off-shell Z boson [249, 261]), two production angles $\theta^*$ and $\Phi_1$, and three decay angles, $\Phi$, $\theta_1$ and $\theta_2$. The production and decay angles are defined as:

- $\theta_1$ and $\theta_2$, the angles between the negative final state lepton and the direction of flight of $Z_1$ and $Z_2$ in the rest frame.

- $\Phi$, the angle between the decay planes of the four final state leptons expressed in the four lepton rest frame.

- $\Phi_1$, the angle defined between the decay plane of the leading lepton pair and a plane defined by the vector of the $Z_1$ in the four lepton rest frame and the positive direction of the proton axis.

- $\theta^*$, the production angle of the $Z_1$ defined in the four lepton rest frame with respect to the proton axis.

These angles are illustrated in Fig. 11.7 [260].

These analyses are sensitive to various $J^P$ hypotheses and in particular discriminate the $0^+$ hypothesis from the $0^−$. In all scenarios investigated, and for both ATLAS and CMS, the data is compatible with the $0^+$ hypothesis. ATLAS and CMS exclude a pure pseudo-scalar nature of the observed boson at $CL_S$ levels of 98% and 99.8% [251].

An individual $ZZ^*$ channel measurement has also been carried out with a partial Run-2 dataset by ATLAS [258]. CMS has performed a CP-mixing analysis of a partial Run-2 dataset of 36 pb$^{-1}$ combined with the full Run-1 data using the $ZZ^*$ channel [262]. In this analysis the CMS experiment sets constraints on the parameters defined in the parametrisation (11.21) of the scattering amplitude.

With the full Run-2 dataset, analyses have started exploiting in combination both the production and decay modes of the Higgs boson as exemplified by the analysis carried out by ATLAS using the $H \to ZZ^*$ $\to 4\ell$ analysis taking advantage of the production and decay angles of the process and VBF production sensitive observables [263]. Very similar analyses have been carried out in CMS as well. The analysis uses optimal observables constructed from matrix elements of the SMEFT. These optimal observables are defined from the ratio of the interference term between the Standard Model and the SMEFT matrix elements (normalised to the Standard Model) for given values of the Wilson coefficients $c_{H\bar{W}}$, $c_{H\bar{W}B}$, and $c_{H\bar{B}}$ below Eq. (11.21). As illustrated in Fig. 11.8, all measurements are in agreement with the Standard Model. ATLAS has also performed a similar analysis with the Higgs boson decaying to two photons analysing the VBF production mode [264] which can then be used to constrain the $c_{H\bar{W}}$ parameter as shown in Fig. 11.8.

An analysis of Higgs boson anomalous couplings $HVV$ (as well as pseudo-scalar Yukawa coupling to the top) in the $H \to 4\ell$ channel was also carried out by CMS [265].

ATLAS has also performed an analysis using optimal observables [266], defined as the ratio of the interference between the CP-odd and the SM contributions normalised to the SM matrix element squared, using the Run-1 data. In this study, the CP-mixing contributions are described in the framework of an effective field theory governed by a single parameter $\tilde{d}$, found to be consistent with its SM value of $\tilde{d} = 0$ and constrained to the interval [-0.11,0.05] at the 68% CL.

CP invariance in the $HVV$ coupling can also be probed with the VBF production process in the $H \to \tau^+\tau^-$ channel or similarly to the D0 analysis in the VH channels. CMS has performed an analysis in this channel and has combined its results with the aforementioned $ZZ^*$ channel using the same dataset [267].

Finally two additional specific measurements as for example in $H \to WW^*jj$ [268] by ATLAS and using the $H \to \tau^+\tau^-$ final state [269] by CMS in several production modes have also been used to set limits on both anomalous CP-even and CP-odd couplings.

These measurements show the complementarity of the study of production and decay modes, separately and in combination, and underscores the importance of a synthetic approach that combines all the available information.

### 11.4.1.6 Probing $H\bar{f}\bar{f}$ coupling CP properties in $pp \to t\bar{t}H$ production

Although EDM limits disfavour large CP-odd effects in the top Yukawa coupling of the Higgs boson, these indirect limits are model dependent. It is therefore important to further probe these

effects directly. This can be achieved using the $pp \to t\bar{t}H$ process. The increased sensitivity to this process at Run 2 in particular in the channels where the Higgs boson decays to a pair of photons provide a good sensitivity to the CP properties of the coupling through kinematics and angular distributions between the Higgs boson and the top quarks produced in the final state.

With the full Run-2 dataset, several Higgs boson decay channels have been investigated. ATLAS [270] excludes CP-mixing angles $\phi_t$ greater than 43° and smaller than -43° at 95% confidence level in the $H \to \gamma\gamma$ channel and has recently published constraints using the $H \to b\bar{b}$ channel with a measurement of $\phi_t = 11^\circ +52^\circ -73^\circ$ [271].

Also using the full Run-2 dataset, CMS [272] measures the CP-odd fraction defined as follows:

$$f_{CP} = \frac{|\kappa_t|^2}{|\kappa_t|^2 + |\kappa_t|^2} \times \text{sign}(\kappa_t/\kappa_t)$$

in the diphoton channel to be $f_{CP} = 0.00 \pm 0.33$. CMS also derived a constraint on the CP-odd fraction using the $H \to 4\ell$ channel [265]. More recently CMS has analysed the $t\bar{t}H$ lepton channel, sensitive both to the $H \to WW$ and $H \to \tau^+\tau^-$ decays, to probe the top Yukawa CP properties [273], and combined its results with the two aforementioned channels to yield the following constraint on the CP-odd fraction:

$$|f_{CP}| = 0.28^{+0.27}_{-0.28}.$$

Both the ATLAS and CMS experiments used multivariate analyses in order to maximize the sensitivity to the measurement of the CP properties of the $H\bar{f}\bar{f}$ coupling. Both experiments used leptonic and hadronic decays of the top quarks.

### 11.4.1.7 Probing $H\bar{f}\bar{f}$ coupling CP properties in $H \to \tau^+\tau^-$ decays

As discussed in Section 11.4.1, it interesting to investigate the CP properties of the $H\bar{f}\bar{f}$ coupling is the Higgs decay to taus through tau polarisation sensitive variables.

CMS performed a first analysis with the full Run-2 dataset [274] of the transverse spin correlations between the tau leptons in the final state which are sensitive to the CP mixing angle or relative component of the scalar and pseudo scalar couplings of the Higgs boson to taus. These transverse spin correlations are in turn related to the direction of the decay products of the taus.

In the case of the two-body decay of a tau to a charged pion and a neutrino, the transverse momentum component of the charged pions are typically anti-aligned for the case of a scalar decay and aligned for the pseudo-scalar case. The angle between the decay planes as illustrated in Fig. 11.9-(left) is therefore sensitive to the CP-mixing. Given that the tau direction cannot be measured directly, the impact parameter direction is used instead as a proxy, where the impact parameter direction is defined as the direction between the interaction point and the point of closest approach of the pion track. This direction in the rest frame of the Higgs boson are denoted $\lambda^\pm$, as shown in Fig. 11.9-(centre). In the case of three-body or more decays of the taus as for example the decays to an intermediate rho or an $a_1(1260)$ vector mesons which subsequently decay to $\rho \to \pi^+\pi^0$ and $a_1 \to \pi^+\pi^0\pi^0$ respectively, the planes are defined as either as the $(\pi^+, \pi^0)$ plane as as shown in Fig. 11.9-(centre) in the case of the $\rho$ decay, or the plane formed by the cahrged pion and a combination of the two neutral pions in the case of the $a_1$ decay. The discrimination obtained, showing the two extreme fully scalar and pseudo-scalar cases are illustrated in Fig. 11.9-(right). ATLAS performed a similar analysis [275]. The results obtained are the following:

$$\text{ATLAS} \quad \varphi_{\tau} = 9^{\circ} \pm 16^{\circ} \quad (0^{\circ} \pm 28^{\circ}) \quad [274]$$

$$\text{CMS} \quad \varphi_{\tau} = -1^{\circ} \pm 19^{\circ} \quad (0^{\circ} \pm 21^{\circ}) \quad [275]$$

It is interesting to note that the precision reached by both experiments is already at the level that was foreseen for HL-LHC by initial studies [276], which yielded a precision ranging between $\pm 18^\circ$ and $\pm 30^\circ$.
<figure><img src="image_35.png" /><figcaption>**Figure 11.8:** The measurement of the $c_{H\bar{W}}$ Wilson coefficient based on the optimal observable sensitive to the Higgs boson production, its decay and their combination (left) for the ATLAS $H \to ZZ^* \to 4\ell$ analysis. The measurement of the $c_{H\bar{W}}$ Wilson coefficient based on optimal observable sensitive to the Higgs boson production in the VBF channel (right) for the ATLAS $H \to \gamma\gamma$.</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>**Figure 11.9:** Sketch of the definition of the $\phi_{CP}$ angle for the decays $\tau^{\pm} \to \pi^{\pm} \nu_{\tau}$ (left) and the definition at reconstruction level of $\phi_{CP}$ observables (center). The distribution weighted per tau decay categories of the $\phi_{CP}$ observables, including the expected distributions for the expected pure scalar or pure pseudo scalar cases.</figcaption></figure>

### 11.4.2 Off-shell couplings of the Higgs boson

In the dominant ggF production mode with a subsequent decay of the Higgs boson into a pair of electroweak vector bosons, the production cross section of an off-shell Higgs boson is known to be sizeable. This follows as a consequence of the enhanced couplings of the Higgs boson to the longitudinal polarization of the massive vector bosons at high energy.

The off-shell to on-shell cross section ratio is approximately 8% in the SM [277]. This part of the 'off-shell' regime corresponding to the exchange of a Higgs boson as a propagator and not as a narrow mass state is very interesting as it is sensitive to physics beyond the SM. Still, the Higgs contribution to $VV$ production at large invariant mass remains small compared to the background. In addition, the difficulty in the off-shell $VV$ analysis, beyond the small signal-to-background ratio, is due to a large negative interference between the signal and the $gg \to VV$ background.

The resulting presence of a SM Higgs boson signal in the far off-shell domain results in a deficit of events with respect to the expectation from background only events. It is only when the off-shell couplings of the Higgs boson are larger than expected in the SM that the presence of a signal appears as an excess over the background expectation. One additional intricacy arises from the precision in the prediction of the rate for $gg \to VV$, a loop process at lowest order, and its interference with the signal.

Besides this 'continuum' $gg \to VV$ background with which the 'off-shell' Higgs process interferes, the largest background comes from the $qq \to VV$ production, making this analysis very challenging.

It is interesting to note that, in this regime, the Higgs boson is

studied as a propagator and not as a particle. The measurement of its off-shell couplings is therefore absolute and does not rely on the knowledge of the total Higgs boson width. The off-shell coupling constraints can then be used to indirectly constrain the width of the Higgs boson, under specific assumptions detailed in Section 11.4.3.3.

This measurement has been carried out in the $H \to ZZ \to 4\ell$, $H \to ZZ \to \ell\ell\nu\nu$ and $H \to WW \to \ell\nu\ell\nu$ channels, via the $ggF$, $VBF$ and $VH$ production modes. To enhance the sensitivity of the analysis, the knowledge of the full kinematics of the events is important. In particular the signal and the background can be further distinguished by the invariant mass of the $VV$ system, which is more accurately accessible in the $H \to ZZ \to 4\ell$ channel. Angular distributions also play an important role in this analysis. For these reasons, the $H \to ZZ^{(*)} \to 4\ell$ channel is significantly more sensitive than $H \to WW^{(*)} \to \ell\nu\ell\nu$.

Limits on the off-shell rates have been reported for the two channels by ATLAS [278] and CMS [279] with Run-1 data. The combined results, assuming that the off-shell rates in the $ZZ$ and $WW$ channels scale equally, are given for two different hypotheses on the VBF production rate: fixing it to its SM value or scaling it as the gluon fusion rate. The observed (expected) limits on the off-shell rate fraction with respect to its SM expectation is 6.7 (9.1) for ATLAS [278] with the VBF rate fixed to its SM value and 2.4 (6.2) for CMS [279] where no assumption is made on the relative production rates of gluon-fusion and VBF. In both cases, the custodial symmetry is assumed and the ratio of the rates in the $ZZ$ and $WW$ decays are fixed to those of the SM. Results without this assumption have also been reported in Ref. [279].
The situation using the Run-2 data has significantly changed. Both ATLAS [280] and CMS [281] have performed off-shell Higgs boson analyses to constrain the off-shell Higgs boson production rates with the full Run-2 datasets using the $H \to ZZ^*$ channel. The results obtained have already reached an impressive sensitivity. Both experiments have considered two independent off-shell contributions, the first corresponding to the $ggF$ production denoted $\mu_{g,\text{off-shell}}$ and the second corresponding the VBF and VH denoted $\mu_{V,\text{off-shell}}$. For simplicity we will report only the results considering $\mu_{g,\text{off-shell}} = \mu_{V,\text{off-shell}} = \mu_{\text{off-shell}}$ here.

The first striking result is that both experiments exclude the background-only hypothesis $\mu_{\text{off-shell}} = 0$ with a significance of $3.2\sigma$ ($2.2\sigma$) observed (expected) for ATLAS [280] and $3.6\sigma$ for CMS [281]. This corresponds to a first evidence for the off-shell production of a Higgs boson, through the measurement of a significant deficit in the overall $gg \to VV$ production.

The measured off-shell signal strengths $\mu_{\text{off-shell}}$ are the following:

$$ (\text{ATLAS}) \quad \mu_{\text{off-shell}} = 1.1^{+0.7}_{-0.6} (\text{obs}) \quad [1.00^{+0.9}_{-0.9} (\text{exp})], \qquad (11.24) $$

$$ (\text{CMS}) \quad \mu_{\text{off-shell}} = 0.74^{+0.56}_{-0.38} (\text{obs}) \quad [1.00^{+1.00}_{-0.84} (\text{exp})]. $$

### 11.4.3 The Higgs boson width

In the SM, the Higgs boson width is very precisely predicted once the Higgs boson mass is known. For a mass of 125 GeV, the Higgs boson has a very narrow width of 4.1 MeV [40]. It is dominated by the fermionic decays partial width at approximately 75%, while the vector boson modes are suppressed and contribute 25% only.

At the LHC or the Tevatron, in all production modes, only the cross sections times branching fractions can be measured. As a consequence, the total width of the Higgs boson cannot be directly derived from the measurements of Higgs boson rates without additional assumptions. Direct constraints on the Higgs boson width are much larger than the expected width of the SM Higgs boson.

#### 11.4.3.1 Direct constraints

Analyses of the reconstructed mass line-shape in the two channels with a good mass resolution, the $H \to \gamma\gamma$ and $H \to ZZ^{(*)} \to 4\ell$ channels, allow for a direct measurement of the width of the SM Higgs boson. The intrinsic mass resolution in these channels is about 1–2 GeV, much larger than the expected width of the SM Higgs boson. As a result, only upper limits on the Higgs boson width have been set by ATLAS [282] and more recently CMS [152]. The two main challenges of direct constraints on the width through the measurement of the line-shape are: (i) the modelling of resolution uncertainties and (ii) the modelling of the interference between the signal and the continuum background which can be sizeable for large widths, in particular in the range where direct constraints are set. These challenges affect mostly the diphoton channel. Given that these interference effects are small with respect to the individual channels sensitivity, they are neglected in deriving constraints on the total width. The combined constraints, however, being more precise, could be affected by the interference. Direct lineshape measurements provide the most unambiguous access to the Higgs boson width, however constraints are still two orders of magnitude larger than the expected SM width. The current results are fully compatible with the SM hypothesis. The most stringent limit is obtained by CMS of 330 MeV at 95% CL [152].

Another direct constraint on the Higgs boson width can be obtained, in the $H \to ZZ^{(*)} \to 4\ell$ channel, from the measurement of the average lifetime of the Higgs boson calculated from the displacement of the four-lepton vertex from the beam spot. This analysis has been carried out by CMS (see references in Ref. [141]), using the measured decay length. The measured $c\tau_H$ is $2_{-2}^{+25} \mu\text{m}$, yielding an observed (and expected) limit at the 95% CL of $c\tau_H < 57(56) \mu\text{m}$. From this upper limit on the lifetime of the Higgs boson, the 95% CL lower limit on its width is $\Gamma_H > 3.5 \times 10^{-12}$ GeV.

#### 11.4.3.2 Indirect constraints from mass shift in the diphoton channel

In the diphoton channel, it was noticed in Ref. [283], that the effect of the interference between the main signal $gg \to H \to \gamma\gamma$ and the continuum irreducible background $gg \to \gamma\gamma$, taking into account detector resolution effects, is responsible for a non negligible mass shift. The size of the mass shift depends on the total width of the Higgs boson and it was suggested that measuring this mass shift could provide a constraint on the width [283]. It was further noticed that the mass shift has a dependence also on the diphoton transverse momentum. The total width of the Higgs boson could therefore be constrained using the diphoton channel alone.

Further studies were performed by ATLAS to estimate the size of the expected mass shift [284]. The expected shift in mass in the diphoton channel is $35 \pm 9$ MeV for the SM Higgs boson. Very preliminary studies of the sensitivity of this method to estimate the width of the Higgs boson in the High-Luminosity regime have been made by ATLAS [285] and yield an expected 95% CL upper limit on the total width of approximately 200 MeV from $3 \text{ ab}^{-1}$ of 14 TeV data.

#### 11.4.3.3 Indirect constraints from on-shell rate in the diphoton channel

In the diphoton channel, it was noticed in Ref. [286], that the interference between the main signal $gg \to H \to \gamma\gamma$ amplitude and the continuum irreducible background $gg \to \gamma\gamma$ amplitude generates non-negligible change in the on-shell cross sections, as a result of the existence of a relative phase between these amplitudes. The size of this on-shell interference effect depends on the total width of the Higgs boson and it was suggested that measuring this on-shell cross section precisely could provide a constraint on the Higgs total width. This interference effect yields a reduction of around 2% for the $gg \to H \to \gamma\gamma$ cross section measurement in the SM. The current evaluation of this interference effect is performed at NLO and has a $^{+50}_{-30}$% uncertainty, due to the fact that the large relative phase is driven by the two-loop $gg \to \gamma\gamma$ background amplitude [283,286]. This on-shell interference effect has a dependence on the $p_T$ of the diphoton system and the photon polar angle in the diphoton rest frame, which can be further exploited to improve the measurement to constrain the Higgs total width.

Taking the ratios of the on-shell cross section of Higgs boson to diphoton channel and the cross section of Higgs boson to four leptons channel where the interference effect is negligible could put a bound on the Higgs boson total width. This ratio is free from many dominant sources of systematic uncertainties for cross section measurements, i.e., PDF uncertainty and luminosity uncertainty, and can be further improved by the accumulation of the LHC data. From this cross section ratio measurement alone, a preliminary estimation of the current limit from this interference effect with current 30% precision puts an upper bound of 800 MeV on the Higgs boson total width and the limit improves to 60 MeV with $3 \text{ ab}^{-1}$ of 14 TeV data [286,287].

#### 11.4.3.4 Indirect constraints from off-shell couplings

Using simultaneously on-shell and off-shell measurements in the $VV$ channels, it was noticed [277,288] that the total width of the Higgs boson could be constrained. This can be illustrated from the parametrisation of the signal strength measurements both on-shell ($\mu_{\text{on-shell}}$) and off-shell ($\mu_{\text{off-shell}}$) as a function of the couplings modifiers $\kappa_g$ and $\kappa_V$ parameterizing the main process $gg \to H \to VV$ (see Section 11.5.1 for the definition of these coupling modifiers). The on-shell and off-shell signal strengths can be written as:

$$ \mu_{\text{on-shell}} = \frac{\kappa_{g, \text{on-shell}}^2 \kappa_{V, \text{on-shell}}^2}{\Gamma_H / \Gamma_{\text{SM}}}, \qquad (11.25) $$

$$ \mu_{\text{off-shell}} = \kappa_{g, \text{off-shell}}^2 \kappa_{V, \text{off-shell}}^2. $$

A bound on the Higgs boson width can then be obtained from the measurements of the on-shell and off-shell signal strengths. This assumes that no new physics alters the Higgs boson couplings in
the off-shell regime, i.e., that the running of its couplings in this energy range does not differ from the Standard Model [289,290].

Both ATLAS [278] and CMS [281] have used their very strong off-shell production analyses described above, with kinematic event characteristics to further gain in sensitivity to discriminate between the signal and background.

In addition, in the CMS analysis, results are also derived allowing for anomalous couplings of the Higgs boson, therefore reducing the discriminating power of the kinematic variables used in the analysis but reducing the model dependence. The sensitivity already reached at Run 2 is remarkable:

$$ (\text{ATLAS}) \quad \Gamma_H = 4.5^{+3.3}_{-2.5} \ [4.1^{+3.8}_{-3.8} \ (\text{exp})] \text{MeV}, \quad (11.26) $$

$$ (\text{CMS}) \quad \Gamma_H = 3.2^{+2.4}_{-1.7} \ [4.1^{+4.0}_{-3.5} \ (\text{exp})] \text{MeV}. \quad (11.27) $$

ATLAS and CMS have also performed a study of the prospects for measuring the Higgs boson width mainly in the four lepton channel. Projecting to a luminosity of $3 ab^{-1}$, it was concluded that, within assumptions similar to the ones mentioned above and assuming the SM central value, the width of the Higgs boson could be constrained with the following precision [118]:

$$ \Gamma_H = 4.1^{+0.7}_{-0.8} \text{MeV}. \quad (11.28) $$

## 11.5 Probing the coupling properties of the Higgs boson

As discussed in Section 11.2, within the SM, all the Higgs boson couplings are fixed unambiguously once all the particle masses are known. Any deviation in the measurement of the couplings of the Higgs boson could therefore signal BSM physics.

Measuring the Higgs boson couplings without relying on the SM assumption requires a general framework treating deviations from the SM coherently at the quantum level in order to provide theoretical predictions for relevant observables to be confronted with experimental data. An attempt in that direction has been formalised in the so-called $\kappa$-formalism [291], following earlier attempts [292] and initial phenomenological studies of the first hints of the existence of the Higgs boson [293]. In this LO-inspired approach, the SM Higgs boson couplings are rescaled by arbitrary factors, $\kappa$'s, keeping the same Lorentz structure of the interactions. This formalism allows for simple interpretation of the signal strengths measured in the various Higgs channels. It has been utilised to test various physics scenarios, like the existence of additional new particles contributing to the radiative Higgs boson production and decays, or to probe various symmetries of the SM itself, as for example the custodial symmetry. It only compares the experimental measurements to their best SM predictions and does not require any new BSM computations per se. And, from a more theoretical perspective, its relevance arises from the fact that it actually fully captures the leading effects in single Higgs processes of well motivated scenarios. Still, the $\kappa$-formalism has obvious limitations and certainly does not capture the most general deformations of the SM, even under the assumptions of heavy and decoupling new physics. A particularly acute shortcoming at the time Higgs physics is entering a precision era is the lack of proficiency to assert the richness of kinematical distributions beyond simple signal strength measurements. Several extensions and alternative approaches are being developed as part of the activities of the LHC Higgs Cross Section Working Group [43].

