# Physics at the e+ e- Linear Collider

G. Moortgat-Pick, H. Baer, M. Battaglia, G. Belanger, K. Fujii, J.

Kalinowski, S. Heinemeyer, Y. Kiyo, K. Olive, F. Simon, et al.

► To cite this version:

G. Moortgat-Pick, H. Baer, M. Battaglia, G. Belanger, K. Fujii, et al., Physics at the e+ e- Linear Collider. European Physical Journal C: Particles and Fields, 2015, 75 (8), pp.371. 10.1140/epjc/s10052-015-3511-9. in2p3-01144792

HAL Id: in2p3-01144792

https://hal.in2p3.fr/in2p3-01144792

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Distributed under a Creative Commons Attribution 4.0 International License
Physics at the $e^{+}e^{-}$ Linear Collider

Editors: G. Moortgat-Pick$^{a,1,2}$, H. Baer$^{3}$, M. Battaglia$^{4}$, G. Belanger$^{5}$, K. Fujii$^{6}$,
J. Kalinowski$^{7}$, S. Heinemeyer$^{8}$, Y. Kiyo$^{9}$, K. Olive$^{10}$, F. Simon$^{11}$, P. Uwer$^{12}$,
D. Wackeroth$^{13}$, P.M. Zerwas$^{2}$

Specific Contributions: A. Arbey$^{38,39,40}$, M. Asano$^{14}$, J. Bagger$^{45,55}$, P. Bechtle$^{15}$,
A. Bharucha$^{16,53}$, J. Brau$^{46}$, F. Brümmer$^{17}$, S.Y. Choi$^{18}$, A. Denner$^{19}$, K. Desch$^{15}$,
S. Dittmaier$^{20}$, U. Ellwanger$^{21}$, C. Englert$^{22}$, A. Freitas$^{23}$, I. Ginzburg$^{24}$, S. Godfrey$^{25}$,
N. Greiner$^{11,2}$, C. Grojean$^{2,26}$, M. Grünewald$^{27}$, J. Heisig$^{28}$, A. Höcker$^{29}$, S. Kanemura$^{30}$,
K. Kawagoe$^{48}$, R. Kogler$^{31}$, M. Krawczyk$^{7}$, A.S. Kronfeld$^{50,54}$, J. Kroseberg$^{15}$,
S. Liebler$^{1,2}$, J. List$^{2}$, F. Mahmoudi$^{38,39,40}$, Y. Mambrini$^{21}$, S. Matsumoto$^{32}$, J. Mnich$^{2}$,
K. Mönig$^{2}$, M.M. Mühlleitner$^{33}$, R. Pöschl$^{43}$, W. Porod$^{19}$, S. Porto$^{1}$, K. Rolbiecki$^{7,34}$,
M. Schmitt$^{35}$, P. Serpico$^{5}$, M. Stanitzki$^{2}$, O. Stål$^{36}$, T. Stefaniak$^{4}$, D. Stöckinger$^{37}$,
G. Weiglein$^{2}$, G.W. Wilson$^{41}$, L. Zeune$^{42}$

LHC contacts: F. Moortgat$^{29}$, S. Xella$^{44}$

Advisory Board: J. Bagger$^{45,55}$, J. Brau$^{46}$, J. Ellis$^{47,29}$, K. Kawagoe$^{48}$, S. Komamiya$^{49}$,
A.S. Kronfeld$^{50,54}$, J. Mnich$^{2}$, M. Peskin$^{51}$, D. Schlatter$^{29}$, A. Wagner$^{2,31}$, H. Yamamoto$^{52}$

<footnote>$^{1}$II. Institute of Theo. Physics, University of Hamburg, D-22761 Hamburg, Germany</footnote>

<footnote>$^{2}$Deutsches Elektronen Synchrotron (DESY), Hamburg und Zeuthen, D-22603 Hamburg, Germany</footnote>

<footnote>$^{3}$Dept. of Physics and Astronomy, University of Oklahoma, Norman, OK73019, USA</footnote>

<footnote>$^{4}$Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, USA</footnote>

<footnote>$^{5}$Laboratoire de Physique Theorique (LAPTh), Université Savoie Mont Blanc, CNRS, B.P.110, Annecy-le-Vieux F-74941, France</footnote>

<footnote>$^{6}$High Energy Accelerator Research Organization (KEK), Tsukuba, Japan</footnote>

<footnote>$^{7}$Faculty of Physics, University of Warsaw, 02093 Warsaw, Poland</footnote>

<footnote>$^{8}$Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain</footnote>

<footnote>$^{9}$Department of Physics, Juntendo University, Inzai, Chiba 270-1695, Japan</footnote>

<footnote>$^{10}$William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA</footnote>

<footnote>$^{11}$Max-Planck-Institut für Physik, 80805 München, Germany</footnote>

<footnote>$^{12}$Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany</footnote>

<footnote>$^{13}$Department of Physics, SUNY at Buffalo, Buffalo, NY 14260-1500, USA</footnote>

<footnote>$^{14}$Physikalisches Institut and Bethe Center for Theoretical Physics Universität Bonn, D-53115 Bonn, Germany</footnote>

<footnote>$^{15}$Physikalisches Institut, University of Bonn, Bonn, Germany</footnote>

<footnote>$^{16}$Physik Department T31, Technische Universität München, D-85748 Garching, Germany</footnote>

<footnote>$^{17}$LUPM, UMR 5299, Université de Montpellier II et CNRS, 34095 Montpellier, France</footnote>

<footnote>$^{18}$Department of Physics, Chonbuk National University, Jeonju 561-756, Republic of Korea</footnote>

<footnote>$^{19}$Universität Würzburg, Institut für Theoretische Physik und Astrophysik, D-97074 Würzburg, Germany</footnote>

<footnote>$^{20}$Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany</footnote>

<footnote>$^{21}$Laboratoire de Physique, UMR 8627, CNRS, Universite de Paris-Sud, 91405 Orsay, France</footnote>

<footnote>$^{22}$SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom</footnote>

<footnote>$^{23}$PITT PACC, Department of Physics & Astronomy, Univ. of Pittsburgh, Pittsburgh, PA 15260, USA</footnote>

<footnote>$^{24}$Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, 630090, Russia</footnote>

<footnote>$^{25}$Ottawa-Carleton Institute for Physics, Department of Physics, Carleton University, Ottawa, Canada K1S 5B6</footnote>

<footnote>$^{26}$ICREA at IFAE, Universitat Autonoma de Barcelona, E-08193 Bellaterra, Spain</footnote>

<footnote>$^{27}$University College Dublin, Dublin, Ireland</footnote>
²⁸Institute for Theoretical Particle Physics and Cosmology, RWTH Aachen University, 52056 Aachen, Germany

²⁹CERN, Geneva, Switzerland

³⁰Department of Physics, University of Toyama, 3190 Gofuku, Toyama, 930-8555, Japan

³¹University of Hamburg, Hamburg, Germany

³²Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan

³³Institute for Theoretical Physics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

³⁴Instituto de Fisica Teorica, IFT-UAM/CSIC, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain

³⁵Department of Physics and Astronomy, Northwestern University, Evanston, IL 60091, USA

³⁶The Oskar Klein Centre, Department of Physics, Stockholm University, SE-106 91 Stockholm, Sweden

³⁷Institut für Kern- und Teilchenphysik, TU Dresden, 01069 Dresden, Germany

³⁸Université de Lyon, Université Lyon 1, 69622 Villeurbanne Cedex, France

³⁹Centre de Recherche Astrophysique de Lyon, CNRS, UMR 5574, 69561 Saint-Genis Laval Cedex, France

⁴⁰Ecole Normale Supérieure de Lyon, France

⁴¹Department of Physics and Astronomy, university of Kansas, Lawrence, KS 66045, USA

⁴²ITFA, University of Amsterdam, Science Park 904, 1018 XE, Amsterdam, The Netherlands

⁴³Laboratoire de L'accelerateur Lineaire (LAL), CNRS/IN2P3, Orsay, France

⁴⁴Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark

⁴⁵Department of Physics and Astronomy, Johns Hopkins University Baltimore, MD 21218, USA

⁴⁶Department of Physics, University of Oregon, Eugene, OR 97403, USA

⁴⁷Theoretical Particle Physics and Cosmology Group, Department of Physics, King's College London, Strand, London WC2R 2LS, U.K

⁴⁸Department of Physics, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, 812-8581 Fukuoka, Japan

⁴⁹Department of Physics, Graduate School of Science, and International Center for Elementary Particle Physics, The University of Tokyo, Tokyo 113-0033, Japan

⁵⁰Theoretical Physics Department, Fermi National Accelerator Laboratory, Batavia, Illinois, USA

⁵¹SLAC, Stanford University, Menlo Park, California 94025 USA

⁵²Department of Physics, Tohoku University, Sendai, Miyagi, Japan

⁵³CNRS, Aix Marseille U., U. de Toulon, CPT, F-13288, Marseille, France

⁵⁴Institute for Advanced Study, Technische Universität München, D-85748 Garching, Germany

⁵⁵TRIUMF, Vancouver, BC V6T 2A3, Canada

Received: 25 March 2015 / Accepted: 9 May 2015

**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 focuses 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 analyzed as well.

**Keywords** Particle Physics · Accelerator · Detector

Contents

<table><tr><td>1</td><td>Executive Summary</td><td>3</td></tr><tr><td>2</td><td>Higgs and Electroweak Symmetry Breaking</td><td>12</td></tr><tr><td>3</td><td>Top and QCD</td><td>82</td></tr><tr><td>4</td><td>Exploring the Quantum Level: Precision Physics in the SM and BSM</td><td>96</td></tr><tr><td>5</td><td>Supersymmetry</td><td>117</td></tr><tr><td>6</td><td>Connection to astroparticle physics and cosmology</td><td>148</td></tr><tr><td>7</td><td>Summary</td><td>157</td></tr></table>

<footnote>ªe-mail: gudrid.moortgat-pick@desy.de</footnote>
# 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 P. Higgs and F. Englert in 2013 [1–4]. 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 a structure. The relevance of such a field not only for our understanding of matter but also for the history of the Universe is immanent.

The discovery of a Higgs boson as the materialization 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 $SU(3)_C \times SU(2)_L \times U(1)_Y$.

However, the next immediate steps are to answer the following questions:

- 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 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 CP-odd component? To which model beyond the Standard Model (BSM) does it point?

In order to definitively establish the mechanism of electroweak symmetry breaking, all Higgs boson properties (mass, width, couplings, quantum numbers) have to be precisely measured and compared with the mass of the corresponding particles.

The LHC has excellent prospects for the future runs<fnref n="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].

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 measureable up to the per mille level. The energy at the LC is tunable which offers to perform precise energy scans and to optimize the kinematic conditions for the different physics processes, respectively. In addition, the beams can be polarized: the electron beam up to about 90%, the positron beam up to about 60%. With such a high degree of polarization, 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 polarization, tunable energy, precisely defined initial state and clear separation of events via excellent detectors. The experimental conditions that are necessary to fulfill the physics requirements have been defined in the LC scope documents [10].

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 to analyze 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, 11, 12], a short and compact overview is given in [13].

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-percent level, the SM cannot be regarded as the final theory describing all aspects of Nature. Astro-physical measurements [14, 15] are consistent with a universe that contains only 4% of the total energy composed of ordinary mass but hypothesize 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 dark matter can easily be embedded within BSM theories as, for instance, supergravity [16]. The strong belief in BSM

<footnote n="1">¹As one example for a recent and comprehensive review of the LHC run-1 results, see [5] and references therein.</footnote>
physics is further supported by the absence of gauge cou-
pling unification in the SM as well as its failure to explain
the observed existing imbalance between baryonic and an-
tibaryonic 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 phys-
ical 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 de-
termine the underlying structure of the model.

A priori there are only two approaches to reveal sig-
nals 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 predic-
tions 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 be-
comes also feasible where the new particles or interactions
can be produced and studied directly.

Obviously, the complementary search strategies at lep-
ton and hadron colliders are predestinated for such success-
ful dual approaches. A successful high-energy LC was al-
ready realized in the 90s with the construction and run-
ning of the SLAC Linear Collider (SLC) that delivered up
to 5 × 10¹⁰ particles per pulse. Applying in addition highly
polarized electrons enabled the SLC to provide the best
single measurement of the electroweak mixing angle with
δ sin² θ<sub>W</sub> ~ 0.00027.

However, such a high precision manifests a still-existing
inconsistency, namely the well-known discrepancy between
the left-right polarization asymmetry at the Z-pole measured
at SLC and the forward-backward asymmetry measured at
LEP [17]. Both values lead to measured values of the elec-
troweak mixing angle sin² θ<sub>eff</sub> that differ by more than 3-σ
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 consis-
tence of the SM as well as BSM models.

Another example for the relevance of highest precision
measurements and their interplay with most accurate theo-
retical predictions at the quantum level is impressively demon-
strated in the interpretation of the muon anomalous moment
g<sub>μ</sub> − 2 [18]. The foreseen run of the g<sub>μ</sub> − 2 experiment at
Fermilab, starting in 2017 [19, 20], will further improve the
current experimental precision by about a factor four and
will set substantial bounds to many new physics models via
their high sensitivity to virtual effects of new particles.

The LC concept has been proposed already in 1965 [21]
for providing electron beams with high enough quality for
collision experiments. In [22] this concept has been pro-
posed for collision experiments at high energies in order to
avoid the energy loss via synchrotron radiation: this energy
loss per turn scales with E⁴/r, where E denotes the beam
energy and r the bending radius. The challenging problems
at the LC compared to circular colliders, however, are the
luminosity and the energy transfer to the beams. The lumi-
nosity is given by

$$
\mathcal{L} \sim \frac{\eta P N_e}{\sigma_{xy} E_{c.m.}}, \qquad (1)
$$

where *P* denotes the required power with efficiency η, *N*<sub>e</sub>
the charge per bunch, *E*<sub>c.m.</sub> the center-of-mass energy and
*σ*<sub>xy</sub> the transverse geometry of the beam size. From eq. (1),
it is obvious that flat beams and high bunch charge allow
high luminosity with lower required beam power *P*<sub>b</sub> = *ε* *P*.
The current designs for a high luminosity *e*<sup>+</sup>*e*<sup>−</sup> collider, ILC
or CLIC, is perfectly aligned with such arguments. One ex-
pects an efficiency factor of about η ~ 20% for the discussed
designs.

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 detec-
tor) and excellent τ<sup>±</sup>- , b, ¯b- and c, ¯c-tagging capabilities [8],
are expected.

As mentioned before, another novelty is the availabil-
ity of the polarization of both beams, which can precisely
project out the interaction vertices and can analyze its chi-
rality directly.

The experimental conditions to achieve such an unprece-
dented precision frontier at high energy are high luminos-
ity (even about three orders of magnitude more particles per
pulse, 5 × 10¹³ than at the SLC), polarized electron/positron
beams, tunable energy, luminosity and beam energy stabil-
ity below 0.1% level [10]. Assuming a finite total overall
running time it is a critical issue to divide up the available
time between the different energies, polarizations and run-
ning options in order to maximize the physical results. Sev-
eral running scenarios are thoroughly studied [23].

In the remainder of this chapter we summarize 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 elec-
troweak high precision physics— chapter 2 provides a com-
prehensive overview about the physics of electroweak sym-
metry breaking. Recent developments in LHC analyses as
well as on the theory side are included, alternatives to the
Higgs models are discussed. Chapter 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 chapter 4 the
impact of electroweak precision observables 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 chapter 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, chapter 6 provides an overview in this regard.

The above mentioned safe physics topics can be realized 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 [23] for technical details. The expected physics results of the high-luminosity LC was studied in different working group reports [24, 25], cf. Sect.2.3.

Such an optimization 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 [23], 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, 26, 27] and references therein.

### 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 Higgsstrahlung 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 $H \to ZZ^*$, $WW^*$ are very sensi-

tive to $m_H$: a change in $m_H$ by 200 MeV shifts $BR(H \to ZZ^*$) 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 $\Gamma_H$ of the Higgs boson. The prediction in the SM is $\Gamma_H = 4.07$ MeV for $m_H = 125$ GeV [28]. 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 $\Gamma_H$ 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 Higgsstrahlung process. At $\sqrt{s} = 500$ GeV, one can derive the total width $\Gamma_H$ with a precision of 5% based on a combination of the $H \to ZZ^*$ 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 [29] in a similar way as at the LHC [30]. 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 $\sigma(ZH)$ [29].

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 [31, 32].

In order to really establish the mechanism of electroweak symmetry breaking it is not only important to measure all couplings but also to measure the Higgs potential:

$$V = \frac{1}{2} m_H^2 \Phi_H^2 + \lambda \nu \Phi_H^3 + \frac{1}{4} \kappa \Phi_H^4,$$

where $\nu = 246$ GeV. It is essential to measure the trilinear coupling rather accurate in order to test whether the observed Higgs boson originates from a field that is in concordance with the observed particle masses and the predicted electroweak symmetry breaking mechanism<fnref n="2" />. Since the cross section for double Higgsstrahlung is small but has a maximum of about 0.2 fb at $\sqrt{s} = 500$ GeV for $m_H = 125$ GeV, this energy stage is required to enable a first measurement of this coupling. The uncertainty scales with $\Delta\lambda/\lambda = 1.8\Delta\sigma/\sigma$.

<footnote n="2">²The quartic coupling will not be accessible either at the LHC or at an LC. Even at the high-luminosity Large Hadron Collider (HL-LHC), i.e. the LHC at $\sqrt{s} = 14$ TeV but with a tenfold increase in luminosity, there does not exist an analysis how to get access to this coupling.</footnote>
New involved analyses methods in full simulations aim at a precision of 20% at $\sqrt{s} = 500$ GeV. Better accuracy one could get applying the full LC programme and going also to higher energy, $\sqrt{s} = 1$ TeV.

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 $\Delta g_{t\bar{t}H}/g_{t\bar{t}H} \sim 10\%$, see Sect.2. This energy stage is close to the threshold of $t\bar{t}H$ 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 $m_t = y_{t\bar{t}H}/\sqrt{2}$. 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 $\sim 20\%$.

Since for a fixed $m_H$ all Higgs couplings are specified in the SM, it is not possible to perform a fit within this model. In order to test the compatibility of the SM Higgs predictions with the experimental data, the LHC Higgs Cross Section Group proposed 'coupling scale factors' [33,34]. These scale factors $\kappa_i$ ($\kappa_i = 1$ corresponds to the SM) dress the predicted Higgs cross section and partial widths. Applying such a $\kappa$-framework, the following assumptions have been made: there is only one 125 GeV state responsible for the signal with a coupling structure identical to the SM Higgs, i.e. a pure CP-even state, and the zero width approximation can be applied. Usually, in addition the theory assumption $\kappa_{W,Z} < 1$ (corresponds to an assumption on the total width) has to be made. Using, however, LC data and exploiting the precise measurement of $\sigma(HZ)$, this theory assumption can be dropped and all couplings can be obtained with an unprecedented precision of at least 1–2%, see Fig.1 [35] and Sect. 2 for further details.

Another important property of the Higgs boson that has to be determined is the CP quantum number. In the SM the Higgs should be a pure CP-even state. In BSM models, however, the observed boson state *a priori* can be any admixture of CP-even and CP-odd states, it is of high interest to determine limits on this admixture. The *HVV* couplings project out only the CP-even components, therefore the degree of CP admixture cannot be tackled via analysing these couplings. The measurements of CP-odd observables are mandatory to reveal the Higgs CP-properties: for instance,

<figure><img src="image_2.png" /><figcaption>**Fig. 1** The achievable precision in the different Higgs couplings at the LHC on bases of 3 ab$^{-1}$ and 50% improvement in the theoretical uncertainties in comparison with the different energy stages at the ILC. In the final LC stage all couplings can be obtained in the 1–2% range, some even better. [35].</figcaption></figure>

the decays of the Higgs boson into $\tau$ leptons provides the possibility to construct unique CP-odd observables via the polarization vector of the $\tau$'s, see further details in Sect.2.

