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# 11. Status of Higgs Boson Physics

Revised August 2023 by M. Carena (FNAL; Chicago U.; Chicago U., Kavli Inst.), C. Grojean (DESY, Hamburg; Physik, Humboldt U.), M. Kado (MPP, Garching) and V. Sharma (UC San Diego).

11.1 Introduction ........................................ 211

11.2 The Standard Model and the mechanism of electroweak symmetry breaking ................................. 212

11.2.1 The SM Higgs boson mass, couplings and quantum numbers ........................................ 212

11.2.2 The SM custodial symmetry ................................. 213

11.2.3 Stability of the Higgs potential ................................. 213

11.2.4 Higgs boson production and decay mechanisms ................................................................. 213

11.3 The experimental profile of the Higgs boson ................................. 218

11.3.1 The principal decay channels of the Higgs boson ................................................................. 218

11.3.2 The Higgs boson decay to vector bosons ................................. 218

11.3.3 Measurement of the Higgs boson mass ................................. 219

11.3.4 Higgs decays to third generation fermions ($b\bar{b}$ and $\tau^{+}\tau^{-}$) ........................................ 220

11.3.5 Higgs boson production in association with top quarks .................................................. 222

11.3.6 Combining the main channels ........................................ 223

11.3.7 Probing the Higgs boson coupling to the second generation ........................................ 225

11.3.8 Searches for rare decays of the Higgs boson ................................. 226

11.3.9 Higgs boson pair production and self-coupling ................................. 228

11.3.10 Beyond the main Higgs boson production modes ................................................................. 230

11.3.11 First results on the main production and decay channels at 13.6 TeV ........................................ 230

11.4 Main quantum numbers and width of the Higgs boson ................................................................. 231

11.4.1 Main quantum numbers $J^{PC}$ ........................................ 231

11.4.2 Off-shell couplings of the Higgs boson ................................. 234

11.4.3 The Higgs boson width .................................................. 235

11.5 Probing the coupling properties of the Higgs boson ................................. 236

11.5.1 Probing coupling properties ........................................ 236

11.5.2 Effective Lagrangian framework ........................................ 240

11.6 New physics models of EWSB in the light of the Higgs boson discovery ........................................ 242

11.6.1 Higgs bosons in the minimal supersymmetric standard model (MSSM) ........................................ 243

11.6.2 Supersymmetry with singlet extensions ........................................ 246

11.6.3 Supersymmetry with extended gauge sectors ................................. 246

11.6.4 Effects of CP violation .................................................. 247

11.6.5 Non-supersymmetric extensions of the Higgs sector .................................................. 248

11.6.6 Composite Higgs models ........................................ 250

11.6.7 Searches for signatures of extended Higgs sectors .................................................. 253

11.7 Summary and outlook ........................................ 258

## 11.1 Introduction

Understanding how the masses of the known elementary particles are generated has been one of the fundamental endeavours in particle physics for several decades. The chiral nature of the weak interactions forbids an intrinsic mass terms for the quarks and leptons at high energy, and their masses can only appear as emergent quantities after the spontaneous symmetry of the electroweak symmetry. The discovery in 2012 by the ATLAS [1] and the CMS [2] collaborations of a new resonance with a mass of approximately 125 GeV and the subsequent studies of its properties with the full data set from Run 1, from 2009 to 2012, with a centre-of-mass energy of 7 TeV and 8 TeV, conclusively provided a first sketch of the Electroweak Symmetry Breaking (EWSB) mechanism. The much larger dataset collected during the LHC Run 2, from 2015 to 2018, with a higher centre-of-mass energy of 13 TeV, demonstrated the compatibility of the measured resonance with the Higgs boson of the Standard Model (SM) [3] and strongly challenged other contenders. The LHC started its Run 3

in 2022 and this data taking phase is expected to comprise of four years of operations until 2026. By then the LHC is expected to deliver proton-proton collision data corresponding to an integrated luminosity of approximately 250 fb⁻¹ per experiment.

In the SM, the electroweak interactions are described by a gauge field theory invariant under the $SU(2)_L \times U(1)_Y$ symmetry group. The mechanism of EWSB [4] provides a general framework to keep untouched the structure of these gauge interactions at high energies and still generate the observed masses of the W and Z gauge bosons as well as those of the quarks and charged leptons. The EWSB mechanism posits a self-interacting complex EW doublet scalar field, whose CP-even neutral component acquires a vacuum expectation value (VEV) $v \approx 246$ GeV, which sets the scale of the symmetry breaking. Three massless Goldstone bosons are generated and are absorbed to give masses to the W and Z gauge bosons. The remaining component of the complex doublet becomes the Higgs boson – a new, and so far unique, fundamental scalar particle. The masses of all fermions are also a consequence of EWSB since the Higgs doublet is postulated to couple to the fermions through Yukawa interactions in a way consistent with all the gauge symmetries of the SM.

The initial measurements during the LHC Run 1 were accessible mainly through production and decay channels related to the couplings of the Higgs boson to the vector gauge bosons. The outstanding performance of the LHC Run 2, made it possible for the ATLAS and CMS experiments to independently and unambiguously establish the couplings of the Higgs boson to the charged fermions of the third generation (the top quark, the bottom quark, and the tau lepton). In all observed production and decay modes measured so far, the rates and differential measurements are found to be consistent, within experimental and theoretical uncertainties, with the SM predictions. In high mass resolution decay channels, such as the ones with four leptons (electrons or muons) or diphoton final states, the mass of the Higgs boson has been measured at the permill precision level. And, from the search for Higgs boson decays to muons, there are tentalising experimental evidences for Yukawa interactions of the Higgs boson to the second generation of fermions. Nevertheless, several Higgs boson decay channels are very challenging experimentally at the LHC. The couplings of the Higgs boson to the first generation of fermions, at the level expected in the Standard Model, remain so far out of reach at the LHC.

Within the current precision, a more complex Higgs sector with additional states, although significantly constrained, is not ruled out. Nor has it been established whether the Higgs boson is an elementary particle or whether it has an internal structure like any other scalar particle observed before it. Therefore, Higgs physics remains a vivid and intense field of research on the experimental and theoretical fronts.

Without the Higgs boson, the calculability of the SM would have been spoiled. In particular, perturbative unitarity [5] would be lost at high energies since the longitudinal W/Z boson scattering amplitude would grow with the centre-of-mass energy. In addition, the radiative corrections to the gauge boson self-energies would exhibit dangerous logarithmic divergences that would be difficult to reconcile with EW precision data. With the discovery of the Higgs boson, the SM is a spontaneously broken gauge theory and, as such, it could a priori be consistently extrapolated well above the masses of the W and Z bosons. Hence, formally there is no need for new physics at the EW scale. However, as the SM Higgs boson is a scalar particle, it has sensitivity to possible new physics scales. Quite generally, the Higgs boson mass is affected by the presence of heavy particles and it receives large quantum corrections which destabilise the weak scale barring a large fine-tuning of unrelated parameters. This is known as the Higgs naturalness or hierarchy problem [6], and it has been the prime argument for expecting new physics right at the TeV scale. To remedy to his unsatisfactory feature, theoretically motivated paradigms have been explored for several decades, such as a new fermion-boson symmetry called supersymmetry (SUSY) [7] (for recent reviews, see Refs. [8, 9]), or the existence of strong interactions at a scale of the order of a TeV from which the Higgs boson would emerge as a composite state [10] (see Refs. [11, 12]
for recent reviews). Alternatively, new agents stabilising the weak scale could also be light but elusive, like in models of neutral naturalness [13, 14]. And it was recently appreciated [15, 16] that the cosmological evolution of the Universe could drive the Higgs boson mass to a value much smaller than the cutoff of the theory, alleviating the hierarchy problem without the need for TeV scale new physics, even though there might still be interesting and spectacular signatures [15, 17]. This new approach spurs a change in perspective invoking our Universe as part of a multiverse that should be treated as a quantum statistical system [18].

The Higgs boson is special and, in the eleven years since its discovery, it became a powerful tool to explore the manifestations of the SM and to probe the physics landscape beyond it.

This review is organised as follows. Section 11.2 is a theoretical introduction to the SM Higgs boson, its properties, production mechanisms and decay rates. The experimental measurements are described in Section 11.3. In Section 11.3.6, the combination of the main Higgs boson production and decay channels is presented. In Section 11.4, measurements of the main quantum numbers and CP properties of the Higgs boson are reported and the bounds on its total width are discussed. In Section 11.5, a general theoretical framework to describe the deviations of the Higgs boson couplings from the SM predictions is introduced and the experimental measurements of these Higgs couplings are reviewed. Measurements of differential cross sections are also outlined. Section 11.6 presents, in detail, some interesting models with an extended Higgs sector, addressing the hierarchy problem or not, and considers their experimental signatures. Section 11.7 provides a short summary and a brief outlook.

## 11.2 The Standard Model and the mechanism of electroweak symmetry breaking

In the SM [3], electroweak symmetry breaking [4] is responsible for generating a mass for the W and Z gauge bosons rendering the weak interactions short ranged. The SM scalar potential reads:

$$V(\Phi) = m^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2 \quad (11.1)$$

with the Higgs field $\Phi$ being a self-interacting SU(2)$_L$ complex doublet (four real degrees of freedom) with weak hypercharge $Y=1$ (the hypercharge is normalised such that $Q = T_{3L} + Y/2$, $Q$ being the electric charge and $T_{3L}$ the eigenvalue of the diagonal generator of SU(2)$_L$):

$$\Phi = \frac{1}{\sqrt{2}} \begin{pmatrix} \sqrt{2}\phi^+ \\ \phi^0 + ia^0 \end{pmatrix}, \quad (11.2)$$

where $\phi^0$ and $a^0$ are the CP-even and CP-odd neutral components, and $\phi^+$ is the complex charged component of the Higgs doublet, respectively. $V(\Phi)$ is the most general renormalisable scalar potential invariant under SU(2)$_L \times U(1)_Y$. If the quadratic term is negative, the neutral component of the scalar doublet acquires a non-zero (real) vacuum expectation value (VEV)

$$\langle \Phi \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 \\ v \end{pmatrix}, \quad (11.3)$$

with $\phi^0 = H + \langle \phi^0 \rangle$ and $\langle \phi^0 \rangle \equiv v$, inducing the spontaneous breaking of the SM gauge symmetry SU(3)$_C \times SU(2)_L \times U(1)_Y$ into SU(3)$_C \times U(1)_\text{em}$. The global minimum of the theory defines the ground state, and spontaneous symmetry breaking implies that there is a (global and/or local) symmetry of the system that is not respected by the ground state. From the four generators of the SU(2)$_L \times U(1)_Y$ SM gauge group, three are spontaneously broken, i.e., they do not leave the ground state invariant. This breaking implies the existence of three massless Goldstone bosons identified with three of the four Higgs field degrees of freedom. The Higgs field couples to the $W_\mu$ and $B_\mu$ gauge fields associated with the SU(2)$_L \times U(1)_Y$ local symmetry through the covariant derivative appearing in the kinetic term of the Higgs Lagrangian,

$$\mathcal{L}_{\text{Higgs}} = (D_{\mu} \Phi)^{\dagger} (D^{\mu} \Phi) - V(\Phi), \quad (11.4)$$

where $D_\mu \Phi = (\partial_\mu + i g a^a W_\mu^a / 2 + i g' Y B_\mu / 2) \Phi$, $g$ and $g'$ are the SU(2)$_L$ and U(1)$_Y$ gauge couplings, respectively, and $\sigma^a$, $a = 1, 2, 3$ are the usual Pauli matrices. As a result, the neutral

and the two charged massless Goldstone degrees of freedom mix with the gauge fields corresponding to the broken generators of SU(2)$_L \times U(1)_Y$ and become, in the unitarity gauge, the longitudinal components of the Z and W physical gauge bosons, respectively. The Z and W gauge bosons acquire masses,

$$m_W^2 = \frac{g^2 v^2}{4}, \quad m_Z^2 = \frac{(g'^2 + g^2)v^2}{4}. \quad (11.5)$$

The fourth generator remains unbroken since it is the one associated to the conserved U(1)$_\text{em}$ gauge symmetry, and its corresponding gauge field, the photon, remains massless. Similarly, the eight color gauge bosons, the gluons, corresponding to the conserved SU(3)$_C$ gauge symmetry with 8 unbroken generators, also remain massless (though confined inside hadrons and mesons as the result of the asymptotic freedom behaviour of Quantum ChromoDynamics (QCD)). Hence, from the initial four degrees of freedom of the Higgs field, two are absorbed by the $W^\pm$ gauge bosons, one by the Z gauge boson, and there is one remaining degree of freedom, H, that is the physical Higgs boson — a new scalar particle first imagined by P. Higgs [4]. The Higgs boson is neutral under the electromagnetic interactions and transforms as a singlet under SU(3)$_C$ and hence does not couple at tree level to the massless photons and gluons.

The fermions of the SM acquire a mass through renormalisable and gauge invariant interactions between the Higgs field and the fermions: the Yukawa interactions,

$$\mathcal{L}_{\text{Yukawa}} = -\hat{h}_{d_{ij}} \bar{q}_{L_i} \Phi d_{R_j} - \hat{h}_{u_{ij}} \bar{q}_{L_i} \tilde{\Phi} u_{R_j} - \hat{h}_{l_{ij}} \bar{L}_{L_i} \Phi e_{R_j} + \text{h.c.}, \quad (11.6)$$

which respect the symmetries of the SM but generate fermion masses once EWSB occurs. In the Lagrangian above, $\tilde{\Phi} = i\sigma_2 \Phi^*$ and $q_L$ ($l_L$) and $u_R$, $d_R$ ($e_R$) are the quark (lepton) SU(2)$_L$ doublets and singlets, respectively, while in each term, $\hat{h}_{f_{ij}}$, $f = u, d, l$ is parametrised by a $3 \times 3$ matrix in family space. The mass term for neutrinos is omitted, but could be added in an analogous manner to the up-type quarks when right-handed neutrinos are supplementing the SM particle content (neutrinos can also acquire Majorana masses via non-renormalisable dimension-5 interactions with the Higgs field [19]). Once the Higgs field acquires a VEV, and after rotation to the fermion mass eigenstate basis that also diagonalises the Higgs-fermion interactions, $\hat{h}_{f_{ij}} \rightarrow h_{f_i} \delta_{ij}$ (the diagonal Yukawa coupling $h_{f_i}$ is often denoted as $y_{f_i}$), all fermions acquire a mass given by $m_{f_i} = h_{f_i} v / \sqrt{2}$. The indices $i, j = 1, 2, 3$ refer to the three families in the up-quark, down-quark or charged lepton sectors. Remarkably, if the Yukawa interactions Eq. (11.6) are indeed fully responsible for the fermion masses, the Higgs interactions do not mediate flavour changing neutral currents at tree-level. It should be further noted that the EWSB mechanism provides no additional insight on possible underlying reasons for the large variety of mass values of the fermions, often referred to as the flavour hierarchy. The fermion masses, accounting for a large number of the free parameters of the SM, are simply translated into Yukawa couplings.

### 11.2.1 The SM Higgs boson mass, couplings and quantum numbers

The SM Higgs boson is a CP-even scalar of spin 0. Its mass is given by $m_H = \sqrt{2\lambda} v$, where $\lambda$ is the self-coupling parameter in $V(\Phi)$. The expectation value of the Higgs field, $v = (\sqrt{2}G_F)^{-1/2} \approx 246$ GeV, is fixed by the Fermi coupling $G_F$, which is determined with a precision of 0.6 ppm from muon decay measurements [20]. The quartic coupling $\lambda$ is a free parameter in the SM, and hence, there is no a priori prediction for the Higgs mass. Moreover the sign of the quadratic coupling, $m^2 = -\lambda v^2$, in the Higgs field potential $V(\Phi)$ has to be negative for the EW symmetry breaking to take place, but there is no a priori understanding of what decides of this sign. The experimentally measured Higgs boson mass, $m_H \simeq 125$ GeV [20], implies that $\lambda \simeq 0.13$ and $|m| \simeq 88.4$ GeV.

The Higgs boson couplings to the fundamental particles are set by their masses. This is a new type of interaction; very weak for light particles, such as up and down quarks, and electrons, but strong for heavy particles such as the W and Z bosons and
the top quark. More precisely, the SM Higgs couplings to fundamental fermions are linearly proportional to the fermion masses, whereas the couplings to bosons are proportional to the square of the boson masses. The SM Higgs boson couplings to gauge bosons and fermions, as well as the Higgs boson self coupling, are summarised in the following Lagrangian:

$$ \mathcal{L} = -g_{Hf\bar{f}}\bar{f}fH + \frac{g_{HHH}}{6}H^3 + \frac{g_{HHHH}}{24}H^4 + \delta_V V_\mu V^\mu \left( g_{HVV}H + \frac{g_{HHVV}}{2}H^2 \right) \quad (11.7) $$

with

$$ g_{Hf\bar{f}} = \frac{m_f}{v}, \quad g_{HVV} = \frac{2m_V^2}{v}, \quad g_{HHVV} = \frac{2m_V^2}{v^2}, \quad (11.8) $$

$$ g_{HHHH} = \frac{3m_H^2}{v}, \quad g_{HHHHH} = \frac{3m_H^2}{v^2}, $$

where $V = W^\pm$ or $Z$ and $\delta_W = 1$, $\delta_Z = 1/2$. As a result, the dominant mechanisms for Higgs boson production and decay involve the coupling of $H$ to $W$, $Z$ and/or the third generation quarks and leptons. The Higgs boson coupling to gluons [21, 22] is induced at leading order by a one-loop process in which $H$ couples to a virtual $t\bar{t}$ pair (with minor contributions from the other lighter quarks). Likewise, the Higgs boson coupling to photons is also generated via loops, although in this case the one-loop graph with a virtual $W^+W^-$ pair provides the dominant contribution [23] and it is interfering destructively with the smaller contribution involving a virtual $t\bar{t}$ pair (as such, the Higgs coupling to photons is sensitive to the relative phase of the interactions between bosons and fermions).

### 11.2.2 The SM custodial symmetry

The SM Higgs Lagrangian, $\mathcal{L}_{Higgs} + \mathcal{L}_{Yukawa}$ of Eq. (11.4) and Eq. (11.6), is, by construction, $SU(2)_L \times U(1)_Y$ gauge invariant, but it also has an approximate global symmetry. In the limit $g' \to 0$ and $h_f \to 0$, a global $SU(2)_R$ symmetry emerges. This symmetry is preserved for non-vanishing Yukawa couplings provided $\hat{h}_u = \hat{h}_d$. Once the Higgs field acquires a VEV, both the $SU(2)_L$ and $SU(2)_R$ symmetry groups are broken but the diagonal subgroup $SU(2)_{L+R}$ remains unbroken and it is this subgroup that defines the custodial symmetry of the SM [24].

In the limit $g' \to 0$, the $W^\pm$ and $Z$ gauge bosons have equal mass and form a triplet of the $SU(2)_{L+R}$ unbroken global symmetry. Using the expressions for the $W$ and $Z$ gauge boson masses in terms of the gauge couplings, one obtains at tree level

$$ \frac{m_W^2}{m_Z^2} = \frac{g^2}{g'^2 + g^2} \equiv \cos^2 \theta_W \quad \text{i.e.} \quad \rho \equiv \frac{m_W^2}{m_Z^2 \cos^2 \theta_W} = 1. \quad (11.9) $$

The custodial symmetry protects the above relation from large radiative corrections. All corrections to the $\rho$ parameter are therefore proportional to terms that break the custodial symmetry. For instance, radiative corrections involving the Higgs boson are proportional to $\sin^2 \theta_W$, $\delta\rho = -11G_Fm_Z^2 \sin^2 \theta_W \log(m_H^2/m_Z^2)/(24\sqrt{2}\pi^2)$, and vanish in the limit $g' \to 0$. Since $m_t \neq m_b$, there are also relevant radiative corrections generated by massive fermions. They are proportional to $m_t^2 + m_b^2 - 2(m_t^2 m_b^2) \log(m_t^2/m_b^2)/(m_t^2 - m_b^2)$ and would indeed vanish for $m_t = m_b$ [25].

