Some new observations for the Georgi-Machacek scenario with triplet Higgs

The Georgi-Machacek model, introducing a complex and a real scalar triplet as additional components of the electroweak symmetry breaking sector, enables substantial triplet contributions to the weak gauge boson masses, subject to the equality of the complex and the real triplet vacuum expectation values (vev) via a custodial SU(2) symmetry. We present an updated set of constraints on this scenario, from collider data (including those from 137/139~fb$^{-1}$ of luminosity at the Large Hadron Collider), available data on the 125-GeV scalar, indirect limits and also theoretical restrictions from vacuum stability and unitarity. It is found that some bounds get relaxed, and the phenomenological potential of the scenario is more diverse, if the doubly charged scalar in the spectrum can decay not only into two like-sign $W$'s but also into one or two singly charged scalars. Other interesting features are noticed in a general approach, such as substantial $\gamma\gamma$ and $Z\gamma$ branching ratios of the additional custodial singlet scalar, and appreciable strength of the trilinear interaction of a charged scalar, the $W$ and the $Z$. Finally,, we take into account the possibility of custodial SU(2) breaking, resulting in inequality of the real and the complex scalar vevs which too in principle may allow large triplet contribution to weak boson masses. Illustrative numerical results on the modified limits and predictions are presented, once more taking into account all the constraints mentioned above.


I. INTRODUCTION
As the electroweak symmetry breaking sector continues to be closely scrutinised both theoretically and experimentally, a query persists far and wide. It is: can the vacuum expectation values (vev) of other scalars, not necessarily SU (2) L doublets, also contribute substantially to the W -and Z-boson masses, besides the 125-GeV particle that arises overwhelmingly out of a doublet? [1] SU (2) triplet scalars are especially interesting in this connection, since (a) they may occur in some Grand Unified Theories (GUT) [2,3] as well as in left-right symmetric scenarios [4][5][6][7] and (b)they offer a mechanism for generating Majorana masses for left-handed neutrinos, called the Type-II seesaw mechanism [8] [9]. However, there is a strong constraint on, say, a single Y=2 complex triplet vev from the ρ-parameter [6] [10], whose tree-level value is given by ρ = m 2 W m 2 Z cos 2 θ W = 1, where θ W is the weak boson mixing angle. The vev of a standalone triplet thus cannot exceed about 4GeV , hence its contribution to the weak boson masses is rather meagre. A frequently discussed model in this context is the one proposed first by Georgi and Machacek (GM), where one complex (χ, Y=2) and one real (ξ,Y=0) triplet were introduced in addition to the doublet Φ of the standard model (SM) [11] [12]. Such a model can be associated with 'composite Higgs' scenarios [13], but it is of sufficient interest on its own. It ensures ρ = 1 at tree-level if the two triplets have equal vev, ensured with the help of a global SU(2) as custodial symmetry [14] 1 . Thus, in the simplest case, one has v χ = v ξ . In this case, s H = In this work, we generalise and update these limits, which enable one to extend the region of the GM parameter space that can be constrained, and also include additional possibilities in the particle spectrum of this kind of a theory. In particular, the new features of our analysis are as follows: 1. Available updated limits from the LHC data have been incorporated. Most important of these are the new limits on doubly charged scalar production via vector boson fusion (VBF) [16] and also Drell-Yan (DY) [17] processes. The latter is particularly important, because it predicts production cross-sections which are four times as much as those for singly-charged scalars.
2. The previous search limits were obtained using the assumption that the doubly charged scalar decays exclusively into two same-sign W -bosons [18][19] [20]. This restricted the analyses to a small region of the parameter space. We, on the other hand, included possibilities where the doubly charged state can also decay into a singly charged state and a W , as also into two singly charged states. We find that some such cases allow higher triplet contributions to the weak boson masses than come with the same-sign W -pair decay alone.
3. We have included the possibility of v χ = v ξ , i.e. of broken custodial SU (2). The consequent shift in parameter values in the scalar potential is not fully calculable unless one knows the UV completion of the theory, which has to be at relatively low scales if the triplet-dominated states have to make any difference to phenomenology [21][22] [23]. Using some phenomenological limits on the parameter shift, we   have computed the changes in contributions to the rates of various collider phenomena, and obtained   modified limits on the 'effective' triplet vev for various masses of the doubly charged states, again allowing for single-channel as well as two-and three-channel decays of the latter.
