bb}+R_2^{bb}\sim 1\).
For \(M_{H_1} \sim 100\) GeV and \(R_1^{bb} \sim 0.1\)–0.25, \(H_1\) can explain the \(\sim 2 \sigma \) excess in the bb final state for this range of Higgs masses at LEP II [321]. Properties of such points in the parameter space of the semiconstrained NMSSM have been studied in [492], amongst others the production cross sections of the various Higgs bosons in various channels at a LC.
However, an additional \({\textit{CP}}\)-even Higgs boson with sizeable coupling \(g_i\) can also be heavier than 125 GeV; such a scenario is motivated by best fits to present LHC and Tevatron data [493].
Other NMSSM-specific scenarios are possible Higgs-to-Higgs decays (see, e.g., [494]). For the 125 GeV Higgs boson, the measured standard model-like decay modes at the LHC indicate that Higgs-to-Higgs decays are not dominant for this state, but branching fractions of \({\mathscr {O}}(10~\%)\) are allowed. In the NMSSM, \(H_{125}\) could decay into pairs of lighter \({\textit{CP}}\)-even or \({\textit{CP}}\)-odd states (if kinematically possible). If these states are heavier than \(\sim 10\) GeV and decay dominantly into bb, such decay modes of \(H_{125}\) into 4b (or \(2b2\tau \)) would be practically invisible at the LHC. At a LC, using the leptonic decays of Z in the ZH Higgs production mode and/or VBF, such unconventional decays can be discovered [490].
In addition, more Higgs-to-Higgs decays involving all three \({\textit{CP}}\)-even states H and both \({\textit{CP}}\)-odd states A (omitting indices for simplicity) like \(H\rightarrow HH\), \(H\rightarrow AA\), \(H\rightarrow ZA\), \(A\rightarrow AH\), \(A\rightarrow ZH\), \(H^\pm \rightarrow W^\pm H\) and \(H^\pm \rightarrow W^\pm A\) are possible whenever kinematically allowed, and visible whenever the “starting point” of the cascade has a sufficiently large production cross section (see, e.g., Fig. 89) and the involved couplings are not too small. Even if a mostly standard model-like Higgs boson at 125 GeV is imposed, the remaining unknown parameters in the Higgs sector of the NMSSM allow for all of these scenarios.
The relevance of a \(\gamma \gamma \) collider for the study of Higgs-to-Higgs decays in the NMSSM has been underlined in [495]. Astonishingly, also pure singlet-like states H and A can be produced in the \(\gamma \gamma \) mode of a LC. In the standard model, a \(H\gamma \gamma \)-vertex is loop-induced with mainly \(W^\pm \) bosons and top-quarks circulating in the loops. In the case of the NMSSM and dominantly singlet-like states \(H_S\) and \(A_S\) (without couplings to \(W^\pm \) bosons or top quarks), higgsino-like charginos can circulate in the loops. The corresponding couplings of \(H_S\) and \(A_S\) to higgsino-like charginos originate from the term \(\lambda \widehat{S} \widehat{H_u} \widehat{H_d}\) in the superpotential (77) and are absent for the MSSM-like \({\textit{CP}}\)-even and \({\textit{CP}}\)-odd Higgs states.
The values of \(R^{\gamma \gamma }\) shown in Fig. 90 correspond to a region in the parameter space of the NMSSM where the Standard Model-like \(H_\mathrm {SM}\) has a mass of \(\sim \)125 GeV and, simultaneously, DM annihilation in the galactic centre can give rise to a 130 GeV photon line [496]. Hence the LSP mass is 130 GeV, \(M_{A_S} (\equiv M_{A_1}) \sim 260\) GeV in order to produce two photons from LSP annihilation with \(A_S\) exchange in the s-channel, and \(M_{H_S} (\equiv M_{H_2}) \approx 260\) GeV such that \(H_S\) exchange in the s-channel gives a relic density compatible with WMAP. \(\lambda \) varies between 0.6 andd 0.65, the wino mass parameter is fixed to \(M_2=300\) GeV, but \(\mu _\mathrm {eff}\) varies from 250–350 GeV. The nature of the chargino\(_1\) varies slightly with \(\mu _\mathrm {eff}\), but is always \(\approx 50\%\) wino and higgsino-like. The values shown in Fig. 90 have been obtained using the code NMSSMTools [497, 498]. We see in Fig. 90 that notably \(R^{\gamma \gamma }(A_S)\) can assume values close to 0.3, leading to a significant production cross section in the \(\gamma \gamma \) mode of a LC.
Finally the NMSSM differs from the MSSM also due to the presence of a fifth neutralino, the fermionic component of the superfield \(\widehat{S}\). Phenomenological analyses of pair production of neutralinos in the NMSSM at \(e^+\,e^-\) colliders at higher energies have been performed in [43, 44, 499, 500, 501, 502, 503]. Since the information on the neutralino sector from the LHC will be quite limited, a \(e^+\,e^-\) collider can be crucial to distinguish the NMSSM neutralino sector from the one of the MSSM [502], although it cannot be guaranteed that the difference is visible if one is close to the decoupling limit \(\lambda ,\ \kappa \rightarrow 0\). This question has also been addressed in the radiative production of the lightest neutralino pair, \(e^+\,e^- \rightarrow \tilde{\chi }^0_1\,\tilde{\chi }^0_1\,\gamma \), at a LC with \(\sqrt{s} = 500\) GeV in [503].
To summarise, the NMSSM is a well-motivated supersymmetric extension of the standard model, notably in view of the discovery of a Higgs boson at 125 GeV and a potentially enhanced branching fraction into \(\gamma \gamma \). Due to their reduced couplings to electroweak gauge bosons it is not clear, however, whether the LHC will be able to verify the extended Higgs and neutralino sectors of the NMSSM. Only a LC will be able to perform measurements of such reduced couplings, correspondingly reduced production cross sections, and possible unconventional decay modes. These incompass both possible Higgs-to-Higgs cascade decays, as well as cascades in the neutralino sector.
2.10 Little Higgs^{24}
The Little Higgs (LH) model [504, 505, 506] is well known to be one of the attractive scenarios for physics beyond the standard model (SM). In this subsection, we review the physics of the model at future linear collider experiments by referring to several studies reported so far.
2.10.1 About the LH model
The cutoff scale of the standard model (SM) is constrained by electroweak precision measurements: If we assume the existence of a \(\sim \)125 GeV SM Higgs-like resonance, the cutoff scale should be higher than roughly 5 TeV [507, 508]. However, such a relatively high cutoff scale requires a fine tuning in the Higgs potential because the Higgs potential receives the quadratic divergent radiative correction.
In LH models, the Higgs boson is regarded as a pseudo Nambu–Goldsone (NG) boson which arises from a global symmetry breaking at high energy, \(\sim \)10 TeV. Although Yukawa and gauge couplings break the global symmetry explicitly, some global symmetry is not broken by one of these couplings: in LH models, the breaking of such a symmetry is achieved only by two or more couplings, which is called “collective” symmetry breaking. Because of the collective symmetry breaking, the quadratic divergence from SM loop diagrams is cancelled by new-particle diagrams at the one-loop level.
As a bottom-up approach, specifying a coset group, we investigate the phenomenology of such a scenario by a non-linear sigma model. In particular, the littlest Higgs (LLH) model [506] described by an SU(5) / SO(5) symmetry breaking and the simplest little Higgs (SLH) model [509] described by an \([SU(3)\times U(1)]^2/ [SU(2)\times U(1)]^2\) symmetry breaking have been studied about its expected phenomenology well so far. Here we review the ILC physics mainly focusing on the LLH model.
The LLH model is based on a non-linear sigma model describing an SU(5) / SO(5) symmetry breaking with the vacuum expectation value \(f \sim \mathscr {O}\)(1) TeV. An [\(SU(2) \times U(1)]^2\) subgroup of the SU(5) is gauged and broken down to the SM \(SU(2)_L\times U(1)_Y\). Fourteen NG bosons arise and it can be decomposed into \(\mathbf{1}_0 \oplus \mathbf{3}_0 \oplus \mathbf{2}_{\pm 1/2} \oplus \mathbf{3}_{\pm 1}\) under the electroweak gauge group. The \(\mathbf{1}_0 \oplus \mathbf{3}_0\) are eaten by heavy gauge bosons \(A_H, Z_H, W_H^\pm \), and \(\mathbf{2}_{\pm 1/2} \oplus \mathbf{3}_{\pm 1}\) are the SM Higgs field h and new triplet Higgs field \(\Phi \), respectively. To realise the collective symmetry breaking, SU(2) singlet vector-like top quark partners, \(T_L\) and \(T_R\), are also introduced. These heavy particles have masses which are proportional to f and depend also on the gauge coupling, charges and Yukawa couplings. The Higgs potential is generated radiatively and it depends also on parameters of UV theory at the cutoff scale \({\Lambda } \sim 4 \pi f\).
Even in the model, the new-particle contributions are strongly constrained at precision measurements.
Pushing new-particle masses up to avoid the constraint, the fine tuning in the Higgs potential is reintroduced. To avoid the reintroducing the fine tuning, implementing of the \(Z_2\) symmetry called T-parity has been proposed [510, 511, 512].^{25}
In the LLH model, the T-parity is defined as the invariance under the exchanging gauged \([SU(2) \times U(1)]_1\) and \([SU(2) \times U(1)]_2\). Then, for all generations of the lepton and squark sector, new heavy fermions are introduced to implement this symmetry. Under the parity, the new particles are assigned to be a minus charge (T odd), while the SM particles have a plus charge (T even). Thus, heavy particles are not mixing with SM particles. Then the tree-level new particle contribution to electroweak precision measurements are forbidden and the new-particle masses can be light.
It has been suggested that the T-parity is broken by anomalies in the typical strongly coupled UV theory [515, 516] and the possibilities of the conserved T-parity scenario and another parity are also studied [517, 518, 519, 520, 521]. If the T-parity is an exact symmetry, the lightest T-odd particle, heavy photon in the LLH model, is stable and provides a DM candidate. Even if the T-parity is broken by anomalies, contribution to electroweak precision measurements are still suppressed, while the lightest T-odd particle would decay at colliders [522, 523].
As described above, top quark partner, new gauge bosons and additional scalar bosons are expected in LH models, while its details strongly depend on models. In the model with T-parity, T-odd quark partners and lepton partners are introduced additionally. The Higgs boson phenomenology would be different from the SM prediction due to the new-particle contributions as well as deviations from the SM coupling which would appear from higher-dimensional operators.
2.10.2 Higgs phenomenology in LH
In LH models, parameters of the Higgs potential cannot be estimated without calculating the contribution of a specifying UV theory. As a phenomenological approach, we consider these parameters as free parameters and these are determined by observables, e.g., Higgs mass. As described here, there are possibilities to change the Higgs boson phenomenology from the SM prediction and it may be checked at the ILC.
Figure 93 shows the range of partial decay widths, \({\varGamma }(h \rightarrow gg)\) and \({\varGamma }(h \rightarrow \gamma \gamma )\), in the LLH model varying model parameters [524]. In the model, the deviation of the top Yukawa coupling suppresses the \({\varGamma }(h \rightarrow gg)\), while contributions from top partner and mixing in the top sector enhance the partial decay width. Totally, these additional top sector contributions suppresses the \({\varGamma }(h \rightarrow gg)\) in Fig. 93. On the other hand, it enhances the \({\varGamma }(h \rightarrow \gamma \gamma )\) because the W boson loop contribution is dominant in the SM and the fermion-loop contributions have a minus sign. The contribution from the heavy gauge bosons suppresses the \({\varGamma }(h \rightarrow \gamma \gamma )\) as well as the deviation of the gauge boson coupling and mixing in the gauge boson sector due to the sign of the \(W_H W_H h\) coupling. The charged Higgs contribution leads to an enhancement. The doubly charged Higgs contribution is small because the coupling to the Higgs boson is suppressed; thus, it is neglected here [524]. In a similar way the \(\gamma Z\) decay would be affected [525].
The expected precision for measurements of the Higgs coupling including \(h \rightarrow \gamma \gamma \) and \(h \rightarrow gg\) branch at ILC are summarised in Sect. 2.3. One of the possibilities to measure the deviation of the \({\varGamma }(h \rightarrow \gamma \gamma )\) is the \(\gamma \gamma \rightarrow h \rightarrow b \bar{b}\) mode in photon collider option [529, 530].
Higgs decay at tree level The deviation of the SM coupling and new particles would also change the Higgs phenomenology at tree level. The deviation of \(ht\bar{t}\) and top partner change the cross section of \(ht\bar{t}\) production [531, 532, 533]. In LHT, production cross section of the \(e^+e^- \rightarrow ht\bar{t}\) normalised to the SM value is about 90 % at \(f=1\) TeV [532].
The deviation of hWW and hZZ couplings (e.g. [534] in LLH model) also change the cross sections of the Higgs-boson production as well as the decay branching ratio.^{26} In some case, the deviation rates of partial decay widths are the same, then the branching ratio of the Higgs decay can be close to the SM prediction [526].
Higgs decay to new particles Another possibility is additional decay branches of Higgs boson into new particles. For example, the lightest new particle in the LHT is the heavy photon which mass is \(\sim \)60 GeV with \(f = 400\) GeV. If it kinematically possible, the Higgs boson also decays into two heavy photons and the value of the branching ratio could be large (\(> \)80 %) in the 125 GeV Higgs boson case because it decays via the gauge coupling [538, 539]. If the T-parity is an exact symmetry, it is the invisible decay. On the other hand, the produced heavy photon decays mainly into SM fermions in such a light Higgs boson case if the T-parity is broken by anomaly. The decay width is about \(10^{-1}\)–\(10^{-2}\) eV [522, 523].
Additional scalar bosons In some models, e.g., simple group models, there could be a pseudo-scalar, \(\eta \), although the mass depends on the models. The Higgs boson could also decay into \(\eta \eta \) and \(Z \eta \) [540] if it is kinematically possible. Furthermore, because the Z–h–\(\eta \) coupling cannot appear in product group models, the measurement at ILC helps to distinguish the kind of LH models [541]. Other phenomenology studies for \(\eta \) can be found in Refs. [542, 543]. As another example of additional scalars, there is the triplet Higgs boson in the LLH model, although these mass is proportional f [544, 545, 546, 547].
Higgs self-coupling The measurement of Higgs self-coupling is one of the important test for the Higgs boson. In the LH models, the triplet and quartet coupling could slightly change from the SM expectation. Study for Zhh process in LLH [548] and the one-loop correction to the hhh coupling from vector-like top quarks [549] have been studied.
2.10.3 Other direct LH signals
Since the LH model is discussed only in this subsection, we also mention here other signals of the model at future liner collider experiments. The signals can be divided into two categories; direct and indirect signals. The direct signals means the direct productions of new particles predicted by the LH model. The indirect signals are, on the other hand, the LH contributions to the processes whose final states are composed only of SM particles. We consider only the direct signals, while we omit to discuss the indirect ones for want of space. Please see references [534, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571] for the indirect signals.
The direct signals can future be divided into two subcategories; the direct productions of coloured particles and non-coloured ones. This is because the LH model requires the cancellation of quadratically divergent corrections to the Higgs mass term from top loop and those of electroweak gauge bosons at one-loop level, and thus the model inevitably predicts both coloured and non-coloured new particles. When the T-parity (or some other Z\(_2\)-symmetry distinguishing SM and new particles) is not imposed on the model like the littlest or the simplest Higgs model, non-coloured new particles will be produced by following two processes: single productions (i.e., \(e^+ e^- \rightarrow V_H\)) [572, 573, 574, 575, 576, 577, 578, 579, 580] and associate productions (i.e., \(e^+ e^- \rightarrow V_H + \gamma (Z)\)) [581, 582, 583, 584, 585], where \(V_H\) is the LH partner of the weak gauge boson (heavy gauge boson). On the other hand, when the T-parity is imposed like the case of the LHT, non-coloured new particles must be produced in pair (i.e., \(e^+ e^- \rightarrow V_H V_H\)) [586, 587, 588, 589, 590, 591]. For the productions of coloured new particles, associate productions (i.e., \(e^+ e^- \rightarrow T + t\)) and pair productions (i.e. \(e^+ e^- \rightarrow f_H f_H\)) are frequently considered to find LH signals [592, 593, 594], where T is the LH partner of the top quark (top partner) and \(f_H\) is the new coloured fermion like the top partner or heavy fermions which are introduced by imposing the T-parity on the model.