## Top quark physics

Top quark physics is another rich field of phenomenology. It opens at $\sqrt{s} = 350$ GeV. The mass of the top quark itself has high impact on the physics analysis. In BSM physics $m_t$ is often the crucial parameter in loop corrections to the Higgs mass. In each model where the Higgs boson mass is not a free parameter but predicted in terms of the other model parameters, the top quark mass enters the respective loop diagrams to the fourth power, see Sect.4 for details. Therefore the interpretation of consistency tests of the electroweak precision observables $m_W, m_Z, \sin^2\theta_{eff}$ and $m_H$ require the most precise knowledge on the top quark mass. The top quark is not an asymptotic state and $m_t$ depends on the renormalization scheme. Therefore a clear definition of the used top quark mass is needed. Measuring the mass via a threshold scan allows to relate the measured mass uniquely to the well-defined $m_t^{\overline{MS}}$ mass, see Fig.2. Therefore, this procedure is advantageous compared to measurements via continuum observables. It is expected to achieve an unprecedented accuracy of $\Delta m_t^{\overline{MS}} = 100$ MeV via threshold scans. This uncertainty contains already theoretical as well as experimental uncertainties. Only such a high accuracy enables sensitivity to loop corrections for electroweak precision observables. Furthermore the accurate determination is also decisive for tests of the vacuum stability within the SM.

A sensitive window to BSM physics is opened by the analysis of the top quark couplings. Therefore a precise determination of all SM top-quark couplings together with the search for anomalous couplings is crucial and can be per-
<figure><img src="image_3.png" /><figcaption>**Fig. 2** Simulated measurement of the background-subtracted $t\bar{t}$ cross section with $10\ \text{fb}^{-1}$ per data point, assuming a top-quark mass of 174 GeV in the 1s scheme with the ILC luminosity spectrum for the CLIC-ILD detector [36].</figcaption></figure>

formed very accurately at $\sqrt{s} = 500$ GeV. Using the form-factor decomposition of the electroweak top quark couplings, it has been shown that one can improve the accuracy for the determination of the couplings [37] by about one order of magnitude at the LC compared to studies at the LHC, see Fig. 3 and Sect. 3.

<figure><img src="image_4.png" /><figcaption>**Fig. 3** Statistical precision on CP-conserving form factors expected at the LHC [38] and at the ILC [37]. The LHC results assume an integrated luminosity of $\mathcal{L} = 300\ \text{fb}^{-1}$. The results for the ILC are based on an integrated luminosity of $\mathcal{L} = 500\ \text{fb}^{-1}$ at $\sqrt{s} = 500$ GeV and a beam polarization of $P_{e^{-}} = \pm 80\%$, $P_{e^{+}} = \mp 30\%$ [37].</figcaption></figure>

## 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 dark matter candidates, stabilizing the Higgs mass, embedding new sources for 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 parametrization 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.

Another striking feature of the LC physics potential is the capability to test predicted properties of new physics candidates. For instance, in SUSY models one essential paradigm is that the coupling structure of the SUSY particle is identical to its SM partner particle. That means, for instance, that the SU(3), SU(2) and U(1) gauge couplings $g_S$, $g$ and $g'$ have to be identical to the corresponding SUSY Yukawa couplings $g_{\tilde{g}}$, $g_{\tilde{W}}$ and $g_{\tilde{B}}$. These tests are of fundamental importance to establish the theory. Testing, in particular, the SUSY electroweak Yukawa coupling is a unique feature of LC physics. Under the assumption that the SU(2) and U(1) parameters have been determined in the gaugino/higgsino sector (see Sect. 5.7), the identity of the Yukawa and the gauge couplings via measuring polarized cross sections can be successfully performed: depending on the electron (and positron) beam polarization and on the luminosity, a per-cent-level precision can be achieved: see Fig. 4.

Another important and unique feature of the LC potential is to test experimentally the quantum numbers of new physics candidates. For instance, a particularly challenging measurement is the determination of the chiral quantum numbers of the SUSY partners of the fermions. These partners are predicted to be scalar particles and to carry the chiral quantum numbers of their Standard Model partners. In $e^{+}e^{-}$ collisions, the associated production reactions $e^{+}e^{-} \to \tilde{e}_{L}^{+}\tilde{e}_{R}^{-}$, $\tilde{e}_{R}^{+}\tilde{e}_{L}^{-}$ occur only via t-channel exchange, where the $e^{\pm}$ are
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Equivalence of the SUSY electroweak Yukawa couplings $g_W$, $g_B$ with the SU(2), U(1) gauge couplings $g$, $g'$. Shown are the contours of the polarized cross sections $\sigma_L(e^+e^- \to \tilde{\chi}_1^0\tilde{\chi}_2^0)$ and $\sigma_R(e^+e^- \to \tilde{\chi}_1^0\tilde{\chi}_2^0)$ in the plane of the SUSY electroweak Yukawa couplings normalized to the gauge couplings, $Y_L = g_W/g$, $Y_R = g_B/g'$ [39] for a scenario with the electroweak spectrum similar to the reference point SPS1a.</figcaption></figure>

directly coupled to their SUSY partners $\tilde{e}^{\pm}$. Separating the associated pairs, the chiral quantum numbers can be tested via the polarization of $e^{\pm}$ since chirality corresponds to helicity in the high energy limit. As can be seen in Fig. 5, the polarization of both beams is absolutely essential to separate the pair $\tilde{e}_L\tilde{e}_R$ [40] and to test the associated quantum numbers.

<figure><img src="image_6.png" /><figcaption>**Fig. 5** Polarized cross sections versus $P_{e^{-}}$ (bottom panel) and $P_{e^{+}}$ (top panel) for $e^{+}e^{-} \to \tilde{e}\tilde{e}$-production with direct decays in $\tilde{\chi}_{1}^{0}e^{+}$ in a scenario where the non-coloured spectrum is similar to a SPS1a-modified scenario but with $m_{\tilde{e}_L} = 200$ GeV, $m_{\tilde{e}_R} = 195$ GeV. The associated chiral quantum numbers of the scalar SUSY partners $\tilde{e}_{L,R}$ can be tested via polarized $e^{\pm}$-beams.</figcaption></figure>

## Dark Matter Physics

Weakly interacting massive particles (WIMPs) are the favourite candidates as components of the cold dark matter. Neutral particles that interact only weakly provide roughly the correct relic density in a natural way. Since there are no candidates for dark matter in the SM, the strong observational evidence for dark matter clearly points to physics beyond the SM. Due to precise results from cosmological observations, for instance [41], bounds on the respective cross section and the mass of the dark matter 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 electroweak symmetry breaking 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 a radiative photons at the LC, that recoil against the WIMPs, for instance, the process $e^{+}e^{-} \to \gamma\chi\chi$ [42], where $\chi$ denotes the WIMP particle in general with a spin $S_{\chi} = 0, \frac{1}{2}, 1$. Such a process can be realized 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 polarization.

The present dark matter density depends strongly on the cross section for WIMP annihilation into SM particles (assuming that there exist only one single WIMP particle $\chi$ and ignoring coannihilation processes between the WIMP and other exotic particles) in the limit when the colliding $\chi$'s are non-relativistic [42], depending on s- or p-wave contributions and on the WIMP mass. Due to the excellent resolution at the LC the WIMP mass can be determined with relative accuracy of the order of 1%, see Fig.6.

Following another approach and parametrizing dark matter interactions in the form of effective operators, a non-relativistic approximation is not required and the derived bounds can be compared with experimental bounds from direct detection. Assuming that the dark matter particles only interact with SM fields via heavy mediators that are kinematically not accessible at the ILC, it was shown in [44] that the ILC could nevertheless probe effective WIMP couplings $G_{\text{max}}^{\text{ILC}} = g_i g_j / M^2 = 10^{-7} \text{ GeV}^{-2}$ (vector or scalar mediator case), or $G_{\text{max}}^{\text{ILC}} = g_i g_j / M = 10^{-4} \text{ GeV}^{-1}$ (fermionic mediator case). The direct detection searches give much stronger bounds on spin independent ('vector') than on spin dependent ('axialvector') interactions under the simplifying assumption that all SM particles couple with the same strength to the dark matter candidate ('universal coupling'). If the WIMP particle is rather light ($< 10$ GeV) the ILC offers a unique opportunity to search for dark matter candidates beyond any other experiment, even for spin independent inter-
<figure><img src="image_7.png" /></figure>

**Fig. 6** WIMP mass as a function of the mass for p-wave ($J_0 = 1$) annihilation and under the assumption that WIMP couplings are helicity- and parity-conserving in the process $e^+e^- \to \gamma\chi\chi$ [42]. With an integrated luminosity of $\mathcal{L} = 500 \text{ fb}^{-1}$ and polarized beams with $P_{e^-} = +80\%$, $P_{e^+} = -60\%$ with $\Delta P/P = 0.25\%$ the reconstructed WIMP mass can be determined with an relative accuracy of the order of 1% [43]. The blue area shows the systematic uncertainty and the red bands the additional statistical contribution. The dominant sources of systematic uncertainties are $\Delta P/P$ and the shape of the beam energy spectrum.

actions, cf. Fig. 7 (upper panel). In view of spin dependent interactions the ILC searches are also superior for heavy WIMP particles, see Fig.7 (lower panel).

## Neutrino mixing angle

Another interesting question is how to explain the observed neutrino mixing and mass patterns in a more complete theory. SUSY with broken R-parity allows to embed and to predict such an hierarchical pattern. The mixing between neutralinos and neutrinos puts strong relations between the LSP branching ratios and neutrino mixing angles. For instance, the solar neutrino mixing angle $\sin^2 \theta_{23}$ is accessible via measuring the ratio of the branching fractions for $\tilde{\chi}_1^0 \to W^\pm \mu^\mp$ and $W^\pm \tau^\mp$. Performing an experimental analysis at $\sqrt{s} = 500$ GeV allows to determine the neutrino mixing angle $\sin^2 \theta_{23}$ up to a percent-level precision, as illustrated in Fig. 8 [45]. This direct relation between neutrino physics and high-energy physics is striking. It allows to directly test whether the measured neutrino mixing angles can be embedded within a theoretical model of high predictive power, namely a bilinear R-parity violation model in SUSY, based on precise measurements of neutralino branching ratios [46, 47] at a future $e^+e^-$ linear collider.

## Beyond Standard Model Physics – “bottom-up”

Electroweak Precision Observables

Another compelling physics case for the LC can be made

<figure><img src="image_8.png" /></figure>

**Fig. 7** Combined limits for fermionic dark matter models. The process $e^+e^- \to \chi\chi\gamma$ is assumed to be detected only by the hard photon. The analysis has been modeled correspondingly to [43] and is based on $\mathcal{L} = 500 \text{ fb}^{-1}$ at $\sqrt{s} = 500$ GeV and $\sqrt{s} = 1$ TeV and different polarizations [44].

<figure><img src="image_9.png" /></figure>

**Fig. 8** Achievable precision on $\sin^2 \theta_{23}$ from bilinear R-parity violating decays of the $\tilde{\chi}_1^0$ as a function of the produced number of neutralino pairs compared to the current precision from neutrino oscillation measurements [45].

for the measurement of Electroweak Precision Observables (EWPO) at $\sqrt{s} \approx 92$ GeV (Z-pole) and $\sqrt{s} \approx 160$ GeV (WW threshold), where a new level of precision can be reached.
Detecting with highest precision any deviations from the SM predictions provides traces of new physics which could lead to groundbreaking discoveries. Therefore, particularly in case no further discovery is made from the LHC data, it will be beneficial to perform such high precision measurements at these low energies. Many new physics models, including those of extra large dimensions, of extra gauge bosons, of new leptons, of SUSY, etc., can lead to measurable contributions to the electroweak mixing angle even if the scale of the respective new physics particles are in the multi-TeV range, i.e. out of range of the high luminosity LHC. Therefore the potential of the LC to measure this quantity with an unprecedented precision, i.e. of about one order of magnitude better than at LEP/SLC offers to enter a new precision frontier. With such a high precision –mandatory are high luminosity and both beams polarized– one gets sensitivity to even virtual effects from BSM where the particles are beyond the kinematical reach of the $\sqrt{s} = 500$ GeV LC and the LHC. In Fig. 9 the prediction for $\sin^2 \theta_{\text{eff}}$ as a function of the lighter chargino mass $m_{\tilde{\chi}_1^+}$ is shown. The MSSM prediction is compared with the prediction in the SM assuming the experimental resolution expected at GigaZ. In this scenario no coloured SUSY particles would be observed at the LHC but the LC could resolve indirect effects of SUSY up to $m_{\tilde{\chi}_1^+} \le 500$ GeV via the measurement of $\sin^2 \theta_{\text{eff}}$ with unprecedented precision at the low energy option GigaZ, see Sect. 4 for details.

<figure><img src="image_10.png" /><figcaption>**Fig. 9** Theoretical prediction for $\sin^2 \theta_{\text{eff}}$ in the SM and the MSSM (including prospective parametric theoretical uncertainties) compared to the experimental precision at the LC with GigaZ option. A SUSY inspired scenario SPS $1a'$ has been used, where the coloured SUSY particles masses are fixed to 6 times their SPS $1a'$ values. The other mass parameters are varied with a common scale factor.</figcaption></figure>

The possibility to run with high luminosity and both beam polarized on these low energies is essential in these regards.

## Extra Gauge Bosons

One should stress that not only SUSY theories can be tested via indirect searches, but also other models, for instance, models with large extra dimensions or models with extra $Z'$, see Fig. 10, where the mass of the $Z'$ boson is far beyond the direct kinematical reach of the LHC and the LC and therefore is assumed to be unknown. Because of the clean LC environment, one even can determine the vector and axial vector coupling of such a $Z'$ model.

<figure><img src="image_11.png" /><figcaption>**Fig. 10** New gauge bosons in the $\mu^+\mu^-$ channel. The plot shows the expected resolution at CLIC with $\sqrt{s} = 3$ TeV and $\mathcal{L} = 1 \text{ ab}^{-1}$ on the ‘normalized’ vector $v_f^n = v'_f \sqrt{s/(m_Z^2 - s)}$ and axial-vector $a_f^n = a'_f \sqrt{s/(m_Z^2 - s)}$ couplings to a 10 TeV $Z'$ in terms of the SM couplings $v'_f$, $a'_f$. The mass of $Z'$ is assumed to be unknown, nevertheless the couplings can be determined up to a two-fold ambiguity. The colours denote different $Z'$ models [9].</figcaption></figure>

## Synopsis

The full Higgs and top quark physics programme as well as the promising programme on dark matter and BSM physics should be accomplished with the higher energy LC set-up at 1 TeV. Model-independent parameter determination is essential for the crucial identification of the underlying model. Accessing a large part of the particle spectrum of a new physics model would nail down the structure of the underlying physics. But measuring already only the light part of the spectrum with high precision and model-independently can provide substantial information. Table 1 gives an overview of the different physics topics and the required energy stages. The possibility of an tunable energy in combination with polarized beams, is particularly beneficial to successfully accomplish the comprehensive physics programme at high-energy physics collider and to fully exploit the complete physics potential of the future Linear Collider.
<table><caption>**Table 1** Physics topics where the $e^{+}e^{-}$-Linear Collider provides substantial results at the different energy stages that are complementary to the LHC. The examples are described in the following chapters as well as in [7–9, 11–13, 23, 24, 26, 27, 48, 49].</caption><thead><tr><th>√s/GeV:</th><th>92,160</th><th>240</th><th>350</th><th>500</th><th>1000</th><th>3000</th><th>threshold scans required</th></tr></thead><tbody><tr><td><strong>Higgs</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>m<sub>H</sub></td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>Γ<sub>tot</sub></td><td>–</td><td>–</td><td>x</td><td>x</td><td></td><td></td><td></td></tr><tr><td>g<sub>c,b</sub></td><td>–</td><td>x</td><td>x</td><td>x</td><td></td><td>x</td><td></td></tr><tr><td>g<sub>ttH</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td></td><td></td></tr><tr><td>g<sub>HHH</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>m<sup>SUSY</sup><sub>H,A</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td><strong>Top</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>m<sup>th</sup><sub>t</sub></td><td>–</td><td>–</td><td>x</td><td></td><td></td><td></td><td>x</td></tr><tr><td>m<sup>cont</sup><sub>t</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>(x)</td><td>(x)</td><td></td></tr><tr><td>A<sup>FB</sup><sub>t</sub></td><td>–</td><td>–</td><td>x</td><td>x</td><td></td><td></td><td></td></tr><tr><td>g<sub>Z,γ</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td></td><td></td><td></td></tr><tr><td>g<sub>FCNC</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>(?)</td><td></td></tr><tr><td><strong>Electroweak Precision Observables</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>sin<sup>2</sup> θ<sub>eff</sub>(Z-pole)</td><td>x</td><td></td><td></td><td></td><td></td><td>(x)</td><td></td></tr><tr><td>m<sup>th</sup><sub>W</sub></td><td>x</td><td></td><td></td><td></td><td></td><td></td><td>x</td></tr><tr><td>m<sup>cont</sup><sub>W</sub></td><td></td><td>x</td><td>x</td><td>x</td><td>(x)</td><td>(x)</td><td></td></tr><tr><td>Γ<sub>Z</sub></td><td>x</td><td></td><td></td><td></td><td></td><td></td><td>x</td></tr><tr><td>A<sub>LR</sub></td><td>x</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>A<sub>FB</sub></td><td>x</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td><strong>SUSY</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>indirect search</td><td>x</td><td>x</td><td>x</td><td></td><td></td><td></td><td>x</td></tr><tr><td>direct search</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>light higgsinos</td><td>–</td><td>–</td><td>x</td><td>x</td><td></td><td></td><td>x</td></tr><tr><td>parameter determination</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td></td><td>x</td></tr><tr><td>quantum numbers</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td></td><td>x</td></tr><tr><td>extrapolations</td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td><strong>ν mixing</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>θ<sup>2</sup><sub>23</sub></td><td>–</td><td>–</td><td>x</td><td>x</td><td></td><td></td><td></td></tr><tr><td><strong>Dark Matter</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>effective-field-theory</td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>non-relativistic</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td><strong>Extra gauge bosons</strong></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>indirect search m<sub>z</sub></td><td>x</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td>x</td></tr><tr><td>v<sup>'</sup><sub>f</sub>, a<sup>'</sup><sub>f</sub></td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td><td>(x)</td><td></td></tr><tr><td>m<sub>W'</sub></td><td>x</td><td>–</td><td>–</td><td>x</td><td>x</td><td>x</td><td></td></tr><tr><td>direct search</td><td>–</td><td>–</td><td>–</td><td>–</td><td>–</td><td>x</td><td>x</td></tr></tbody></table>
## 2 Higgs and Electroweak Symmetry Breaking<fnref n="3" />

After a brief description of the physical basis of the Higgs mechanism, we summarize 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: partial 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<fnref n="5" />

The Brout-Englert-Higgs mechanism [1–4, 50] 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 Standard Model (SM) [51–53] 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 [54,55] 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) [56–59].