The Higgs Pseudo-Observable (HPO) approach [294] allows one to report the data in terms of a finite set of on-shell form factors parametrising amplitudes of physical processes subject to constraints from Lorentz invariance and other general requirements like analyticity, unitarity, and crossing symmetry. These form factors are expanded in powers of kinematical invariants of the process around the known poles of SM particles, assuming that poles from BSM particles are absent in the relevant energy regime. A set of HPOs have been proposed to characterise both the Higgs boson decays and the EW Higgs boson production channels, thus exploring different kinematical regimes. Prospective studies concluded that these HPOs can be measured/bounded at the percent level at the HL-LHC and could therefore be used to constrain some explicit models of new physics.

Another systematic and popular approach to characterise the possible Higgs boson coupling deviations induced by BSM physics is the use of Effective Field Theories (EFT) [295,296]. This approach assumes again that the new physics degrees of freedom are sufficiently heavy to be integrated out, and they give rise to effective interactions among the light SM particles. By construction, an EFT approach cannot systematically account for deviations in Higgs physics induced by light degrees of freedom unless they are added themselves as extra fields in the effective Lagrangians, requiring to know about these light fields beforehand. In Section 11.6, several examples of models with light degrees of freedom affecting Higgs boson production and decay rates will be presented. Like in explicit UV models and unlike the $\kappa$-approach, the main advantage of EFTs is their prowess to relate different observables in different sectors and at different energies to constrain a finite set of effective interactions among the SM degrees of freedom. In an EFT, the SM Lagrangian is extended by a set of higher-dimensional operators, and it reproduces the low-energy limit of a more fundamental UV description. It will be assumed that the Higgs boson is part of a CP-even EW doublet, $\Phi$, and that the Lagrangian is an analytic function of the gauge invariant $\Phi^\dagger\Phi$. This scenario is commonly referred as SMEFT. Even though it is not fully established experimentally, this set-up is motivated by the measurements of the Higgs couplings to the different SM particles that show an alignment with their masses, such an alignment naturally follows under this assumption of a linear realisation of the $SU(2)_L \times U(1)_Y$ symmetry of the SM. General Lagrangians bypassing this linear assumption have been explicitly written down, see for instance Ref. [297], and connections with explicit UV models made [298,299]. They rely on a chiral expansion with a specific power-counting, effectively resuming the expansion in powers of the Higgs field, usually referred as HEFT as opposed to SMEFT.

### 11.5.1 Probing coupling properties

#### 11.5.1.1 From effective Lagrangians to Higgs observables

The $\kappa$ framework, described in detail in Ref. [39,291], facilitates the characterisation of Higgs coupling properties in terms of a series of Higgs coupling strength modifier parameters $\kappa_i$, which are defined as the ratios of the couplings of the Higgs bosons to particles $i$ to their corresponding SM values. The $\kappa$ framework assumes a single narrow resonance so that the zero-width approximation can be used to decompose the cross section as a product of two factors characterising the production and the decay of the Higgs boson. The $\kappa$ parameters are introduced by expressing each of the these factors as their SM expectation multiplied by the square of a coupling strength modifier for the corresponding process at leading order:

$$ \begin{aligned} (\sigma \cdot \text{BR})(i \to \text{H} \to f) &= \frac{\sigma_i^{\text{SM}} \kappa_i^2 \cdot \Gamma_f^{\text{SM}} \kappa_f^2}{\Gamma_H^{\text{SM}} \kappa_H^2} \quad \to_i \\ \to_i \mu_i^f &\equiv \frac{\sigma \cdot \text{BR}}{\sigma_{\text{SM}} \cdot \text{BR}_{\text{SM}}} = \frac{\kappa_i^2 \cdot \kappa_f^2}{\kappa_H^2}, \end{aligned} \quad (11.29) $$

where $\mu_i^f$ is the rate relative to the SM expectation and $\kappa_H^2$ is an expression that adjusts the SM Higgs width to take into account the modifications induced by the deformed Higgs boson couplings. When all $\kappa_i$ are set to 1, the SM is reproduced. For loop-induced processes, e.g. $H \to \gamma\gamma$, there is a choice of either resolving the coupling strength modification in its SM expectation, i.e., $\kappa_\gamma(\kappa_t, \kappa_W)$ or keeping $\kappa_\gamma$ as an effective coupling strength parameter.

The $\kappa$-framework is the simplest parametrisation directly related to experimental measurements of the Higgs boson production and decay modes. For this reason, it has been widely used by the community. It can also be connected to the SMEFT formalism as follows. Restricting to the EFT directions not probed outside Higgs physics [300], the Higgs boson couplings are written
in the unitary gauge as:

$$ 
\begin{aligned}
\mathcal{L} ={}& \kappa_Z \frac{m_Z^2}{v} Z_\mu Z^\mu H + \kappa_W \frac{2m_W^2}{v} W_\mu^+ W^{-\mu} H \\
& + \kappa_{VV} \frac{\alpha}{2\pi v} (\cos^2 \theta_W Z_{\mu\nu} Z^{\mu\nu} + 2 W_{\mu\nu}^+ W^{-\mu\nu}) H \\
& + \kappa_g \frac{\alpha_s}{12\pi v} G_{\mu\nu}^a G^{a\mu\nu} H + \kappa_\gamma \frac{2\alpha}{9\pi v} A_{\mu\nu} A^{\mu\nu} H + \kappa_{Z\gamma} \frac{\alpha}{\pi v} A_{\mu\nu} Z^{\mu\nu} H \\
& - \sum_F \kappa_F \frac{m_F}{v} \bar{F} F H + \kappa_3 \frac{m_H^2}{2v} H^3 + \dots
\end{aligned} 
\quad (11.30) 
$$

The exact correspondence between the effective coefficients of the dimension-6 operators and the $\kappa$'s can be found for instance in Ref. [40]. In the SM, the Higgs boson does not couple to massless gauge bosons at tree level, hence $\kappa_g = \kappa_\gamma = \kappa_{Z\gamma} = 0$. Nonetheless, the contact operators are generated radiatively by SM particle loops. In particular, the top quark gives a contribution to the 3 coefficients $\kappa_g, \kappa_\gamma, \kappa_{Z\gamma}$ that does not decouple in the infinite top mass limit (for instance, in that heavy limit, the top quark contribution amounts to $\Delta\kappa_\gamma = \Delta\kappa_g = 1$ [21, 22, 301], while the $W^\pm$ also contributes to $\kappa_\gamma$). Depending on the nature of new physics, these effective couplings can be generated at tree level or only at the radiative/loop level, see Refs. [295,302–304] for further discussions.

The coefficient for the contact interactions of the Higgs boson to the $W$ and $Z$ field strengths is not independent but obeys the relation

$$ (1 - \cos^4 \theta_W) \kappa_{VV} = \sin 2\theta_W \kappa_{Z\gamma} + \sin^2 \theta_W \kappa_{\gamma\gamma}. \quad (11.31) $$

This relation is a general consequence of the custodial symmetry [305], which also imposes $\kappa_Z = \kappa_W$ at leading order ($\kappa_Z/\kappa_W - 1$ is a measure of custodial symmetry breaking and, as such, is already constrained by electroweak precision data and the bounds on anomalous gauge couplings). When the Higgs boson is part of an SU(2)$_L$ doublet, the custodial symmetry in the bosonic sector could only be broken by the $O_T = \frac{1}{2}(\Phi^\dagger \overleftrightarrow{D}^\mu \Phi)^2$ operator at the level of dimension-6 operators and it is accidentally realised among the interactions with four derivatives, like the contact interactions considered.

The coefficient $\kappa_3$ is the trilinear Higgs boson self-coupling parameter which corresponds to the more generic $\kappa_\lambda$ self-coupling parameter discussed in Section 11.3.9.1. Before the associated production of a Higgs boson with a pair of top quarks was observed, the Higgs boson coupling to the top quark was only probed indirectly via the one-loop gluon fusion production or the radiative decay into two photons. However, these two processes are only sensitive to the combinations of couplings ($\kappa_t + \kappa_g$) and ($\kappa_t + \kappa_\gamma$) and not to the individual couplings. Therefore a deviation in the Higgs boson coupling to the top quark can in principle always be masked by new contact interactions to photons and gluons (and this is precisely what is happening in minimal incarnations of composite Higgs models [306]). The current and still limited sensitivity, of the order of 20%, in the $t\bar{t}H$ channel leaves elongated ellipses in the direction $\kappa_g = \kappa_\gamma = 1 - \kappa_t$.

The operators already bounded by EW precision data and the limits on anomalous gauge couplings modify in general the Lorentz structure of the Higgs couplings and hence induce some modifications of the kinematical differential distributions [307]. A promising way to have a direct access to the effective coefficients of these operators in Higgs physics is to study the $VH$ associated production with a $W$ or a $Z$ at large invariant mass of the $VH$ system [308]. These differential distributions could also be a way to test the hypothesis that the Higgs boson belongs to a SU(2)$_L$ doublet together with the longitudinal components of the massive electroweak gauge bosons.

### 11.5.1.2 Measurements of the Higgs boson coupling properties

As described in Section 11.3, a framework was developed by ATLAS and CMS [153], individually and together, to combine the very large number of exclusive categories aimed at reconstructing the five main decay modes and the five main production modes

of the Higgs boson. The general conclusion of this combination, illustrating the compatibility of the observation with the SM expectations, is given in Section 11.3. The same framework with its master formula, Eq. (11.14), can be used to further measure coupling properties of the Higgs boson under specific additional assumptions.

#### i. Interpretations of the experimental data

The measurements of the coupling properties of the Higgs boson are entirely based on the formalism of the effective Lagrangian described above. Measurements of coupling properties in this framework implies assessing the parameters of the model Eq. (11.30) or combinations of these parameters with different sets of assumptions.

These measurements are carried out with the combination framework described in Section 11.3.6, where the $\mu_i$ and $\mu_f$ signal strength parameters are further interpreted in terms of modifiers of the SM couplings $\kappa_k$ where $k \in \{Z, W, f, g, \gamma, Z\gamma\}$ as in Eq. (11.30). The number of signal events per category for the various production modes are typically estimated at higher orders in the analyses but are scaled by these single LO-inspired factors, thus not taking into account possible intricacies and correlations of these parameters through the higher-order corrections. This approximation is valid within the level of precision of current results and their compatibility with the SM expectation.

In this formalism, further assumptions are explicitly made: (i) the signals observed in the different search channels originate from a single narrow resonance with a mass of 125 GeV; (ii) similarly to the combination described in Section 11.3.6, the narrow width approximation is assumed (to allow the decomposition of signal yields into products of production and decay signal strengths); (iii) the tensor structure of the couplings is assumed to be the same as that of a SM Higgs boson. This means in particular that the observed state is assumed to be a CP-even scalar as in the SM.

Loop-level couplings such as the $gg \to H, H \to \gamma\gamma$ and $H \to Z\gamma$ can either be treated effectively, with $\kappa_g, \kappa_\gamma$ and $\kappa_{Z\gamma}$ as free parameters in the fit or these parameters can be expressed in terms of the known SM field content and as a function of the SM coupling modifiers, in the following way [309] by either scaling cross sections (for $\kappa_g$ given for a centre-of-mass energy of 13 TeV and a Higgs boson mass of 125 GeV) or partial widths (for $\kappa_\gamma$ and $\kappa_{Z\gamma}$ for a Higgs boson mass of 125 GeV):

$$ 
\begin{aligned}
\kappa_g^2(\kappa_t, \kappa_b, \kappa_c) &= 1.042 \kappa_t^2 - 0.040 \kappa_t \kappa_b + 0.002 \kappa_b^2 \\
&\quad - 0.005 \kappa_t \kappa_c + 0.0005 \kappa_b \kappa_c + 0.00002 \kappa_c^2, \\
\kappa_\gamma^2(\kappa_t, \kappa_W) &= 1.59 \kappa_W^2 - 0.66 \kappa_W \kappa_t + 0.07 \kappa_t^2, \\
\kappa_{Z\gamma}^2(\kappa_t, \kappa_W) &= 1.12 \kappa_V^2 - 0.15 \kappa_V \kappa_F + 0.03 \kappa_F^2.
\end{aligned} 
\quad (11.32) 
$$

It should be noted that the parametrisation of the gluon coupling modifier could also be used to scale the $H \to gg$ partial width and would be different from the cross section parametrisation. However when considered effectively, i.e., without resolving the content of the loop in terms of SM fields, it can be assumed that the scaling behaviour of the production cross section and partial width will be very similar.

The parametrisations are given for a Higgs boson mass hypothesis of 125 GeV (and in the last two expressions, all the Higgsfermion couplings are assumed to be rescaled by an universal multiplicative factor $\kappa_F$). It can be noted from the expression of $\kappa_\gamma$ that the coupling of the Higgs boson to photons is dominated by the loop of $W$ bosons, and it is affected by the top quark loop mostly through its interference with the $W$ loop. The sensitivity of the current measurements to the relative sign of the fermion and vector boson couplings to the Higgs boson is due to this large negative interference term. The $\kappa_g$ parameter is expressed in terms of the scaling of production cross sections and therefore also depends on the pp collisions centre-of-mass energy. The parametrisations of $\kappa_\gamma$ and $\kappa_{Z\gamma}$ are obtained from the scaling of partial widths and are only dependent on the Higgs boson mass hypothesis. Experiments use a more complete parametri-
sation with the contributions from the b-quarks, τ-leptons in the loops [39, 291].

The global fit is then performed expressing the $\mu_i$ and $\mu_f$ parameters in terms of a limited number of $\kappa_k$ parameters or their ratios, under various assumptions. The parametrisation for the main production modes are: (i) $\mu_{ggF} = \kappa_g^2$ for the gluon fusion and an effective coupling of the Higgs boson to the gluons; (ii) $\mu_{VBF,VH} = \kappa_V^2$ for the VBF and VH processes when the W and Z couplings are assumed to scale equally, and $\mu_{VBF}^2(\kappa_W, \kappa_Z) = (\kappa_W^2\sigma_{WWH} + \kappa_Z^2\sigma_{ZZH})/(\sigma_{WWH} + \sigma_{ZZH})$, when the couplings to the W and Z bosons are varied independently ($\sigma_{WWH}$ and $\sigma_{ZZH}$ denote the VBF cross sections via the fusion of a W and a Z boson respectively, the small interference term is neglected); (iii) $\mu_{t\bar{t}H} = \kappa_t^2$ for the $t\bar{t}H$ production mode. Numerically the production modes signal strengths as a function of the coupling modifiers to the SM fields for a centre-of-mass energy of 13 TeV and Higgs boson mass of 125 GeV are:

$$ 
\begin{aligned}
\mu_{ggF} &= 1.040\kappa_t^2 + 0.002\kappa_b^2 - 0.038\kappa_t\kappa_b, \quad \text{and} \\
\mu_{VBF} &= 0.73\kappa_W^2 + 0.27\kappa_Z^2.
\end{aligned}
\qquad (11.33) $$

The decay mode signal strengths are parametrised as $\mu_k = \kappa_k^2/\kappa_H^2$ where $k \in \{Z, W, f, g, \gamma, Z\gamma\}$ denotes the decay mode and $\kappa_H$, the overall modifier of the total width that affects all the signal yields. $\kappa_H$ is a priori an independent parameter. However, when it is assumed that the Higgs boson cannot decay to new particles beyond those of the SM, $\kappa_H$ can also be treated as an effective parameter and expressed in terms of the coupling modifiers to the SM field content. Its general expression is:

$$ 
\begin{aligned}
\kappa_H^2 = & 0.57\kappa_b^2 + 0.06\kappa_\tau^2 + 0.03\kappa_c^2 + 0.22\kappa_W^2 + 0.03\kappa_Z^2 + 0.09\kappa_g^2 \\
& + 0.0023\kappa_\gamma^2.
\end{aligned}
\qquad (11.34) $$

The general expression of the total width of the Higgs boson can be written as follows:

$$ \Gamma_H = \frac{\kappa_H^2 \Gamma_H^{\text{SM}}}{1 - \text{BR}_{\text{BSM}}} \qquad (11.35) $$

where $\Gamma_H^{\text{SM}}$ is the total width of the SM Higgs boson and $\text{BR}_{\text{BSM}}$ is the branching fraction of the Higgs boson to new particles beyond the SM.

It is worth reminding that, in the absence of a measurement of the Higgs width which can be accessed through Off-Shell couplings measurements as discussed in Section 11.4.2, LHC requires some hypotheses in order to extract the Higgs couplings from the measured Higgs signal strengths. In particular, one can assume either (i) no BSM particles in the decay, or (ii) any BSM particles in the decay would give rise to invisible signatures and thus be detectable through invisible Higgs boson channels (or in other terms that there is no BSM undetected contribution), or (iii) $|\kappa_V| \le 1$. These scenarios will be examined below.

Specific parametrisations will be made in order to address the following aspects of the coupling properties of the Higgs boson under different assumptions, depending essentially on two assumptions concerning either the natural width of the Higgs boson, whether it consists of partial width from Standard Model particles only or not, or the effective couplings of the Higgs boson to photons, gluons, $Z\gamma$ and $\gamma^*\gamma$, whether these contributions are parametrised as function of the Standard Model field content in the loops or whether these couplings are considered as effective.

Negative values of couplings are not excluded a priori, but would imply the existence of new physics at a light scale and would also raise questions about the validity of the perturbative treatment of the SM deformations and also about the stability of the vacuum [310]. Since the results are only sensitive to the relative sign of vector bosons with respect to fermions, by convention negative coupling values will only be considered for vector boson couplings. Among the five main Higgs boson decay channels, only the $\gamma\gamma$ is sensitive to the sign of $\kappa_F$ (or $\kappa_V$) through the interference of the W and t loops as shown in Eq. (11.32). The current global fit disfavours a negative value of $\kappa_F$ at more than

five standard deviations. A specific analysis for the Higgs boson production in association with a single top quark has been proposed in order to more directly probe the sign of $\kappa_F$ (see references in Ref. [141]). All available experimental data show a fair agreement of the SM prediction of the couplings of the Higgs boson to fermions and gauge bosons.

## ii. Coupling measurements and probing BSM physics in loops and in the decay

Measuring Higgs boson couplings to other Standard Model fields requires either assumptions on the total Higgs boson width, or an assumption on the coupling of the Higgs boson to vector bosons, or a direct constraint on the Higgs boson width as the one obtained from the ratio of off-shell to on-shell Higgs boson couplings. These three cases will be discussed briefly. With the precision reached by analyses discussed in Section 11.3.1, strong assumptions on the BSM content of the loop-induced couplings of the Higgs boson to vector bosons as Hgg, Hγγ or HZγ are not needed and these couplings can be considered as effective and provide a stringent probe of new fields potentially entering the loop.

Using the full Run-2 dataset and the formalism described in Section i., ATLAS [183] and CMS [184], have performed interpretations of the results presented in Section 11.3.6 to derive measurements of the Higgs boson couplings. These results are summarised in Table 11.8.

The results above are obtained under the assumption that the Higgs boson decays only to SM particles. The degeneracy that originates from the fact that at LHC only production cross sections times branching fractions are measured is embodied in the degeneracy between all $\kappa$ factors and $\kappa_H$. It is interesting to note that in order to measure all couplings to be in agreement with the Standard Model model value requires that any deviation in the Higgs boson total width or $\kappa_H$ be precisely compensated by all couplings to Standard Model fields in the same fashion. This is often referred to a flat direction, meaning that all the couplings are scaled by a factor $\alpha$ and the Higgs boson width by a factor $\alpha^2$. A model realising such flat direction is given in [311]. It requires the introduction of a scalar field which decays for instance light hadrons which would not be detectable in the current experimental setups and are referred to as 'undetected'. It cannot be 'invisible' as it would otherwise be constrained by direct invisible Higgs boson constraints. The model parameters would also have to lead to couplings of the Higgs to Standard Model fields to be larger than in the Standard Model and a coupling of the Higgs boson to the scalar which will have to match the scaling of the Standard Model couplings. The former condition is exotic and the later requires a 'conspiracy'.

As we have previously discussed, the flat direction can however be removed in two ways. The first is to impose $\kappa_V \le 1$. This will induce a constraint on $\kappa_H$ which can be seen for instance from processes such as VBF Higgs production with subsequent decays to two vector bosons which are proportional to $\kappa_V^4/\kappa_H^2$. As was discussed, BSM scenarios yielding $\kappa_V > 1$ can be considered exotic. The measurements using this constraints from the ATLAS experiment are shown in Fig. 11.10 (left). The second is to use the Higgs boson width derived from the off-shell Higgs boson production measurements. In this specific interpretation, the flat off-shell Higgs measurement constrain the visible, invisible and undetected branching fractions. The 'visible' branching fraction corresponds to the Higgs branchings to Standard Model fields. The invisible branching fraction is constrained from 'invisible' Higgs searches. The flat direction is therefore represented by the 'undetected' branching fraction which is then constrained by the off-shell Higgs boson measurements. This constraint was used in previous Higgs boson couplings measurements (see references for example the partial Run-2 combination performed in [312]).