### 11.2.3 Stability of the Higgs potential

The discovery of the Higgs boson with $m_H \approx 125$ GeV has far reaching consequences within the SM framework. In particular, the precise value of $m_H$ determines the value of the quartic coupling $\lambda$ at the electroweak scale and makes it possible to study its behavior up to high energy scales. A larger value of $m_H$ would have implied that the self coupling $\lambda$ would become non-perturbative at some scale $\Lambda$ that could be well below the Planck scale [26].

However, for the value of Higgs boson mass experimentally measured, the EW vacuum of the Higgs potential is most likely metastable [27]. The high-energy evolution of $\lambda$ shows that it becomes negative at energies $\Lambda = O(10^{11})$ GeV (even though $\lambda$

could remain positive till higher energy, maybe all the way to the Planck scale, if the top quark mass exceeds its current measured value by $3\sigma$). When this occurs, the SM Higgs potential develops an instability and the long term existence of the EW vacuum is challenged. This behaviour may call for new physics at an intermediate scale before the instability develops, i.e., below $M_{Planck}$, even though new physics at $M_{Planck}$ could influence the stability of the EW vacuum and possibly modify this conclusion [28]. The consequences of the instability of the EW vacuum on high-scale inflation have been discussed in Ref. [29]. It was also noticed that Higgs field fluctuations during inflation could seed the formation of primordial black holes, possibly making up the Dark Matter relic abundance [30] or they could produce a stochastic background of gravitational waves with characteristic structures [31], offering a probe of the EW vacuum near criticality.

The lifetime of the EW metastable vacuum is determined by the rate of quantum tunnelling from this vacuum into the true vacuum of the theory (for the most recent computation of the EW vacuum lifetime within the SM, see Ref. [32]). Within the SM, the running of the Higgs self coupling slows down at high energies with a cancellation of its $\beta$-function at energies just one to two orders of magnitude below the Planck scale [33]. This slow evolution of the quartic coupling is responsible for saving the EW vacuum from premature collapse. It might also help the Higgs boson to play the role of an inflaton [34] (see, however, Ref. [35] for potential issues with this Higgs-boson-as-an-inflaton idea).

### 11.2.4 Higgs boson production and decay mechanisms

Comprehensive reviews of the SM Higgs boson properties and phenomenology, with an emphasis on the impact of loop corrections to the Higgs boson decay rates and cross sections, can be found in Refs. [36–40]. The main results are summarised here.

#### 11.2.4.1 Production mechanisms at hadron colliders

The main production mechanisms at the Tevatron collider and the LHC are gluon fusion (ggF), weak-boson fusion (VBF), associated production with a gauge boson ($VH$), and associated production with a $t\bar{t}$ pair ($t\bar{t}H$) or with a single top quark ($tHq$). Figure 11.1 depicts representative diagrams for these dominant Higgs boson production processes.

The state-of-the-art of the theoretical calculations in the main different production channels is summarised in Table 11.1.

<table><caption>**Table 11.1:** State-of-the-art of the theoretical calculations in the main Higgs boson production channels in the SM, and, when publicly available, the major MC tools used in the simulations. Furthermore, all the Higgs-boson production modes have been interfaced with parton-shower event generators at NLO QCD like Poweg-Box, MG5_aMC@NLO or Sherpa (the entries “—” indicate cases when the results are implemented in codes that are not yet public). For ggF and $VH$ NNLO matched simulations now also exist. Differential NNLO QCD results exist for $t\bar{t}H$ [41] (though the exact 2-loop virtual corrections are still missing) and have been implemented in MATRIX [42].</caption><thead><tr><th>ggF</th><th>VBF</th><th>VH</th><th>t<span style="text-decoration: overline;">t</span>H</th></tr></thead><tbody><tr><td><strong>Inclusive:</strong><br>N3LO QCD + NLO EW<br>(iHixs)<br>N3LO QCD<br>ggHiggs<br><strong>Differential:</strong><br>NNLO+N3LL QCD<br>(Radish+NNLOJET)<br>N3LO+N3LL' QCD<br>(TROLL)</td><td><strong>Inclusive:</strong><br>N3LO QCD<br>proVBFH</td><td><strong>Inclusive:</strong><br>N3LO QCD<br>(n3loxs)<br>NNLO QCD<br>(VH@NNLO)<br><strong>Differential:</strong><br>NNLO QCD</td><td><strong>Inclusive:</strong><br>NNLO QCD<br>—<br>—<br><strong>Differential:</strong><br>NLO QCD+EW<br>(MG5_aMC@NLO)</td></tr></tbody></table>

The cross sections for the production of a SM Higgs boson as a function of $\sqrt{s}$, the center of mass energy, for pp collisions, including bands indicating the theoretical uncertainties, are summarised in Fig. 11.2 (left) [43]. A detailed discussion, including uncertainties in the theoretical calculations due to missing higher-order effects and experimental uncertainties on the determination
<figure><img src="image_28.png" /><figcaption>**Figure 11.1:** Main leading order Feynman diagrams contributing to the single Higgs boson production in (a) gluon fusion, (b) Vector-boson fusion, (c) Higgs-strahlung (or associated production with a gauge boson at tree level from a quark-quark interaction), (d) associated production with a gauge boson (at loop level from a gluon-gluon interaction), (e) associated production with a pair of top/bottom quarks, (f-g) production in association with a single top quark, and double Higgs production through (h) a top- and bottom-quark loop, (i) the self coupling of the Higgs boson.</figcaption></figure>

of SM parameters involved in the calculations, can be found in Refs. [37–40]. These references also contain state-of-the-art discussions on the impact of PDF uncertainties, QCD scale uncertainties and uncertainties due to different procedures for including higher-order corrections matched to parton shower simulations, as well as uncertainties due to hadronisation and parton-shower events.

Table 11.2 summarises the Higgs boson production cross sections and relative uncertainties for a Higgs boson mass of 125 GeV, for $\sqrt{s} = 7, 8, 13, 13.6$ and 14 TeV.

### i. Gluon fusion production mechanism

At high-energy hadron colliders, the Higgs boson production mechanism with the largest cross section is the gluon-fusion process, $gg \to H + X$, mediated by the exchange of a virtual, heavy top quark [45]. Contributions from lighter quarks propagating in the loop are suppressed proportionally to $m_q^2$. QCD radiative corrections to the gluon-fusion process are very important and have been studied in detail. Including the full dependence on the (top, bottom, charm) quark and Higgs boson masses, the cross section has been calculated at the next-to-leading order (NLO) in $\alpha_s$ [46, 47]. To a very good approximation, the leading top-quark contribution can be evaluated in the limit $m_t \to \infty$ by matching the SM to an effective theory. The gluon-fusion amplitude is then evaluated from an effective Lagrangian containing a local $HG^{a}_{\mu\nu}G^{a\mu\nu}$ operator [21, 22]. In this approximation, the cross section is known at next-to-next-to-next-to-leading order in $\alpha_s$ expansion (N3LO QCD) [48]. The validity of the effective theory with infinite $m_t$ is greatly enhanced by rescaling the result by the exact LO result: $\sigma = (\sigma_{m_t^{\text{LO}}}/\sigma_{m_t^{\text{\textit{∞}}}}) \times \sigma_{m_t^{\text{\textit{∞}}}}$ [40]. The large top-quark mass approximation, after this rescaling of the cross section, yields a next-to-next-to-leading order (NNLO) QCD result that has been established to be at the percent level accuracy [49].

Further progress was made to include full top mass dependence at NNLO QCD [50]. And the bottom and light quark contributions to the Higgs production have been reassessed in Refs. [51–53].

The LO contribution and NLO QCD corrections [54] amount to about 80% of the total N3LO QCD cross section. The NNLO QCD corrections [55] further enhance the cross section by approximately 30% of the LO plus NLO result (at $\mu_f = \mu_r = m_H/2$). Electroweak radiative corrections have been computed at NLO and increase the LO cross section by about 5% for $m_H \simeq 125$ GeV [56]. Mixed QCD-EW corrections are now being investigated with encouraging results on the computation of the exact 3-loop amplitude [57] complementing the results obtained in either limit of heavy [58] or massless [59] gauge bosons. The mixed QCD-EW corrections still have to be evaluated at large $p_T$.

At N3LO QCD, the perturbation series is rather stable with a mere enhancement of 3% of the total cross section, with a central value quite insensitive to threshold resummation effects with the scale choice mentioned above ($\mu_f = \mu_r = m_H/2$) [40, 48, 60, 61]. At the LHC with a center-of-mass energy of 13 TeV, the most up-to-date value for the production cross section of a 125 GeV Higgs boson amounts to [40]

$$ \sigma_{\text{ggF}}^{\text{N3LO}} = 48.6^{+2.2}_{-3.3} (4.6\%) \text{ (theory)} \pm 1.6 (3.2\%) \text{ (PDF} + \alpha_s) \text{ pb.} \quad (11.10) $$

This cross-section was first computed in terms of an expansion around the production threshold of the Higgs boson. This approximation was overcome in Ref. [61]. The PDF uncertainty of order 1-2% quoted above has been questioned as the first global analysis of PDFs at approximate N3LO QCD was presented, suggesting an unexpected decrease of the cross-section central value by more than 5% [62].

Besides considering the inclusive Higgs boson production cross section at the LHC, it is important to study differential distri-
<figure><img src="image_29.png" /><figcaption>**Figure 11.2:** (Left) The SM Higgs boson production cross sections as a function of the center of mass energy, $\sqrt{s}$, for pp collisions [43]. The VBF process is indicated here as $qqH$. (Right) The branching ratios for the main decays of the SM Higgs boson near $m_H = 125$ GeV [39, 40]. The theoretical uncertainties are indicated as bands.</figcaption></figure>

butions in order to probe the properties of the Higgs boson in a detailed way. The Higgs boson rapidity distribution has been computed at the N3LO QCD order in Ref. [63]. And fully differential N3LO QCD predictions for Higgs production in gluon fusion have been obtained by combining N3LO QCD predictions for Higgs rapidity distribution with NNLO QCD predictions for $H+jet$ [64]. A more exclusive account of Higgs boson production is also required because experimental analyses often impose cuts on the final states in order to improve the signal-to-background ratio. Using soft-collinear effective theory, large fiducial power corrections induced by fiducial cuts have been resummed at N3LO+N3LL' QCD accuracy, i.e. at next-to-next-to-next-leading-log, incorporating the complete $O(\alpha_s^3)$ singular structure for $q_T \to 0$ allowing one to consistently match to N3LO QCD [65]. This is the most accurate result that can be directly compared with fiducial measurements. Different fiducial cuts are explored to suppress these fiducial effects. In addition, progress has been made to improve the calculation of the Higgs boson production cross section with a jet veto (the “0-jet bin” or in the presence of a veto bounding the transverse momentum of the hardest accompanying jet) [66], reaching NNLL accuracy matched to N3LO QCD. These accurate predictions for the jet-veto cross section are required, e.g., to suppress the $t\bar{t}$ background in the $H \to WW$ channel [67].

## ii. Vector boson fusion production mechanism

The SM Higgs boson production mode with the second-largest cross section at the LHC is vector boson fusion. At the Tevatron collider, VBF also occurred, but for $m_H = 125$ GeV had a smaller cross section than Higgs boson production in association with a $W$ or $Z$ boson. Higgs boson production via VBF, $qq \to qqH$, proceeds by the scattering of two (anti-)quarks, mediated by $t-$ or $u-$channel exchange of a $W$ or $Z$ boson, with the Higgs boson radiated off the weak-boson propagator. The scattered quarks give rise to two hard jets in the forward and backward regions of the detector with a large rapidity gap [68]. Because of the colorsinglet nature of the weak-gauge boson exchange, gluon radiation from the central-rapidity regions is strongly suppressed. These characteristic features of VBF processes can be exploited to distinguish them from overwhelming QCD backgrounds, including gluon-fusion induced Higgs boson + 2 jet production, and from s-channel $WH$ or $ZH$ production with a hadronically decaying weak gauge boson. After the application of specific selection cuts,

the VBF channel provides a clean environment, not only for the Higgs boson searches originally performed, but also for the subsequent determination of Higgs boson couplings at the LHC.

The cross-section is known at N3LO QCD at the inclusive level [69] and at NNLO enforcing the VBF cuts [70] with a residual uncertainty of the order of few permill. However, this result is obtained in the DIS/factorized approximation [71] where the fusing gauge bosons are emitted from the two quark legs independently. While, the exact NNLO QCD VBF calculation will remain out-of-reach in the near future, the leading non-factorisable contributions with two forward jets have been estimated [72] and the impact of non-factorizable effects have been studied in Ref. [73]. They give some corrections, also of the order of few permill, to inclusive quantities, but they are an order of magnitude larger for differential observables. Overall, the residual uncertainty is of the order of a few percent but is quite sensitive to the tagging jet cuts and jet radius modelling [74]. Extensive studies on VBF at fixed order and with parton shower matched computations can be found in Ref. [75]. In addition, EW corrections have been included and the VBF process is known at NLO QCD+EW order for some time [76, 77].

## iii. WH and ZH associated production mechanism

The next most relevant Higgs boson production mechanisms are associated production with $W$ and $Z$ gauge bosons. The cross sections for the associated production processes, $pp \to VH+X$, with $V = W^{\pm}, Z$ receive contributions at NLO QCD given by NLO QCD corrections to the Drell-Yan cross section [78, 79] and from NLO EW corrections. The latter, unlike the QCD corrections, do not respect the factorisation into Drell-Yan production since there are irreducible box contributions already at one loop [76, 80]. At NNLO QCD, the Drell-Yan-like corrections to $WH$ production also give the bulk of the corrections to $ZH$ production [81]. For $ZH$ production there are, however, gluon-gluon induced contributions that do not involve a virtual $Z$ gauge boson but are such that the $Z$ gauge boson and $H$ boson couple to gluons via top-quark loops [82], see diagram (d) in Fig. 11.1. NLO QCD virtual corrections to the partonic cross section for $gg \to ZH$, in the high-energy and large top mass limits are available in Refs. [83]. In addition, $WH$ and $ZH$ production receive non Drell-Yan-like corrections in the $q\bar{q}'$ and $q\bar{q}$ initiated channels, respectively, at the NNLO level in QCD, where the Higgs boson is radiated off
<table><caption>**Table 11.2:** The SM Higgs boson production cross sections for $m_H = 125$ GeV in $pp$ collisions ($p\bar{p}$ collisions at $\sqrt{s} = 1.96$ TeV for the Tevatron), as a function of the center of mass energy, $\sqrt{s}$. The predictions for the LHC energies are taken from Refs. [37–40, 43], the ones for the Tevatron energy are the ones used in Ref. [44]. The predictions for the $ggF$ channel at the LHC include the latest N3LO QCD + NLO EW results. The uncertainties are estimated assuming no correlation between $\alpha_S$ and PDF uncertainties. The uncertainties on the total cross sections are dominated by the ones on the gluon fusion process.</caption><thead><tr><th rowspan="2">&radic;<i>s</i><br>(TeV)</th><th colspan="6">Production cross section (in pb) for <i>m</i><sub>H</sub> = 125 GeV</th></tr><tr><th>ggF</th><th>VBF</th><th>WH</th><th>ZH</th><th>ttH</th><th>total</th></tr></thead><tbody><tr><td>1.96</td><td>0.95<sup>+17%</sup><sub>-17%</sub></td><td>0.065<sup>+8%</sup><sub>-7%</sub></td><td>0.13<sup>+8%</sup><sub>-8%</sub></td><td>0.079<sup>+8%</sup><sub>-8%</sub></td><td>0.004<sup>+10%</sup><sub>-10%</sub></td><td>1.23<sup>+15%</sup><sub>-15%</sub></td></tr><tr><td>7</td><td>16.9<sup>+5.5%</sup><sub>-7.6%</sub></td><td>1.24<sup>+2.2%</sup><sub>-2.2%</sub></td><td>0.58<sup>+2.2%</sup><sub>-2.3%</sub></td><td>0.34<sup>+3.1%</sup><sub>-3.0%</sub></td><td>0.09<sup>+5.6%</sup><sub>-10.2%</sub></td><td>19.1<sup>+5%</sup><sub>-7%</sub></td></tr><tr><td>8</td><td>21.4<sup>+5.4%</sup><sub>-7.6%</sub></td><td>1.60<sup>+2.1%</sup><sub>-2.1%</sub></td><td>0.70<sup>+2.1%</sup><sub>-2.2%</sub></td><td>0.42<sup>+3.4%</sup><sub>-2.9%</sub></td><td>0.13<sup>+5.9%</sup><sub>-10.1%</sub></td><td>24.2<sup>+5%</sup><sub>-7%</sub></td></tr><tr><td>13</td><td>48.6<sup>+5.6%</sup><sub>-7.4%</sub></td><td>3.78<sup>+2.1%</sup><sub>-2.1%</sub></td><td>1.37<sup>+2.0%</sup><sub>-2.0%</sub></td><td>0.88<sup>+4.1%</sup><sub>-3.5%</sub></td><td>0.50<sup>+6.8%</sup><sub>-9.9%</sub></td><td>55.1<sup>+5%</sup><sub>-7%</sub></td></tr><tr><td>13.6</td><td>52.2<sup>+5.6%</sup><sub>-7.4%</sub></td><td>4.1<sup>+2.1%</sup><sub>-1.5%</sub></td><td>1.46<sup>+1.8%</sup><sub>-1.9%</sub></td><td>0.95<sup>+4.0%</sup><sub>-3.6%</sub></td><td>0.57<sup>+6.9%</sup><sub>-9.9%</sub></td><td>59.2<sup>+5%</sup><sub>-7%</sub></td></tr><tr><td>14</td><td>54.7<sup>+5.6%</sup><sub>-7.4%</sub></td><td>4.28<sup>+2.1%</sup><sub>-2.1%</sub></td><td>1.51<sup>+1.8%</sup><sub>-1.9%</sub></td><td>0.99<sup>+4.1%</sup><sub>-3.7%</sub></td><td>0.61<sup>+6.9%</sup><sub>-9.8%</sub></td><td>62.1<sup>+5%</sup><sub>-7%</sub></td></tr></tbody></table>

top-quark loops [84]. The full QCD corrections up to NNLO order, the NLO EW corrections and the NLO corrections to the gluon-gluon channel are available in VH@NNLO [85]. NNLO QCD corrections to $pp \to WH + jet$ are important for signal modelling. NLO corrections to cross section inclusive in the number of jets show excellent convergence while NNLO QCD corrections to exclusive single jet cross section are much more significant [86].

As neither the Higgs boson nor the weak gauge bosons are stable particles, their decays also have to be taken into account. Providing full kinematical information for the decay products can furthermore help in the suppression of large QCD backgrounds. Differential distributions for the processes $pp \to WH \to \bar{\nu}_e \ell \ell H$ and $pp \to ZH \to \ell^+ \ell^- H / \nu_e \bar{\nu}_e H$, including NLO QCD and EW corrections, have been presented in Ref. [87]. The NNLO QCD corrections to differential observables for $WH$ production at the LHC, including the leptonic decays of the $W$ boson and the decay of the Higgs boson into a $b\bar{b}$ pair, are presented in Ref. [88]. Calculations at the same level, including also the $ZH$ process have been performed [89]. The $WH$ production mode has also been matched to a parton shower at NNLO QCD accuracy [90]. Full NNLO QCD results for both the production and decay are available [91] and show a large impact of radiation from the final-state bottoms. The $WH$ and $ZH$ production modes, especially in the boosted regime, provide a relatively clean environment for studying the decay of the Higgs boson into bottom quarks [92]. Bottom mass effects have been computed at NNLO [93, 94] and tend to increase the rates by 6-7%.

