Probing the Higgs sector of $Y=0$ Higgs Triplet Model at LHC

In this paper, we investigate the Higgs Triplet Model with hypercharge $Y_{\Delta}=0$ (HTM0), an extension of the Standard model, caracterised by a more involved scalar spectrum consisting of two CP even Higgs $h^0, H^0$ and two charged Higgs bosons $H^\pm$. We first show that the parameter space of HTM0, usually delimited by combined constraints originating from unitarity and BFB as well as experimental limits from LEP and LHC, is severely reduced when the modified Veltman conditions at one loop are also imposed. Then, we perform an rigorous analysis of Higgs decays either when $h^0$ is the SM-like or when the heaviest neutral Higgs $H^0$ is identified to the observed $125$ GeV Higgs boson at LHC. In these scenarios, we perform an extensive parameter scan, in the lower part of the scalar mass spectrum, with a particular focus on the Higgs to Higgs decay modes $H^0 \to h^0h^0, H^\pm\,H^\mp$ leading predominantly to invisible Higgs decays. Finally, we also study the scenario where $h^0, H^0$ are mass degenerate. We thus find that consistency with LHC signal strengths favours a light charged Higgs with a mass about $176\sim178$ GeV. Our analysis shows that the diphoton Higgs decay mode and $H \to Z \gamma$ are not always positively correlated as claimed in a previous study. Anti-correlation is rather seen in the scenario where $h$ is SM like, while correlation is sensitive to the sign of the potential parameter $\lambda$ when $H$ is identified to $125$ GeV Higgs.

theoretical (BFB, unitarity and Veltman conditions) as well as the experimental ones, and scrutinised via HiggsBounds v4.2.1 [17] which we use to check agreement with all 2σ exclusion limits from LEP, Tevatron and LHC Higgs searches. Our conclusion is drawn in section 5, while some technical details are postponed into appendices.
2 Review of the HTM0 model 2

.1 Lagrangian and Higgs masses
The Higgs triplet model with hypercharge Y ∆ = 0 can be implemented in the Standard Model by adding a colourless scalar field ∆ transforming as a triplet under the SU (2) L gauge group with hypercharge Y ∆ = 0. The most general gauge invariant and renormalisable SU (2) L ×U (1) Y Lagrangian of the scalar sector is given by, where the covariant derivatives are defined by, (W a µ , g), and (B µ , g ) are respectively the SU (2) L and U (1) Y gauge fields and couplings and T a ≡ σ a /2, where σ a (a = 1, 2, 3) denote the Pauli matrices. The potential V (H, ∆) can be expressed as [10], where T r is the trace over 2 × 2 matrices. Last, L Yukawa contains all the Yukawa sector of the SM plus an extra Yukawa term that leads after spontaneous symmetry breaking to (Majorana) mass terms for the neutrinos, without requiring right-handed neutrino states.
Defining the electric charge as usual, Q = I 3 + Y 2 where I denotes the isospin, we write the two Higgs multiplets in components as: For later convenience, the vacuum expectation values v d and v t are supposed positive values.
Assuming that spontaneous electroweak symmetry breaking (EWSB) is taking place at some electrically neutral point in the field space, and denoting the corresponding VEVs by one finds, after minimisation of the potential Eq.(2.4), the following necessary conditions : where λ a = λ 1 + λ 4 2 and λ b = λ 2 + λ 3 2 . The 7 × 7 squared mass matrix, can be cast, thanks to Eqs.(2.8, 2.9), into a block diagonal form of three 2 × 2 matrices, denoted in the following by M 2 ± , M 2 CPeven , and one odd eigenstate corresponding to the neutral Goldstone boson G 0 . The mass-matrix for singly charged field given by, is diagonalised by a 2 × 2 rotation matrix R θ ± , where θ ± is a rotation angle. Among the two eigenvalues of M 2 ± , one is equal to zero indentifying the charged Goldstone boson G ± , while the other one corresponds to the mass of singly charged Higgs bosons H ± given by, The mass-eigenstate H ± and G ± are rotated from the Lagrangian fields φ ± , δ ± as follows : Diagonalization of M 2 ± leads to the following relations involving the rotation angle θ ± : since the Goldstone boson G ± is massless. These three equations have a unique solution for sin θ ± and cos θ ± up to a global sign ambiguity. Indeed, Eq. (2.14) implies µ > 0 in order to forbid tachyonic H ± state, since our convention uses v t > 0. Hence, from Eq. (2.15), sin θ ± and cos θ ± should have different signs; one gets : with a sign freedom = ±1, which leads to negative tan θ ± .
As to the neutral scalar, its mass matrix reads: This symmetric matrix is also diagonalised by a 2 × 2 rotation matrix R α , where α denotes the rotation angle in the CP even sector.
After diagonalization of M 2 CPeven , one gets two massive even-parity physical states h 0 and H 0 defined by, Their masses are given by the eigenvalues of M 2 CPeven : Note that the lighter state h 0 is not necessarily the lightest of the Higgs sector. Furthermore, the only odd eigenstate leads to one massless Goldstone boson G 0 defined by G 0 = z 1 .
Once we know the above eigenmasses for the CP even , one can determine the rotation angle α which controls the field content of the physical states. One has : While they will have the same sign and tan α > 0 for most of the allowed µ and λ 1 , λ 4 ranges, there will be a small but interesting domain of small µ values and tan α < 0.