Interpretations which are less sensitive to modelling systematic uncertainties and require no constraints on the natural width of the Higgs bosons have been considered, either through the ratio of cross section and branching ratios (see results in Ref. [141]) or through a more generic approach to avoid the degeneracy in the measurement of the coupling modifiers, probing the coupling properties of the Higgs boson through ratio of couplings. In the
<table><caption>**Table 11.8:** Coupling modifier combined measurements assuming the absence of perceptible new physics in the decay of the Higgs boson. No assumption is made for the loop level couplings of the Higgs boson to gluons and photons which are considered as effective. The last column gives the expected precision at the HL-LHC [118].</caption><thead><tr><th></th><th>LHC Run 1</th><th>ATLAS Run 2</th><th>CMS Run 2</th><th>HL-LHC (expected)</th></tr></thead><tbody><tr><td>κ<sub>γ</sub></td><td>0.87<sup>+0.14</sup><sub>-0.09</sub></td><td>1.01 ± 0.06</td><td>1.10 ± 0.08</td><td>1.8%</td></tr><tr><td>κ<sub>W</sub></td><td>0.87<sup>+0.13</sup><sub>-0.09</sub></td><td>1.05 ± 0.06</td><td>1.02 ± 0.08</td><td>1.7%</td></tr><tr><td>κ<sub>Z</sub></td><td>-0.98 ± 0.10</td><td>0.99 ± 0.06</td><td>1.04 ± 0.07</td><td>1.5%</td></tr><tr><td>κ<sub>g</sub></td><td>0.78<sup>+0.13</sup><sub>-0.10</sub></td><td>0.95 ± 0.07</td><td>0.92 ± 0.08</td><td>2.5%</td></tr><tr><td>κ<sub>t</sub></td><td>1.40<sup>+0.24</sup><sub>-0.21</sub></td><td>0.94 ± 0.11</td><td>1.01 ± 0.11</td><td>3.4%</td></tr><tr><td>κ<sub>b</sub></td><td>0.49<sup>+0.27</sup><sub>-0.15</sub></td><td>0.89 ± 0.11</td><td>0.99 ± 0.16</td><td>3.7%</td></tr><tr><td>κ<sub>τ</sub></td><td>0.84<sup>+0.15</sup><sub>-0.11</sub></td><td>0.93 ± 0.07</td><td>0.92 ± 0.08</td><td>1.9%</td></tr><tr><td>κ<sub>μ</sub></td><td>—</td><td>1.06<sup>+0.25</sup><sub>-0.30</sub></td><td>1.12 ± 0.21</td><td>4.3%</td></tr><tr><td>κ<sub>Zγ</sub></td><td>—</td><td>1.38<sup>+0.31</sup><sub>-0.36</sub></td><td>1.65 ± 0.34</td><td>9.8%</td></tr></tbody></table>

<figure><img src="image_37.png" /></figure>

<figure><img src="image_38.png" /><figcaption>**Figure 11.10:** ATLAS [183] (left) and CMS [184] (right) combined measurements of coupling modifiers with various assumptions. The ATLAS measurements also use the off-shell Higgs measurements.</figcaption></figure>

latter model, the cross section times branching fraction of the $gg \to H \to ZZ$ process is parametrised as a function of a single coupling modifier:

$$ \kappa_{gZ} = \kappa_g \times \frac{\kappa_Z}{\kappa_H}. \qquad (11.36) $$

Then all combination signals can be parametrised with the following ratios of coupling modifiers: (i) the $\lambda_{Zg} = \kappa_Z/\kappa_g$ ratio which is mainly probed by the measurements of the VBF and ZH production; (ii) the $\lambda_{tg} = \kappa_t/\kappa_g$ ratio constrained by the $t\bar{t}H$ production process; (iii) the $\lambda_{WZ} = \kappa_W/\kappa_Z$ ratio mainly probed by the WW and ZZ decay modes; (iv) the $\lambda_{\tau Z} = \kappa_{\tau}/\kappa_Z$ ratio constrained by the $\tau^+\tau^-$ channel; (v) the $\lambda_{bZ} = \kappa_b/\kappa_Z$ ratio probed mainly by the $VH(\bar{b})$ channels; and (vi) the $\lambda_{\gamma Z} = \kappa_{\gamma}/\kappa_Z$ ratio constrained by the diphoton channel. In this parametrisation, the ZZ channel plays an important normalisation role (the results are discussed in detail in the previous edition of this review [141]).

### 11.5.1.3 Differential cross sections

To further characterise the production and decay properties of the Higgs boson, with the increase in size of the LHC datasets, measurements of fiducial and differential cross sections are being carried out by ATLAS and CMS both at Run 1 (the references can be found in the previous edition of this review [141]) and Run 2 and in several channels [313–316]: the diphoton, the four leptons, the WW, $b\bar{b}$ and $\tau^+\tau^-$ channels.

The definition of a fiducial volume as close as possible to the reconstruction level selection criteria is very important, as it will minimise the model dependence from possible variations in the signal reconstruction efficiencies. Minimising model dependence of unfolded fiducial differential cross section measurements is also key to ensure their usefulness to further probe and tune more accurate models in the future.

While strict fiducial requirements are key to minimise model dependence, if the decay products are part of the fiducial volume definition, these make combinations of decay channels impossible. To gain precision in the measurement of the production properties
of the Higgs boson, the fiducial volume can be defined in terms of the Higgs boson distributions, while its decay channels and properties are fully integrated. This has been used to combine differential cross section for instance in the transverse momentum of the Higgs boson. Such hybrid approaches are also discussed in Section 11.5.1.5.

A large number of observables have been studied aiming at probing the accuracy of the modelling of the Higgs boson production simulations as well as measuring the properties of the Higgs boson.

The measured differential cross section in the Higgs boson transverse momentum by ATLAS and CMS using the full Run-2 datasets are illustrated in Fig. 11.11 [314, 315, 317].

The Higgs boson transverse momentum distribution is of particular importance to probe new physics in the Higgs production. An example of interpretations that can be made using these measurements is the constraints on the b and charm quark Yukawa couplings through their impact on the production, both via the loop-level gluon fusion contribution or the $gg \to b\bar{b}H, c\bar{c}H$ processes. These results are also combined with the direct search for the $VH(b\bar{b})$ process and shown in Fig. 11.12. It is interesting to compare the constraint, $|\kappa_c| < 8.5$, from the $VH(c\bar{c})$ production analysis alone and the limit obtained using only the differential distribution which is approximately the double of the direct search limit as well as the limit that also uses the normalisation which yields $|\kappa_c| < 2.3$. The latter indirect limits assume that $\kappa_b = 1$. This shows the already good strength of the shape analysis as well as the limit coming from the parametrisation of the normalisation in the $H \to \gamma\gamma, 4\ell$ decay channels. The constraint coming from the normalisation is due to the modification of the total width from from the charm and b quark couplings [314].

### 11.5.1.4 Constraints on non-SM Higgs boson interactions in an effective Lagrangian

An example of the possible use of differential cross sections in constraining non-SM Higgs boson couplings in an EFT is given by ATLAS [318]. In this analysis, differential cross section measured in the diphoton channel are used to constrain an effective Lagrangian where the SM is supplemented by dimension six CP-even operators of the Strongly Interacting Light Higgs (SILH) formulation [295] and corresponding CP-odd operators. The diphoton differential cross sections are mainly sensitive to the operators that affect the Higgs boson interactions with gauge bosons. CMS has also recently analysed [313] the Higgs boson transverse momentum distribution to constrain the Higgs boson couplings to top, bottom, and charm quarks as well as the effective coupling to gluons. This analysis is, however, not performed in an EFT framework.

The differential distributions used in this combination are: (i) the transverse momentum of the Higgs boson, (ii) the number of reconstructed jets produced in association with the diphoton pair, (iii) the invariant mass of the diphoton system and (iv) the difference in azimuthal angle of the leading and sub-leading jets in events with two or more jets. This analysis shows how differential information significantly improves the sensitivity to dimension-6 operators.

### 11.5.1.5 Simplified Template Cross Sections (STXS)

An overarching subject of discussion between the theory and experimental communities in the field of Higgs physics has been how experimentalists could best communicate their results to be most efficiently used by others for further interpretation. The commonly used practice is that results are given at particle level within a well-defined fiducial volume of phase space. The fiducial volume is usually defined close enough to the experimental reconstruction to minimise the possible variations of the reconstruction efficiency within the particle level fiducial volume. In this way, results minimise their dependence on theoretical uncertainties.

ATLAS and CMS have produced fiducial and unfolded cross sections based on all objects reconstructed in the events. These measurements could be used for further interpretation. However, performing a proper combination of channels taking into account all experimental systematic uncertainties is non trivial and given the number of channels, combinations are very involved. A pro-

posal [40, 319] was made by the LHC Higgs Working Group to produce results in each decay channel with a well defined fiducial phase space of the Higgs boson (and not its decay products) and for other associated objects pertaining to all channels, such as jets and missing transverse momentum (MET). The definition of the fiducial regions is motivated by maximising the experimental sensitivity, isolating possible BSM effects, and minimising the dependence on theoretical uncertainties. The number of regions is also minimised to avoid the loss of experimental sensitivity. The observables that are measured in this approach are still the standard production cross sections (the gluon fusion, the vector boson fusion, the $VH$ and $t\bar{t}H$ associated production modes) within the defined fiducial volumes.

This approach is hybrid. It is fiducial on specific objects to reduce the theory dependence and inclusive in the Higgs kinematics in order to allow for a more straightforward combination. This approach also allows the use of multivariate techniques to enhance the sensitivity within given fiducial regions, at the expense of a greater extrapolation and therefore increased model dependence.

The currently used Simplified Template Cross Sections (STXS) scheme covers, with a limited number of bins, the $ggF$ process in four categories in number of jets (0, 1, 2 and 2 VBF-like jets, where VBF-like means a selection of two high invariant mass jets with large pseudo rapidity difference) further subdivided in four transverse momentum categories covering the full spectrum with the last bin being inclusive for $p_T > 200$ GeV. The $VH$ process is subdivided in two categories depending on the number of reconstructed charged leptons corresponding to the decays of either a W boson or a Z boson, and two bins in transverse momentum. VBF and hadronic $VH$ categories are defined using jet cuts and two bins in transverse momentum.

Measurements in this framework have been made in various decay channels. The first measurements have been performed in the main Higgs boson discovery channels. ATLAS and CMS have produced measurements of the diphoton and the $4\ell$ channels with Run-2 data [258, 314, 320–325].

The ATLAS and CMS collaborations have also performed measurements of the STXS in the $H \to W^+W^-$ decay channel [326, 327].

CMS has also carried out a measurement of the STXS in the $H \to \tau^+\tau^-$ decay channel targeting the high transverse momentum of the Higgs boson [156], in particular in the channel where the Higgs boson is produced with one jet of transverse energy in excess of 200 GeV.

ATLAS [328] has made a measurement of the STXS aiming at the $VH$ production mode in the $H \to b\bar{b}$ decay mode at high transverse momentum of the vector boson above 250 GeV, where the discrimination of the background further increases.

A combination of STXS across decay channels has also been carried out by ATLAS with the full Run-2 dataset [183, 329, 330].

### 11.5.2 Effective Lagrangian framework

The SMEFT has the same field content and it respects the same linearly-realised $SU(3)_C \times SU(2)_L \times U(1)_Y$ local symmetry as the SM. The difference is the presence of operators with canonical mass-dimension $d$ larger than 4. These are organised in a systematic expansion in $d$, where each consecutive term is suppressed by a larger power of a high mass scale. Assuming baryon and lepton number conservation, the most general Lagrangian takes the form

$$ \mathcal{L}_{\text{eff}} = \mathcal{L}_{\text{SM}} + \sum_{i} c_{i}^{(6)} \mathcal{O}_{i}^{(6)} + \sum_{j} c_{j}^{(8)} \mathcal{O}_{j}^{(8)} + \dots \quad (11.37) $$

The contribution of the higher order operators of dimension $d$ to physical amplitudes is suppressed by $(E/\Lambda)^{d-4}$, where $E$ is the relevant energy scale of the process and $\Lambda$ is the energy scale suppressing the higher-dimensional operators. The Wilson coefficients $c_i^{(d)}$ encode the virtual effects of the heavy new physics in low-energy observables. Their precise forms in terms of masses and couplings of the new particles can be obtained via matching with the ultraviolet (UV) completion of the SM [331–334], or inferred using specific power-counting rules [295, 335].

The list of dimension-6 operators was first classified in a systematic way in Ref. [336] after the works of Ref. [337]. Subsequent
<figure><img src="image_39.png" /><figcaption>**Figure 11.11:** (Left) Fiducial differential cross sections in Higgs boson transverse momentum in the $H \to \gamma\gamma$ channel from the CMS experiment [315]. (Right) Partially fiducial combined cross sections using the $H \to \gamma\gamma$ and $H \to 4\ell$ channels from the ATLAS experiment [314].</figcaption></figure>

<figure><img src="image_40.png" /><figcaption>**Figure 11.12:** Measured likelihood contours for the $\kappa_b$ and $\kappa_c$ parameters from a simultaneous fit of the direct search for $VH(\bar{b}\bar{b}, c\bar{c})$ analyses and the measurement of the differential cross section in transverse momentum in the $H \to \gamma\gamma, 4\ell$ channels, using the overall normalisation of the cross sections parametrised as a function of the $\kappa_b$ and $\kappa_c$ couplings and assuming no non-Standard Model contributions to the total width of the Higgs boson (left) and using only the differential shape in transverse momentum [314].</figcaption></figure>

analyses pointed out the presence of redundant operators, and a minimal and complete list of operators was finally provided in Ref. [338]$^3$. For a single family of fermions, there are 76 real ways to deform the SM generated by 59 independent operators. With the 3 families of fermions of the SM, flavour indices can be added to these 59 operators, and furthermore, new operator structures, that have been dismissed by means of Fierz transformations in the single family case, have to be considered, for a total of 2499 real deformations [341].

The numbers of operators to constrain can be reduced by imposing suitable flavour assumptions. Inspired by flavour physics, a $U(2)_q \times U(2)_u \times U(2)_d \times U(3)_l \times U(3)_e$ symmetry can be enforced to impose a flavour symmetry in the first two quark generations

and all three lepton generations. ATLAS performed a detailed analysis of Higgs and EW data from LEP/SLD and LHC. Under the previous flavour assumption, 22 operators are affecting Higgs observables alone, 22 operators are affecting EW precision observables alone and 11 operators are affecting both sets of observables simultaneously. Dedicated samples representing SM, linear, and quadratic effects of dimension-six operators are generated at leading order using MadGraph5_aMC@NLO and SMEFTsim. Events are interfaced to Pythia 8 to simulate parton shower and hadronization. The decays of bottom and charm hadrons are performed by EvtGen. For the loop-induced processes, calculations are performed with SMEFTatNLO, which allows for the calculation of NLO QCD corrections, instead of SMEFTsim. Effects of dimension-six operators on background processes are usually not taken into account except in the analysis of VBF Z measurement and of WW production. The results are displayed in Fig. 11.14 [342]. The current measurements cannot currently constrain simultaneously

<footnote>$^3$Complete enumerations of $d=8$ operators have been obtained [339] and some preliminary constraints on peculiar subsets of these operators have been derived from experimental measurements [340]. Still, in this review, the EFT Lagrangians will be truncated at the level of dimension-6 operators.</footnote>
<figure><img src="image_41.png" /><figcaption>**Figure 11.13:** Simultaneous combined measurement of simplified template cross sections assuming SM branching fractions of the Higgs boson [183].</figcaption></figure>

<figure><img src="image_42.png" /><figcaption>**Figure 11.14:** Constraints on Wilson coefficients from the combined LHC+EWPO analysis, presented in four blocks with different x-axis ranges. Right-hand side panel shows the contribution of each input measurement group to the eigenvector constraint in the Gaussian approximation.</figcaption></figure>

all SMEFT Wilson coefficients as numerous degrees of freedom are not constrained by the data. One option is to fit a limited (typically one) coefficient while fixing the others. Another approach as shown in Fig. 11.14 is to perform a Principal Component Analysis (PCA) which will identify the sensitive and the flat directions of the likelihood and compute a modified basis corresponding to a linear combination of the SMEFT coefficients [342]. For this reason the results displayed in Fig. 11.14 illustrate coefficients with several indices, which correspond to the main Wilson coefficients represented in a given direction of the PCA analysis. The superscript represent a numbering of different linear combinations with similar coefficients. These results show the excellent agreement between measurements and the Standard Model expectations projected onto SMEFT sensitive directions. The largest discrepancy is observed for the $c_{HVV,Vff}^4$ coefficient which is driven by the longstanding discrepancy in the $A_{FB}^{0,b}$ measurements at LEP. In this paper [342], ATLAS has performed both fits in the linear and quadratic parametrisations of SMEFT effects corresponding to taking into account the interference between SM and SMEFT only or the entire SMEFT effect (including the quadratic SMEFT contributions) respectively.

Phenomenological studies [343, 344] have also combined Higgs, diboson, and top quark production and decay measurements from the LHC and LEP to produce global fit of SMEFT at the dimension-6 order. Automated codes like or Matchete [334] or Matchmaker effort [333] allows one to match explicit UV models onto SMEFT and hence to interpret the global EFT fit to directly constrain these UV models.

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

The discovery of a light scalar with couplings to gauge bosons and fermions that are consistent with SM predictions, together with the slow running of the Higgs boson self-coupling at high energies allow one to consider the SM as a valid perturbative description of nature all the way to the Planck scale. This picture is admittedly very attractive, but it posits that the Higgs boson is an elementary scalar field, whose mass has quantum sensitivity to possible new physics scales. This EW/Higgs naturalness problem [6] has become much more definite after the Higgs boson discovery.
There are two broad classes of models addressing the natural-
ness problem<fnref n="4" />. One is based on SUSY [7] (for recent reviews,
see Refs. [8,9]). This is a weakly coupled approach to EWSB,
maintaining the perturbativity of the SM, and, the Higgs boson
remains elementary and the corrections to its mass are screened
at the scale at which SUSY is broken so the value of the weak
scale remains insensitive to the details of the physics at higher
scales. These theories predict at least three neutral Higgs parti-
cles and a pair of charged Higgs particles [23]. One of the neutral
Higgs bosons, most often the lightest CP-even one, has properties
that can resemble those of the SM Higgs boson (at least in some
regions of the parameter space referred to as the alignment limit
[346,347], which in the case of sufficiently heavy additional Higgs
bosons is dubbed the decoupling limit [348,349]). Such scalar is
referred to as a SM-like Higgs boson, meaning that its couplings
are close to the ones predicted in the SM. The other approach
invokes the existence of strong interactions at a scale of the or-
der of one TeV or above and these new interactions induce the
breaking of the electroweak symmetry [350]. In the original in-
carnation, dubbed technicolor, the strong interactions themselves
trigger EWSB without the need of a Higgs boson. Another pos-
sibility, more compatible with the ATLAS and CMS discovery, is
that the strong interactions produce four light resonances identi-
fied with the Higgs doublet and EWSB proceeds through vacuum
misalignment [10] (see Refs. [11,12] for recent reviews). In that
case, the Higgs boson itself has a finite size and thus never feels the
UV degrees of freedom that would otherwise have dragged its mass
to much higher scales. The Higgs boson could also correspond to
the Goldstone boson associated with the spontaneous breaking
of scale invariance, see Ref. [351] and references therein. How-
ever, this dilaton/radion scenario now requires a jumbled model-
building to be consistent with the constraints from the coupling
measurements. All these BSM scenarios can have important ef-
fects on the phenomenology of the Higgs boson. Also, in each
case, the role of the Higgs boson in the unitarisation of scattering
amplitudes is shared by other particles which remain targets of
experimental searches.

The realisation of SUSY at low energies has many good quali-
ties that render it attractive as a model of new physics. First of
all since, for every SM degree of freedom, there is a superpartner
of different spin but of equal mass and effective coupling to the
SM-like Higgs boson, in the case of exact SUSY, an automatic
cancellation of quantum corrections to the Higgs mass parame-
ter holds. In practice, it is known that SUSY must be broken
since no superpartners of the SM particles have been observed so
far. The mass difference between the precise value of the mass of
any particle and that of its corresponding superpartner is propor-
tional to the correlated soft SUSY breaking parameter, generically
called M<sub>SUSY</sub>. The quantum corrections to the Higgs boson mass
parameter are proportional to M<sup>2</sup><sub>SUSY</sub>, and provided M<sub>SUSY</sub> is
of order of a few TeV, the low energy mass parameters of the
Higgs sector become insensitive to physics at the GUT or Planck
scale. Another interesting feature of SUSY theories is related to
the dynamical generation of EWSB [352]. In the SM, a negative
Higgs mass parameter, m<sup>2</sup>, needs to be inserted by hand to in-
duce EWSB, see Eq. (11.1). In SUSY, instead, even if the relevant
Higgs mass parameter is positive in the ultraviolet, it may become
negative and induce EWSB radiatively through the strong effect
of the top quark-Higgs boson coupling in its renormalisation group
evolution [352].

In the following, the Higgs sector will be explored in specific
SUSY models. In all of them, it is often possible to find regions
of the parameter space that accommodate one neutral Higgs bo-
son with properties that resemble those of the SM Higgs boson,
whereas additional neutral and charged Higgs bosons are also
predicted and are intensively being sought for at the LHC (see
Section 11.6.7). In the simplest SUSY model, accommodating a
SM-like Higgs boson mass of about 125 GeV results in constraints
on the stop sector, with at least one stop mass in the few TeV
mass range. In non-minimal SUSY extensions of the SM (details

and related references can be found in the 2018 edition of this
review [141]), a SM-like Higgs boson with mass of 125 GeV can
be accommodated with less restrictions on the stop sector. While
naturalness dictates relatively light stops and - at the two loop
level - also gluinos, the first and second generation of squarks
and sleptons couple weakly to the Higgs sector and may be heavy.
Moreover, small values of the μ parameter and therefore light Hig-
sinos, the fermionic superpartners of the Higgs bosons, would
be a signature of a natural realization of electroweak symmetry
breaking [353]. Such SUSY spectra, consisting of TeV range stop
masses and light Higgsinos, continue to be under intense scrutiny
by the experimental collaborations [354], in order to understand
if such natural SUSY scenarios endure and can explain why the
Higgs boson remains light.

In the context of weakly coupled models of EWSB, one can
also consider multiple Higgs SU(2)$_{L}$ doublets as well as additional
Higgs singlets, triplets or even more complicated multiplet struc-
tures, with or without low energy SUSY. In general, for such
models, one needs to take into account experimental constraints
from precision measurements and flavour changing neutral cur-
rents. The LHC signatures of such extended Higgs sectors are
largely shaped by the role of the exotic scalar fields in EWSB.

The idea that the Higgs boson itself could be a composite bound
state emerging from a new strongly-coupled sector has been recon-
sidered thanks to the insights gained from the AdS/CFT duality.
The composite Higgs boson idea is an incarnation of EWSB via
strong dynamics that smoothly interpolates between the standard
technicolor approach and the true SM limit. To avoid the usual
conflict with EW data, it is sufficient, if not necessary, that a mass
gap separates the Higgs resonance from the other resonances of the
strong sector. Such a mass gap can naturally follow from dynam-
ics if the strongly-interacting sector exhibits a global symmetry,
G, broken dynamically to a subgroup H at the scale f, such that,
in addition to the three Nambu-Goldstone bosons of SO(4)/SO(3)
that describe the longitudinal components of the massive W and
Z, the coset G/H contains a fourth Nambu-Goldstone boson that
can be identified with the physical Higgs boson. Simple examples
of such a coset are SU(3)/SU(2) or SO(5)/SO(4), the latter be-
ing favoured since it is invariant under the custodial symmetry.
It is also possible to have non-minimal custodial cosets with ex-
tra Goldstone bosons leading to additional Higgs bosons in the
spectrum, see for instance Ref. [355]. Modern incarnations of
composite Higgs models have been recently investigated in the
framework of 5D warped models where, according to the princi-
ples of the AdS/CFT correspondence, the holographic composite
Higgs boson then originates from a component of a gauge field
along the 5th dimension with appropriate boundary conditions.