#### 2.1.1 Zeroing in on the Higgs particle of the SM

Within the SM the Higgs mechanism is realized by introducing a scalar weak-isospin-doublet. Three Goldstone degrees of freedom are absorbed for generating the longitudinal components of the massive electroweak $W^{\pm}, Z$ bosons, and one degree of freedom is realized as a scalar physical particle unitarizing the theory properly. After the candidate particle has been found, three steps are necessary to establish the relation with the Higgs mechanism:

- *The mass, the lifetime (width) and the spin/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.*

When the mass of the Higgs particle is fixed, all its properties are pre-determined. The spin/CP assignment $J^{CP} = 0^{++}$ is required for an isotropic and C,P-even vacuum. Gauge interactions of the vacuum Higgs-field with the electroweak bosons and Yukawa interactions with the leptons/quarks generate the masses which in turn determine the couplings of the Higgs particle to all SM particles. Finally, the self-interaction potential, which leads to the non-zero vacuum value $v$ of the Higgs field, being responsible for breaking the electroweak symmetries, is determined by the Higgs mass, and, as a result, the trilinear and quadrilinear Higgs self-interactions are fixed.

Since the Higgs mechanism provides the closure of the SM, the experimental investigation of the mechanism, connected with precision measurements<fnref n="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

<footnote n="3">³Editors: K. Fujii, S. Heinemeyer, P.M. Zerwas⁴
Contributing authors: M. Asano, K. Desch, U. Ellwanger, C. Englert, I. Ginzburg, C. Grojean, S. Kanemura, M. Krawczyk, J. Kroseberg, S. Matsumoto, M.M. Mühlleitner, M. Stanitzki</footnote>

<footnote n="4">⁴Cooperation, including Résumé, in early phase of the report.</footnote>

<footnote n="5">⁵Keisuke Fujii, Sven Heinemeyer, Peter M. Zerwas</footnote>

<footnote n="6">⁶Experimental results and simulations quoted in this introduction, as well as the large *corpus* of original theoretical studies in this field, are referenced properly in the review articles included subsequently in this section.</footnote>
could hold essential clues to this underlying theory.

The SM Higgs boson can be produced through several channels in pp collisions at LHC, with gluon fusion providing by far the maximum rate for intermediate masses. In $e^{+}e^{-}$ collisions the central channels [60–64] are

$$ \text{Higgs-strahlung : } e^{+}e^{-} \rightarrow ZH \qquad (2) $$

$$ \text{W-boson fusion : } e^{+}e^{-} \rightarrow \bar{\nu}_{e}\nu_{e} + H, \qquad (3) $$

with cross sections for a Higgs mass $M_H = 125$ GeV as shown in Tab.2 for the LC target energies of 250 GeV, 500 GeV, 1 TeV and 3 TeV. By observing the Z-boson in Higgs-

<table><caption>**Table 2** 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 polarizations: $P(e^{-}, e^{+}) = (-0.8, +0.3)$ 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.</caption><thead><tr><th></th><th>250 GeV</th><th>500 GeV</th><th>1 TeV</th><th>3 TeV</th></tr></thead><tbody><tr><td>σ[e<sup>+</sup>e<sup>-</sup> → ZH]</td><td>318</td><td>95.5</td><td>22.3</td><td>2.37</td></tr><tr><td>σ[e<sup>+</sup>e<sup>-</sup> → ν<sub>e</sub>ν<sub>e</sub>H]</td><td>36.6</td><td>163</td><td>425</td><td>862</td></tr></tbody></table>

strahlung, cf. Fig.11, the properties of the Higgs boson in the recoil state can be studied experimentally in a model-independent way.

### a) Higgs particle: mass and $J^{CP}$

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 [17], above the lower LEP2 limit of 114.4 GeV [65] (for a review see [66]). The mass of the new particle observed close to 125 GeV at LHC, agrees nicely with this expectation.

The final accuracy for direct measurements of an SM Higgs mass of 125 GeV is predicted at LHC/HL-LHC and LC in the bands

$$ \text{LHC / HL-LHC : } M_H = 125 \pm 0.1/0.05 \text{ GeV} \qquad (4) $$

$$ \text{LC : } M_H = 125 \pm 0.03 \text{ GeV.} \qquad (5) $$

Extrapolating the Higgs self-coupling associated with this mass value to the Planck scale, a value remarkably close to zero emerges [67–69].

Various methods can be applied for confirming the $J^{CP} = 0^{++}$ quantum numbers of the Higgs boson. While $C = +$ follows trivially from the $H \rightarrow \gamma\gamma$ decay mode, correlations among the particles in decay final states and between initial and final states, as well as threshold effects in Higgs-strahlung [70], cf. Fig.12 (upper plot), can be exploited for

<figure><img src="image_12.png" /><figcaption>**Fig. 11** Upper plot: Event in Higgs-strahlung $e^{+}e^{-} \rightarrow ZH \rightarrow (\mu^{+}\mu^{-})(jet)$ for a Higgs mass of 125 GeV at a collider energy of 500 GeV; lower plot: Distribution of the recoiling Higgs decay jets.</figcaption></figure>

measuring these quantum numbers.

### b) Higgs couplings to SM particles

Since the interaction between SM particles $x$ and the vacuum Higgs-field generates the fundamental SM masses, the coupling between SM particles and the physical Higgs particle, defined dimensionless, is determined by their masses:

$$ g_{Hxx} = [\sqrt{2}G_F]^{\frac{1}{2}} M_x, \qquad (6) $$

the coefficient fixed in the SM by the vacuum field $v = [\sqrt{2}G_F]^{-\frac{1}{2}}$. This fundamental relation is a cornerstone of the Higgs mechanism. It can be studied experimentally by measuring production cross sections and decay branching ratios.
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 12** Upper plot: Threshold rise of the Cross section for Higgs-strahlung $e^+e^- \to ZH$ corresponding to Higgs spin = 0, 1, 2, complemented by the analysis of angular correlations; lower plot: Measurements of Higgs couplings as a function of particle masses.</figcaption></figure>

At hadron colliders the twin observable $\sigma \times BR$ is measured for narrow states, and ratios of Higgs couplings are accessible directly. Since in a model-independent analysis BR potentially includes invisible decays in the total width, absolute values of the couplings can only be obtained with rather large errors. This problem can be solved in $e^+e^-$ colliders where the invisible Higgs decay branching ratio can be measured directly in Higgs-strahlung. Expectations for measurements at LHC (HL-LHC) and linear colliders are collected in Tab.3. The rise of the Higgs couplings with the masses is demonstrated for LC measurements impressively in Fig.12 (lower plot).

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.

<table><caption>**Table 3** Expected accuracy with which fundamental and derived Higgs couplings can be measured; the deviations are defined as $\kappa := g/g_{SM} = 1 \pm \Delta$ compared to the SM at the LHC/HL-LHC, LC and in combined analyses of the HL-LHC and LC [25]. 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% C.L. upper limit of potential couplings to invisible channels is also given.</caption><thead><tr><th>Coupling</th><th>LHC</th><th>HL-LHC</th><th>LC</th><th>HL-LHC + LC</th></tr></thead><tbody><tr><td>HWW</td><td>4-6%</td><td>2-5%</td><td>0.3%</td><td>0.1%</td></tr><tr><td>HZZ</td><td>4-6%</td><td>2-4%</td><td>0.5%</td><td>0.3%</td></tr><tr><td>Htt</td><td>14-15%</td><td>7-10%</td><td>1.3%</td><td>1.3%</td></tr><tr><td>Hbb</td><td>10-13%</td><td>4-7%</td><td>0.6%</td><td>0.6%</td></tr><tr><td>Hττ</td><td>6-8%</td><td>2-5%</td><td>1.3%</td><td>1.2%</td></tr><tr><td>Hγγ</td><td>5-7%</td><td>2-5%</td><td>3.8%</td><td>3.0%</td></tr><tr><td>Hgg</td><td>6-8%</td><td>3-5%</td><td>1.2%</td><td>1.1%</td></tr><tr><td>Hinvis</td><td>—</td><td>—</td><td>0.9%</td><td>0.9%</td></tr></tbody></table>

From the cross section measured in WW-fusion the partial width $\Gamma[WW^*]$ can be derived and, at the same time, from the Higgs-strahlung process the decay branching ratio $BR[WW^*]$ can be determined so that the total width follows immediately from

$$ \Gamma_{tot}[H] = \Gamma[WW^*]/BR[WW^*]. \qquad (7) $$

Based on the expected values at LC, the total width of the SM Higgs particle at 125 GeV is derived as $\Gamma_{tot}[H] = 4.1 \text{ MeV}[1 \pm 5\%]$. Measurements based on off-shell production of Higgs bosons provide only a very rough upper bound on the total width.

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 $g_H = g_H^{SM} [1 + v^2/\Lambda_*^2]$, 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 [71]. 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 $\sim v^2/m_\Lambda^2$. 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

The self-interaction of the Higgs field,

$$V = \lambda [ |\phi|^2 - v^2/2 ]^2, \qquad (8)$$

is responsible for electroweak symmetry breaking by shifting the vacuum state of minimal energy from zero to $v/\sqrt{2} \simeq 174$ GeV. The quartic form of the potential, required to render the theory renormalizable, generates trilinear and quadrilinear self-couplings when $\phi \to [v+H]/\sqrt{2}$ is shifted to the physical Higgs field $H$. The strength of the couplings are determined uniquely by the Higgs mass, with $M_H^2 = 2\lambda v^2$:

$$\lambda_3 = M_H^2/2v, \quad \lambda_4 = M_H^2/8v^2 \quad \text{and} \quad \lambda_{n>4} = 0. \qquad (9)$$

The trilinear Higgs coupling can be measured in Higgs pair-production [72]. Concerning the LHC, the cross section is small and thus the high luminosity of HL-LHC is needed to achieve some sensitivity to the coupling. Prospects are brighter in Higgs pair-production in Higgs-strahlung and W-boson fusion of $e^+e^-$ collisions, i.e. $e^+e^- \to Z+H^* \to Z+HH$, etc. In total, a precision of

$$\text{LC : } \lambda_3 = 10 - 13\% \qquad (10)$$

may be expected. On the other hand, the cross section for triple Higgs production is so small, $\mathcal{O}(ab)$, that the measurement of $\lambda_4$ values near the SM prediction will not be feasible at either type of colliders.

### d) Invisible Higgs decays

The observation of cold dark matter 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, $\tilde{Y} = \eta |\phi_{SM}|^2 |\phi_{hid}|^2$, in a form compatible with all standard symmetries. Thus a portal could be opened from the SM to the hidden sector [73,74]. 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 facility. 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 [75,76] 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. fourteen 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 explanation of the Higgs mechanism. It should be noted that supersymmetrization 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 realization of cold dark matter. 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 absorbed to provide masses to the electroweak gauge bosons while five degrees of freedom are realized as new physical fields, corresponding to two neutral CP-even scalar particles $h^0, H^0$; one neutral CP-odd scalar particle $A^0$; and a pair of charged $H^\pm$ scalar particles [77–80].
Since the quadrilinear Higgs couplings are predetermined 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 ~130 GeV by radiative corrections, adding a contribution of order $3M_t^4/2\pi^2 v^2 \log M_t^2/M_t^2 + \text{mix}$ for large top and stop masses. To reach a value of 125 GeV, large stop masses and/or large trilinear couplings are required in 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.

The heavy Higgs boson quartet is difficult to search for at LHC. In fact, these particles cannot be detected in a blind wedge which opens at 200 GeV for intermediate values of the mixing parameter $\tan\beta$ and which covers the parameter space for masses beyond 500 GeV. At the LC, Higgs-strahlung $e^+e^- \to Zh^0$ is supplemented by Higgs pair-production:

$$e^+e^- \to A^0H^0 \text{ and } H^+H^- \quad (11)$$

providing a rich source of heavy Higgs particles in $e^+e^-$ collisions for masses $M < \sqrt{s}/2$, cf. Fig.13. Heavy Higgs masses come with ZAH couplings of the order of gauge couplings so that the cross sections are large enough for copious production of heavy neutral CP even/odd and charged Higgs boson pairs.

Additional channels open in single Higgs production $\gamma\gamma \to A^0, H^0$, 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 quadrilinear 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 [81, 82], the next-to-minimal model (NMSSM). This extension augments the Higgs spectrum by two additional physical states, CP-even and CP-odd, which mix with the corresponding MSSM-type states.

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13 Upper plot: Reconstructed 2-jet invariant mass for associated production: $e^+e^- \to AH \to b\bar{b}b\bar{b}$ for a Higgs mass of 900 GeV at a collider energy of 3 TeV; lower plot: Similar plot for $e^+e^- \to H^+H^- \to t\bar{b}\bar{b}$.</figcaption></figure>

The bound on the mass of the lightest MSSM Higgs particle is alleviated by contributions from the trilinear 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 $pp \to Higgs \to \gamma\gamma$, for instance, can be accommodated 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 [83, 84]. 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 order.

Such a scenario [85] can be realized, 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 5-dimensional Anti-de-Sitter space. The AdS/CFT correspondence relates this theory to a 4-dimensional strongly coupled theory, the 5-th component of the gauge fields interpreted as Goldstone modes in the strongly coupled 4-

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 [86], either universally or non-universally with alternating signs for vector bosons and fermions.

### 2.2 The SM Higgs at the LHC: Status and Prospects<fnref n="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, $H \to \gamma\gamma$, $H \to ZZ \to 4l$ and $H \to WW \to 2l2v$ ($l = e, \mu$), 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 [55, 87–89].

Since then, the LHC experiments have concluded their first phase of data taking ("Run1") and significantly larger data sets 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.

While the following discussion is restricted to analyses within the framework of the SM, the consistency of the observed Higgs boson candidate with SM expectations (as evaluated in [34, 90, 91] and references therein) does not exclude that extensions of the SM with a richer Higgs sector are realised in nature and might show up experimentally at the LHC. Thus, both the ATLAS and CMS collaborations have been pursuing a rich program of analyses that search for deviations from the SM predictions and for additional Higgs bosons in the context of models beyond the SM. A review of this work is, however, beyond the scope of this section.

<footnote n="7">⁷Jürgen Kroseberg</footnote>
<figure><img src="image_17.png" /><figcaption>**Fig. 14** Displays of example Higgs boson candidate events. Top: $H \to ZZ \to 2\mu2e$ candidate in the ATLAS detector; bottom: VBF $H \to \gamma\gamma$ candidate in the CMS detector.</figcaption></figure>

### 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 [92, 93]. 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.

#### Relevant decay channels

For all decay channels described below, the analysis strategies have evolved over time in similar ways. Early searches were based on inclusive analyses of the Higgs boson decay products. With larger data sets, these were replaced by analyses in separate categories corresponding to different event characteristics and background composition. Such categorisation significantly increases the signal sensitivity and can also be used to separate different production processes, which is relevant for the current and future studies of the Higgs boson couplings discussed below. Also, with larger data sets and higher complexity of the analyses, it became increasingly important to model the background contributions from data control regions instead of relying purely on simulated events. Another common element is the application of multivariate techniques in more recent analyses. Still,

the branching ratios, detailed signatures and relevant background processes for different decays differ substantially; two example Higgs boson production and decay candidate event displays are shown in Fig. 14. Therefore, the experimental approaches and resulting information on the 125 GeV Higgs boson vary as well:

- $H \to \gamma\gamma$: 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 [94] and CMS [97] analyses yield signals with significances of $5.2\sigma$ and $5.7\sigma$, respectively, where $4.6\sigma$ and $5.2\sigma$ are expected.

- $H \to ZZ \to 4\ell$: 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, $Z+\text{jets}$ and $t\bar{t}$ production processes. ATLAS [98] and CMS [95] report observed (expected) signal significances of $8.1\sigma$ ($6.2\sigma$) and $6.8\sigma$ ($6.7\sigma$).

- $H \to WW \to 2\ell2\nu$: 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 [96] and CMS [99] signals have significances of $6.1\sigma$ ($5.8\sigma$) and $4.3\sigma$ ($5.8\sigma$).

Fig. 15 shows reconstructed Higgs candidate mass distributions from ATLAS and CMS searches for $H \to \gamma\gamma$ and $H \to ZZ \to 4\ell$, respectively, as well as the ATLAS $H \to WW \to 2\ell2\nu$ transverse mass distribution. Other bosonic decay modes are searched for as well but these analyses are not yet sensitive to a SM Higgs boson observation.

- $H \to b\bar{b}$: 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.

- $H \to \tau\tau$: 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 $Z \to \tau\tau$ decays; further background contributions arise from processes with a vector boson and jets, top and diboson production.
While searches for $H \to b\bar{b}$ decays [100, 101] have not yet resulted in significant signals, first evidence for direct Higgs boson decays to fermions has been reported by both ATLAS and CMS following analyses of $\tau\tau$ final states. The CMS results [102] are predominantly based on fits to the reconstructed $\tau\tau$ invariant mass distributions, whereas the ATLAS analysis [103] uses the output of boosted decision trees (BDTs) throughout for the statistical analysis of the selected data. ATLAS (CMS) find signals with a significance of $4.5\sigma$ (3.5$\sigma$), where $3.4\sigma$ (3.7$\sigma$) are expected, cf. Fig. 16. In [104] CMS present the combination of their $H \to \tau\tau$ and $H \to b\bar{b}$ analyses yielding an observed (expected) signal significance of $3.8\sigma$ (4.4$\sigma$). Searches for other fermionic decays are performed as well but are not yet sensitive to the observation of the SM Higgs boson.

<figure><img src="image_18.png" /><figcaption>**Fig. 15** Reconstructed distributions of the Higgs boson candidate decay products for the complete 2011/12 data, expected backgrounds, and simulated signal from top: the ATLAS $H \to \gamma\gamma$ [94], centre: the CMS $H \to ZZ \to 4\ell$ [95], and bottom: the ATLAS $H \to WW \to 2\ell/2\nu$ [96] analyses.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 16** Evidence for the decay $H \to \tau\tau$. Top: CMS observed and predicted $m_{\tau\tau}$ distributions [102]. The distributions obtained in each category of each channel are weighted by the ratio between the expected signal and signal-plus-background yields in the category. The inset shows the corresponding difference between the observed data and expected background distributions, together with the signal distribution for a SM Higgs boson at $m_H=125$ GeV; bottom: ATLAS event yields as a function of $\log(S/B)$, where S (signal yield) and B (background yield) are taken from the corresponding bin in the distribution of the relevant BDT output discriminant [103].</figcaption></figure>
In the following, we summarise the status of SM Higgs boson analyses of the full 2011/12 data sets with ATLAS and CMS. The discussion is based on preliminary combinations of ATLAS and published CMS results collected in [105] and [106], respectively; an ATLAS publication of Higgs boson mass measurements [107]; ATLAS [108] and CMS [109] constraints on the Higgs boson width; studies of the Higgs boson spin and parity by CMS [110] and ATLAS [58, 111, 112]; and other results on specific aspects or channels referenced later in this section.

## Signal strength

For a given Higgs boson mass, the parameter $\mu$ is defined as the observed Higgs boson production strength normalised to the SM expectation. Thus, $\mu = 1$ reflects the SM expectation and $\mu = 0$ corresponds to the background-only hypothesis.

Fixing the Higgs boson mass to the measured value and considering the decays $H \to \gamma\gamma$, $H \to ZZ \to 4\ell$, $H \to WW \to 2\ell 2\nu$, $H \to b\bar{b}$, and $H \to \tau\tau$, ATLAS report [105] a preliminary overall production strength of

$$ \mu = 1.18^{+0.15}_{-0.14}; $$

<figure><img src="image_20.png" /><figcaption>Fig. 17 Higgs boson signal strength as measured by ATLAS for different decay channels [105].</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>Fig. 18 Higgs boson production strength, normalised to the SM expectation, based on CMS analyses [106], for a combination of analysis categories related to different production modes.</figcaption></figure>

the separate combination of the bosonic and fermionic decay modes yields $\mu = 1.35^{+0.21}_{-0.20}$ and $\mu = 1.09^{+0.36}_{-0.32}$, respectively. The corresponding CMS result [106] is

$$ \mu = 1.00 \pm 0.13. $$

Good consistency is found, for both experiments, across different decay modes and analyses categories related to different production modes, see Figs. 17 and 18.