The inclusive cross section for associated Higgs boson production with a massive gauge boson is recently known up to N3LO QCD too [95].

iv. Higgs boson production in association with $t\bar{t}$

Higgs boson radiation off top quarks, $pp \to t\bar{t}H$, provides a direct probe of the top-Higgs Yukawa coupling. The LO cross section for this production process was computed in Ref. [96]. Later, the NLO QCD [97] and NLO EW corrections [98] were evaluated yielding a moderate increase in the total cross section of at most 20%, but significantly reducing the scale dependence of the inclusive cross section. The EW corrections can be enhanced by large electroweak Sudakov logarithms in particular in the boosted regime often used in the phenomenological analyses [99]. The re-

summation of soft gluon contributions close to the partonic kinematical threshold are considered in Refs. [100] to reach a result at NLO+NNLL QCD order. Full off-shell calculations with decaying top quarks are computed at NLO QCD [101] and NLO QCD + EW [102] order, respectively. The fixed-order NLO QCD calculation has been interfaced with the standard parton shower Monte-Carlo generators in Refs. [103], allowing an accurate description of the $t\bar{t}H$ signal, from the energy scale of the hard scattering to the hadronisation energy scale. These programs provide the most flexible tools to date for the computation of differential distributions, including experimental selection cuts and vetoes on the final-state particles and their decay products. The exploitation of this $t\bar{t}H$ channel requires, however, a proper description of the background, in particular $t\bar{t}b\bar{b}$, which exhibits a huge k-factor<fnref n="1" /> enhancement from shower effects, see Ref. [40] for a detailed discussion. While the $H \to b\bar{b}$ decay mode was first considered, the focus has been broadened and other modes have been studied too, in particular the multilepton channel and other backgrounds such as $t\bar{t}W$ and $t\bar{t}Z$ have become relevant and intensively analyzed.

The total theoretical uncertainties, estimated by combining the ones from factorisation and renormalisation scales, strong gauge coupling, and parton distributions, amount to 10–15% of the corresponding inclusive cross section. To match the experimental precision envisioned at the end of HL-LHC, NNLO QCD predictions are required. Using the $q_T$ subtraction formalism for $t\bar{t}H$ production, Ref. [104] presented the first quantitative results at NNLO. The calculation is accurate at NLO in QCD and agrees with the results obtained using the conventional subtraction methods. The NNLO corrections are computed for the flavour off-diagonal partonic channels but their contribution turn out to be at the permill level. The results for the diagonal channels have been obtained in Ref. [41]. The two-loop amplitudes are not yet known, but they have been reliably evaluated by using a soft Higgs boson approximation. At the 13 TeV centre-of-mass energy, the NNLO corrections increase the NLO result for the total cross section by about 4% and lead to a significant reduction of perturbative uncertainties. Also recently, the fragmentation and splitting functions for the production of a Higgs boson have been computed at order $y_t^2 \alpha_s$ and they can be used to compute dif-

<footnote n="1">¹the k-factor is defined as the ratio of a physical quantity with and without radiative corrections included.</footnote>
ferential cross sections with arbitrary top-quark and Higgs-boson masses directly from the results in the massless limit [105].

### v. Other single Higgs boson production mechanisms at the LHC

The Higgs boson production in association with a single top quark, though subdominant, can bring valuable information, in particular regarding the sign of the top Yukawa coupling. This is due to an almost totally destructive interference between two large contributions, one where the Higgs boson couples to a space-like W boson and the other where it couples to the top quark. This process has been computed at NLO QCD in a five-flavour scheme [106] and amounts to about 90 fb at $\sqrt{s} = 14$ TeV (with the opposite sign of the top Yukawa coupling, the cross section increases by one order of magnitude while the cross section for associated production with a pair of top quarks is unaffected). Recently, the $tHj$ (and $tZj$) process has been computed at NLO QCD+EW accuracy [107]. The NLO EW correction is found to be within the NLO QCD theory uncertainties if the 4 vs 5 flavour scheme uncertainty is taken into account. The EW corrections reduce the total cross section by 3-4%.

The Higgs boson production in association with a pair of bottom quarks ($b\bar{b}H$) has been computed at the N3LO QCD order in the 5-flavour scheme [108], while the production with one/two b-jets, defined by specific kinematic cuts, is known respectively at NLO QCD in the 4-flavour scheme and at NNLO QCD in the 5-flavour scheme [109–111]. The coupling of the Higgs boson to a b-quark is suppressed in the SM by the bottom-quark mass over the Higgs VEV, $m_b/v$, implying that associated production of a SM Higgs boson with b-quarks is small at the LHC. Yet, at high energy, large logarithms are present and need to be resummed, leading to an enhancement of the inclusive cross section. At $\sqrt{s} = 14$ TeV, the $b\bar{b}H$ cross section can be as large as 550 fb, still two orders of magnitude below the ggF production cross section. Furthermore, the overwhelming $ZH$ and VBF backgrounds (at the inclusive but also differential levels) make it challenging to extract genuine bottom Yukawa signal in a traditional cut-based analysis [112] but innovative machine-learning techniques are a promising avenue to obtain information complementary to the ones extracted from $gg \to h \to b\bar{b}$, in particular for what concerns the phase of the bottom Yukawa coupling [113]. In a two Higgs doublet model or a SUSY model, which will be discussed in Section 11.6, this coupling is proportional to the ratio of neutral Higgs boson vacuum expectation values, $\tan\beta$, and can be significantly enhanced for large values of this ratio. Consequently, the $b\bar{b}H$ mode can even become the dominant production process for the Higgs boson, unlike in the SM.

The Higgs boson production in association with charm quarks is also known at NNLO QCD and its cross section is approximately 85 fb at $\sqrt{s} = 13$ TeV [40].

### vi. Double Higgs boson production at the LHC

The main interest in the double Higgs boson production is that it can provide invaluable information on the Higgs potential. In particular, it gives access to the Higgs trilinear self coupling. The dominant production is via gluon fusion $gg \to HH$ (slightly more than 31 fb at 13 TeV) and it accounts for more than 90% of the total inclusive cross-section, the sub-leading production mechanisms are VBF $HHjj$ (around 1.7 fb at 13 TeV), $HHW$ (0.50 fb), $HHZ$ (0.36 fb) and $t\bar{t}HH$ (0.8 fb). The fixed order QCD corrections, computed in the infinite top mass limit, are large, typically doubling the cross section from LO to NLO QCD [114], further enhancing it by 20% and reducing the scale uncertainty by a factor 2-3 from NLO to NNLO QCD [115], with a milder increase of order 3% at N3LO [116] to finally reach at 13 TeV

$$ \sigma(gg \to HH)_{ggF, m_t \to \infty}^{\text{N3LO}} = 31.31_{-2.8\%}^{+0.66\%} \text{ fb.} \quad (11.11) $$

Recently, the complete NLO corrections with all top quark mass effects also became available numerically [117], intriguingly revealing a k-factor much less flat than predicted in the large top mass approximations. The non-trivial dependence of the results on the renormalisation scheme and scale for the top quark mass

questions the assessment of the scale uncertainty and would warrant a proper NNLO computation that will however remain out of reach for quite some time. At the differential level, the destructive interference between the box and the triangle contributions complicates the predictions made in the infinite top mass limit for both the $HH$ invariant mass and the leading Higgs boson $p_T$ distributions. With an inclusive cross section of about 35 fb at $\sqrt{s} = 13$ TeV and a difficult signal vs background discrimination, the double Higgs boson production remains a challenging channel to probe and will greatly benefit from the high-luminosity run of the LHC and the combination of various decay modes [118]. The double Higgs channel is definitively a prime target on the Higgs physics agenda and the forthcoming analyses will greatly benefit from the solidification of the fixed order predictions and the detailed uncertainty budget estimate that is underway [119].

### 11.2.4.2 Production mechanisms at $e^+e^-$ colliders

The dominant Higgs boson production cross sections at an $e^+e^-$ collider are from the Higgs-strahlung process [21, 120], $e^+e^- \to ZH$, and the WW fusion process [121], $e^+e^- \to \bar{\nu}_e\nu_e W^*W^* \to \bar{\nu}_e\nu_e H$. The cross-section for the Higgs-strahlung process scales as the inverse of the collision centre-of-mass squared energy, $s$, and is predominant at low energies, while the cross-section for the WW fusion process scales as $\ln(s/m_H^2)$ and dominates at high energies [122]. The ZZ fusion mechanism, $e^+e^- \to e^+e^-Z^*Z^* \to e^+e^-H$, also contributes to the Higgs boson production, with a cross-section suppressed by an order of magnitude with respect to that of WW fusion. The process $e^+e^- \to t\bar{t}H$ [123] becomes important for $\sqrt{s} \ge 500$ GeV. For a more detailed discussion of Higgs boson production properties at lepton colliders, see for example Refs. [124]. Interesting progress has been made recently [125] to accurately account for beam-strahlung and initial state radiation using the factorisation approach primarily developed for hadron colliders and this method is being implemented in Monte Carlo event generators. Further information about Higgs physics at an $e^+e^-$ collider can be found in Ref. [126].

### 11.2.4.3 SM Higgs boson branching ratios and total width

For the understanding and interpretation of the experimental results, the computation of all relevant Higgs boson decay widths is essential, including an estimate of their uncertainties and, when appropriate, the effects of Higgs boson decays into off-shell particles with successive decays into lighter SM ones. A Higgs boson mass of about 125 GeV allows to explore the Higgs boson couplings to many SM particles. In particular the dominant decay modes are $H \to b\bar{b}$ and $H \to WW^*$, followed by $H \to gg$, $H \to \tau^+\tau^-$, $H \to c\bar{c}$ and $H \to ZZ^*$. With much smaller rates follow the Higgs boson decays into $H \to \gamma\gamma$, $H \to \gamma Z$ and $H \to \mu^+\mu^-$. Since the decays into gluons, diphotons and $Z\gamma$ are loop induced, they provide indirect information on the Higgs boson couplings to $WW$, $ZZ$ and $t\bar{t}$ in different combinations. The uncertainties in the branching ratios include the missing higher-order corrections in the theoretical calculations as well as the errors in the SM input parameters, in particular fermion masses and the QCD gauge coupling, involved in the decay. In the following the state-of-the-art of the theoretical calculations compiled by the LHC Higgs Working Group [43] will be discussed and the reader is referred to Refs. [37–40, 127] for detail.

The evaluation of the radiative corrections to the fermionic decays of the SM Higgs boson are implemented in HDECAY [128] at different levels of accuracy. The computations of the $H \to b\bar{b}$ and $H \to c\bar{c}$ decays include the complete massless QCD corrections up to N4LO, with a corresponding scale dependence of about 0.1% [129]. Both the electroweak corrections to $H \to b\bar{b}$, $c\bar{c}$ as well as $H \to \tau^+\tau^-$ are known at NLO [130] providing predictions with an overall accuracy of about 1–2% for $m_H \simeq 125$ GeV.

The loop induced decays of the SM Higgs boson are known fully at NLO and partially beyond that approximation. For $H \to gg$, the QCD corrections are known up to N3LO in the limit of heavy top quarks [47,131] and the uncertainty from the scale dependence is about 3%. For the $H \to \gamma\gamma$, the full NLO QCD corrections are available [47, 132] and the three-loop QCD corrections have also been evaluated [133]. The NLO electroweak corrections to
$H \to gg$ and $H \to \gamma\gamma$ have been computed in Ref. [134]. All these corrections are implemented in HDECAY [128]. For $m_H \simeq 125$ GeV, the overall impact of known QCD and EW radiative effects turns out to be well below 1%. In addition, the contribution of the $H \to \gamma e^+ e^-$ decay via virtual photon conversion has been computed in Ref. [135]. The partial decay width $H \to Z\gamma$ is only implemented at LO in HDECAY, including the virtual W, top-, bottom-, and $\tau$-loop contributions. The QCD corrections have been calculated and are at the percent level [136]. The theoretical uncertainty due to unknown electroweak corrections is estimated to be less than 5%, an accuracy that will be hard to achieve in the measurement of this process at the LHC.

<table><caption>**Table 11.3:** The branching ratios and the relative uncertainty for a SM Higgs boson with $m_H = 125$ GeV [39, 40].</caption><thead><tr><th>Decay channel</th><th>Branching ratio</th><th>Rel. uncertainty</th></tr></thead><tbody><tr><td>$H \to \gamma\gamma$</td><td>2.27 × 10<sup>-3</sup></td><td>2.1%</td></tr><tr><td>$H \to ZZ$</td><td>2.62 × 10<sup>-2</sup></td><td>±1.5%</td></tr><tr><td>$H \to W^{+}W^{-}$</td><td>2.14 × 10<sup>-1</sup></td><td>±1.5%</td></tr><tr><td>$H \to \tau^{+}\tau^{-}$</td><td>6.27 × 10<sup>-2</sup></td><td>±1.6%</td></tr><tr><td>$H \to b\bar{b}$</td><td>5.82 × 10<sup>-1</sup></td><td>+1.2%<br/>-1.3%</td></tr><tr><td>$H \to c\bar{c}$</td><td>2.89 × 10<sup>-2</sup></td><td>+5.5%<br/>-2.0%</td></tr><tr><td>$H \to Z\gamma$</td><td>1.53 × 10<sup>-3</sup></td><td>±5.8%</td></tr><tr><td>$H \to \mu^{+}\mu^{-}$</td><td>2.18 × 10<sup>-4</sup></td><td>±1.7%</td></tr></tbody></table>

The decays $H \to WW/ZZ \to 4f$ can be simulated with the Prophecy4f Monte-Carlo generator [137] that includes complete NLO QCD and EW corrections for Higgs decays into any possible four-fermion final state. All calculations are consistently performed with off-shell gauge bosons, without any on-shell approximation. For the SM Higgs boson, the missing higher-order corrections are estimated to be roughly 0.5%. Such uncertainties will have to be combined with the parametric uncertainties, in particular those associated to the bottom-quark mass and the strong gauge coupling, to arrive at the full theory uncertainty. A detailed treatment of the differential distributions for a Higgs boson decay into four charged leptons in the final state is discussed in Refs. [39, 138].

The total width of a 125 GeV SM Higgs boson is $\Gamma_H = 4.07 \times 10^{-3}$ GeV, with a relative uncertainty of $+4.0\%$ or $-3.9\%$. The branching ratios for the most relevant decay modes of the SM Higgs boson as a function of $m_H$, including the most recent theoretical uncertainties, are shown in Fig. 11.2 (right) and listed for $m_H = 125$ GeV in Table 11.3. Further details of these calculations can be found in the reviews [37–40] and references therein.

## 11.3 The experimental profile of the Higgs boson

The observation [1, 2] at the LHC of a narrow resonance with a mass of about 125 GeV was an important landmark in the decades-long direct search [44, 139] for the SM Higgs boson. This observation is being followed by a detailed exploration of properties of the Higgs boson at the different runs of the LHC at $\sqrt{s} = 7, 8, 13$ and 13.6 TeV.

The dataset at $\sqrt{s}=13$ TeV collected during the Run-2 phase of the LHC operation is the largest analysed dataset so far and corresponds to an integrated luminosity of about $160 \text{ fb}^{-1}$ delivered to the ATLAS and CMS experiments, see Table 11.4 [140]. The datasets effectively useful for analysis need to take into account the data-taking efficiency with fully operational detectors and the data quality efficiency. The typical total inefficiency for both ATLAS and CMS is approximately 10%, where approximately half

is due to the data acquisition inefficiency and the other half from data quality requirements.

In this section, most of the references for the Run-1 measurements that have been updated with the data delivered in Run 2 are given in the previous version of this review [141] and are not repeated herein.

<table><caption>**Table 11.4:** The LHC pp collision centre-of-mass energies and approximate collision samples delivered to ATLAS & CMS experiments.</caption><thead><tr><th>Year</th><th>√s (TeV)</th><th>∫ L.dt (fb<sup>-1</sup>)</th><th>Period</th></tr></thead><tbody><tr><td>2010</td><td>7.0</td><td>0.04</td><td>Run 1</td></tr><tr><td>2011</td><td>7.0</td><td>6</td><td>Run 1</td></tr><tr><td>2012</td><td>8.0</td><td>23</td><td>Run 1</td></tr><tr><td>2015</td><td>13.0</td><td>4</td><td>Run 2</td></tr><tr><td>2016</td><td>13.0</td><td>39</td><td>Run 2</td></tr><tr><td>2017</td><td>13.0</td><td>51</td><td>Run 2</td></tr><tr><td>2018</td><td>13.0</td><td>61</td><td>Run 2</td></tr><tr><td>2022</td><td>13.6</td><td>39</td><td>Run 3</td></tr><tr><td>2023</td><td>13.6</td><td>32</td><td>Run 3</td></tr></tbody></table>

### 11.3.1 The principal decay channels of the Higgs boson

For a given $m_H$, the sensitivity of a channel depends on the production cross section of the Higgs boson, its decay branching fraction, the reconstructed mass resolution, the selection efficiency and the level of background in the final state. It is defined typically as the expected significance for the observation of a given process or in terms of expected upper limit at 95% Confidence Level (CL). At the LHC, for a Higgs boson with a mass close to 125 GeV, five decay channels play an important role. In the $H \to \gamma\gamma$ and $H \to ZZ^* \to 4\ell$ channels, all final state particles can be very precisely measured and the reconstructed $m_H$ resolution is excellent (typically 1-2%). The $H \to W^{\pm}W^{\mp(\pm)} \to \ell^{+}\ell^{-}\nu_{\ell}\nu_{\ell'}$ channel has a relatively large branching fraction. However, the presence of neutrinos, which are not reconstructed in the final state, means that the $m_H$ resolution, obtained through observables sensitive to the Higgs boson mass such as the transverse mass, is poor (approximately 20%). The $H \to b\bar{b}$ and the $H \to \tau^{+}\tau^{-}$ (with the tau subsequently decaying to electrons or muons and two neutrinos $\tau \to \ell\nu\bar{\nu}$ or to hadrons and one neutrino $\tau \to \text{hadrons}\nu$) channels suffer from large backgrounds and lead to an intermediate mass resolution of about 10% and 15% respectively.

With the increase in the size of datasets, measurements in the most sensitive channels are now carried out differentially or in exclusive modes depending on specific production characteristics. These measurements are discussed in Section 11.5.1.5.

The candidate events in each Higgs boson decay channel are split into several mutually exclusive categories (or event tags) based on the specific topological, kinematic or other features present in the event. The categorization of events increases the sensitivity of the overall analysis and allows a separation of different Higgs boson production processes. Most categories are dominated by signal from one Higgs boson decay mode but contain an admixture of various Higgs boson production processes. For example, a typical VBF selection requires Higgs boson candidates to be accompanied by two energetic jets ($\ge 30$ GeV) with a large dijet mass ($\ge 400$ GeV) and separated by a large pseudo-rapidity ($\Delta\eta_{jj} \ge 3.5$) [142]. While such a category is enriched in Higgs bosons produced via VBF, the contamination from the ggF or VH production mechanisms can be significant. Hence, a measurement of the signal rate in the VBF category is not equivalent to a measurement of VBF production cross section since one cannot resolve the contamination from ggF. Simulations are used to determine the relative contributions of the various Higgs boson production modes in each of the specific categories.

### 11.3.2 The Higgs boson decay to vector bosons

#### 11.3.2.1 $H \to \gamma\gamma$

In the diphoton channel, the measurement of a narrow peak over a smoothly falling background in the invariant mass distribution of two high $p_T$ photons is performed. The background in
this channel is conspicuous and stems from prompt γγ processes for the irreducible backgrounds, and the γ+jet and dijet processes for the reducible backgrounds where one jet fragments typically into a leading π⁰. In order to optimise search sensitivity and also to separate the various Higgs boson production modes, ATLAS and CMS split events into several mutually exclusive categories. Diphoton events containing high p<sub>T</sub> muons or electrons, or missing energy (E<sub>T</sub><sup>miss</sup>) consistent with the decay of a W or Z boson, are tagged in the VH production category. Diphoton events containing energetic dijets with a large mass and pseudo-rapidity difference are assigned to the VBF production category, and events with two jets compatible with the hadronic decay of a W or a Z are considered in the VH category. While the leptonic VH category is relatively pure, the VBF category has significant contamination from the gluon fusion process. Finally more complex topologies aiming at selecting events from the t<span style="text-decoration: overline">t</span>H production mode in the semi-leptonic and fully hadronic top decays are also considered and discussed in more detail in Section 11.3.5.1. Events are also further categorised according to their expected m<sub>γγ</sub> resolution and signal-to-background ratio. Categories with good m<sub>H</sub> resolution and larger signal-to-background ratio contribute most to the sensitivity of the search.