A last crucial ingredient in the construction of viable composite
Higgs boson models is the concept of partial compositeness [356],
i.e., the idea that there are only linear mass mixings between ele-
mentary fields and composite states. After diagonalisation of the
mass matrices, the SM particles, fermions and gauge bosons, are
admixtures of elementary and composite states and thus they in-
teract with the strong sector, and in particular with the Higgs
boson, through their composite component. This setup has im-
portant consequences on the flavour properties, chiefly the sup-
pression of large flavour changing neutral currents involving light
fermions. It also plays an important role in dynamically gen-
erating a potential for the would-be Goldstone bosons. Partial
compositeness also links the properties of the Higgs boson to the
spectrum of the fermionic resonances, i.e., the partners of the top
quark. As in the MSSM, these top partners are really the agents
that trigger the EWSB and also generate the mass of the Higgs
boson that otherwise would remain an exact Goldstone boson and
hence massless. The bounds from the direct searches for the top
partners, in addition to the usual constraints from EW precision
data, force the minimal composite Higgs models into some unnat-
ural corners of their parameter spaces [357].

11.6.1 Higgs bosons in the minimal supersymmetric stan-
dard model (MSSM)

The particle masses and interactions in a SUSY theory are
uniquely defined as a function of the superpotential and the Käh-
ler potential [9]. A fundamental theory of SUSY breaking, how-

<footnote n="4">⁴Another solution to the naturalness problem is to lower the fundamen-
tal scale of quantum gravity, like for instance in models with large extra-
dimensions, see Ref. [345].</footnote>
ever, is unknown at this time. Nevertheless, one can parametrise the low-energy theory in terms of the most general set of soft SUSY-breaking operators [9]. The simplest realistic model of low-energy SUSY is the minimal SUSY extension of the SM (MSSM) [9, 358], that associates a SUSY partner to each gauge boson and chiral fermion of the SM, and provides a realistic model of physics at the weak scale. However, in this minimal model, with the most general set of soft SUSY-breaking terms, more than 100 new parameters are introduced. Nevertheless, only a subset of these parameters impact the Higgs boson phenomenology either directly at tree-level or through quantum effects.

The MSSM contains the particle spectrum of a two-Higgs-doublet model (2HDM) extension of the SM and the corresponding SUSY partners. Two Higgs doublets, $\Phi_1$ and $\Phi_2$, with hypercharge $Y = -1$ and $Y = 1$, respectively, are required to ensure an anomaly-free SUSY extension of the SM and to generate mass for down-type quarks/charged leptons ($\Phi_1$) and up-type quarks ($\Phi_2$) [23]. The Higgs potential reads

$$V = m_1^2 \Phi_1^\dagger \Phi_1 + m_2^2 \Phi_2^\dagger \Phi_2 - m_3^2 (\Phi_1^T i \sigma_2 \Phi_2 + \text{h.c.}) \\ + \frac{1}{2} \lambda_1 (\Phi_1^\dagger \Phi_1)^2 + \frac{1}{2} \lambda_2 (\Phi_2^\dagger \Phi_2)^2 + \lambda_3 (\Phi_1^\dagger \Phi_1) (\Phi_2^\dagger \Phi_2) \\ + \lambda_4 |\Phi_1^T i \sigma_2 \Phi_2|^2 + \frac{1}{2} \lambda_5 [(\Phi_1^T i \sigma_2 \Phi_2)^2 + \text{h.c.}] \\ + [\lambda_6 (\Phi_1^\dagger \Phi_1) + \lambda_7 (\Phi_2^\dagger \Phi_2)] \Phi_1^T i \sigma_2 \Phi_2 + \text{h.c.}, \quad (11.38)$$

where $m_i^2 = \mu^2 + m_{H_i}^2$ ($i=1,2$), with $\mu$ being the supersymmetric Higgsino mass parameter and $m_{H_i}$ the soft supersymmetric breaking mass parameters of the two Higgs doublets; $m_2^2 \equiv B\mu$ is associated to the B-term soft SUSY breaking parameter; and $\lambda_i$, for $i=1$ to 7, are all the Higgs quartic couplings.

After the spontaneous breaking of the electroweak symmetry, five physical Higgs particles are left in the MSSM spectrum: one charged Higgs pair, $H^\pm$, one CP-odd neutral scalar, $A$, and two CP-even neutral states, $H$ and $h$, with $h$ being the lightest.<fnref n="5" /> The Higgs sector at tree level depends on the electroweak gauge coupling constants and the vacuum expectation value $v$ – or equivalently the $Z$ gauge boson mass – and is determined by only two free parameters: $\tan \beta$ – the ratio of the two Higgs doublets’ vacuum expectation values $v_2/v_1$ – and one Higgs boson mass, conventionally chosen to be the CP-odd Higgs boson mass, $m_A$. The other tree-level Higgs boson masses are then given in terms of these parameters. The tree level value of $m_h$ is maximised not only for $m_A \gg m_Z$ but also for $\tan \beta \gg 1$. For $m_A \gg m_Z$ it acquires a maximum value $m_h = m_Z \cos 2\beta$.

Radiative corrections have a significant impact on the values of Higgs boson masses and couplings in the MSSM. The dominant radiative effects to the SM-like Higgs boson mass arise from the incomplete cancellation between top and scalar-top (stop) loops and at large $\tan \beta$ also from sbottom and stau loops. The stop, sbottom and stau masses and mixing angles depend on the SUSY Higgsino mass parameter $\mu$ and on the soft-SUSY-breaking parameters [9, 358]: $M_Q, M_U, M_D, M_L, M_E$, and $A_t, A_b, A_\tau$. The first three of these are the left-chiral and the right-chiral top and bottom scalar quark mass parameters. The next two are the left-chiral stau/sneutrino and the right-chiral stau mass parameters, and the last three are the trilinear parameters that enter in the off-diagonal squark/slepton mixing elements: $X_t \equiv A_t - \mu \cot \beta$ and $X_{b,\tau} \equiv A_{b,\tau} - \mu \tan \beta$. At one-loop, the electroweak gauginos yield a small contribution to the Higgs boson mass, and at the two-loop level, the masses of the gluinos also enter in the calculations. Radiative corrections to the Higgs boson masses have been computed using a number of techniques, with a variety of approximations, see Ref. [359] for a recent review.

<figure><img src="image_43.png" /><figcaption>**Figure 11.15:** Values of the SUSY mass scale $M_{\text{SUSY}} = M_S$ versus the stop mixing parameter normalised by the SUSY mass scale $X_t/M_{\text{SUSY}}$, for fixed $\tan \beta = 20$, $\mu = 200$ GeV and $A_t = A_b = A_\tau$. The solid black line corresponds to $m_h = 125$ GeV while in the grey band $m_h$ varies by $\pm 1$ GeV. The red dotted lines are iso-values of the stop mass in GeV. This figure is based on Ref. [360].</figcaption></figure>

The discovered SM-like Higgs boson, if interpreted as the lightest MSSM Higgs boson with a mass of about 125 GeV, provides information on the possible MSSM parameter space, see Fig. 11.15.

The phenomenology of the Higgs sector depends on the couplings of the Higgs bosons to gauge bosons and fermions. At tree-level, the couplings of the two CP-even Higgs bosons to $W$ and $Z$ bosons are given in terms of the angles $\alpha$, that diagonalises the CP-even Higgs boson squared-mass matrix, and $\beta$

$$g_{hVV} = g_V m_V \sin(\beta - \alpha), \quad g_{hVV} = g_V m_V \cos(\beta - \alpha), \quad (11.39)$$

where $g_V \equiv 2m_V/v$, for $V = W$ or $Z$ ($g_V m_V$ is the SM $hVV$ coupling). Observe that in the limit $\cos(\beta - \alpha) \to 0$, the lightest CP-even Higgs boson $h$ behaves as the SM Higgs boson. This situation is called alignment and is achieved in specific regions of parameter space for $m_A \ge m_Z$ [348] or in the large $m_A \gg m_Z$ limit, in which alignment is achieved through decoupling [348, 349]. There are no tree-level couplings of $A$ or $H^\pm$ to $VV$. The couplings of the $Z$ boson to two neutral Higgs bosons are given by $g_{\phi AZ}(p_\phi - p_A)$, where $\phi = H$ or $h$, the momenta $p_\phi$ and $p_A$ point into the vertex, and

$$g_{hAZ} = g_Z \cos(\beta - \alpha)/2, \quad g_{hAZ} = -g_Z \sin(\beta - \alpha)/2. \quad (11.40)$$

The expressions of the couplings between a charged Higgs boson, a neutral Higgs boson and the $W$ boson as well as the expressions of the four-point couplings of vector bosons and Higgs bosons can be found in Ref. [23].

The tree-level Higgs boson couplings to fermions obey the following property: the neutral components of one Higgs doublet, $\Phi_1$, couple exclusively to down-type fermion pairs while the neutral components of the other doublet, $\Phi_2$, couple exclusively to up-type fermion pairs [23]. This Higgs-fermion coupling structure defines the Type-II 2HDM. In the MSSM, fermion masses are generated when both neutral Higgs components acquire a vacuum expectation value, and the relations between Yukawa couplings and fermion masses are (in third-generation notation)

$$h_{b,\tau} = \sqrt{2} m_{b,\tau} / (v \cos \beta), \quad h_t = \sqrt{2} m_t / (v \sin \beta). \quad (11.41)$$

<footnote n="5">⁵Observe that in the SM sections of this review, $H$ denotes the SM Higgs boson, whereas in the sections about SUSY, or extensions of the SM with two Higgs doublets, $H$ is used for the heaviest CP-even Higgs boson, since this is the standard notation in the literature, and the 125 GeV SM-like light Higgs boson will be denoted by $h$. Generically, in the MSSM, the lightest CP-even Higgs boson is indeed SM-like and thus it is naturally identified with the 125 GeV Higgs boson discovered by ATLAS and CMS, while in general 2HDM extensions, with or without SUSY, there could still be lighter scalar states below 125 GeV.</footnote>
The couplings of the neutral Higgs bosons to $f\bar{f}$, relative to their SM values, $gm_f/(2m_W)$, are therefore given by

$$ 
\begin{aligned}
hb\bar{b} & : -\sin\alpha/\cos\beta, & ht\bar{t} & : \cos\alpha/\sin\beta, \\
Hb\bar{b} & : \cos\alpha/\cos\beta, & Ht\bar{t} & : \sin\alpha/\sin\beta, \\
Ab\bar{b} & : \gamma_5 \tan\beta, & At\bar{t} & : \gamma_5 \cot\beta.
\end{aligned} 
\quad (11.42) $$

In each relation above, the factor listed for $b\bar{b}$ also pertains to $\tau^+\tau^-$. The charged Higgs boson couplings to fermion pairs, normalised to $g/(\sqrt{2}m_W)$, are given by

$$ 
\begin{aligned}
g_{H-t\bar{b}} & : m_t \cot \beta \frac{1+\gamma_5}{2} + m_b \tan \beta \frac{1-\gamma_5}{2}, \\
g_{H-\tau+\nu} & : m_\tau \tan \beta \frac{1-\gamma_5}{2}.
\end{aligned} 
\quad (11.43) $$

The non-standard neutral Higgs bosons have significantly enhanced couplings to down-type fermions at sizeable $\tan\beta$. Radiative corrections can modify significantly the values of the Higgs boson couplings to fermion pairs and to vector boson pairs, through a radiatively-corrected value for $\cos(\beta - \alpha)$ as well as from the one-loop vertex corrections to tree-level Higgs-fermion Yukawa couplings, see Ref. [9] and references therein, for a detailed discussion.

### 11.6.1.1 MSSM Higgs boson phenomenology

The MSSM parameters have to be arranged such that the mass, the CP properties, the decay and production properties of one of the neutral Higgs bosons agree with the LHC Higgs data. Given that present data allows only for moderate departures from the SM predictions, it implies that some degree of alignment is necessary.

The SM-like branching ratios of $h$ can be modified if decays into SUSY particles are kinematically allowed, and, in particular, decays into a pair of the lightest SUSY particles – i.e., the lightest neutralinos, $\tilde{\chi}_1^0$ – can become dominant and would be invisible if R-parity is conserved [361]. Moreover, if light superpartners exist and couple to photons and/or gluons, the h loop-induced coupling to gg and $\gamma\gamma$ could deviate sizeably from the corresponding SM predictions (see for instance the review [362]), and would be in conflict with present data (see Section 11.3). For the heavier Higgs states, there are two possibilities to be considered<fnref n="6" />:

i) Alignment triggered by decoupling, hence $m_A \ge$ several hundred GeV: The HWW and HZZ couplings are very small. The dominant $H, A$ decay branching ratios strongly depend on $\tan\beta$. The decay modes $H, A \to b\bar{b}, \tau^+\tau^-$ dominate when $\tan\beta$ is large (this holds even away from decoupling). For small $\tan\beta$, the $t\bar{t}$ decay mode dominates above its kinematic threshold. For the charged Higgs boson, $H^+ \to t\bar{b}$ dominates.

ii) Some degree of alignment without decoupling, hence $m_A \le$ a few hundred GeV: The main difference with the previous case is that, in the low $\tan\beta$ regime ($\tan\beta \le 5$), additional decay channels may be allowed which involve decays into the lightest SM-like Higgs boson; $A \to Zh, H \to hh$ as well as $H \to WW/ZZ$ decay modes are available (they are suppressed in the strict alignment limit). When kinematically open, the decays $A/H \to t\bar{t}$ become relevant or even dominant for sufficiently small $\tan\beta$. For the charged Higgs boson, $H^+ \to \tau^+\nu_\tau$ dominates below the $tb$ threshold, and also $H^\pm \to W^\pm h$ may be searched for.

In both cases i) and ii), the heavier Higgs states, $H, A$ and $H^\pm$, are roughly mass degenerate (with masses $\pm 20$ GeV or less apart). If kinematically allowed, the heavy Higgs boson decays into charginos, neutralinos and third-generation squarks and sleptons can be important [365].

At hadron colliders, the dominant neutral Higgs boson production mechanism at moderate values of $\tan\beta$ is gluon fusion, mediated by loops containing heavy top and bottom quarks and the corresponding SUSY partners. The effect of light stops that may

contribute to the gluon fusion production can be partially cancelled by mixing effects. Higgs boson radiation off bottom quarks becomes important for large $\tan\beta$, where at least two of the three neutral Higgs bosons have enhanced couplings to bottom-type fermions [366, 367]. Detailed discussions of the impact of radiative corrections in these search modes are presented for instance in Ref. [368]. The vector boson fusion and Higgs-strahlung production of the CP-even Higgs bosons as well as the associated production of neutral Higgs bosons with top quark pairs have lower production cross sections by at least an order of magnitude with respect to the dominant ones, depending on the precise region of MSSM parameter space [37–40]. Higgs boson pair production of non-standard MSSM Higgs bosons has been studied in Ref. [369]. For a discussion of charged Higgs boson production at LHC, see Refs. [38, 39, 370].

Strong production of a heavy neutral Higgs boson followed by its decay into top-quark pairs is a challenging channel, only most recently being searched for by ATLAS and CMS. Interference effects between the signal and the SM $t\bar{t}$ background need to be carefully taken into account [371, 372].

Summarising, the additional Higgs bosons are sought for mainly via the channels:

$$ 
\begin{aligned}
& pp \to A/H \to \tau^+\tau^- \text{ (inclusive),} \\
& b\bar{b}A/H, A/H \to \tau^+\tau^- \text{ (with b-tag),} \\
& b\bar{b}A/H, A/H \to b\bar{b} \text{ (with b-tag),} \\
& pp \to t\bar{t} \to H^\pm W^\mp b\bar{b}, H^\pm \to \tau^+\nu_\tau (\tau^-\bar{\nu}_\tau) \text{ or } t\bar{b} (b\bar{t}), \\
& gb \to H^-t \text{ or } g\bar{b} \to H^+\bar{t}, H^\pm \to \tau^+\nu_\tau (\tau^-\bar{\nu}_\tau) \text{ or } t\bar{b} (b\bar{t}).
\end{aligned} 
\quad (11.44) $$

After the Higgs boson discovery, updated MSSM benchmarks scenarios have been defined to highlight interesting conditions for the MSSM Higgs boson searches [39, 364] and are scrutinised by the LHC Higgs Working Group [373]. The latest benchmark scenarios update [364], partly based in MSSM parameter space discussions in Ref. [374], considers six benchmarks to illustrate different aspects of Higgs phenomenology in the MSSM. They include one case with complex parameters, but they all assume R-parity conservation and no flavour mixing. Each scenario contains one CP-even scalar with mass around 125 GeV and SM-like couplings. These scenarios include a $M_{H}^{125}$ scenario with relatively heavy superparticles, so the Higgs phenomenology at the LHC resembles that of a 2HDM with MSSM-inspired Higgs boson couplings. Other two scenarios are characterised by some of the superparticles – staus or electroweakinos – being relatively light, that in turn is of relevance for heavy neutral Higgs boson searches. In particular, the traditional $A/H \to \tau^+\tau^-$ search channel varies depending on the values of $\mu$ and $M_2$, that may enable the $A/H$ decays into electroweakinos. Another two scenarios are characterised by the phenomenon of alignment without decoupling, in which one of the two neutral CP-even scalars has SM-like couplings independently of the mass spectrum of the remaining Higgs bosons, hence allowing for all the Higgs bosons to have relatively low mass values (about few hundred GeV). Finally, there is one scenario which incorporates CP violation in the Higgs sector and gives rise to a strong admixture of the two heavier neutral states. All the above scenarios assume all parameters in the mass range from 1 to a few TeV, hence they are not applicable for values of $\tan\beta$ of order a few, for which a Higgs boson mass value of 125 GeV is out of reach. An additional study [375], relying on an EFT approach in the MSSM, focusses on two scenarios specifically designed for the low $\tan\beta$ region and ensures a 125 GeV Higgs boson mass in almost the entire parameter space by employing a flexible supersymmetric mass scale, reaching values of up to $10^{16}$ GeV.

An alternative approach to reduce the large number of parameters relevant to the Higgs sector is to consider that, in the Higgs basis, the only important radiative corrections are those affecting the Higgs boson mass [376]. This approximation is called hMSSM and works well in large regions of parameter space but it breaks down for sizeable values of $\mu$ and $A_t$, and moderate values of $\tan\beta$, for which the radiative corrections to the mixing between the two CP even eigenstates become relevant. The effect of such radiative corrections is to allow for alignment for small to intermediate

<footnote n="6">⁶In very special regions of the parameter space, there is still the possibility that the heavier CP-even Higgs state is identified with the 125 GeV Higgs boson discovered by ATLAS and CMS, see for instance the discussion in Ref. [363] and the benchmark $M_H^{125}$ defined in Ref. [364].</footnote>
values of tan $\beta$, independent of the specific value of $m_A$ [346]. In addition, the hMSSM assumption that the right value of the Higgs boson mass may be obtained for all values of $m_A$ and tan $\beta$ is in conflict with the MSSM predictions for the Higgs boson mass for small values of $m_A$ and tan $\beta \simeq O(1)$. The recent $M_h^{125}$ [364] and EFTMSSM benchmarks [375], are designed to address the limitations of the hMSSM, in particular the low tan $\beta$ region for the EFTMSSM.

The compatibility between the predicted and measured Higgs boson mass sets stringent constraints on the parameter space of BSM models. The predictions are illustrated in Fig. 11.16 for two concrete scenarios. Note that to use the predicted Higgs boson mass as a constraint (exclusion at nearly constant tan $\beta$ at high $M_A$ in the ($M_A$,tan $\beta$) plane), it is important to account for the theoretical uncertainty on the prediction which is in excess of an order of magnitude larger than the experimental uncertainty on the measured mass of the Higgs boson. The theoretical uncertainty depends itself on the specific SUSY spectrum for a given MSSM parameter set and should be estimated accordingly, however, a more generic estimate of $\pm 3$ GeV is made and found to be a conservative choice [359].

Reviews of the properties and phenomenology of the Higgs bosons of the MSSM can be found for example in Refs. [9,362,378]. Future precision measurements of the Higgs boson couplings to fermions and gauge bosons together with information on heavy Higgs boson searches will provide powerful information on the SUSY parameter space [379].

Improvements in our understanding of B-physics observables put indirect constraints on additional Higgs bosons in mass ranges that would be accessible in direct LHC searches. In particular, $BR(B_s \to \mu^+\mu^-)$, $BR(b \to s\gamma)$, and $BR(B_u \to \tau\nu)$ play an important role within minimal flavour-violating (MFV) models [380], in which flavour effects proportional to the CKM matrix elements are induced as in the SM [381].

### 11.6.2 Supersymmetry with singlet extensions

The Higgs mass parameter $\mu$ is a SUSY parameter, and as such, it should naturally be of order $M_{GUT}$ or $M_{Planck}$. The fact that phenomenologically it is required that $\mu$ be at the electroweak/TeV scale is known as the $\mu$ problem [382]. SUSY models with additional singlets can provide a solution to the $\mu$ problem, by promoting the $\mu$ parameter to a dynamical singlet superfield $S$ that only interacts with the MSSM Higgs doublets through a coupling $\lambda_S$ at the level of the superpotential. An effective $\mu$ is generated when the real scalar component of $S$ acquires a vacuum expectation value $v_S$, yielding $\mu_{eff} = \lambda_S v_S$. After the minimization of the Higgs potential, the vacuum state relates the vacuum expectation values of the three CP-even neutral scalars, $v_1$, $v_2$ and $v_S$, to the scalar doublet and singlet soft SUSY breaking masses, hence, one expects that these VEVs should all be of order $M_{SUSY}$ and therefore the $\mu$ problem is solved.

The addition of a singlet superfield to the MSSM may come along with additional symmetries imposed to the theory. Depending on such symmetries, different models with singlet extensions of the MSSM (xMSSM) have been proposed, see Ref. [383] for a general review. Among the most studied examples are the NMSSM with an additional discrete $Z_3$ symmetry (first introduced in Ref. [384]), the Nearly-Minimal SUSY SM (nMSSM), with additional discrete $Z_R^\pm$, and $Z_R^\mp$ symmetries [385], and the U(1)$_Y$-extended MSSM (UMSSM) [386]. A Secluded U(1)$_Y$-extended MSSM (sMSSM) [387] contains three singlets in addition to the standard UMSSM Higgs boson singlet; this model is equivalent to the nMSSM in the limit that the additional singlet VEV's are large, and the trilinear singlet coupling, $\lambda_S$, is small [388]. The non-zero neutrino masses provide also a motivation for a particular extension of the MSSM, $\mu\nu$SSM [389], which also happens to address the $\mu$ problem. The computation of the Higgs boson spectrum in these various models has been reviewed in Ref. [359].