\(M_{A_H}\) | \(M_{W_H}\) | \(M_{Z_H}\) | \(M_{e_H}\) | \(M_{\nu _{eH}}\) | |
---|---|---|---|---|---|
Mass (GeV) | 81.9 | 368 | 369 | 410 | 400 |
The above mass spectrum has been obtained by choosing the vacuum expectation value of the global symmetry f and the Yukawa coupling of the heavy electron \(\kappa _e\) to be 580 GeV and 0.5, respectively.^{27} Flavour-changing effects caused by the heavy lepton Yukawa couplings are implicitly assumed to be negligibly small.
\(M_{A_H}\) | \(M_{W_H}\) | \(M_{Z_H}\) | \(M_{e_H}\) | \(M_{\nu _{eH}}\) | |
---|---|---|---|---|---|
Accuracy (%) | 1.3 | 0.20 | 0.56 | 0.46 | 0.1 |
Since the relevant physics of the LHT model is described with only two model parameters f and \(\kappa _e\), the masses of non-coloured new particles are also given by the parameters. Performing these model-independent mass measurements therefore provides strong evidence that the discovered new particles are indeed LHT particles. The parameters f and \(\kappa _e\) are eventually extracted from the measurements very accurately; f and \(\kappa _e\) are extracted at accuracies of 0.16 and 0.01 %.
\(e^+ e^- \rightarrow \) | \(A_H Z_H\) | \(Z_H Z_H\) | \(e_H^+ e_H^-\) | \(\nu _{eH} \bar{\nu }_{eH}\) | \(W_H^+ W_H^-\) |
---|---|---|---|---|---|
Accuracy (%) | 7.70 | 0.859 | 2.72 | 0.949 | 0.401 |
Only \(Z_H A_H\) process has been analysed with 500 fb\(^{-1}\) data at \(\sqrt{s} =\) 500 GeV running, while others have been done with the same luminosity at 1 TeV running.
We next consider the direct productions of coloured new particles. Among several coloured new particles, the most important one is the top partner T (and its T-parity partner \(T_-\)), because it is responsible for the cancellation of the quadratically divergent correction to the Higgs mass term from top loop. Since the top partner has a colour-charge, it is expected to be constrained by the LHC experiment when its mass is not heavy. Thus we summarise the current status of the constraint before going to discuss the physics of the top partner at future linear collider experiments.
The most severe limit on the mass of the top partner comes from its pair production process followed by the decay \(T \rightarrow b W\) [595]. The limit is \(m_T >\) 650 GeV at 95 % CL with assuming BR(\(T \rightarrow bW\)) = 1. Since the top partner has other decay channels like \(T \rightarrow tZ/T \rightarrow th\) and the branching fraction to bW is typically about 40 %, the actual limit on the mass is \(m_T >\) 500 GeV. On the other hand, the T-parity partner of the top partner \(T_-\) decays into \(tA_H\) with BR(\(T_- \rightarrow t A_H\)) \(\simeq \) 1. The most severe limit on its mass again comes from its pair production process, which gives \(m_{T_-} >\) 420 GeV at 95 % CL when \(A_H\) is light enough [596].
The physics of the top partner at future linear collider experiments has been discussed in some details in reference [594]. When \(m_T \simeq \) 500 GeV, the cross section of its pair production process (\(e^+ e^- \rightarrow T \bar{T}\)) is \({\mathscr {O}}(100)\) fb, while that of the associate production process (\(e^+ e^- \rightarrow t \bar{T} + \bar{t} T\)) is \({\mathscr {O}}(1\)–10) fb with appropriate centre-of-mass energy. It has been shown that the Yukawa coupling of the top partner and the coupling of the interaction between h, t, and T can be precisely measured with use of the threshold productions of these processes. Since these couplings are responsible for the cancellation of the quadratically divergent correction to the Higgs mass term from top loop, these measurements will give a strong test of the LH model.
The physics of the T-parity partner \(T_-\) at future LC experiments has been discussed in some details in reference [593]. When \(m_{T_-} \simeq \) 500 GeV, the cross section of its pair production process (\(e^+ e^- \rightarrow T_- \bar{T}_-\)) is \({\mathscr {O}}(100)\) fb with appropriate centre-of-mass energy. Since \(T_-\) decays into \(t A_H\), the masses of both \(T_-\) and \(A_H\) can be precisely measured using the energy distribution of reconstructed top quarks, which will provide an excellent test of the LHT model by comparing this signal with those of non-coloured new particles. Furthermore, it has also been pointed out that the process can be used to discriminate new physics models at the TeV scale. This is because many new physics models predict similar processes, a new coloured particle decaying into t and an invisible particle like a squark decaying into t and a neutralino in the MSSM.
As a recent review and recent studies for current status of new particles and DM in LHT, please see also [269, 597, 598, 599].
2.11 Testing Higgs physics at the photon linear collider^{28}
In a nominal LC option, i.e. with the electron-beam energy of 250 GeV, the geometric luminosity \(L_\mathrm{geom}=12\cdot 10^{34} \mathrm{cm}^{-2}\,\mathrm{s}^{-1}\) can be obtained, which is about four times higher than the expected \(e^+ e^-\) luminosity. Still, the luminosity in the high-energy \(\gamma \gamma \) peak (see Fig. 96) corresponds to about \(\frac{1}{3}\) of the nominal \(e^+ e^-\) luminosity – so we expect \(L_{\gamma \gamma }(\sqrt{s_{\gamma \gamma }}> 0.65 \cdot \sqrt{s_{ee}})\) equal to about 100 \(\mathrm{fb}^{-1}\) per year (400 \(\mathrm{fb}^{-1}\) for a whole energy range) [603, 604]. Adjusting the initial electron-beam energy and direction of polarisations of electrons and laser photons at fixed laser photon energy one can vary a shape of the \(\gamma \gamma \) effective-mass spectrum.
At a \(\gamma \gamma \) collider the neutral C-even resonance with spin 0 can be produced, in contrast to C-odd spin 1 resonances in the \(e^+e^-\) collision. Simple change of signs of polarisations of incident electron and laser photon for one beam transforms PLC to a mode with total helicity 2 at its high-energy part. It allows one to determine degree of possible admixture of state with spin 2 in the observed Higgs state. The s-channel resonance production of \(J^{PC}=0^{+\!+}\) particle allows to perform precise measurement of its properties at PLC.
In summer 2012 a Higgs boson with mass about 125 GeV has been discovered at LHC [94]. We will denote this particle as \(\mathscr {H}\). The collected data [605, 606] allow one to conclude that the SM-like scenario, suggested e.g. in [607, 608], is realised [609]: all measured \(\mathscr {H}\) couplings are close to their SM values in their absolute value. Still the following interpretations of these data are discussed: A) \({\mathscr {H}}\) is Higgs boson of the SM. B) We deals with phenomenon beyond SM, with \({\mathscr {H}}\) being some other scalar particle (e.g. one of neutral Higgs bosons of Two Higgs Doublet Model (2HDM) – in particular MSSM, in the \({\textit{CP}}\)-conserving 2HDM that are h or H). In this approach the following opportunities are possible: (1) measured couplings are close to SM values; however, some of them (especially the ttH coupling) with a “wrong” sign. (2) In addition some new heavy charged particles, like \(H^\pm \) from 2HDM, can contribute to the loop couplings. (3) The observed signal is not due to one particle but it is an effect of two or more particles, which were not resolved experimentally – the degenerated Higgses. Each of these opportunities can lead to the enhanced or suppressed, as compared to the SM predictions, \({\mathscr {H}}\gamma \gamma \), \({\mathscr {H}}gg\) and \({\mathscr {H}}Z\gamma \) loop-coupling.
The case with the observed Higgs-like signal being due to degenerated Higgses \(h_i\) demands a special effort to diagnose it. In this case the numbers of events with production of some particle x are proportional to sums like \(\sum _i ({\varGamma }^x_i/{\varGamma }^{\mathrm{tot}}_i){\varGamma }^{gg}_i\). Data say nothing about couplings of the individual Higgs particles and there are no experimental reasons in favour of the SM-like scenario for one of these scalars. In such case each of degenerated particles have low total width, and there is a hope that the forthcoming measurements at PLC can help to distinguish different states due to much better effective-mass resolution. The comparison of different production mechanisms at LHC, \(e^+e^-\) LC and PLC will give essential impact in the problem of resolution of these degenerated states. Below we do not discuss the case with degenerated Higgses with masses \(\sim \)125 GeV in more detail, concentrating on the case when observed is one Higgs boson \(\mathscr {H}\), for which the SM-like scenario is realised.
In the discussion we introduce useful relative couplings, defined as ratios of the couplings of each neutral Higgs boson \(h^{(i)}\) from the considered model, to the gauge bosons W or Z and to the quarks or leptons (\(j=V (W,Z),u,d,\ell \ldots \)), to the corresponding SM couplings: \( \chi _j^{(i)}=g_j^{(i)}/g_j^\mathrm{SM}\). Note that all couplings to EW gauge bosons \(\chi _V^{(i)}\) are real, while the couplings to fermions are generally complex. For \({\textit{CP}}\)-conserving case of 2HDM we have in particular \(\chi _j^h\), \(\chi _j^H\), \(\chi _j^A\) (with \(\chi _V^A=0\)), where couplings of fermions to h and H are real, while couplings to A are purely imaginary.
The SM-like scenario for the observed Higgs \({\mathscr {H}}\), to be identified with some neutral \(h^{(i)}\), corresponds to \(|\chi _j^{\mathscr {H}}|\approx 1\). Below we assume this scenario is realised at present.
It is well known already since a long time ago that the PLC is a very good observatory of the scalar sector of the SM and beyond SM, leading to important and in many cases complementary to the \(e^+e^-\) LC case tests of the EW symmetry breaking mechanism [610, 611, 612]. The \(e^+e^-\) LC, together with its PLC options (\(\gamma \gamma \) and \(e \gamma \)), is very well suited for the precise study of properties of this newly discovered \({\mathscr {H}}\) particle, and other scalars. In particular, the PLC offers a unique opportunity to study resonant production of Higgs bosons in the process \(\gamma \gamma \rightarrow \mathrm{Higgs}\), which is sensitive to charged fundamental particles of the theory. In principle, PLC allows one to study also resonant production of heavier neutral Higgs particles from the extension of the SM. Other physics topic which could be studied well at PLC is the \({\textit{CP}}\) property of Higgs bosons. Below we discuss the most important aspects of the Higgs physics which can be investigated at PLC. Our discussion is based on analyses done during last two decades and takes into account also some recent “realistic” simulations supporting those results.
2.11.1 Studies of 125-GeV Higgs \({\mathscr {H}}\)
Several NLO analyses of the production at the PLC of a light SM-Higgs boson \(H_\mathrm{{SM}}\) decaying into the \(b \bar{b}\) final state were performed, including the detector simulation, e.g. [614, 615, 616, 617]. These analyses demonstrate a high potential of this collider to measure accurately the Higgs two-photon width. By combining the production rate for \(\gamma \gamma \rightarrow H_\mathrm{{SM}}\rightarrow b \bar{b}\) (Fig. 97), to be measured with 2.1 % accuracy, with the measurement of the \(\mathrm{BR}(H_{\mathrm{SM}}\rightarrow bb)\) at \(e^+e^-\) LC, with accuracy \(\sim \) 1 %, the width \({\varGamma }(H_{\mathrm{SM}} \rightarrow \gamma \gamma )\) for \(H_{\mathrm{SM}}\) mass of 120 GeV can be determined with precision \(\sim \)2 %. This can be compared to the present value of the measured at LHC signal strength for 125 GeV \(\mathscr {H}\) particle, which ratio to the expected signal for SM Higgs with the same mass (approximately equal to the ratio of \(|g_{\gamma \gamma {\mathscr {H} }}|^2/|g_{\gamma \gamma {H_{\mathrm{SM}} }}|^2\)), are 1.17\(\pm \)0.27 and 1.14\(^{+0.26}_{-0.23}\) from ATLAS [101] and CMS [618], respectively.
The process \(\gamma \gamma \rightarrow {\mathscr {H}}\rightarrow \gamma \gamma \) is also observable at the PLC with reasonable rate [617]. This measurement allows one to measure directly two-photon width of Higgs without assumptions as regards unobserved channels, couplings, etc.
Neutral Higgs resonance couples to photons via loops with charged particles. In the Higgs \(\gamma \gamma \) coupling the heavy charged particles, with masses generated by the Higgs mechanism, do not decouple. Therefore the \({\mathscr {H}}\rightarrow \gamma \gamma \) partial width is sensitive to the contributions of charged particles with masses even far beyond the energy of the \(\gamma \gamma \) collision. This allows one to recognise which type of extension of the minimal SM is realised. The \(H^+\) contribution to the \({\mathscr {H}} \gamma \gamma \) loop coupling is proportional to \({\mathscr {H}} H^+H^-\) coupling, which value and sign can be treated as free parameters of model.^{29} The simplest example gives a 2HDM with type II Yukawa interaction (2HDM II). For a small \(m_{12}^2\) parameter, see Sect. 2.6, the contribution of the charged Higgs boson \(H^+\) with mass larger than 400 GeV leads to 10% suppression in the \({\mathscr {H}}\rightarrow \gamma \gamma \) decay width as compare to the SM one, for \(M_{\mathscr {H}}\) around 120 GeV [607, 608], Table 24 (solution A). The enhancement or decreasing of the \({\mathscr {H}} \gamma \gamma \) coupling is possible, as discussed for 2HDM with various Yukawa interaction models in [276, 619] as well in the inert doublet model^{30} [620, 621].
Solution | Basic couplings | \(|\chi _{gg}|^2\) | \(|\chi _{\gamma \gamma }|^2\) | \(|\chi _{Z\gamma }|^2\) |
---|---|---|---|---|
\(A_{{\mathscr {H}}}\) | \(\chi _V\approx \chi _b\approx \chi _t\approx \pm 1\) | 1.00 | 0.90 | 0.96 |
\(B_{{\mathscr {H}} b}\) | \(\chi _V\approx -\chi _b\approx \chi _t\approx \pm 1\) | 1.28 | 0.87 | 0.96 |
\(B_{{\mathscr {H}} t}\) | \(\chi _V\approx \chi _b\approx -\chi _t\approx \pm 1\) | 1.28 | 2.28 | 1.21 |
The observed Higgs particle can have definite \({\textit{CP}}\) parity or can be admixture of states with different \({\textit{CP}}\) parity (CP mixing). In the latter case the PLC provides the best among all colliders place for the study of such mixing. Here, the opportunity to simply vary polarisation of photon beam allows one to study this mixing via dependence of the production cross section on the incident photon polarisation [623, 624, 625, 626, 627, 628, 629, 630]. In particular, the change of sign of circular polarisation (\(+\!+ \leftrightarrow --\)) results in variation of production cross section of the 125-GeV Higgs in 2HDM by up to about 10 %, depending on a degree of \({\textit{CP}}\)-admixture. Using mixed circular and linear polarisations of photons gives opportunity for more detailed investigations [631].
