Constraining the Georgi-Machacek Model with a Light Higgs

In this work, we investigate the viability of a light Higgs ($\eta$) scenario in the Georgi-Machacek (GM) model, where we consider all theoretical and experimental constraints such as the perturbativity, vacuum stability, unitarity, electroweak precision tests, the Higgs di-photon and undetermined decays and the Higgs total decay width. In addition, we consider more recent experimental bounds from the searches for doubly-charged Higgs bosons in the VBF channel $H_{5}^{++}\rightarrow W^{+}W^{+}$, Drell-Yan production of a neutral Higgs boson $pp\rightarrow H_{5}^{0}(\gamma\gamma)H_{5}^{+}$, and for the light scalars at LEP $e^{-}e^{+}\rightarrow Z\eta$, and at ATLAS and CMS in different final states such as $pp\rightarrow\eta\rightarrow2\gamma$ and $pp\rightarrow h\rightarrow\eta\eta\rightarrow4\gamma,2\mu2\tau,2\mu2b,2\tau2b$. By combining these bounds together, we found a parameter space region that is significant as the case of the SM-like Higgs to be the light CP-even eigenstate, and this part of the parameter space would be tightened by the coming analyses.

electroweak phase transition strength [27]. The GM scalar sector has been confronted with the existing data [28], where direct search constraints for extra Higgs bosons and measurements of the SM-like Higgs properties are considered. In addition, the authors derived bounds from the negative searches of the doubly-charged Higgs bosons in the VBF channel H ++ 5 → W + W + ; and the Drell-Yan production of a neutral Higgs boson pp → H 0 5 (γγ)H + 5 . In [29], we have investigated the GM parameter space where the SM-like Higgs is considered to be the light CP-even eigenstate h = h 125 ; and the eigenstate η is a heavier scalar m η > m h . We have considered all the known theoretical and experimental constraints, including those from the negative searches of the heavy scalar via pp → η → hh, τ τ, ZZ; and we have found that a significant part of the parameter space is viable and could be probed soon by future analyses with more data. One has to mention that it turns out that two thirds of the parameter space allowed by the constraints described in the literature, are excluded by some possibly existing scalar potential minima (that either preserve or violate the CP and electric charge symmetries) that are deeper than the electroweak (EW) vacuum [29]. Here, we aim to investigate other part of the parameter space where the SM-like Higgs is the heavy CP-even eigenstate, i.e., h = h 125 and m η < m h .
In section II, we review the GM model and present the constraints described in the literature.
We discuss the physics of a light CP-even scalar at colliders in section III; and discuss our numerical results in section IV. In section V, we give our conclusion.
It has been shown in [29] that the scalar potential (2) could acquire some minima that could violate the CP-symmetry and/or electric charge, where they could be deeper than the electroweak vacuum {υ φ , √ 2υ ξ , υ ξ }. Then, this part of the parameter space would be ignored. Here, we impose the constraints from (1) vacuum stability, (2) unitarity, (3) the electroweak precision tests, (4) the di-photon and undetermined Higgs branching ratios and total decay width; in addition to (5) the constraints from negative searches for light scalar resonances at LEP [38]. For the constraints (1-3), we used the results described in [29].
In this setup, the SM-like Higgs h (the CP-even scalar m h = 125.18 GeV) decays mainly into pairs of fermions (cc, µµ, τ τ, bb) and gauge bosons W W * and ZZ * , in addition to a pair of light scalars ηη when kinematically allowed. Since the Higgs couplings to SM fields are scaled by the coefficients then, its total decay width can be written as where the last term represents the partial decay width Γ(h → ηη), Γ SM h = 4.08 MeV [39] and B SM (h → XX) are the SM values for total decay width and the branching ratios for the Higgs, respectively. Here, g GM hηη is the scalar triple coupling hηη. Since the light scalar η can be seen at detectors via its decay to light fermions η → ff , then, the Higgs decay h → ηη does not match any of the known SM final states, and hence called undetermined channel, which is constrained by ATLAS as B und < 0.22 [40,41]. The total Higgs decay width recent measurements based on the off-shell Higgs production in the final state . In previous analysis of the GM parameter space [29], it has been shown that the measurements of the Higgs signal strength modifiers imply constraints on the coefficients κ F,V . Here, another factor is constrained in addition, which is the undetermined Higgs decay; and consequently the scalar η mass and the triple coupling g GM hηη . Then, the partial Higgs signal strength modifier for the channel h → XX (X = f, V ) can be simplified in this setup as where For the di-photon Higgs strength modifier, it can be written as where i = H + 3 , H + 5 , H ++ 5 stands for all charged scalars inside the loop diagrams, Q i is the electric charge of the field i in units of |e|, g GM hii are the Higgs triple couplings to the charged scalars; and the functions A γγ i are given in the literature [43]. According to the partial Higgs strength modifier formulas in (7) and (8), one expects that the experimental constraints would enforce the coefficients κ F,V to lie around unity. In our numerical scan, we consider the recent experimental values [44] for the partial Higgs strength modifiers (7) and (8).
Besides the above mentioned constraints, the negative searches for doubly-charged Higgs bosons in the VBF channel H ++ 5 → W + W + ; and from Drell-Yan production of a neutral Higgs boson give strong bounds on the parameter space [28]. It has been shown in [28], that the doubly-charged Higgs bosons in the VBF channel leads to a constraint from CMS on s 2 [45]. Clearly, non vanishing values for the branching ratios B(H ++ , will significantly help to relax this bounds. While the relevant quantity for the constraints on H 0 5 → γγ is the fiducial cross section times branching ratio , that is constrained by ATLAS at 8 TeV [46] and at 13 TeV [47]. Here, we used the decay rate formulas, the cross section and efficiency values used in [28] to include these constraints in our numerical analysis. Since part of the charged triplet H ± 3 is coming from the SM doublet as shown in (3), then it should couple the up and down quarks as the W gauge bosons does. This interaction leads to flavor violating processes such as the b → s transition ones, which depend only on the charged triplet mass m 3 and the mixing angle β (and consequently the triplet VEV υ χ ). In our numerical scan, we consider the bounds on the m 3 -υ χ plan shown in [19].