Both ATLAS and CMS have studied in detail the calibration of the energy response of photons, in particular using $Z \to e^+e^-$, $Z \to \mu^+\mu^-\gamma$ and the response of muons in the calorimeter (for ATLAS) from $Z \to \mu^+\mu^-$ events. This information is used to correct the simulated signal mass line-shapes. In each category, parametric signal models are adjusted to these line-shapes to provide a functional form for the signal. Simple functional forms of the backgrounds are determined by a fit to the $m_{\gamma\gamma}$ distribution in each category (typically exponential, Bernstein polynomials, Laurent series or power laws). All categories are fitted simultaneously to determine the signal yield at a given Higgs boson mass hypothesis.

The results are expressed in terms of signal strength, which can be defined as $\mu = (\sigma \cdot \text{BR})_{\text{obs}} / (\sigma \cdot \text{BR})_{\text{SM}}$, the observed product of the Higgs boson production cross section ($\sigma$) and its branching ratio (BR) normalised to the corresponding SM values. A more detailed definition is given in Section 11.3.6 and Eq. 11.14. The signal strength overall for this channel and for specific production processes are summarised in Fig. 11.3. All results are in agreement with the SM expectation within the overall measurement precisions of approximately 9% for both experiments.

$$11.3.2.2 \quad H \to ZZ^* \to \ell^+\ell^-\ell'^+\ell'^-$$

In the $H \to ZZ^* \to \ell^+\ell^-\ell'^+\ell'^-$ channel, a search is performed for a narrow mass peak over a small continuous background dominated by non-resonant $ZZ^*$ production from $q\bar{q}$ annihilation and $gg$ fusion processes. The contribution and the shape of this irreducible background is taken from simulation. The subdominant and reducible backgrounds stem from $Z+b\bar{b}$, $t\bar{t}H$ and $Z+jets$ events. Their contribution is suppressed by requirements on lepton isolation and lepton impact parameter and their yield is estimated from control samples in data.

To distinguish the Higgs boson signal from the dominant non-resonant $ZZ^*$ background, both ATLAS and CMS use multivariate techniques as well as a matrix element likelihood approach to construct a kinematic discriminant built for each $4\ell$ event based on the ratio of complete leading-order matrix elements $|M_{\text{sig}}^2/M_{\text{bkg}}^2|$ for the signal ($gg \to H \to 4\ell,qq \to VH(\to 4\ell)$ and $qq \to qqH(\to 4\ell)$) and background ($q\bar{q} \to ZZ \to 4\ell$) hypotheses.

To improve the sensitivity to more exclusive production processes such as VBF, VH, and $t\bar{t}H$, the experiments divide $4\ell$ events into mutually exclusive categories. Events are categorised in terms of the number of reconstructed jets, the number of additional leptons (from the decay of a vector boson in the associated production mode), number of jet tagged as containing a b-hadron, the transverse momentum of the Higgs boson (or e.g. its associated vector boson) and missing transverse momentum. Dijets with a large mass and pseudo-rapidity difference populate the VBF category. ATLAS requires the presence of an additional lepton in the VH category. In events with less than two jets, CMS uses the $p_T^\ell$ to distinguish between production via the gluon fu-

sion and the VH/VBF processes.

Since the $m_{4\ell}$ resolutions and the reducible background levels are different in the $4\mu, 4e$ and $2e2\mu$ sub-channels, they are analysed separately and statistically combined to produce the final results.

The results for this channel, overall and in specific production processes are summarised in Fig. 11.3. All results are in agreement with the SM expectation within the overall measurement precisions of approximately 9% and 12% for the ATLAS and CMS experiments respectively.

$$11.3.2.3 \quad H \to W^+W^- \to \ell^+\nu\ell^-\bar{\nu}$$

In this challenging channel, experiments search for an excess of events with two leptons (electrons or muons) of opposite charge accompanied by missing energy and/or jets. A typical event selection is described below in order to give an idea of the main intricacies. Specific selections vary between experiments and between Run-1 and Run-2 analyses. Events are divided into several categories depending on the lepton flavour combination ($e^+e^-$, $\mu^+\mu^-$ and $e^\pm\mu^\mp$) and the number of accompanying jets ($N_{jet} = 0, 1, \ge 2$). The $N_{jet} \ge 2$ category is optimised for the VBF production process by selecting two leading jets with a large pseudo-rapidity difference and with a large mass (typically $m_{jj} > 500$ GeV).

Backgrounds contributing to this channel are numerous and depend on the category of selected events. Reducing them and accurately estimating the remainder is a major challenge in this analysis. For events with opposite-flavour leptons and no accompanying high $p_T$ jets, the dominant background stems from non-resonant WW production. Events with same-flavour leptons suffer from large Drell-Yan contamination (note that also the opposite-flavour leptons analysis has Drell-Yan $\tau\bar{\tau}$ background in 0-jet category). The $t\bar{t}$, $tW$ and $W+jets$ (with the jet misidentified as a lepton) events contaminate all categories. Non-resonant WZ, ZZ and $W\gamma$ processes also contribute to the background at a sub-leading level.

A requirement of large missing transverse energy ($E_T^{\text{miss}}$) is used to reduce the Drell-Yan and multi-jet backgrounds. In the $e^+e^-$ and $\mu^+\mu^-$ categories, events with $m_{\ell\ell}$ consistent with the $Z$ mass are vetoed. The $t\bar{t}$ background is suppressed by a veto against identified b-jets and tight isolation requirements diminish the $W+jets$ background. The scalar nature of the Higgs boson and the $V-A$ nature of the W boson decay implies that the two charged leptons in the final state are preferentially emitted at small angles with respect to each other. Therefore the dilepton invariant mass ($m_{\ell\ell}$) and the azimuthal angle difference between the leptons ($\Delta\phi_{\ell\ell}$) are used to discriminate between the signal and non-resonant WW events [143]. The transverse mass, constructed from the dilepton $p_T$ ($p_T^\ell$), $E_T^{\text{miss}}$ and the azimuthal angle between $E_T^{\text{miss}}$ and $p_T^\ell$, is defined as $m_T = \sqrt{2p_T^\ell E_T^{\text{miss}}(1 - \cos \Delta\phi_{E_T^{\text{miss}}\ell\ell})}$ and serves as an effective discriminant against backgrounds. The transverse mass variable also tracks the Higgs boson mass but with a poor mass resolution. Background rates except for the small contributions typically from non-resonant WZ, ZZ and $W\gamma$ are evaluated from data control samples with floating normalization.

ATLAS and CMS have also searched for the associated Higgs boson production in this channel. The signal consists of up to three (WH) or four (ZH) high $p_T$ isolated leptons with missing transverse energy and low hadronic activity. The major backgrounds stem from triboson and diboson production where each boson decays leptonically.

Measurements by ATLAS and CMS, in terms of signal strengths, for the five largest Higgs boson production modes are displayed in Fig. 11.3. All measurements are in agreement with the SM expectation within the overall measurement precisions of approximately 12% and 9% for ATLAS and CMS respectively.

### 11.3.3 Measurement of the Higgs boson mass

To measure the mass of the Higgs boson, ATLAS and CMS collaborations rely on the two high mass-resolution channels, $\gamma\gamma$ and $ZZ^*/4\ell$. The ATLAS and CMS approaches are very similar in these two analyses with small differences on the usage of cat-
egories, additional discriminating variables and per-event errors. In these two channels, the mass resolutions range from 1.4 GeV to 2 GeV for ATLAS and from 1.0 GeV to 2.8 GeV for CMS (see Ref. [144] and the reconstruction-performance references therein). An excellent mass resolution is obtained for both experiments in the diphoton channel for central diphoton pairs (typically for events where both photons are not converted). However, the best precision in this measurement at Run 2 is obtained using the four lepton channel and in particular in the sub-channel of four muons (followed, in precision, by the two-electrons and two muons channel where the mass resolution is driven by the reconstruction of the lower mass di-lepton pair with typically lower transverse momentum leptons).

Both ATLAS and CMS have produced several Higgs boson mass measurements, including the individual and combined Run-1 measurements [144] and several Run-2 measurements by ATLAS [145,146] and CMS [147,148] for both the 4ℓ and the diphoton channels. The measurements performed by ATLAS include the full Run-2 dataset whereas in the case of CMS, the mass measurements were done with a fraction of the Run-2 data corresponding to the data collected in 2016. An overall combination of all available measurements has not yet been performed. The Run-1 combination including both ATLAS and CMS measurements yielded [144]:

$$m_H = 125.09 \pm 0.21 \text{ (stat.)} \pm 0.11 \text{(syst.) GeV.}$$

The current most precise di-photon channel result is obtained by ATLAS using the full Run-2 dataset and a combination with the Run-1 measurements in the $H \to \gamma\gamma$ yielding a measurement of [149]:

$$m_H = 125.22 \pm 0.11 \text{ (stat.)} \pm 0.09 \text{(syst.) GeV.}$$

The systematic uncertainty of 90 MeV is worth noting, as it results from a very challenging calibration and estimates of the amount of material upstream of the calorimeter to extrapolate the calorimeter energy response calibration from electrons in Z boson decays to photons. This result is promising for prospects in precision measurements in this channel. In Ref. [149], ATLAS has also estimated the expected effect of the interference between the signal and the gluon fusion continuum di-photon background to be of 24 MeV. This ATLAS measurement is significantly more precise than the full Run-2 measurement in the 4ℓ channel, however the latter is dominated by the statistical uncertainty of 170 MeV with a systematic uncertainty of 30 MeV [150]. The ATLAS diphoton channel measurement reached the same sensitivity as the early Run-2 and Run-1 combined measurement of CMS, which uses both channels but with a much smaller amount of Run-2 data, corresponding only to the 2016 LHC run, which yielded [148]:

$$m_H = 125.38 \pm 0.11 \text{ (stat.)} \pm 0.08 \text{(syst.) GeV.}$$

ATLAS has also combined its results in the diphoton and the four-lepton channels yielding the current most precise measurement with a precision of 0.09% [151]:

$$m_H = 125.11 \pm 0.09 \text{ (stat.)} \pm 0.06 \text{(syst.) GeV.}$$

The most precise single channel result has very recently been obtained by CMS in the 4ℓ channel with the full Run 2 dataset [152]:

$$m_H = 125.08 \pm 0.12 \text{ GeV.}$$

The analysis of the entire Run-2 dataset and the LHC combination of the two experiments will be of great interest and important for the combination of all Higgs boson measurements across experiments.

### 11.3.4 Higgs decays to third generation fermions ($b\bar{b}$ and $\tau^+\tau^-$)

In the SM, fermions acquire a mass through gauge invariant interactions with the Higgs field which is also responsible for the electroweak symmetry breaking and thus for generating the masses of gauge bosons (see Section 11.2 for more details). While

this minimal solution is very elegant, there is no fundamental reason for it to be the case, and probing the couplings of the Higgs boson to fermions is therefore of fundamental importance, in particular since BSM physics can largely change the SM predictions.

The discovery of the Higgs boson was made essentially through the bosonic final states. These decays probed mostly the couplings of the Higgs boson to vector bosons (the decay of the Higgs boson to photons occurring only through loops is also dominated in the SM by the coupling of the Higgs boson to W bosons). However, the predominant Higgs boson production mode is the gluon fusion, occurring only through loops dominated by the coupling of the Higgs boson to fermions. The observation of the Higgs boson in the two photons or two gluons decay modes is also an indirect evidence for the coupling of the Higgs boson to fermions (and in particular to the top quark). Nevertheless, the observation of either decays to fermions or production modes which unambiguously proceed through fermion couplings provide direct probes of the coupling of the Higgs boson to fermions and is thus of fundamental importance.

At hadron colliders, the most promising channel for probing the coupling of the Higgs field to the quarks and leptons are $H \to b\bar{b}$ and $H \to \tau^+\tau^-$, respectively. For a Higgs boson with $m_H \approx 125$ GeV, the branching fraction to $b\bar{b}$ is about 58% and to $\tau^+\tau^-$ is about 6%. Nevertheless, the presence of very large backgrounds makes the measurement of a Higgs boson signal in these channels challenging.

One of the most prominent goals of the LHC Run-2 physics program was the direct observation of the Yukawa coupling of the Higgs boson to fermions of the third generation (bottom quarks, tau leptons and top quarks). This goal has been reached independently by both ATLAS and CMS and with only partial Run-2 datasets.

#### 11.3.4.1 $H \to \tau^+\tau^-$

In the $H \to \tau^+\tau^-$ search, $\tau$ leptons decaying to electrons ($\tau_e$), muons ($\tau_\mu$) and hadrons ($\tau_{had}$) are considered. The $\tau^+\tau^-$ invariant mass ($m_{\tau\tau}$) is reconstructed from a kinematic fit of the visible products from the two $\tau$ leptons and the missing energy observed in the event. Due to the presence of missing neutrinos, the $m_{\tau\tau}$ resolution is poor ($\approx 10\% - 15\%$). As a result, a broad excess over the expected background in the $m_{\tau\tau}$ distribution is searched for. The major sources of background stem from Drell-Yan $Z \to \tau^+\tau^-$ and $Z \to e^+e^-$, W+jets, $t\bar{t}$ and multijet production. Events in all sub-channels are divided into categories based on the number and kinematic properties of additional energetic jets in the event and the transverse momentum of the reconstructed Higgs boson and the distance $\Delta R$ distance between the two $\tau$'s. The sensitivity of the search is generally higher for categories with one or more additional jets. The VBF category, consisting of a $\tau$ pair with two energetic jets separated by a large pseudo-rapidity, has the best signal-to-background ratio and search sensitivity, followed by the $\tau^+\tau^- + 1$ jet category. The signal to background discrimination relies in part on the $m_{\tau\tau}$ resolution, which improves with the boost of the Higgs boson. The non-VBF categories are further subdivided according to the observed boost of the $\tau^+\tau^-$ system. ATLAS and CMS combine various kinematic properties of each event categories with multivariate techniques to build the final discriminant.

Searches for $H \to \tau^+\tau^-$ decays in the VH production mode are also performed in final states where the W or Z boson decays into leptons or jets. The irreducible background in this search arises from non-resonant WZ and ZZ diboson production. The reducible backgrounds originate from W, Z, and $t\bar{t}$ events that contain at least one fake lepton in the final state due to a misidentified jet. The shape and yield of the major backgrounds in each category are estimated from control samples in data. Contributions from non-resonant WZ and ZZ diboson production are estimated from simulations but corrected for reconstruction efficiency using control samples formed from observed data.

The first clear evidence of direct decays of the higgs boson to taus was achieved by ATLAS and CMS in combination [153], with the observation of an excess with a significance of 5.5$\sigma$ and the combined signal strength is $\mu = 1.11_{-0.22}^{+0.24}$, consistent with the SM expectation. The early Run-2 data fully corroborated these
results to establish an unambiguous observation in each experiment independently. ATLAS [154] and CMS [155, 156] analysed an early Run-2 dataset to provide independent observations, with significances of $4.4\sigma$ and $4.9\sigma$ respectively.

Both ATLAS [157] and CMS [158, 159] have then completed the analyses of the full Run-2 dataset in various production channels. These results are summarised in Fig. 11.3. These results perform simultaneously measurements of the VBF production, the gluon fusion mode where the Higgs boson has a sizeable boost, and the associated production with a vector boson as well as the associated production with a pair of top quarks. It is interesting to note that the with the current measurements the gluon fusion channels have a sensitivity which is very close to the VBF production.

$$11.3.4.2 \quad H \to \bar{b}b$$

To measure the decays of the Higgs boson to a pair of b-quarks, the most sensitive production modes are the associated WH and ZH processes allowing use of the leptonic W and Z decays which are helpful for triggering the events and to reject QCD backgrounds. The W bosons are reconstructed via their leptonic decay $W \to \ell\bar{\nu}_\ell$ where $\ell = e, \mu$ or $\tau$. The Z bosons are reconstructed via their decay into $e^+e^-$, $\mu^+\mu^-$ or $\nu\bar{\nu}$. The Higgs boson candidate mass is reconstructed from two b-tagged jets in the event. Backgrounds arise from production of W and Z bosons in association with gluon, light and heavy-flavoured jets (V+jets), $t\bar{t}$, diboson (ZZ and WZ with $Z \to b\bar{b}$) and QCD multi-jet processes. Due to the limited $m_{b\bar{b}}$ mass resolution, a SM Higgs boson signal is expected to appear as a relatively broad enhancement in the reconstructed dijet mass distribution. The crucial elements in this search are b-jet tagging with high efficiency and low fake rate, accurate estimate of b-jet momentum and estimate of backgrounds from various signal depleted control samples constructed from data.

At the Tevatron, the $H \to \bar{b}b$ channel contributed the majority of the Higgs boson search sensitivity below $m_H = 130$ GeV. To separate signal from background, CDF and D0 used multivariate analysis (MVA) techniques that combined several discriminat- ing variables into a single final discriminant. Each channel was divided into exclusive sub-channels according to various lepton, jet multiplicity, and b-tagging characteristics in order to group events with similar signal-to-background ratio and thus optimise the overall search sensitivity. The combined CDF and D0 data showed [44, 160] an excess of events with respect to the predicted background in the 115-140 GeV mass range in the most sensitive bins of the discriminant distributions suggesting the potential presence of a signal. At $m_H = 125$ GeV, the observed signal strength was $\mu = 1.59^{+0.69}_{-0.72}$.

At the LHC, in order to reduce the dominant V+jets background, following Ref. [92], experiments select a region in the V H production phase space where the vector boson is significantly boosted and recoils from the $H \to \bar{b}b$ candidate with a large azimuthal angle $\Delta\phi_{V H}$. For each channel, events are categorised into different $p_T(V)$ regions with varying signal/background ratios. Events with higher $p_T(V)$ have smaller backgrounds and better $m_{b\bar{b}}$ resolution. Experiments use MVA classifiers based on kinematic, topological and quality of b-jet tagging and trained on different values of $m_H$ to separate Higgs boson signal in each category from backgrounds. The main discriminating variables in this analysis are the transverse momentum of the vector boson and the di-b-jet invariant mass.

The direct observation of the Higgs boson decaying to a pair of b-quarks, a major result of Run 2, was obtained by both ATLAS and CMS independently after the update of their search with similar analyses as those performed at Run 1 but with a larger dataset of approximately 80 fb$^{-1}$ of data collected in 2015, 2016 and 2017. The increase in signal cross sections of nearly a factor of 3 at the centre-of-mass energy of 13 TeV with respect to 7 TeV, has also been instrumental in bringing the two experiments to the required sensitivity to claim an evidence for this decay mode in the V H production mode (in the high transverse momentum of the vector boson fiducial region of interest for this channel). The expected significance for a SM Higgs boson was $4.3\sigma$ for ATLAS [161] and $4.9\sigma$ for CMS [162]. Both ATLAS and CMS observed significant excesses corresponding to $4.9\sigma$ and $4.8\sigma$ respectively with Run-2

data only. When combined with results obtained in Run 1, the observed (expected) significance of the excesses were $5.4\sigma$ ($5.5\sigma$) and $5.6\sigma$ ($5.5\sigma$) respectively. These results provided direct and unambiguous observation for the $H \to b\bar{b}$ decay through the V H production mode.

Since these important observations, both experiments have completed the analysis of the full Run-2 dataset using two reconstruction techniques. The first is based on the reconstruction of the Higgs boson with two jets is referred to as resolved. The second is based on the reconstruction of the Higgs boson as a large radius jet and is referred to as boosted. These measurements expand the range of measurements of the Higgs boson decays to b-quarks in association with a vector boson at higher transverse momentum, above 400 GeV. These results are summarised in Fig. 11.3. It should be noted that the sensitivity of these analyses are already limited by systematic uncertainties.

