Cornering variants of Georgi-Machacek model using Higgs precision data

We show that in the absence of trilinear terms in the scalar potential of Georgi-Machacek model, heavy charged scalars do not necessarily decouple from the $h \to \gamma \gamma$ decay amplitude. In such scenarios, measurement of the Higgs to diphoton signal strength can place stringent constraints in the parameter space. Using the projected precisions at the High Luminosity LHC (HL-LHC) and the ILC, we find that the upper bound on the triplet vacuum expectation value can be as low as 10 GeV. We also found that when combined with the theoretical constraints from perturbative unitarity and stability, such variants may be ruled out altogether.

of our paper, which runs contrary to the common perception [12] is that the charged scalar contributions to the h → γγ decay amplitude do not necessarily decouple in certain variants of the GM models. Consequently, for such variants, the bounds on the model parameters, especially v t , extracted using the data for Higgs signal strengths will not get washed away even when the nonstandard masses are in the multi-TeV range. To our knowledge, this subtle but interesting feature has not been emphasized earlier in the context of GM models.
We start by revisiting the scalar potential of the GM model. As stated earlier, GM model [3][4][5] extends the scalar sector of the SM by adding one real SU (2) L triplet ξ and one complex SU (2) L triplet χ with hypercharges Y = 0 and Y = 2 respectively. The scalar fields of such a model can be concisely represented in the forms of a bi-doublet Φ and a bi-triplet X as In order to preserve the custodial symmetry, both the real and complex triplet should acquire the same VEV. Accordingly, the VEVs of the scalar multiplets are defined by Φ = (v d / √ 2)1 2×2 and X = v t 1 3×3 . Comparing the W and Z boson masses with their corresponding SM expressions, we obtain the formula for the electroweak (EW) VEV as Following the convention of Refs. [12,[21][22][23], we now write down the most general scalar potential for the GM model as follows: where, τ a ≡ σ a /2, (a = 1, 2, 3) with σ a s being the Pauli matrices and t a s correspond to the generators of triplet representation of SU (2) and are expressed as The matrix U appearing in the cubic terms of Eq. (3) is given by, In the limit when both the triplets receive the same VEV, we will have two independent minimization conditions, using which we can trade the bilinear coefficients, µ 2 2 and µ 2 3 in favor of v d and v t as follows: After SSB, the neutral component of the fields are expanded around the minima as The mass terms in the scalar potential, then, can be diagonalized to obtain a custodial quintuplet (H ++ 5 , H + 5 , H 0 5 , H − 5 , H −− 5 ) with common mass m 5 , a custodial triplet (H + 3 , H 0 , H − 3 ) with common mass m 3 and two custodial singlets h and H with masses m h and m H respectively. 3 In particular, to diagonalize the CP -even scalar sector, we need to introduce an angle α which is defined as follows In what follows, we assume h to be the lightest among the CP -even scalars, which has been discovered at the LHC with mass m h ≈ 125 GeV.
At this point it is instructive to count the number of independent parameters in the potential of Eq. (3). There are a total of nine parameters consisting of two bilinears (µ 2 2 and µ 2 3 ), five quartic couplings (λ i , i = 1, . . . , 5) and two cubic couplings (M 1 and M 2 ). Among these, µ 2 2 and µ 2 3 have already been traded in favor of v d and v t as in Eq. (6). The five quartic couplings can now be exchanged in favor of four physical masses, m 5 , m 3 , m H and m h and the mixing angle, α. The relevant formulas connecting the λ i -s with the physical masses and mixings are given below: where, we have defined Note that Eq. (9) will be extremely useful in translating the scalar couplings as well as our final results in terms of the physical parameters which are directly accessible at the experiments.
Before moving on to the main part of the paper, let us first examine the theoretical bounds coming from perturbative unitarity and stability of the scalar potential. The relevant constraints have been summarized in Refs. [12,21]. In Fig. 1 we display our main results in the limit M 1 , M 2 = 0. It is quite intriguing to note that there arises a lower bound on v t ( 27 GeV) from unitarity and stability. It is not very difficult to find a qualitative justification for such a lower bound as we will explain below.