ATLAS and CMS have also studied the relative contributions from production mechanisms mediated by vector bosons (VBF and VH processes) and gluons (ggF and $t\bar{t}H$ processes), respectively. For example, Fig. 19 shows ATLAS

<figure><img src="image_22.png" /><figcaption>Fig. 19 Likelihood for the ratio $\mu_{VBF}/\mu_{ggF+ttH}$ obtained by ATLAS for the combination of the $H \to \gamma\gamma$, $ZZ \to 4\ell$ and $WW \to 2\nu 2\ell$ channels and $m_H = 125.5$ GeV [105].</figcaption></figure>
results constituting a 4.3σ evidence that part of the Higgs boson production proceeds via VBF processes [105].

## 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, CP-even scalar state, its couplings are tested by introducing free parameters $κ_X$ for each particle X, such that the SM predictions for production cross sections and decay widths are modified by a multiplicative factor $κ_X^2$. This includes effective coupling modifiers $κ_g$, $κ_γ$ for the loop-mediated interaction with gluons and photons. An additional scale factor modifies the total Higgs boson width by $κ_H^2$.

Several different set of assumptions, detailed in [33] and [34], form the basis of such coupling analyses. For example, a fit to the ATLAS data [105] assuming common scale factors $κ_F$ and $κ_V$ for all fermions and bosons, respectively, yields the results depicted in Fig. 20.

<figure><img src="image_23.png" /><figcaption>**Fig. 20** Preliminary ATLAS results of fits for a two-parameter benchmark model that probes different coupling strength scale factors common for fermions ($κ_F$) and vector bosons ($κ_V$), respectively, assuming only SM contributions to the total width. Shown are 68% and 95% C.L. contours of the two-dimensional fit; overlaying the 68% CL contours derived from the individual channels and their combination. The best-fit result (x) and the SM expectation (+) are also indicated [105].</figcaption></figure>

Within the SM, $λ_{WZ} = κ_W/κ_Z = 1$ is implied by custodial symmetry. Agreement with this prediction is found by both CMS, see Fig. 21, and ATLAS. Similar ratio analyses are performed for the couplings to leptons and quarks ($λ_{lq}$) as well as to down and up-type fermions ($λ_{du}$).

Within a scenario where all modifiers $κ$ except for $κ_g$ and $κ_γ$ are fixed to 1, contributions from beyond-SM particles to the loops that mediate the $ggH$ and $Hγγ$ interactions can be constrained; a corresponding CMS result [106] is shown in Fig. 22.

<figure><img src="image_24.png" /><figcaption>**Fig. 21** Test of custodial symmetry: CMS likelihood scan of the ratio $λ_{WZ}$, where SM coupling of the Higgs bosons to fermions are assumed [106].</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. 22** Constraining BSM contributions to particle loops: CMS 2d likelihood scan of gluon and photon coupling modifiers $κ_g$, $κ_γ$ [106].</figcaption></figure>

Summaries of CMS results [106] 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 [106] yield the same conclusions. It should be noted, however, that this does not yet constitute a complete, unconstrained analysis of the Higgs boson couplings.

For the fit assuming that loop-induced couplings follow the SM structure as in [34] without any BSM contributions to Higgs boson decays or particle loops, ATLAS, see Fig. 24, and CMS also demonstrate that the results follow the predicted relationship between Higgs boson couplings and the SM particle masses.
<figure><img src="image_26.png" /><figcaption>Fig. 23 Summary plot of CMS likelihood scan results [106] for the different parameters of interest in benchmark models documented in [34]. The inner bars represent the 68% CL confidence intervals while the outer bars represent the 95% C.L. confidence intervals.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>Fig. 24 ATLAS summary of the fits for modifications of the SM Higgs boson couplings expressed as function of the particle mass. For the fermions, the values of the fitted Yukawa couplings for the Hf$\bar{f}$ vertex are shown, while for vector bosons the square-root of the coupling for the HVV vertex divided by twice the vacuum expectation value of the Higgs boson field [105].</figcaption></figure>

### Mass

Current measurements of the Higgs boson mass are based on the two high-resolution decay channels $H \to \gamma\gamma$ and $H \to ZZ \to 4\ell$. Based on fits to the invariant diphoton and four-lepton mass spectra, ATLAS measures [107] $m_H = 125.98 \pm 0.42(\text{stat}) \pm 0.28(\text{sys})$ and $m_H = 124.51 \pm 0.52(\text{stat}) \pm 0.06(\text{sys})$, respectively. A combination of the two results, which are consistent within 2.0 standard deviations,

yields $m_H = 125.36 \pm 0.37(\text{stat}) \pm 0.18(\text{sys})$. An analysis [106] of the same decays by CMS find consistency between the two channels at $1.6\sigma$, see Fig. 25. The combined result $m_H = 125.02_{-0.27}^{+0.26}(\text{stat})_{-0.15}^{+0.14}(\text{sys})$ agrees well with the corresponding ATLAS measurement.

A preliminary combination [113] of both experiments gives a measurement of the Higgs boson mass of

$$m_H = 125.09 \pm 0.21(\text{stat}) \pm 0.11(\text{sys}),$$

with a relative uncertainty of 0.2%.

<figure><img src="image_28.png" /><figcaption>Fig. 25 CMS mass measurements [106] in the $\gamma\gamma$ and $ZZ \to 4\ell$ final states and their combinations. The vertical band shows the combined uncertainty. The horizontal bars indicate the $\pm 1$ standard deviation uncertainties for the individual channels.</figcaption></figure>

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 $m_H = 128_{-5}^{+7}$ GeV and $m_H = 122 \pm 7$ GeV from the analysis of WW [99] and $\tau\tau$ [102] 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 $\Gamma_H$ is about 4 MeV.

Analyses of ZZ and WW events in the mass range above the $2m_{Z,W}$ threshold provide an alternative approach [30, 114], which was first pursued by CMS [109] based on the $ZZ \to 4\ell$ and $ZZ \to 2\ell2\nu$ channels; a later ATLAS analysis [108] included also the $WW \to e\nu\mu\nu$ final state. The studied distributions vary between experiments and channels; for example, Fig. 26 shows the high-mass $ZZ \to 2\ell2\nu$ transverse mass distribution observed by ATLAS with the expected background contributions and the predicted signal
<figure><img src="image_29.png" /><figcaption>**Fig. 26** Observed transverse mass distributions for the ATLAS ZZ → 2ℓ2ν analysis [108] in the signal region compared to the expected contributions from ggF and VBF Higgs production with the decay H* → ZZ SM and with μ<sub>off-shell</sub> = 10 (dashed) in the 2e2ν channel. A relative gg → ZZ background K-factor of 1 is assumed.</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>**Fig. 27** CMS likelihood scan versus Γ<sub>H</sub>. Different colours refer to: combination of 4ℓ low-mass and high-mass (ochre), combination of 4ℓ low-mass and 2ℓ/2ν high-mass and combination of 4ℓ low-mass and both channels at high-mass (blue). Solid and dashed lines represent observed and expected limits, respectively [109].</figcaption></figure>

for different assumptions for the off-shell $H \to ZZ$ signal strength $\mu_{\text{off-shell}}$. The resulting constraints on $\mu_{\text{off-shell}}$, together with the on-shell $H \to ZZ \to 4\ell$ $\mu_{\text{on-shell}}$ measurement, can be interpreted as a limit on the Higgs boson width if the relevant off-shell and on-shell Higgs couplings are assumed to be equal.<fnref n="8" />

Combining ZZ and WW channels, ATLAS find an observed (expected) 95% C.L. limit of

$$5.1(6.7) < \mu_{\text{off-shell}} < 8.6(11.0)$$

when varying the unknown K-factor ratio between the $gg \to ZZ$ continuum background and the $gg \to H^* \to ZZ$ signal between 0.5 and 2.0. This translates into

$$4.5(6.5) < \Gamma_H / \Gamma_H^{\text{SM}} < 7.5(11.2)$$

if identical on-shell and off-shell couplings are assumed.

Fig. 27 illustrates the results of a corresponding CMS analysis, yielding observed (expected) 95% C.L. limit of $\Gamma_H / \Gamma_H^{\text{SM}} < 22(33)$ MeV or $\Gamma_H / \Gamma_H^{\text{SM}} < 5.4(8.0)$.

### Spin and parity

Within the SM, the Higgs boson is a spin-0, 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: $H \to \gamma\gamma$, $H \to ZZ \to 4\ell$, and $H \to WW \to 2\ell2\nu$.

In the $H \to \gamma\gamma$ analysis, the $J^P = 0^+$ and $J^P = 2^+$ hypothesis can be distinguished via the Collins-Soper angle $\theta^*$ of the photon system. Since there is a large non-resonant diphoton background, the spin information is extracted from a simultaneous fit to the $|\cos\theta^*|$ and $m_{\gamma\gamma}$ distributions. The

<figure><img src="image_31.png" /><figcaption>**Fig. 28** Top: Final state observables sensitive to the spin and parity of the decaying resonance in $ZZ^* \to 4\ell$ final states. Bottom: $\cos\theta_1$ distribution for ATLAS data (point with errors), the backgrounds (filled histograms) and several spin hypotheses (SM solid line and alternatives dashed lines) [112].</figcaption></figure>

<footnote n="8">⁸ However, the relation between the off-shell and on-shell couplings can be severely affected by new-physics contributions, in particular via threshold effects. In fact, such effects may be needed to give rise to a Higgs-boson width that differs from the one of the SM by the currently probed amount, see also the discussion in [115]. In this sense, these analyses currently provide a consistency test of the SM rather than model-independent bounds on the total width.</footnote>
<figure><img src="image_32.png" /><figcaption>**Fig. 29** Distributions of the test statistic $q = -2\ln(\mathcal{L}_{J^P}/\mathcal{L}_{0^+})$ for the spin-one and spin-two JP models tested against the SM Higgs boson hypothesis in the combined $X \to ZZ$ and WW analyses [110]. The expected median and the 68.3%, 95.4%, and 99.7% C.L. regions for the SM Higgs boson (orange, the left for each model) and for the alternative $J^P$ hypotheses (blue, right) are shown. The observed $q$ values are indicated by the black dots.</figcaption></figure>

charged-lepton kinematics and the missing transverse energy in $H \to WW \to e\nu_e\mu\nu_\mu$ candidate decays are combined in multivariate analyses to compare the data to the SM and three alternative ($J^P = 2^+, 1^+, 1^-$) hypotheses. The $H \to ZZ \to 4\ell$ analysis combines a high signal-to-background ratio with a complete final state reconstruction. This makes it possible to perform a full angular analysis, c.f. Fig. 28, albeit currently still with a rather limited number of events. Here, in addition to the spin-parity scenarios discussed above, also the $J^P = 0^-$ hypothesis is tested.

Including the spin-1 hypotheses in the analyses of the decays into vector bosons provides a test independent of the $H \to \gamma\gamma$ 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-two 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 $H \to ZZ \to 4\ell$ and $H \to WW \to 2\ell 2\nu$ channels [110]. 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 [58, 111, 112] 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 $H \to \gamma\gamma$ [116] and $H \to ZZ$ [117] events and searches for rarer decays, such as $H \to \mu\mu$ [118, 119], $H \to ee$ [119], $H \to Z\gamma$ [120, 121], decays to heavy quarkonia states and a photon [122], and invisible modes [123, 124].

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 [125–128] 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 [129] and [130], 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.

## Underlying assumptions

CMS extrapolates the results of current Run1 measurements to $\sqrt{s} = 14$ TeV data samples corresponding to 300 fb$^{-1}$ and 3000 fb$^{-1}$ assuming that the upgraded detector and trigger systems will provide the same performance in the high-luminosity environment as the current experiments during 2012, i.e. the signal and background event yields are scaled according to the increased luminosities and cross sections. Results based on two different assumptions concerning the systematic uncertainties are obtained: a first scenario assumes no changes with respect to 2012, while in a second scenario theoretical uncertainties are reduced by a factor of two and other uncertainties scaled according to the square root of the integrated luminosities.

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.

### Signal strength

Both experiments study expectations for the experimentally most significant SM Higgs boson decay modes $H \to \gamma\gamma$, $H \to ZZ \to 4\ell$, $H \to WW \to 2\ell 2\nu$, $H \to \tau\tau$, and $H \to b\bar{b}$ but also include analyses of additional sub-modes as well as rare decays to $Z\gamma$, $\mu\mu$, and invisible final states. Fig. 30 shows two examples for expected mass signals based on ATLAS simulations of SM Higgs boson decays to two photons (after a VBF selection) and two muons, respectively.

The expected relative uncertainties on the signal strength for CMS and ATLAS are shown in Table 4 and Fig. 31, indicating that for the most sensitive channels, experimental uncertainty around 5% should be reachable with 3000 fb$^{-1}$.

<figure><img src="image_33.png" /><figcaption>**Fig. 30** Projected (a) diphoton mass distribution for the SM Higgs boson signal and background processes after VBF selection and (b) background-subtracted dimuon mass distribution based on ATLAS simulations assuming an integrated luminosity of 3000 fb$^{-1}$ [131].</figcaption></figure>

Combining different final states and again assuming SM branching ratios, projections on the sensitivity to individual Higgs boson production can be obtained; the corresponding ATLAS results are summarised in Table 5. For 3000 fb$^{-1}$, the expected experimental uncertainties on the signal strength range from about 4% for the dominant ggF production to about 10% for the rare $t\bar{t}H$ production mode. Fig. 31 and Table 5 also indicate the contribution of current theoretical uncertainties, showing that reducing them further will be impossible.

<table><caption>**Table 4** Relative uncertainty on the determination of the signal strength expected for the CMS experiment for integrated luminosities of 300 fb$^{-1}$ and 3000 fb$^{-1}$ [130] and the two uncertainty scenarios described in the text.</caption><thead><tr><th rowspan="2">L<br>Scenario</th><th colspan="2">300 fb<sup>-1</sup></th><th colspan="2">3000 fb<sup>-1</sup></th></tr><tr><th>2</th><th>1</th><th>2</th><th>1</th></tr></thead><tbody><tr><td>γγ</td><td>6%</td><td>12%</td><td>4%</td><td>8%</td></tr><tr><td>WW</td><td>6%</td><td>11%</td><td>4%</td><td>7%</td></tr><tr><td>ZZ</td><td>7%</td><td>11%</td><td>4%</td><td>7%</td></tr><tr><td>b<span style="text-decoration: overline;">b</span></td><td>11%</td><td>14%</td><td>5%</td><td>7%</td></tr><tr><td>ττ</td><td>8%</td><td>14%</td><td>5%</td><td>8%</td></tr><tr><td>Zγ</td><td>62%</td><td>62%</td><td>20%</td><td>24%</td></tr><tr><td>μμ</td><td>40%</td><td>42%</td><td>14%</td><td>20%</td></tr></tbody></table>

<figure><img src="image_34.png" /><figcaption>**Fig. 31** Relative uncertainty on the signal strength determination expected for the ATLAS experiment [129]. assuming a SM Higgs boson with a mass of 125 GeV and 300 fb$^{-1}$ and 3000 fb$^{-1}$ of 14 TeV data. The uncertainty pertains to the number of events passing the experimental selection, not to the particular Higgs boson process targeted. The hashed areas indicate the increase of the estimated error due to current theory systematic uncertainties.</figcaption></figure>
portant to fully exploit the HL-LHC for Higgs boson precision studies.

<table><caption>**Table 5** 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 [129], assuming a SM Higgs boson with a mass of 125 GeV and branching ratios as in the SM.</caption><thead><tr><th rowspan="2">L<br>Uncertainties</th><th colspan="2">300 fb<sup>-1</sup></th><th colspan="2">3000 fb<sup>-1</sup></th></tr><tr><th>All</th><th>No theory</th><th>All</th><th>No theory</th></tr></thead><tbody><tr><td>gg → H</td><td>12%</td><td>6%</td><td>11%</td><td>4%</td></tr><tr><td>VBF</td><td>18%</td><td>15%</td><td>15%</td><td>9%</td></tr><tr><td>WH</td><td>41%</td><td>41%</td><td>18%</td><td>18%</td></tr><tr><td>qqZH</td><td>80%</td><td>79%</td><td>28%</td><td>27%</td></tr><tr><td>ggZH</td><td>371%</td><td>362%</td><td>147%</td><td>138%</td></tr><tr><td>ttH</td><td>32%</td><td>30%</td><td>16%</td><td>10%</td></tr></tbody></table>

## Couplings to other particles

The individual channels are combined to obtain projections on the experimental sensitivity concerning Higgs boson couplings to other elementary bosons and fermions. Following the same formalism and set of assumptions used for the current Run1 results described above, coupling scale factors $κ_X$ are extracted. Fig. 32, for example, shows the projected ATLAS results of the minimal coupling fit constrained to common scale factors $κ_F$ and $κ_V$ for all fermions and bosons, respectively, and assuming SM values for both; cf. Fig. 20 for the corresponding Run1 results. Fig. 33 gives an overview of the precision on the extraction of individual coupling scale factors expected for the CMS experiment.

The $κ_X$ 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

<figure><img src="image_35.png" /><figcaption>**Fig. 32** Expected ATLAS 68% and 95% C.L. likelihood contours for $κ_V$ and $κ_F$ in a minimal coupling fit for an integrated luminosity of 300 fb⁻¹ and 3000 fb⁻¹ [129].</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>**Fig. 33** CMS projected relative uncertainty on the measurements of $κ_γ$, $κ_V$, $κ_g$, $κ_b$, $κ_t$, and $κ_τ$ assuming $\sqrt{s} = 14$ TeV and an integrated luminosity 300 and 3000 fb⁻¹. The results are shown for two uncertainty scenarios described in the text [130].</figcaption></figure>

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, 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 elec-

<table><caption>**Table 6** 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 [130] and the two uncertainty scenarios described in the text.</caption><thead><tr><th rowspan="2">L<br>Scenario</th><th colspan="2">300 fb<sup>-1</sup></th><th colspan="2">3000 fb<sup>-1</sup></th></tr><tr><th>2</th><th>1</th><th>2</th><th>1</th></tr></thead><tbody><tr><td>κ<sub>γ</sub> · κ<sub>Z</sub> / κ<sub>H</sub></td><td>4%</td><td>6%</td><td>2%</td><td>5%</td></tr><tr><td>κ<sub>W</sub> / κ<sub>Z</sub></td><td>4%</td><td>7%</td><td>2%</td><td>3%</td></tr><tr><td>λ<sub>g</sub> = κ<sub>t</sub> / κ<sub>g</sub></td><td>13%</td><td>14%</td><td>6%</td><td>8%</td></tr><tr><td>λ<sub>bZ</sub> = κ<sub>b</sub> / κ<sub>Z</sub></td><td>8%</td><td>11%</td><td>3%</td><td>5%</td></tr><tr><td>λ<sub>τZ</sub> = κ<sub>τ</sub> / κ<sub>Z</sub></td><td>6%</td><td>9%</td><td>2%</td><td>4%</td></tr><tr><td>λ<sub>μZ</sub> = κ<sub>μ</sub> / κ<sub>Z</sub></td><td>22%</td><td>23%</td><td>7%</td><td>8%</td></tr><tr><td>λ<sub>Zg</sub> = κ<sub>Z</sub> / κ<sub>g</sub></td><td>6%</td><td>9%</td><td>3%</td><td>5%</td></tr><tr><td>λ<sub>γZ</sub> = κ<sub>γ</sub> / κ<sub>Z</sub></td><td>5%</td><td>8%</td><td>2%</td><td>5%</td></tr><tr><td>λ<sub>(Zγ)Z</sub> = κ<sub>Zγ</sub> / κ<sub>Z</sub></td><td>40%</td><td>42%</td><td>12%</td><td>12%</td></tr></tbody></table>
ATLAS Simulation Preliminary

<figure><img src="image_37.png" /><figcaption>**Fig. 34** Relative uncertainty expected for the ATLAS experiment on the determination of coupling scale factor ratios $\lambda_{XY} = \kappa_X / \kappa_Y$ from a generic fit [129], assuming a SM Higgs boson with a mass of 125 GeV and 300 fb$^{-1}$ and 3000 fb$^{-1}$ of 14 TeV data. The hashed areas indicate the increase of the estimated error due to current theory uncertainties.</figcaption></figure>

troweak bosons to 5-8% for the ratios involving gluons and fermions outside the first generation.