A singlet extended SUSY Higgs sector opens new avenues for discovery. Since the singlet pseudoscalar particle may be identified as the pseudo-Goldstone boson of a spontaneously broken Peccei-Quinn symmetry, it may become naturally light [390]. Generally, there is mixing of the singlet sector with the MSSM Higgs sector, and for a sufficiently light, singlet-dominated scalar or pseudoscalar, $h_S$ or $A_S$, respectively, the SM-like Higgs boson $h$ may decay to pairs of $h_S$ or $A_S$. The light scalar and/or pseudoscalar may subsequently decay to $\tau\tau$ or $b\bar{b}$ pairs. Such cascade decays are more difficult to detect than in standard searches due to the potentially soft decay products. There is also a rich phenomenology for the decays of the heavy CP-even and CP-odd doublets, $H$ and $A$ into two lighter Higgs bosons such as $H \to hh_S$, $hh$, $h_S h_S$ or $A \to A_S h_S$, $A_S h$ as well as into a light Higgs boson and a gauge boson: $H \to A_S Z$; $A \to h_S Z$, $hZ$. If kinematically allowed, the heavy Higgs bosons decay into $t\bar{t}$. If the singlet-dominated scalar or pseudoscalar are somewhat heavier, the decays $h_S \to WW$ or $A_S \to h_S Z$ will be allowed.

In addition, the light singlet scenario in the NMSSM or nMSSM is typically associated with a light singlino-dominated neutralino. The 125 GeV SM-like Higgs boson can then decay to pairs of this neutralino [391], opening an invisible decay mode that is not excluded by present data. All of the Higgs bosons can decay into electroweakinos depending on kinematics and on the singlino or Higgsino composition of the electroweakinos.

In models with extended singlets, at low $\tan\beta$, it is possible to trade the requirement of a large stop mixing by a sizeable trilinear Higgs-singlet Higgs coupling $\lambda_S$, rendering more freedom on the requirements for gluon fusion production. As in the MSSM, mixing in the Higgs sector – additionally triggered by the extra new parameter $\lambda_S$ – can produce variations in the Higgs–$b\bar{b}$ and Higgs–$\tau^-\tau^+$ couplings that can alter the Higgs to $ZZ/WW$ and to diphoton rates. Light charginos at low $\tan\beta$ can independently contribute to enhance the di-photon rate, without altering any other of the Higgs boson decay rates, see for instance Ref. [392].

There is much activity in exploring the NMSSM phenomenology in the light of the 125 GeV Higgs boson as well as in defining benchmark scenarios with new topologies including Higgs decay chains, see Refs. [40, 393] and references therein. An analytic understanding of the alignment condition in the NMSSM is presented in Ref. [347]. The NMSSM with a Higgs boson of mass 125 GeV can be compatible with stop masses of order of the electroweak/TeV scale, thereby reducing the degree of fine-tuning necessary to achieve electroweak symmetry breaking. Interestingly, the alignment conditions point toward a more natural region of parameter space for electroweak symmetry breaking, while allowing for perturbativity of the theory up to the Planck scale and yielding a rich and interesting Higgs boson phenomenology at the LHC.

### 11.6.3 Supersymmetry with extended gauge sectors

In the MSSM, the tree-level value of the lightest CP-even Higgs boson mass originates from the D-term dependence of the scalar potential that comes from the SUSY kinetic terms in the Kähler potential. The D-terms lead to tree-level quartic couplings which are governed by the squares of the gauge couplings of the weak interactions, under which the Higgs boson has non-trivial charges. Hence, the lightest Higgs mass is bounded to be smaller than $M_Z$. In the presence of new gauge interactions at the TeV scale, and if the Higgs fields had non-trivial charges under them, new D-term contributions would lead to an enhancement of the tree-level Higgs boson mass value. Since the low energy gauge interactions reduce to the known $SU(3)_c \times SU(2)_L \times U(1)_Y$ ones, in order for this mechanism to work, the extended gauge and Higgs sectors should be integrated out in a non-SUSY way. This means that there must be SUSY breaking terms that are of the order of, or larger than, the new gauge boson masses. The tree-level quartic couplings would then be enhanced through their dependence on the square of the gauge couplings of the extended Higgs sector. This effect will be suppressed when the heavy gauge boson masses are larger than the SUSY breaking scale and will acquire its full potential only for large values of this scale.

One of the simplest possibilities is to extend the weak interactions to a $SU(2)_1 \times SU(2)_2$ sector, such that the known weak interactions are obtained after the spontaneous breaking of these groups to $SU(2)_L$ [394]. This example is briefly summarised in the previous editions of this review [141]. Assuming SUSY breaking terms of the order of the new gauge boson masses, enhancements of order 50% of the MSSM D-term contribution to the Higgs boson mass may be obtained. Such enhancements are sufficient to obtain
<figure><img src="image_44.png" /><figcaption>Figure 11.16: The 95% CL exclusion contours in the $(M_A, \tan \beta)$ parameter space for the $M_h^{125}$ (right) and $M_h^{125}(\tilde{\chi})$ (left) benchmark scenarios [377]. The $M_h^{125}$ benchmark assumes that super partners are heavy, so that the phenomenology of the observed Higgs boson is not altered except in its couplings due to the existence of another doublet. The $M_h^{125}(\tilde{\chi})$ scenario on the other hand considers light electroweakinos and therefore the heavy Higgs bosons H and A can have sizeable decay rates to charginos and neutralinos, consequently suppressing the $\tau^+\tau^-$ decay rate. The nearly vertical dotted line illustrates the lower limit on the mass of the A boson from precision measurements of the Higgs boson couplings and the close-to horizontal dotted line represents the limit on $\tan \beta$ from the compatibility of the measured mass of the observed Higgs boson and the prediction using radiative corrections (mostly from the stop sector).</figcaption></figure>

the measured Higgs mass value without the need for very heavy stops or large stop mixing parameters. This gauge extension leads to new, heavy gauge and Higgs bosons, as well as new neutralinos and charginos, that depending on the region of parameter space can induce novel phenomenology at the LHC. Gauge extensions including new Abelian gauge groups have also been considered.

Gauge extensions of the MSSM can also lead to an enhancement of the Higgs boson mass value by modifying the renormalisation group evolution of the Higgs quartic coupling to low energies. In the MSSM, the evolution of the quartic coupling is governed by the top-quark Yukawa interactions and depends on the fourth power of the top-quark Yukawa coupling. The neutralino and chargino contributions, which depend on the fourth power of the weak gauge couplings, are small due to the smallness of these couplings. Depending on the values of the soft SUSY breaking parameters in the gaugino and Higgsino sectors, the $SU(2)_1$ gauginos may become light, with masses of the order of the weak scale. Since the $SU(2)_1$ coupling may be significantly larger than the $SU(2)_L$ one, for small values of the Higgsino mass parameter $\mu$, the associated charginos and neutralinos may modify the evolution of the quartic coupling in a significant way [395]. This may lead to a significant increase of the lightest CP-even Higgs boson mass, even for small values of $\tan \beta \simeq 1$ for which the D-term contributions become small. Radiative corrections should be properly taken into account in this scenario as they might modify the tree-level result.

### 11.6.4 Effects of CP violation

SUSY scenarios with CP-violation (CPV) phases are theoretically appealing, since additional CPV beyond that observed in the K, D, and B meson systems is required to explain the observed cosmic matter-antimatter asymmetry. In the MSSM, CP-violation effects in the Higgs sector appear at the quantum level, while in singlet extensions of the MSSM CP-violation effects can already be effective at tree level. In general, CP-violation effects in the Higgs sector have significant constraints from electric dipole moments data [396].

In the MSSM, the gaugino mass parameters ($M_{1,2,3}$), the Higgsino mass parameter, $\mu$, the bilinear Higgs squared-mass parameter, $m_{12}^2$, and the trilinear couplings of the squark and slepton fields to the Higgs fields, $A_f$, may carry non-trivial phases. The two parameter combinations $\arg[\mu A_f(m_{12}^2)^*]$ and $\arg[\mu M_i(m_{12}^2)^*]$ are invariant under phase redefinitions of the MSSM fields [397, 398]. Therefore, if one of these quantities is non-zero, there would be new sources of CP-violation affecting the Higgs sector through radiative corrections, see Ref. [399] and

references therein. The mixing of the neutral CP-odd and CP-even Higgs boson states is no longer forbidden. Hence, $m_A$ is no longer a physical parameter. However, the charged Higgs boson mass $m_{H^\pm}$ is still physical and can be used as an input for the computation of the neutral Higgs boson spectrum of the theory. For large values of $m_{H^\pm}$, corresponding to the decoupling limit, the properties of the lightest neutral Higgs boson state approach those of the SM Higgs boson. In particular, the upper bound on the lightest neutral Higgs boson mass takes the same value as in the CP-conserving case [398]. Nevertheless, there still can be significant mixing between the two heavier neutral mass eigenstates. For a detailed study of the Higgs boson mass spectrum and parametric dependence of the associated radiative corrections, see Ref. [399] and references therein.

Major variations to the Higgs boson phenomenology occur in the presence of explicit CPV phases. In the CPV case, vector boson pairs couple to all three neutral Higgs boson mass eigenstates, $H_i$ ($i = 1, 2, 3$), with couplings

$$ \begin{aligned} g_{H_i V V} &= \cos \beta \mathcal{O}_{1i} + \sin \beta \mathcal{O}_{2i}, \\ g_{H_i H_j Z} &= \mathcal{O}_{3i} (\cos \beta \mathcal{O}_{2j} - \sin \beta \mathcal{O}_{1j}) \\ &\quad - \mathcal{O}_{3j} (\cos \beta \mathcal{O}_{2i} - \sin \beta \mathcal{O}_{1i}), \end{aligned} \quad (11.45) $$

where the $g_{H_i V V}$ couplings are normalised to the analogous SM coupling and the $g_{H_i H_j Z}$ have been normalised to $g_Z^{SM}/2$. The orthogonal matrix $\mathcal{O}_{ij}$, only defined in the $p^2 \to 0$ limit, is relating the weak eigenstates to the mass eigenstates. It has non-zero off-diagonal entries mixing the CP-even and CP-odd components of the weak eigenstates. Moreover, CPV phases imply that all neutral Higgs bosons can couple to both scalar and pseudoscalar fermion bilinear densities. The couplings of the mass eigenstates $H_i$ to fermions depend on the loop-corrected fermion Yukawa couplings (similarly to the CP conserving (CPC) case), on $\tan \beta$ and on $\mathcal{O}_{ji}$ [400].

The production processes of neutral MSSM Higgs bosons in the CPV scenario are similar to those in the CPC scenario. Regarding the decay properties, the lightest mass eigenstate, $H_1$, predominantly decays to $b\bar{b}$ if kinematically allowed, with a smaller fraction decaying to $\tau^+\tau^-$. If kinematically allowed, a SM-like neutral Higgs boson, $H_2$ or $H_3$ can decay predominantly to $H_1 H_1$ leading to many new interesting signals both at lepton and hadron colliders; otherwise it will decay preferentially to $b\bar{b}$.

The discovery of a 125 GeV Higgs boson has put strong constraints on the realisation of the CPV scenario within the MSSM. This is partly due to the fact that the observed Higgs boson rates
are close to the SM values, and a large CP-violating component would necessarily induce a large variation in the rate of the SM-like Higgs boson decays into the weak gauge bosons $W^±$ and $Z$. The measured Higgs mass imposes additional constraints on the realisation of this scenario. Once all effects are considered, the CP-odd Higgs boson A component of the lightest Higgs boson tends to be smaller than about 10% [401]. This restriction can be alleviated in the NMSSM or more general two Higgs doublet models. CP-violating effects can still be significant in the heavy Higgs sector. For instance, the Higgs bosons $H_2$ and $H_3$ may be admixtures of CP-even and CP-odd scalars, and therefore both may be able to decay into pairs of weak gauge bosons. The observation of such decays would be a clear signal of CP-violation. In the MSSM, the proximity of the masses of $H_2$ and $H_3$ makes the measurement of such effect quite challenging, but in generic two Higgs doublet models, the mass splitting between the two heavy mass eigenstates may become larger, facilitating the detection of CP-violating effects at collider experiments [402].

### 11.6.5 Non-supersymmetric extensions of the Higgs sector

There are many ways to extend the minimal Higgs sector of the SM. In the preceding sections the phenomenology of SUSY Higgs sectors is considered, which at tree level implies a constrained type-II 2HDM (with restrictions on the Higgs boson masses and couplings). In the following discussion, more generic 2HDM's are presented (for some comprehensive reviews, see the reviews [403]). These models are theoretically less compelling since they do not provide an explanation for the SM Higgs naturalness problem, but can lead to different patterns of Higgs-fermion couplings, hence, to different phenomenology. It is also possible to consider models with a SM Higgs boson and one or more additional scalar $SU(2)_L$ doublets that acquire no VEV and hence play no role in the EWSB mechanism. Such models are dubbed Inert Higgs Doublet Models (IHD) [404]. Without a VEV associated to it, a Higgs boson from an inert doublet has no tree-level coupling to gauge bosons and hence cannot decay into a pair of them. Moreover, imposing a $Z_2$ symmetry that prevents them from coupling to the fermions, it follows that, if the lightest inert Higgs boson is neutral, it becomes a good DM candidate with interesting associated collider signals. Various studies of IHD models in the light of a 125 GeV Higgs boson have been performed, see for instance Ref. [405], showing an interesting interplay between collider and direct DM detection signals.

An interesting type of 2HDMs are those in which an Abelian flavour symmetry broken at the electroweak scales creates the fermion mass hierarchies and mixing angles [406]. This idea is based on the Froggatt–Nielsen model [407], where a flavon field couples differently to the SM fermions of different flavour charges. Such flavon acquires a vacuum expectation value, breaking the flavour symmetry but leaving both the flavour breaking and the new physics scales undetermined. In Refs. [408], it was proposed to relate the flavour breaking scale to the electroweak scale by identifying the flavon with the modulus square of the Higgs field. A 2HDM, however, provides a more compelling realisation of the electroweak scale flavour breaking idea. In the most ambitious constructions of two Higgs doublet flavour models (2HDFM), the textures of the Yukawa couplings are a result of an Abelian flavour symmetry that only allows renormalisable Yukawa couplings of the top quark to the Higgs bosons. All other Yukawa couplings are generated by higher dimensional operators that produce hierarchical entries of the Yukawa matrices, explaining the observed quark masses and mixing angles. Flavour observables, LHC Higgs signal strength measurements, electroweak precision measurements, unitarity and perturbativity bounds, as well as collider searches for new scalar resonances result in precise predictions for the parameters of these 2HDFMs. In particular, correlated departures from SM Higgs boson couplings, as well as additional Higgs bosons with masses < 700 GeV must be observed at the LHC. Other incarnations of 2HDFMs can aim at only partially explaining the fermion mass hierarchies but are therefore less restrictive.

Other extensions of the Higgs sector can include multiple copies of $SU(2)_L$ doublets [409], additional Higgs singlets [410], triplets or more complicated combinations of Higgs multiplets. It is also possible to enlarge the gauge symmetry beyond $SU(2)_L \times U(1)_Y$

along with the necessary Higgs field structure to generate gauge boson and fermion masses. There are two main experimental constraints on these extensions: (i) precision measurements which constrain $\rho = m_W^2/(m_Z^2 \cos^2\theta_W)$ to be very close to 1 and (ii) flavour changing neutral current (FCNC) effects. In electroweak models based on the SM gauge group, the tree-level value of $\rho$ is determined by the Higgs multiplet structure. By suitable choices for the hypercharges, and in some cases the mass splitting between the charged and neutral Higgs sector or the vacuum expectation values of the Higgs fields, it is possible to obtain a richer combination of singlets, doublets, triplets and higher multiplets compatible with precision measurements. Concerning the constraints coming from FCNC effects, the Glashow–Weinberg (GW) criterion [411] states that, in the presence of multiple Higgs doublets, the tree-level FCNC's mediated by neutral Higgs bosons will be absent if all fermions of a given electric charge couple to no more than one Higgs doublet. An alternative way of suppressing FCNC in a two Higgs doublet model has been considered in Ref. [412], where it is shown that it is possible to have tree level FCNC completely fixed by the CKM matrix, as a result of an Abelian symmetry.

There is a lot of activity on the study of non-supersymmetric models with an extended Higgs sector with an intense collaboration between theorists and experimentalists, in particular inside the LHC Higgs Working Group [373].

#### 11.6.5.1 Two-Higgs-doublet models

General two Higgs doublet models [403] can have a more diverse Higgs-fermion coupling structure than in SUSY, and can be viewed as a simple extension of the SM to realise the spontaneous breakdown of $SU(2)_L \times U(1)_Y$ to $U(1)_{em}$. Quite generally, if the two Higgs doublets contain opposite hypercharges, the scalar potential will contain mixing mass parameters of the kind $m_{12}^2 \Phi_1^T i\sigma_2 \Phi_2 + h.c..$ In the presence of such terms, both Higgs doublets will acquire vacuum expectation values, $v_1/\sqrt{2}$ and $v_2/\sqrt{2}$, respectively, and the gauge boson masses will keep their SM expressions with the Higgs VEV $v$ replaced by $\sqrt{v_1^2 + v_2^2}$. Apart from the mass terms, the most generic renormalisable and gauge invariant scalar potential for two Higgs doublets with opposite hypercharges contains seven quartic couplings, as presented in Eq. (11.38).

Just as in the MSSM case, after electroweak symmetry breaking and in the absence of CP-violation, the physical spectrum contains a pair of charged Higgs bosons $H^±$, a CP-odd Higgs boson A and two neutral CP-even Higgs bosons, $h$ and $H$. The angles $\alpha$ and $\beta$ diagonalise the CP-even, and the CP-odd and charged Higgs sectors, respectively. The complete 2HDM is defined only after considering the interactions of the Higgs fields to fermions. Yukawa couplings of the generic form

$$ -h_{ij}^a \bar{\psi}_L^i H_a \Psi_R^j + h.c. \quad (11.46) $$

may be added to the renormalisable Lagrangian of the theory. Contrary to the SM, the two Higgs doublet structure does not ensure the alignment of the fermion mass terms $m_{ij} = h_{ij}^a v_a / \sqrt{2}$ with the Yukawa couplings $h_{ij}^a$. This implies that quite generally the neutral Higgs boson will mediate flavour changing interactions between the different mass eigenstates of the fermion fields. Such flavour changing interactions should be suppressed in order to describe properly the Kaon, D and B meson phenomenology. Based on the Glashow–Weinberg criterion, it is clear that the simplest way of avoiding such transitions is to assume the existence of a symmetry that ensures the couplings of the fermions of each given quantum number (up-type and down-type quarks, charged and neutral leptons) to only one of the two Higgs doublets. Different models may be defined depending on which of these fermion fields couple to a given Higgs boson, see Table 11.9. Models of type-I are those in which all SM fermions couple to a single Higgs field. In type-II models, down-type quarks and charged leptons couple to a common Higgs field, while the up-type quarks and neutral leptons couple to the other. In models of type-III (lepton-specific), quarks couple to one of the Higgs bosons, while leptons couple to the other. Finally, in models of type-IV (flipped), up-type quarks and charged leptons couple to one of the Higgs fields
while down-quarks and neutral leptons couple to the other.

<table><caption>**Table 11.9:** Higgs boson couplings to up, down and charged lepton-type $SU(2)_L$ singlet fermions in the four discrete types of 2HDM models that satisfy the Glashow–Weinberg criterion.</caption><thead><tr><th colspan="5">Model 2HDM I 2HDM II 2HDM III 2HDM IV</th></tr><tr><th>u</th><th>Φ<sub>2</sub></th><th>Φ<sub>2</sub></th><th>Φ<sub>2</sub></th><th>Φ<sub>2</sub></th></tr></thead><tbody><tr><th>d</th><th>Φ<sub>2</sub></th><th>Φ<sub>1</sub></th><th>Φ<sub>2</sub></th><th>Φ<sub>1</sub></th></tr><tr><th>e</th><th>Φ<sub>2</sub></th><th>Φ<sub>1</sub></th><th>Φ<sub>1</sub></th><th>Φ<sub>2</sub></th></tr></tbody></table>

The two Higgs doublet model phenomenology depends strongly on the size of the mixing angle $\alpha$ and therefore on the quartic couplings. For large values of $m_A$, $\sin\alpha \to -\cos\beta$, $\cos\alpha \to \sin\beta$, $\cos(\beta - \alpha) \to 0$, and the lightest CP-even Higgs boson $h$ behaves as the SM Higgs boson. The same behaviour is obtained if the quartic couplings are such that $M_{12}^2 \sin\beta = -(M_{11}^2 - m_h^2) \cos\beta$. The latter condition represents a situation in which the couplings of $h$ to fermions and weak gauge bosons become the same as in the SM, without decoupling the rest of the non-standard scalars and it is of particular interest due to the fact that the discovered Higgs boson has SM-like properties. This situation will be referred to as alignment, as in the MSSM case.

In analogy to the effects of CP violation in the SUSY 2HDM, some parameters of the Higgs potential can be complex and one has a model that is explicitly CP violating. The three neutral mass eigenstates mixed with each other and the Higgs phenomenology is analogous to the one described for the SUSY case above, with the caveat that when considering the neutral Higgs boson couplings to the scalar and pseudoscalar fermion bilinear densities, the proper weight should be considered for the respective 2HDM's.

In type-II Higgs doublet models, at large values of $\tan\beta$ and moderate values of $m_A$, the non-standard Higgs bosons $H$, $A$ and $H^\pm$ couple strongly to bottom quarks and $\tau$ leptons. Hence, the decay modes of the non-standard Higgs bosons tend to be dominated by the b-quark and $\tau$-lepton modes, including top quarks or neutrinos in the case of the charged Higgs boson. However, for large and negative values of $\lambda_4$, the charged Higgs boson mass may be sufficiently heavy to allow on-shell decays $H^\pm \to W^\pm + (H, A)$, via a trilinear coupling

$$g_{H^\pm W \mp H, A} \approx \frac{M_W}{v} \sin(\beta - \alpha) (p_{H^\pm} - p_{H, A}), \quad (11.47)$$

where $p_{H^\pm}$ and $p_{H, A}$ are the charged and neutral scalar Higgs boson momenta pointing into the vertex. On the other hand, for large and positive values of $\lambda_5$, the above charged Higgs boson decay into a $W^\pm$ and the CP-odd Higgs boson may be allowed, but the heavy Higgs boson $H$ may be sufficiently heavy to decay into a CP-odd Higgs boson and an on-shell $Z$, $H \to Z + A$, via

$$g_{HZA} \approx \frac{M_Z}{v} \sin(\beta - \alpha) (p_H - p_A). \quad (11.48)$$

The decay $H^\pm \to W^\pm + H$, on the other hand may be allowed only if $\lambda_4 < -\lambda_5$. The couplings controlling all the above decay modes are proportional to $\sin(\beta - \alpha)$ and therefore they are unsuppressed in the alignment limit. Moreover, these could still be the dominant decay modes at moderate values of $\tan\beta$, offering a way to evade the current bounds obtained assuming a dominant decay into b-quarks or $\tau$-leptons.