The smaller but interesting effects are expected in \(e\gamma \rightarrow e{\mathscr {H}}\) process with \(p_{\bot e}> 30\) GeV, where \({\mathscr {H}}Z\gamma \) vertex can be extracted with reasonable accuracy [635].
2.11.2 Studies of heavier Higgses, for 125 GeV \({{\mathscr {H}}=h^{(1)}}\)
- (i)For an arbitrary Yukawa interaction there are sum rules for coupling of different neutral Higgses to gauge bosons \(V=W,\,Z\) and to each separate fermion f (quark or lepton)The first sum rule (to the gauge bosons) was discussed e.g. in [353, 636]. The second one was obtained only for Models I and II of Yukawa interaction [637], however, in fact it holds for any Yukawa sector [638].$$\begin{aligned}&\sum \limits _{i=1}^{3} (\chi _V^{(i)})^2=1. \quad \sum \limits _{i=1}^{3}(\chi _f ^{(i)})^2=1. \end{aligned}$$(82)
In the first sum rule all quantities \(\chi _V^{(i)}\) are real. Therefore, in SM-like case (i.e. at \(|\chi _V^{(1)}|\approx 1\)) both couplings \(|\chi _V^{2,3}|\) are small. The couplings entering the second sum rule (for fermions) are generally complex. Therefore this sum rule shows that for \(|\chi _f^{(1)}|\) close to 1, either \( \left| \chi _f^{(2)}\right| ^2\) and \(\left| \chi _f^{(3)}\right| ^2\) are simultaneously small, or \( \left| \chi _f^{(2)}\right| ^2 \approx \left| \chi _f^{(3)}\right| ^2\).
- (ii)For the 2HDM I there are simple relations, which in the \({\textit{CP}}\) conserved case are as follows:$$\begin{aligned}&\chi _u^{(h)}=\chi _d^{(h)}\,,\qquad \chi _u^{(H)}=\chi _d^{(H)}\,. \end{aligned}$$(83)
- (iii)In the 2HDM II following relations hold:
- (a)
- (b)For each neutral Higgs boson \(h^{(i)}\) one can write a horizontal sum rule [641]:$$\begin{aligned}&|\chi _u^{(i)}|^2\sin ^2\beta +|\chi _d^{(i)}|^2\cos ^2\beta =1\,. \end{aligned}$$(84b)
- (a)
Total width (in MeV) of H, A in some benchmark points for the SM-like h scenario (\(M_h=125\) GeV) in the 2HDM (\(\chi _V^h\approx 0.87 \), \(|\chi _V^H|=0.5\) and \(|\chi _t^h|=1\)). Results for \(\tan \beta =1/7, \, 1 \ \ \mathrm{and} \ \ 7\) are shown
\(M_{H,A}\) | \(\tan \beta =1/7\) | \(\tan \beta =1\) | \(\tan \beta =7\) | |||
---|---|---|---|---|---|---|
\({\varGamma }_H\) | \({\varGamma }_A\) | \({\varGamma }_H\) | \({\varGamma }_A\) | \({\varGamma }_H\) | \({\varGamma }_A\) | |
200 | 0.35 | \(8 \times 10^{-5}\) | 0.35 | \(4\times 10^{-3}\) | 0.4 | 0.2 |
300 | 2.1 | \(1.2\times 10^{-4}\) | 2.1 | \(6\times 10^{-3}\) | 0.75 | 0.3 |
400 | 138 | 132 | 8.8 | 2.7 | 2.5 | 0.45 |
500 | 537 | 524 | 22.8 | 10.7 | 6.1 | 0.7 |
In the SM-like h scenario it follows from the sum rule (82) that the W-contribution to the \(H\gamma \gamma \) width is much smaller than that of would-be heavy SM Higgs, with the same mass, \(M_{H_\mathrm{SM}}\approx M_H\). At the large \(\tan \beta \) also \(H\rightarrow tt\), \(A\rightarrow tt\) decay widths are extremely small, so that the total widths of H, A become very small.^{33}
Moreover, in MSSM with \(M_h=125\) GeV we can have heavy and degenerate H and A, \(M_H\approx M_A\). At large \(\tan \beta \) the discovery channel of H / A at LHC is \(gg\rightarrow b\bar{b}\rightarrow b\bar{b}H/A\). Nevertheless, in some region of parameters, at intermediate \(\tan \beta \), these \(H{\, \mathrm{and}\,}\,A\) are elusive at LHC. That is the so-called LHC wedge region [644]; see the latest analysis [645]. The PLC allows one to diminish this region of elusiveness, since here the H and A production is generally not strongly suppressed and the \(b\bar{b}\) background is under control [274, 642, 643, 646]. Figure 99 show that PLC allows one to observe joined effect of \(H,\,A\) within this wedge region. Precision between 11 and 21 % for \(M_A\) equal to 200–300 GeV, \(\tan \beta \) = 7 of the Higgs-boson production measurement (\(\mu \) =200 GeV (the Higgs mixing parameter) and \(A_f=1500\) GeV (the tri-linear Higgs-sfermion couplings)) can be reached after one year [643]. To separate these resonances even in the limiting case \(\chi _V^H=0\) is a difficult task, since the total number of expected events is small.
At \(\chi _V^H\ne 0\), taking \(\chi _V^H \sim \)0.3–0.4 as an example (what is allowed by current LHC measurement of couplings of \({\mathscr {H}} = h\) to ZZ), an observation of \(H\rightarrow ZZ\) decay channel can be good method for the H discovery in 2HDM. The signal \(\gamma \gamma \rightarrow H\rightarrow WW, ZZ\) interferes with background of \(\gamma \gamma \rightarrow WW, ZZ\), what results in irregular structure in the effective-mass distribution of products of reaction \(\gamma \gamma \rightarrow WW, ZZ\) (this interference is constructive and destructive below and above resonance, respectively). The study of this irregularity seems to be the best method for discovery of heavy Higgs, decaying to \(WW,\, ZZ\) [647], and to measure the corresponding \(\phi _{\gamma \gamma }\) phase, provided it couples to ZZ / WW reasonably strong.^{34}
Just as it was described above for the observed 125-GeV Higgs, PLC provides the best among colliders place for the study of spin and the \({\textit{CP}}\) properties of heavy \(h^{(2)}\), \(h^{(3)}\). That are \({\textit{CP}}\) parity in the \({\textit{CP}}\) conserved case [with (\(h^{(2)}\), \(h^{(3)}\) = (\(H,\,A\))], and (complex) degree of the admixtures of states with different \({\textit{CP}}\) parity, if \({\textit{CP}}\) is violated. This admixture determines dependence on the Higgs production cross section on direction of incident photon polarisation [624, 626, 627, 628, 629, 630, 650]. These polarisation measurements are useful in the study of the case when the heavy states \(h^{(2)}\), \(h^{(3)}\) (\(H,\,A\)) are degenerated in their masses. A study [631] shows that the 3-years operation of PLC with linear polarisation of photons, the production cross section of the H and A corresponding to the LHC wedge for MSSM (with mass \(\sim 300\) GeV) can be separately measured with precison 20 %. Pure scalar versus pure pseudoscalar states can be distinguished at the \(\sim \)4.5 \(\sigma \) level.
- (a)
Instrumental degeneracy when \(|M_B-M_A|>{\varGamma }_B+{\varGamma }_A\), with mass difference within a mass resolution of detector. This effect can be resolved with improving of a resolution of the detector.
- (b)
Physical degeneracy when \(|M_B-M_A|<{\varGamma }_B+{\varGamma }_A\).
Another method for study of \({\textit{CP}}\) content of a produced particle provides the measurement of angular distribution of decay products [623, 651, 652]. In the \(t \bar{t}\) decay mode one can perform a study of the \({\textit{CP}}\)-violation, exploiting fermion polarisation. The interference between the Higgs exchange and the continuum amplitudes can be sizeable for the polarised photon beams, if helicities of the top and antitop quarks are measured. This enables to determine the \({\textit{CP}}\) property of the Higgs boson completely [649, 653], Fig. 100.
The discovery of charged Higgses \(H^\pm \) will be a crucial signal of the BSM form of the Higgs sector. These particles can be produced both at the LC (\(e^+e^-\rightarrow H^+H^-\)) and at the PLC (\(\gamma \gamma \rightarrow H^+H^-\)). These processes are described well by QED. The \(H^+H^-\) production process at PLC has a worse energy-threshold behaviour than the corresponding process at the LC, but a higher cross section. On the other hand, the process \(e^+e^-\rightarrow H^+H^-\) can be analysed at LC better by measurements of decay products due to known kinematics. At the PLC the variation of a initial-beam polarisation could be used for checking up the spin of \(H^\pm \) [654]. See also the analysis for flavour violation models in [655, 656].
After a \(H^\pm \) discovery, the observation of the processes \(e^+e^-\rightarrow H^+H^-h\) and \(\gamma \gamma \rightarrow H^+H^-h\), \(H^+H^-H\), \(H^+H^-A\) may provide direct information on a triple Higgs (\(H^+H^-h\)) coupling \(\lambda \), with cross sections in both cases \(\propto \alpha ^2\lambda ^2\). The \(\gamma \gamma \) collisions are preferable here due to a substantially higher cross section and the opportunity to study polarisation effects in the production process via a variation of the initial photon polarisations.
Synergy of LHC, LC and PLC colliders may be useful in the determination of the Higgs couplings, as different production processes dominating at these colliders lead to different sensitivities to the gauge and Yukawa couplings. For example LC Higgs-strahlung leads to a large sensitivity to the Higgs coupling to the EW gauge bosons, while at PLC \(\gamma \gamma \) and \(Z\gamma \) loop couplings depend both on the Higgs gauge and Yukawa couplings, as well as on coupling with \(H^+\); see the results both for the \({\textit{CP}}\)-conserving/\({\textit{CP}}\)-violating cases in e.g. [652, 657, 658].
3 Top and QCD^{35}
3.1 Introduction
The experimental studies of electron–positron annihilation into hadrons were historically essential to establish Quantum Chromodynamics (QCD) as the theory of the strong interaction: from the measurement of the R-ratio \({\sigma _{\text{ had }}/\sigma _t}\) the number of colours could be determined, the discovery of three-jet events at PETRA provided the first direct indication of the gluon, and the measurement of the Bengtson–Zerwas and Nachtmann–Reiter angles illustrated the non-abelian gauge structure of QCD – to name only a few milestones on the road to develop the theory of the strong interactions.
At the Large Electron Positron Collider (LEP) the experimental tests of QCD were further refined. Three-, four-, and even five-jet rates were measured with unprecedented accuracy. These measurements provided important input to constrain the structure constants of the underlying non-abelian gauge group and to determine the QCD coupling constant \(\alpha _s\) with high precision. The R-ratio and the forward–backward asymmetry were studied in detail including precise investigations of the flavour (in-)dependence. At SLD the measurements were extended to polarised electrons in the initial state. The tremendous experimental effort has been complemented over the time by a similar effort on the theory side: Next-to-leading order (NLO) calculations have been performed for event-shape observables and jet-rates involving jets originating from massless as well as massive quarks. New jet-algorithms with an improved theoretical behaviour were developed. Very recently theoretical predictions for three-jet rates have been extended to next-to-next-to-leading order (NNLO) accuracy. For inclusive hadron production the theoretical predictions have been extended to N\(^3\)LO accuracy in QCD. Beyond fixed order perturbation theory also power corrections and soft gluon resummation have been considered. All this effort has paved the way to establish QCD as the accepted theory of the strong interaction.
Today QCD is a mature theory and no longer the primary target of experimental studies. Assuming QCD as the underlying theory of strong interaction the precision measurements possible in \(e^+e^-\) annihilation can be used to determine fundamental parameters like coupling constants and particle masses. For example three-jet rates at LEP have been used to measure the QCD coupling constant and the b-quark mass. Since the small b-quark mass leads only to effects of the order of 5 % at the Z-resonance (compared to massless b-quarks), this example nicely illustrates the impressive theoretical and experimental precision reached. The steadily increasing experimental accuracy together with LHC as a “QCD machine” and the perspective of a future linear collider have kept QCD a very active field, where significant progress has been achieved in the last two decades. Conceptually effective field theories have been further developed with specific realisations for dedicated applications. For example, soft collinear effective theory (SCET) is nowadays used to systematically improve the quality of the perturbative expansion through the resummation of logarithmically enhanced contributions. SCET may also help to deepen our current understanding of factorisation of QCD amplitudes. Applications to the production of top-quark pair production have also demonstrated the power of this approach to assess the impact of non-perturbative corrections. Non-relativistic QCD (NRQCD) provides the well-established theoretical framework to analyse the threshold production of top-quark pair production where binding effects between top quarks are important. The theoretical description of unstable particles in the context of effective field theories have demonstrated another successful application of effective field theories. Theoretical predictions for a future Linear Collider will profit from the improved theoretical understanding in terms of an increased precision. Recently we have witnessed a major breakthrough in the development of technologies for one-loop calculations. One-loop calculations involving multiplicities of five or even more particles in the final state – which were a major bottleneck over several years in the past – are today regularly performed for a variety of different processes. The new techniques have also led to an increased automation of the required calculations. Various programmes are now publicly available to generate NLO matrix elements. Furthermore a standardised interface allows the phase-space integration within MC event generators like for example Sherpa. Also the two-loop technology has seen important progress and is now a continuously growing field. The description of threshold effects in the production of heavy particles notably heavy quarks has been further improved to include higher order corrections in the perturbative expansion.
The detailed understanding of QCD achieved today has been proven essential for the current interpretation of LHC results and the very precise measurements performed so far. Evidently LHC data can also be used for QCD studies in the TeV regime. However, owing to the complicated hadronic environment it will be difficult to reach accuracies at the per-cent level or even below. In contrast \(e^+e^-\) Linear Colliders allows one to test QCD at the sub per-cent level at energies above the Z resonance. The reachable precision of any measurement involving strongly interacting particles will depend on the ability of making accurate predictions within QCD. QCD studies will thus continue to play an important role at a future Linear Collider. Since non-perturbative effects are intrinsically difficult to assess, the highest accuracy – and thus the most precise tests of the underlying theory – can be reached for systems, where these effects are believed to be small or even negligible. A particular interesting example is provided by top-quark physics. With a mass almost as heavy as a Gold atom the top quark is the heaviest elementary fermion discovered so far.