Among the many unitarity constraints, we draw the reader's attention to the following ones [12,21] in particular: After using Eq. (9) to substitute for the λ i -s and putting M 1 , M 2 = 0, the above conditions read Figure 1: Allowed region in the v t -m H plane from perturbative unitarity and stability in the limit M 1 , M 2 = 0.
Note that a lower limit on v t emerges from these theoretical constraints.
where we have used the following abbreviation: Clearly, for very small v t , i.e., sin β → 0, the following correlations will be imposed: Moreover, the following unitarity conditions will also be relevant [12,21]: Using the triangle inequality we may write, In the above inequality, the terms involving λ 3 and λ 4 will give rise to the dominant contributions on the left hand side for small v t . Again, using Eq. (9) one can check that the limit sin β → 0 will entail the following correlation: It is easy to verify that no nontrivial solution set {m 2 5 , m 2 3 , m 2 0 } exists such that Eqs. (16), (17) and (20) are satisfied simultaneously. This explains why there should be a lower limit on v t (or equivalently, on tan β) so that the correlations of Eqs. (16), (17) and (20) do not become too strong.
Let us now proceed by defining the Higgs coupling modifiers as: where x stands for the gauge bosons and the massive fermions. In the GM models, these coupling modifiers can be calculated to be Using the current limits on κ f and κ V [24], we display the 2σ allowed region (light gray shade) in the sin α-v t plane in Fig. 2. In the same plot we also show how this allowed region will shrink if the Higgs data continue to drift towards the SM expectations with the projected accuracy at the HL-LHC (2%) (red shade) and at the ILC (< 1%) (green shade) [25]. Clearly, with the projected accuracies at the HL-LHC and the ILC, we can hope to constrain v t to be smaller than 20 GeV. in the sin α-v t plane using the current [24] (shaded in gray) and projected [25] (shaded in red and green) accuracies in the measurements of κ V and κ f . While extracting bounds using future precisions, the central values of all the κ x parameters have been assumed to be 1, i.e., consistent with the SM. The horizontal line corresponds to the lower limit on v t arising from the theoretical constraints. Figure 3: Shrinking of the allowed region in Fig. 2 when combined with the constraints from κ γ in the limit M 1 , M 2 = 0. The shaded regions in gray and red characterize the bounds arising from current [24] and future [25] measurements of κ γ respectively. In drawing this plot, a mass hierarchy m 3 ≈ m 5 m h has been assumed. The horizontal line corresponds to the lower limit on v t arising from the theoretical constraints.
A crucial observation of our paper is that the allowed region in Fig. 2 from κ f and κ V can be considerably reduced further by superimposing the constraints from h → γγ even in the limit when the charged scalars are super heavy. To illustrate, let us first write down the effective hγγ coupling as follows: where F µν = ∂ µ A ν − ∂ ν A µ is the usual electromagnetic field tensor. Now we define the hγγ coupling modifier as whose analytic expression for the GM model can be written as 4 where, Q f and N f c denote the electric charge and the color factor respectively for the fermion, f , and, defining τ x = (2m x /m h ) 2 , the loop functions are given by [26], In Eq. (26) the function f (τ ) is defined as The expressions for κ 3 and κ 5 appearing in Eq. (25), which encapsulate the contributions from the charged scalars that are members of the custodial triplet and fiveplet respectively, are given by 5 Figure 4: Allowed regions at 95% C.L. in the m 3 -m 5 plane using the current (shaded in blue) [24] and projected (shaded in magenta) [25] accuracies in the measurement of κ γ for different sets of {sin α, v t } values. The dark blue points cover the region that passes the theoretical constraints of perturbative unitarity. 4 The factor 5 in front of the scalar loop contribution A S (τ 5 ) arises from the sum of the singly and doubly charged scalar contributions from the custodial fiveplet. 