The quartic couplings are restricted by the condition of stability of the effective potential as well as by the restriction of obtaining the proper value of the lightest CP-even Higgs boson mass. Close to the alignment limit, the lightest CP-even Higgs boson mass becomes

$$m_h^2 \approx v^2(\lambda_1 \cos^4 \beta + \lambda_2 \sin^4 \beta + 2\tilde{\lambda}_3 v^2 \cos^2 \beta \sin^2 \beta + 4\lambda_6 \cos^3 \beta \sin \beta + 4\lambda_7 \sin^3 \beta \cos \beta), \quad (11.49)$$

where $\tilde{\lambda}_3 = \lambda_3 + \lambda_4 + \lambda_5$.

The stability conditions imply the positiveness of all masses, as well as the avoidance of run-away solutions to large negative values of the fields in the scalar potential. These conditions imply

$$\lambda_1 \ge 0, \quad \lambda_2 \ge 0, \quad \lambda_3 + \lambda_4 - |\lambda_5| \ge -\sqrt{\lambda_1 \lambda_2}, \quad \lambda_3 \ge -\sqrt{\lambda_1 \lambda_2}, \\ 2|\lambda_6 + \lambda_7| < \frac{\lambda_1 + \lambda_2}{2} + \tilde{\lambda}_3, \qquad (11.50)$$

where the first four conditions are necessary and sufficient conditions in the case of $\lambda_6 = \lambda_7 = 0$, while the last one is a necessary condition in the case all couplings are non-zero. Therefore, to obtain the conditions that allow the decays $H^\pm \to W^\pm H, A$ and $H \to ZA$, $\lambda_3$ should take large positive values in order to compensate for the effects of $\lambda_4$ and $\lambda_5$. For more detailed discussions about 2HDM phenomenology, see for example Refs. [40, 403].

### 11.6.5.2 Higgs triplets

Electroweak triplet scalars are the simplest non-doublet extension of the SM that can participate in the spontaneous breakdown of $SU(2)_L \times U(1)_Y$ to $U(1)_{em}$. Two types of model have been developed in enough detail to make a meaningful comparison to LHC data: the Higgs triplet model (HTM) [413] and the Georgi–Machacek model (GM) [414].

The Higgs triplet model extends the SM by the addition of a complex $SU(2)_L$ triplet scalar field $\Delta$ with hypercharge $Y = 2$, and a general gauge-invariant renormalisable potential $V(\Phi, \Delta)$ for $\Delta$ and the SM Higgs doublet $\Phi$. The components of the triplet field can be parameterised as

$$\Delta = \frac{1}{\sqrt{2}} \begin{pmatrix} \Delta^+ & \sqrt{2}\Delta^{++} \\ v_\Delta + \delta + i\xi & -\Delta^+ \end{pmatrix}. \quad (11.51)$$

where $\Delta^+$ is a singly-charged field, $\Delta^{++}$ is a doubly-charged field, $\delta$ is a neutral CP-even scalar, $\xi$ is a neutral CP-odd scalar, and $v_\Delta$ is the triplet VEV. The general scalar potential mixes the doublet and triplet components. After electroweak symmetry breaking there are seven physical mass eigenstates, denoted $H^{\pm\pm}$, $H^\pm$, $A$, $H$, and $h$.

A distinguishing feature of the HTM is that it violates the custodial symmetry of the SM; thus the $\rho$ parameter deviates from 1 even at tree level. Letting $x$ denote the ratio of triplet and doublet VEVs, the tree level expression is

$$\rho = \frac{1 + 2x^2}{1 + 4x^2}. \quad (11.52)$$

The measured value of the $\rho$ parameter then limits the triplet VEV to be quite small, $x \lesssim 0.03$, or $v_\Delta < 8$ GeV. This constraint severely limits the role of the triplet scalar in the EWSB mechanism.

The small VEV of the Higgs triplet in the HTM is a virtue from the point of view of generating neutrino masses without the necessity for introducing right-handed neutrino fields. The gauge invariant dimension four interaction

$$h_{\nu_{ij}} \ell_i^T C^{-1} i\sigma_2 \Delta \ell_j, \quad (11.53)$$

where $\ell_i$ are the lepton doublets, $C$ is the charge conjugation matrix, and $h_{\nu_{ij}}$ is a complex symmetric coupling matrix, generates a Majorana mass matrix for the neutrinos:

$$m_{\nu_{ij}} = \sqrt{2} h_{\nu_{ij}} v_{\Delta}. \quad (11.54)$$

This can be combined with the usual neutrino seesaw to produce what is known as the type-II seesaw [415].

The HTM suggests the exciting possibility of measuring parameters of the neutrino mass matrix at the LHC. If the doubly-charged Higgs boson is light enough and/or its couplings to $W^+W^+$ are sufficiently suppressed, then its primary decay is into same-sign lepton pairs: $H^{++} \to \ell_i^+ \ell_j^+$; from Eq. (11.53) and Eq. (11.54), it is apparent that these decays are in general lepton-flavour violating with branchings proportional to elements of the neutrino mass matrix [416].
Precision electroweak data constrain the mass spectrum as well as the triplet VEV of the HTM [417]. These constraints favour a spectrum where $H^{++}$ is the lightest of the exotic bosons, and where the mass difference between $H^+$ and $H^{++}$ is a few hundred GeV. The favoured triplet VEV is a few GeV, which also favours $H^{++}$ decays into $W^+W^+$ over same-sign dileptons.

The GM model addresses the $\rho$ parameter constraint directly by building in a custodial symmetry (it was however argued that large corrections to $\rho = 1$ are generated at one-loop [418], requiring some fine-tuning to preserves agreement with EW precision data). Writing the complex scalar doublet of the SM as a $(2,2)$ under $SU(2)_L \times SU(2)_R$, it is obvious that the next simplest construction respecting custodial symmetry is a scalar transforming like a $(3,3)$ [419]. These nine real degrees of freedom correspond to a complex electroweak triplet combined with a real triplet, with the scalar potential required to be invariant under $SU(2)_R$. Under the custodial $SU(2)_{L+R}$, they transform as $1 \oplus 3 \oplus 5$, with a CP-even neutral scalar as the custodial singlet (thus matching the SM Higgs boson), a CP-odd neutral scalar in the custodial triplet, and another CP-even neutral scalar in the custodial 5-plet.

The scalar components can be decomposed as

$$ \Xi = \begin{pmatrix} \chi_3^* & \xi_1 & \chi_1 \\ -\chi_2^* & \xi_2 & \chi_2 \\ \chi_1^* & -\xi_1^* & \chi_3 \end{pmatrix}, \quad (11.55) $$

where $\xi_2$ is a real scalar and the others are complex scalars. Linear combinations of these scalars account for the neutral custodial singlet, a neutral and singly-charged field making up the custodial triplet, and neutral, singly-charged, and doubly-charged fields making up the custodial 5-plet.

When combined with the usual SM doublet field $\Phi$, the electroweak scale $v$ is now related to the doublet and triplet VEVs by

$$ v^2 = v_{\Phi}^2 + 8v_{\Xi}^2. \quad (11.56) $$

Note that the GM triplets by themselves are sufficient to explain electroweak symmetry breaking and the existence of a 125 GeV neutral boson along with a custodial triplet of Goldstone bosons; the complex doublet field in the GM model is required to generate fermion masses via the usual dimension four Yukawa couplings. This raises the question of whether one can rule out the possibility that the 125 GeV boson is the neutral member of a custodial 5-plet rather than a custodial singlet, without invoking decays to fermions. A conclusive answer is given by observing that the ratio of the branching fractions to W versus Z bosons is completely determined by the custodial symmetry properties of the boson. For a custodial 5-plet, the ratio of the signal strength to WW over that to ZZ is predicted to be 1/4 that of a SM Higgs boson [419], and thus already ruled out by the experimental results presented in Section 11.5.

Another interesting general feature of Higgs triplet models is that, after mixing, the SM-like neutral boson can have stronger couplings to WW and ZZ than predicted by the SM [420]; this is in contrast to mixing with additional doublets and singlet, which can only reduce the WW and ZZ couplings versus the SM. This emphasises that LHC Higgs data cannot extract model independent coupling strengths for the Higgs boson [291].

Because of the built-in custodial symmetry, the triplet VEV in the GM model can be large compared to the doublet VEV. The custodial singlet neutral boson from the triplets mixes with the neutral boson from the doublet. Two interesting special cases are (i) the triplet VEV is small and the 125 GeV boson is SM-like except for small deviations, and (ii) the 125 GeV boson is mostly the custodial singlet neutral boson from the electroweak triplets. The phenomenology of the doubly-charged and singly-charged bosons is similar to that of the HTM. The constraints on the GM model from precision electroweak data, LEP data, and current LHC data are summarised in Ref. [40].

### 11.6.6 Composite Higgs models

Within the SM, EWSB is posited but has no dynamical origin. Furthermore, the Higgs boson appears to be unnaturally light. A scenario that remedies these two catches is to consider the Higgs boson as a bound state of new dynamics becoming strong around

the weak scale. The Higgs boson can be made significantly lighter than the other resonances of the strong sector if it appears as a pseudo-Nambu-Goldstone boson, see Refs. [11] for reviews.

#### 11.6.6.1 Little Higgs models

The idea behind the Little Higgs boson models [421] is to identify the Higgs doublet as a (pseudo) Nambu-Goldstone boson while keeping some sizeable non-derivative interactions, in particular a largish Higgs quartic interaction. By analogy with QCD where the pions $\pi^{\pm,0}$ appear as Nambu-Goldstone bosons associated to the breaking of the chiral symmetry $SU(2)_L \times SU(2)_R/SU(2)$, switching on some interactions that break explicitly the global symmetry will generate masses for the would-be massless Nambu-Goldstone bosons of the order of $g\Lambda_{G/H}/(4\pi)$, where g is the coupling of the symmetry breaking interaction and $\Lambda_{G/H} = 4\pi f_{G/H}$ is the dynamical scale of the global symmetry breaking $G/H$. In the case of the Higgs boson, the top Yukawa interaction or the gauge interactions themselves will certainly break explicitly (part of) the global symmetry since they act non-linearly on the Higgs boson. Therefore, obtaining a Higgs boson mass around 125 GeV would demand a dynamical scale $\Lambda_{G/H}$ of the order of 1 TeV, which is known to lead to too large oblique corrections. Raising the strong dynamical scale by at least one order of magnitude requires an additional selection rule to ensure that a Higgs boson mass is generated at the 2-loop level only

$$ m_H^2 = \frac{g^2}{16\pi^2} \Lambda_{G/H}^2 \rightarrow m_H^2 = \frac{g_1^2 g_2^2}{(16\pi^2)^2} \Lambda_{G/H}^2. \quad (11.57) $$

The way to enforce this selection rule is through a “collective breaking” of the global symmetry:

$$ \mathcal{L} = \mathcal{L}_{G/H} + g_1 \mathcal{L}_1 + g_2 \mathcal{L}_2. \quad (11.58) $$

Each interaction $\mathcal{L}_1$ or $\mathcal{L}_2$ individually preserves a subset of the global symmetry such that the Higgs boson remains an exact Nambu-Goldstone boson whenever either $g_1$ or $g_2$ is vanishing. A mass term for the Higgs boson can be generated only by diagrams involving simultaneously both interactions. At one-loop, such diagrams are not quadratically divergent, so the Higgs boson mass is not UV sensitive. Explicitly, the cancellation of the SM quadratic divergences is achieved by a set of new particles around the Fermi scale: gauge bosons, vector-like quarks, and extra massive scalars, which are related, by the original global symmetry, to the SM particles with the same spin. Contrary to SUSY, the cancellation of the quadratic divergences is achieved by same-spin particles. These new particles, with definite couplings to SM particles as dictated by the global symmetries of the theory, are perfect goals for the LHC.

The simplest incarnation of the collective breaking idea, the so-called littlest Higgs boson model, is based on a non-linear $\sigma$-model describing the spontaneous breaking $SU(5)$ down to $SO(5)$. A subgroup $SU(2)_1 \times U(1)_1 \times SU(2)_2 \times U(1)_2$ is weakly gauged. This model contains a weak doublet, that is identified with the Higgs doublet, and a complex weak triplet whose mass is not protected by collective breaking. Other popular little Higgs models are based on different coset spaces: minimal moose ($SU(3)^2/SU(3)$), the simplest little Higgs ($SU(3)^2/SU(2)^2$), the bestest little Higgs ($SO(6)^2/SO(6)$). For comprehensive reviews, see Ref. [422].

Generically, oblique corrections in Little Higgs models are reduced either by increasing the coupling of one of the gauge groups (in the case of product group models) or by increasing the masses of the W and Z partners, leading ultimately to a fine-tuning of the order of a few percents (see for instance Ref. [423] and references therein). The compatibility of Little Higgs models with experimental data is significantly improved when the global symmetry involves a custodial symmetry as well as a T-parity [424] under which, in analogy with R-parity in SUSY models, the SM particles are even and their partners are odd. Such Little Higgs models would therefore appear in colliders as jet(s) with missing transverse energy [425] and the ATLAS and CMS searches for squarks and gluinos can be recast to obtain limits on the masses of the heavy vector-like quarks. The T-even top partner, with an expected mass below 1 TeV to cancel the top loop quadratic divergence without too much fine-tuning, would decay dominantly into
a $t+Z$ pair or into a $b+W$ pair or even into $t+H$. The latest CMS and ATLAS direct searches [426] for vector-like top partners put a lower bound around 1.1–1.3 TeV (for various branching fraction combinations), excluding the most natural region of the parameter space of these models, i.e., imposing a fine-tuning below the percent level.

The motivation for Little Higgs models is to solve the little hierarchy problem, i.e., to push the need for new physics (responsible for the stability of the weak scale) up to around 10 TeV. Per se, Little Higgs models are effective theories valid up to their cut-off scale $\Lambda_{G/H}$. Their UV completions could either be weakly or strongly coupled.

### 11.6.6.2 Models of partial compositeness

Even in composite models, the Higgs boson cannot appear as a regular resonance of the strong sector without endangering the viability of the setup when confronted to data. The way out is that the Higgs boson appears as a pseudo Nambu–Goldstone boson: the new strongly coupled sector is supposed to be invariant under a global symmetry $G$ spontaneously broken to a subgroup $H$ at the scale $f$ (the typical mass scale of the resonances of the strong sector is $m_\rho \sim g_\rho f$ with $g_\rho$ the characteristic coupling of the strong sector). To avoid conflict with EW precision measurements, the strong interactions themselves should better not break the EW symmetry. Hence the SM gauge symmetry itself should be contained in $H$. See Table 11.10 for a few examples of coset spaces.

<table><caption>**Table 11.10:** Global symmetry breaking patterns and the corresponding Goldstone boson contents of the SM, the minimal composite Higgs model, the next to minimal composite Higgs model, and the minimal composite two Higgs doublet model. Note that the SU(3) model does not have a custodial invariance. *a* denotes a CP-odd scalar while *h* and *H* are CP-even scalars.</caption><thead><tr><th>Model</th><th>Symmetry Pattern</th><th>Goldstones</th></tr></thead><tbody><tr><td>SM</td><td>SO(4)/SO(3)</td><td>W<sub>L</sub>, Z<sub>L</sub></td></tr><tr><td>–</td><td>SU(3)/SU(2)×U(1)</td><td>W<sub>L</sub>, Z<sub>L</sub>, H</td></tr><tr><td>MCHM</td><td>SO(5)/SO(4)</td><td>W<sub>L</sub>, Z<sub>L</sub>, H</td></tr><tr><td>NMCHM</td><td>SO(6)/SO(5)</td><td>W<sub>L</sub>, Z<sub>L</sub>, H, a</td></tr><tr><td>MC2HM</td><td>SO(6)/SO(4)×SO(2)</td><td>W<sub>L</sub>, Z<sub>L</sub>, h, H, H<sup>±</sup>, a</td></tr></tbody></table>

The SM (light) fermions and gauge bosons cannot be part of the strong sector itself since LEP data have already put stringent bounds on the compositeness scale of these particles far above the TeV scale. The gauge bosons couple to the strong sector by a weak gauging of a SU(2)×U(1) subgroup of the global symmetry $G$. Inspiration for the construction of such models comes from the AdS/CFT correspondence: the components of a gauge field along an extra warped space dimension can be interpreted as the Goldstone bosons resulting from the breaking of global symmetry of the strong sector. The couplings of the SM fermions to the strong sector could a priori take two different forms:

(i) a bilinear coupling of two SM fermions to a composite scalar operator, $\mathcal{O}$, of the form $\mathcal{L} = y \bar{q}_L u_R \mathcal{O} + h.c.$, in simple analogy with the SM Yukawa interactions. This is the way fermion masses were introduced in technicolor theories and it generically comes with severe flavour problems and calls for extended model-building gymnastics [12] to circumvent them;

(ii) a linear mass mixing with fermionic vector-like operators: $\mathcal{L} = \lambda_L \bar{q}_L Q_R + \lambda_R U_L u_R$. $Q$ and $U$ are two fermionic composite operators of mass $M_Q$ and $M_U$.

Being part of the composite sector, the composite fermionic operators can have a direct coupling of generic order $Y_*$ to the Higgs boson. In analogy with the photon-$\rho$ mixing in QCD, once the linear mixings are diagonalised, the physical states are a linear combination of elementary and composite fields. Effective Yukawa couplings are generated and read for instance for the up-type quark

$$y = Y_* \sin \theta_L \sin \theta_R \qquad (11.59)$$

where $\sin \theta_i = \lambda_i / \sqrt{M_{Q,U}^2 + \lambda_i^2}$, $i = L, R$, measure the amount of compositeness of the SM left- and right-handed up-type quark. If the strong sector is flavour-anarchic, i.e., if the couplings of the Higgs boson to the composite fermions does not exhibit any particular flavour structure, the relation Eq. (11.59) implies that the light fermions are mostly elementary states ($\sin \theta_i \ll 1$), while the third generation quarks need to have a sizable degree of compositeness. The partial compositeness paradigm offers an appealing dynamical explanation of the hierarchies in the fermion masses. In fact, assuming the strong sector to be almost conformal above the confinement scale, the low-energy values of the mass-mixing parameters $\lambda_{L,R}$ are determined by the (constant) anomalous dimension of the composite operator they mix with. If the UV scale at which the linear mixings are generated is large, then $\mathcal{O}(1)$ differences in the anomalous dimensions can generate naturally large hierarchies in the fermion masses via renormalisation group running [427]. While the introduction of partial compositeness greatly ameliorated the flavour problem of the original composite Higgs models, nevertheless, it did not solve the issue completely, at least in the case where the strong sector is assumed to be flavour-anarchic [428]. While the partial compositeness set-up naturally emerges in models built in space-times with extra dimensions, no fully realistic microscopic realisation of partial compositeness has been proposed in the literature.

Another nice aspect of the partial compositeness structure is the dynamical generation of the Higgs potential that is not arbitrary like in the SM. The Higgs boson being a pseudo-Nambu–Goldstone boson, its mass does not receive any contribution from the strong sector itself but it is generated at the one-loop level via the couplings of the SM particles to the strong sector since these interactions are breaking the global symmetries under which the Higgs doublet transforms non-linearly. Obtaining $v \ll f$, as required phenomenologically, requires some degree of tuning, which scales like $\xi \equiv v^2/f^2$. A mild tuning of the order of 10% ($\xi \approx 0.1$) is typically enough to comply with electroweak precision constraints. This is an important point: in partial compositeness models, the entire Higgs potential is generated at one loop, therefore the separation between $v$ and $f$ can only be obtained at a price of a tuning. This marks a difference with respect to the Little Higgs models which realise a parametric hierarchy between the quartic and mass terms through the collective symmetry breaking mechanism. In fact in Little Higgs models, the quartic coupling is a tree-level effect, leading to a potential

$$V(H) \approx \frac{g_{\mathrm{SM}}^2}{16\pi^2} m_{\rho}^2 H^2 + g_{\mathrm{SM}}^2 H^4, \qquad (11.60)$$

where $g_{\mathrm{SM}}$ generically denotes the SM couplings. The minimisation condition reads $v^2/f^2 \sim g_\rho^2/(16\pi^2)$, therefore $v$ is formally loop suppressed with respect to $f$. This is the major achievement of the Little Higgs constructions, which however comes at the price of the presence of sub-TeV vectors carrying EW quantum numbers and therefore giving rise generically to large oblique corrections to the propagators of the $W$ and the $Z$ gauge bosons.

After minimisation, the dynamically generated potential leads to an estimate of the Higgs boson mass as

$$m_H^2 \approx g_\rho^3 y_t 2\pi^2 v^2. \qquad (11.61)$$

It follows that the limit $f \to \infty$, i.e., $\xi \to 0$, is a true decoupling limit: all the resonances of the strong sector become heavy but the Higgs boson whose mass is protected by the symmetries of the coset $G/H$. When compared to the experimentally measured Higgs boson mass, this estimate puts an upper bound on the strength of the strong interactions: $g_\rho \lesssim 2$. In this limit of not so large coupling, the Higgs potential receives additional contributions. In particular, the fermionic resonances in the top sector which follow from the global symmetry structure of the new physics sector can help raising the Higgs boson mass. Using some dispersion relation techniques, the mass of the Higgs is connected to the resonance masses. In the minimal SO(5)/SO(4) model, it was shown [429] that a 125 GeV mass can be obtained if at least one of the fermionic resonances is lighter than $\sim 1.4 f$. As in SUSY scenarios, the top sector is playing a crucial role in the
dynamics of EWSB and can provide the first direct signs of new physics. The direct searches for these top partners, in particular the ones with exotic electric charges 5/3, are already exploring the natural parameter spaces of these models [430].