III. THE LIGHT SCALAR η IN THE COLLIDER
After the discovery of the Higgs boson with m h = 125.18 GeV, efforts have been devoted to search for light neutral scalar boson through different channels over a wide range of mass. Such results can also be used to impose constraints on models with many neutral scalars such as the GM model. Here, the light scalar η has similar couplings as the SM Higgs, but modified with the factors then, the partial decay width of the light scalar η into SM final states can be written as Γ Thus, its total decay width can be written as Here, we will consider the constraints from the negative searches for pp → η → γγ at CMS at 8+13 TeV [30], and at ALTAS at 13 TeV with integrated luminosity 80 fb −1 [31]; and at 138 fb −1 [32].
At LEP, many searches for Higgs at low mass range m h < 100 GeV have been performed, and bounds on the form factor [38]; that can be simplified in our setup as σ SM (pp→η)×B SM (η→γγ) . In our setup, the parameter κ η γγ can be simplified as with i = H ± 3 , H ± 5 , H ±± 5 , g GM ηii are the scalar triple couplings of the scalar η to the charged scalars; and the functions A γγ 0,1,1/2 are given in the literature [43].  [37]. All these analyses were performed using the ggF Higgs production mode at the LHC.

IV. NUMERICAL ANALYSIS AND DISCUSSION
Here, we have considered the heavy CP-even scalar to be the 125 GeV SM-like Higgs; and have taken into account the different theoretical and experimental constraints described in Sections II and III, such as the constraints from perturbativity, vacuum stability [18,49], electroweak precision tests [19] , the di-photon and undetermined Higgs decays, the total Higgs decay width; and the B physics flavor constraints. In addition, we have considered also the constraints from the fact that the EW vacuum (υ φ , √ 2υ ξ , υ ξ ) must be the deepest among possible minima that may preserve or violate the CP and electric charge symmetries as described in [29]. As a first step of this numerical study, we perform a full numerical scan over the GM model parameter space, then, in a next step we will imposes the constraints from the negative searches for doubly-charged Higgs bosons in the VBF channel H ++ 5 → W + W + ; and from Drell-Yan production of a neutral Higgs boson pp → H 0 5 (γγ)H + 5 ; negative searches for a light Higgs at LEP e − e + → η Z [38], the direct search for a light resonance at the LHC [30] and the indirect searches for light resonance via the final states h → ηη → XXY Y [33][34][35][36][37].
Here, the negative values of t β are considered for the following reason. In the GM model, there exists an invariance under the transformation (υ ξ , µ 1,2 ) → (−υ ξ , −µ 1,2 ), which means V (Φ, ∆, µ 1,2 ) = V (Φ, −∆, −µ 1,2 ). Consequently, the scalar mass matrix elements also remain invariant under this transformation. However, because the physical scalar eigenstates are mixtures of the components of Φ and ∆, most of the physical vertices that involves scalars are not invariant under (υ ξ , µ 1,2 ) → (−υ ξ , −µ 1,2 ). This means that these vertices change; and therefore any two benchmark points (BPs) with the same input parameters but with different signs of (±t β , ±µ 1,2 ) are physically different. This can be seen in the scaling factors (5) and (9). This makes the BPs with negative t β values in (13) independent part of the parameter space that should not be ignored.
After combining all the first step constraints, we show in Fig. 1 the viable parameters space and the different physical observables using 34.7k BPs. Here, one has to mention that most of the allowed light scalar mass values are for m η > m h /2 due to the conflict between the constraints from the undetermined (h → ηη) and di-photon (h → γγ) Higgs decays.