Also, the LHCb collaboration has performed a search for the V H production with subsequent decay of the Higgs boson to a pair of b-quarks [163] with $1.98 \text{ fb}^{-1}$ of data taken at a centre-of-mass energy of 8 TeV. The final state is required to have two reconstructed b quarks and one lepton in the LHCb acceptance of $2 < \eta < 5$. The sensitivity of this search is an expected 95% CL exclusion of 84 times the SM production rate. This analysis is also used to set a limit on the V H production with the subsequent decay of the Higgs boson in a pair of c quarks with a 95% CL limit at $6.4 \times 10^3$ times the SM production rate, while the expected sensitivity corresponds to an exclusion of $7.9 \times 10^3$ times the SM production rate.

ATLAS and CMS have also searched for $H \to b\bar{b}$ in the VBF production mode. The event topology consists of two VBF-tagging energetic light-quark jets in the forward and backward direction relative to the beam direction and two b-tagged jets in the central region of the detector. Due to the electroweak nature of the process, for the signal events, no additional energetic jet activity (excluding that from the Higgs boson) is expected in the rapidity gap between the two VBF-tagging jets. The dominant background in this search stems from QCD production of multi-jet events and the hadronic decays of vector bosons accompanied by additional jets. A contribution of Higgs boson events produced in the ggF process but with two or more associated jets is expected in the signal sample. The signal is expected as a relatively broad enhancement in the $m_{b\bar{b}}$ distribution over the smoothly falling contribution from the SM background processes. Both ATLAS [164] and CMS [165] have produced results in this channel with Run-1 data, but with limited sensitivity. Both experiments performed a similar analysis with Run-2 data [166, 167].

Two of the main difficulties for the VBF production mode are the large QCD background and the difficulty in triggering fully hadronic events. Both difficulties are addressed, by the proposal made in Ref. [168], where the requirement of an additional photon in the final state reduces the background through an interference effect and enhances the possibilities for triggering. This analysis has been carried out by ATLAS at Run 2 [169] and its result combined with the standard VBF channel. The sensitivity in this channel is smaller than in the fully hadronic mode. Through the combination of these channels ATLAS has reached a first evidence at 3 standard deviations of the Higgs boson decaying to a pair of b quarks in the VBF production mode. The results are reported below

$$\text{ATLAS } \mu_{VBF(\gamma)}^{b\bar{b}} = 0.99_{-0.30}^{+0.30} (\text{stat})_{-0.16}^{+0.18} (\text{syst}), \quad (11.12)$$

$$\text{CMS } \mu_{VBF}^{b\bar{b}} = 1.01 \pm 0.36 (\text{stat})_{-0.27}^{+0.40} (\text{syst}).$$

The sensitivity in the inclusive search for the Higgs boson in the ggF production mode with $H \to b\bar{b}$ is limited by the overwhelming background from the inclusive production of $pp \to b\bar{b} + X$ via the strong interaction. With the increase in centre-of-mass energy to 13 TeV, and taking advantage of the harder transverse momentum spectrum of the $gg \to H$ production mode with respect to the QCD background, a search for high $p_T$ Higgs boson decaying to a pair of b quarks in association with an energetic Initial State
Radiation (ISR) jet. Considerable efforts have been made using state-of-the-art deep learning techniques to improve the performance of the identification of a Higgs boson decaying to b quarks and reconstructed as a large radius jet at high transverse momentum. In order to gain sensitivity in this channel and reconstruct the Higgs boson within one jet of radius 0.8 or 1.0 a lower cut at 450 GeV of transverse momentum is required. This analysis was performed by both ATLAS [170] and CMS [171] with the entire Run-2 data dataset. As in the case of VH production mode, this analysis is sensitive also to the inclusive $Z \to b\bar{b}$ production, which is an important step in the validation of the analysis chain. The $Z \to b\bar{b}$ decay is observed with a significance in excess of five standard deviations, in agreement with the expected rate from the Standard Model. This estimate has a non negligible uncertainty from the precise estimate of the fiducial signal cross section in the specific acceptance of this analysis. CMS observes an excess at $m_H = 125$ GeV of $1.2\sigma$ [171] with a sensitivity of $0.9\sigma$. It should be noted that the estimates of the signal expectation in this remote region of phase space at high transverse momentum for the production are highly non-trivial and subject to fairly large theoretical and statistical uncertainties:

$$ \text{ATLAS } \mu_{\text{inclusive}}^{b\bar{b}} = -0.1 \pm 3.5 \ (1.0 \pm 3.4 \text{ expected}), \quad (11.13) $$

$$ \text{CMS } \mu_{\text{inclusive}}^{b\bar{b}} = 2.1^{+1.9}_{-1.7}. $$

In addition to the inclusive production, in [171] CMS has also performed a search sensitive to the VBF production in the boosted Higgs regime, yielding a sensitivity of $0.9\sigma$. In this channel a broad excess has been observed which results in an observed excess with a significance of $3.0\sigma$. These results are particularly interesting as these measurements will bring a strong complement to the main channels in the high transverse momentum regime.

Another important production mode sensitive to the decay of the Higgs boson to bottom quarks, is the associated production with a pair of top quarks. The results of the searches for this process have been combined with the channels described above, to provide an additional constraint on the Yukawa coupling of the Higgs boson to bottom quarks. The channels corresponding to this production mode are described in Section 11.3.5.

## 11.3.5 Higgs boson production in association with top quarks

### 11.3.5.1 The associated production with top quark pairs

As discussed in Section 11.2, the coupling of the Higgs boson to top quarks plays a special role in the electroweak symmetry breaking mechanism in the SM, as well as in its possible extensions. Substantial indirect evidence of this coupling is provided by the compatibility of observed rates of the Higgs boson in the principal discovery channels, given that the main production process – gluon fusion – is dominated by a top quark loop. Direct evidence of this coupling at the LHC and the future $e^+e^-$ colliders will be mainly available through the $t\bar{t}H$ final state and will permit a clean measurement of the top quark-Higgs boson Yukawa coupling. The $t\bar{t}H$ production cross section at the LHC is small in comparison with the ggF or even VH production modes. The production cross section for a 125 GeV Higgs boson in $p\bar{p}$ collisions at $\sqrt{s} = 8$ TeV of about 130 fb made it challenging to measure the $t\bar{t}H$ process with the LHC Run-1 dataset. However, at Run 2, the increase in cross section at $\sqrt{s} = 13$ TeV is substantial, reaching approximately 500 fb.

The analysis channels for such complex final states can be separated in four classes according to the decays of the Higgs boson. In each of these classes, most of the decay final states of the top quarks are considered (fully hadronic, semi-leptonic and dilepton decay final states).

The first analysis is the search for $t\bar{t}H$ production in the $H \to \gamma\gamma$ channel. This analysis relies on the search for a narrow mass peak in the $m_{\gamma\gamma}$ distribution. The background is estimated from the $m_{\gamma\gamma}$ sidebands. The sensitivity in this channel is mostly limited by the available statistics.

The second analysis is the search for the Higgs boson decaying to $ZZ^*$ and subsequently to four leptons (electrons and/or muons). This channel is currently limited by the low statistics

due to the small branching fraction of the $Z$ decays to leptons.

In the third analysis the $W^+W^-$, $\tau^+\tau^-$ and $ZZ^*$ final states are searched for inclusively in multilepton event topologies (but not including the resonant $H \to 4\ell$ channel that is covered in the specific analysis described in Section 11.3.2). The corresponding $t\bar{t}H$ modes can be decomposed in terms of the decays of the Higgs boson and those of the top quarks as having two b-quarks and four W bosons (or two W and two taus, or two W and two Z) in the final state.

The fourth channel is the search in the $H \to b\bar{b}$ channel. This search is intricate due to the large backgrounds, both physical and combinatorial in resolving the $b\bar{b}$ system from the Higgs boson decay, in events with six jets and four b-tagged jets. Already with the Run-1 dataset, the sensitivity of this analysis was strongly impacted by the systematic uncertainties on the background predictions.

Prior to its start, it was unclear if this process would at all be observable at the LHC. A first milestone was reached with the completion of the Run-1 analyses by ATLAS and CMS with a complete set of results in these channels and their combination [172,173]. These results are reported in Table 11.5. However the sensitivity remained very limited. The breakthrough in the search for this process of fundamental importance came with the Run-2 data at a higher centre-of-mass energy of 13 TeV and the increase of a factor of nearly four of the production cross section of the $t\bar{t}H$ associated production process.

Using only a partial Run-2 dataset, ATLAS and CMS were able to independently make an unambiguous observation of this process. For these results ATLAS and CMS had different strategies. ATLAS relied mostly in the $t\bar{t}(H \to \gamma\gamma)$ channel where a narrow peak over a continuous background is searched for. It is interesting to note that the rapid evolution of the sensitivity in the diphoton channel owes not only to the large increase in cross section, but also to the fact that high mass diphoton events in an environment with additional leptons, jets and b-tagged jets the search for a pair of photon at high mass has a low background. In contrast with the inclusive search for $H \to \gamma\gamma$, the analyses in this channel reached a signal-to-background ratio in excess of unity. This channel, which is characterised by a narrow peak over a continuous background, also has the advantage of being less prone to systematic uncertainties than channels where the assessment of backgrounds cannot be made through the use of sidebands. ATLAS used up to $79.8 \text{ fb}^{-1}$ of Run-2 data to reach an expected sensitivity of $4.9\sigma$ and an observed significance of $5.8\sigma$ with the Run-2 partial dataset alone, and $6.3\sigma$ (with $5.1\sigma$ expected) in combination with the Run-1 results [174]. CMS instead relied on a smaller dataset of $35.9 \text{ fb}^{-1}$ and relied on all channels. The most sensitive in this case was the $t\bar{t}H(bb)$ whose sensitivity was limited by systematic uncertainties. CMS reached a sensitivity of $4.2\sigma$ and observed an excess with respect to the background-only hypothesis of $5.2\sigma$, combining the Run-1 and Run-2 results [175].

This direct observation of the Yukawa coupling of the Higgs boson to top quarks was particularly important in comparison to the indirect evidence through the gluon fusion production process dominated by the top quark loop

Since the observation of this important process, with few exceptions, all channels have been updated with the full Run-2 dataset, by both experiments.

For the principal decays of the Higgs boson in the diphoton and four lepton channels, the measurements are done in the framework of 'Simplified Template Cross Section' discussed in Section 11.5.1.5). The diphoton channels were updated by both the ATLAS and CMS experiments in Refs. [176,177]. The four lepton decays of the Higgs boson in the associated production with a pair of top quarks is a search for a narrow peak in the busy $t\bar{t}H$ environment. It was also performed by ATLAS [176] and CMS [177] with the full Run-2 datasets. These results are reported in Table 11.5. This channel has currently very low statistic but also very high purity, reaching signal-to-background ratios in excess of 10.

To address the $WW$, $ZZ$ and $\tau\tau$ decay modes, an approach based on multiple channels with electrons, muons and taus decaying hadronically, the so-called "multi-lepton", is adopted by both
experiments. This analysis excludes the $H \to 4\ell$ channel. CMS has analysed the Full Run-2 dataset [178] and ATLAS only part of it [179]. In a preliminary release of this analysis using a larger, but not full Run-2 dataset corresponding to an integrated luminosity of approximately 80 fb⁻¹, the ATLAS experiment found that the normalisation of the $t\bar{t}W$ background was larger than expected from SM calculations by factors ranging from 1.3 to 1.7 and modelling issues were observed in analysis regions where the $t\bar{t}W$ process is dominant [179]. The search for Higgs boson decays to tau leptons in association with a pair of top quarks was carried out by ATLAS with the full Run-2 dataset and discussed in 11.3.4.1 [180]. Using the full dataset in this channel CMS obtained an observation of the Higgs boson production in association with top quarks with an observed significance of 4.7 standard deviations (5.2σ expected). The results are reported in Table 11.5. The rate of the $t\bar{t}W$ background in this analysis is also high by a factor of approximately 1.4.

Both ATLAS and CMS have performed the analysis of Higgs bosons decaying to b-quarks in the $t\bar{t}H$ production mode with the full Run-2 dataset. In ATLAS, this analysis was performed in the semi-leptonic and di-leptonic channels [181], including a resolved mode where the Higgs boson is reconstructed as two standard jets and a boosted mode where the Higgs boson is reconstructed as a single large radius jet. CMS also performed this analysis with the full Run-2 dataset, including in addition the fully hadronic mode where both W bosons from top decays subsequently decay hadronically [182]. The results reported in Table 11.5. It should be noted that in this channel a deficit in observed Higgs boson decays is observed. This deficit is rather significant and in tension with the observation of the other $t\bar{t}H$ modes as well as the observation of the Higgs boson in the $b\bar{b}$ channels.

The discrepancies observed in both the multilepton and the $H \to b\bar{b}$ channels will require further studies.

A combination of all available channels has been performed by ATLAS [183] and CMS [184] and is reported in Fig. 11.3. It should be noted that the latest $t\bar{t}H(\to b\bar{b})$ channel from CMS with the full Run 2 data has not yet been included in the combination. A fit of individual decay modes in the $t\bar{t}H$ production is also given therein.

### 11.3.5.2 The associated production with a single top quark

An additional production mode of the Higgs boson in association with a top quark is the single top associated production mode. There is an interesting similarity between this production mode and the $H \to \gamma\gamma$ decay mode. Both processes proceed through either the top Yukawa coupling or the interaction of the Higgs boson with the W boson, with a negative interference between the two. Representative Feynman diagrams for this production process are shown in Fig. 11.1. Contrary to the diphoton decay channel, in this production mode the interference occurs at the tree level and is dominant. This process can therefore be used to further discriminate a negative relative sign between the couplings of the Higgs boson to fermions and its couplings to gauge bosons [185].

ATLAS and CMS have produced specific searches for the $tH$ production mode with the Run-1 and Run-2 data exploiting a variety of Higgs boson decay modes resulting in final states with photons, bottom quarks, and multiple charged leptons, including tau leptons. In particular, with the Run-2 data, CMS has searched for multi-leptonic decay signatures from the $H \to WW^*$, $H \to \tau^+\tau^-$ and $H \to ZZ^*$ modes [186]. This analysis restricts values of $\kappa_t$, the top-Higgs coupling normalised to its SM value, to [-1.25, 1.60] at 95% CL. CMS has also performed an analysis of the 2015 dataset to search for the $H \to b\bar{b}$ mode [187], yielding much less stringent constraints.

The diphoton channel has also been used to search specifically for this production mode by ATLAS using Run-1 data, yielding the restricted range of allowed values of $\kappa_t$ at the 95% CL to [-1.3, 8].

The strongest constraint on the negative (relative) sign of $\kappa_t$ was obtained by CMS using the recent analysis of the full dataset [178] in the multilepton ($H \to WW, H \to ZZ, H \to \tau\tau$) discussed in 11.3.5.1, negative values of $\kappa_t$ are disfavoured and the

only non excluded negative values of $\kappa_t$ at 95% CL. range between -0.9 and -0.7.

### 11.3.6 Combining the main channels

The analysis strategy used by the LHC experiments to perform the searches for the Higgs boson has been based on the Higgs boson decay modes. It is a natural choice given that it focusses on the decay products of the object searched for. However, for each channel, exclusive sub-channels have been defined to target different production modes and kinematic regimes, and these sub-channels have been combined. The natural extension of this approach in order to probe further the production and decay modes of the Higgs boson is to combine the analysis channels together. Such a combination is also used in Section 11.5 to further measure the coupling properties of the Higgs boson.

At the LHC, the total cross section cannot be measured independently of the decay branching fractions. As a consequence, the total width of the Higgs boson cannot be directly measured. However, a combined measurement of the large number of categories described above with different sensitivities to various production and decay modes, permits a wide variety of measurements of the production, decay and coupling properties. These measurements, however, rely on assumptions.

Combining Higgs observation channels requires that measurements in all channels are done under the same Higgs boson mass hypothesis. This is always the case within experiments but is not necessarily the case across experiments. At Run 1, prior to combining the measurements of rates in ATLAS and CMS, a common and thus combined Higgs boson mass measurement was made. Only thereafter were the ATLAS and CMS measurements combined [153]. The Run 1 full combination results were derived by the two collaborations, taking rigorously into account all correlations in the systematic uncertainties and in the large number of channels and their categories. For a review of the Run-1 combined results see former PDG reviews in [188] and references therein.

It is important to note that, between the Run-1 and the Run-2 results, the signal theoretical systematic uncertainties have improved significantly.

At Run 2, ATLAS [183] and CMS [184] have already produced combined measurements of the coupling properties and production cross section ratios of the Higgs boson.

In this section, only the results on the main Higgs boson production and decay modes will be discussed. Only a brief presentation of the combination framework is given here (a more detailed description is given in Ref. [189]). This framework will also be used in Section 11.5 to discuss the measurements of the coupling properties of the Higgs boson.

#### 11.3.6.1 Principles of the combination

The combination of the Higgs boson analysis channels in each experiment and for the two experiments together was done using a fit of a signal and background model to the data. As described above, the data was made of a large number of categories, aiming at reconstructing exclusive production and decay modes. In the combination of ATLAS and CMS [153], there were approximately 600 categories. The combination was a simultaneous fit to the data across all these categories, using a reduced number of parameters of interest and the Higgs boson mass fixed to its measured value at Run 1 [144]. The much larger number of categories present in the ATLAS and CMS combination [153] is due to additional separation in terms of finer exclusive production regions, decay channels of the Z and the W bosons, and taus, control regions where little-to-no signal is present, and different center-of-mass energies.

In their Run-2 individual combinations, ATLAS and CMS have not considered the $Z\gamma$ channel. Both ATLAS and CMS experiment have included the $\mu\mu$ channel based on a partial Run-2 dataset corresponding to an integrated luminosity of 36 fb⁻¹ only.

The key to understanding how the combination of channels works relies on the combination master formula, which expresses for each category, indexed by c, of a given channel (typically a category covers mostly one decay mode, but possibly various production modes), the measured number of signal events $n_s^c$ as a
<table><caption>**Table 11.5:** Summary of the results of searches for a Higgs boson in association with a top quark pair by ATLAS and CMS. The results are given in terms of a measured signal strength.</caption><thead><tr><th>t<span style="text-decoration: overline;">t</span>H</th><th>ATLAS Run 1</th><th>CMS Run 1</th><th>ATLAS Run 2</th><th>CMS Run 2</th></tr></thead><tbody><tr><td>H → γγ</td><td>1.3 <sup>+2.6</sup><sub>-1.7</sub> <sup>+2.5</sup><sub>-1.7</sub></td><td>1.2 <sup>+2.5</sup><sub>-1.7</sub> <sup>+2.6</sup><sub>-1.8</sub></td><td>1.38 <sup>+0.33</sup><sub>-0.31</sub> <sup>+0.26</sup><sub>-0.18</sub></td><td>2.27 <sup>+0.86</sup><sub>-0.74</sub></td></tr><tr><td>H → 4ℓ</td><td>—</td><td>—</td><td>1.2 <sup>+1.4</sup><sub>-0.8</sub></td><td>0.00 <sup>+0.91</sup><sub>-0.00</sub></td></tr><tr><td>H → WW/ττ/ZZ</td><td>1.4 ± 0.6 ± 1.0</td><td>3.3 ± 1.4</td><td>1.56 <sup>+0.30</sup><sub>-0.29</sub> <sup>+0.30</sup><sub>-0.27</sub></td><td>0.92 <sup>+0.26</sup><sub>-0.23</sub></td></tr><tr><td>H → b<span style="text-decoration: overline;">b</span></td><td>1.4 ± 1.0 ± 0.6</td><td>1.6 <sup>+1.6</sup><sub>-1.5</sub></td><td>0.35 ± 0.20 ± 0.30</td><td>0.33 ± 0.16 ± 0.21</td></tr><tr><td>Combination</td><td>1.7 ± 0.5 ± 0.8</td><td>2.6 <sup>+1.0</sup><sub>-0.9</sub></td><td>0.97 ± 0.14 <sup>+0.14</sup><sub>-0.13</sub></td><td>0.94 ± 0.20 <sup>+0.09</sup><sub>-0.14</sub></td></tr></tbody></table>

function of a limited number of parameters as follows:

$$n_s^c = \left( \sum_{i,f} \mu_i \sigma_i^{\text{SM}} \times A_{if}^c \times \varepsilon_{if}^c \times \mu_f \text{BR}_f^{\text{SM}} \right) \times \mathcal{L}^c. \quad (11.14)$$

The production index is defined as $i \in \{\text{ggF}, \text{VBF}, \text{ZH}, \text{WH}, \text{t\bar{t}H}, \text{tH}, \dots\}$ (including specific modes and regions of phase space) and the decay index is defined as $f \in \{\gamma\gamma, \text{WW}, \text{ZZ}, \text{bb}, \tau\tau, \mu\mu, \text{Z}\gamma\}$, while $\sigma_i^{\text{SM}}$ and $\text{BR}_f^{\text{SM}}$ are the corresponding production cross sections and decay branching fractions, estimated as described in Section 11.2, assuming that the Higgs boson is that of the SM. $A_{if}^c$ and $\varepsilon_{if}^c$ are the signal acceptance and the reconstruction efficiency for the given production and decay modes in the category c. $\mathcal{L}^c$ is the integrated luminosity used for that specific category. For the purpose of this review, these parameters can be considered as fixed<fnref n="2" />.