5 Note that the equality g hH  pair. The loop functions in Eq. (26) saturate to finite nonzero values once the particles circulating in the loop become much heavier than m h . Therefore, the (non)decoupling behaviors of the charged scalars will be captured by the quantities κ 3 and κ 5 . In the limit M 1 , M 2 = 0 [4,5,27,28], we can calculate sin α cos 2 β sin β + cos α sin 2 β cos β , (29a) Evidently, in the limit m 3 ≈ m 5 m h , both κ 3 and κ 5 assume nonzero values, which implies that contrary to the common perception [12] the charged scalars do not decouple. Note that for M 1 , M 2 = 0 but v t = 0, the charged scalars receive their masses purely from the SSB sources and consequently, such nondecoupling behaviors are to be expected [29]. Because of this, the allowed region in Fig. 2 can be further constrained by superimposing the bounds from κ γ , which has been shown in Fig. 3. From Fig. 3 we can easily see that, for M 1 , M 2 = 0, using the projected combined accuracy for the HL-LHC and the ILC [25], we can restrict the upper limit on v t to be as low as 10 GeV at 95% C.L. From Fig. 3 it is also clear that the values of sin α and v t can be strongly correlated if the measurement of κ γ agrees with the SM with the expected combined precision at the HL-LHC and the ILC. Consequently, the bound on v t will foster a strong upper limit on sin α (| sin α| < 0.1). Quite clearly, when combined with the lower limit on v t stemming from perturbative unitarity, we can potentially rule out the variants of GM models with M 1 , M 2 = 0 using future Higgs precision data. Finally, in Fig. 4, we display the allowed parameter space at 95% C.L. in the m 3 -m 5 plane using the current as well as future precisions in κ γ for different sets of {sin α, v t } values. We find that as the accuracy in κ γ improves, the allowed region shrinks rapidly to the extent that with the future precision of about 1% at the HL-LHC and the ILC [25], the allowed area reduces to a thin sliver in the parameter space, thereby allowing us to strongly correlate m 3 and m 5 as well. In the same figure we also show how perturbative unitarity entails a different correlation in the m 3 -m 5 plane (dark blue points) in the limit M 1 , M 2 = 0. This implies that if the measurement of h → γγ signal strength continues to agree with the corresponding SM expectation with higher precision, the GM models with M 1 , M 2 = 0 will be increasingly disfavored. We have checked that the allowed region from κ γ in Fig. 4 does not crucially depend on the value of v t as long as it is nonzero.
To summarize, we have revisited the scalar sector of the GM model in the light of Higgs precision data. The GM model has long been preferred over minimal triplet model due to its attribution towards the preservation of custodial symmetry at the tree-level and thus lifting the stringent bound on the triplet scalar VEV. In this paper, we have demonstrated that, for certain variants of the GM models, the triplet VEV can still be tightly constrained using the Higgs precision data, especially the measurement of the h → γγ signal strength. More specifically, we have shown that when the trilinear couplings in the scalar potential of Eq. (3) vanish, the charged scalars do not decouple from the h → γγ decay amplitude. Consequently, using the projected precision in the measurement of κ γ , we have been able to set an upper limit on the triplet VEV, v t , which can be as low as 10 GeV. This nondecoupling behavior of the charged scalars in the context of the GM model does not appear to be a widespread knowledge and therefore constitutes a new finding of our paper. The upshot of our analysis is that, unlike the bound on v t obtained using the flavor data [7,8], our bounds derived from κ γ do not get washed away when the charged scalars are super heavy. We have also demonstrated how, using a precise measurement of κ γ , the masses of the custodial triplet and fiveplet scalars can be correlated. Moreover, when these bounds are used in conjunction with the theoretical constraints from perturbative unitarity and stability, we have shown how the variants of GM models with M 1 , M 2 = 0 can fall out of favor in the near future. Thus our current study highlights the importance of the measurement of Higgs to diphoton signal strength in the future experiments in the context of GM models.