Probing the Higgs boson through Yukawa force

The ATLAS and CMS collaborations of the LHC have observed that the Higgs boson decays into the bottom quark-antiquark pair, and have also established that the Higgs coupling with the top quark-antiquark pair is instrumental in one of the modes for Higgs production. This underlines the discovery of the Yukawa force at the LHC. We demonstrate the impact of this discovery on the Higgs properties that are related to the dynamics of electroweak symmetry breaking. We show that these measurements have considerably squeezed the allowed window for new physics contributing to the Higgs couplings with the weak gauge bosons and the third generation quarks. The expected constraints at the HL-LHC are also shown. We project these constraints on the parameter space of a few motivated scenarios beyond the Standard Model. We picked them under two broad categories, namely, the composite Higgs and its RS dual, as well as various types of multi-Higgs models. The latter category includes models with singlet scalars, Type I, II and BGL-type two-Higgs doublet models, and models with scalar triplets a la Georgi and Machacek.


Introduction
Since the discovery of the Higgs boson in 2012 at the CERN Large Hadron Collider (LHC) [1,2], one of the most notable achievements by the ATLAS and CMS Collaborations has been the measurements of the Yukawa force between the Higgs boson (h) and the third generation quarks (t and b). Although Yukawa interaction was postulated long back in the context of the pion-nucleon scattering, advent of Quantum Chromodynamics showed that it is but an artefact of the strong gauge force. Do the present measurements of hbb [3,4] and htt [5,6] couplings constitute a discovery of a fundamental Yukawa force, or, it is again a low energy manifestation of some unknown UV dynamics? Even if the Higgs boson is an elementary object, is it the only neutral scalar that Nature offered us? Precision measurements of these Yukawa couplings can shed important light on both these questions. In the Standard Model (SM), the Yukawa couplings are precisely known in terms of the fermion masses. Any departure would indicate physics beyond the SM (BSM) triggering electroweak symmetry breaking [7,8]. In this paper, we review the status of some BSM physics in the light of the LHC data armed with the new measurements of the Yukawa forces. Since the flavor changing couplings of the 125 GeV Higgs boson are already too constrained, increasingly precise measurements of the flavor diagonal couplings at the LHC are essential to probe the Yukawa structure. In order to quantify the BSM window we employ a χ 2 -analysis using the Higgs signal strength data from the ATLAS and CMS Collaborations. The Run 2 data [9, 10] with improved measurements of the hbb and htt couplings, compared to what Run 1 could achieve [11][12][13][14][15][16][17], penetrate rather deep into the BSM parameter space, leading to new constraints. To show the future prospects, we also give projections for these measurements at the high luminosity runs of the LHC (HL-LHC) [18]. For our purpose, we employ a simple model independent phenomenological Lagrangian upto two-derivative order, which essentially captures modification of the Higgs couplings [18][19][20]. We also translate the limits of our model independent parameter space to In the present analysis, we use a simple model independent phenomenological Lagrangian, in the broken phase of electroweak symmetry, which captures the modifications of the Higgs couplings arising from both the above sources [18][19][20]. We expand the terms in the Lagrangian in powers of h as well as in the number of derivatives. Since our primary interest lies in the production and decay of a single Higgs boson, we will only keep terms up to a single insertion of h. The Lagrangian involving the SM fields after the electroweak symmetry breaking, up to two-derivative terms is given below: where the lowest order in derivative L (0) is given as The two-derivative terms, which may arise by integrating out the BSM states, are given by The coefficients c i are free parameters capturing the impact of BSM physics, and to be constrained by the experimental data. In the SM, c V = c f = 1 and c γγ = c Zγ = c gg = 0. We also assume those coefficients to be real, i.e. we assume the 125 GeV Higgs boson to be CP even. Implications of CP odd Higgs couplings have been discussed in [32,33]. The Higgs production cross sections and decay widths, normalized to their SM values, can be expressed solely in terms of these coefficients. Throughout this paper, we fix c Zγ = 0, since the h → Zγ data is too constrained from the electroweak precision observables and, not unexpectedly, is still unobserved at the LHC [34,35]. We will also assume that c τ = c b and c c = c s = c t , to simplify the analysis.