Top quarks have unique properties, making them a highly interesting research topic on their own right. The large mass leads to an extremely short life time such that top quarks decay before they can form hadronic bound states. This simple observation has several important consequences. First of all the finite width essentially cuts off non-perturbative physics such that top-quark properties can be calculated with high accuracy in perturbative QCD. Top-quark physics thus allows one to study the properties of a ‘bare quark’. In the standard model top quarks decay almost exclusively through electroweak interactions into a W-boson and a b-quark. The parity-violating decay offers the possibility to study the polarisation of top quarks through the angular distribution of the decay products. Polarisation studies, which are difficult in the case of the lighter quarks since hadronisation usually dilutes the spin information, offer an additional opportunity for very precise tests of the underlying interaction. This is of particular interest since top-quark physics is controlled in the standard model by only ‘two parameters’: The top-quark mass and the relevant Cabbibo–Kobayashi–Maskawa matrix elements. Once these parameters are known top-quark interactions are predicted through the structure of the standard model. In particular all the couplings are fixed through local gauge invariance. Top-quark physics thus allows one to test the consistency of the standard model with high precision. A prominent example is the relation between the top-quark mass and the mass of the W-boson. Obviously the accuracy of such tests is connected to the precision with which the top-quark mass – as a most important input parameter – can be determined. While the LHC achieved already an uncertainty in the mass measurements of one GeV, it is expected that a Linear Collider will improve this accuracy by an order of magnitude down to 100 MeV or even below. Using top quarks to test the standard model with high precision and search for new physics is very well motivated. In addition to the high experimental and theoretical accuracy achievable in top-quark measurements, top-quarks provide a particular sensitive probe to search for standard model extensions. Due to their large mass, top quarks are very sensitive to the mechanism of EWSB. In many extensions of the standard model which aim to present an alternative mechanism of EWSB top quarks play a special role. It is thus natural to ask whether the top-quark mass, being so much larger than the masses of the lighter quarks, is indeed produced by the Englert–Brout–Higgs–Guralnik–Hagen–Kibble mechanism. A detailed measurement of the top-quark Yukawa coupling to the Higgs boson, which is very difficult to assess at a hadron collider, will provide a crucial information to answer this question. In the past top quarks have been extensively studied at the Tevatron and the LHC. With exception of the forward–backward charge asymmetry studied at the Tevatron the measurements are in very good agreement with the standard model predictions. However, it should be noted that due to the complex environment at a hadron collider the accuracy is often limited. The top-quark mass which is now measured with sub per cent accuracy represents an important exception. While the measurements at the Tevatron and the LHC are perfectly consistent the precise interpretation of the measured mass value in terms of a renormalised parameter in a specific scheme is still unclear. The mass which is determined from a kinematical reconstruction of the top-quark decay products is assumed to be close to the pole mass. Since precise theoretical predictions for the measured observable are lacking the exact relation between the measured mass and the pole mass has not been quantified so far. An alternative method in which the mass is determined from cross section measurements where the renormalisation is uniquely fixed through a higher order calculation gives consistent results. However, the experimental uncertainties of this method are quite large owing to the weak sensitivity of the total cross section with respect to the top-quark mass. A new method using top-quark pair production in association with an additional jet represents an interesting alternative but will most likely also be limited in precision to one GeV. Although it is not better in precision, the advantage of this method lies in the fact that the method gives a clear interpretation of the measured value in a specific renormalisation scheme. Given the importance of a precise determination of the top-quark mass, going significantly below one GeV may remain the task of a future Linear Collider.
In the following we shall briefly describe in Sect. 3.2 recent progress in QCD with a special emphasis on \(e^+e^-\) annihilation. In Sect. 3.3 we summarise new developments in top-quark physics in particular concerning the theoretical understanding of top-quark production at threshold. In the last Section we briefly comment on the physics potential of a future linear collider with respect to QCD and top-quark physics. In particular the prospects of a precise measurement of the top-quark mass are discussed.
3.2 Recent progress in QCD
3.2.1 Inclusive hadron production
3.2.2 Three-jet production at NNLO
3.2.3 NLO QCD corrections to 5-jet production and beyond
At the LEP experiments exclusive production of jet multiplicities up to five jets were studied experimentally. However, until very recently only NLO results for four-jet production were available due to the tremendous growth in complexity of the theoretical calculations. In Ref. [676] the NLO QCD corrections to five-jet production are presented.
The virtual corrections were calculated using generalised unitarity (for more details as regards this method we refer to Sect. 3.2.4), relying to a large extent on amplitudes calculated in Ref. [677] where one-loop corrections to \(W^+ + \text{3-jet }\) production in hadronic collisions were studied. The real corrections are calculated using MadFKS [678] – an implementation of the Frixione–Kunszt–Signer (FKS) subtraction scheme [679] into Madgraph. The Durham jet algorithm is used to define the jets. Results for the five-jet rate, differential with respect to the parameter \(y_{45}\), which determines the \(y_{\text{ cut }}\)-value at which a five-jet event becomes a four-jet event, are shown. Furthermore the five-jet rate as function of the jet resolution parameter \(y_{\text{ cut }}\) is presented. In addition hadronisation corrections are analysed using the Sherpa event generator. At fixed order in perturbation theory it is found that the scale uncertainty is reduced from about \([-30~\%,+45~\%]\) in LO to about \([-20~\%,+25~\%]\) in NLO. In this analysis the renormalisation scale has been chosen to be \(\mu =0.3\sqrt{s}\) and variations up and down by a factor of 2 were investigated. The central scale is chosen smaller than what is usually used for lower jet multiplicities. The reasoning behind this is that for increasing multiplicities the average transverse momentum per jet becomes smaller. This is taken into account by using \(\mu =0.3\sqrt{s}\) instead of the more common setting \(\mu =\sqrt{s}\). It would be interesting to compare with a dynamical scale like \(H_T\), the sum of the ‘transverse energies’, which has been proven in four- and five-jet production at hadron colliders to be a rather useful choice [680, 681, 682]. Using in LO \(\alpha _s\)= 0.130 and in NLO \(\alpha _s\)= 0.118 NLO corrections of the order of 10–20 % are found. It is noted that using the same value of \(\alpha _s\) in LO and NLO would amount to corrections at the level of 45–60 %. Including hadronisation corrections through Sherpa the theoretical results are used to extract \(\alpha _s\) from the experimental data. As final result \(\alpha _s(M_Z) = 0.1156^{+0.0041}_{-0.0034}\) is quoted which is well consistent with the world average and also shows the large potential of \(\alpha _s\) measurements using jet rates for high multiplicities: The uncertainty is similar to the \(\alpha _s\) determinations from three-jet rates using NNLO + NLLA predictions [675]. As an interesting observation it is also pointed out in Ref. [676] that hadronisation corrections calculated with standard tools like HERWIG, PYTHIA and ARIADNE are typically large and uncertain unless the tools are matched/tuned to the specific multi-jet environment. It is suggested to use in such cases event generators like SHERPA which incorporates high-multiplicity matrix elements through CKKW matching.
Recently an alternative method to calculate one-loop corrections has been used to calculate the NLO corrections for six- and seven-jet production. The method developed in [683, 684, 685, 686, 687, 688, 689, 690] combines the loop integration together with the phase-space integration. Both integrations are done together using Monte Carlo integration. Since the analytic structure of the one-loop integrand is highly non-trivial special techniques have to be developed to enable a numerical integration. In Ref. [691] this technique has been applied to the NLO calculation of the six- and five-jet rate in leading colour approximation. No phenomenological studies are presented. It is, however, shown that the method offers a powerful alternative to existing approaches.
3.2.4 Progress at NLO
An essential input for NLO calculations are the one-loop corrections. Four momentum conservation at each vertex attached to the loop does not fix the momentum inside the loop. As a consequence an additional integration over the unconstrained loop momentum is introduced. Since the loop momenta appears not only in the denominator through the propagators but also in the numerator in general tensor integrals have to be evaluated. The traditional method to deal with these tensor integrals is the so-called Passarino–Veltman reduction which allows one to express the tensor integrals in terms of a few basic scalar one-loop integrals [692]. All relevant scalar integrals have been calculated and can be found for example in Refs. [693, 694, 695]. In practical applications the Passarino–Veltman reduction procedure may lead to large intermediate expressions when applied analytically to processes with large multiplicities or many different mass scales. An alternative to overcome this problem is to apply the reduction procedure numerically. In this case, however, numerical instabilities may appear in specific phase-space regions where the scalar one-loop integrals degenerate for exceptional momentum configurations. Approaching these exceptional momentum configurations the results behave as “0/0”. Evaluating the limit analytically one finds a well-defined result. The numerical evaluation, however, will typically lead to instabilities unless special precautions are taken to deal with these configurations. In the past various approaches have been developed to stabilise the numerical evaluation of exceptional momentum configurations. Details can be found for example in Refs. [696, 697, 698, 699, 700, 701, 702, 703, 704, 705] and references therein. With the steadily increasing computing power of modern CPUs today an alternative approach is frequently used: instead of stabilising the numerical evaluation it is checked during the numerical evaluation whether instabilities were encountered. If this is the case the numerical evaluation of the respective phase-space point is repeated using extended floating point precision. The price to pay in this approach is a slight increase of computing time which is, however, affordable as long as the fraction of points needed to be recomputed remains small.
Beside the numerical evaluation of tensor integrals the significant increase in complexity when studying virtual corrections for processes with large multiplicities is another major bottleneck of one-loop calculations. Here the recently developed method of generalised unitarity may provide a solution. The starting point of this method is the observation that any one-loop amplitude can be written in terms of scalar one-point, two-point, three-point and four-point one-loop integrals. No higher point scalar integrals are required. This observation is a direct consequence of the Passarion–Veltman reduction procedure. Starting from this observation one can reformulate the problem of one-loop calculations: How do we calculate most efficiently the coefficients in this decomposition? One answer to this question is the method proposed by Ossola, Papadopoulos, Pittau (OPP) [706]. The idea of this method is to perform a decomposition at the integrand level: the integrand is decomposed into contributions which integrate to zero or lead to scalar integrals. To derive the decomposition at integrand level internal propagators are set on-shell. As a consequence the integrand factorises into a product of on-shell tree amplitudes. For more details as regards the method of generalised unitarity we refer to the recent review of Ellis, Kunszt, Melnikov and Zanderighi [707]. From the practical point of view the important result is that the algorithm can be implemented numerically and requires as input only on-shell tree amplitudes. For on-shell tree amplitudes very efficient methods to calculate them, like for example the Berends-Giele recursion, exist [708]. In principle it is also possible to use analytic results for the tree-level amplitudes or apply on-shell recursions à la Britto, Cachazo, Feng, and Witten ((BCFW) see for example Ref. [709]). Using tree amplitudes instead of individual Feynman diagrams helps to deal with the increasing complexity of processes for large multiplicities. It may also lead to numerically more stable results since the tree amplitudes are gauge invariant and gauge cancellation – usually occurring in Feynman diagramatic calculations – are avoided. The enormous progress made recently is well documented in the increasing number of publicly available tools to calculate one-loop amplitudes, see for example Refs. [710, 711, 712, 713, 714, 715]. As can be seen from recent work e.g. Refs. [691, 716, 717] further progress can be expected in the near future (for the method discussed in Ref. [691] see also the discussion at the end of the previous section). As mentioned already the calculation of real emission processes can be considered as a solved problem since very efficient algorithms to calculate the required Born matrix elements are available. In principle also the cancellation of the infrared and collinear singularities appearing in one-loop amplitudes as well as in the real emission processes can be considered as solved. General algorithms like Catani–Seymour subtraction method [718] or FKS subtraction [679] exist to perform the required calculation. Also here significant progress has been obtained in the recent past towards automation. The required subtractions can now be calculated with a variety of publicly available tools [678, 719, 720, 721, 722]. While most of the aforementioned tools have been applied recently to LHC physics it is evident that an application to \(e+e-\) annihilation is also possible. It can thus be assumed that for a future Linear Collider all relevant NLO QCD corrections will be available.
3.3 Recent progress in top-quark physics
3.3.1 Top-quark decays at next-to-next-to-leading order QCD
3.3.2 Two-loop QCD corrections to heavy quark form factors and the forward–backward asymmetry for heavy quarks
The measurements of the forward–backward asymmetry \(A_{\mathrm{FB}}^b\) for b-quarks differ significantly from the standard model predictions [734]. The theoretical predictions take into account NNLO QCD corrections, however, the b-quark mass has been neglected at NNLO. The forward–backward asymmetry for massive quarks may be calculated from the fully differential cross section. As far as the two-loop QCD corrections are concerned this requires the calculation of the two-loop form factor for heavy quarks. These corrections have been calculated recently. In Ref. [735] the NNLO QCD corrections for the vector form factor are calculated. In Ref. [736] the results are extended to the axial-vector form factor. The anomaly contribution has been studied in Ref. [737]. The two-loop corrections need to be combined with the one-loop corrections for real emission and the Born approximation for double real emission. All individual contributions are of order \(\alpha _s^2\) and thus contribute. The cancellation of the collinear and soft singularities encountered in the different contributions is highly non-trivial. In Refs. [738, 739] ‘antenna functions’ are derived, which match the singular contributions in the double real emission processes. As an important result also the integrated antenna functions are computed in Refs. [738, 739]. In principle all building blocks are now available to calculate the differential cross section for heavy quark production in NNLO accuracy in QCD. Evidently these results, once available, can also be applied to top-quark pair production.
3.3.3 Threshold cross section
Threshold production of top-quark pairs in electron–positron annihilation is an unique process where one can extract the top-quark mass through a threshold scan by measuring the total cross section \(\sigma (e^+e^-\rightarrow t\bar{t})\). It is a counting experiment of the production rate of the colour singlet \(t\bar{t}\) bound state. Therefore the measurement of the threshold cross section for \(e^+ e^-\rightarrow t\bar{t}\) is very clean experimentally as well as theoretically concerning QCD non-perturbative effects.
Electroweak corrections and effect of unstable top In early studies of the \(e^+ e^- \rightarrow t\bar{t}\) threshold it was recognised [740, 741] that the effect of the top quark width can be consistently incorporated into the computation of the total cross section by the replacement \(E\rightarrow E+i{\varGamma }_t\). This prescription works well up to NLO, but it turns out that in NNLO an uncancelled ultraviolet divergence appears, which is proportional to the top-quark width (in dimensional regularisation an example of such a term is \(R_{t\bar{t}}\sim \alpha _s {\varGamma }_t/\epsilon \)). This is a signal of an improper treatment of electroweak effects, and the solution of this problem is to abandon the amplitude \(e^+ e^- \rightarrow t\bar{t}\) where the unstable \(t\bar{t}\) is treated as a final state of the S-matrix. Physical amplitudes should treat stable particles as final states of S-matrix, i.e. \(e^+ e^- \rightarrow t\bar{t} \rightarrow (bW^-) (\bar{b}W^+)\)^{36} and the unstable particles can appear only as intermediate states.
Electroweak corrections to the production vertex \(t\bar{t}-\gamma /Z\) were first described in [764] and re-derived in [765, 766]. In the later refence it is readily realised that amplitudes for single top production, e.g. \(e^+ e^- \rightarrow t b W\), and even no-top quark production \(e^+ e^- \rightarrow b W^+ \bar{b} W^-\) can contribute to (or mix with) the top-pair production because the physical final state is the same.
The top-quark width is generated by the EW interaction, \(t\rightarrow b W\), therefore the effects of the top-quark finite width are intimately related to the EW corrections of the process. To take into account certain electroweak non-resonant effects a method referred to as phase-space matching was introduced in [767, 768].
This idea has been further developped and rephrased in the framework of an effective theory for unstable particle [769, 770]. (See Refs. [771, 772] for an application of the method to W-pair production in \(e^+ e^-\) annihilation.) A systematic analysis of the electroweak effects in top-quark pair production has started rather recently, and NLO electroweak non-resonant contributions were computed [773], e.g. \(R(e^+e^-\rightarrow t\bar{b}W^-)\sim \alpha _\mathrm{EW}\), where resonant (on-shell) top quarks decay and the final state \((b W^+) (\bar{b} W^+)\) is measured assuming stable W-bosons and b-quarks. In this work invariant mass cuts on the top-quark and antitop-quark decay products are implemented. It is found that the non-resonant correction results in a negative 5 % shift of the total cross section which is almost energy independent, in agreement with Ref. [768]. The dominant NNLO non-resonant corrections were computed in Refs. [774, 775] and it was shown that the single resonant amplitudes (e.g. \(e^+ e^- \rightarrow t (\bar{b} W^-) g\)) provide the counter terms for the uncancelled ultraviolet divergence \(\alpha _s {\varGamma }/\epsilon \) discussed previously for the double resonant \(e^+ e^- \rightarrow t\bar{t}\) amplitude at NNLO QCD. Therefore, the non-resonant corrections provide together with NNLO QCD a consistent treatment of top quark width effects.
In Fig. 107 the forward–backward asymmetry is plotted as a function of energy E. Top and bottom panels show the dependence on \({\varGamma }_t\) and \(\alpha \), respectively. As discussed above the asymmetry \(A_{\mathrm{FB}}\) is an effect of \(\gamma \) and Z-boson interference. Therefore, the asymmetry provides useful information on the mechanism of top-quark production near threshold.