Constraints in the HTM0
The full experimental validation of the HTM0 would require not only evidence for the neutral and charged Higgs states but also the experimental values for the various field couplings in the gauge and matter sectors of the model. Crucial tests would then be driven by the predicted correlations among these measurable quantities. For instance, the µ and λ's parameters can be easily expressed in terms of the physical Higgs masses and the mixing angle α as well as the VEV's v d , v t , using equations (2.11), (2.24 -2.26). One finds The remaining two Lagrangian parameters m 2 H and M 2 ∆ are then related to the physical parameters through the EWSB conditions Eqs. (2.8, 2.9).
In the Standard Model the custodial symmetry ensures that the ρ parameter, ρ ≡ (2.33) Hence the modified form of the ρ parameter is ρ = Since we are interested in the limit v t v d , we rewrite GeV. From a global fit to EWPO one obtains the 1σ result [18], Consequently, in what follows, we adopt the bound The first equation is actually always satisfied thanks to the positivity of µ and the boundedness from below conditions for the potential. The second equation, quadratic in µ, will lead to new constraints on µ in the form of an allowed range The full expressions of µ ± are given by Let us discuss their behaviours in the favoured regime v t v d . In this case one finds a vanishingly small µ − given by and a large µ + given by Depending on the signs and magnitudes of the λ's, lower bound µ > 0 (positivity of Eq.(2.11)) or µ − will overwhelm the others. Moreover, these no-tachyon bounds will have eventually to be amended by taking into account the existing experimental exclusion limits. This is straightforward for the charged Higgs boson H ± , thus we define for later reference : where (m H ± ) exp denotes the experimental lower exclusion limit for the charged Higgs boson mass. So µ must be larger than µ min in order for the mass to satisfy this exclusion limit.
Upon use of Eqs.(2.7, 2.8, 2.9) in Eq.(2.4) one readily finds that the value of the potential at the electroweak minimum, V EWSB , is given by: Since the potential vanishes at the gauge invariant origin of the field space, V H=0,∆=0 = 0, then spontaneous electroweak symmetry breaking would be energetically disfavoured if V EWSB > 0.
One can thus require as a first approximation the naive bound on µ Always with the theoretical considerations, these are the requirements for tree-level perturbative unitarity, namely that the eigenvalues of the 2 → 2 scalar scattering matrix are below an absolute upper value given by 8π, and boundedness form below (BFB), which means that the potential in Eq. (2.4) has to be bounded from below. Obviously, at large field values, this potential is generically dominated by its quartic part : In Appendices A and B respectively, we demonstrate all the necessary and sufficient conditions for the BFB, and we give an introduction explaining the various sub-matrices for the unitarity in this model. These constraints are, BFB: Unitarity [19]: where we introduced the parameter κ that takes the values κ = 8, since we choose |Re(a 0 )| ≤ 1 In order to make the space parameter more compact, and working out analytically these two sets of BFB and unitarity constraints, one can reduce them to a more compact system where the allowed ranges for the λ's are easily identified. One can obtain a necessary domain for λ, λ b that does not depend on λ a , by considering simultaneously Eqs.(2.50 -2.52) together with Eq. (2.47), (the first two lines) We stress here that the above constraints define the largest possible domain for λ, λ b for any set the dependence on λ a has been explicitly separated from that on λ, λ b .
The reduced couplings g Hf f and g HV V of the Higgs bosons to fermions and W bosons are given in Tab.1, while the trilinear couplings to charged Higgs bosons can be extracted from the Lagrangian as L = g HH ± H ∓ HH + H − + g ZH ± H ∓ Z(∂ µ H + )H − + . . . . We will use the reduced HTM0 trilinear coupling of H and Z to H ± given by: relative to the SM Higgs couplings. α and θ ± are the mixing angles respectively in the CP-even and charged Higgs sectors, e is the electron charge, m W the W gauge boson mass and s W the sinus of the weak mixing angle.
The trilinear coupling g h 0 H + H − for the light CP-even Higgs boson is given by : The couplings for the heavy Higgs boson are obtained from the previous ones by simple sub-