Analyzing the LHC data
The Higgs signal strength (µ) of a specific process i → h → f is conventionally defined as where σ i , Γ f and B f denote the cross section of the i th production mode of the Higgs boson, the partial decay width of the Higgs into a final state f , and the corresponding branching ratio, respectively. In the total decay width of the Higgs, Γ h , we shall generally assume that the Higgs can decay only to the SM particles. Towards the end, however, we shall comment on the possibility of the Higgs boson having a non-vanishing branching fraction to invisible decay modes. In terms of the 'κ-framework' [36,37], we can express the cross-sections and decay widths normalized to their SM values as The mapping between the κ-framework and the coefficients c i can be found in [38]. To put limits on c i 's, we define a χ 2 -function using the individual signal strengths. We use ATLAS Run 2 data with 80 f b −1 luminosity [9] and CMS Run 2 data with 137 f b −1 luminosity [10]. For the purpose of comparison, we also show the results obtained from the combined ATLAS and CMS Run 1 data [11]. Note that, in case of the CMS Run 2 data, due to the unavailability of the full correlation matrix, we have assumed that the Higgs signal strength measurements are all independent. As for the HL-LHC projections, with luminosity 3000 f b −1 , we use the SM predictions as central values, and the uncertainties are obtained from [18]. Some crucial observations regarding the present data are the following. First, the processes involving tth production mode have been measured with unprecedented precision at Run 2. Similarly, the errors for the hbb decay channels have got significantly reduced, in particular in the associated Higgs production channel. Besides, gg → h → γγ and gg → h → ZZ * processes, which were already measured with less than 30% errors in the Run 1 phase, now stand better with around 15% errors after the Run 2 data were analyzed.
We also define new variables by normalizing all the signal strengths by that of the gold-plated gg → h → ZZ * process, measured with maximum precision. This way the inherent uncertainties in the total decay width of the Higgs coming from possible invisible modes get eliminated. The other advantage is that, if we assume only the SM particles are running inside the loops for processes like gg → h and h → γγ, all the ratios can be expressed in terms of only two variables, viz. c t /c V and c b /c V . Then the constraints from the Higgs signal strength measurements can be represented in a two-dimensional Admittedly, even if the Γ h dependence is eliminated in this approach, the errors and correlations among the ratios of signal strengths get slightly jacked up compared to the approach where we have analyzed individual signal strengths.

Results
It has been shown in [19,39,40] that the LEP data admit around 10% − 20% deviation in c V from its SM value at 95% CL. In the present analysis we have observed that the present Higgs signal strength data provide competitive, if not better, limits on c V .
The parameter c t receives major constraints from the gluon fusion and tth production modes of the Higgs boson as well as from its diphoton decay channel. On the other hand, constraints on c b primarily     arise from the h → bb decay (58% branching ratio). Moreover, since we have assumed c b = c τ in our analysis, data from the h → τ + τ − channel also contribute to the limits on c b . We show in the left panel of Fig. 1 the allowed region in the c b /c V − c t /c V plane, obtained using the ratios of the signal strengths (µ f i /µ ZZ * gg ). The clear improvement from Run 1 to Run 2 is a direct consequence of more precise measurements of htt and hbb couplings.
In the right panel of Fig. 1, we use the conventional approach of using the individual signal strengths to extract the limits. Here we assume c t = c b = c τ = c f to show the allowed region in the c f −c V plane. In Table 1 we display the allowed ranges of parameters at 95% CL. Two major points are worth noting here. First, the limits on c V from the Run 2 data are already competitive to those obtained from the electroweak precision tests. This happened primarily due to the increasingly precise measurements of the gg → h → ZZ * and gg → h → W W * processes. Second, the window for new physics through has significantly narrowed down, only 10% − 15% deviation is allowed from the SM reference point. This improvement in Yukawa force measurement helps discriminate various BSM scenarios. We note that the combined Run 1 + Run 2 data improve the limits obtained from Run 2 data alone by at most 2% -3%. In obtaining the above constraints we have assumed c gg = c γγ = 0. The inherent assumption is that any new BSM particle(s) which might have contributed to the triangle loops creating the effective ggh and hγγ vertices are sufficiently heavy and decoupled.
Then we go to the other extreme. Keeping c b = c t = c τ = c V = 1, we display the limits in c gg − c γγ plane in Fig. 2. Here we capture the effects of the new BSM particles floating in the triangle loops, e.g. if the SM is extended with additional colored and electrically charged particles. The solid lines represent the contributions from colored particles, transforming as triplets of SU(3) c and having electric charges Q = 1/3 (cyan), Q = 2/3 (purple) and Q = 5/3 (brown), respectively. The exact location of a model-point on each straight line, however, depends on the mass and model-dependent couplings of the new particles with the Higgs boson [19].
If a non-vanishing branching fraction for the invisible decay mode (Br inv ) of the Higgs boson is admitted, all the individual signal strengths of the Higgs boson receive a scaling by an overall factor of (1 − Br inv ). Assuming c f = c V = 1 and c gg = c γγ = 0, we observe that the Run 1 (Run 2) data exclude Br inv 18% (7%), while the HL-LHC would exclude Br inv 3%. Admittedly, these limits will relax considerably, if deviations in c i parameters are allowed (e.g. the Particle Data Group excludes a rather conservative Br inv 24% [41]).