The parameters of interest in the master formula are the signal strength parameters $\mu_i$ and $\mu_f$. It is important to note that the formula relies on the factorisation of the production cross section and decay branching fraction, which assumes the narrow width approximation. The width of the Higgs boson will be discussed in Section 11.4, however, for the precision needed here, the fact that the Higgs boson has been observed in decay channels with high mass resolution as a resonance is sufficient to validate this hypothesis. It is also manifest in the above equation that the ten parameters for the production modes ($\mu_i$) and decay modes ($\mu_f$) cannot be determined simultaneously. This illustrates that total cross sections or branching fractions cannot be measured without further assumptions in this fit.

The master formula also illustrates an important caveat to the measurement of signal strength parameters. In case these are interpreted as scale factors of the production cross sections or branching fractions, then all the other quantities such as the acceptances and efficiencies, $A_{if}^c$ and $\varepsilon_{if}^c$, need to be assumed as independent and fixed to their estimated values for the SM Higgs boson. An additional important caveat to note concerning these combined results is that only the normalisation is varied, while the discriminating variables for the signal are not modified and are still used in the fit. These caveats are of particular importance in the use of the combination to measure the coupling properties of the Higgs boson, as discussed in Section 11.5. For relatively small perturbations of the couplings of the Higgs boson from the SM values, this hypothesis is valid.

However, the products $\mu_i \times \mu_f$ can be considered as free parameters and in principle measurable (if there is sufficient sensitivity from specific categories). Measuring the products of signal strengths can be viewed as measuring the cross sections times the branching fraction, $\sigma \cdot \text{BR}$. Measurements by ATLAS and CMS of $\sigma \cdot \text{BR}$, normalised to the SM predictions, for the five largest Higgs boson production modes and five major Higgs decay modes are displayed in Fig. 11.3.

Other fits involving ratios of cross sections, which are less sensitive to theory uncertainties, are performed and reported in Ref. [189].

The most constrained fit in the combination allows for only one single parameter to vary, i.e., $\forall(i, f)$, $\mu_i = \mu_f = \mu$. This global-signal-strength model provides the simplest probe of the compatibility of the signal with the SM Higgs boson. Indeed, it is sensitive to any deviation from the SM Higgs boson couplings provided that these deviations do not cancel overall. The full Run-1 combination determines the global signal strength to be

$$\mu = 1.09 \pm 0.11 = 1.09 \pm 0.07 \text{ (stat.)} \pm 0.04 \text{ (expt.)} \\ \pm 0.03 \text{ (th. bkg.)} \pm 0.07 \text{ (th. sig.),} \quad (11.15)$$

where the statistical, experimental uncertainties as well as the theoretical uncertainties on the background and on the signal are reported separately. The ATLAS Run-2 combination of the global signal strength yields [183]:

$$\mu = 1.05 \pm 0.06 = 1.05 \pm 0.03 \text{ (stat.)} \pm 0.03 \text{ (exp.)} \\ \pm 0.02 \text{ (th. bkg.)} \pm 0.04 \text{ (th. sig.),} \quad (11.16)$$

while the CMS Run-2 combination yields [184] $\mu = 1.002 \pm 0.057$.

These overall signal strengths are fully compatible with the SM expectation, $\mu = 1$, with a precision of 6%. It is interesting to note that the main uncertainty in these measurements arises from the limited precision in the theoretical predictions for the signal production processes. The precision reached with the individual experiments combinations using partial Run-2 data sets have already exceeded the full Run-1 ATLAS and CMS combination precision.

#### 11.3.6.2 Main decay modes

Despite the large number of decay channels, since the cross sections cannot be independently measured, from the measurements described in this section it is impossible to measure the decay branching fractions without a loss of generality. The simplest assumption that can be made is that the production cross sections are those of the SM, which is equivalent to assume that, for all $i$ indices, $\mu_i = 1$. All branching fractions $\mu_f$ can then be measured in a simple 7 parameter fit. Results of these measurements are reported in Fig. 11.3.

#### 11.3.6.3 Main production modes

Most analysis channels are divided into exclusive categories allowing for an increased overall sensitivity and permitting to access the various Higgs boson production modes. The cross sections of the main production modes can be measured assuming that the branching fractions are those of the SM, i.e., for all $f$ indices $\mu_f = 1$. These assumptions lead to a 6 parameter combination.

The gluon fusion production process is the dominant production mode. Although no numerical estimate of combined significance of observation for this process has been given by the experiments, it is considered as established due to the overwhelming evidence from the three main discovery channels. At the LHC, all production modes have been unambiguously observed. The intricate VH production mode has been unambiguously observed by ATLAS

<footnote n="2">²In the combination performed by ATLAS and CMS, the systematic uncertainties on these parameters are taken into account by allowing these parameters to vary within their constraints in the fit.</footnote>
<figure><img src="image_30.png" /><figcaption>**Figure 11.3:** Combined measurements by ATLAS [183] and CMS [184] of the products $\sigma \cdot \text{BR}$, normalised to the SM predictions, for the five main production and five main decay modes.</figcaption></figure>

and CMS independently (as discussed in Section 11.3.4) through the $V(H \to b\bar{b})$ channel. The results of these measurements are reported in Fig. 11.3.

With the Run-2 data, all the main production processes have been established, and in particular the $pp \to t\bar{t}H$ process, which provides direct evidence of the coupling of the Higgs boson to top quarks. This is another milestone in the LHC physics program.

### 11.3.7 Probing the Higgs boson coupling to the second generation

#### 11.3.7.1 First evidence for $H \to \mu^+\mu^-$

The branching fraction in the $H \to \mu^+\mu^-$ channel for a 125 GeV SM Higgs boson is $2.2 \times 10^{-4}$, about ten times smaller than that for $H \to \gamma\gamma$. The major challenge in this channel is the extremely small, permil level, signal-to-background ratio. The dominant and irreducible background arises from the $Z/\gamma^* \to \mu^+\mu^-$ process which has a rate several orders of magnitude larger than that from the SM Higgs boson signal. Due to the precise muon momentum measurement achieved by ATLAS and CMS, the $m_{\mu^+\mu^-}$ mass resolution is very good ($\approx 2-4\%$ for ATLAS and $\approx 1-3\%$ for CMS depending on the selected categories; a better resolution is expected for CMS due its higher magnetic field strength in the inner detector). A search is performed for a narrow peak over a large but smoothly falling background. For optimal search sensitivity, events are divided into several categories. Either taking advantage of the superior muon momentum measurement in the central region, events can be subdivided by the pseudo-rapidity of the muons, or designing selections aiming at specific production processes such in particular as the vector boson fusion.

ATLAS and CMS analyses target the main production modes such as gluon fusion, associated production with a vector boson (including in the vector boson leptonic decay modes), vector boson fusion and associated production with a top-quark pair. The analyses use multivariate and deep machine learning methods to discriminate signal from background in these Higgs production categories. The signal in both experiments is extracted from a fit to the di-muon invariant mass spectrum and the background is modeled using a suite of analytical functions, except in the case of the most sensitive channel (VBF) in the CMS analysis where the distribution of a discriminant Deep Neural Network output is used.

With the Run-2 dataset, ATLAS measured [190] a signal strengths of $\mu = 1.2 \pm 0.6$ with an observed (expected) significance of $2.0(1.7)\sigma$. The 95% CL limit on $pp \to H \to \mu^+\mu^-$ was set at 2.2 times the SM prediction while the expected limit on a $H \to \mu^+\mu^-$ signal assuming the absence (presence) of a SM signal was 1.1 (2.0). CMS reported [191] the first evidence for $H \to \mu^+\mu^-$ at $3\sigma$ level with a signal strength of $\mu = 1.19 \pm 0.40$ (stat.) $\pm 0.15$ (syst.) and an expected analysis sensitivity of $2.5\sigma$. These measurements provide the first direct evidence for the decay $H \to \mu^+\mu^-$ and thus of the Yukawa coupling of the Higgs boson to second generation fermions.

#### 11.3.7.2 Constraints on Higgs boson coupling to charm quarks

The possibility of probing the Yukawa coupling to the charm has been discussed early in Ref. [192] where indirect bounds are estimated from a combined fit to the Higgs data and the importance of using charm tagging is emphasised. A SM decay rate in standard production was long thought to be out of reach at the LHC. However recent results, in particular from CMS have shown that a different perspective might be emerging [193].

The channels which are currently foreseen to be most sensitive are similar to those aiming at the b-quark Yukawa coupling, and pertain to the $VH$ production mode. As was discussed in [188], the typical sensitivity of these analysis using the LHC Run-2 data on the production times branching fraction was approximately a factor of 30 times the expected rate in the Standard Model.

The key aspect to bring sensitivity to the charm coupling is the ability of flavour tagging algorithms to discriminate charm quark initiated jets from both b quark and light quark jets. Significant progress was recently made in development of parton flavour taggers using state-of-the-art machine learning techniques [194] and additional information to these algorithms. Novel algorithms have also been used to identify pairs of charm quarks resulting from the decay of a Higgs boson, in particular in a regime where the Higgs boson is reconstructed as a large radius jets with a distinctive substructure. These new techniques have resulted in impressive improvements which have led to the first observation of Z boson decays to charm quarks at a hadron collider with a significance of 5.7 standard deviations [193] and an expected sensitivity to the exclusion of a Higgs boson signal decaying to a pair of charm quarks of 7.6 times the Standard Model rate. The observed limit
for CMS is at 14 times the SM rate. This led CMS to set limits on the absolute value of $\kappa_c$, the Yukawa coupling of the Higgs boson to charm quarks with respect to its value in the Standard Model.

The expected 95% CL interval is $|\kappa_c| < 3.4$. The observed 95% CL interval is the following:

$$1.1 < |\kappa_c| < 5.5 \qquad (11.17)$$

This is the most stringent constraint today. The ATLAS analysis [195], in contrast, is still relying on traditional techniques and sets an observed (expected) limit at 26 (31) times the SM rate, using the VH production mode where the vector boson decays to leptons.

CMS complemented this search in the VH production mode with an inclusive search for the Higgs boson produced at high transverse momentum (in excess of 450 GeV) and decaying to a pair of charm quarks [196]. The observed (expected) limit at 95% CL is 47 (39) times the SM rate, which is also an impressive result as it is almost comparable to the previous results obtained in the more sensitive VH channel.

With this striking improvement in sensitivity, CMS reappraised their prospect for the search of Higgs boson decays to charm quarks at the HL-LHC and showed that a CMS could reach a relative precision on the production rate of approximately 80%. In combination with ATLAS and potential improvements in the analysis, this is bringing the Yukawa coupling of the Higgs boson closer to being at reach at the HL-LHC.

### 11.3.8 Searches for rare decays of the Higgs boson

#### 11.3.8.1 Combined evidence for $H \to Z\gamma$ and the first evidence for the Dalitz $H \to \ell^+\ell^-\gamma$ decay

The search for $H \to Z\gamma$ is performed in the final states where the Z boson decays into opposite sign and same flavour leptons ($\ell^+\ell^-$), $\ell$ here refers to $e$ or $\mu$. While the branching fraction for $H \to Z\gamma$ is comparable to $H \to \gamma\gamma$ (about $10^{-3}$) at $m_H = 125$ GeV, the observable signal yield is brought down by the small branching ratio of $Z \to (e^+e^- + \mu^+\mu^-) = 6.7 \times 10^{-2}$. In these channels, the $m_{\ell\ell\gamma}$ mass resolution is excellent (1–3%), therefore the analyses search for a narrow mass peak over a continuous background. The major backgrounds arise from the $Z+\gamma$ final state radiation in Drell–Yan decays and from the $Z + \text{jets}$ processes where a jet is misidentified as a photon. The ratio of signal over background in this channel is typically of the order of 0.5%. In a narrow window of a few GeV around 125 GeV, several hundreds of events are expected in a Run-2 dataset corresponding to approximately 36 fb$^{-1}$. Events are divided into mutually exclusive categories on the basis either of the expected $m_{Z\gamma}$ resolution or exclusive production mode categories.

With the full Run-2 dataset, ATLAS observes an excess with a significance of $2.2\sigma$ while an excess with a significance of $1.2\sigma$ is expected. CMS observes an excess with a significance of $2.6\sigma$ while $1.2\sigma$ is expected. ATLAS and CMS have jointly published a combination of these results, yielding a first evidence for the Higgs boson decay to a Z boson and a photon at the LHC with an observed significance of $3.4\sigma$ and an expected sensitivity of $1.6\sigma$ [197]. This combined result is compatible with the SM hypothesis at $1.9\sigma$.

Both ATLAS and CMS experiments have extended the search for the so-called Dalitz Higgs boson decays $H \to \gamma^*\gamma \to \ell^+\ell^-\gamma$ in the low mass $\gamma^*$ range of approximately 20–30 GeV. This decay mode has a substantially larger branching fraction compared to the $Z\gamma$ decay, as $\Gamma(H \to \gamma^*\gamma \to e^+e^-\gamma) \sim 3.5\% \times \Gamma(H \to \gamma\gamma)$ and $\Gamma(H \to \gamma^*\gamma \to \mu^+\mu^-\gamma) \sim 1.7\% \times \Gamma(H \to \gamma\gamma)$, while $\Gamma(H \to Z\gamma) = 2.3\% \times \Gamma(H \to \gamma\gamma)$ (which does not account for the subsequent decay of the Z boson to electrons or muons). With the Run-1 dataset, CMS observes an upper limit of 6.7 times the SM branching ratio [198]. With a partial Run-2 dataset, in the $\gamma^*\gamma$ CMS obtained a much more stringent limits on the cross section times the corresponding branching fractions of 1.4 (6.1) times the SM cross section [199]. CMS also performed a combination of this mode with $H \to Z\gamma$, obtaining a combined observed (expected) limit of 3.9 (2.0) times the SM branching fractions.

Using the full Run-2 dataset corresponding to 139 fb$^{-1}$, ATLAS obtained the first evidence for Higgs boson decays to a low-mass

dilepton pair and a photon with a significance of 3.2 (2.1) standard deviations (expected) [200]. The invariant mass requirement on the di-lepton is $m_{\ell\ell} < 30$ GeV. The corresponding signal strength is $\mu = 1.5 \pm 0.5$, compatible with the SM expectation.

#### 11.3.8.2 Higgs coupling to first generation, search for $H \to e^+e^-$

Probing the Yukawa couplings to the first generation fermions is extremely challenging at the LHC given the overwhelming backgrounds and very small signal rates.

A search similar to the $H \to \mu^+\mu^-$ is performed in the non-VBF categories by CMS in the di-electron channel [201]. In this search, the contribution from the peaking background from Higgs boson decays to diphotons mis-identified as di-electrons (when converted photons are faking electrons) needs to be carefully assessed. The sensitivity to the SM Higgs decays is negligible given the extremely small branching fraction to $e^+e^-$, approximately 40,000 times smaller than the branching fraction to dimuons. It is nevertheless interesting to probe this decay channel to search for potential large anomalous couplings. Assuming a SM Higgs boson production cross section, the observed limit on the $H \to e^+e^-$ branching fraction at the 95% CL is $3.0 \times 10^{-4}$, more than five orders of magnitude larger than the expected SM prediction. ATLAS has also performed a search for the $H \to e^+e^-$ decay mode with the Run-2 dataset, placing a limit of $3.6 \times 10^{-4}$ on the branching fraction [202].

It is important to note that processes not depending on the electron Yukawa coupling such as the $H \to e^+e^-\gamma$ (where the photon is soft), are sizeably larger than the direct Yukawa coupling process, but also much smaller than the current constraints, making any interpretation in terms of constraint on the electron Yukawa couplings far from straightforward.

#### 11.3.8.3 Invisible decays of the Higgs boson

The discovery of the Higgs boson immediately raised the question of its couplings to DM and how it could be used to reveal at colliders the existence of a dark sector coupled to the SM via the Higgs boson portal, see Ref. [203] and references therein. If kinematically accessible and with a sufficiently large coupling to the Higgs boson, DM particles, such as, e.g., neutralinos in SUSY models, graviscalarins in models with extra dimensions or heavy neutrinos in the context of four-generation fermion models, would manifest themselves as invisible decays of the Higgs boson, thus strongly motivating searches for the invisible decays of the Higgs boson.

It should be noted that in the Standard Model the branching fraction of a Higgs boson into invisible decays through the $H \to ZZ^* \to \nu\bar{\nu}\nu\bar{\nu}$ process has a branching fraction of 0.12%.

To identify an invisibly decaying Higgs boson at the LHC, it must be produced in association with other particles. Searches for invisible decays of the Higgs boson at the LHC have been carried out in the three associated production modes of the Higgs boson with the highest SM cross sections and target events with large missing energy.

The ggF production mode has the largest SM cross section but it usually results in the Higgs boson being created alone and hence leaving no characteristic signature in the detector of its invisible decay. One way to search for invisible decays in ggF production mode is to look for events with the monojet topology arising from initial state gluon radiation and containing missing energy. The major irreducible background in such searches stems from $Z + \text{jets}$ events where the Z boson decays into a pair of neutrinos [204]. The analysis with the best sensitivity targets the VBF production topology but it suffers from large backgrounds arising from events with two jets and large missing energy. The VH mode has much smaller cross section but the presence of a W or Z boson allows a variety of final states that can be tagged with relatively low background.

ATLAS and CMS have searched for such final states at Run 1 and have observed no significant excess over the predicted backgrounds (for references, see the previous edition of this review [141]). Table 11.6 summarises the 95% CL limits on the invisible decays of the Higgs boson assuming a SM Higgs boson production cross section and the corresponding detector acceptances.
<table><caption>**Table 11.6:** Summary of the channels searched for and the corresponding 95% CL limits from ATLAS and CMS on the branching fraction for the Higgs boson decay to invisible particles assuming a SM Higgs boson production cross section. The results in parentheses are the expected exclusions.</caption><thead><tr><th></th><th>ATLAS (Run 1)</th><th>ATLAS (Run 2)</th><th>CMS (Run 1)</th><th>CMS (Run 2)</th></tr></thead><tbody><tr><td>ggF (monojet); H → inv.</td><td>—</td><td>33 (38) %</td><td>67 (71) %</td><td>49 (32)%</td></tr><tr><td>VBF; H → inv.</td><td>28 (31) %</td><td>14.5 (10.3) %</td><td>57 (40) %</td><td>18 (10) %</td></tr><tr><td>ZH; Z → ℓ<sup>+</sup>ℓ<sup>−</sup>; H → inv.</td><td>75 (62)%</td><td>18.5 (18.5) %</td><td>75 (91) %</td><td>—</td></tr><tr><td>VH; Z, W → jj; H → inv.</td><td>78 (86)%</td><td>83 (58) %</td><td>—</td><td>21 (18)%</td></tr><tr><td>ZH; Z → b̄b; H → inv.</td><td>—</td><td>—</td><td>182 (189)%</td><td>—</td></tr><tr><td>t<span style="text-decoration: overline;">t</span>H; H → inv.</td><td>—</td><td>38 (30)%</td><td>—</td><td>26 (30)%</td></tr><tr><td>Run 1 & 2 Combination</td><td>10.7 (7.7)%</td><td></td><td>15 (8)%</td><td></td></tr></tbody></table>

ATLAS and CMS have performed the search for invisible decays of the Higgs boson at Run 2 with the full dataset in all the aforementioned channels [205–207]. The results are summarised in Table 11.6.