Veltman conditions
To derive the Veltman conditions (VC), one just has to collect the quadratic divergencies [20]. There are various ways to do that, and to be on a safer side, we use the dimensional regularisation because this procedure ensures gauge as well as Lorentz invariances. To work out these quadratic divergencies, we follow exactly the procedure of calculations used in our previous work on the Higgs Triplet Model with hypercharge Y = 2 [9]. Moreover, it is worth to note that the main difference with [9] is the absence of the CP odd neutral Higgs A 0 and the doubly charged Higgs H ±± , from HTM0 spectrum. Also we have calculated the quadratic divergencies of the CP-neutral Higgs H 0 and h 0 tadpoles in a general linear R ξ gauge respectively, leading to results which are independent of the ξ parameters but depending on the model mixing angles.
As noted in [9], it is more convenient to combine these two results to get the tadpoles quadratic divergencies of the real neutral components of the doublet (h 1 ) and triplet (h 2 ) which are free of any mixing angles. After their VEV shifts, one finds, for the doublet: and for the triplet: where the simplified notations c w = cos θ W einberg and v = v 2 d + v 2 t have been used. Notice that the quadratic divergencies of the Standard Model are easily recovered in T d when the λ 1 and λ 4 couplings vanish, implying λ a = 0. Now to proceed with the implementation of the two VC's in the parameter space and the subsequent scans, we usually assume that the deviations δT t and δT d should not exceed the Higgs mass scale. In our analysis, we will allow them to vary within the reduced conservative range from 0.1 to 10 GeV. In addition to the theoretical constraints shown in Eqs.2.48-2.52, namely the unitarity, BFB and R γγ from LHC measurements, if the supplementary VC constraints are imposed as well, we see that the allowed region of the parameter space dramatically reduces and its extent depends on the value given to the deviation δT . This salient feature is illustrated in Fig.1, which exhibits the allowed domains in the (λ a , λ b ) plan. Our analysis shows that naturalness constraint is stronger than the other theoretical conditions and that deviations δT should be larger than 3 GeV in order to keep a viable model. Moreover, taken those constraints together, one might see that λ a will be restricted around ∼ 1.2.

Results and Discussions
Since HTM0 spectrum contains two CP even Higgs boson h 0 and H 0 , either h 0 or H 0 can be identified as the observed SM-like boson with mass ≈ 125GeV. Therefore, we are facing two  For each benchmark scenario, we investigate the allowed parameters space by the 1σ limit of the current Higgs data after run-II in the gg → H → γγ channel, reported by ATLAS µ γγ = 0.85 +0.22 −0.20 [21][22][23] and CMS µ γγ = 1.11 +0.19 −0.18 [24], which are consistent with the Standard Model expectation either for ATLAS or for CMS at 1σ. Furthermore, we can see that the errors reported are smaller from those reported at 7 ⊕ 8 TeV.   According to the ratio definition adopted in [25], we display the deficit of R γγ (h 0 ) in the left panel of Fig.5 as a function of H ± mass for various values of λ a and with m H 0 ≥ 140 GeV. As it can be seen, a mass about 255 GeV and above is allowed for H ± within +1σ of ATLAS value for λ a = −1.4. Once λ a increases, this lower bound decreases consistently to reach its lowest value around ∼ 197 GeV, given λ a > −0.5. This situation is exactly the opposite for CMS, where only the range 200 ≤ m H ± ≤ 250 (GeV) is excluded for λ a = 1.4. Besides, R γγ (h 0 ) tends towards its standard value for λ a = 0, and to 1 for large m H ± whatever the variation of λ a .

h 0 SM-like
In this scenario, the anti-correlation between R γγ (h 0 ) and R γ Z (h 0 ) is displayed in the left panel of Fig.5, taking into account the experimental tests at 1σ. At first sight, the R γ Z (h 0 ) deviation is almost nul relatively to its standard value, and contrary to what has been claimed in [12], R γγ (h 0 ) and R γ Z (h 0 ) are always anti-correlated, independently of λ a sign.