4. One characteristic feature of such scenarios is the existence of non-vanishing trilinear interactions involving a charged scalar, a W and a Z, something that is not permissible with scalar doublets alone [24]. Such interactions, essentially related to the triplet vev value, bring in additional collider phenomenology [6][25]- [29]. We have indicated the upper limits on the strength of such interactions, for both v χ = v ξ and v χ = v ξ .
5. In addition to various experimental limits, direct as well as indirect, we have also included the theoretical limits (arising mostly from unitarity and occasionally from vacuum stability) on the triplet vev, for both We present a brief outline of the GM scenario in section 2. Sections 3 and 4 are devoted to the experimental and theoretical limits and some related features of scenarios with v χ = v ξ and v χ = v ξ , respectively. We summarise and conclude in section 5.

II. BRIEF SUMMARY OF THE SCENARIO
The scalar sector of the Georgi-Machacek model [11] [12] consists of a Y = 2 complex triplet χ = (χ ++ , χ + , χ 0 ) and a Y = 0 real triplet ξ = (ξ + , ξ 0 , ξ − ) alongwith the usual Standard model doublet. The most general potential preserving a global SU (2) L × SU (2) R is given by [18] [19][34] where, The angle α depends on the 2 × 2 CP-even custodial-singlet scalar mass matrix. The elements of the mass matrix are, Here we have set h to be the 125 GeV scalar and denoted the mass of H as m H which can be larger as well as smaller than 125 GeV depending on the parameters of the scalar potential. Since the most stringent constraint on the parameter space of this model comes from the collider searches of the doubly charged Higgs, its branching ratios in different channels play a crucial role. Hence we will treat the mass of the 5-plet state, 3-plet state and the 125 GeV scalar as our input parameter and will trade off three potential parameters in terms of them. Thus the final set of input parameter for our study is m 5 , m 3 , s H , λ 2 , λ 3 , λ 4 , M 2 .
In this model the 5-plet states couple to vector boson pairs as opposed to the 3-plet states which only couple to fermions. Let us in this context define the quantites, ζ W , ζ Z and ζ f , which reflect the strengths of the W, Z and fermionic coupling, of H normalized to the similar couplings of the SM Higgs respectively [24].
where β is the matrix that rotates (h, H, H 0 5 ) to (φ r , χ r , ξ r ) and summation is over all the scalar multiplets participating in EWSB. In eq. (11) and (12) i = 2. n r k = 3 for the single real triplet ξ, while v c k and v r k are the vevs of the complex and real multipltes respectively. As long as the custodial symmetry is preserved at both the potential and the vacuum level, vχ v and breaking of custodial symmetry will introduce a splitting between them. Also with custodial symmetry β 12 = sin α and β 13 = 0.
Another remarkable feature of this model is the presence of the non-zero H ± W ∓ Z coupling. With custodial symmetry intact only H ± 5 has this kind of coupling and when the symmetry is broken a non-zero H ± 3 W ∓ Z vertex strength starts to appear. For the custodial symmetric case, this coupling strength is a simple scaling to s H by a factor of where c W is the cosine of the Weinberg angle. But when the custodial symmetry is broken this coupling takes a complicated form. For this case let us define two quantities, where, g N C for custodial symmetric case. The expressions of g H ± 5 W ∓ Z , following a general derivation given [24], are where α is the 3 × 3 matrix that diagonalises the singly charged scalar mass matrix in (φ + , χ + , ξ + ) basis, Here f c k corresponds to k = 1, 2 comprising φ and χ, with n c k = 2 and 3 respectively. Similarly f r k corresponds to ξ with k = 3 and n r k = 3. In the situation where v χ = v ξ the H ± W ∓ Z coupling is possible for H + 5 only and then the expression in eq. (16) reduces to, It should be noted that the assertion v χ = v ξ , based on eq. (1) and the invariance of the potential under the custodial SU (2), can be exactly ensured at the tree level only. However, the kinetic energy terms explicitly breaks the custodial SU (2) via U (1) Y interactions. This at higher orders violates the custodial symmetry of the potential, resulting v χ = v ξ (Unless there is fine-tuning). It is therefore useful to also estimate the modifications to the constraints on the parameter space, when the real and complex triplet vevs are unequal. In particular, the various points itemized in the previous section needs to be revisited for such a situation. The higher order modifications , however, leads to large contributions to mass parameters in the potential, which in general shift them to the upper limit of validity of the GM scenario. Therefore in case the triplet sector is expected to have bearing on accessible phenomenology, the GM model will have to pass the baton to some over-seeing scenario not too far above the TeV scale, which controls the divergent corrections [37] - [39]. This essentially brings in new interactions, symmetries or degrees of freedom. Thus it is difficult to compute the modified potential in terms of the GM parameters alone. Keeping this in mind, we have studied the constraints for v χ = v ξ in section 4, using some illustrative scenarios introduced in a phenomenological manner.