The main physics properties of a pseudo Nambu-Goldstone Higgs boson can be captured in a model-independent way by a small number of higher-dimensional operators. Indeed, the strong dynamics at the origin of the composite Higgs boson singles out a few operators among the complete list discussed earlier in Section 11.5: these are the operators that involve extra powers of the Higgs doublets, and they are therefore generically suppressed by a factor $1/f^2$ as opposed to the operators that involve extra derivatives or gauge bosons that are suppressed by a factor $1/(g_\rho^2 f^2)$. The relevant effective Lagrangian describing a strongly interacting light Higgs boson is:

$$ \mathcal{L}_{\text{SILH}} = \frac{c_H}{2f^2} (\partial_\mu (\Phi^\dagger \Phi))^2 + \frac{c_T}{2f^2} (\Phi^\dagger \overleftrightarrow{D}^\mu \Phi)^2 - \frac{c_6 \lambda}{f^2} (\Phi^\dagger \Phi)^3 \\ + \left( \sum_f \frac{c_f y_f}{f^2} \Phi^\dagger \Phi \bar{f}_L \Phi f_R + \text{h.c.} \right). \qquad (11.62) $$

Typically, these new interactions induce deviations in the Higgs boson couplings that scale like $O(v^2/f^2)$. Hence, the measurements of the Higgs boson couplings can be translated into some constraints on the compositeness scale, $4\pi f$, of the Higgs boson. The peculiarity of these composite models is that, due to the Goldstone nature of the Higgs boson, the direct couplings to photons and gluons are further suppressed and generically the coupling modifiers scale like

$$ \begin{aligned} \kappa_{W,Z,f} &\sim 1 + \mathcal{O}\left(\frac{v^2}{f^2}\right), & \kappa_{Z\gamma} &\sim \mathcal{O}\left(\frac{v^2}{f^2}\right), \\ \kappa_{\gamma,g} &\sim \mathcal{O}\left(\frac{v^2}{f^2} \times \frac{y_t^2}{g_\rho^2}\right), & & \end{aligned} \qquad (11.63) $$

where $g_\rho$ denotes the typical coupling strength among the states of the strongly coupled sector and $y_t$ is the top Yukawa coupling, the largest interaction that breaks the Goldstone symmetry. The $\kappa_{Z\gamma,\gamma,g}$ coupling modifiers are not generated by the strong coupling operators of Eq. (11.62) but by some subleading form-factor operator generated by loops of heavy resonances of the strong sector. The coupling modifiers also receive additional contributions from the other resonances of the strong sector, in particular the fermionic resonances of the top sector that are required to be light to generate a 125 GeV Higgs boson mass. Some indirect information on the resonance spectrum could thus be inferred by a precise measurement of the Higgs boson coupling deviations. However, it was realised, see in particular Ref. [306], that the task is actually complicated by the fact that, in the minimal models, these top partners give a contribution to both $\kappa_t$ (resulting from a modification of the top Yukawa coupling) and $\kappa_\gamma$ and $\kappa_g$ (resulting from new heavy particles running into the loops) and the structure of interactions is such that the net effect vanishes for inclusive quantities like $\sigma(gg \to H)$ or $\Gamma(H \to \gamma\gamma)$ as a consequence of the Higgs low energy theorem [21, 22, 301]. So, one would need to rely on differential distribution, like the Higgs boson $p_T$ distribution discussed in Section 11.2.4.1, to see the top partner effects in Higgs data [431]. The off-shell channel $gg \to H^* \to 4\ell$ [290] and the double Higgs boson production $gg \to HH$ [432] can also help to resolve the gluon loop and separate the top and top-partner contributions.

### 11.6.6.3 Minimal composite Higgs models

The minimal composite Higgs models (MCHM) are concrete examples of the partial compositeness paradigm. The Higgs doublet is described by the coset space SO(5)/SO(4) where a subgroup SU(2)$_L \times$ U(1)$_Y$ is weakly gauged and under which the four Goldstone bosons transform as a doublet of hypercharge 1. There is some freedom on how the global symmetry is acting on the SM fermions: in MCHM4 the quarks and leptons are embedded into spinorial representations of SO(5), while in MHCM5

they are part of fundamental representations (it might also be interesting phenomenologically to consider larger representations like MCHM14 [433] with the SM fermions inside a representation of dimension 14). It is also possible to consider that fermions of different chirality and flavour are in different representations of SO(5), leading to a more varied phenomenology [434]. The non-linearly realised symmetry acting on the Goldstone bosons leads to general predictions of the coupling of the Higgs boson to the EW gauge bosons. For instance, it can be shown that the quadratic terms in the W and Z bosons read

$$ m_W^2(H) \left( W_\mu W^\mu + \frac{1}{2 \cos^2 \theta_W} Z_\mu Z^\mu \right), \qquad (11.64) $$

with $m_W(H) = \frac{g_f}{2} \sin \frac{H}{f}$. Expanding around the EW vacuum, the expression of the weak scale is $v = f \sin(\langle H \rangle/f)$. And the values of the modified Higgs boson couplings to the W and Z become:

$$ g_{HVV} = \frac{2m_V^2}{v} \sqrt{1 - v^2/f^2}, \qquad g_{HHVV} = \frac{2m_V^2}{v^2} (1 - 2v^2/f^2). \qquad (11.65) $$

Note that the Higgs boson couplings to gauge bosons is always suppressed compared to the SM prediction. This is a general result [435] that holds as long as the coset space is compact.

The Higgs boson couplings to the fermions depend on the representation which the SM fermions are embedded into. The most commonly used embeddings consider all fermion doublets and singlets in the same representations. While, in MCHM4 and MCHM5, the modifications of the couplings depend only on the Higgs boson compositeness scale, in MCHM14 the leading corrections depend also on the mass spectrum of the resonances [433]. This is due to the fact that more than one SO(5) invariant gives rise to SM fermion masses. The $(\kappa_V, \kappa_f)$ experimental fit of the Higgs boson couplings can be used to derive a lower bound on the Higgs boson compositeness scale $4\pi f \gtrsim 9$ TeV, which is less stringent than the indirect bound obtained from EW precision data, $4\pi f \gtrsim 15$ TeV [436] but more robust and less subject on assumptions [437].

### 11.6.6.4 Twin Higgs models

In all composite models presented above, the particles responsible for canceling the quadratic divergences in the Higgs boson mass are charged under the SM gauge symmetries. In particular, the top partner carries color charge, implying a reasonably large minimal production cross section at the LHC. An alternative scenario, which is experimentally quite challenging and might explain the null result in various new physics searches, is the case nowadays referred to as “neutral naturalness” [13, 14], where the particles canceling the 1-loop quadratic divergences are neutral under the SM. The canonical example for such theories is the Twin Higgs model of Ref. [13]. This is an example of a pseudo-Goldstone boson model with an approximate global SU(4) symmetry broken to SU(3). The Twin Higgs model is obtained by gauging the SU(2)$_A \times$ SU(2)$_B$ subgroup of SU(4), where SU(2)$_A$ is identified with the SM SU(2)$_L$, while SU(2)$_B$ is the twin SU(2) group. Gauging this subgroup breaks the SU(4) symmetry explicitly, but quadratically divergent corrections do not involve the Higgs boson when the gauge couplings of the two SU(2) subgroups are equal, $g_A = g_B$. The SU(4) $\to$ SU(3) breaking will also result in the breaking of the twin SU(2)$_B$ group and, as a result, three of the seven Goldstone bosons will be eaten, leaving 4 Goldstone bosons corresponding to the SM Higgs doublet. In fact, imposing the $Z_2$ symmetry on the full model will ensure the cancellation of all 1-loop quadratic divergences to the Higgs boson mass. Logarithmically divergent terms can, however, arise for example from gauge loops, leading to a Higgs boson mass of order $g^2 f / 4\pi$, which is of the order of the physical Higgs boson mass for $f \sim 1$ TeV. The quadratic divergences from the top sector can be eliminated if the $Z_2$ protecting the Higgs boson mass remains unbroken by the couplings that result in the top Yukawa coupling. This can be achieved by introducing top partners charged under a twin SU(3)$_C$. In this case, the quadratic divergences are cancelled by top partners that are neutral under the SM gauge symmetries.
Twin Higgs models are low-energy effective theories valid up to a cutoff scale of order $\Lambda \sim 4\pi f \sim 5-10$ TeV, beyond which a UV completion has to be specified. The simplest such possibility is to also make the Higgs boson composite, and to UV complete the twin Higgs model via gauge and top partners at masses of the order of a few TeV. A concrete implementation is the holographic twin Higgs model [438], which also incorporates a custodial symmetry to protect the T-parameter from large corrections. It is based on a warped extra dimensional theory with a bulk SO(8) gauge group, which incorporates the SU(4) global symmetry discussed above enlarged to contain the $SU(2)_L \times SU(2)_R$ custodial symmetry. In addition, the bulk contains either a full SU(7) group or an $SU(3) \times SU(3) \times U(1) \times U(1) \times Z_2$ subgroup of it to incorporate the QCD, its twin, and the hypercharge local symmetries. The breaking on the UV brane is to the SM symmetries and their twin symmetries, while on the IR brane $SO(8) \to SO(7)$, giving rise to the 7 Goldstone bosons, three of which will be again eaten by the twin W, Z. The main difference compared to ordinary composite Higgs models is that, in composite twin Higgs models, the cancellation of the one-loop quadratic divergences is achieved by the twin partners. They have a mass of order 700 GeV–1 TeV and they are uncharged under the SM gauge group. This allows the IR scale of the warped extra dimension to be raised to the multi-TeV range without reintroducing the hierarchy problem. The role of the composite partners is to UV complete the theory, rather than to cancel the one-loop quadratic divergences. For more details about the composite twin Higgs models, see Refs. [439].

### 11.6.7 Searches for signatures of extended Higgs sectors

The measurements described in Sections 11.3 to 11.5 have established the existence of one state of the electroweak symmetry breaking sector, compatible with a SM Higgs boson, but not that it is the only one. As was discussed above, several classes of models beyond the SM require extended Higgs sectors. The searches are typically designed to be as model-independent as possible<fnref n="7" /> and can be categorised in the classes summarised as follows:

(i) the search for an additional CP-even state mostly in the high mass domain decaying to vector bosons, which would correspond either to the heavy CP-even state in a generic 2HDM where the light state would be the discovered Higgs boson at 125 GeV or to a generic additional singlet;

(ii) the search for a state in the high mass domain decaying to pairs of fermions, which would correspond to the CP-odd A or the heavy CP-even state H in a generic 2HDM;

(iii) the search for charged Higgs bosons, which also appear in generic 2HDMs;

(iv) the search for a CP-odd state $a$ in the low mass region which appears in the NMSSM in a variety of final states, e.g., with one or two $a$ bosons decaying to pairs of photons, muons, taus, and b-quarks;

(v) the search for doubly charged Higgs bosons which are expected in extensions of the Higgs sector with triplets.

Below is a concise description of the most recent searches performed at the LHC and elsewhere. A summary of these searches in terms of final states is given in Table 11.11 where the corresponding references are given for more details.

#### 11.6.7.1 Searches for an additional CP-even state

##### (a) Exclusion limits from LEP

The searches for the SM Higgs boson at LEP provided an absolute lower limit of 114 GeV on its mass. These searches are also relevant for non-SM Higgs bosons. These searches were interpreted as 95% CL upper bounds on the ratio of the coupling $g_{HZZ}$ to its SM prediction as a function of the Higgs boson mass [139, 523]. These results have an impact on MSSM benchmarks such as the low-$m_H$ scenario where the heavy CP-even Higgs boson is the discovered 125 GeV boson [374], which is also nearly ruled out by

current direct constraints and charged Higgs boson limits from LHC. These results also impact scenarios of light CP-even Higgs boson of the NMSSM which are constrained to project predominantly onto the EW singlet component. Additional interest for these scenarios is due to the slight excess observed at LEP [139] at a Higgs boson mass hypothesis of approximately 98 GeV, bolstered by a possible excess in the diphoton channel at 96 GeV in Run-2 data of CMS [445].

##### (b) Searches at the LHC

At the LHC, the searches for the SM Higgs boson before the 2012 discovery covered a wide range of mass hypotheses up to approximately 1 TeV. After the discovery, the SM Higgs boson searches have been reappraised to search for a heavy CP-even state, extending progressively the search mass range beyond 1 TeV. This state could be the heavy CP-even Higgs boson of a 2HDM, or a generic additional singlet. In both cases, the natural width of the additional H state can be very different from that of the SM Higgs boson. To preserve unitarity of the longitudinal vector boson scattering and the longitudinal vector boson scattering into fermion pairs, the couplings of the additional CP-even Higgs boson to gauge bosons and fermions should not be too large and should constrain the natural width to be smaller than that of a unique Higgs boson at high mass with couplings to fermions and gauge bosons as predicted by the SM (and provided that tri-linear and quartic couplings are not too large and that no new state affects the heavy state total width). It is therefore reasonable to consider total widths for the high mass CP-even state smaller than the equivalent SM width. Two specific cases have been considered: (i) the SM width using the complex pole scheme (CPS), and (ii) the narrow width approximation. For the sake of generality, these searches are now done as a function of the Higgs boson mass and total width.

Searches for the Higgs boson in the channels $H \to \gamma\gamma$, $H \to Z\gamma$, $H \to WW^{(*)}$ leptonic and semi-leptonic, and in the $H \to ZZ^{(*)}$ searches in the $4\ell$, $\ell\bar{q}\bar{q}$ and $\ell\ell\nu\nu$ channels have also been done, but some of them are simple reinterpretations of the SM Higgs boson search in the CPS scheme. References for these searches are summarised in Table 11.11.

##### (c) Searches for an additional resonance decaying to a pair of Higgs bosons

In addition to the rare and expected Higgs boson pair production mode, high mass CP-even Higgs bosons can be searched for in the resonant double Higgs boson mode. Searches for such processes, where the Higgs boson is used as a tool for searches for BSM phenomena, have been carried out in a variety of distinct modes depending on the subsequent decays of each Higgs bosons. ATLAS and CMS have searched for the $H \to hh \to b\bar{b}\tau\tau$, $b\bar{b}\gamma\gamma$, $H \to hh \to 4b$, $H \to hh \to \gamma\gamma WW^*$, $H \to hh \to b\bar{b}WW^*$, $H \to hh \to WW^* WW^*$ and $H \to hh \to b\bar{b}ZZ^*$ final states. For mass hypotheses of an additional Higgs boson below 500 GeV, the two dominant search channels are the $b\bar{b}\gamma\gamma$ and the $b\bar{b}\tau\tau$ channels. For masses above 500 GeV, the most powerful search is with the $4b$ final state. As illustrated in Fig. 11.17, these searches provide useful limits in the low $\tan\beta$ and high mass domain. The list of references for these searches is given in Table 11.11.

##### (d) Searches for an additional state with the presence of the Higgs boson

In the post-discovery era, analyses searching for additional Higgs bosons need to take into account the presence of the 125 GeV Higgs boson. For searches with sufficiently high mass resolution to disentangle the additional states which are not degenerate in mass, the strength of the observed state and limits on the signal strength of a potential additional state can be set independently, as discussed in the next section. However, in some cases where channels do not have a sufficiently fine mass resolution to resolve states nearly degenerate in mass, specific analyses need to be designed. There are two examples of such analyses: (i) the search for an additional state in the $H \to WW^{(*)} \to \ell\nu\ell\nu$ channel in ATLAS, and (ii) the search for nearly degenerate states in the $H \to \gamma\gamma$ channel with the CMS detector.

<footnote n="7">⁷Still, most non-SUSY models are likely to include further states and dynamics above the weak scale to stabilise the scalar sector and this new and unknown physics may influence the searches described in this section in a way difficult to estimate.</footnote>
<table><caption>**Table 11.11:** Summary of references to the searches for additional states from extended Higgs sectors. (BBr) denotes the BaBar experiment and (TeV), the Tevatron experiments. V denotes either the W or the Z boson. Only Run-2 searches references are indicated except when searches have been carried out using Run-1 data only. References for Run-1 searches are available in Ref. [141].</caption><thead><tr><th></th><th>ATLAS</th><th>CMS</th><th>Other experiments</th></tr></thead><tbody><tr><td colspan="4"><strong>CP-even H</strong></td></tr><tr><td>H → γγ</td><td>[440, 441]</td><td>[442, 443]</td><td>—</td></tr><tr><td>H → γγ (low mass)</td><td>[444]</td><td>[445]</td><td>—</td></tr><tr><td>H → Zγ</td><td>[446, 447]</td><td>[448]</td><td>—</td></tr><tr><td>H → ZZ → 4ℓ</td><td>[449]</td><td>[450]</td><td>—</td></tr><tr><td>H → ZZ → ℓℓνν</td><td>[449]</td><td>[451]</td><td>—</td></tr><tr><td>H → WZ → ℓℓℓν</td><td>—</td><td>[452]</td><td>—</td></tr><tr><td>H → ZZ → ℓℓq<span style="text-decoration: overline">q</span></td><td>[453]</td><td>[454, 455]</td><td>—</td></tr><tr><td>H → ZZ → ννq<span style="text-decoration: overline">q</span></td><td>[456]</td><td>—</td><td>—</td></tr><tr><td>H → WW → ℓνℓν</td><td>[457]</td><td>[458]</td><td>—</td></tr><tr><td>H → WW → ℓνqq'</td><td>[459]</td><td>[458]</td><td>—</td></tr><tr><td>H → VV → qq'<span style="text-decoration: overline">q</span>q'(JJ)</td><td>[460, 461]</td><td>—</td><td>—</td></tr><tr><td>H → VV combination</td><td>[462]</td><td>—</td><td>—</td></tr><tr><td>H → hh → b<span style="text-decoration: overline">b</span>ττ, b<span style="text-decoration: overline">b</span>γγ, 4b, γγWW*, b<span style="text-decoration: overline">b</span>WW*, WW*WW*, b<span style="text-decoration: overline">b</span>ZZ*</td><td>[463–465]</td><td>[455, 466, 467]</td><td>—</td></tr><tr><td colspan="4"><strong>CP-odd A (and/or CP-even H)</strong></td></tr><tr><td>H, A → τ<sup>+</sup>τ<sup>−</sup></td><td>[468]</td><td>[469, 470]</td><td>[471, 472] (TeV)<br>[473] (LHCb)</td></tr><tr><td>A → τ<sup>+</sup>τ<sup>−</sup> (low mass)</td><td>—</td><td>[474]</td><td>—</td></tr><tr><td>H, A → μ<sup>+</sup>μ<sup>−</sup></td><td>[475]</td><td>[476]</td><td>—</td></tr><tr><td>H → μτ, eτ LFV</td><td>—</td><td>[477]</td><td>—</td></tr><tr><td>b<span style="text-decoration: overline">b</span>μ<sup>+</sup>μ<sup>−</sup> (low μ<sup>+</sup>μ<sup>−</sup> mass)</td><td>[478]</td><td>[479]</td><td>—</td></tr><tr><td>H, A → t<span style="text-decoration: overline">t</span></td><td>[461, 480]</td><td>[481]</td><td>—</td></tr><tr><td>H, A → b<span style="text-decoration: overline">b</span></td><td>[482]</td><td>[483]</td><td>[484, 485] (TeV)</td></tr><tr><td>A → hV → b<span style="text-decoration: overline">b</span>qq', b<span style="text-decoration: overline">b</span>ℓν, b<span style="text-decoration: overline">b</span>ℓℓ, ℓℓττ, νν<span style="text-decoration: overline">b</span>b</td><td>[486, 487]</td><td>[488, 489]</td><td>—</td></tr><tr><td>A → ZH</td><td>[490]</td><td>[491]</td><td>—</td></tr><tr><td>H → ZA → b<span style="text-decoration: overline">b</span>ℓ<sup>+</sup>ℓ<sup>−</sup></td><td>[213]</td><td>[492]</td><td>—</td></tr><tr><td colspan="4"><strong>Charged H<sup>±</sup></strong></td></tr><tr><td>H<sup>±</sup> → τ<sup>±</sup>ν</td><td>[493, 494]</td><td>[495]</td><td>—</td></tr><tr><td>H<sup>±</sup> → cs</td><td>[496]</td><td>[497]</td><td>—</td></tr><tr><td>H<sup>±</sup> → tb</td><td>[498]</td><td>[499]</td><td>—</td></tr><tr><td>H<sup>±</sup> → W<sup>±</sup>Z</td><td>[500]</td><td>[501, 502]</td><td>—</td></tr><tr><td>H<sup>±</sup> → W<sup>±</sup>A</td><td>—</td><td>[503]</td><td>—</td></tr><tr><td>H<sup>±</sup> → HW<sup>±</sup></td><td>—</td><td>[504]</td><td>—</td></tr><tr><td>H<sup>±</sup> → cb</td><td>—</td><td>[505]</td><td>—</td></tr><tr><td colspan="4"><strong>CP-odd a</strong></td></tr><tr><td>a → μ<sup>+</sup>μ<sup>−</sup></td><td>[506]</td><td>[507]</td><td>—</td></tr><tr><td>h → aa → 4μ, 4τ, 2μ2τ, 4γ</td><td>[508]</td><td>[509–512]</td><td>[513] (LEP)</td></tr><tr><td>[514] (TeV) h → aa → b<span style="text-decoration: overline">b</span>μμ, b<span style="text-decoration: overline">b</span>ττ, 4b</td><td>[515, 516]</td><td>[517, 518]</td><td>—</td></tr><tr><td>Υ<sub>1s,3s</sub> → aγ</td><td>—</td><td>—</td><td>[519, 520] (BBr)</td></tr><tr><td>Doubly charged H<sup>±±</sup></td><td>[521]</td><td>[502, 522]</td><td>—</td></tr></tbody></table>

In the $H \to WW^{(*)} \to \ell\nu\ell\nu$ channel, the search for an additional state is done using a boosted decision tree combining several discriminating kinematic characteristics to separate both the signal from the background and a high mass signal $H$ from the lower mass state $h$ [524]. A simultaneous fit of the two states $h$ and $H$ is then made to test the presence of an additional state. In this case, the usual null hypothesis of background includes the SM signal.

The CMS search for nearly degenerate mass states decaying to a pair of photons [525] is more generic and could for instance apply to CP-odd Higgs bosons as well. It consists of a fit to the diphoton mass spectrum using two nearly degenerate mass templates.

### (e) Type I 2HDM and fermiophobia

The measurements of coupling properties of the 125 GeV Higgs boson directly establish its couplings to fermions. However, the presence of an additional fermiophobic state, as predicted by Type I 2HDMs, is not excluded. Prior to the discovery, ATLAS and CMS have performed searches for a fermiophobic Higgs boson, i.e., produced through couplings with vector bosons only (VBF and VH) and decaying in two photons. CMS has further combined these results with searches in the $W^+W^-$ and ZZ channels, assuming fermiophobic production and decay. This way,

CMS excluded a fermiophobic Higgs boson in the range 110 GeV < $m_H$ < 188 GeV at the 95% CL. References for these Run-1 measurements can be found in Ref. [141].

### 11.6.7.2 Searches for additional neutral states ($\phi \equiv h, H, A$) decaying to fermions

#### (a) Exclusion limits from LEP

In $e^+e^-$ collisions, around the centre-of-mass energies reached by LEP, the main production mechanisms of the neutral MSSM Higgs bosons were the Higgs-strahlung processes $e^+e^- \to hZ$, $HZ$ and the pair production processes $e^+e^- \to hA, HA$, while the vector boson fusion processes played a marginal role. Higgs boson decays to $b\bar{b}$ and $\tau^+\tau^-$ were used in these searches.