Composite Higgs models
In generic composite Higgs scenario, the modification in the hV V coupling is universal [42,43] c V = 1 − ξ , (4.1) where ξ = v 2 /f 2 parametrizes the hierarchy between the electroweak scale and the composite scale f . The Yukawa couplings, however, depend on the details of the particular model. In the minimal composite Higgs model, with coset SO(5)/SO(4) [44][45][46], the Yukawa coupling modifiers are controlled by the specific representations of SO (5) in which the SM quarks and leptons are embedded. A generic parametrization for c f in such cases can be given as Here we discuss three specific cases for which we have obtained new limits: : This is an oft-quoted example when both the left-and right-chiral top quark are kept in 5 of SO (5). In this case, the χ 2 -function depends on a single parameter ξ. We obtain f 1.2 TeV at 95% CL using the Run 2 data, while in HL-LHC we expect f 1.8 TeV.
• ∆ b = ∆ τ = −3/2: Here, we keep ∆ t as a free parameter, which implies either the left-or the right-handed top quark is embedded in 14 of SO(5). The allowed region at 95% CL in the ∆ t − ξ plane is shown in the left panel of Fig. 3. Clearly, the constraint on f gets relaxed, as alluded in [58]. For a generic value of ∆ t , we obtain the most conservative limit f 660 GeV after inclusion of the Run 2 data.
• ξ = constant: We fix two representative values of ξ = 0.1 and 0.06, to put simultaneous limits in ∆ t − ∆ b plane as shown in the right panel of Fig. 3. We observe that while the present data have not yet gathered enough strength to discriminate between the choices of representations in which the top and bottom quarks are embedded, future measurements with better statistical significance can do the job.
We have kept c gg = c γγ = 0. This is motivated by the observation that in the composite pseudo-Goldstone Higgs scenario the top partner loop contribution cancels against the contribution of the anomalous dimension of the top quark [53,59].
Composite Higgs models are often seen as dual to some variants of the weakly coupled warped extra dimensional models using the AdS / CFT correspondence [60]. We take a custodial Randall-Sundrum (RS) setup with the Higgs boson localized near the IR brane to study the constraints on the scale of the Kaluza-Klein states (M KK ) [61][62][63][64][65]. Adapting the expressions for the Higgs coupling modifiers from [65], including the Run 2 data, we obtain a conservative lower limit on the mass of the first excited KK-gluon, M g 9 TeV (which translates into M KK 3.7 TeV). The projected limit from HL-LHC is M g 12 TeV.