These constraints can then be further used to probe Higgs portal models to DM [203], where an additional weakly interacting particle $\chi$ with mass lower than $m_H/2$ is introduced as DM candidate and where the Higgs boson is considered as the only mediator between the SM particles and DM. In this model, it is interesting to express the limit on the invisible branching fraction in terms of strength of interaction of DM with standard matter, i.e., in terms of it interaction cross section with nucleons $\sigma_{\chi-N}$. In this model, the couplings of the Higgs boson to SM particles are assumed to be those of the SM and the interaction of the Higgs boson with the nucleon is parametrised in a Higgs-Nucleon form factor estimated using lattice QCD calculations [203]. The results for the ATLAS limits are shown in Fig. 11.4.

### 11.3.8.4 Exclusive Higgs boson decays

A possibility to access the charm Yukawa coupling has been discussed in Ref. [208]. It relies on the decays of the Higgs boson to a final state with quarkonia. Searches have been performed in several channels including $H \to \Upsilon\gamma, J/\Psi\gamma, \phi\gamma, \rho\gamma$ these processes can provide indirect constraints on the Higgs boson Yukawa coupling to b, charm, strange and light quarks. The limits obtained so far in these channels [209, 210] are reported in the Higgs particle listings [188]. Limits on branching fractions range between $10^{-3}-10^{-4}$.

CMS has also performed a search of the decays of the Higgs boson in the $J/\Psi J/\Psi$ and $\Upsilon\Upsilon$ final states to cover the cases where the photon in the $J/\Psi\gamma$ decay is virtual and transforms into a $J/\Psi$ meson. These decays provide an additional channel potentially sensitive to BSM phenomena [211].

More recently a search for Higgs boson decays to mesons and a photon in the channels $H \to \omega\gamma$ and $H \to K^*\gamma$ have been carried out by the ATLAS experiment [212]. The limits on these processes are of $1.5 \times 10^{-4}$ and $8.5 \times 10^{-5}$ on the $\omega\gamma$ and the $K^*\gamma$ respectively.

Another class of searches for exclusive decays of the Higgs boson is $H \to ZQ$ where $Q$ is a quarkonium state. This decay results from the interference of two contributions, one where the Higgs boson couples to a quark-anti-quark pair which forms a quarkonium and radiates a Z boson at leading order, and another one from the off-shell decay $H \to ZZ^*$ or $H \to Z\gamma^*$ and the conversion of the off-shell Z or photon into a quarkonium. A search was carried out by ATLAS in the $ZJ/\Psi$ and $Z/\eta_c$ channel in the $Z \to \ell^+\ell^-$ and a light hadronically decaying resonance signature [213] and by CMS in the $ZJ/\Psi$ and $ZJ/\Upsilon$ channels in fully leptonic final states [214].

### 11.3.8.5 Lepton flavour violating (LFV) Higgs boson decays

Given the Yukawa suppression of the couplings of the Higgs boson to quarks and leptons of the first two generations and the small total width of the Higgs boson, new physics decay modes could easily have sizeable branching fractions. One very interesting possibility is the Lepton Flavour Violating (LFV) decays of the Higgs boson, in particular in the $\tau\mu$ and $\tau e$ modes. These decays are suppressed in the SM but they could easily be enhanced in theories such as two-Higgs-doublet models (discussed

in Section 11.6). Indirect constraints on the LFV Yukawa couplings $|Y_{\tau\mu}|$ can be obtained from channels such as the $\tau \to 3\mu$ or $\tau \to \mu\gamma$, or a re-interpretation of the search for Higgs boson decays to $\tau^+\tau^-$. A direct search at the LHC, however, complements these indirect limits.

The search for LFV decays in the $\tau\mu, \tau e$ and $\mu e$ channels were carried out with the Run-1 and early Run-2 datasets in several channels according to the subsequent decay of the $\tau$ [215–220]. The limits on LFV decays of the Higgs boson are reviewed in this volume of the particle listings [188] and reached 0.15% in the $\tau\mu$ channel and 0.22% in the $\tau e$ channel.

ATLAS [221] has performed the analysis with the full Run-2 dataset, yielding limits of 0.20% and 0.18% on the $e\tau$ and $\mu\tau$ channels respectively. CMS [222] has performed analyses with the full Run-2 dataset of the $e\mu$ channel reaching a limit of $4.4 \times 10^{-5}$ at 95% CL.

The CMS search in the $e\mu$ channel was done not only at the specific Higgs boson mass, but in a wider mass domain between 110 GeV and 160 GeV. A mild excess with a global significance of 2.8 standard deviations is observed for a mass hypothesis of approximately 146 GeV.

### 11.3.8.6 Flavour changing neutral current decays of the top quark

The discovery of the Higgs boson at a mass smaller than the top quark mass opened a new decay channel for the top quark. The decays of the top quark to a Higgs boson and a charm or an up quark proceed through a Flavour Changing Neutral Current (FCNC) which are forbidden in the Standard Model at tree level and suppressed at higher orders through the Glashow-Lliopoulos-Maiani (GIM) mechanism [3]. The SM prediction for these branching fractions is $BR(t \to Hc) = 10^{-15}$ and two orders of magnitude less for the $H\nu$ final state. These decay channels of the top quark are, therefore, very interesting to probe possible FCNC interactions in the Yukawa couplings to the quark sector, see Section 11.6.

ATLAS has searched for FCNC top decays specifically in channels involving a Higgs boson that subsequently decays into a pair of b quarks, two photons, a tau pair or two W bosons [223]. It has also reinterpreted a search for the $t\bar{t}H$ production in the multilepton final state (discussed in Section 11.3.5.1) [173]. The latter channel covers Higgs boson decays to a pair of W bosons and a pair of taus. No significant excess was observed in any of the specific channels (as discussed in Section 11.3.5.1, a slight excess is observed in the $t\bar{t}H$ multilepton channel) and 95% CL upper limits are set on $BR(t \to Hc) < 0.46\%$ with an expected sensitivity of 0.25% and $BR(t \to H\nu) < 0.45\%$ with an expected sensitivity of 0.29%. CMS has performed a search for these FCNC top decays in the diphoton and multi-lepton channels [224], placing a 95% CL upper limit on $BR(t \to Hc) < 0.40\%$ with an expected sensitivity of 0.43%.

From these limits on branching fractions, constraints on non-flavour-diagonal Yukawa couplings of a FCNC Lagrangian of the form:

$$ \mathcal{L}_{\text{FCNC}} = \lambda_{t c H} \bar{t} H c + \lambda_{t u H} \bar{t} H u + \text{h.c.} \quad (11.18) $$

can be derived. The 95% CL observed (expected) upper limits from ATLAS on the $|\lambda_{t c H}|$ and $|\lambda_{t u H}|$ couplings are 0.13 (0.10) and 0.13 (0.10), respectively.
<figure><img src="image_31.png" /><figcaption>**Figure 11.4:** 90% CL upper limits on the spin-independent WIMP-nucleon scattering cross section as a function of the DM particle mass for direct detection experiments and the interpretation of the $H \to$ invisible combination result in models where the Higgs boson is a portal between the SM matter and the Dark Sector. For the vector UV-model, $m_2$ are two representative values of the predicted Dark Higgs particle with mixing angle $\alpha$. The regions above the limit contours are excluded in the range shown in the plot. From Ref. [205].</figcaption></figure>

The results above are derived from the combination of several channels for searches performed with Run-1 data. Both ATLAS and CMS have produced updates of individual channels with Run-2 data. ATLAS has searched for FCNC top decays with subsequent decays of the Higgs boson to a pair of photons [225], yielding a 95% CL upper limit on $BR(t \to Hc) < 0.22\%$ with an expected sensitivity of 0.16%. CMS has searched for FCNC top decays with subsequent decays of the Higgs boson to a pair of b-quarks [226], yiedling a 95% CL upper limit on $BR(t \to Hu) < 0.079\%$ with an expected sensitivity of 0.011% and on $BR(t \to Hc) < 0.094\%$ with an expected sensitivity of 0.086%.

#### 11.3.8.7 Exotic Higgs boson decays

The 125 GeV Higgs boson serves not only as a probe for potential DM candidates, but also to search for other exotic particles arising from fields associated with a low-mass hidden sector (for a review, see for instance Ref. [227]). Such hidden sectors are composed of fields that are singlet under the SM gauge group $SU(3) \times SU(2) \times U(1)$. These models are referred to as hidden valley models [228]. Since a light Higgs boson is a particle with a narrow width, even modest couplings to new states can give rise to a significant modification of the Higgs boson phenomenology through exotic decays. Simple hidden valley models exist in which the Higgs boson decays to an invisible fundamental particle, which has a long lifetime to decay back to SM particles through a small mixing with the SM Higgs boson, see Ref. [228] for a concrete example. The Higgs boson may also decay to a pair of hidden valley “v-quarks,” which subsequently hadronise in the hidden sector, forming “v-mesons.” These mesons often prefer to decay to the heaviest state kinematically available, so that a possible signature is $H \to 4b$. Some of the v-mesons may be stable, implying a mixed missing energy plus heavy flavour final state. In other cases, the v-mesons may decay to leptons, implying the presence of low mass lepton resonances in high-$H_T$ events [229]. Other scenarios have been studied [230] in which the Higgs boson decays predominantly into light hidden sector particles, either directly, or through light SUSY states, and with subsequent cascades that increase the multiplicity of hidden sector particles. In such scenarios, the high-multiplicity hidden-sector particles, after decaying back into the SM, appear in the detector as clusters of collimated leptons known as “lepton jets”.

A variety of models have been investigated searching for final states involving dark photons and hidden valley scalars. The resulting topologies typically have leptons or light hadrons which in some cases can be prompt (i.e., originating from the hard process interaction point) or not and are in some cases collimated and reconstructed as jets [231, 232], and long-lived weakly interacting particles. The latter occur not only in hidden valley scenarios, but also in gauge-mediated extensions of the minimal SUSY standard model (MSSM), the MSSM with R-parity violation, and inelastic DM scenarios [233]. Finally, CMS has performed a search for pair production of light bosons [234]. Such a scenario can occur in SUSY models with additional hidden (or dark) valleys.

### 11.3.9 Higgs boson pair production and self-coupling

Higgs boson pair production in the SM is a rare but very important mode to measure and search for. The measurement of Higgs boson pair production is essential to directly constrain the trilinear Higgs boson self coupling and the search for Higgs boson pair resonances is key in a variety of BSM models. The latter searches are discussed in Section 11.6.7.

In the SM, the main non-resonant production mode of two Higgs bosons proceeds through a loop, mainly of top quarks, see Fig. 11.1 (h). Another production mode is via the trilinear coupling of the Higgs boson, see Fig. 11.1 (i), whose amplitude is not negligible compared to the former. These diagrams interfere negatively, making the overall production rate smaller than what would be expected in the absence of a trilinear coupling.

#### 11.3.9.1 Searches for Higgs boson pair production

The searches for Higgs boson pair production both resonant and non-resonant are important probes for a variety of BSM theories, and they can be done in a large number of Higgs boson decay channels. At Run 1, ATLAS and CMS have searched for both resonant and non resonant Higgs boson pair production in the following channels: (i) $HH \to b\bar{b}\gamma\gamma$; (ii) $HH \to b\bar{b}\tau^+\tau^-$; (iii) $HH \to b\bar{b}b\bar{b}$; (iv) $HH \to b\bar{b}4l$; (v) $HH \to WW^*\gamma\gamma$; (vi) in final states containing multiple leptons (electrons or muons) covering the $WW^*WW^*$, $WW^*ZZ^*$, $ZZ^*ZZ^*$, $ZZ^*\tau^+\tau^-$, $WW^*\tau^+\tau^-$, $ZZ^*b\bar{b}$, $\tau^+\tau^-\tau^+\tau^-$ channels; and (vi) $\gamma\gamma\tau^+\tau^-$ channels.

At Run 2, similarly to the $t\bar{t}H$ production process, the di-Higgs production profits from a significant increase in production cross section of a factor in excess of 3 from 8 TeV to 13 TeV, and most
of these channels have been updated both by ATLAS [235] and CMS [184, 236]. The detailed description of the various analyses used can be found in the references therein. All the results and their combinations are summarised in Table 11.7 and in Fig. 11.5 (left). The channels (i)-(iii) are the most sensitive to the HH production. These three channels have been updated with the full Run-2 dataset except the $HH \to b\bar{b}\tau^+\tau^-$ channel by CMS and the $HH \to b\bar{b}b\bar{b}$ channel by ATLAS which have both been analysed using a partial Run-2 dataset.

In several of these analysis channels the VBF production mode is searched for separately providing sensitivity to the coupling involving two vector bosons and two Higgs bosons HHVV which is an expected coupling in the Standard Model.

<table><caption>**Table 11.7:** Summary of the final states investigated in the search for Higgs boson pair production by ATLAS and CMS, the results reported in bold are based on the full Run-2 dataset. For ATLAS, the result indicated by (*) uses mostly the $b\bar{b}W^+W^-$ channel. Results are 95% CL upper limits on the observed (expected) signal strengths.</caption><thead><tr><th>Channel</th><th>ATLAS</th><th>CMS</th></tr></thead><tbody><tr><td>b<span style="text-decoration: overline;">b</span>γγ</td><td><strong>4.2</strong> (5.7)</td><td><strong>8.4</strong> (5.5)</td></tr><tr><td>b<span style="text-decoration: overline;">b</span>b<span style="text-decoration: overline;">b</span></td><td><strong>5.4</strong> (8.1)</td><td><strong>7.2</strong> (4.2)</td></tr><tr><td>b<span style="text-decoration: overline;">b</span>τ<sup>+</sup>τ<sup>-</sup></td><td><strong>4.7</strong> (3.9)</td><td><strong>3.3</strong> (5.2)</td></tr><tr><td>b<span style="text-decoration: overline;">b</span>4ℓ</td><td>-</td><td><strong>32</strong> (40)</td></tr><tr><td>W<sup>+</sup>W<sup>-</sup>W<sup>+</sup>W<sup>-</sup></td><td>160 (120)</td><td>-</td></tr><tr><td>W<sup>+</sup>W<sup>-</sup>γγ</td><td>230 (170)</td><td><strong>97</strong> (52)</td></tr><tr><td>b<span style="text-decoration: overline;">b</span>VV</td><td>305 (305)*</td><td><strong>14</strong> (18)</td></tr><tr><td>Multi-lepton</td><td>-</td><td><strong>19</strong> (21)</td></tr><tr><td>Combination</td><td><strong>2.4</strong> (2.9)</td><td><strong>3.4</strong> (2.5)</td></tr></tbody></table>

The current single channel most stringent limits obtained on the HHVV coupling modifier denoted $\kappa_{2V}$ are obtained from the analysis performed by CMS [237] in the $b\bar{b}b\bar{b}$:

$$-0.1 < \kappa_{2V} < 2.2 \text{ (observed), } -0.4 < \kappa_{2V} < 2.5 \text{ (expected)}$$

and for the overall most stringent constraints are obtained through the combination of channels in ATLAS, yielding [235]:

$$-0.1 < \kappa_{2V} < 2.0 \text{ (observed), } 0.0 < \kappa_{2V} < 2.1 \text{ (expected)}$$

### 11.3.9.2 The Higgs boson self coupling

The Higgs boson self coupling is an extremely important direct probe of the Higgs potential with implications on our understanding of the electroweak phase transition. Constraints on the trilinear self coupling from HH processes is an outstanding long term goal of the LHC and the reach in sensitivity has been reappraised in the light of the recent HH analyses from ATLAS and CMS, shedding a different light on the achievable sensitivity [118]. Constraints from the HHH final state on the quartic Higgs boson self coupling are out of reach at the LHC due mostly to the very small production rates and intricate final states.

In the SM, the Higgs boson pair production through the trilinear Higgs boson self coupling has an on-shell component and a large off-shell component. The on-shell $H \to H^*H^*$ is strongly disfavoured, requiring two off-shell Higgs bosons in the final state. The sensitivity region to the trilinear coupling production as in Fig. 11.1 (i), is mainly in the kinematic region where the two Higgs boson in the final state are on-shell and the Higgs boson acts as a propagator (off-shell). As discussed in the introduction to this section, this process interferes negatively with the background Higgs

boson pair production (Fig. 11.1 (h)). The extraction of the trilinear component of the di-Higgs production measurements is very similar to the extraction of the off-shell component of the Higgs coupling to vector bosons discussed in Section 11.4.2.

The measurement of the trilinear coupling requires separating the contributions of the diagram of Fig. 11.1 (i) from the box diagram of Fig. 11.1 (h), and therefore a precise knowledge of the top-Yukawa coupling is needed. Each diagram alone would produce a rather distinct $m_{HH}$ distribution. And, for values of the trilinear coupling close to the SM value, an additional discriminating feature of the signal with respect to one obtained with the box contribution alone is a deficit in the number of events. With large variations of the trilinear coupling, an excess of events over the SM prediction would be observed (for a value of the trilinear coupling about 6 times larger than its SM value, the number of events is equal to the SM expectation). Additional sensitivity to the trilinear coupling is also obtained from the kinematical distributions of the signal taking in particular into account the effect of the HH mass distribution which discriminates the main contributions of Fig. 11.1. This further discrimination is instrumental in resolving the degeneracy in the total cross section mentioned above. Bounds on the trilinear coupling are obtained from limits on the HH production as a function of the trilinear coupling factor $\kappa_\lambda$ as illustrated in Fig. 11.5. As the sensitivity in this channel grows it will be interesting to more closely look at the combined and individual likelihoods as a function of the trilinear coupling to quantify the exclusion of the secondary minimum induced by the degeneracy between the value of $\kappa_\lambda$ yielding the same overall HH rate as the Standard Model (i.e. $\kappa_\lambda = 1$). The results obtained by ATLAS [235] and CMS [184] are the following:

$$ (\text{ATLAS}) - 0.6 < \kappa_{\lambda} < 6.6 \text{ (observed),} \\ - 2.1 < \kappa_{\lambda} < 7.8 \text{ (expected),} \qquad (11.19) $$

$$ (\text{CMS}) - 1.2 < \kappa_{\lambda} < 6.5 \text{ (observed),} $$

where $\kappa_\lambda$ is the ratio between the trilinear coupling value left free in the fit and its expected value in the SM ($\kappa_\lambda = 1$ corresponds to the SM). For the CMS result the expected excluded range is not specified in [184], however its illustration in the exclusion figure therein shows that it is compatible with the observation. The so-called $\kappa$ formalism is discussed in more detail in Section 11.5.1.1. These results are also illustrated in Fig. 11.5.

Single Higgs production can indirectly also be used to further constrain the trilinear coupling of the Higgs boson, both using inclusive total or differential cross sections. A combination of di-Higgs and single-Higgs boson production results has been performed by the ATLAS collaboration yielding the following limits [235]:

$$-0.4 < \kappa_{\lambda} < 6.3 \text{ (observed),}$$

It is interesting to note that currently the direct ($-0.6 < \kappa_\lambda < 6.6$) and indirect constraints ($-4.0 < \kappa_\lambda < 10.3$) on the Higgs boson trilinear self-coupling are of comparable strength. However, the double Higgs boson measurements are dominated by statistical uncertainties and are expected to improve more rapidly than the precision on single Higgs boson measurements. It should also be stressed that the constraints on the trilinear self-coupling obtained via the NLO fit of single Higgs boson data are model-dependent since the NLO effects induced by a shift of the trilinear self-coupling compete with possible LO effects sourced by the deviations of the Higgs boson couplings to the other SM particles. The different effects can be disentangled by the measurements of various kinematical differential distributions in addition to the study of the inclusive rates [238], but the expected sensitivity obtained in such global fits are not as strong as when only the Higgs boson self-coupling is allowed to deviate from its SM value [119]. It should also be noted that in this fit the parameters $\kappa_t, \kappa_b, \kappa_\tau$ and $\kappa_V$ are fit simultaneously.