3.3.4 Top-quark production in the continuum
3.4 Physics potential
The excellent possibilities for precision top-quark measurements at \(e^+e^-\) colliders have been confirmed by experimental studies of the physics potential of linear colliders, which, in particular in the framework of recent reports of the CLIC and ILC physics and detector projects, often are based on full detector simulations. Particular emphasis has been placed on the measurement of the top-quark mass, which has been studied both at and above threshold, and on the study of the \(t\bar{t}Z/\gamma ^*\) vertex through the measurement of asymmetries. For all of these measurements, precise flavour tagging and excellent jet reconstruction are crucial to identify and precisely reconstruct top-quark pair events. The detectors being developed for linear colliders provide these capabilities, and, together with the rather modest background levels in \(e^+e^-\) collisions, allow one to acquire high-statistics high-purity top-quark samples. In the following, the most recent published results from simulation studies of top-quark mass measurements are discussed. The studies of top-quark couplings, which make use of the possibilities for polarised beams at linear colliders, are still on-going. Preliminary results indicate a substantially higher precision than achievable at hadron colliders.
3.4.1 Top-quark mass measurement at threshold
Since the cross section depends not only on the top-quark mass, but also on \(\alpha _s\), those two values are determined simultaneously with a two-dimensional fit, resulting in a statistical uncertainty of 27 MeV on the mass and 0.0008 on \(\alpha _s\). Assuming the CLIC luminosity spectrum, which is characterised by a somewhat more pronounced beamstrahlung tail and a larger energy spread, the uncertainties increase to 34 MeV and 0.0009, respectively. Systematic uncertainties from the theoretical cross-section uncertainties, from the precision of the background description and the understanding of the detector efficiency as well as from the absolute knowledge of the beam energy are expected to be of similar order as the statistical uncertainties. Thus, the differences between different linear collider concepts for a top threshold scan are negligible, and total uncertainties of below 100 MeV on the mass are expected [40]. For a phenomenological interpretation, the measured 1S mass typically has to be converted into the standard \(\overline{\text{ MS }}\) mass. This incurs additional uncertainties of the order of 100 MeV, depending on the available precision of \(\alpha _s\) [747].
As discussed in detail in Ref. [787], in addition to the mass and the strong coupling constant, also the top-quark width can be determined in a threshold scan. The use of additional observables such as the top-quark momentum distribution and the forward–backward asymmetry has the potential to further reduce the statistical uncertainties. The cross section around threshold is also sensitive to the top-quark Yukawa coupling, as discussed above. However, its effect on the threshold behaviour is very similar to that of the strong coupling constant, so an extraction will only be possible with a substantially improved knowledge of \(\alpha _s\) compared to the current world average uncertainty of 0.0007, and with reduced theoretical uncertainties on the overall cross section.
3.4.2 Top-quark mass measurement in the continuum
For both CLIC and ILC this measurement has been studied using full detector simulations with all relevant physics backgrounds at an energy of 500 GeV. In the case of the CLIC study, also the influence of background from hadron production in two-photon processes was included, which is more severe at CLIC than at ILC due to the very high bunch-crossing frequency. The reconstructed invariant mass after background rejection and kinematic fitting for the fully hadronic final state at CLIC is shown in Fig. 110. The figure also illustrates the high purity achievable for top quarks at linear colliders. For an integrated luminosity of 100 fb\(^{-1}\), combined statistical precisions of 70 and 80 MeV are obtained for ILC [207] and CLIC [40], respectively. The CLIC study showed that it is expected that systematic uncertainties due to the jet energy scale can be limited to below the statistical uncertainty by constraining the light jet-energy scale through the direct reconstruction of the W bosons in the top-quark decay. The b jet energy scale in turn can be determined in a similar way from \(Z\rightarrow b\bar{b}\) decays. Also other experimental systematics, such as the knowledge of the beam energy, which enters in the kinematic fit, and uncertainties from colour reconnection effects are expected to be small.
However, in contrast to the measurement via a threshold scan, the mass determined by direct reconstruction is theoretically not well defined. Rather, it is obtained in the context of the event generator used to determine the detector and reconstruction effects on the measured invariant mass. At present, no conversion of this invariant mass value to the \(\overline{\text{ MS }}\) mass exists. This leads to additional uncertainties in the interpretation of the result, which potentially far exceed the experimental accuracy of the invariant mass measurement.
3.4.3 Measurement of coupling constants
For precise test of the standard model and New Physics searches a precise determination of the standard model couplings together with the search for anomalous couplings is important. In the following we try to review the prospects of a future Linear Collider and compare where possible with the LHC. From top-quark pair production at hadron collider the top-quark coupling to gluons is already constrained. As mentioned in Sect. 3.4.1 the threshold studies can be used to measure the top-quark mass together with \(\alpha _s\). Top-quark pairs produced in association with an additional jet can be used to search directly for anomalous top-gluon couplings. This can be done independent of the production mechanism in hadronic collisions as well as in electron–positron annihilation. For hadronic \(t\bar{t} + \text {1-Jet}\) production dedicated NLO calculations are available [788, 789, 790, 791]. For electron–positron annihilation the corresponding calculations for massive b-quarks [792, 793, 794, 795, 796] can be applied by adjusting the coupling constants. A dedicated analysis of top-quark pair \(+\) 1-jet production at a future Linear Collider can be found in Ref. [797]. Since anomalous couplings will show up more likely in the couplings to the weak gauge bosons no detailed study of the sensitivity to anomalous top-gluon couplings has been performed so far for a future Linear Collider.
The Wtb-coupling can be probed through top-quark decay and single-top-quark production. A detailed measurement of this coupling is interesting because the \(V-A\) structure of the vertex can be tested. Furthermore the existence of a fourth family – if not yet ruled out by other measurements – could significantly change the SM predictions for the respective coupling. Tevatron and LHC measurements constrain the coupling already through the measurement of the top-quark width [798] and the measurements of the W-boson helicity fractions [799, 800, 801]. A measurement of the top-quark width from threshold studies can be used to indirectly constraint the coupling in electron–positron annihilation. A direct measurement of the Wtb coupling at a Linear Collider is difficult [779]. In top-quark pair production close to the threshold the coupling enters only through the branching ratio for \(t\rightarrow Wb\), which is expected to be very close to one and thus does not lead to a strong dependence on the Wtb coupling. Measurements using single-top-quark production are difficult owing to sizeable backgrounds. In Ref. [554] it has been argued that using \(e^+e^-\rightarrow W^+bW^-\bar{b}\) events below the \(t\bar{t}\) threshold the coupling can be measured at ILC with an accuracy of about 3 % using an integrated luminosity of about 100/fb.
Given that the top quark is so much heavier than the next heavy quark it seems reasonable to question whether the mechanism to generate the top-quark mass is the same as for the lighter quarks. In this context the measurement of the \(t\bar{t}H\) Yukawa coupling is of great importance. At the LHC this coupling can be accessed through the measurement of top-quark pair production in association with a Higgs boson. A recent study of the sensitivity where the subsequent decay \(H\rightarrow b\bar{b}\) has been used can be found for example in Ref. [806]. In Ref. [268] it has been estimated that the ttH coupling can be measured at the LHC with an accuracy of about 15 % assuming an integrated luminosity of 300/fb at 14 TeV centre-of-mass energy. With an increased luminosity of 3000/fb a measurement at the level of 7–14 % may become feasible. Due to the large mass of the final state it is difficult to improve this measurement significantly at a linear collider operating at 500 GeV. For an integrated luminosity of 1000/fb at 500 GeV centre-of-mass energy an uncertainty of 10 % has been estimated [268]. Increasing the energy to 1 TeV (ILC) or even 1.4 TeV (CLIC) will help to improve the situation: In both cases a precision of 4 % seems to be feasible. Using the ILC design at 1 TeV would require 1000/fb of integrated luminosity, while at 1.4 TeV 1500/fb would be required.
Very recently it has been argued in Ref. [807] that the ttH coupling could also be inferred at the LHC from single-top-quark production in association with an additional Higgs. Since the cross section of this process is below 100 fb such a measurement will be challenging. In the standard model the cross section is reduced through an accidental cancellation. As a consequence BSM models may show sizeable deviations compared to the Standard Model prediction.
3.4.4 The top-quark polarisation
4 Exploring the quantum level: precision physics in the SM and BSM^{37}
We review the LC capabilities to explore the electroweak (EW) sector of the SM at high precision and the prospects of unveiling signals of BSM physics, either through the presence of new particles in higher-order corrections or via direct production of extra EW gauge bosons. We discuss the experimental and theory uncertainties in the measurement and calculation of EWPO, such as the W boson mass, Z pole observables, in particular the effective weak mixing angle, \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\), and the anomalous magnetic moment of the muon, \(a_\mu \). We concentrate on the MSSM to illustrate the power of these observables for obtaining indirect information on BSM physics. In particular, we discuss the potential of two key EWPOs at a LC, \(M_W\) and \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\), to provide a stringent test of the SM and constraints on the MSSM parameter space. Naturally, the recent discovery of a Higgs-like particle at the LHC has a profound impact on EW precision tests of the SM. We present a study of the impact of this discovery on global EW fits, and also include a discussion of the important role of the top-quark mass in performing these high precision tests of the SM. Finally, we review the anticipated accuracies for precision measurements of triple and quartic EW gauge boson couplings, and how deviations from SM gauge boson self interactions relate to different BSM scenarios. These observables are of special interest at a LC, since they have the potential of accessing energy scales far beyond the direct kinematical reach of the LHC or a LC. We conclude with a discussion of the LC reach for a discovery of extra EW gauge bosons, \(Z'\) and \(W'\), and the LC’s role for pinning down their properties and origin, once discovered.
4.1 The role of precision observables
The SM cannot be the ultimate fundamental theory of particle physics. So far, it succeeded in describing direct experimental data at collider experiments exceptionally well with only a few notable exceptions, e.g., the left–right (\(A_\mathrm{LR}^e\)(SLD)) and forward–backward (\(A_\mathrm{FB}^b\)(LEP)) asymmetry (see Sect. 4.3.3), and the muon magnetic moment \(g_\mu -2\) (see Sect. 4.6). However, the SM fails to include gravity, it does not provide cold DM, and it has no solution to the hierarchy problem, i.e. it does not have an explanation for a Higgs-boson mass at the electroweak scale. On wider grounds, the SM does not have an explanation for the three generations of fermions or their huge mass hierarchies. In order to overcome (at least some of) the above problems, many new physics models (NPM) have been proposed and studied, such as supersymmetric theories, in particular the MSSM, two-Higgs-doublet models (THDM), technicolour, little Higgs models, or models with (large, warped, or universal) extra spatial dimensions. So far, the SM has withstood all experimental tests at past and present collider experiments, such as the LEP and SLC \(e^+ e^-\) colliders, the HERA ep, Tevatron \(p \bar{p}\), and LHC pp collider. Even the recently discovered Higgs-like particle at the LHC, after analysing the 2012 data agrees with the SM Higgs boson expectation, albeit more precise measurements of its properties will be needed to pin down its identity. Measurements of precision observables and direct searches for NPM particles succeeded to exclude or set stringent bounds on a number of these models. The direct search reach is going to be significantly extended in the upcoming years, when the LHC is scheduled to run at or close to its design energy of 14 TeV. Future \(e^+e^-\) colliders, such as the ILC or CLIC, have good prospects for surpassing the LHC direct discovery reach, especially in case of weakly interacting, colourless NPM particles (see, e.g., Sect. 4.8).
Even if a direct discovery of new particles is out of reach, precision measurements of SM observables have proven to be a powerful probe of NPM via virtual effects of the additional NPM particles. In general, precision observables (such as particle masses, mixing angles, asymmetries etc.) that can be predicted within a certain model, including higher order corrections in perturbation theory, and thus depending sensitively on the other model parameters, and that can be measured with equally high precision, constitute a test of the model at the quantum-loop level. Various models predict different values of the same observable due to their different particle content and interactions. This permits to distinguish between, e. g., the SM and a NPM, via precision observables. Naturally, this requires a very high precision of both the experimental results and the theoretical predictions. The wealth of high-precision measurements carried out at the Z pole at LEP and SLC, the measurement of the W boson at LEP and the Tevatron [21, 822, 824], as well as measurements at low-energy experiments, such as \(a_\mu =(g_\mu -2)/2\) at the “Muon \(g-2\) Experiment” (E821) [818], are examples of EWPOs that probe indirect effects of NPM particles. These are also examples where both experiment and theory have shown that they can deliver the very high precision needed to fully exploit the potential of these EWPOs for detecting minute deviations from the SM. The most relevant EWPOs in which the LC plays a key role are the W boson mass, \(M_W\), and the effective leptonic weak mixing angle, \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\). In the MSSM, the mass of the lightest \({\mathscr {CP}}\)-even MSSM Higgs boson, \(M_h\), constitutes another important EWPO [819]. Note that in these examples, the top quark mass plays a crucial role as input parameter.
Also EWPOs that cannot be measured at a LC can be very relevant in the assessment of its physics potential. A prominent role in this respect plays the muon magnetic moment, \((g_\mu -2)\). It already provides some experimental indication for NPM particles in reach of a LC, and its role in constraining NPM and its complementarity to the LC is summarised in Sect. 4.6.
Another type of PO is connected to the self interactions of EW gauge bosons in multiple EW gauge boson production, i.e. they directly probe the triple and quartic EW gauge boson couplings. Deviations from SM predictions would indicate new physics, entering either through loop contributions or are due to new heavy resonances, which at low energy manifest themselves as effective quartic gauge boson couplings. Precision measurements of these POs could provide information as regards NPM sectors far beyond the kinematic reach of the LHC and LC.
As discussed above, in this report we focus our discussion on the EWPO, i.e. (pseudo-) observables like the W-boson mass, \(M_W\), the effective leptonic weak mixing angle, \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\), and the anomalous magnetic moment of the muon. Since in the literature virtual effects of NPM particles are often discussed in terms of effective parameters instead of the EWPO we briefly discuss this approach in the following.
A widely used set of effective parameters are the S, T, U parameters [820]. They are defined such that they describe the effects of new physics contributions that enter only via vacuum-polarisation effects (i.e. self-energy corrections) to the vector boson propagators of the SM (i.e. the new physics contributions are assumed to have negligible couplings to SM fermions). The S, T, U parameters can be computed in different NPMs as certain combinations of one-loop self-energies, and then can be compared to the values determined from a fit to EW precision data, i.e. mainly from \(M_W, M_Z\) and \({\varGamma }_Z\) (see, e.g., the review in [821]). A non-zero result for S, T, U indicates non-vanishing contributions of new physics (with respect to the SM reference value). According to their definition, the S, T, U parameters are restricted to leading order contributions of new physics. They should therefore be applied only for the description of small deviations from the SM predictions, for which a restriction to the leading order is permissible. Examples of new physics contributions that can be described in the framework of the S, T, U parameters are contributions from a fourth generation of heavy fermions or effects from scalar quark loops to the W- and Z-boson observables. A counter example, i.e. where the S, T, U framework is not sufficicent, are SUSY corrections to the anomalous magnetic moment of the muon. Due to these restrictions of this effective description of BSM effects in W and Z boson observables, in this report we decided to only present investigations of these effects in the EWPO themselves.