We start by updating and extending the existing constraints on the parameter space of the GM scenario.
Before stating explicitly where we have gone beyond the studies already performed in this direction [ , keeping of course in view other considerations going beyond direct searches.
Among the channels listed above, H ++ 5 → + + is undoubtedly the most spectacular signal, where one notices a peak in the invariant mass distribution of the same-sign dileptons [41]. However, given the potential contribution of this scenario to neutrino masses via the Type-2 seesaw mechanism, a significant branching ratio for H ++ 5 → + + will require v χ ≤ 10 −10 GeV [42] [43]. This however is a situation far removed from those which are our main focus here, namely, the viability of substantial role of triplets in EWSB. We therefore keep the ± ± decay channel out of our consideration.
Among the three remaining channels mentioned above, H ± 3 W ± and H ± 3 H ± 3 will have the 3-plet charged scalars mostly decaying into quarks , in which case the leptons, even if produced, are likely to be degraded in cascades.
Their detection is therefore relatively inefficient, and no analysis exists on these two channels so far. They, however, may serve to suppress the branching ratio into W ± W ± , thereby weakening the limits and allowing higher values of s H for a given m 5 , at least from direct search at the LHC. We therefore extend the existing analyses for not only the single decay channel H ++ 5 → W ± W ± but also with either one or two of the additional channels mentioned above. In addition, the following constraints are taken into account: • Indirect constraints [44], primarily from the experimental bounds on the rates for b → sγ [45] and B s → µ + µ − [46], and also from oblique electroweak parameters (of which the limit from T , or equivalently the ρ-parameter, is explicitly used at every step).
• Theoretical constraints such as perturbative unitarity, vacuum stability, and the requirement that the custodial symmetry preserving vacuum corresponds to the global minimum of the potential [47] [48].
• Whatever constraints come withinthe scope of the package GMCALC [49] have been made use of. We have updated them, using our own code for VBF with 137 f b −1 and DY with 139 f b −1 . The consistency of the code developed by us has been checked against GMCALC in the appropriate limits.
• All otherconstraints from the searches for additional neutral searches obtained using the code HiggsBounds [50].
• The requirement that the signal strength of the 125-GeV scalar in all channels is consistent with LHC data. For this, the code HiggsSignals [51] has been made use of.
In obtaining the scatter plots presented in Figures 1-3, the quantities whose specific values have entered as independent variables are m 5 , m 3 and s H , all other parameters in the GM scalar potential being subjected to an unbiased scan, subject to the requirements of unitarity, vacuum stability and the demand that the EWSB vacuum corresponds to global minimum of the potential. The mass parameter (m 5 ) corresponding to the custodial 5-plet has been varied in the range 200 GeV -2 TeV, subject to the constraints mentioned above. The 3-plet mass m 3 is also subjected similar constraints, with its maximum allowed values being set according the different kinematic situations described in the next paragraph. In cases where the 3-plet is never produced in    Figures 1(a), (b) Fig. 2, the left, central and right coloumns fall in these three kinematic categories.  decay becomes more stringent compared to that found in [19], especially for masses exceeding 500 GeV.
• When the decay channel into H + 3 W + opens up, the limits from VBF are relatively relaxed, and can go up to s H ≈ 0.4 at 500 GeV, where the earlier limit is 0.28 [18] [19].