The searches and limits from the four LEP experiments are described in Refs. [526]. The combined LEP data did not contain any excess of events which would imply the production of a Higgs boson. Combined limits were derived [523]. For $m_A \gg M_Z$, the limit on $m_h$ is nearly that of the SM searches, as $\sin^2(\beta - \alpha) \approx 1$. For high values of $\tan\beta$ and low $m_A$ ($m_A \le m_h^{max}$), the $e^+e^- \to hA$ searches become the most important, and the lightest Higgs boson $h$ is non SM-like. In this region, the 95% CL mass bounds are $m_h > 92.8$ GeV and $m_A > 93.4$ GeV. In the $m_h^{max}$
scenario [527], values of tan $\beta$ from 0.7 to 2.0 are excluded taking $m_t = 174.3$ GeV, while a much larger tan $\beta$ region is excluded for other benchmark scenarios such as the no-mixing one.

A flavour-independent limit for Higgs bosons in the Higgs-strahlung process at LEP has also been set at 112 GeV [528].

Neutral Higgs bosons may also be produced by Yukawa processes $e^+e^- \to f\bar{f}\phi$, where the Higgs particle $\phi \equiv h, H, A$, is radiated off a massive fermion ($f \equiv b$ or $\tau^\pm$). These processes can be dominant at low masses, and whenever the $e^+e^- \to hZ$ and $hA$ processes are suppressed. The corresponding ratios of the $f\bar{f}h$ and $f\bar{f}A$ couplings to the SM coupling are $-\sin\alpha/\cos\beta$ and $\tan\beta$, respectively. The LEP data have been used to search for $b\bar{b}b\bar{b}, b\bar{b}\tau^+\tau^-$, and $\tau^+\tau^- \tau^+\tau^-$ final states [529]. Regions of low mass and high enhancement factors are excluded by these searches.

The searches for the Higgs boson at LEP also included the case where it does not predominantly decay to a pair of b quarks. All four collaborations conducted dedicated searches for the Higgs boson with reduced model dependence, assuming it is produced via the Higgs-strahlung process, and not addressing its flavour of decay, a lower limit on the Higgs boson mass of 112.9 GeV is set by combining the data of all four experiments [528].

Using an effective Lagrangian approach and combining results sensitive to the $h\gamma\gamma, hZ\gamma$ and $hZZ$ couplings, an interpretation of several searches for the Higgs boson was made and set a lower limit of 106.7 GeV on the mass of a Higgs boson that can couple anomalously to photons [528].

(b) Searches at the Tevatron and the LHC

The best sensitivity is in the regime with low to moderate $m_A$ and with large $\tan\beta$ which enhances the couplings of the Higgs bosons to down-type fermions. The corresponding limits on the Higgs boson production cross section times the branching ratio of the Higgs boson into down-type fermions can be interpreted in MSSM benchmark scenarios [530]. The most promising channels at the Tevatron are the inclusive $p\bar{p} \to \phi \to \tau^+\tau^-$ process ($\phi = A, H, h$), with contributions from both $gg \to \phi$ and $b\bar{b}\phi$ production, and $b\bar{b}\phi, \phi \to \tau^+\tau^-$ or $\phi \to b\bar{b}$, with $b\tau\tau$ or three tagged b-jets in the final state, respectively. Although the Higgs boson production via gluon fusion has a higher cross section in general than via associated production, it cannot be used to study the $\phi \to b\bar{b}$ decay mode since the signal is overwhelmed by the QCD background.

CDF and D0 have searched for neutral Higgs bosons produced in association with bottom quarks and which decay into $b\bar{b}$ [484, 485], or into $\tau^+\tau^-$ [471, 472]. The most recent searches in the $b\bar{b}\phi$ channel with $\phi \to b\bar{b}$ analyse approximately 2.6 fb$^{-1}$ (CDF) and 5.2 fb$^{-1}$ (D0) of data, seeking events with at least three b-tagged jets. The cross section is defined such that at least one b quark not from $\phi$ decay is required to have $p_T > 20$ GeV and $|\eta| < 5$. The invariant mass of the two leading jets as well as b-tagging variables are used to discriminate the signal from the backgrounds. The QCD background rates and shapes are inferred from data control samples, in particular, the sample with two b-tagged jets and a third, untagged jet. Separate-signal hypotheses are tested and limits are placed on $\sigma(pp \to b\bar{b}\phi) \times \text{BR}(\phi \to b\bar{b})$. A local excess of approximately 2.5$\sigma$ significance has been observed in the mass range of 130–160 GeV, but D0’s search is more sensitive and sets stronger limits. The D0 result had a $\mathcal{O}(2\sigma)$ local upward fluctuation in the 110 to 125 GeV mass range. These results have been superseded by the LHC searches and the excess seen by D0 has not been confirmed elsewhere.

A substantially larger sensitivity in the search for the $\phi \to \tau^+\tau^-$ is obtained with the ATLAS and CMS analyses. The higher centre-of-mass energy reached at the Run 2 brings a substantial, though not excessively large, increase in sensitivity due to the intermediate masses probed. Both ATLAS and CMS have reported the result of their searches in this important channel with the full 2016 dataset. The searches are performed in categories of the decays of the two tau leptons: $e\tau_{\text{had}}, \mu\tau_{\text{had}}, e\mu$, and $\mu\mu$, where $\tau_{\text{had}}$ denotes a tau lepton which decays to one or more hadrons plus a tau neutrino, $e$ denotes $\tau \to e\nu\nu$, and $\mu$ denotes $\tau \to \mu\nu\nu$. The dominant background comes from $Z \to \tau^+\tau^-$ decays, although $t\bar{t}$,

$W$+jets and $Z$+jets events contribute as well. Separating events into categories based on the number of b-tagged jets improves the sensitivity in the MSSM. The $b\bar{b}$ annihilation process and radiation of a Higgs boson from a b quark gives rise to events in which the Higgs boson is accompanied by a $b\bar{b}$ pair in the final state. Requiring the presence of one or more b-jets reduces the background from $Z$+jets. Data control samples are used to constrain background rates. The rates for jets to be identified as a hadronically decaying tau lepton are measured in dijet samples, and $W$+jets samples provide a measurement of the rate of events that, with a fake hadronic tau, can pass the signal selection requirements. Lepton fake rates are measured using samples of isolated lepton candidates and same-sign lepton candidates. Constraints from the ATLAS searches are shown in Fig. 11.17 (left) in the hMSSM approximation defined in Ref. [376]. The neutral Higgs boson searches consider the contributions of both the CP-odd and CP-even neutral Higgs bosons with enhanced couplings to bottom quarks, similarly to what was done for the Tevatron results. In Fig. 11.17, decays of the charged Higgs boson into $\tau\nu$ and decays of the heavy Higgs boson into a pair of SM-like Higgs bosons or gauge bosons, or decays of A into $hZ$ are also being constrained. In addition, decays of the neutral Higgs bosons into muon pairs are also being explored. In the $m_h^{\text{mod}+}$ scenario, the region of $\tan\beta$ lower than 5 does not allow for a Higgs boson mass $m_h$ close to 125 GeV. For the hMSSM scenario, instead, the SM-like Higgs boson mass is fixed as an input and hence the requirement that it is close to 125 GeV is always fulfilled, although this may imply other limitations as discussed in Section 11.6.1.1.

A search for $\phi \to \mu^+\mu^-$ has also been performed by ATLAS [475] and CMS [476] and no significant deviations from the SM expectation were observed, allowing to set model-independent limit on the product of the branching fraction for the decay into a muon pair and the cross section for the production of a scalar neutral boson, either via gluon fusion, or in association with b quarks, in the mass range from 130 to 1000 GeV.

Finally, searches for a resonance decaying to a top quark pair were done by ATLAS [480, 531] and CMS [481, 532]. These searches were interpreted as searches for scalar resonances by ATLAS [480], however, an important component of these searches is an accurate treatment of the interference effects between the signal and the continuum background. These effects can yield a dip and peak structure instead of a simple peak [372]. ATLAS has performed a search for a high mass state decaying to a pair of top quarks taking into account the deformation in mass shape of the signal in the presence of the continuum background [533].

The LHC has the potential to explore a broad range of SUSY parameter space through the search for non-SM-like Higgs bosons. As illustrated in Fig. 11.17, the parameter space corresponding to large $\tan\beta$ values and large masses of the A boson are covered mostly by the searches in the $A, H \to \tau^+\tau^-$ channel. A projection of the combined sensitivity of ATLAS and CMS at the HL-LHC has been performed in Ref. [118], showing that, compared to the current sensitivity, the full HL-LHC luminosity can expand the exclusion domain by nearly 1 TeV. In the low $\tan\beta$ limit, the parameter space spanning large A boson masses is best excluded indirectly from the observed Higgs boson measurements. This is illustrated in the $M_h^{125}$ scenario by the nearly horizontal exclusion which is due to the compatibility of the Higgs boson mass measurement with its prediction from radiative corrections (mostly from the stop sector). Nevertheless, Fig. 11.17 (right) shows a broad region with intermediate $\tan\beta$ and large values of $m_A$ that is not accessed by current searches, and in which the most promising channel is the very difficult search for $t\bar{t}$ decays with its aforementioned intricacies [379]. In this region of parameter space, it is possible that only the SM-like Higgs boson can be within the LHC’s reach. If no other state of the EWSB sector than the 125 GeV state is discovered, it may be challenging to determine only from the Higgs sector whether there is a SUSY extension of the SM in nature.

11.6.7.3 Searches for a CP-odd state decaying to hZ

Similarly, to the search for a CP-even high mass Higgs boson decaying to a pair of Higgs bosons, the search for a CP-odd states decaying to $hZ$ was carried out at the LHC by ATLAS and CMS
<figure><img src="image_45.png" /><figcaption>**Figure 11.17:** The 95% CL exclusion contours in the $(M_A, \tan\beta)$ parameter space for: (left) a summary of CMS Run-2 searches in the hMSSM and (right) in the $M_h^{125}$ scenario.</figcaption></figure>

in various channels: (i) $(Z \to \ell\ell)(h \to b\bar{b})$, (ii) $(Z \to \nu\nu)(h \to b\bar{b})$, (iii) $(Z \to \ell\ell)(h \to \tau\tau)$, and (iv) $(Z \to \ell\ell)(h \to \tau\tau)$. The searches where the A boson decays to a pair of b quarks have been performed both in the regime where both b-jets are resolved and in the boosted regime where the two b-jets are merged in a single larger radius jet. These searches have been used to constrain the parameter space of 2HDMs. In the MSSM, these searches place limits on small values of $\tan\beta$ for masses of A between 220 GeV and 360 GeV.

### 11.6.7.4 Searches for low mass states

Searches for pseudo-scalar Higgs boson at intermediate to low masses, below the Z mass (in the 25 GeV to 80 GeV mass range) have been performed by CMS both in the $\tau^+\tau^-$ [534] and the $\mu^+\mu^-$ [535] decay channels. A light pseudo-scalar in this mass range is excluded by current direct constraints in the MSSM but not in general 2HDMs [536]. These searches are done in the decay channels where the pseudo-scalar Higgs boson is produced in association with a pair of b-quarks and decays into a pair of taus or muons.

CMS has also reported an anomaly observed in the search for $\mu^+\mu^-$ resonances produced with one jet tagged as containing a b-hadron and a forward jet in the Run-1 data. A mild excess appeared in the di-muon mass distribution at approximately 28 GeV. Another very mild excess was then also found in the 2016 Run-2 data [479]. ATLAS then performed a similar analysis with the full Run-2 dataset corresponding to an integrated luminosity of approximately 139 fb⁻¹, and no significant excess was found [478].

Searches for low mass Higgs bosons were also performed in the diphoton channel by both ATLAS and CMS [537, 538] at Run 1. ATLAS and CMS have updated the results of this search with Run-2 data [440, 442]. A modest excess has been observed by CMS at a mass of 95.4 GeV with a local significance of 2.9σ (the corresponding global significance is 1.3σ). A slight excess was also seen by ATLAS at the same mass of 95.4 GeV with a local significance of 1.7σ. No significant excess has been observed in this region by ATLAS at Run 1. This mildly significant excess also coincides in mass with the excess observed at LEP and discussed in Section 11.6.7.1. It is interesting to note that the sensitivity of searches at masses very near that of the Z boson, require a very strong rejection of events where electrons are misidentified as photons. Dedicated experimental methods have been developed to reduce the vulnerability of the analysis in this specific mass region to this background. It has therefore raised interest and speculations on its possible nature, see for instance Ref. [539] and references therein.

### 11.6.7.5 Searches for charged Higgs bosons $H^{\pm}$

At $e^+e^-$ colliders, charged Higgs bosons can be pair produced in the s-channel via $\gamma$ or Z boson exchange. This process is dominant in the LEP centre-of-mass energies range, i.e., up to 209 GeV. At higher centre-of-mass energies, other processes can play an important role such as the production in top quark decays via $t \to b + H^+$ if $m_{H^\pm} < m_t - m_b$ or via the one-loop process $e^+e^- \to W^\pm H^\pm$ [540, 541], which allows the production of a charged Higgs boson with $m_{H^\pm} > \sqrt{s}/2$, even when $H^+H^-$ production is kinematically forbidden. Other single charged Higgs boson production mechanisms include $t\bar{b}H^-/t\bar{b}H^+$ production [123], $\tau^+\nu H^-/\tau^- \bar{\nu} H^+$ production [542], and a variety of processes in which $H^\pm$ is produced in association with a one or two other gauge and/or Higgs bosons [543].

At hadron colliders, charged Higgs bosons can be produced in several different modes depending on the value of its mass with respect to the top-quark mass. For light values of the charged Higgs boson mass, defined by Higgs boson masses smaller than the mass of the top quark (with experimental analyses typically considering masses up to $m_{H^\pm} \le 160$ GeV), the top-quark decay $t \to Hb$ is allowed and the charged Higgs boson is light enough so that top-quark off-shell effects can be neglected. The cross section for the production of a light charged Higgs boson is simply given by the product of the top-pair production cross section and the branching ratio of a top quark into a charged Higgs boson. The top-pair production cross section is known up to NNLO in perturbative QCD [544], and relevant QCD and SUSY-QCD including NLO corrections to the branching ratio for $t \to H^+b$ have been computed in the literature, see Refs. [545–547] and references therein. At present, the theoretical accuracy for the production of a light charged Higgs boson is at the few percent level. For the intermediate mass range, values of $m_{H^\pm}$ near $m_t$, the finite top-width effects as well as the interplay between top-quark resonant and non-resonant diagrams cannot be neglected. Hence, the full process $pp \to H^\pm W^\mp b\bar{b}$ (with massive b-quarks) must be considered to perform a reliable perturbative calculation of the charged Higgs boson production cross section [547]. For heavy charged Higgs boson scenarios, with charged Higgs boson masses larger than the top-quark mass (typically above 180 GeV), the dominant charged Higgs boson production channel is the associated production with a top quark/antiquark and a (possibly low transverse momentum) bottom antiquark/quark. Theoretical calculation at NLO have been computed both at the inclusive and fully-differential level in the five-flavour scheme and in the four-flavour scheme, see Ref. [40] and references therein. Charged Higgs bosons can also be produced via associated production with $W^\pm$ bosons through $b\bar{b}$ annihilation and $gg$-fusion annihilation [548].

For charged Higgs boson production cross section predictions
for the Tevatron and the LHC, see Refs. [38, 39, 370].

(a) *Exclusion limits from LEP*

Charged Higgs bosons have been searched for at LEP, where the combined data of the four experiments, ALEPH, DELPHI, L3, and OPAL, were sensitive to masses of up to about 90 GeV [523] in two decay channels, $\tau\nu$ and $c\bar{s}$. The combined LEP data exclude, at 95% CL, charged Higgs bosons with mass below 80 GeV (Type II scenario) or 72.5 GeV (Type I scenario) [549].

(b) *Exclusion limits from Tevatron*

Compared to the mass domain covered by LEP searches, the Tevatron covered a complementary range of charged Higgs boson masses. CDF and D0 have also searched for charged Higgs bosons in top quark decays with subsequent decays to $\tau\nu$ or to $c\bar{s}$ [550]. For the $H^+ \to c\bar{s}$ channel, the limits on BR($t \to H^+b$) from CDF and D0 are $\approx 20\%$ in the mass range $90 \text{ GeV} < m_{H^+} < 160 \text{ GeV}$ and assuming a branching fraction of 100% in this specific final state. $H^+ \to \tau^+\nu_\tau$ channel, D0's limits on BR($t \to H^+b$) are also $\approx 20\%$ in the same mass range and assuming a branching fraction of 100% in this final state. These limits are valid in general 2HDMs, and they have also been interpreted in terms of the MSSM [550].

(c) *Exclusion limits from LHC*

Similarly, to the Tevatron, at the LHC, light charged Higgs bosons can be searched for in the decays of top quarks. The main initial production mode for light charged Higgs bosons ($m_{H^+} < m_t - m_b$) is top pair production. The subsequent decay modes of the charged Higgs boson for these searches are $\tau\nu$ and $c\bar{s}$. More recently, ATLAS and CMS have also searched for higher mass charged Higgs bosons ($m_{H^+} > m_t + m_b$) in $H^+ \to t\bar{b}$. The main production modes are the associated production of a charged Higgs boson in association with a top and a bottom quark or in association with a top quark only.

The decay $H^+ \to \tau^+\nu_\tau$ is searched typically in three final state topologies:

(i) lepton+jets: with $t\bar{t} \to b\bar{W}H^+ \to b\bar{b}(q\bar{q}')(\tau_{lep}\nu)$, i.e., the W boson decays hadronically and the tau decays into an electron or a muon, with two neutrinos;

(ii) $\tau$ + lepton: with $t\bar{t} \to b\bar{W}H^+ \to b\bar{b}(l\nu)(\tau_{had}\nu)$, i.e., the W boson decays leptonically (with $l = e, \mu$) and the tau decays hadronically;

(iii) $\tau$+jets: $t\bar{t} \to b\bar{W}H^+ \to b\bar{b}(q\bar{q}')(\tau_{had}\nu)$, i.e., both the W boson and the $\tau$ decay hadronically.

CMS has also searched for the charged Higgs boson in the decay products of top quark pairs: $t\bar{t} \to H^\pm W^\mp b\bar{b}$ and $t\bar{t} \to H^+ H^- b\bar{b}$ as well. Three types of final states with large missing transverse energy and jets originating from b-quark hadronisation have been analysed: the fully-hadronic channel with a hadronically decaying tau in association with jets, the dilepton channel with a hadronically decaying tau in association with an electron or muon and the dilepton channel with an electron-muon pair. The results of the searches at the LHC are illustrated in Figure 11.17.

Both ATLAS and CMS have also searched for high mass charged Higgs bosons decaying to a top and bottom quarks. The main production mode for this search is the associated production with one top quark (5-flavour scheme) or a top quark and bottom quark (4-flavour scheme) in the final state. The s-channel production mode where the charged Higgs boson is produced alone in the final state at tree level is also considered. This search is particularly intricate and it is sensitive to the modelling of the top pair production background produced in association with additional partons and in particular b-quarks. No excess was found and the results are expressed in terms of exclusion limits of cross section times branching fractions.

ATLAS and CMS have also searched for charged Higgs bosons in top quark decays assuming BR($H^+ \to c\bar{s}$) = 100% [496, 551], and sets limits of $\approx 20\%$ on BR($t \to H^+b$) in the $90 \text{ GeV} < m_{H^+} < 160 \text{ GeV}$ mass range.

In 2HDMs, the decay of the charged Higgs boson to a W and a Z boson is allowed only at loop level and is therefore suppressed.

However the $H^\pm \to W^\pm Z$ decay channel is allowed in Higgs triplet models. ATLAS [500] has searched for such decays, requiring that the charged Higgs boson is produced through the fusion of vector bosons. No excess with respect to the SM backgrounds has been observed in this channel, and the results are interpreted in the Georgi–Machacek model [414] discussed in Section 11.6.5.2.

At the LHC, various other channels still remain to be explored, in particular searches involving additional neutral scalars in particular in the WH, WA channels (A is the pseudo-scalar MSSM Higgs boson), and in the Wa channel (a is the light CP-odd scalars of the NMSSM).

### 11.6.7.6 Interpretation of the measurements of the coupling properties of the Higgs boson

The 125 GeV Higgs boson being part of any hypothetically extended EWSB sector, it can be used through the compatibility of its measured couplings and mass with those predicted in specific models to provide constraints on these specific models parameters.

As discussed in Section 11.6.1.1, the mass of the Higgs boson limits drastically the MSSM parameter space and can be used to set limits on specific MSSM benchmarks. This is the case for the Mh125 scenario as illustrated in Fig. 11.16 and in Fig. 11.17, corresponding approximately to a lower limit on $\tan\beta$ in this model [118].

The measurements of the Higgs boson couplings, discussed in Section 11.5, can be interpreted in the framework of a constrained model where the couplings of the Higgs boson to vector bosons, up-type quarks, down-type quarks and leptons, are varied. In 2HDMs, these couplings are functions of the mixing angle $\alpha$ between the observed Higgs boson and the heavy CP-even neutral scalar, and of the ratio of the vacuum expectation values of the two doublets, $\tan\beta$. In the case of the MSSM, the two parameters are the A boson mass and $\tan\beta$ (the sole two parameters needed to describe the MSSM Higgs sector at tree level). The coupling measurements have been interpreted both by ATLAS [312] and CMS [552] in specific MSSM benchmarks and in 2HDMs. The exclusion contour in the hMSSM for the ATLAS combination [312] is illustrated in Fig. 11.17.

### 11.6.7.7 Searches for a light CP-odd Higgs boson

A light pseudo-scalar boson a is present in any two Higgs doublet model enhanced with an additional singlet field. A prominent example is the NMSSM. The theoretical motivations for singlet extensions of the MSSM are discussed in Section 11.6.2. There is also a variety of other models with light additional spin-0 bosons such as two Higgs doublet models with a scalar, Little Higgs models or light scalar mediator to a dark sector.

In the framework of the NMSSM, the searches now focus on the low a mass region for several reasons:

(i) in the NMSSM, the light pseudo-scalar a boson can, as a pseudo-Goldstone boson, be a natural candidate for an axion;

(ii) scenarios where $m_a > 2m_b$ and a CP-even state h decaying to a pair of a ($m_h > 2m_a$) are excluded by direct searches at LEP in the four b's channel [514, 523, 553];

(iii) in the pre-discovery era, LEP limits on a CP-even Higgs boson resulted in fine-tuning MSSM constraints [554] which could be evaded through non standard decays of the Higgs boson to aa;

(iv) in the NMSSM, a CP-odd a boson with a mass in the range 9.2–12 GeV can also account for the difference observed between the measured anomalous muon magnetic moment and its prediction [555].

The benchmark scenarios have also changed in the light of the Higgs boson discovery. The 125 GeV state could be the lightest or the next-to-lightest of the three CP-even states of the NMSSM. Light pseudo-scalar scenarios are still very interesting in particular for the potential axion candidate. There are three main types of direct searches for the light a boson:

(i) for masses below the $\Upsilon$ resonance, the search is for radiative decays $\Upsilon \to a\gamma$ at B-factories;

(ii) the inclusive search in high energy pp collisions at the LHC;