Some of these 34.7k BPs are in agreement all the above mentioned constraints, including those are considered in the second step of our analysis. For instance, we show in Fig. 2 [30], where the palette shows the factor (12) that represents the enhancement effect on the decay η → γγ due to the coupling with charged scalars. The yellow (blue) region corresponds to 68% (95%) CL.
From Fig. 2-left and -right, one learns that the first and second islands in Fig. 1-top-right is in agreement with the OPAL bounds (i.e., the lower green island in Fig. 2-left that corresponds to |ζ V | 0.7). The third island is also in agreement with OPAL since it corresponds to 80 GeV ≤ m η ≤ 120 GeV, i.e., the pink island in Fig. 2 → W + W + ) is significantly smaller than unity, as shown in Fig. 2 In this setup, the light scalar η can be probed due to production modes both at the HL-LHC [51] and at future lepton colliders such as the ILC [52], CLIC [53], the CEPC [54] and the FCCee [55] running at the Z-pole and above the Zh production threshold. In future searches, more severe constraints on the parameters m η , κ V,F , ζ V,F would be established. Distinguish the GM model among other similar models that involves a light scalar such by just relying on the Higgs measurements and the η-discovery (or η negative searches) is not trivial. However, in the GM model, a correlation between the scalargauge and scalar-fermions couplings is absent unlike most of the SM multi-scalar extensions, i.e., no correlation between κ V − κ F and ζ V − ζ F exists. This feature could help to probe the GM model in some cases where both ratios κ V /κ F and ζ V /ζ F are significantly different than unity. In addition, the custodial triplet and fiveplet members have many production modes, which makes the GM model more interesting for any search.

V. CONCLUSION
In this work, we studied the GM scalar sector in the case where the SM-like Higgs corresponds to the heavy CP-even eigenstate. We have shown the viability of an important region of the parameter space, that is significant as the case of the light CP-even scalar to be the SM-like Higgs. In our analysis, we considered the constraints from perturbativity, unitarity, boundness from below, the electroweak precision tests, the di-photon and undetermined Higgs decays; and the total Higgs decay width. For this we generated around 34.7k BPs that fulfill all the previously mentioned constraints. In addition, we have imposed more bounds from the searches for (1) 3 . It is important if the future searches for doubly charged scalars would consider masses below 200 GeV to probe this scenario. In the near future, the coming analyses with more data will make the parameter space more constrained.
The light scalar η can be produced and detected at future hadron/lepton colliders, which is the case of many light scalars in multi-scalar models. Then, more severe constraints on the parameters m η , κ V,F , ζ V,F would be established. However, distinguishing the GM light scalar among other scalars involved in other multi-scalar models could be possible; and relies on the measurement of the η couplings to gauge bosons (ζ V ) and fermions (ζ F ). The reason for this is that, unlike in many multi-scalar models, a correlation between the scalar-gauge and scalar-fermions couplings is absent in the GM model. Although, considering some future neutral/charged scalar searches may put more constraints on the custodial triplet and fiveplet members, which makes the GM model more interesting.