This review of precision physics in the SM and BSM at the LC is organised as follows: in Sect. 4.2 we concentrate on \(M_W\) from both the experimental and the theoretical view points, and then turn to a discussion of Z pole observables, in particular \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\), in Sect. 4.3. The relevance of the top-quark mass in EW precision physics is briefly summarised in Sect. 4.4, before we present the prospects of extracting information as regards the SM Higgs-boson mass from a global EW fit in Sect. 4.5. We close our discussion of EWPOs with an overview of predictions for the muon magnetic moment in NPM in Sect. 4.6. An overview of possible parametrisations of non-standard EW gauge boson couplings, available calculations and the experimental prospects for precision measurements of these couplings is presented in Sect. 4.7. Finally, in Sect. 4.8 we present an overview of studies of new gauge bosons at the LC.
4.2 The \(\varvec{W}\) boson mass
4.2.1 Experimental prospects for a precision measurement of \({M_W}\) a the ILC^{38}
The ILC facility^{39} can contribute decisively by making several complementary measurements of the W mass using \(e^{+}e^{-}\) collisions at centre-of-mass energies spanning from near WW threshold to as high as 1 TeV. Data samples consisting of between 10 and 100 million W decays can be produced, corresponding to an integrated luminosity of about 250fb\(^{-1}\) at \(\sqrt{s} = 250\) GeV (and correspondingly lower integrated luminosity at higher energies).
The main production channels of W bosons at ILC are pair production, \(e^{+} e^{-} \rightarrow W^+ W^-\) and single-W production, \(e^{+} e^{-} \rightarrow W e \overline{\nu }_e\), which proceeds mainly through \(\gamma -W\) fusion. Pair production dominates at lower centre-of-mass energies, while single-W production dominates over other \(e^+e^-\) sources of hadronic events at the higher energies.
Polarised threshold scan of the \(W^+W^-\) cross section as discussed in [825].
Kinematically constrained reconstruction of \(W^+W^-\) using constraints from four-momentum conservation and optionally mass-equality as was done at LEP2.^{40}
Direct measurement of the hadronic mass. This can be applied particularly to single-W events decaying hadronically or to the hadronic system in semileptonic \(W^+W^-\) events.
We first give an outline of statistical considerations for \(M_W\) measurements and then outline the strategies considered for being able to make use of this considerable statistical power in experimentally robust ways.
The statistical errors on a W mass determination at ILC are driven by the cross sections, the intrinsic width of the W (\({\varGamma }_W \approx 2.08 \mathrm {GeV}\,\)), the potential integrated luminosity, the availability of polarised beams, and where appropriate the experimental di-jet mass resolution, event selection efficiencies and backgrounds. The width is the underlying fundamental issue. This broadens the turn-on of the W-pair cross section near threshold, decreasing its dependence on \(M_W\). It also broadens the W line-shape, diluting the statistical power of mass measurements for both kinematically constrained reconstruction and direct mass reconstruction. For the detectors envisaged at ILC, hadronically decaying Ws should be measured with mass resolutions in the 1–2 \(\mathrm {GeV}\,\) range.
Statistical errors from a single cross-section measurement near threshold (\(\sqrt{s} \approx 2 M_W+ 0.5\;\mathrm {GeV}\,\)) are discussed in [826]. The statistical sensitivity factor on \(M_W\) for an optimised single cross-section measurement assuming unpolarised beams, 100 % efficiency and no backgrounds is \(0.91\,\mathrm {MeV}\,/\sqrt{{\mathscr {L}}_\mathrm{int} [\mathrm {ab}^{-1}]}\). For an integrated luminosity of \({\mathscr {L}}_\mathrm{int} = 100~\mathrm {fb}^{-1}\) this translates to 2.9 \(\mathrm {MeV}\,\). However experimental systematic errors on such a single cross-section measurement of \(\sim \)0.25 % enter directly and would give a corresponding 4.2 \(\mathrm {MeV}\,\) experimental systematic uncertainty. At the ILC, the statistical sensitivity factor can be further improved using polarised beams colliding with the appropriate helicities corresponding effectively for practical polarisation values (80–90, 40–60%) to a factor of up to 3 WW-production luminosity upgrade.
The method of a polarised threshold scan is discussed in some detail in [825] based on conservative extrapolations from the measurements using the LEP detectors. The idea is to use the measurement of the threshold dependence of the cross section to determine \(M_W\). The study is based on 100 fb\(^{-1}\) allocated to 5 scan points near threshold and 1 scan point at 170 \(\mathrm {GeV}\,\). Data are collected mostly with \(e^{-}_{L} e^{+}_{R}\) but other combinations of two-beam, single-beam and no beam polarisation are used to control the backgrounds and polarisation systematics. The \(170\,\mathrm {GeV}\,\) point has little sensitivity to \(M_W\) but helps to constrain the efficiency systematics. The overall experimental error on the W mass (excluding beam-energy systematic and eventual theoretical errors) is estimated to be \(5.2~\mathrm {MeV}\,\).
A critical external input needed to interpret the threshold dependence of the cross section in terms of \(M_W\) is knowledge of the centre-of-mass energies. Various measurements sensitive to the centre-of-mass energy can be made using \(e^+e^- \rightarrow \ell \ell \gamma \) (\(\ell = e, \mu \)) events. From knowledge of the polar angles of the leptons, under the assumption of a 3-body final state, one can measure statistically the luminosity-weighted centre-of-mass energy with an error of 31 ppm for the proposed scan. This translates into a \(M_W\) error of \(2.5\,\mathrm {MeV}\,\) per 100 fb\(^{-1}\) polarised scan. A related method using the momenta of the two leptons (particularly the muons) can determine the centre-of-mass energy with much better statistical precision. The tracker momentum scale needs to be controlled – this is feasible using Z’s – and potentially with other particles with well-measured masses.
In summary, it is estimated that \(M_W\) can be measured to \(6\,\mathrm {MeV}\,\) experimental accuracy using this method which uses dedicated running near threshold. This number includes also the anticipated uncertainties from the beam energy (\(\sim \)1.9 \(\mathrm {MeV}\,\)) and from theory (\(\sim \)2.5 \(\mathrm {MeV}\,\)), where the corresponding theoretical issues will be discussed in the next subsection.
Much of the ILC programme is likely to take place at energies significantly above the WW threshold in a regime where both WW production and single-W production are prevalent. Consequently, a direct reconstruction of the hadronic mass can be very important. One can use WW events with one W decaying leptonically (\(e, \mu , \tau \)) and the other decaying hadronically, and also single-W events with the W decaying hadronically to measure \(M_W\) from the measured hadronic mass. Beam polarisation can be used to enhance the cross sections. The critical issue is being able to control the jet energy scale. A number of approaches are plausible and should be pursued. One approach consists of using Z(\(\gamma \)) radiative return events where the Z decays hadronically and the photon is unmeasured within or close to the beam-pipe. Another approach attempts to do a jet-energy calibration from first principles using the individual components that make up the measured jet energy, namely using the calibration of the tracker momentum scale and the calorimeter energy scales at the individual particle level determined from for example calibration samples of well-known particles (\(J/\psi \), \(K^0_S\), \({\Lambda }\), \(\pi ^0\) etc.). The latter has the advantage that it does not rely directly on the Z mass. Other calibration possibilities are using ZZ, Zee and Z\(\nu \nu \) events. Assuming a sample of \(5\,10^6\) hadronic Zs for calibration one should be able to approach a jet-energy scale related statistical error of around \(2.0\,\mathrm {MeV}\,\) for \(M_W\). Systematic limitations in the Z-based methods is the knowledge of the Z mass (currently \(2.1\,\mathrm {MeV}\,\)) – and any residual quark-flavour related systematics that make the detector response of hadronic Ws different from hadronic Zs. It seems plausible to strive for an overall error of \(5\,\mathrm {MeV}\,\) from these methods.
A kinematically constrained reconstruction of WW pairs was the work-horse of LEP2 – but has received little attention to date for ILC studies related to W mass measurement. By imposing kinematic constraints, the LEP2 experiments were able to compensate for modest jet-energy resolution. At ILC, the constraints are no longer as valid (beamstrahlung) the detector resolution is much better (of the same order as \({\varGamma }_W\)), and until recently, it seemed that the beam energy could not be determined with adequate precision at high energy. Lastly, at the order of precision that is being targeted, it seems unwise to bank on the fully hadronic channel where it is quite possible that final-state interactions such as colour reconnection may cause the mass information to be corrupted. So it seems that the kinematically constrained reconstruction method is most pertinent to the \(q \bar{q} e \nu _e\) and \(q \bar{q} \mu \nu _\mu \) channels.
Recent work exploring the reconstruction of the centre-of-mass energy using the measured muon momenta in \(e^+e^- \rightarrow \mu ^+ \mu ^- (\gamma )\) events indicates that it is very feasible to measure the luminosity-weighted centre-of-mass energy with high precision, and that this approach is promising also at relatively high centre-of-mass energies.
In addition, given the impetus for potentially running the ILC at a centre-of-mass energy of around \(250\,\mathrm {GeV}\,\), not far above LEP2, there seems a clear potential to improve the \(M_W\) measurement by including information from the leptons in the mass estimate. This lower energy regime should be the most favourable for beamstrahlung and beam-energy determination outlook. Probably by performing kinematically constrained fits that build on the existing methods one would be able to get complementary information, which would be significantly uncorrelated in several of the main systematics with the direct reconstruction method. This deserves more study – but errors at the \(5\,\mathrm {MeV}\,\) level or less may be achievable.
To summarise, the ILC facility has three principal ways of measuring \(M_W\). Each method can plausibly measure \(M_W\) to a precision in the \(5\,\mathrm {MeV}\,\) range. The three methods are largely uncorrelated. If all three methods do live up to their promise, one can target an overall uncertainty on \(M_W\) in the 3–4 \(\mathrm {MeV}\,\) range.
4.2.2 Theory aspects concerning the WW threshold scan^{41}
While in the previous subsection the experimental precision for the W boson mass measurement at the LC was discussed, this subsection deals with the correspondingly required theory calculations and precisions, in particular for the WW threshold scan.
The theoretical uncertainty (TU) for the direct mass reconstruction at LEP2 has been estimated to be of the order of \(\sim \)5–10 \(\mathrm {MeV}\,\) [827, 828], based on results of YFSWW [829] and RacoonWW [830], which used the double-pole approximation (DPA) for the calculation of the NLO corrections. This is barely sufficient for the accuracies aimed at a LC. These shortcomings of the theoretical predictions have been cured by dedicated calculations.
In [831, 832] the total cross section for the charged-current four-fermion production processes \(e^+e^-\rightarrow \nu _\tau \tau ^+\mu ^-\bar{\nu }_\mu \), \(u\bar{d}\mu ^-\bar{\nu }_\mu \), \({u}\bar{d}s\bar{c}\) was presented including the complete electroweak NLO corrections and all finite-width effects. This calculation was made possible by using the complex-mass scheme for the description of the W-boson resonances and by novel techniques for the evaluation of the tensor integrals appearing in the calculation of the one-loop diagrams. The full \({\mathscr {O}}(\alpha )\) calculation, improved by higher-order effects from ISR, reduced the remaining TU due to unknown electroweak higher-order effects to a few 0.1 % for scattering energies from the threshold region up to \(\sim \)500 \(\mathrm {GeV}\,\); above this energy leading high-energy logarithms, such as Sudakov logarithms, beyond one loop have to be taken into account to match this accuracy [833]. At this level of accuracy, also improvements in the treatment of QCD corrections to semileptonic and hadronic \(e^+e^-\rightarrow 4f\) processes are necessary. The corrections beyond DPA, were assessed by comparing predictions in DPA from the generator RacoonWW to results from the full four-fermion calculation [831, 832], as coded in the follow-up program Racoon4f (which is not yet public). This comparison revealed effects on the total cross section without cuts of \(\sim \)\(0.3\,\% (0.6\,\%)\) for CM energies ranging from \(\sqrt{s}\sim 200\,\mathrm {GeV}\,\) (\(170\,\mathrm {GeV}\,\)) to \(500\,\mathrm {GeV}\,\). The difference to the DPA increases to 0.7–1.6 % for \(\sqrt{s}\sim 1{-}2\,{\text {TeV}}\). At threshold, the full \({\mathscr {O}}(\alpha )\) calculation corrects the IBA by about 2 %. While the NLO corrections beyond DPA have been calculated only for the processes \(e^+e^-\rightarrow \nu _\tau \tau ^+\mu ^-\bar{\nu }_\mu \), \(u\bar{d}\mu ^-\bar{\nu }_\mu \), \({u}\bar{d}s\bar{c}\) so far, the effect for the other four-fermion processes, which interfere with ZZ production, should be similar. Once the corrections to those channels are needed, they can be calculated with the available methods.
Using methods from effective field theory, the total cross section for 4-fermion production was calculated near the W pair production threshold [771, 772]. These calculations used unstable-particle effective field theory to perform an expansion in the coupling constants, \({\varGamma }_W/M_W\), and in the non-relativistic velocity v of the W boson up to NLO in \({\varGamma }_W/M_W\sim \alpha _\mathrm{ew}\sim v^2\). In [771] the theoretical error of an \(M_W\) determination from the threshold scan has been analysed. As a result, the resummation of next-to-leading collinear logarithms from initial-state radiation is mandatory to reduce the error on the W mass from the threshold scan below \(30\,\mathrm {MeV}\,\). It was found that the remaining uncertainty of the pure NLO EFT calculation is \(\delta M_W\approx 10{-}15\,\mathrm {MeV}\,\) and is reduced to about \(5\,\mathrm {MeV}\,\) with additional input from the NLO four-fermion calculation in the full theory. In order to reduce this error further, in [772] the (parametrically) dominant next-to-next-to-leading order (NNLO) corrections (all associated with the electromagentic Coulomb attraction of the intermediate W bosons) in the EFT have been calculated leading to a shift of \(\delta M_W\sim 3\,\mathrm {GeV}\,\) and to corrections to the cross section at the level of 0.3 %. The effect of typical angular cuts on these corrections was shown to be completely negligible. Thus, one may conclude that the inclusive partonic four-fermion cross section near the W-pair production threshold is known with sufficient precision.
In summary, all building blocks for a sufficiently precise prediction of the W-pair production cross section in the threshold region are available. They require the combination of the NLO calculation of the full four-fermion cross section with the (parametrically) dominant NNLO corrections, which are calculated within the EFT. For the precise determination of the cross section at energies above \(500\,\mathrm {GeV}\,\) the leading two-loop (Sudakov) corrections should be included in addition to the full NLO corrections. Combining the theoretical uncertainties with the anticipated precision from a threshold scan (see the previous subsection) a total uncertainty of \(7\,\mathrm {MeV}\,\) can be estimated [834].
4.2.3 Theory predictions for \(M_W\) in the SM and MSSM^{42}
The precise measurement of the W boson mass can be used to test NPM via their contribution to quantum corrections to \(M_W\). However, this requires a precise prediction of \(M_W\) in the respective models. Here we will concentrate on the prediction of \(M_W\) in the MSSM.
The prediction of \(M_W\) in the MSSM depends on the masses, mixing angles and couplings of all MSSM particles. Sfermions, charginos, neutralinos and the MSSM Higgs bosons enter already at one-loop level and can give substantial contributions to \(M_W\). Consequently, it is expected to obtain restrictions on the MSSM parameter space in the comparison of the \(M_W\) prediction and the experimental value of Eq. (110).