Fig. 35 gives the ATLAS projection for the precision of the Higgs boson couplings to other elementary SM particles as a function of the particle masses obtained from fits assuming no BSM contributions to Higgs boson decays or particle loops; see Fig. 24 for corresponding CMS Run1 results.

Higgs self-coupling

One of the most important long-term goals of the SM Higgs physics program is the measurement of the trilinear 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, [132] presents the ATLAS prospects for the search for Higgs pair production in the channel $H(\to \gamma\gamma)H(\to b\bar{b})$, which combines the large $H \to b\bar{b}$ 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

<figure><img src="image_38.png" /><figcaption>**Fig. 35** Fit results for the reduced coupling scale factors for weak bosons and fermions as a function of the particle mass, assuming 300/fb or 3000/fb of 14 TeV data and a SM Higgs boson with a mass of 125 GeV [129].</figcaption></figure>

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$^{-1}$.

<figure><img src="image_39.png" /><figcaption>**Fig. 36** Projected diphoton mass distribution for signal and background processes based on ATLAS simulations for a search for Higgs boson pair production with subsequent decays $H \to b\bar{b}$ and $H \to \gamma\gamma$ assuming an integrated luminosity of 3000 fb$^{-1}$ [132]. The simulated distributions are scaled to match the expected event yields but do not necessarily reflect the corresponding statistical fluctuations.</figcaption></figure>
## 2.3 Higgs at ILC: prospects<fnref n="9" />

### 2.3.1 Introduction

The success of the Standard Model (SM) is a success of gauge principle. It is the success of the transverse components of W and Z identified as gauge fields of the electroweak (EW) gauge symmetry. Since explicit mass terms for W and Z are forbidden by the gauge symmetry, it must be spontaneously broken by something condensed in the vacuum which carries EW charges ($I_3$ and $Y$ denoting the third component of the weak iso-spin and the hyper charge, respectively),

$$ \langle 0 | I_3, Y | 0 \rangle \neq 0 \text{ while } \langle 0 | I_3 + Y | 0 \rangle = 0. \qquad (12) $$

We are hence living in a weak-charged vacuum. This something provides three longitudinal modes of W and Z:

$$ \text{Goldstone modes: } \chi^+, \chi^-, \chi_3 \rightarrow W_L^+, W_L^-, Z_L. \qquad (13) $$

It should be emphasized that we do not know the nature of these longitudinal modes which stem from the something. The gauge symmetry also forbids explicit mass terms for matter fermions, since left- ($f_L$) and right-handed ($f_R$) matter fermions carry different EW charges, hence, as long as the EW charges are conserved, they cannot mix. Their Yukawa interactions with some weak-charged vacuum can compensate the EW-charge difference and hence allow the $f_L-f_R$ mixing. In the SM, the same something is responsible for the $f_L-f_R$ mixing, thereby generating masses and inducing flavor-mixings among generations. To form gauge-invariant Yukawa interaction terms, we need a complex doublet scalar field, which has four real components. In the SM, three of them are identified with the three Goldstone modes and are used to supply the longitudinal modes of W and Z. The remaining one is the physical Higgs boson. There is no reason for this simplicity of the symmetry breaking sector of the SM. The symmetry breaking sector (hear after called the Higgs sector) can well be much more complicated. The something could be composite instead of being elementary. We know it's there around us with a vacuum expectation value of 246 GeV. But this was about all we knew concerning the something until July 4th, 2012.

Since the July 4th, the world has changed! The discovery of the 125 GeV boson ($X(125)$) at the LHC could be called a quantum jump [135]. The observation of $X(125) \rightarrow \gamma\gamma$ decay implies X is a neutral boson having a spin not equal to 1 (Landau-Yang theorem). We know that the 125 GeV boson decays also to $ZZ^*$ and $WW^*$, indicating the existence

of XVV couplings, where $V = W/Z$, gauge bosons. There is, however, no gauge coupling like XVV, see Fig. 37. There

<figure><img src="image_40.png" /><figcaption>Fig. 37 The origin of XVV coupling and its relation to the mass term of V.</figcaption></figure>

are only XXVV and XXV. The XVV coupling is hence most probably from XXVV with one X replaced by its vacuum expectation value $\langle X \rangle \neq 0$, namely $\langle X \rangle XVV$. Then there must be $\langle X \rangle \langle X \rangle VV$, a mass term for V, meaning that X is at least part of the origin of the masses of $V = W/Z$. This is a great step forward to uncover the nature of the something in the vacuum but we need to know whether $\langle X \rangle$ saturates the SM vev of 245 GeV. The observation of the $X \rightarrow ZZ^*$ decay means that X can be produced via $e^+e^- \rightarrow Z^* \rightarrow ZX$, since by attaching an $e^+e^-$ pair to the $Z^*$ leg and rotate the whole diagram we can get the X-strahlung diagram as shown in Fig.38. By the same token, $X \rightarrow WW^*$ means that X can

<figure><img src="image_41.png" /><figcaption>Fig. 38 $X \rightarrow ZZ^*$ decay and $e^+e^- \rightarrow ZX$ process.</figcaption></figure>

be produced via the WW-fusion process: $e^+e^- \rightarrow v\bar{v}X$. So we now know that the major Higgs production processes in $e^+e^-$ collisions are indeed available at the ILC, which can be regarded as a *no lose theorem* for the ILC. The 125 GeV is the best place for the ILC, where variety of decay modes are accessible. We need to check the 125GeV boson in detail to see if it has indeed all the required properties of the something in the vacuum.

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.

Our mission is the bottom-up model-independent reconstruction of the electroweak symmetry breaking (EWSB) sector through the coupling measurements. We need to determine the multiplet structure of the Higgs sector by answering questions like: Is there an additional singlet or dou-

<footnote n="9">⁹Keisuke Fujii</footnote>

<footnote>The presented materials were prepared for the ILC TDR physics chapter and for the Snowmass Higgs white paper in collaboration with the members of the ILC physics working group [133, 134] and the members of the ILC physics panel. The author would like to thank them for useful discussions, especially M. Peskin, Y. Okada, S. Kanemura, H. Haber, T. Barklow, A. Miyamoto, J. Tian, H. Ono, and T. Tanabe.</footnote>
<figure><img src="image_42.png" /><figcaption>Fig. 39 Mass-coupling relation [136].</figcaption></figure>

blet or triplet? What about the underlying dynamics? Is it weakly interacting or strongly interacting? In other words, is the Higgs boson elementary or composite? We should also try to investigate its possible relation to other questions of particle physics such as dark matter, electroweak baryogenesis, neutrino masses, and inflation.

There are many possibilities and different models predict different deviation patterns in the mass-coupling relation. An example is given in Table 7, where a model with an extra singlet and four types of two-Higgs doublet models (2HDM) are compared. The four types of 2HDMs differ in the assignment of a $Z_2$ charge to the matter fermions, which protects them from inducing dangerous flavor-changing neutral currents [137].

<table><caption>**Table 7** 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 $\cos(\beta - \alpha) > 0$.</caption><thead><tr><th>Model</th><th>μ</th><th>τ</th><th>b</th><th>c</th><th>t</th><th>g<sub>V</sub></th></tr></thead><tbody><tr><td>Singlet mixing</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td></tr><tr><td>2HDM-I</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td></tr><tr><td>2HDM-II (SUSY)</td><td>↑</td><td>↑</td><td>↑</td><td>↓</td><td>↓</td><td>↓</td></tr><tr><td>2HDM-X (Lepton-specific)</td><td>↑</td><td>↑</td><td>↓</td><td>↓</td><td>↓</td><td>↓</td></tr><tr><td>2HDM-Y (Flipped)</td><td>↓</td><td>↓</td><td>↑</td><td>↓</td><td>↓</td><td>↓</td></tr></tbody></table>

Notice that though both singlet mixing and 2HDM-I with $\cos(\beta - \alpha) < 0$ give downward deviations, they are quantitatively different: the singlet mixing reduces the coupling constants universally, while 2HDM-I reduces them differently for matter fermions and gauge bosons. In these models, $g_V < 1$ is guaranteed because of the sum rule for the vacuum expectation values of the SM-like Higgs boson and the additional doublet or singlet. When a doubly charge Higgs boson is present, however, $g_V > 1$ is possible. The size of any of

these deviations is generally written in the following form due to the decoupling theorem:

$$ \frac{\Delta g}{g} = \mathcal{O}\left(\frac{v^2}{M^2}\right) \qquad (14) $$

where $v$ is the SM vev and $M$ is the mass scale for the new physics. Since there is no hint of new physics beyond the SM seen at the LHC, $M$ should be rather large implying small deviations. In order to detect possible deviations and to fingerprint the BSM physics from the deviation pattern, we hence need a % level precision, which in turn requires a 500 GeV linear collider such as the International Linear Collider (ILC) and high precision detectors that match the potential of the collider.

The ILC, being an $e^+e^-$ collider, inherits all of its traditional merits: cleanliness, democracy, detail, and calculability. The two detector concepts proposed for the ILC: ILD and SiD (see Fig.40) take advantage of these merits.

<figure><img src="image_43.png" /><figcaption>Fig. 40 Two proposed detector concepts for the ILC: ILD (left) and SiD (right) [138].</figcaption></figure>

Moreover, they are designed with an ambitious goal of reconstructing all the events in terms of fundamental particles such as quarks, leptons, gauge bosons, and Higgs bosons, thereby viewing events as viewing Feynman diagrams. This requires a thin and high resolution vertex detector that enables identification of b- and c-quarks by detecting secondary and tertiary vertices, combination of a high resolution charged particle tracker and high granularity calorimeters optimized for Particle Flow Analysis (PFA) to allow identification of W, Z, t, and H by measuring their jet invariant masses, and hermeticity down to $\mathcal{O}(10 \text{ mrad})$ or better for indirect detection of a neutrino as missing momentum. Notice that both ILD and SiD put all the calorimeters inside the detector solenoidal magnets to satisfy the requirement of hermeticity and high performance PFA. Furthermore, the power of beam polarizations should be emphasized. Consider the $e^+e^- \to W^+W^-$ process. At the energies explored by the ILC, $SU(2)_L \otimes U(1)_Y$ symmetry is approximately recovered and hence the process can be regarded as taking place through two diagrams: s-channel $W_3$ exchange and t-channel $v_e$ exchange. Since both $W_3$ and $v_e$ couple only to a left-handed electron (and
right-handed positrons), right-handed electrons will not contribute to the process. This is also the case for one of the most important Higgs production process at the ILC: $e^{+}e^{-} \to \nu_{e}\bar{\nu}_{e}H$ (WW-fusion single Higgs production). If we have an 80% left-handed electron beam and a 30% right-handed positron beam the Higgs production cross section for this WW-fusion process will be enhanced by a factor of 2.34 as compared to the unpolarized case. Beam polarization hence plays an essential role.

## Why 250 to 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, see Fig. 41.

<figure><img src="image_44.png" /><figcaption>Fig. 41 Why 250-500 GeV? The three thresholds.</figcaption></figure>

The first threshold is at around $\sqrt{s} = 250$ GeV, where the $e^{+}e^{-} \to Zh$ 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 $h \to b\bar{b}$, $c\bar{c}$, $\tau\bar{\tau}$, $gg$, $WW^{*}$, $ZZ^{*}$, $\gamma\gamma$, 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 $m_t(\overline{MS})$ 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 [139]. 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: $\gamma\gamma \to hh$ is possible, which can be used to study the Higgs self-coupling [140]. Notice also that at $\sqrt{s} = 350$ GeV and above, the WW-fusion Higgs production process, $e^{+}e^{-} \to \nu\bar{\nu}h$, becomes sizable 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, $e^{+}e^{-} \to Zhh$ attains

its cross section maximum, which can be used to access the Higgs self-coupling. At $\sqrt{s} = 500$ GeV, another important process, $e^{+}e^{-} \to t\bar{t}h$, 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 $\sqrt{s} = 250$ to 500 GeV, we will hence be able to 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

The first threshold is at around $\sqrt{s} = 250$ GeV, where the $e^{+}e^{-} \to Zh$ (Higgs-strahlung) process attains its cross section maximum (see Fig.42).

<figure><img src="image_45.png" /><figcaption>Fig. 42 Cross sections for the three major Higgs production processes as a function of center of mass energy.</figcaption></figure>

The most important measurement at this energy is that of the recoil mass for the process: $e^{+}e^{-} \to Zh$ followed by $Z \to \ell^{+}\ell^{-}$ ($\ell = e, \mu$) decay. By virtue of the $e^{+}e^{-}$ collider, we know the initial state 4-momentum. We can hence calculate the invariant mass of the system recoiling against the lepton pair from the $Z$ decay by just measuring the momenta of the lepton pair:

$$M_X^2 = (p_{CM} - (p_{\ell^+} + p_{\ell^-}))^2. \qquad (15)$$

The recoil mass distribution is shown in Fig. 43 for a $m_h = 125$ GeV Higgs boson with $250 \text{ fb}^{-1}$ at $\sqrt{s} = 250$ GeV. A very clean Higgs peak is sticking out from small background. Notice that with this recoil mass technique even invisible decay is detectable since we do not need to look at the Higgs
<figure><img src="image_46.png" /><figcaption>**Fig. 43** Recoil mass distribution for the process: $e^{+}e^{-} \to Zh$ followed by $Z \to \mu^{+}\mu^{-}$ decay for $m_{h} = 125$ GeV with $250 \text{ fb}^{-1}$ at $\sqrt{s} = 250$ GeV [141].</figcaption></figure>

decay at all [142]. This way, we can determine the Higgs mass to $\Delta m_h = 30$ MeV and the production cross section to $\Delta\sigma_{Zh}/\sigma_{Zh} = 2.6\%$, and limit the invisible branching ratio to 1% at the 95% confidence level [143, 144]. This is the flagship measurement of the ILC at 250 GeV that allows absolute measurement of the hZZ coupling thereby unlocking the door to completely model-independent determinations of various couplings of the Higgs boson as well as its total width as we will see below.

Before moving on to the coupling determinations, let us discuss here the determination of the spin and CP properties of the Higgs boson. The LHC observed the $h \to \gamma\gamma$ decay, which fact alone rules out the possibility of spin 1 and restricts the charge conjugation C to be positive. The more recent LHC analysis strongly prefer the $J^P = 0^+$ assignment over $0^-$ or $2^\pm$ [145]. By the time of the ILC the discrete choice between different spin and CP-even or odd assignments will certainly be settled, assuming that the 125 GeV boson is a CP eigen state. Nevertheless, it is worth noting that the ILC also offers an additional, orthogonal, and clean test of these assignments. The threshold behavior of the Zh cross section has a characteristic shape for each spin and each possible CP parity. For spin 0, the cross section rises as $\beta$ near the threshold for a CP-even state and as $\beta^3$ for a CP-odd state. For spin 2, for the canonical form of the coupling to the energy-momentum tensor, the rise is also $\beta^3$. If the spin is higher than 2, the cross section will grow as a higher power of $\beta$. With a three-20 fb$^{-1}$-point threshold scan of the $e^{+}e^{-} \to Zh$ production cross section we can separate these possibilities [146] as shown in Fig. 44. The discrimination of more general forms of the coupling is possible by the use

<figure><img src="image_47.png" /><figcaption>**Fig. 44** Threshold scan of the $e^{+}e^{-} \to Zh$ process for $m_{h} = 120$ GeV, compared with theoretical predictions for $J^{P} = 0^{+}, 1^{-}$, and $2^{+}$ [146].</figcaption></figure>

<figure><img src="image_48.png" /><figcaption>**Fig. 45** Determination of CP-mixing with $1-\sigma$ bands expected at $\sqrt{s} = 350$ GeV and $500 \text{ fb}^{-1}$ [148].</figcaption></figure>

of angular correlations in the boson decay; this is discussed in detail in [147].

The power of the ILC manifests itself when we ask more subtle questions. There is no guarantee that the $h$ is a CP eigenstate. It can rather be a mixture of CP-even and CP-odd components. This happens if CP is violated in the Higgs sector. A small CP-odd contribution to the hZZ coupling can affect the threshold behavior. Figure 45 shows the determination of the small CP-odd component $\eta$ at $\sqrt{s}=350$ GeV from the value of the total cross section and from an appropriately defined optimal observable [148]. The hZZ coupling is probably not the best tool to study possible CP admixture, since in many scenarios the CP-odd hZZ coupling is only
generated through loops. It is, hence, more effective to use a coupling for which the CP-even and CP-odd components are on the same footing as in the $h$ coupling to $\tau^+\tau^-$, given by

$$ \Delta\mathcal{L} = -\frac{m_\tau}{v} h \bar{\tau}(\cos\alpha + i\sin\alpha\gamma^5)\tau \quad (16) $$

for a Higgs boson with a CP-odd component. The polarizations of the final state $\tau$s can be determined from the kinematic distributions of their decay products; the CP-even and odd components interfere in these distributions [149, 150]. In [151], it is estimated that the angle $\alpha$ can be determined at the ILC to an accuracy of 6°.

The $e^+e^- \to Zh$ process can also be used to measure various branching ratios for various Higgs decay modes. For this purpose $Z \to q\bar{q}$ and $v\bar{v}$ decays can be included in our analysis to enhance the statistical precision. We should stress here that as with similar Higgs-related measurements at the LHC what we can actually measure is NOT branching ratio (BR) itself but the cross section times branching ratio ($\sigma \times \text{BR}$). The crucial difference is the recoil mass measurement at the ILC that provides $\sigma$ to extract BR from $\sigma \times \text{BR}$ model-independently. Table 8 summarizes the expected precisions for the $\sigma \times \text{BR}$ measurements together with those for the extracted BRs [152–159].

<table><caption>**Table 8** Expected relative errors for the $\sigma \times \text{BR}$ measurements at $\sqrt{s} = 250$ GeV with $250 \text{ fb}^{-1}$ for $m_h = 125$ GeV.</caption><thead><tr><th>process</th><th>decay mode</th><th>Δ(σBR)/(σBR)</th><th>ΔBR/BR</th></tr></thead><tbody><tr><td rowspan="7">Zh</td><td>h → b<span style="text-decoration: overline;">b</span></td><td>1.2%</td><td>2.9%</td></tr><tr><td>h → c<span style="text-decoration: overline;">c</span></td><td>8.3%</td><td>8.7%</td></tr><tr><td>h → gg</td><td>7.0%</td><td>7.5%</td></tr><tr><td>h → WW<sup>*</sup></td><td>6.4%</td><td>6.9%</td></tr><tr><td>h → τ<span style="text-decoration: overline;">τ</span></td><td>4.2%</td><td>4.9%</td></tr><tr><td>h → ZZ<sup>*</sup></td><td>19%</td><td>19%</td></tr><tr><td>h → γγ</td><td>34%</td><td>34%</td></tr></tbody></table>

Notice that the cross section error, $\Delta\sigma_{Zh}/\sigma_{Zh} = 2.5\%$, 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.