• All the three aforementioned decays open up for a rather limited region of the parameter space, which ends around m 5 ≈ 300GeV . This is because it is not possible beyond this mass to satisfy equations 5 and 6 simultaneously, without creating an unacceptable tension between the potential minimisation conditions and unitarity limits on the quartic couplings. However, as expected, higher values of s H (upto about 0.44) get allowed within this range, as compared to both the single-and double-channel decay situations.
Looking at it from another angle, the higher m 5 is, the more difficult it is to keep m 3 small enough for the above channel to open up. • The Drell-Yan channel has also been taken into account for 139 fb −1 of integrated luminosity. For the single channel case, this practically excludes the mass range 200 -300 GeV for the doubly charged scalar goes down proportionally with the squared branching ratio into W + W + .
• The code HiggsBounds has been used to constrain the particle spectrum from neutral Higgs searches at the LHC as well as LEP. However, the limits thus obtained are weaker than those from VBF[16] [52]. As for the constraints from HiggsSignals, the allowed regions shown the Fig. 1a and 1b are consistent at 95% C.L. However, the low H + 3 masses relevant to the three-channel case tend to create more tension with the measurements of the diphoton signal strength for the 125-GeV scalar. This explains, for instance, the paucity of allowed points in Fig. 2c where 95% C.L. consistency with HiggsSignals results are explicitly demanded.
• Fig. 2 shows the modifications of the couplings for the custodial-singlet neutral scalar H to fermion and gauge boson pairs, as compared to those for the SM Higgs boson. As is particularly evident in Fig. 2d-

IV. CONSTRAINTS FOR vχ = v ξ
So far we have discussed the constraints corresponding to v χ = v ξ which is a direct consequence of the custodial SU (2) symmetry of the scalar potential . Although this symmetry has been shown to be preserved in scalar loop corrections [12], it is in general liable to be broken on inclusion of gauge couplings. This is because hypercharge interactions in the scalar covariant derivatives break the custodial SU (2). We have studied the constraints on model parameters, and associated issues, in such a situation as well, as reported below.
The higher-order corrections mentioned above entail quadratically divergent terms [22]. If the triplet scalar mass terms in the GM scenario have to be around the TeV-scale or less (which is the situation where the GM model is phenomenologically significant), then some additional inputs have to come in the form of a cut-off for the GM theory. This has prompted us to use some phenomenological inputs for v χ = v ξ .
Our investigation pertains to cases where the theoretical limits (which in principle requires some knowledge of the overseeing theory controlling the divergent contributions) are not substantially different from those in the corresponding cases with v ξ = v χ . A similar consideration applies to indirect constraints. 3 We have thus treated v χ and v ξ as phenomenogical inputs, with their difference not exceeding 30% in our randomly generated points. In a similar fashion, the parameters in all scalar mass-squared matrices have been subjected to random variation, without differing by more than 30% with respect to values yielding v χ = v ξ .
After diagonalizing the mass matrices, the couplings of the physical scalar states have been calculated using Feynrules [55]. It has been checked that replacing 30% by 50% in the random number generation criteria do not make any qualitative change in the final results presented below. Apart from these, the general guidelines followed in obtaining the scatter plots (Figures 4 -9) are similar to those used in the previous section.
The quantity s H , which in the previous cases uniquely parameterised the triplet contribution to the 'effective vev' breaking the electroweak symmetry, is not similarly applicable when v χ = v ξ . We instead make use of the where is the effective vev. Since the custodial symmetry is broken, the physical states no more constitute the irreducible representations under the custodial SU (2) group. Hence all the H 5 states will now have non-zero component coming from the SU (2) L doublet Φ 4 . Similar effects will be seen in states belonging to the other representations of the custodial SU (2).The contribution of φ 0 r in H after this is denoted by β 12 . It is to be noted in this 3 It should be remembered that the inequality of the two kinds of triplet vevs results in H + 3 W − Z interactions, too, which alters the limits from precision electroweak observables. Similarly, H ± 5 now develops small fermion couplings. 4 The mixing between the custodial 3-plet and the 5-plet will in principle also allow H ++ ) decay mode. However the unitarity of such mixing implies that the constraints will not be strengthened further context that the W W and ZZ coupling strengths for the state H are liable to differ for v χ = v ξ , a feature that affects the modification of interaction strengths in comparison with those for the SM Higgs. The allowed branching ratios for H → γγ, Zγ are shown in Fig. 8. Fig. 9 demonstrates the allowed points in the space spanned by the values of the κ H + 5 W − Z and κ H + 3 W − Z , following the expressions given in section 2.  For three allowed channels, however, the constraints are relatively relaxed here, as the allowed inequality  • H, the custodial singlet neutral scalar, can now have a larger number of allowed points with substantial  doublet content, especially for cases with double and triple channel decays allowed, as seen from Fig. 5-7.