Parameter ranges. All parameters with mass dimension are given in GeV
Parameter | Minimum | Maximum |
---|---|---|
\(\mu \) | \(-\)2000 | 2000 |
\(M_{\tilde{E}_{1,2,3}}=M_{\tilde{L}_{1,2,3}}\) | 100 | 2000 |
\(M_{\tilde{Q}_{1,2}}=M_{\tilde{U}_{1,2}}=M_{\tilde{D}_{1,2}}\) | 500 | 2000 |
\(M_{\tilde{Q}_{3}}\) | 100 | 2000 |
\(M_{\tilde{U}_{3}}\) | 100 | 2000 |
\(M_{\tilde{D}_{3}}\) | 100 | 2000 |
\(A_e=A_{\mu }=A_{\tau }\) | \(-\)3\(\,M_{\tilde{E}}\) | 3\(\,M_{\tilde{E}}\) |
\(A_{u}=A_{d}=A_{c}=A_{s}\) | \(-\)3\(\,M_{\tilde{Q}_{12}}\) | 3\(\,M_{\tilde{Q}_{12}}\) |
\(A_b\) | \(-\)3 max(\(M_{\tilde{Q}_{3}},M_{\tilde{D}_{3}}\)) | 3 max(\(M_{\tilde{Q}_{3}},M_{\tilde{D}_{3}}\)) |
\(A_t\) | \(-\)3 max(\(M_{\tilde{Q}_{3}},M_{\tilde{U}_{3}}\)) | 3 max(\(M_{\tilde{Q}_{3}},M_{\tilde{U}_{3}}\)) |
\(\tan \beta \) | 1 | 60 |
\(M_3\) | 500 | 2000 |
\(M_A\) | 90 | 1000 |
\(M_2\) | 100 | 1000 |
The evaluation of \(M_W\) includes the full one-loop result and all known higher order corrections of SM- and SUSY-type, for details see [835, 840] and references therein. The results for \(M_W\) are shown in Fig. 112 as a function of \(m_{t}\,\). In the plot the green region indicated the MSSM \(M_W\) prediction assuming the light \({\mathscr {CP}}\)-even Higgs h in the region \(125.6 \pm 3.1\,\mathrm {GeV}\,\). The red band indicates the overlap region of the SM and the MSSM. The leading one-loop SUSY contributions arise from the stop sbottom doublet. However, requiring \(M_h\) in the region \(125.6 \pm 3.1\,\mathrm {GeV}\,\) restricts the parameters in the stop sector [248] and with it the possible \(M_W\) contribution. Large \(M_W\) contributions from the other MSSM sectors are possible, if either charginos, neutralinos or sleptons are light.
The grey ellipse indicates the current experimental uncertainty, see Eqs. (110), (120), whereas the red ellipse shows the anticipated future ILC/GigaZ precision. While at the current level of precision SUSY might be considered as slightly favoured over the SM by the \(M_W\)–\(m_{t}\,\) measurement, no clear conclusion can be drawn. The small red ellipse, on the other hand, indicates the discrimination power of the future ILC/GigaZ measurements. With the improved precision a small part of the MSSM parameter space could be singled out. The comparison of the SM and MSSM predictions with the ILC/GigaZ precision could rule out either of the models.
4.3 \(\varvec{Z}\) pole observables
Other important EWPOs are the various observables related to the Z boson, measured in four-fermion processes, \(e^+ e^- \rightarrow \gamma ,Z \rightarrow f \bar{f}\), at the Z boson pole. We review the theoretical precision of SM predictions for various Z boson pole observables and the anticipated experimental precision at GigaZ. As for \(M_W\), we also review the potential of a precise measurement and prediction of \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) to obtain information as regards the MSSM parameter space.
4.3.1 Theoretical prospects^{44}
Equation (111) explicitly spells out the leading Z-pole contribution, while additional effects from photon exchange and box corrections are included in the remainder \(\sigma _{\text {non-res}}\).
The ratio of \(g_{Vf}\) and \(g_{Af}\) is commonly parametrised through the effective weak mixing angle \(\sin ^2\theta ^f_\mathrm{eff}\). It can be determined from the angular distribution with respect to \(\cos \theta \) or from the dependence on the initial electron polarisation \({\mathscr {P}}_e\). On the other hand, the partial and total widths are determined from the total cross section \(\sigma (s)\) for different values of s and from branching ratios (see the previous subsection).
For leptonic final states, the effective weak mixing angle \(\sin ^2\theta ^\ell _\mathrm{eff}\) has been calculated in the SM to the complete two-loop order [842, 843, 844, 845, 846, 847, 848, 849], and three- and four-loop corrections of order \({\mathscr {O}}(\alpha \alpha _s^2)\) [850, 851, 852, 853] and \({\mathscr {O}}(\alpha \alpha _{s}^3)\) [854, 855, 856] are also known. Furthermore, the leading \({\mathscr {O}}(\alpha ^3)\) and \({\mathscr {O}}(\alpha ^2\alpha _s)\) contributions for large values of \(m_t\) [857, 858] or \(m_H\) [859, 860] have been computed.
The current uncertainty from unknown higher orders is estimated to amount to about \(4.5\times 10^{-5}\) [849], which mainly stems from missing \({\mathscr {O}}(\alpha ^2\alpha _s)\) and \({\mathscr {O}}(N_f^2\alpha ^3,\,N_f^3\alpha ^3)\) contributions beyond the leading \(m_t^4\) and \(m_t^6\) terms, respectively. (Here \(N_f^n\) denotes diagrams with n closed fermion loops. Based on experience from lower orders, the \({\mathscr {O}}(\alpha ^3)\) diagrams with several closed fermion loops are expected to be dominant.) The calculation of these corrections requires three-loop vertex integrals with self-energy sub-loops and general three-loop self-energy integrals, which realisitically can be expected to be worked out in the forseeable future. The remaining \({\mathscr {O}}(\alpha ^3)\) and four-loop terms should amount to \(\sim 10^{-5}\).^{46}
For quark final states, most two-loop corrections to \(\sin ^2\theta ^q_\mathrm{eff}\) have been computed [849, 861, 862, 863], but only the \({\mathscr {O}}(N_f\alpha ^2)\) and \({\mathscr {O}}(N_f^2\alpha ^2)\) contributions are known for the electroweak two-loop corrections, while the diagrams without closed fermion loops are still missing. However, based on experience from the leptonic weak mixing angle, they are expected to amount to \(\lesssim \)10\(^{-5}\). However, the \({\mathscr {O}}(\alpha \alpha _s^2)\) also not known in this case, leading to an additional theory error of \(\sim 2\times 10^{-5}\). The calculation of the missing \({\mathscr {O}}(\alpha \alpha _s^2)\) corrections, as well as the \({\mathscr {O}}(\alpha ^2\alpha _s)\) corrections, involves general three-loop vertex corrections to \(Z \rightarrow q\bar{q}\), which will only be possible with serious progress in calculational techniques.
When extracting \(\sin ^2\theta ^\ell _\mathrm{eff}\) from realistic observables [left–right (LR) and forward–backward (FB) asymmetries, see the next subsection], the initial- and final-state QED radiator functions \({\mathscr {R}}_i\) must be taken into account. In general, the QED corrections are known to \({\mathscr {O}}(\alpha )\) for the differential cross section and to \({\mathscr {O}}(\alpha ^2)\) for the integrated cross section (see Ref. [864] for a summary). However, for the LR asymmetry they complete cancel up to NNLO [865, 866], while for the FB asymmetry they cancel if hard-photon contributions are excluded, i.e. they cancel up to terms of order \(E_\gamma /\sqrt{s}\) [865, 866, 867, 868, 869]. Therefore, a sufficiently precise result for the soft-photon contribution with \(E_\gamma < E_\gamma ^\mathrm{cut}\) can be obtained using existing calcations for small enough \(E_\gamma ^\mathrm{cut}\), while the hard-photon contribution (\(E_\gamma > E_\gamma ^\mathrm{cut}\)) can be evaluated with numerical Monte-Carlo methods. A similar procedure can be carried out for final-state QCD effects for \(\sin ^2\theta ^q_\mathrm{eff}\) although the corrections beyond NLO are not fully implemented in existing programs (see below).
For the branching fraction \(R_b = {\varGamma }_b/{\varGamma }_\mathrm{had}\) and the total width \({\varGamma }_Z\), two-loop corrections of \({\mathscr {O}}(\alpha \alpha _s)\), \({\mathscr {O}}(N_f\alpha ^2)\), and \({\mathscr {O}}(N_f^2\alpha ^2)\) are known [862, 863, 870, 871, 872]. Assuming geometric progression of the perturbative series, the remaining higher-order contributions are estimated to contribute at the level of \(\sim 1.5\times 10^{-4}\) and 0.5 MeV, respectively. As before, the contribution from electroweak two-loop diagrams without closed fermion loops is expected to be small. The dominant missing contributions are the same as for \(\sin ^2\theta ^q_\mathrm{eff}\).
Some of the most important precision observables for Z-boson production and decay (first column), their present-day estimated theory error (second column), the dominant missing higher-order corrections (third column), and the estimated improvement when these corrections are available (fourth column). In many cases, the leading parts in a large-mass expansion are already known, in which case the third column refers to the remaining pieces at the given order. The numbers in the last column are rough order-of-magnitude guesses. Entries in [italics] indicate contributions that probably will require very significant improvements in calculational techniques to be completed
Quantity | Cur. theo. error | Lead. missing terms | Est. improvem. |
---|---|---|---|
\(\sin ^2\theta ^\ell _\mathrm{eff}\) | \(4.5\times 10^{-5}\) | \({\mathscr {O}}(\alpha ^2\alpha _s)\), \({\mathscr {O}}(N_f^{\ge 2}\alpha ^3)\) | Factor 3–5 |
\(\sin ^2\theta ^q_\mathrm{eff}\) | \(5\times 10^{-5}\) | \({\mathscr {O}}(\alpha ^2)\), \({\mathscr {O}}(N_f^{\ge 2}\alpha ^3)\) | Factor 1–1.5 |
[\({\mathscr {O}}(\alpha \alpha _s^2)\), \({\mathscr {O}}(\alpha ^2\alpha _s)\)] | [Factor 3–5] | ||
\(R_b\) | \(\sim \)\(1.5\times 10^{-4}\) | \({\mathscr {O}}(\alpha ^2)\), \({\mathscr {O}}(N_f^{\ge 2}\alpha ^3)\) | Factor 1–2 |
[\({\mathscr {O}}(\alpha \alpha _s^2)\), \({\mathscr {O}}(\alpha ^2\alpha _s)\)] | [Factor 3–5] | ||
\({\varGamma }_Z\) | \(0.5\,\text {MeV}\) | \({\mathscr {O}}(\alpha ^2)\), \({\mathscr {O}}(N_f^{\ge 2}\alpha ^3)\) | Factor 1–2 |
[\({\mathscr {O}}(\alpha \alpha _s^2)\), \({\mathscr {O}}(\alpha ^2\alpha _s)\)] | [Factor 3–5] |
The known corrections to the effective weak mixing angles and the leading corrections to the partial widths are implemented in programs such as Zfitter [864, 873] and Gfitter [874] (see also Sect. 4.5), while the incorporation of the recent full fermionic two-loop corretions is in progress. However, these programs are based on a framework designed for NLO but not NNLO corrections. In particular, there are mismatches between the electroweak NNLO corrections to the \(Zf\bar{f}\) vertices and QED/QCD corrections to the external legs due to approximations and factorisation assumptions. Another problem is the separation of leading and sub-leading pole terms in Eq. (111) [849]. While these discrepancies may be numerically small, it would be desirable to construct a new framework that treats the radiative corrections to Z-pole physics systematically and consistently at the NNLO level and beyond. Such a framework can be established based on the pole scheme [875, 876], where the amplitude is expanded about the complex pole \(s=M_Z^2-i M_Z{\varGamma }_Z\), with the power counting \({\varGamma }_Z/M_Z\sim \alpha \).
4.3.2 Experimental prospects^{47}
The largest possible uncertainty comes from the knowledge of the beam energy. \(\sqrt{s}\) must be known with \(1\,\mathrm{MeV}\) relative to the Z-mass. The absolute precision can be calibrated in a Z-scan, however, a spectrometer with a relative precision of \(10^{-5}\) is needed not to be dominated by this uncertainty. Similarly the beamstrahlung must be known to a few per-cent relative between the calibration scans and the pole running. However, both requirements seem to be possible.
Apart from \(\sin ^2\theta _{\mathrm {eff}}^{\ell }\) also some other Z-pole observables can be measured at a LC. Running at the Z peak gives access to the polarised forward–backward asymmetry for b-quarks which measures \(\sin ^2\theta _{\mathrm {eff}}^{b}\) and the ratio of the b to the hadronic partial width of the Z-boson \(R_{b}^0={\varGamma }_{b \overline{b}}/{\varGamma }_{\mathrm{had}}\). Both quantities profit from the large statistics and the much improved b-tagging capabilities of an ILC detector compared to LEP.
\(R_{b}^0\) can be measured using the same methods as at LEP. The statistical error will be almost negligible and the systematic errors shrink due to the better b-tagging. In total \({\varDelta } R_{b}^0= 0.00014\) can be reached which is an improvement of a factor 5 compared to the present value [877].
\(\sin ^2\theta _{\mathrm {eff}}^{b}\) can be measured from the left–right–forward–backward asymmetry for b-quarks, \(A_\mathrm{FB,LR}^b = 3/4 {\mathscr {P}} {\mathscr {A}}_{b}\). \({\mathscr {A}}_{b}\) depends on \(\sin ^2\theta _{\mathrm {eff}}^{b}\) as shown in Eq. (114), however, in general one has \({g_{V_f}}/{g_{A_f}} = 1 - 4 q_f \sin ^2\theta _{\mathrm {eff}}^{f}\) and due to the small b-charge the dependence is very weak. At present \(\sin ^2\theta _{\mathrm {eff}}^{b}\) is known with a precision of 0.016 from \(A_\mathrm{FB,LR}^b\) measured at the SLC and the forward–backward asymmetries for b-quarks at LEP combined with \(\sin ^2\theta _{\mathrm {eff}}^{\ell }\) measurements at LEP and SLC [879]. Using the left–right–forward–backward asymmetry only at the ILC an improvement by more than a factor 10 seems realistic [877].
The total Z-width \({\varGamma }_Z\) can be obtained from a scan of the resonance curve. The statistical error at GigaZ will be negligible and the systematic uncertainty will be dominated by the precision of the beam energy and the knowledge of beamstrahlung. If a spectrometer with a precision of \(10^{-5}\) can be built, \({\varGamma }_Z\) can be measured with \(1\,\mathrm{MeV}\) accuracy [877]. However, no detailed study on the uncertainty due to beamstrahlung exists.
4.3.3 Constraints to the MSSM from \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\)^{48}
As for \(M_W\) we review examples showing how the MSSM parameter space could be constrained by a precise measurement of \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\). We also discuss the relevance of this measurement in a combined \(M_W\)–\(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) analysis.
In the first example it is investigated whether the high accuracy achievable at the GigaZ option of the LC would provide sensitivity to indirect effects of SUSY particles even in a scenario where the (strongly interacting) superpartners are so heavy that they escape detection at the LHC [880].
We consider in this context a scenario with very heavy squarks and a very heavy gluino. It is based on the values of the SPS 1a’ benchmark scenario [881], but the squark and gluino mass parameters are fixed to 6 times their SPS 1a’ values. The other masses are scaled with a common scale factor except \(M_A\), which we keep fixed at its SPS 1a’ value. In this scenario the strongly interacting particles are too heavy to be detected at the LHC, while, depending on the scale factor, some colour-neutral particles may be in the LC reach. In Fig. 113 we show the prediction for \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) in this SPS 1a’ inspired scenario as a function of the lighter chargino mass, \(m_{\tilde{\chi }^\pm _{1}}\). The prediction includes the parametric uncertainty, \(\sigma ^{\text {para-LC}}\), induced by the LC measurement of \(m_{t}\,\), \(\delta m_{t}\,= 100\,\mathrm {MeV}\,\) (see Sect. 3), and the numerically more relevant prospective future uncertainty on \({\varDelta }\alpha ^{(5)}_{\text {had}}\), \(\delta ({\varDelta }\alpha ^{(5)}_{\text {had}})=5\times 10^{-5}\). The MSSM prediction for \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) is compared with the experimental resolution with GigaZ precision, \(\sigma ^\mathrm{LC} = 0.000013\), using for simplicity the current experimental central value. The SM prediction (with \(M_H^\mathrm{SM}=M_h^\mathrm{MSSM}\)) is also shown, applying again the parametric uncertainty \(\sigma ^{\text {para-LC}}\).