H 0 SM-like invisible decays
This section investigates the possible existence of a scalar state h 0 lighter than H 0 , with M 0 H ≈ 125. Such a scenario has attracted attention within a plethora of theoretical frameworks dealing with new physics beyond standard model [26,27], particularly those considering enlargement of the Higgs sector of the SM via doublet or triplet fields. However, to our knowledge, it has not yet been addressed in the HTM0. should not be of the same order of magnitude, indeed, to fulfil such situation, we request v t to be equal or slightly higher than 1 GeV for a given µ below 1 GeV. As a results, the parameter space is quite restricted offering many new interesting features. Indeed, the Higgs charged is very light with an upper bound on its mass about 180 GeV, as can been seen from Eq. (2.11). Also, for such small values of µ, the lighter CP-even state h 0 is mostly dominated by a triplet component and is typically very light mass as shown in Eq. (2.22). It is worth to notice that, according to Then, we plot in Fig.7 the branching ratios of the H 0 decays into bb, cc, W + W − ZZ, and into the invisible decay modes h 0 h 0 and H ± H ∓ . We clearly see that the branching ratios into h 0 h 0 and H ± H ∓ become dominant for non-vanishing values of |λ a |, as can be seen from Eq. (4.2) where the corresponding couplings get substancially large values. However, once λ a approaches zero, these decay channels fade away.
By the following, we fix v t = 1 GeV and λ b = 1, we present in Fig.8   From the left side of Fig.9, the ratio R γγ (H 0 ) reaches its SM-like value for λ a ≈ 0 and for the charged Higgs mass in the range 40 ∼ 160 GeV, while an excess up to 20% can be achieved for negative values of λ a . If ATLAS/CMS exclusions data at 1σ, is taken into account, then this excess is largely reduced to less than 10%. As a byproduct, this analysis sets up a lower limit on the m H ± of order ∼ 115 GeV (for λ a = −0.2). In addition, R γγ (H 0 ) remains below it SM value when λ a > 0, even for m H ± above this lower value. At last, we study correlation of consistent with ATLAS and CMS data. A first analysis has been performed in [12]. This analysis used an intriguing and unjustified hypothesis considering the charged Higgs mass equals to the neutral ones. In this model, this possibility is excluded by theoretical constraint as we will show shortly. But first, we will demonstrate that the parameter space is restricted further by an additional constraint, induced by the Higgs mass degeneracy, and leading to a severe control of the potential parameters.
The two eigenvalues m ± (with m − = m 2 h 0 < m + = m 2 H 0 ), representing the squared masses of h 0 and H 0 , are : Then where ∆M , the difference of masses between the two neutral Higgs H 0 and h 0 is set to about 1 GeV, corresponding to the detector inability to resolve two nearly Higgs signals, and M ex is the experimental Higgs boson mass ≈ 125 GeV. Taking into account these considerations one gets (A − C) 2 + 4B 2 ≤ 2M ex ∆M , that obviously leads to two constraints: |B| ≤ M ex ∆M and |A − C| ≤ 2 M ex ∆M .
The first constraint reads as: while, for small ratio of the two vevs vt v d , the second constraint reduces to, Since the ratio 2 Mex∆M v d is about 1 GeV, these two relations simplify to |2 λ a v t − µ| ≤ √ 2 GeV and µ λ ≈ 4 v t , providing strict bounds to the three potential parameters µ, λ and λ a , hence severely reducing the allowed regions in the parameter space, as it is illustrated in Fig.10.
This feature has a dramatic effect on the discrepancy between the neutral and charged Higgs masses as can be seen from Fig.2. In such figure, the Higgs bosons masse behaviours are plotted as a function of the µ parameter; these values satisfy the above resulting relation in the degenerate case. The seemingly constant m 2 h 0 for µ > µ c and constant m 2 H 0 for µ < µ c are clearly achieved around the critical value µ c ≈ 2.1 GeV. Contrary to what one might think, if we take the Higgs bosons masses as inputs [12], such a situation matches a splitting between the charged Higgs boson mass and the H (= h 0 = H 0 ) degenerate state mass in the range of ∆ m = m H ± − m H ≈ 51 GeV. Hereafter we define the diphoton signal strength R γγ by the following quantity, and by the same way R γ Z is introduced. In this scenario, the charged Higgs boson loops are included with the g Hww , g Hf f couplings given by Table.1.
Finally, we display in Fig. 12, we have plotted R γγ versus R γ Z in mass degenerate scenario for various values of λ a . From this plot one can see that the correlation is always positive whatever the value of λ a . We also note that no noticeable enhancement can be achieved, since most part of the parameter space is drastically constrained by a constant charged Higgs mass at about m H ± ∼ 176 GeV; as shown form Fig.2.

Conclusion
In this paper, we have discussed some features of the Higgs triplet model with null hypercharge (HTM0), an extension of the SM with a larger scalar sector. First, we have shown that the space parameter of HTM0 generally constrained by unitarity and boundedness from below, is severely reduced when the modified Veltman conditions are imposed. Then, we have investigated some Higgs decays, including Higgs to Higgs decays, in light of LHC data, either when h 0 is the SMlike Higgs or when the heaviest neutral Higgs H 0 is identified to the 125 observed GeV Higgs.
In addition, we have analysed the degenerate scenario and shown that LHC signal strengths    Fig.11, except for λ a .