In order to extract couplings from branching ratios, we need the total width, since the coupling of the Higgs boson to a particle $A$, $g_{hAA}$, squared is proportional to the partial width which is given by the total width times the branching ratio:

$$ g_{hAA}^2 \propto \Gamma(h \to AA) = \Gamma_h \cdot \text{BR}(h \to AA). \quad (17) $$

Solving this for the total width, we can see that we need at least one partial width and corresponding branching ratio to determine the total width:

$$ \Gamma_h = \Gamma(h \to AA)/\text{BR}(h \to AA). \quad (18) $$

In principle, we can use $A = Z$ or $A = W$, for which we can measure both the BRs and the couplings. In the first case, $A = Z$, we can determine $\Gamma(h \to ZZ^*)$ from the recoil mass measurement and $\text{BR}(h \to ZZ^*)$ from the $\sigma_{Zh} \times \text{BR}(h \to ZZ^*)$ measurement together with the $\sigma_{Zh}$ measurement from the recoil mass. This method, however, suffers from the low statistics due to the small branching ratio, $\text{BR}(h \to ZZ^*) = \mathcal{O}(1\%)$, A better way is to use $A = W$, where $\text{BR}(h \to WW^*)$ is subdominant and $\Gamma(h \to WW^*)$ can be determined by the WW-fusion process: $e^+e^- \to v\bar{v}h$. The measurement of the WW-fusion process is, however, not easy at $\sqrt{s} = 250$ GeV since the cross section is small. Nevertheless, we can determine the total width to $\Delta\Gamma_h/\Gamma_h = 11\%$ with $250 \text{ fb}^{-1}$ [160, 161]. Since the WW-fusion process becomes fully active at $\sqrt{s} = 500$ GeV, a much better measurement of the total width is possible there, as will be discussed in the next subsection.

### 2.3.3 ILC at 500 GeV

At $\sqrt{s} = 500$ GeV, the WW-fusion process $e^+e^- \to v\bar{v}h$ already starts dominating the higgsstrahlung process: $e^+e^- \to Zh$. We can use this WW-fusion process for the $\sigma \times \text{BR}$ measurements as well as to determine the total width to $\Delta\Gamma_h/\Gamma_h = 5\%$ [161]. Table 9 summarizes the $\sigma \times \text{BR}$ measurements for various modes. We can see that the $\sigma_{v\bar{v}h} \times \text{BR}(h \to b\bar{b})$

<table><caption>**Table 9** Expected relative errors for the $\sigma \times \text{BR}$ measurements at $\sqrt{s} = 250$ GeV with $250 \text{ fb}^{-1}$ and at $\sqrt{s} = 500$ GeV with $500 \text{ fb}^{-1}$ for $m_h = 125$ GeV and $(e^-, e^+) = (-0.8, +0.3)$ beam polarization. The last column of the table shows the relative errors on branching ratios. Then numbers in the parentheses are as of $250 \text{ fb}^{-1}$ at $\sqrt{s} = 250$ GeV alone.</caption><thead><tr><th></th><th colspan="2">Δ(σ·BR)/(σ·BR)</th><th>ΔBR/BR</th></tr></thead><tbody><tr><td>energy (GeV)</td><td>250</td><td>500</td><td>250+500</td></tr><tr><td>mode</td><td>Zh</td><td>Zh</td><td>v<span style="text-decoration: overline;">v</span>h</td><td>combined</td></tr><tr><td>h → b<span style="text-decoration: overline;">b</span></td><td>1.2%</td><td>1.8%</td><td>0.66%</td><td>2.2 (2.9)%</td></tr><tr><td>h → c<span style="text-decoration: overline;">c</span></td><td>8.3%</td><td>13%</td><td>6.2%</td><td>5.1 (8.7)%</td></tr><tr><td>h → gg</td><td>7.0%</td><td>11%</td><td>4.1%</td><td>4.0 (7.5)%</td></tr><tr><td>h → WW<sup>*</sup></td><td>6.4%</td><td>9.2%</td><td>2.4%</td><td>3.1 (6.9)%</td></tr><tr><td>h → τ<sup>+</sup>τ<sup>-</sup></td><td>4.2%</td><td>5.4%</td><td>9.0%</td><td>3.7 (4.9)%</td></tr><tr><td>h → ZZ<sup>*</sup></td><td>19%</td><td>25%</td><td>8.2%</td><td>7.5 (19)%</td></tr><tr><td>h → γγ</td><td>29-38%</td><td>29-38%</td><td>20-26%</td><td>17(34)%</td></tr></tbody></table>

can be very accurately measured to better than 1% and the $\sigma_{v\bar{v}h} \times \text{BR}(h \to WW^*)$ to a reasonable precision with $500 \text{ fb}^{-1}$ at $\sqrt{s} = 500$ GeV. The last column of the table shows the results of $\Delta\text{BR}/\text{BR}$ from the global analysis combining all the
measurements including the total cross section measurement using the recoil mass at $\sqrt{s} = 250$ GeV (2.6%) and 500 GeV (3%). The numbers in the parentheses are with the 250 GeV data alone. We can see that the $\Delta\text{BR}(h \to b\bar{b})/\text{BR}(h \to b\bar{b})$ is already limited by the recoil mass measurements.

Perhaps more interesting than the branching ratio measurements is the measurement of the top Yukawa coupling using the $e^+e^- \to t\bar{t}h$ process [162–164], since it is the largest among matter fermions and not yet directly observed. Although its cross section maximum is reached at around $\sqrt{s} = 800$ GeV as seen in Fig.46, the process is accessible already at $\sqrt{s} = 500$ GeV, thanks to the QCD bound-state effects (non-relativistic QCD correction) that enhance the cross section by a factor of two [163, 165–170]. Since the background diagram where a Higgs boson is

radiated off the s-channel Z boson makes negligible contribution to the signal process, we can measure the top Yukawa coupling by simply counting the number of signal events. The expected statistical precision for the top Yukawa coupling is then $\Delta g_Y(t)/g_Y(t) = 9.9\%$ for $m_h = 125$ GeV with $1\text{ab}^{-1}$ at $\sqrt{s} = 500$ GeV [38, 171–175]. Notice that if we increase the center of mass energy by 20 GeV, the cross section doubles. Moving up a little bit hence helps significantly.

Even more interesting is the measurement of the trilinear Higgs self-coupling, since it is to observe the force that makes the Higgs boson condense in the vacuum, which is an unavoidable step to uncover the secret of the EW symmetry breaking. In other words, we need to measure the shape of the Higgs potential. There are two ways to measure the trilinear Higgs self-coupling. The first method is to use the double higgsstrahlung process: $e^+e^- \to Zhh$ and the second is by the double Higgs production via WW-fusion: $e^+e^- \to \nu\bar{\nu}hh$. The first process attains its cross section maximum at around $\sqrt{s} = 500$ GeV, while the second is negligible there but starts to dominate at energies above $\sqrt{s} \approx 1.2$ TeV, as seen in Fig.47. In any case the signal cross sections are very

<figure><img src="image_49.png" /><figcaption>**Fig. 46** Cross sections for the signal $t\bar{t}h$ process with and without the non-relativistic QCD (NRQCD) correction together with those for the background processes: $t\bar{t}Z$, $t\bar{t}g(g \to b\bar{b})$ and $t\bar{t}$ (upper plot). The invariant mass distribution for the $t\bar{t}$ subsystem with and without the NRQCD correction (lower plot).</figcaption></figure>

<figure><img src="image_50.png" /><figcaption>**Fig. 47** Cross sections for the double Higgs production processes, $e^+e^- \to Zhh$ and $e^+e^- \to \nu\bar{\nu}hh$, as a function of $\sqrt{s}$ for $m_h = 120$ GeV.</figcaption></figure>

small (0.2 fb or less) and as seen in Fig.48 irreducible background diagrams containing no self-coupling dilute the contribution from the self-coupling diagram, thereby degrading the sensitivity to the self-coupling, even if we can control the relatively huge SM backgrounds from $e^+e^- \to t\bar{t}$, WWZ, ZZ, $Z\gamma$, ZZZ, and $ZZh$. See Fig.49 for the sensitivity factors for $e^+e^- \to Zhh$ at $\sqrt{s} = 500$ GeV and $e^+e^- \to \nu\bar{\nu}hh$ at $\sqrt{s} = 1$ TeV, which are 1.66 (1.80) and 0.76 (0.85), respectively, with (without) weighting to enhance the contribution from the signal diagram. Notice that if there were no background diagrams, the sensitivity factor would be 0.5. The self-coupling measurement is very difficult even in the clean environment of the ILC and requires a new flavor tagging algorithm that precedes jet-clustering, sophisticated neural-
<figure><img src="image_51.png" /><figcaption>**Fig. 48** Diagrams contributing to (a) $e^+e^- \to Zhh$ and (b) $e^+e^- \to v\bar{v}hh$.</figcaption></figure>

<figure><img src="image_52.png" /><figcaption>**Fig. 49** (Upper plot) cross section for $e^+e^- \to Zhh$ at $\sqrt{s} = 500$ GeV normalized by that of the SM as a function of the self-coupling normalized by that of the SM. (Lower plot) a similar plot for $e^+e^- \to v\bar{v}hh$ at $\sqrt{s} = 1$ TeV.</figcaption></figure>

net-based data selection, and the event weighting technique [72, 176–180]. The current state of the art for the Zhh data selection is summarized in Table 10.

Combining all of these three modes, we can achieve Zhh excess significance of $5\sigma$ and measure the production cross section to $\Delta\sigma/\sigma = 27\%$, which translates to a relative precision of 44(48)% for the self-coupling with (without) the event weighting for $m_h = 120$ GeV at $\sqrt{s} = 500$ GeV with $2\text{ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.3)$ beam polarization [176]. The expected precision is significantly worse than that of

<table><caption>**Table 10** The number of remaining events for the three event selection modes: $Zhh \to (\ell\bar{\ell})(b\bar{b})(b\bar{b})$, $(v\bar{v})(b\bar{b})(b\bar{b})$, 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 $2\text{ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.3)$ beam polarization.</caption><thead><tr><th rowspan="2">mode</th><th rowspan="2">signal</th><th rowspan="2">BG</th><th colspan="2">significance</th></tr><tr><th>excess</th><th>meas.</th></tr></thead><tbody><tr><td rowspan="2">Zhh → (ℓℓ)(b̄b)(b̄b)</td><td>3.7</td><td>4.3</td><td>1.5σ</td><td>1.1σ</td></tr><tr><td>4.5</td><td>6.0</td><td>1.5σ</td><td>1.2σ</td></tr><tr><td>Zhh → (vV̄)(b̄b)(b̄b)</td><td>8.5</td><td>7.9</td><td>2.5σ</td><td>2.1σ</td></tr><tr><td rowspan="2">Zhh → (q̄q)(b̄b)(b̄b)</td><td>13.6</td><td>30.7</td><td>2.2σ</td><td>2.0σ</td></tr><tr><td>18.8</td><td>90.6</td><td>1.9σ</td><td>1.8σ</td></tr></tbody></table>

the cross section because of the background diagrams. Since the sensitivity factor for the $e^+e^- \to v\bar{v}hh$ process is much closer to the ideal 0.5 and since the cross section for this WW-fusion double Higgs production process increases with the center of mass energy, $\sqrt{s} = 1$ TeV is of particular interest, as will be discussed in the next subsection.

### 2.3.4 ILC at 1000 GeV

As we already pointed out the WW-fusion processes become more and more important at higher energies. In addition the machine luminosity usually scales with the center of mass energy. Together with the better sensitivity factor we can hence improve the self-coupling measurement significantly at $\sqrt{s} = 1$ TeV, using the $e^+e^- \to v\bar{v}hh$ process. Table 11 summarizes a full simulation result for the numbers of expected signal and background events before and after selection cuts with corresponding measurement significance values.

<table><caption>**Table 11** 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 $2\text{ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.2)$ beam polarization.</caption><thead><tr><th>mode</th><th>no cut</th><th>after cuts</th></tr></thead><tbody><tr><td>vV̄hh (WW-fusion)</td><td>272</td><td>35.7</td></tr><tr><td>vV̄hh (Zhh)</td><td>74.0</td><td>3.88</td></tr><tr><td>BG (t̄/vV̄Zh)</td><td>7.86 × 10<sup>5</sup></td><td>33.7</td></tr><tr><td>meas. significance</td><td>0.30</td><td>4.29</td></tr></tbody></table>

With $2\text{ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.2)$ beam polarization at $\sqrt{s} = 1$ TeV, we would be able to determine the cross section for the $e^+e^- \to v\bar{v}hh$ process to $\Delta\sigma/\sigma = 23\%$, corresponding to the self-coupling precision of $\Delta\lambda/\lambda = 18(20)\%$ with (without) the event weighting to enhance the contribution from the signal diagram for $m_h = 120$ GeV [176].
According to preliminary results from a on-going full simulation study [181], adding $hh \to WW^*b\bar{b}$ would improve the self-coupling measurement precision by about 20% relatively, which means $\Delta\lambda/\lambda = 21\%$ for $m_h = 125$ GeV with the baseline integrated luminosity of $1\text{ ab}^{-1}$ at 1 TeV.

At $\sqrt{s} = 1$ TeV, the $e^+e^- \to t\bar{t}h$ 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 $e^+e^- \to t\bar{t}h$ 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 $\Delta g_Y(t)/g_Y(t) = 3.1\%$ with $1\text{ ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.2)$ beam polarization at $\sqrt{s} = 1$ TeV for $m_h = 125$ GeV [182].

However, obvious but most important advantage of higher energies in terms of Higgs physics is its higher mass reach to extra Higgs bosons expected in extended Higgs sectors and higher sensitivity to $W_LW_L$ scattering to decide whether the Higgs sector is strongly interacting or not. In any case thanks to the higher cross section for the WW-fusion $e^+e^- \to v\bar{v}h$ process at $\sqrt{s} = 1$ TeV, we can expect significantly better precisions for the $\sigma \times \text{BR}$ measurements (see Table 12), which also allows us to access very rare decays such as $h \to \mu^+\mu^-$ [183, 184].

<table><caption>**Table 12** Independent Higgs measurements using the higgsstrahlung (Zh) and the WW-fusion ($\bar{v}\bar{v}h$) processes for $m_h = 125$ GeV at three energies: $\sqrt{s} = 250$ GeV with $250\text{ fb}^{-1}$, $500$ GeV with $500\text{ fb}^{-1}$ both with $(e^-, e^+) = (-0.8, +0.3)$ beam polarization, $\sqrt{s} = 1$ TeV with $1\text{ ab}^{-1}$ and $(e^-, e^+) = (-0.8, +0.2)$ beam polarization.</caption><thead><tr><th>√s</th><th>250 GeV</th><th>500 GeV</th><th>1 TeV</th></tr></thead><tbody><tr><td>lumi.</td><td>250 fb<sup>-1</sup></td><td>500 fb<sup>-1</sup></td><td>1 ab<sup>-1</sup></td></tr><tr><td>process</td><td>Zh v̄V̄h Zh v̄V̄h v̄V̄h</td><td>Δσ/σ</td><td></td></tr><tr><td></td><td>2.6% - 3.0%</td><td>-</td><td>-</td></tr><tr><td>mode</td><td colspan="3">Δ(σ·BR)/(σ·BR)</td></tr><tr><td>h → b̄b</td><td>1.2%</td><td>10.5%</td><td>1.8%</td><td>0.66%</td><td>0.5%</td></tr><tr><td>h → c̄c</td><td>8.3%</td><td></td><td>13%</td><td>6.2%</td><td>3.1%</td></tr><tr><td>h → gg</td><td>7.0%</td><td></td><td>11%</td><td>4.1%</td><td>2.3%</td></tr><tr><td>h → WW<sup>*</sup></td><td>6.4%</td><td></td><td>9.2%</td><td>2.4%</td><td>1.6%</td></tr><tr><td>h → τ<sup>+</sup>τ<sup>-</sup></td><td>4.2%</td><td></td><td>5.4%</td><td>9.0%</td><td>3.1%</td></tr><tr><td>h → ZZ<sup>*</sup></td><td>18%</td><td></td><td>25%</td><td>8.2%</td><td>4.1%</td></tr><tr><td>h → γγ</td><td>34%</td><td></td><td>34%</td><td>23%</td><td>8.5%</td></tr><tr><td>h → μ<sup>+</sup>μ<sup>-</sup></td><td>100%</td><td>-</td><td>-</td><td>-</td><td>31%</td></tr></tbody></table>

### 2.3.5 ILC 250+500+1000: global fit for couplings

The data at $\sqrt{s} = 250, 500$, and $1000$ GeV can be combined to perform a global fit to extract various Higgs couplings [185]. We have $33\sigma \times \text{BR}$ measurements: 31 shown in Table 12 plus two $\sigma(t\bar{t}h) \times \text{BR}(h \to b\bar{b})$ measurements at $\sqrt{s} =$

500 and $1000$ GeV. The key is the recoil mass measurement that unlocks the door to a fully model-independent analysis. Notice that such a fully model-independent analysis is impossible at the LHC. As shown in Table 12, we can measure the recoil mass cross section at $\sqrt{s} = 250$ and $500$ GeV. Altogether we have 35 independent measurements: $33\sigma \times \text{BR}$ measurements ($Y_i : i = 1 \cdots 33$) and $2\sigma(Zh)$ measurements ($Y_{34,35}$). We can then define a $\chi^2$ function:

$$ \chi^2 = \sum_{i=1}^{35} \left( \frac{Y_i - Y'_i}{\Delta Y_i} \right)^2 \qquad (19) $$

where

$$ Y'_i := F_i \cdot \frac{g_{hA_iA_i}^2 g_{hB_iB_i}^2}{\Gamma_0} \quad (i = 1, \dots, 33) \qquad (20) $$

with $A_i$ being $Z, W$, or $t$, and $B_i$ being $b, c, \tau, \mu, g, \gamma, Z$, and $W, \Gamma_0$ denoting the total width and

$$ F_i = S_i G_i \qquad (21) $$

with

$$ S_i = \left( \frac{\sigma_{Zh}}{g_{hZZ}^2} \right), \quad \left( \frac{\sigma_{v\bar{v}h}}{g_{hWW}^2} \right), \quad \text{or} \quad \left( \frac{\sigma_{t\bar{t}h}}{g_{htt}^2} \right) \\ G_i = \left( \frac{\Gamma_i}{g_i^2} \right). \qquad (22) $$

Cross section calculations ($S_i$) do not involve QCD ISR unlike with the LHC. Partial width calculations ($G_i$), being normalized by the coupling squared, do not need quark mass as input. We are hence confident that the goal theory errors for $S_i$ and $G_i$ will be at the 0.1% level at the time of ILC running. The free parameters are 9 coupling constants: $g_{hbb}, g_{gcc}, g_{h\tau\tau}, g_{h\mu\mu}, g_{hgg}, g_{h\gamma\gamma}, g_{hZZ}, g_{hWW}$, and 1 total width: $\Gamma_0$. Table 13 summarizes the expected coupling precisions for $m_h = 125$ GeV with the baseline integrated luminosities of $250\text{ fb}^{-1}$ at $\sqrt{s} = 250$ GeV, $500\text{ fb}^{-1}$ at $500$ GeV both with $(e^-e^+) = (-0.8, +0.3)$ beam polarization, and $1\text{ ab}^{-1}$ at 1 TeV with $(e^-e^+) = (-0.8, +0.2)$ beam polarization. The expected coupling precisions are plotted in the mass-coupling plot expected for the SM Higgs sector in Fig.50. The error bars for most couplings are almost invisible in this logarithmic plot.