• As compared to the case with exact custodial symmetry, one can have larger mass splitting allowed between two custodial multiplets. It is thus possible to have allowed regions in the parameter space with the state H as high as a TeV, with the erstwhile 3-and 5-plet state lying as low as 500 GeV.
• Here too, there are allowed regions in all three cases with enhanced branching ratios in the γγ and Zγ channels for H-decay. In general, for the parameter points where such enhancement occurs, the deciding factor turns out to be not only the vev split but also their absolute values, a fact that may not be visible in the scatter plots.
• Because of the added freedom of differencing v χ and v ξ , the maximum permissible strength of the H + 5 W − Z interaction can be stronger than that with v χ = v ξ over a non-negligible region of the parameter space.
However, such interaction is found to be appreciably enhanced with respect to the case with v χ = v ξ , when H ++ 5 exclusively decays to W + W + that is to say in the single channel case. This effect is less pronounced for the two and three channel cases where the limit is already relaxed for v χ = v ξ .
• In practically all cases Fig. 4 onwards, the allowed parameter regions answering to three decay channels for H ++ 5 are rather restricted, largely due to the interconnected nature of parameters and the proliferation of conditions to satisfy. For the same reason, the 'allowed' or 'disallowed' label of any point in a plot may get altered upon minute variation in parameter values. For example such status may be effected s H altered in the second/third place after decimal.The broadly allowed regions are nonetheless represented faithfully.
• The possibility of mixture of the 5-and 3-plets allows tree-level H + 3 W − Z interaction in this case. Such coupling is less abundant for the triple channel H ++ 5 decay, since this mass hierarchy allow very few points which simultaneously pass through the checks imposed by HiggsBounds and HiggsSignals.

V. SUMMARY AND CONCLUSION
We have made an extensive analysis of the constraints on the parameter space of the GM scenario, based on existing data. These includes collider data (including VBF/Drell-Yan data at the LHC with integrated luminosity of 137/139 fb −1 ), those on the SM-like 125-GeV scalar, indirect limits including those from rare heavy flavour decays, and also all theoretical guidelines such as vacuum stability and unitarity. Searches for the doubly-charged scalar constitute the most spectacular way of probing such a scenario. We have gone beyond the usually adopted idea that the W + W + decay channel is the only significant one when it comes to situations with substantial triplet contributions to the weak gauge boson masses. Thus we have carried detailed scans of the parameter space where the W + H + 3 and H + 3 H + 3 decay modes also open up, thus eating into the branching ratio share of the W + W + mode.
It is found that, with all possibilities and constraints included, the the upper limit on the value of s H , a measure of the triplet contribution to the W -and Z-masses, goes up to about 0.4 in with mass around 500 GeV, in contrast to studies based on earlier data where similar values are attainable for masses close to a TeV only, and with constraints coming from data with lower luminosity. We also note that the constraints from unitarity tend to suppress the maximum value of s H for large (≥ 1.5 TeV) m H ±± 5 , thanks to the relations among physical masses and quartic couplings. The H + W − Z couplings correspondingly have the scope of enhancement when two-and three-channel decays of the doubly charged scalar are allowed. We also note that, in such situations, the γγ and Zγ branching ratios for the custodial singlet scalar H can sometimes be much higher than that of the SM-like scalar state.
While the above conclusions apply to the case with the custodial SU(2) symmetry intact, we extend our study to situations where the real and complex triplets have unequal vev's. In a phenomenological approach to parameter scans, we have demonstrated that a larger number of parameter points can become allowed, subject to all constraints, and the features outlined above are more widely visible. In addition the charged scalar H + 3 , too, has W − Z interaction here. Thus the GM scenario still admits of interesting phenomenology, subject to the various constraints that tend to tie it up.