The analyses performed at Run2 bring substantial improvements from those of Run 1, and they were first used to reappraise the sensitivity of the LHC in the High Luminosity regime in the framework of the update of the European Strategy for Particle Physics [118]. The result in terms of bounds on the trilinear coupling indicated that the significance of the observation of the HH
<figure><img src="image_32.png" /><figcaption>**Figure 11.5:** (Left) Upper limits obtained by CMS on the total $pp \to HH$ production cross section. (Top right) Upper limits obtained by ATLAS on the total $pp \to HH$ production cross section as a function of the trilinear coupling modifier $\kappa_\lambda$. The variation of the limit corresponds to variations in the signal acceptance. The expected total production cross section is also illustrated (red). (Bottom right) Likelihood contours for the ATLAS di-Higgs and single-Higgs combined results as a function of $\kappa_\lambda$ and $\kappa_t$.</figcaption></figure>

process reaches $4\sigma$. It is also apparent that the degeneracy of secondary minimum at intermediate values of $\kappa_\lambda$ is resolved by the use of the kinematic discriminants. This secondary minimum was expected to be excluded at 99.4% CL and at HL-LHC, the foreseen precision on $\kappa_\lambda$ was estimated to reach approximately 50%. These studies were based on early LHC Run-2 analyses. Then in the framework of the Snowmass2021 studies [239] and more recently using the latest HH searches results, these projections have been updated. Both experiments have reappraised their studies at HL-LHC [240], to reach individually a sensitivity of approximately 3.5$\sigma$, a significant improvement compared to the sensitivities estimated between 2.5$\sigma$ and 3$\sigma$ in [118]. These studies indicate that a sensitivity of 5$\sigma$ in combination of the two experiments should be reached at HL-LHC.

Higher precisions can be reached at pp colliders (and $e^+e^-$ colliders) at higher centre-of-mass energies. The foreseen precision for a High-Energy (HE) LHC at a centre-of-mass energy of 27 TeV is expected to be within 10% to 20% [118]. At a very large hadron collider at a centre-of-mass energy of 100 TeV, a 5% sensitivity is expected to be reached, provided that the theoretical and parametric uncertainties are kept at the 1% level [241].

One of the main open questions in particle physics is related to the precise history of the EWSB mechanism. This in turn can be of relevance to provide a mechanism for generating the matter - antimatter asymmetry of the universe at the electroweak scale, the so called electroweak baryogenesis (EWBG) (see Refs. [242, 243] for some reviews). In the SM EWBG is ruled out, but new physics models, in particular those additional scalar particles coupled/mixed with the Higgs boson, can provide a successful alternative. Given that di-Higgs production will become an excellent probe of trilinear Higgs coupling modifications, it will provide a unique window into the history of EWSB and will become a powerful test of EWBG models.

### 11.3.10 Beyond the main Higgs boson production modes

Most of the production modes illustrated in Fig.11.1 have been investigated at the LHC so far. These processes are often referred to as the 'main production modes'. Beyond these production modes, two additional SM processes involving the Higgs boson

have been investigated and are worth noting.

The first is the four top production mode which is closely related to the $t\bar{t}H$ mode. This process has recently been unambiguously observed at the LHC by both ATLAS [244] and CMS [245]. This production mode has a non trivial contribution from the exchange of a Higgs boson. A first limit on the Yukawa coupling of the Higgs boson has thus been derived using this four top process, $|\kappa_t| < 1.8(1.6)$, under the assumption that the Yukawa coupling of the Higgs boson to top quarks in CP even [244].

Another process beyond the main production modes is the associated WH production through Vector Boson Fusion (VBF) yielding a topology of events with two jets in the forward region with a large rapidity gap, a W boson and a Higgs boson. This process is particularly interesting as it is sensitive to the relative sign of the coupling of the Higgs boson to the W and Z boson through the interference of the processes where the Higgs boson couples either to the W or the Z bosons in the vector boson fusion process. ATLAS has carried out analyses targeting at the decay of the Higgs boson to $b\bar{b}$ and $W^+W^-$ for this process, resulting in the exclusion of the opposite sign hypothesis by 8 standard deviations and an upper limit corresponding to 11.2 times the cross section expected in the Standard Model [246].

### 11.3.11 First results on the main production and decay channels at 13.6 TeV

After a long shutdown between 2019 and 2022, devoted to the upgrades of the experiments and the machine to prepare for the High Luminosity phase, in the Spring 2022 the LHC delivered pp collisions data at an enhanced centre-of-mass energy of 13.6 TeV. During this high-energy run, ATLAS and CMS have collected datasets corresponding to integrated luminosities of approximately 60 fb⁻¹ each. The first preliminary measurements of Higgs boson production at this increased centre-of-mass energy using the two high-resolution channels $H \to \gamma\gamma$ and $H \to ZZ^*$ → $\ell^+\ell^-\ell^+\ell^-$ have been carried out by ATLAS [247], see Fig. 11.6. The measurements are in good agreement within the precision with the theoretical prediction which display the expected increase of approximately 7% in inclusive cross section from the increase in centre-of-mass energy.
<figure><img src="image_33.png" /><figcaption>Figure 11.6: Higgs boson production cross sections as a function of the pp centre-of-mass energy. The SM predictions are shown by the shaded band.</figcaption></figure>

## 11.4 Main quantum numbers and width of the Higgs boson

### 11.4.1 Main quantum numbers $J^{PC}$

Probing the Higgs boson quantum numbers is essential to further unveiling its coupling properties. The measurements of the signal event yields in all the channels discussed in Section 11.3 and their compatibility with the SM Higgs boson predictions, give a qualitative but, nonetheless, compelling indication of its nature. For the observed particle not to be of spin 0 and +1 parity would require an improbable conspiracy of effects. It is nevertheless important to test this hypothesis independently, in particular since the measurements of coupling properties of the Higgs boson assume that it is a CP-even state.

This qualitative picture is further complemented by the implications of the observation of the particle in the diphoton channel. According to the Landau-Yang theorem [248], the observation made in the diphoton channel excludes the spin-1 hypothesis and restricts possibilities for the spin to 0 or 2.

The Landau-Yang theorem does not apply if the observed state is not decaying to a pair of photons but to a pair of scalars subsequently decaying to two very collimated pairs of photons (as for example in the case of $H \to a_1a_1 \to 4\gamma$). This possibility has not been rigorously excluded but is not experimentally favoured since tight selection criteria are applied on the electromagnetic shower shapes of the reconstructed photons. A more systematic analysis of shower shapes and the fraction of conversions could be performed to further discriminate between the single prompt photon and the two overlapping photons hypotheses. There are also potential theoretical loopholes concerning the applicability of the Landau-Yang theorem, such as off-shell vector boson decays.

The spin 1 hypothesis has therefore been investigated quantitatively with the LHC Run-1 dataset and has been excluded at more than 95% CL through the analysis of the Higgs boson decays to $WW^*$ and $ZZ^*$ channels. These results are reported in previous versions of this review and will not be covered herein. [188].

#### 11.4.1.1 Charge conjugation

The charge conjugation quantum number is multiplicative, therefore given that the Higgs-like particle is observed in the $H \to \gamma\gamma$ channel, and given that photons are C-odd eigenstates, assuming C conservation, the observed neutral particle should be C-even.

#### 11.4.1.2 Spin and parity

To probe the spin and parity quantum numbers of the discovered particle, a systematic analysis of its production and decay processes is performed in several analyses. These analyses are designed to be independent of the measured event yields and they rely instead on the production and the decay angles, and on the threshold distributions as long as a significant signal is observed,

i.e., in situations when an excess over the expected background can be used to further discriminate between signal hypotheses. These analyses are based on probing various alternative models of spin and parity [249]. These models can be expressed in terms of an effective Lagrangian [250] or in terms of helicity amplitudes [251]. The two approaches are equivalent.

### i. Spin-0 models

The interaction Lagrangian relevant for the analysis of spin-0 particle interaction with a pair of W or Z bosons with either fixed or mixed SM and BSM CP-even couplings or CP-odd couplings, is the following [252]:

$$ \mathcal{L}_0^{W,Z} \supset \left\{ \begin{aligned} & \cos(\alpha) \kappa_{SM} \left[ \frac{1}{2} g_{HZZ} Z_\mu Z^\mu + g_{HWW} W_\mu^+ W^{-\mu} \right] \\ & - \frac{1}{4\Lambda} \left[ \cos(\alpha) \kappa_{HZZ} Z_{\mu\nu} Z^{\mu\nu} + \sin(\alpha) \kappa_{AZZ} Z_{\mu\nu} \tilde{Z}^{\mu\nu} \right] \\ & - \frac{1}{2\Lambda} \left[ \cos(\alpha) \kappa_{HWW} W_{\mu\nu}^{+} W^{-\mu\nu} + \sin(\alpha) \kappa_{AWW} W_{\mu\nu}^{+} \tilde{W}^{-\mu\nu} \right] \end{aligned} \right\} H, \quad (11.20) $$

where $V^\mu = Z^\mu, W^+\mu$ are the vector boson fields, $V^{\pm\mu\nu}$ are the reduced field tensors, $\tilde{V}^{\pm\mu\nu} = 1/2 \epsilon^{\mu\nu\rho\sigma} V_{\rho\sigma}$ are the dual tensor fields, and $\Lambda$ defines an effective theory energy scale. The factors $\kappa_{SM}, \kappa_{HZZ}, \kappa_{HWW}, \kappa_{AZZ}, \kappa_{AWW}$ denote the coupling constants corresponding to the coupling of the SM and BSM CP-even and CP-odd components of the Higgs boson to the W and Z fields. The mixing angle $\alpha$ allows for the production of a CP-mixed state and the CP-symmetry is broken when $\alpha \neq 0, \pi$.

This formalism can be used to probe both CP-mixing for a spin-0 state, as discussed in Section 11.4.1.5 or specific alternative hypotheses, as discussed below, such as a pure CP-odd state ($J^P = 0^-$) corresponding to $\alpha = \pi/2$, $\kappa_{SM} = \kappa_{HVV} = 0$ and $\kappa_{AVV} = 1$. A BSM CP-even state $J^P = 0^+$ corresponds to $\alpha = 0, \kappa_{AVV} = 0, \kappa_{HVV} = 1$ and $\kappa_{SM} = 1$. This formalism has been adopted by the ATLAS experiment.

<figure><img src="image_34.png" /><figcaption>Figure 11.7: Definition of the production and decay angles defined for the $H \to ZZ^{(*)} \to 4\ell$ final state.</figcaption></figure>

A different parametrisation of anomalous couplings of a spin-zero boson with two gauge bosons $VV$ can also be expressed in the
general form of the scattering amplitude A :

$$
\begin{equation}
\begin{split}
A \sim & \left[ a_1^{VV} - \frac{\kappa_1^{VV} q_1^2 + \kappa_2^{VV} q_2^2}{(\Lambda_1^{VV})^2} - \frac{\kappa_3^{VV} (q_1 + q_2)^2}{(\Lambda_Q^{VV})^2} \right] m_{V1}^2 \varepsilon_{V1}^* \varepsilon_{V2}^* \\
& + a_2^{VV} f_{\mu\nu}^{*(1)} f^{*(2)\mu\nu} + a_3^{VV} f_{\mu\nu}^{*(1)} \tilde{f}^{*(2)\mu\nu}
\end{split}
\tag{11.21}
\end{equation}
$$

where $\epsilon_i$ is the polarization vector of the boson $V_i$, $f_{\mu\nu}^{*(i)} = \epsilon_i^\mu q^\nu - \epsilon_i^\nu q^\mu$ is a scalar tensor constructed from the vector boson $V_i$ polariza-
tion and four momentum, $\tilde{f}_{\mu\nu}^{*(i)} = \frac{1}{2}\epsilon_{\mu\nu\rho\sigma}f^{*(i)\rho\sigma}$ is the corre-
sponding pseudo-scalar tensor. $\Lambda_1$ and $\Lambda_Q$ are new physics scales,
$a_{1,2,3}$ are coupling strength modifiers and $|\kappa_{1,2,3}^{VV}| = 0$ or 1. This
parametrisation can be used both for gluons, photons and elec-
troweak vector bosons, as well as the $Z\gamma$ coupling. In the case of
the W and Z bosons, the custodial symmetry would require that
$a^{WW} = a^{ZZ}$ and, at tree-level, the only non-zero contributions
would come from the $a_1$ term. It is fully equivalent to the inter-
action Lagrangian approach described above. This formalism has
been adopted by the CMS experiment.

Finally, CP-odd interactions can also be expressed in the frame-
work of the Standard Model Effective Field Theory (SMEFT) in
various possible bases (as discussed in Section 11.5.2). This for-
malism is more in line with analyses of a broader ensemble of
observable probing the Standard Model. In this case the effec-
tive Lagrangian is defined as a function of the $\Phi^\dagger\Phi$ gauge invari-
ant and the gauge field tensors before the electroweak symme-
try breaking $W^I_{\mu\nu}$ and $B_{\mu\nu}$ with three operators, $\Phi^\dagger\Phi\tilde{W}^I_{\mu\nu}W^{I\mu\nu}$,
$\Phi^\dagger\Phi\tilde{W}^I_{\mu\nu}B^{\mu\nu}$, and $\Phi^\dagger\Phi\tilde{B}_{\mu\nu}B^{\mu\nu}$, and their corresponding Wilson
coefficients $c_{H\bar{W}}$, $c_{H\bar{W}B}$, and $c_{H\bar{B}}$ respectively.

It is important to note that, as is apparant in Eq. (11.20), CP-odd couplings of the Higgs boson to vector boson are not allowed at tree level and are therefore suppressed at higher order by the scale Λ. Instead, CP-odd couplings of the Higgs boson to fermions are allowed at tree level, these can be parametrised in a somewhat simpler fashion with the following Lagrangian which accounts only for an additional pseudo-scalar term:

$$
\mathcal{L}_0^{\psi f} \supset \frac{\lambda_f}{\sqrt{2}} \left( \kappa_f H \bar{\psi}_f \psi_f + i \bar{\kappa}_f H \bar{\psi}_f \gamma_5 \psi_f \right). \quad (11.22)
$$

This formulation is equivalent to a definition a CP-mixing angle
$\varphi_f = \tan \frac{\bar{\kappa}_f}{\kappa_f}$.

It is interesting to note that CP-odd Higgs couplings to fermions could have a significant impact on the very clean and precisely measured low energy observables as the electron EDM [253]. Owing to its large Yukawa couplings, the CP-odd top Yukawa is strongly constrained by the stringent electron EDM limits to better than $\bar{\kappa}_t < 0.001$ [253]. This makes it very challenging at the LHC to surpass. However, one should keep in mind that EDM experiments provide an indirect constraint on the SM Yukawa couplings, in particular in this case they constrain the product $\kappa_e \times \bar{\kappa}_t$, and one needs to make assumptions on the value of $\kappa_e$, a quantity that remains out of reach experimentally. The limits on the tau Yukawa coupling from the electron EDM constraints are much less stringent and at the level of $\bar{\kappa}_\tau < 0.3$ [253]. This makes the possibility of measuring CP mixing of the fermion Higgs Yukawa couplings particularly interesting in the tau channels since these can be accessed from the Higgs decays directly using tau polarisation sensitive variable and will be discussed herein. For more details on the searches for CP-odd couplings of the Higgs boson to vector bosons please see previous versions of this review [188].

ii. Spin-2 model

The graviton-inspired interaction Lagrangian for a spin-2 boson $X^{\mu\nu}$ that does not carry any color, weak and electromagnetic charge and that uniquely interacts with the energy momentum tensor $T^{V,f}$ of vector bosons $V$ or fermions $f$, can be written as follows [252]:

$$
\mathcal{L}_2 \supset \frac{1}{\Lambda} \left[ \sum_V \xi_V \mathcal{T}_{\mu\nu}^V X^{\mu\nu} + \sum_f \xi_f \mathcal{T}_{\mu\nu}^f X^{\mu\nu} \right], \quad (11.23)
$$

where the strength of the interaction is determined by the cou-
plings ξ_V and ξ_f. The simplest scenarios, referred to as universal
couplings (UC), correspond to ξ_V = ξ_f. They predict a large
branching ratio to photons (of approximately 5%) and negligi-
ble couplings to massive gauge bosons (W and Z). They are
therefore disfavoured, and other models are investigated where
the couplings of the W, Z and γ are assumed to be independent.
Universality of the couplings refers to ξ_g = ξ_q. Two other scenar-
ios are considered: ξ_q = 0 and ξ_q = 2ξ_g. In these scenarios, a large
enhancement of the tail of the transverse momentum of the spin-2
state is expected and requires a further selection requirement in
order to probe the models within the range of validity of the effec-
tive field theory. Two requirements are considered, p_T^X < 300 GeV
and p_T^X < 125 GeV [250].

11.4.1.3 The VH production at D0

The mass of the VH system is a powerful discriminant to dis-
tinguish a $J^P = 0^+$, with a threshold behaviour in $d\sigma/dM^2 \sim$
$\beta, \beta^3, \beta^5$ from a $0^+, 0^-$ and $2^+$ state, respectively [254]. The VH
mass observable not only discriminates signal hypotheses, but also
has an increased separation between the $0^-$ and $2^+$ hypotheses
with respect to the backgrounds, thus allowing, with a small and
not yet significant signal yield, to exclude that the observed state
is $0^-$ at 98% CL [255] and $2^+$ at the 99.9% CL [256], assuming
a signal produced with their best fit signal strength (which was
$\mu = 1.23$).

11.4.1.4 Probing the spin in the γγ channel at the LHC

In the $H \to \gamma\gamma$ channel, the analysis is performed inclusively using the production angle $\cos\theta_{CS}^*$ and the transverse momentum of the diphoton pair [250]. The polar angle in the rest frame is defined with respect to the bisector axis of the momenta of the incoming protons and is referred to as the polar angle in the Collins-Soper frame [257]. The SM Higgs boson signal distribution is expected to be uniform with a cutoff due to the selection requirements on the photons transverse momentum. The $H \to \gamma\gamma$ channel is mostly sensitive to the gluon-initiated spin-2 production scenarios, which yield a $\cos\theta_{CS}^*$ distribution peaking at values close to 1. The ATLAS limits are derived from a fit of the signal in bins of $\cos\theta_{CS}^*$ and diphoton transverse momentum. The data is compatible with the SM $0^+$ hypothesis and contributes strongly to the exclusion of several spin-2 scenarios. The conclusions are the same from CMS results [251].

11.4.1.5 Probing CP-mixing and anomalous HVV couplings

The careful study of the kinematic properties of angular distributions in events observed in the $H \to ZZ^{(*)} \to 4\ell$ and $H \to WW^{(*)} \to \ell\nu\ell\nu$ channel, as well as the analysis of production observables sensitive to anomalous HVV interactions in the VBF or VH production have been used to set limits wither on CP-mixing parameters, amplitude terms ot Wilson coefficients in the framework of SMEFT. These analysis assume that the observed particle is a spin-0 state, several results were obtained by the ATLAS [258] and CMS [251] collaborations, all show a good agreement with the Standard Model CP-even hypothesis. Recent specific examples are given here.

In the $H \to WW^{(*)} \to \ell\nu\ell\nu$ channel, the production and decay
angles cannot be easily reconstructed due to the presence of neu-
trinos in the final state, however, sensitivity arises from the $V-A$
structure of the decay of the $W$ bosons. A scalar state thus yields
a clear spin correlation pattern that implies that the charged lep-
tons $e$ or $\mu$ from the decays of the $W$ bosons are produced close to
one another in the transverse plane. This feature impacts observ-
ables such as the azimuthal angle between the two leptons $\Delta\Phi_{\ell\ell}$
or their invariant mass $m_{\ell\ell}$ in addition to the threshold behaviour
of the decay. It can be used to discriminate between various spin
and parity hypotheses.

The $H \to ZZ^{(*)} \to 4\ell$ coupling analysis, as described in Sec-
tion 11.3, also uses a discriminant based on the $0^+$ nature of the
Higgs boson to further separate signal and background. In this
analysis, this feature is used to discriminate between signal hy-
potheses. The observables sensitive to the spin and parity have
first been identified in Ref. [249, 259, 260] the masses of the two
Z bosons (due to the threshold dependence of the mass of the