Multi-Higgs models
Here we deal with theories involving multiple Higgs bosons with non-trivial SU(2) L × U(1) Y charges. The question is whether the 125 GeV Higgs boson discovered at the LHC is the only one of its genre. Since a long time, searches for additional Higgs multiplets are going on in colliders including the LHC. The most trivial extension of the SM is the addition of a gauge singlet CP-even scalar boson [67,68]. Due to the ensuing doublet-singlet scalar mixing, parametrized by an angle α, the 125 GeV Higgs couplings pick up a factor of cos α. At 95% CL, from Run 1 (Run 2) data we obtain sin α 0.31 (0.18), while the HL-LHC expectation is sin α 0.12.
Now we focus on two-Higgs doublet models (2HDM) [69][70][71][72][73][74][75]. The hV V coupling modifications in 2HDM depends on two mixing angles as Above, the angle β parametrizes the mixing between the two doublets, while α is a measure of massmixing between the two CP even neutral scalars. In Type-II 2HDM, which also forms the basis of constructing the minimal supersymmetric standard model, the Yukawa coupling modifiers are given by Note that, c t = c b in this case. We have shown the limits on tan β and cos(β − α) in the top-left panel of Fig. 4. The narrow window of allowed region around the alignment limit β − α = π/2 have shrunk considerably with respect to earlier data. In Type-I 2HDM, however, the top and bottom Yukawa couplings are modified by the same factor as c t = c b = cos α sin β . In this case, constraints are displayed in the top-right panel of Fig. 4. The results we found for both Type-I and Type-II 2HDM are compatible with those reported in [9,66]. A special category of 2HDM postulated by Branco, Grimus and Lavoura (the BGL scenario) admits flavor changing neutral current interactions at the tree level, suppressed by the elements of the Cabibbo-Kobayashi-Maskawa (CKM) matrix [76][77][78]. In some variants of the BGL model (t-type), the expression for c t resembles that of Type-I 2HDM, while c b receives an additional contribution proportional to (tan β + cot β) as follows [77]: In the low tan β 1 regime, the constraints on the BGL (t-type) model follow that of the Type-I scenario (see the bottom-left panel of Fig. 4). But with the increasing tan β 10, owing to the second term in c b in Eq. (4.6), tighter limits are obtained compared to the Type-I model. Notably, the LHC data provide complementary constraints in the low tan β region, which is otherwise less sensitive to the flavor observables [77].
In this model the custodial symmetry is protected by the tree level scalar potential, even if the triplets receive a vev (v t ). Without going into the details of the model, we give the expressions for c V , c t and c b for this case as The limits obtained using the Run 1 and Run 2 data from the LHC and the HL-LHC projections are shown in the v t − sin α plane in the bottom-right panel of Fig. 4.
A few comments on our analysis are in order. First, in deriving these constraints for 2HDM and the GM model, we assumed that the contribution of the charged Higgs bosons decouple in the hγγ decay width and thus can be neglected. However, as shown in [87,88] the decoupling of the charged Higgs contribution to diphoton decay channel depends on the details of the particular model in question. Indeed, our limits would change accordingly. Second, the limits are obtained assuming only renormalizable interactions. The presence of higher dimensional operators [31,58,[89][90][91][92][93][94] would lead to further modifications of all the couplings in addition to what comes out of the mixing in the renormalizable setup. As shown in [95] in the context of 2HDM and in [94] for the GM model, these additional modifications would leave indelible imprint on the ranges of the model parameters.

Conclusions and outlook
We summarize below the important points raised in this paper. The LHC Run 2 data contain a significantly improved information on the Yukawa couplings. Their inclusion has allowed us to extract important limits.
• The ATLAS and CMS Collaborations have made an important breakthrough in getting a grip on the Yukawa force for the first time. If the Higgs boson turns out to be elementary, then it signifies the observation of a new fundamental force. The experimental measurements have made a huge impact in constraining the allowed region of BSM physics manifesting through modified Yukawa couplings. The Run 2 data are particularly instrumental in squeezing the 2σ BSM window in the Yukawa couplings from 25% to 15% around their SM values when compared to the performance of the Run 1 data. HL-LHC would bring it down to within 5%. The limits on hV V (V = W, Z) couplings from the LHC are now competitive with those obtained from electroweak precision tests. The Run 1 (Run 2) data allow not more than 18% (7%) of the total branching fraction of the Higgs boson in the invisible channel. However, larger leak into invisible mode can be accommodated if the hV V and hff couplings substantially deviate from their SM reference points.
• We consider a few motivated BSM scenarios and recast the constraints from our model independent analysis on the parameter space of those specific models using the latest Higgs data. We have observed that, in the context of the SO ( We have shown how the future HL-LHC data would further sharpen the limits. In the RS scenarios with the Higgs boson localized near the IR brane, the first excited KK-gluon weighs more than O(10) TeV. The exact limit depends on the details of the model parameters.
• The amount of mixing between the SM Higgs with any additional scalar singlet is observed to be rather constrained by the present data, given by sin α 0.18. For Type-II 2HDM, only a narrow region around the alignment limit is acceptable, while for the Type-I case a considerable area in the large tan β region is still allowed. In the BGL (t-type) model the constraints in the low tan β 1 region is in the same ballpark as in the Type-I scenario, while for tan β 10 the BGL (t-type) receives stronger constraints than Type-I. We have also shown that for the Georgi-Machacek model v t 48 GeV and −0.3 sin α 0.5 are allowed by the present data. If data continue to push the Higgs couplings towards the SM-like limits, certain scenarios might still accommodate additional light scalars; however, their hunt at the LHC would require special strategies.
• Once the HL-LHC data become available, a better handle on the Yukawa couplings, including those involving other fermions (e.g. τ lepton), would unravel even inner layers of underlying dynamics.