### 2.3.6 Synergy: LHC + ILC

So far we have been discussing the precision Higgs physics expected at the ILC. It should be emphasized, however, that the LHC is expected to impose significant constraints on possible deviations of the Higgs-related couplings from their SM values by the time the ILC will start its operation, even though fully model-independent analysis is impossible with the LHC alone. Nevertheless, Refs. [186, 187] demonstrated that with a reasonably weak assumption such as the $hWW$ and $hZZ$ couplings will not exceed the SM values the LHC
<table><caption>**Table 13** 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 $250 \text{ fb}^{-1}$, $500$ GeV with $500 \text{ fb}^{-1}$ both with $(e^{-}, e^{+}) = (-0.8, +0.3)$ beam polarization, $\sqrt{s} = 1$ TeV with $2 \text{ ab}^{-1}$ and $(e^{-}, e^{+}) = (-0.8, +0.2)$ beam polarization, cf. [25] and Scen. ‘Snow’ in [23]. Values with (*) assume inclusion of $hh \to WW^{*}b\bar{b}$ decays.</caption><thead><tr><th rowspan="2">coupling</th><th colspan="3">√s (GeV)</th></tr><tr><th>250</th><th>250+500</th><th>250 + 500 + 1000</th></tr></thead><tbody><tr><td>hZZ</td><td>1.3%</td><td>1.0%</td><td>1.0%</td></tr><tr><td>hWW</td><td>4.8%</td><td>1.1%</td><td>1.1%</td></tr><tr><td>hbb</td><td>5.3%</td><td>1.6%</td><td>1.3%</td></tr><tr><td>hcc</td><td>6.8%</td><td>2.8%</td><td>1.8%</td></tr><tr><td>hgg</td><td>6.4%</td><td>2.3%</td><td>1.6%</td></tr><tr><td>hττ</td><td>5.7%</td><td>2.3%</td><td>1.6%</td></tr><tr><td>hγγ</td><td>18%</td><td>8.4%</td><td>4.0%</td></tr><tr><td>hμμ</td><td>91%</td><td>91%</td><td>16%</td></tr><tr><td>Γ<sub>0</sub></td><td>12%</td><td>4.9%</td><td>4.5%</td></tr><tr><td>htt</td><td>-</td><td>14%</td><td>3.1%</td></tr><tr><td>hhh</td><td>-</td><td>83%(*)</td><td>21%(*)</td></tr></tbody></table>

<figure><img src="image_53.png" /><figcaption>**Fig. 50** Expected mass-coupling relation for the SM case after the full ILC program.</figcaption></figure>

can make reasonable measurements of most Higgs-related coupling constants except for the *hcc* coupling. Figure 51 shows how the coupling measurements would be improved by adding, cumulatively, information from the ILC with $250 \text{ fb}^{-1}$ at $\sqrt{s} = 250$, $500 \text{ fb}^{-1}$ at $500$ GeV, and $1 \text{ ab}^{-1}$ at $1$ TeV to the LHC data with $300 \text{ fb}^{-1}$ at $14$ TeV.

The figure tells us that the addition of the $250$ GeV data, the $hZZ$ coupling in particular, from the ILC allows the absolute normalization and significantly improves all the couplings. It is interesting to observe the synergy for the measurement of the $h\gamma\gamma$ coupling, whose precision significantly

<figure><img src="image_54.png" /><figcaption>**Fig. 51** Comparison of the capabilities of the LHC and the ILC, when the ILC data in various stages: ILC1 with $250 \text{ fb}^{-1}$ at $\sqrt{s} = 250$, ILC: $500 \text{ fb}^{-1}$ at $500$ GeV, and ILCTeV: $1 \text{ ab}^{-1}$ at $1$ TeV are cumulatively added to the LHC data with $300 \text{ fb}^{-1}$ at $14$ TeV [187].</figcaption></figure>

exceeds that of the ILC alone. This is because the LHC can precisely determine the ratio of the $h\gamma\gamma$ coupling to the $hZZ$ coupling, while the ILC provides a precision measurement of the $hZZ$ coupling from the recoil mass measurement. The addition of the $500$ GeV data from the ILC further improves the precisions, this time largely due to the better determination of the Higgs total width. Finally as we have seen above, the addition of the $1$ TeV data from the ILC improves the top Yukawa coupling drastically with even further improvements of all the other couplings except for the $hWW$ and $hZZ$ couplings which are largely limited by the cross section error from the recoil mass measurement at $\sqrt{s} = 250$ GeV. This way we will be able to determine these couplings to $\mathcal{O}(1\%)$ or better. The SFitter group performed a similar but more model-independent analysis and obtained qualitatively the same conclusions [188]. This level of precision matches what we need to fingerprint different BSM scenarios, when nothing but the $125$ GeV boson would be found at the LHC (see Table 14). These numbers can be understood from the

<table><caption>**Table 14** Maximum possible deviations when nothing but the $125$ GeV boson would be found at the LHC [189].</caption><thead><tr><th></th><th>ΔhVV</th><th>Δh<span style="text-decoration: overline;">t</span>t</th><th>Δh<span style="text-decoration: overline;">b</span>b</th></tr></thead><tbody><tr><td>Mixed-in Singlet</td><td>6%</td><td>6%</td><td>6%</td></tr><tr><td>Composite Higgs</td><td>8%</td><td>10%</td><td>10%</td></tr><tr><td>Minimal SUSY</td><td>&lt; 1%</td><td>3%</td><td>10%<sup>a</sup>, 100%<sup>b</sup></td></tr><tr><td>LHC 14 TeV, 3 ab<sup>-1</sup></td><td>8%</td><td>10%</td><td>15%</td></tr></tbody></table>

following formulas for the three different models in the decoupling limit (see [138] for definitions and details),

Mixing with singlet:

$$ \frac{g_{hVV}}{g_{hSMVV}} = \frac{g_{hff}}{g_{hSMff}} = \cos \theta \approx 1 - \frac{\delta^2}{2} $$
Composite Higgs:

$$ \frac{g_{hVV}}{g_{hSMVV}} \approx 1 - 3\% \left( \frac{1 \text{ TeV}}{f} \right)^2 $$ $$ \frac{g_{hff}}{g_{hSMff}} \approx \begin{cases} 1 - 3\% \left( \frac{1 \text{ TeV}}{f} \right)^2 & (\text{MCHM4}) \\ 1 - 9\% \left( \frac{1 \text{ TeV}}{f} \right)^2 & (\text{MCHM5}). \end{cases} $$

Supersymmetry:

$$ \frac{g_{hbb}}{g_{hSMbb}} = \frac{g_{h\tau\tau}}{g_{hSM\tau\tau}} \approx 1 + 1.7\% \left( \frac{1 \text{ TeV}}{m_A} \right)^2 . $$

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

As mentioned above, the LHC needs some model assumption to extract Higgs couplings. If we use stronger model assumptions we may have higher discrimination power at the cost of loss of generality. As an example of such a model-dependent analysis, let us consider here a 7-parameter global fit with the following assumptions:

$$ \kappa_c = \kappa_l =: \kappa_{\mu}, $$

$$ \kappa_s = \kappa_b =: \kappa_d, $$

$$ \kappa_{\mu} = \kappa_{\tau} =: \kappa_{\ell}, $$

and

$$ \Gamma_{\text{tot}} = \sum_{i \in \text{SM} \text{ decays}} \Gamma_i^{\text{SM}} \kappa_i^2, \qquad (23) $$

where $\kappa_i := g_i/g_i(\text{SM})$ is a Higgs coupling normalized by its SM value. The first three of these constrain the relative deviations of the up-type and down-type quark Yukawa couplings as well as that of charged leptons to be common in each class, while the last constraint restricts unknown decay modes to be absent. The results of the global fits assuming projected precisions for the LHC and the ILC are summarized in Table 15 [185]. Figures 52 and 53 compare the model discrimination power of the LHC and the ILC in the $\kappa_{\ell}-\kappa_d$ and $\kappa_{\ell}(\kappa_d)-\kappa_{\mu}$ planes for the four types of two-Higgsdoublet model discussed in 2.3.1 [134, 190]. Figure 54 is a similar plot in the $\kappa_V-\kappa_F$ plane showing the discrimination power for four models: doublet-singlet model, 2HDM-I, Georgi-Machacek model, and doublet-septet model, all of which naturally realize $\rho = 1$ at the tree level [134, 190].

### 2.3.8 High Luminosity ILC?

We have seen the crucial role played by the recoil mass measurement for the model-independent coupling extraction. We have also pointed out that because of this the recoil mass measurement would eventually limit the coupling

<table><caption>**Table 15** Expected Higgs precisions on normalized Higgs couplings ($\kappa_i := g_i/g_i(\text{SM})$) for $m_h = 125$ GeV from model-dependent 7-parameter fits for the LHC and the ILC, where $\kappa_c = \kappa_l =: \kappa_{\mu}$, $\kappa_s = \kappa_b =: \kappa_d$, $\kappa_{\mu} = \kappa_{\tau} =: \kappa_{\ell}$, and $\Gamma_{\text{tot}} = \sum_i \Gamma_i^{\text{SM}} \kappa_i^2$ are assumed.</caption><thead><tr><th>Facility</th><th>LHC</th><th>HL-LHC</th><th>ILC500</th><th>ILC1000</th></tr></thead><tbody><tr><td>√s(GeV)</td><td>1,400</td><td>14,000</td><td>250/500</td><td>250/500/1000</td></tr><tr><td>∫L dt (fb<sup>-1</sup>)</td><td>300/exp</td><td>3000/exp</td><td>250+500</td><td>250+500+1000</td></tr></tbody></table><table><tbody><tr><td>κ<sub>Y</sub></td><td>5-7%</td><td>2-5%</td><td>8.3%</td><td>3.8%</td></tr><tr><td>κ<sub>g</sub></td><td>6-8%</td><td>3-5%</td><td>2.0%</td><td>1.1%</td></tr><tr><td>κ<sub>W</sub></td><td>4-6%</td><td>2-5%</td><td>0.39%</td><td>0.21%</td></tr><tr><td>κ<sub>Z</sub></td><td>4-6%</td><td>2-4%</td><td>0.49%</td><td>0.50%</td></tr><tr><td>κ<sub>ℓ</sub></td><td>6-8%</td><td>2-5%</td><td>1.9%</td><td>1.3%</td></tr><tr><td>κ<sub>d</sub></td><td>10-13%</td><td>4-7%</td><td>0.93%</td><td>0.51%</td></tr><tr><td>κ<sub>μ</sub></td><td>14-15%</td><td>7-10%</td><td>2.5%</td><td>1.3%</td></tr></tbody></table>

<figure><img src="image_55.png" /><figcaption>**Fig. 52** Comparison of the model-discrimination capabilities of the LHC and the ILC [190].</figcaption></figure>

precisions achievable with the ILC. Given the situation, let us now consider the possibility of luminosity upgrade. As a matter of fact, the ILC technical design report (TDR) [191] describes some possible luminosity and energy upgrade scenarios, which are sketched in Fig.55 as blue boxes. 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 [134] (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 [192], 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 four luminosity upgrade at $\sqrt{s} = 250$ GeV. Let us now assume that after the baseline program at $\sqrt{s} = 250, 500$, and 1000 GeV we will run at the
<figure><img src="image_56.png" /><figcaption>**Fig. 53** Comparison of the model-discrimination capabilities of the LHC and the ILC [190].</figcaption></figure>

<figure><img src="image_57.png" /><figcaption>**Fig. 54** Comparison of the model-discrimination capabilities of the LHC and the ILC [190].</figcaption></figure>

same three energies with the luminosity upgrade, thereby achieving $1150 \text{ fb}^{-1}$ at 250 GeV, $1600 \text{ fb}^{-1}$ at 500 GeV, and $2500 \text{ fb}^{-1}$ at 1000 GeV.

The expected precisions for the independent Higgs-related measurements are summarized in Table 16 for the full data after the luminosity upgraded running. Corresponding expected precisions for various Higgs couplings are tabulated in Table 17. The table shows that with the luminosity upgrade we can achieve sub-% level precisions for most of the Higgs couplings even with the completely model-independent analysis.

<figure><img src="image_58.png" /><figcaption>**Fig. 55** Possible machine upgrade scenarios for the ILC [134, 191].</figcaption></figure>

<table><caption>**Table 16** Similar table to Table 12 but with the luminosity upgrade described in the text: $1150 \text{ fb}^{-1}$ at 250 GeV, $1600 \text{ fb}^{-1}$ at 500 GeV, and $2500 \text{ fb}^{-1}$ at 1 TeV.</caption><thead><tr><th>√s</th><th colspan="2">250 GeV</th><th colspan="2">500 GeV</th><th>1 TeV</th></tr></thead><tbody><tr><td>lumi.</td><td colspan="2">1150 fb<sup>-1</sup></td><td colspan="2">1600 fb<sup>-1</sup></td><td>2.5 ab<sup>-1</sup></td></tr><tr><td>process</td><td>Zh</td><td>v̄h</td><td>Zh</td><td>v̄h</td><td>v̄h</td></tr><tr><td></td><td colspan="4">Δσ/σ</td></tr><tr><td></td><td>1.2%</td><td>-</td><td>1.7%</td><td>-</td><td>-</td></tr><tr><td>mode</td><td colspan="5">Δ(σ·BR)/(σ·BR)</td></tr><tr><td>h → b̄b</td><td>0.56%</td><td>4.9%</td><td>1.0%</td><td>0.37%</td><td>0.3%</td></tr><tr><td>h → c̄c</td><td>3.9%</td><td></td><td>7.2%</td><td>3.5%</td><td>2.0%</td></tr><tr><td>h → gg</td><td>3.3%</td><td></td><td>6.0%</td><td>2.3%</td><td>1.4%</td></tr><tr><td>h → WW*</td><td>3.0%</td><td></td><td>5.1%</td><td>1.3%</td><td>1.0%</td></tr><tr><td>h → τ<sup>+</sup>τ<sup>-</sup></td><td>2.0%</td><td></td><td>3.0%</td><td>5.0%</td><td>2.0%</td></tr><tr><td>h → ZZ*</td><td>8.4%</td><td></td><td>14%</td><td>4.6%</td><td>2.6%</td></tr><tr><td>h → γγ</td><td>16%</td><td></td><td>19%</td><td>13%</td><td>5.4%</td></tr><tr><td>h → μ<sup>+</sup>μ<sup>-</sup></td><td>46.6%</td><td>-</td><td>-</td><td>-</td><td>20%</td></tr></tbody></table>

### 2.3.9 Conclusions

The primary goal for the next decades is to uncover the secret of the electroweak symmetry breaking. This will open up a window to BSM and set the energy scale for the energy frontier machine that will follow the LHC and the ILC 500. Probably the LHC will hit systematic limits at $O(5-10\%)$ for most of $\sigma \times \text{BR}$ measurements, being insufficient to see the BSM effects if we are in the decoupling regime. The recoil mass measurements at the ILC unlocks the door to a fully model-independent analysis. To achieve the primary goal we hence need a 500 GeV linear collider for self-contained precision Higgs studies to complete the mass-coupling plot, where we start from $e^{+}e^{-} \to Zh$ at $\sqrt{s} = 250 \text{ GeV}$, then $t\bar{t}$ at around 350 GeV, and then $Zhh$ and $t\bar{t}h$ at 500 GeV. The ILC to cover up to $\sqrt{s} = 500 \text{ GeV}$ is an ideal machine to carry out this mission (regardless of BSM scenarios) and we can do this completely model-independent with staging starting from $\sqrt{s} \approx 250 \text{ GeV}$. We may need more data at this energy depending on the size of the deviation, since the recoil mass measurement eventually limits the coupling
<table><caption>**Table 17** Similar table to Table 13 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, cf. [25] and Scen. 'Snow' in [23]. Values with (*) assume inclusion of hh → WW* b̄b decays.</caption><thead><tr><th rowspan="2">coupling</th><th colspan="3">√s (GeV)</th></tr><tr><th>250</th><th>250+500</th><th>250 + 500 + 1000</th></tr></thead><tbody><tr><td>hZZ</td><td>0.6%</td><td>0.5%</td><td>0.5%</td></tr><tr><td>hWW</td><td>2.3%</td><td>0.6%</td><td>0.6%</td></tr><tr><td>hbb</td><td>2.5%</td><td>0.8%</td><td>0.7%</td></tr><tr><td>hcc</td><td>3.2%</td><td>1.5%</td><td>1.0%</td></tr><tr><td>hgg</td><td>3.0%</td><td>1.2%</td><td>0.93%</td></tr><tr><td>hττ</td><td>2.7%</td><td>1.2%</td><td>0.9%</td></tr><tr><td>hγγ</td><td>8.2%</td><td>4.5%</td><td>2.4%</td></tr><tr><td>hμμ</td><td>42%</td><td>42%</td><td>10%</td></tr><tr><td>Γ₀</td><td>5.4%</td><td>2.5%</td><td>2.3%</td></tr><tr><td>htt</td><td>-</td><td>7.8%</td><td>1.9%</td></tr><tr><td>hhh</td><td>-</td><td>46%(*)</td><td>13%(*)</td></tr></tbody></table>

precisions. Luminosity upgrade possibility should be always kept in our scope. If we are lucky, some extra Higgs boson or some other new particle might be within reach already at the ILC 500. Let's hope that the upgraded LHC will make another great discovery in the next run from 2015. If not, we will most probably need the energy scale information from the precision Higgs studies. Guided by the energy scale information, we will go hunt direct BSM signals, if necessary, with a new machine. Eventually we will need to measure $W_L W_L$ scattering to decide if the Higgs sector is strongly interacting or not.

## 2.4 Higgs at CLIC: prospects<fnref n="10" />

### 2.4.1 Introduction

The CLIC accelerator [193] offers the possibility to study $e^+e^-$ collisions at center-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 [194], the pulse length is significantly shorter (150 ns) with an bunch spacing of just 0.5 ns and a repetition rate of 50 Hz.

The detectors used for the CLIC physics and detector studies [9] are based on the SiD [195, 196] and ILD [196, 197] detectors proposed for the ILC. They have been adapted for the more challenging environment of running at $\sqrt{s} =$

3 TeV. The most significant changes for both CLIC_SID and CLIC_ILD (see Fig. 56) is the use of tungsten in the hadronic calorimeter and an increase of the depth of hadronic calorimeter to 7.5 $A_{int}$.

<figure><img src="image_59.png" /><figcaption>**Fig. 56** Longitudinal cross section of the top quadrant of CLIC_SiD (left) and CLIC_ILD (right) [9].</figcaption></figure>

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].

An entire bunch train at CLIC roughly deposits around 20 TeV in the detector, which is predominantly coming from $\gamma\gamma \rightarrow$ hadrons events. By applying tight cuts on the reconstructed particles this number can be reduced to about 100 GeV. Using hadron-collider type jet clustering algorithms, which treat the forward particles in a similar way than an underlying event this can be even further improved [9]. The impact of this approach is illustrated with a reconstructed $e^+e^- \rightarrow H^+H^- \rightarrow t\bar{b}t\bar{b}$ event in the CLIC_ILD detector (see Fig. 57).

<figure><img src="image_60.png" /><figcaption>**Fig. 57** Reconstructed particles in a simulated $e^+e^- \rightarrow H^+H^- \rightarrow t\bar{b}t\bar{b}$ event at $\sqrt{s}=3$ TeV in the CLIC_ILD detector including the background from $\gamma\gamma \rightarrow$ hadrons before (left) and after(right) applying tight timing cuts on the reconstructed cluster times [9].</figcaption></figure>

This section focuses on the production of heavy Higgs bosons ($H,A,H^\pm$), which are predicted in extended models

<footnote n="10">¹⁰Marcel Stanitzki: The materials presented in this subsection were prepared for the CLIC Conceptual Design Report.</footnote>
like the 2HDM or supersymmetric models. The CLIC capabilities for studying light, Standard Model (SM)-like Higgs bosons are summarized elsewhere [9, 198].

### 2.4.2 Searches for heavy Higgs Bosons

In many supersymmetric scenarios, the Higgs sector consists of one light Higgs boson $h$, consistent with a SM Higgs boson while the remaining four Higgs bosons are almost mass-degenerate and have masses way beyond 500 GeV, see Sect. 2.5. These scenarios are consistent with current results from ATLAS and CMS on the Higgs boson [199,200]. If this scenario for the Higgs sector has been realized, it will be extremely challenging to discover these additional final states at the LHC, especially in the low $\tan\beta$ regime, where e.g. the reach for the pseudoscalar $A$ can be as low as 200 GeV (See Fig. 58).