In Fig. 114 we compare the SM and the MSSM predictions for \(M_W\) and \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) as obtained from the scatter data. The predictions within the two models give rise to two bands in the \(M_W\)–\(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\) plane with only a relatively small overlap region [indicated by a dark-shaded (blue) area]. The parameter region shown in the SM [the medium-shaded (red) and dark-shaded (blue) bands] arises from varying the mass of the SM Higgs boson, from \(M_H^\mathrm{SM} = 114\,\mathrm {GeV}\,\), the old LEP exclusion bound [882] [lower edge of the dark-shaded (blue) area], to \(400\,\mathrm {GeV}\,\) [upper edge of the medium-shaded (red) area], and from varying \(m_{t}\,\) in the range of \(m_{t}\,= 165 \ldots 175\,\mathrm {GeV}\,\). The value of \(M_H^\mathrm{SM} \sim 125.5\,\mathrm {GeV}\,\) corresponds roughly to the dark-shaded (blue) strip. The light shaded (green) and the dark-shaded (blue) areas indicate allowed regions for the unconstrained MSSM, where no restriction on the light \({\mathscr {CP}}\)-even Higgs mass has been applied. The decoupling limit with SUSY masses, in particular of scalar tops and bottoms, of \({\mathscr {O}}(2\,{\text {TeV}})\) yields the upper edge of the dark-shaded (blue) area. Including a Higgs mass measurement into the MSSM scan would cut out a small part at the lower edge of the light shaded (green) area.
4.4 The relevance of the top-quark mass^{49}
The top quark could play a special role in/for EWSB.
The experimental uncertainty of \(m_{t}\,\) induces the largest parametric uncertainty in the prediction for EWPO [819, 884] and can thus obscure new physics effects.
In SUSY models the top-quark mass is an important input parameter and is crucial for radiative EWSB and unification.
Little Higgs models contain “heavier tops”.
The relevance of the \(m_{t}\,\) precision as parametric uncertainty has been discussed for the W boson mass, \(M_W\), in Sect. 4.2, and for the effective leptonic weak mixing angle, \(\sin ^2{\theta ^\ell }_{\mathrm {eff}}\), in Sect. 4.3.
Another issue that has to be kept in mind here (in SUSY as in any other model predicting \(M_H\)) is the intrinsic theoretical uncertainty due to missing higher-order corrections. Within the MSSM currently the uncertainty for the lightest \({\mathscr {CP}}\)-even Higgs is estimated to \(\delta M_h^\mathrm{intr,today} \approx 2\)–\(3\,\mathrm {GeV}\,\) [226, 819].^{50} In the future one can hope for an improvement down to \(\lesssim 0.5\,\mathrm {GeV}\,\) or better [819], i.e. with sufficient effort on higher-order corrections it should be possible to reduce the intrinsic theoretical uncertainty to the level of \(\delta M_H^\mathrm{exp, LHC}\).
Confronting the theoretical prediction of \(M_H\) with a precise measurement of the Higgs-boson mass constitutes a very sensitive test of the MSSM (or any other model that predicts \(M_H\)), which allows one to obtain constraints on the model parameters. However, the sensitivity of the \(M_H\) measurement cannot directly be translated into a prospective indirect determination of a single model parameter. In a realistic situation the anticipated experimental errors of all relevant SUSY parameters have to be taken into account. For examples including these parametric errors see Refs. [491, 884].
4.5 Prospects for the electroweak fit to the SM Higgs mass^{51}
Input values and fit results for the observables and parameters of the global electroweak fit in a hypothetical future scenario. The first and second columns list respectively the observables/parameters used in the fit, and their experimental values or phenomenological estimates (see text for references). The subscript “theo” labels theoretical error ranges. The third column indicates whether a parameter is floating in the fit and in the fourth column the fit results are given without using the corresponding experimental or phenomenological estimate in the given row
Parameter | Input value | Free in fit | Predicted fit result |
---|---|---|---|
\(M_{H}\) [GeV] | \(125.8 \pm 0.1\) | Yes | \(125.0^{\,+12}_{\,-10}\) |
\(M_{W}\) [GeV] | \(80.378\pm 0.006\) | – | \(80.361\pm 0.005\) |
\({\varGamma }_{W}\) [GeV] | – | – | \(2.0910\pm 0.0004\) |
\(M_{Z}\) [GeV] | \(91.1875\pm 0.0021\) | Yes | \(91.1878 \pm 0.0046\) |
\({\varGamma }_{Z}\) [GeV] | – | – | \(2.4953\pm 0.0003\) |
\(\sigma _\mathrm{had}^{0}\) [nb] | – | – | \(41.479\pm 0.003\) |
\(R^{0}_{\ell }\) | \(20.742\pm 0.003\) | – | – |
\(A_\mathrm{FB}^{0,\ell }\) | – | – | \(0.01622 \pm 0.00002 \) |
\(A_\ell \) | – | – | \(0.14706 \pm 0.00010 \) |
\(\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\) | \(0.231385\pm 0.000013\) | – | \(0.23152\pm 0.00004 \) |
\(A_{c}\) | – | – | \(0.66791\pm 0.00005 \) |
\(A_{b}\) | – | – | \(0.93462\pm 0.00002 \) |
\(A_\mathrm{FB}^{0,c}\) | – | – | \(0.07367\pm 0.00006 \) |
\(A_\mathrm{FB}^{0,b}\) | – | – | \(0.10308\pm 0.00007 \) |
\(R^{0}_{c}\) | – | – | \(0.17223\pm 0.00001 \) |
\(R^{0}_{b}\) | – | – | \(0.214746\pm 0.000004 \) |
\(\overline{m}_c\,\) [GeV] | \(1.27^{+0.07}_{-0.11}\) | Yes | – |
\(\overline{m}_b\,\) [GeV] | \(4.20^{+0.17}_{-0.07}\) | Yes | – |
\(m_{t}\) [GeV] | \(173.18\pm 0.10\) | Yes | \(173.3\pm 1.2 \) |
\({\varDelta }\alpha _\mathrm{had}^{(5)}(M_Z^2)\,\)\(^{(\bigtriangleup )}\) | \(2757.0\pm 4.7\) | Yes | \(2757 \pm 10\) |
\(\alpha _s(M_{Z}^{2})\) | – | Yes | \(0.1190\pm 0.0005\) |
\(\delta _\mathrm{th}\,M_W\) [MeV] | \([-2.0,2.0]_\mathrm{theo}\) | Yes | – |
\(\delta _\mathrm{th}\,\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\)\(^{(\bigtriangledown )}\) | \([-1.5,1.5]_\mathrm{theo}\) | Yes | – |
For the study aiming at a comparison of the accuracies of the measured and predicted electroweak observables, the central values of the input observables are chosen to agree with the SM prediction for a Higgs mass of 125.8 GeV. Total experimental uncertainties of 6 MeV for \(M_W\), \(1.3 \cdot 10^{-5}\) for \(\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\), \(4\cdot 10^{-3}\) for \(R^{0}_{\ell }\), and 100 MeV for \(m_t\) (interpreted as pole mass) are used. The exact achieved precision on the Higgs mass is irrelevant for this study. For the hadronic contribution to the running of the QED fine structure constant at the Z pole, \({\varDelta }\alpha _\mathrm{had}^{(5)}(M_Z^2)\,\), an uncertainty of \(4.7\cdot 10^{-5}\) is assumed (compared to the currently used uncertainty of \(10\cdot 10^{-5}\) [890, 891]), which benefits below the charm threshold from the completion of BABAR analyses and the on-going programme at VEPP-2000, and at higher energies from improved charmonium resonance data from BES-3, and a better knowledge of \(\alpha _s\) from the \(R^{0}_{\ell }\) measurement and reliable lattice QCD predictions. The other input observables to the electroweak fit are taken to be unchanged from the current settings [890].
For the theoretical predictions, the calculations detailed in [888] and references therein are used. They feature among others the complete \(\mathscr {O}(\alpha _s^4)\) calculation of the QCD Adler function [661, 662] and the full two-loop and leading beyond-two-loop prediction of the W mass and the effective weak mixing angle [848, 849, 892]. An improved prediction of \(R^0_b\) is invoked that includes the calculation of the complete fermionic electroweak two-loop (NNLO) corrections based on numerical Mellin–Barnes integrals [870]. The calculation of the vector and axial-vector couplings in Gfitter relies on accurate parametrisations [893, 894, 895, 896].
Table 28 gives the input observables and values used (first and second columns) and the predictions obtained from the fit to all input data except for the one that is predicted in a given row (last column). It allows one to compare the accuracy of direct and indirect determinations. To simplify the numerical exercise the Z-pole asymmetry observables are combined into a single input \(\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\), while for the reader’s convenience the fit predictions are provided for all observables.
The indirect prediction of the Higgs mass at 125 \(\mathrm {GeV}\) achieves an uncertainty of \(^{+12}_{-10}\,\mathrm {GeV}\,\). For \(M_W\) the prediction with an estimated uncertainty of 5 \(\mathrm {MeV}\) is similarly accurate as the (assumed) measurement, while the prediction of \(\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\) with an uncertainty of \(4\cdot 10^{-5}\) is three times less accurate than the experimental precision. The fit would therefore particularly benefit from additional experimental improvement in \(M_W\). It is interesting to notice that the accuracy of the indirect determination of the top mass (1.2 \(\mathrm {GeV}\) ) becomes similar to that of the present experimental determination. An improvement beyond, say, 200 \(\mathrm {MeV}\) uncertainty cannot be exploited by the fit. The input values of \(M_Z\) and \({\varDelta }\alpha _\mathrm{had}^{(5)}(M_Z^2)\,\) are twice more accurate than the fit predictions, which is sufficient to not limit the fit but further improvement would certainly be useful.
Keeping the present theoretical uncertainties in the prediction of \(M_W\) and \(\sin \!^2\theta ^{\ell }_{\mathrm{eff}}\) would worsen the accuracy of the \(M_H\) prediction to \(^{+20}_{-17}\,\mathrm {GeV}\,\), whereas neglecting theoretical uncertainties altogether would improve it to \(\pm \)7 \(\mathrm {GeV}\) . This emphasises the importance of the required theoretical work.
Profiles of \({\varDelta }\chi ^2\) as a function of the Higgs mass for present and future electroweak fits compatible with an SM Higgs boson of mass 125.8 and 94 \(\mathrm {GeV}\) , respectively, are shown in Fig. 116 (see caption for a detailed description). The measured Higgs-boson mass is not used as input in these fits. If the experimental input data, currently predicting \(M_H=94^{+25}_{-22}\,\mathrm {GeV}\,\) [890], were left unchanged with respect to the present values, but had uncertainties as in Table 28, a deviation of the measured \(M_H\) exceeding \(4\sigma \) could be established with the fit (see right-hand plot in Fig. 116). Such a conclusion does not strongly depend on the treatment of the theoretical uncertainties (Rfit versus Gaussian) as can be seen by comparison of the solid yellow and the long-dashed yellow \({\varDelta }\chi ^2\) profiles.
A similar result has also been obtained by the LEPEWWG, as can be seen in Fig. 117 [21]. The \({\varDelta }\chi ^2\) profile of their fit is shown as a function of the Higgs mass. The blue band shows the current result with a best-fit point at \(\sim \)94 \(\mathrm {GeV}\) with an uncertainty of \(\sim \pm 30 \mathrm {GeV}\,\). The pink parabola shows the expected improvement under similar assumptions to Fig. 116. This confirms that a strong improvement of the fit can be expected taking into account the anticipated future LC accuracy for the electroweak precision data.
4.6 The muon magnetic moment and new physics^{52}
One of the prime examples of precision observables sensitive to quantum effects are the magnetic moments \((g-2)\) of the electron and muon. In particular after the measurements at Brookhaven [22], the muon magnetic moment \(a_\mu =(g_\mu -2)/2\) has reached a sensitivity to all sectors of the SM and to many NPM. The currently observed deviation between the experimental value and the SM prediction is particularly well compatible with NPM which can also be tested at a LC. Before the startup of a future LC, new \(a_\mu \) measurements are planned at Fermilab [23] and J-PARC [24]. For these reasons it is of interest to briefly discuss the conclusions that can be drawn from current and future \(a_\mu \) results on LC physics.
Like many LC precision observables, \(a_\mu \) is a flavour- and \({\textit{CP}}\)-conserving quantity; unlike the former it is chirality-flipping and therefore particularly sensitive to modifications of the muon Yukawa coupling or more generally the muon mass-generation mechanism. A simple consideration, however, demonstrates that like a LC, \(a_\mu \) is generically sensitive to NPM with new weakly interacting particles at the weak scale [899].
For models with new weakly interacting particles (e.g. \(Z'\), \(W'\), see Sect. 4.7, little Higgs or universal extra dimension models) one typically obtains perturbative contributions to the muon mass \(C={\mathscr {O}}(\alpha /4\pi )\). Hence, for weak-scale masses these models predict very small contributions to \(a_\mu \) and might be challenged by the future more precise \(a_\mu \) measurement, see e.g. [901, 902]. Models of this kind can only explain a significant contribution to \(a_\mu \) if the new particles interact with muons but are otherwise hidden from the searches. An example is the model with a new gauge boson associated to a gauged lepton number \(L_\mu -L_\tau \) [903, 904], where a gauge boson mass of \({\mathscr {O}}(100 \text{ GeV })\) is viable, If this model is the origin of the observed \(a_\mu \) deviation it would be highly desirable to search for the new \(Z'\), corresponding to the \(L_\mu -L_\tau \)-symmetry. This would be possible at the LHC in part of the parameter space but also at the LC in the process \(e^+e^-\rightarrow \mu ^+\mu ^-Z'\) [903, 904].
- For SUSY models one obtains an additional factor \(\tan \beta \), the ratio of the two Higgs vacuum expectation values, see e.g. [905] and references therein. A numerical approximation for the SUSY contributions is given bywhere \(M_\mathrm{SUSY}\) denotes the common superpartner mass scale and \(\mu \) the Higgsino mass parameter. It agrees with the generic result Eq. (123) for \(C={\mathscr {O}}(\tan \beta \times \alpha /4\pi )\) and is exactly valid if all SUSY masses are equal to \(M_\mathrm{SUSY}\). The formula shows that the observed deviation could be explained e.g. for relevant SUSY masses (smuon, chargino and neutralino masses) of roughly 200 GeV and \(\tan \beta \sim 10\) or SUSY masses of 500 GeV and \(\tan \beta \sim 50\). This is well in agreement with current bounds on weakly interacting SUSY particles and in a very interesting range for a LC. This promising situation has motivated high-precision two-loop calculations of \(a_\mu ^\mathrm{SUSY}\) [906, 907], which depend on all sfermion, chargino and neutralino masses and will benefit particularly from precise SUSY mass measurements at a LC.$$\begin{aligned} a_\mu ^\mathrm{SUSY} \approx 13\times 10^{-10}\left( \frac{100\,\mathrm GeV}{M_\mathrm{SUSY}}\right) ^2\, \tan \beta \ \text{ sign }(\mu ), \end{aligned}$$(125)
Models with large \(C\simeq 1\) are of interest since there the muon mass is essentially given by new physics loop effects. Some examples of such radiative muon mass-generation models are given in [899]. For examples within SUSY see e.g. [908, 909]. In such models \(a_\mu \) can be large even for particle masses at the TeV scale, potentially beyond the direct reach of a LC. The possibility to test such models using precision observables at the LC has not yet been explored in the literature.