<figure><img src="image_61.png" /><figcaption>**Fig. 58** Search reach in the $m_A - \tan\beta$ plane for LHC and CLIC. The left-most colored regions are current limits from the Tevatron with ~7.5 $fb^{-1}$ of data at $\sqrt{s} = 1.96$ TeV and from ~1 $fb^{-1}$ of LHC data at $\sqrt{s} = 7$ TeV. The black line is projection of search reach at LHC with $\sqrt{s} = 14$ TeV and 300 $fb^{-1}$ of luminosity [201]. The right-most red line is search reach of CLIC in the HA mode with $\sqrt{s} = 3$ TeV. This search capacity extends well beyond the LHC [9].</figcaption></figure>

The pair production processes $e^{+}e^{-} \to H^{+}H^{-}$ and $e^{+}e^{-} \to HA$ will give access to these heavy Higgs bosons almost up to the kinematic limit [202, 203]. Two separate scenarios have recently been studied [9], 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 $HA \to b\bar{b}b\bar{b}$ and $H^{+}H^{-} \to t\bar{b}t\bar{b}$. 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 $\gamma\gamma \to hadrons$.

The resulting di-jet mass distributions including the background processes are shown in Fig. 59(Model I) and Fig. 60(Model II). The achievable accuracy on the Higgs boson mass using a dataset of $2 ab^{-1}$ at $\sqrt{s}=3$ TeV is about 0.3% [9] and the width can be determined with an accuracy

of 17-31% for the $b\bar{b}b\bar{b}$ final state and 23-27% for the $t\bar{b}t\bar{b}$ final state, showing the excellent physics capabilities of CLIC for studying heavy Higgs bosons.

<figure><img src="image_62.png" /><figcaption>**Fig. 59** Di-jet invariant mass distributions for the $e^{+}e^{-} \to HA \to b\bar{b}b\bar{b}$ (left) and the $e^{+}e^{-} \to H^{+}H^{-} \to t\bar{b}t\bar{b}$ (right) signal together with the individual background contributions for model I [9].</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Fig. 60** Di-jet invariant mass distributions for the $e^{+}e^{-} \to HA \to b\bar{b}b\bar{b}$ (left) and the $e^{+}e^{-} \to H^{+}H^{-} \to t\bar{b}t\bar{b}$ (right) signal together with the individual background contributions for model II [9].</figcaption></figure>

## 2.5 Prospects for MSSM Higgs bosons<fnref n="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 ~125 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<fnref n="12" />. The NMSSM will be covered in Sect. 2.9

### 2.5.1 The Higgs sector of the MSSM at tree-level

Contrary to the SM, in the MSSM [206] two Higgs doublets are required (since the superpotential is a holomorphic function of the superfields). The Higgs potential

$$V = m_1^2 |\mathcal{H}_1|^2 + m_2^2 |\mathcal{H}_2|^2 - m_{12}^2 (\epsilon_{ab} \mathcal{H}_1^a \mathcal{H}_2^b + \text{h.c.}) \\ + \frac{1}{8} (g_1^2 + g_2^2) [|\mathcal{H}_1|^2 - |\mathcal{H}_2|^2]^2 + \frac{1}{2} g_2^2 |\mathcal{H}_1^\dagger \mathcal{H}_2|^2, \quad (24)$$

<footnote n="11">¹¹Sven Heinemeyer</footnote>

<footnote n="12">¹²Analyses with complex parameters can be found in Refs. [204, 205] and references therein.</footnote>
contains $m_1, m_2, m_{12}$ as soft SUSY breaking parameters; $g_2$ and $g_1$ are the $SU(2)$ and $U(1)$ gauge couplings, respectively, and $\epsilon_{12} = -1$.

The doublet fields $\mathcal{H}_1$ and $\mathcal{H}_2$ are decomposed in the following way:

$$ 
\begin{aligned}
\mathcal{H}_1 &= \begin{pmatrix} \mathcal{H}_1^0 \\ \mathcal{H}_1^- \end{pmatrix} = \begin{pmatrix} v_1 + \frac{1}{\sqrt{2}}(\phi_1^0 - i\chi_1^0) \\ -\phi_1^- \end{pmatrix}, \\
\mathcal{H}_2 &= \begin{pmatrix} \mathcal{H}_2^+ \\ \mathcal{H}_2^0 \end{pmatrix} = \begin{pmatrix} \phi_2^+ \\ v_2 + \frac{1}{\sqrt{2}}(\phi_2^0 + i\chi_2^0) \end{pmatrix},
\end{aligned} 
\quad (25) $$

where $\phi_{1,2}^0$ denote the $\mathcal{CP}$-even fields, $\chi_{1,2}^0$ the $\mathcal{CP}$-odd fields and $\phi_{1,2}^{\pm}$ the charged field components. The potential (24) can be described with the help of two independent parameters (besides $g_2$ and $g_1$): $\tan\beta = v_2/v_1$ (with $v_1^2 + v_2^2 =: v^2 \approx (246 \text{ GeV})^2$) and $M_A^2 = -m_{12}^2(\tan\beta + \cot\beta)$, where $M_A$ is the mass of the $\mathcal{CP}$-odd Higgs boson A.

The diagonalization of the bilinear part of the Higgs potential, i.e. of the Higgs mass matrices, is performed via orthogonal transformations, introducing the mixing angle $\alpha$ for the $\mathcal{CP}$-even part (with $m_h$ denoting the tree-level value of the light $\mathcal{CP}$-even Higgs, see below),

$$ \tan \alpha = \left[ \frac{-(M_A^2 + M_Z^2) \sin \beta \cos \beta}{M_Z^2 \cos^2 \beta + M_A^2 \sin^2 \beta - m_h^2} \right], \quad -\frac{\pi}{2} < \alpha < 0. \quad (26) $$

One gets the following Higgs spectrum:

2 neutral bosons, $\mathcal{CP} = +1 : h, H$

1 neutral boson, $\mathcal{CP} = -1 : A$

2 charged bosons: $H^+, H^-$

$$ \text{3 unphysical Goldstone bosons: } G, G^+, G^- . \quad (27) $$

At tree level the masses squares are given by

$$ m_{H,h}^2 = \frac{1}{2} \left[ M_A^2 + M_Z^2 \pm \sqrt{(M_A^2 + M_Z^2)^2 - 4M_Z^2 M_A^2 \cos^2 2\beta} \right] \quad (28) $$

$$ m_{H^{\pm}}^2 = M_A^2 + M_W^2 . \quad (29) $$

In the decoupling limit [207], $M_A \gg M_Z$, the light $\mathcal{CP}$-even Higgs becomes SM-like, i.e. all its couplings approach their SM value.

### 2.5.2 The relevance of higher-order corrections

Higher-order corrections give large contributions to the Higgs sector predictions in the MSSM [208]. Most prominently, they affect the prediction of the Higgs boson masses in terms of the other model parameters. In the MSSM, in particular, the light $\mathcal{CP}$-even Higgs boson mass receives

higher-order contributions up to $\mathcal{O}(100\%)$ [209]. The very leading one-loop correction reads

$$ \Delta M_h^2 = \frac{3 g_2^2 m_t^4}{8 \pi^2 M_W^2} \left[ \log \left( \frac{M_S^2}{m_t^2} \right) + \frac{X_t^2}{M_S^2} \left( 1 - \frac{X_t^2}{12 M_S^2} \right) \right], \quad (30) $$

where $M_S = (m_{\tilde{t}_1} + m_{\tilde{t}_2})/2$ denotes the average of the two scalar top masses, and $m_t X_t$ is the off-diagonal element in the scalar top mass matrix. Via this kind of higher-order corrections the light Higgs mass is connected to all other sectors of the model and can serve as a precision observable. The missing higher-order uncertainties have been estimated to be at the level of $\sim 2-3 \text{ GeV}$ [210, 211].

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 $\mathcal{CP}$-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 $\mathcal{CP}$-even mixing angle, $\alpha \to \alpha_{\text{eff}}$ [212]. For large higher-order corrections which drive $\alpha_{\text{eff}} \to 0$ the decay widths $\Gamma(h \to b\bar{b})$ and $\Gamma(h \to \tau^+\tau^-)$ could be substantially smaller than in the SM [213], altering the available search modes for such a Higgs boson.

The relation between the bottom-quark mass and the Yukawa coupling $h_b$, which controls also the interaction between the Higgs fields and the sbottom quarks, is also affected by higher-order corrections, summarized in the quantity $\Delta_b$ [214–216]. These, often called threshold corrections, are generated either by gluino–sbottom one-loop diagrams (resulting in $\mathcal{O}(\alpha_b\alpha_s)$ corrections), or by chargino–stop loops (giving $\mathcal{O}(\alpha_b\alpha_t)$ corrections). Analytically one finds $\Delta_b \propto \mu \tan\beta$. The effective Lagrangian is given by [215]

$$ \mathcal{L} = \frac{g_2}{2M_W} \frac{\bar{m}_b}{1+\Delta_b} \left[ \begin{aligned} & \tan\beta A i \bar{b} \gamma_5 b + \sqrt{2} V_{tb} \tan\beta H^+ \tilde{t}_L b_R \\ & + \left( \frac{\sin\alpha}{\cos\beta} - \Delta_b \frac{\cos\alpha}{\sin\beta} \right) h \bar{b}_L b_R \\ & - \left( \frac{\cos\alpha}{\cos\beta} + \Delta_b \frac{\sin\alpha}{\sin\beta} \right) H \bar{b}_L b_R \end{aligned} \right] + \text{h.c.} $$

Large positive (negative) values of $\Delta_b$ lead to a strong suppression (enhancement) of the bottom Yukawa coupling. For large $M_A$ the decoupling of the light $\mathcal{CP}$-even Higgs boson to the SM bottom Yukawa coupling is ensured in Eq. (31). Effects on the searches for heavy MSSM Higgs bosons via $\Delta_b$ have been analyzed in Refs. [217, 218].

Deviations from the SM predictions can also be induced by the appearance of light virtual SUSY particles in loop-induced processes. Most prominently a light scalar top can have a strong impact on the prediction of $gg \to h$. The additional contributions can interfere negatively with the top
prediction of these models [231]. These predictions take into account the current SUSY search limits (but no direct light Higgs search limits), as well as the relevant electroweak precision observables, *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 ~1.5GeV on the evaluation of $M_h$. The green columns indicate the range of the newly discovered particle mass.

<figure><img src="image_64.png" /><figcaption>**Fig. 62** Fit for the light *CP*-even Higgs mass in the CMSSM (left) and NUHM1 (right) [231]. Direct searches for the light Higgs boson are not included.</figcaption></figure>

- Parameter scans in the MSSM with 19 free parameters (pMSSM-19 [230]) are naturally compatible with a light Higgs boson around $M_h \sim 125$ GeV, as has been analyzed in Ref. [232] (see also Ref. [233] for a more recent analysis in the pMSSM-15 and Ref. [234] 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 analyzed in the upcoming LHC runs. Also the effects on the various production cross sections and branching ratios were analyzed, where it was confirmed

that light particles can modify in particular the decay rate to photons [221].

- 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 $\mathcal{O}(10^7)$ points; but they have the disadvantage of potentially not including all relevant phenomenology of the MSSM. The pMSSM-7 fits to the full set of Higgs data (and several low-energy observables) [235] allow to show an enhancement of the BR($h \to \gamma\gamma$), 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. [236]). In particular, these scans (while not incorporating the latest data) demonstrate that light scalar top masses are compatible with $M_h \sim 125$ GeV (see also Ref. [227]). 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 trilinear 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 *CP*-even Higgs boson corresponds to the new state at ~125 GeV. The colored points passed the Higgs exclusion bounds (obtained using HiggsBounds [229]). The red (yellow) points correspond to the best fit points with a $\Delta\chi^2 < 2.3(5.99)$, see Ref. [235] 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 $M_h \sim 125$ GeV (unless $m_{\tilde{q}_3}$ is extremely large, see, e.g., Ref. [237]), and that values of $|X_t/m_{\tilde{q}_3}|$ between ~1 and ~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 $m_{\tilde{\tau}_1} \sim 200$GeV, where also the regions favored by the fit to the Higgs rates start, in the case of $X_t$ positive. Such a light $\tilde{\tau}_1$ is accompanied by a somewhat heavier $\tilde{\tau}_2$, as can be seen in the right plot of Fig. 63. Values of $m_{\tilde{\tau}_1} \sim 200$GeV are realized for $m_{\tilde{\tau}_2} \sim 600$GeV, which would mean that both stop masses are rather light, offering interesting possibilities for the LHC. The highest favoured $m_{\tilde{\tau}_1}$ values we find are ~1.4 TeV. These are the maximal values reached in the scan in Ref. [235], 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.

Searches for the other Higgs bosons of the MSSM have so far not been successful. This applies to the heavy Higgs bosons of the MSSM as well as to a potentially light *CP*-even Higgs bosons in the MSSM in the case that the new state at ~125 GeV is interpreted as the heavy *CP*-even Higgs boson, see Sect. 2.2.
<figure><img src="image_65.png" /><figcaption>**Fig. 63** Stop mixing parameter $X_t/m_{\tilde{q}_3}$ vs. the light stop mass (left), and the light vs. heavy stop masses (right), see text.</figcaption></figure>

### 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 program is devoted to this kind of analyses.

The prospects for the SM Higgs boson in this respect are the following [238–240]:

- The Higgs boson mass can be determined down to a level of $\mathcal{O}(200 \text{ MeV})$.

- 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. [242, 243]. 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 ~10% level can be achieved. Without any assumptions only ratios of couplings can be determined (see also Ref. [71] 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. [244] for a discussion.

- It can be expected that the spin 2 hypothesis can be rejected using LHC data.

- A pure $\mathcal{CP}$-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 $\mathcal{CP}$-even case). However, the prospects for the LHC to determine a certain level of $\mathcal{CP}$-odd admixture to the Higgs state are less clear [245].

In the case that the light $\mathcal{CP}$-even MSSM Higgs boson is identified with the new state at ~125 GeV, as can be seen in Fig. 61, the decoupling region, $M_A \gg M_Z$ is a viable option. In this case the SM Higgs analyses can be taken over directly to the MSSM case – and will yield (nearly) identical results. Only light SUSY particles in the loops mediating the gluon fusion process or the decay to two photons might result in somewhat different predictions. However, depending on the actual values of the SUSY mass scales, these differences might easily remain unobservable with the anticipated LHC precision. Furthermore, in the decoupling regime the heavy MSSM Higgs bosons can easily be too heavy to be discovered at the LHC, in particular for medium or lower values of $\tan\beta$.

Only in the lower allowed range for $M_A$ in this scenario larger deviations from the phenomenology of the light $\mathcal{CP}$-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

As outlined in the previous subsection, identifying the light $\mathcal{CP}$-even Higgs with the new state at ~125 GeV can easily result in a scenario where the LHC can neither distinguish the $h$ from the SM Higgs boson, nor be able to discover additional Higgs bosons. In this case the analyses at an LC offer good prospects to reveal the non-SM nature of the Higgs particle. The anticipated experimental precisions for couplings to SM particles, the self-coupling etc., as given in detail in Sect. 2.3. In particular, the following improvements over the anticipated LHC precision/potential can be expected:

- The mass of a SM-like Higgs boson at ~125 GeV can be determined at the level of 50MeV.

- 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 Tab. 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 $\mathcal{CP}$ decomposition can be determined, in particular, using the channel $e^+e^- \to t\bar{t}H$ [247].

<footnote>$^{14}$ A recent analysis from CMS using the Higgs decays to ZZ far off-shell yielded an upper limit on the total width about four times larger than the SM width [241]. However, these constraints on the total width rely on the assumption of the equality of the on-shell and off-shell couplings of the Higgs boson. The relation between those couplings can be severely affected by new physics contributions, in particular via threshold effects, which on the other hand would be needed to give rise to a Higgs-boson width that differs from the SM one by the currently probed amount, see the discussion in Ref. [115].</footnote>
<table><caption>**Table 18** Examples of the precision of SM-like Higgs observables at a $\sqrt{s} = 500$GeV LC assuming a Higgs boson mass of 125GeV. The results are based on the ILC set-up. For the direct measurements, an integrated luminosity of $\mathcal{L}^{\text{int}} = 500 \text{ fb}^{-1}$ is assumed. For the indirect measurements at GigaZ, a running time of approximately one year is assumed, corresponding to $\mathcal{L} = \mathcal{O}(10 \text{ fb}^{-1})$. Taken from Ref. [246].</caption><thead><tr><th>Observable</th><th>Expected precision</th></tr></thead><tbody><tr><td>M<sub>H</sub> [GeV]</td><td>0.03%</td></tr><tr><td>g<sub>HWW</sub></td><td>1.4%</td></tr><tr><td>g<sub>HZZ</sub></td><td>1.4%</td></tr><tr><td>g<sub>Hbb</sub></td><td>1.4%</td></tr><tr><td>g<sub>Hcc</sub></td><td>2.0%</td></tr><tr><td>g<sub>Hππ</sub></td><td>2.5%</td></tr><tr><td>g<sub>Htt</sub></td><td>10%</td></tr><tr><td>g<sub>HHH</sub></td><td>40%</td></tr><tr><td>BR(H → γγ)</td><td>25%</td></tr><tr><td>BR(H → gg)</td><td>5%</td></tr><tr><td>BR(H → invisible)</td><td>0.5%</td></tr></tbody></table>

- 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 [248]).

An indirect determination of $M_A$ can be performed via a precise measurement of the Higgs couplings, where a sensitivity up to 800GeV was found [249].

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 realized [250] (see also Ref. [251]).

Another measurement at the LC can turn out to be crucial for Higgs physics in the MSSM: the determination of $m_t$ from a threshold scan. As can be seen in Eq. (30), the theory prediction of $M_h$ depends strongly on $m_t$. Only the LC determination of a well defined top quark mass can yield a theory prediction that matches the LHC precision in $M_h$. More details can be found in Sect. 4.4.

## 2.6 General multi-Higgs structures<fnref n="15" />

### 2.6.1 Introduction

We here give a review of extended Higgs sectors and their collider phenomenology. In the Standard Model (SM), one isospin doublet scalar field $\Phi$ is simply introduced as the minimum form. Under the requirement of the renormalizability its potential can be uniquely written as

$$V(\Phi) = +\mu^2 |\Phi|^2 + \lambda |\Phi|^4. \qquad (31)$$

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) $v (= (\sqrt{2}G_F)^{-1/2} \simeq 246 \text{ GeV})$. 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, dark matter 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 we can 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 flavor changing neutral currents (FCNCs).

The electroweak $\rho$-parameter is calculated at the tree level for a Higgs sector with $N$ multiplets by

$$\rho = \frac{m_W^2}{m_Z^2 \cos^2 \theta_W} = \frac{\sum_i \{4T_i(T_i + 1) - Y_i^2\} |v_i|^2 c_i}{\sum_i 2Y_i^2 |v_i|^2}, \qquad (32)$$

where $T_i$ and $Y_i$ ($i = 1, \dots, N$) are isospin and hyper charges of the $i$-th multiplet field ($Q_i = T_i + Y_i/2$), and $c_i = 1/2$ for real fields ($Y_i = 0$) and 1 for complex fields. The data shows that $\rho = 1.0004^{+0.0003}_{-0.0004}$ [252]. Higgs sectors with additional doublets ($T_i, Y_i$) = (1/2, 1) (and singlets with $Y_i = 0$) predict $\rho = 1$ at the tree level, like the SM Higgs sector. Thus, multi-doublet structures would be a natural extension of the Higgs

<footnote n="15">¹⁵Shinya Kanemura</footnote>