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Theory and Phenomenology of Two Higgs Doublet Type-II Seesaw Model at the LHC Run-2

We study the most popular scalar extension of the Standard Model, namely the Two Higgs doublet model, extended by a complex triplet scalar (2HDMcT). Such considering model with a very small vacuum expectation value, provides a solution to the massive neutrinos through the so-called type II seesaw mechanism. We show that the 2HDMcT enlarged parameter space allow for a rich and interesting phenomenology compatible with current experimental constraints. In this paper the 2HDMcT is subject to a detailed scrutiny. Indeed, a complete set of tree level unitarity constraints on the coupling parameters of the potential is determined, and the exact tree-level boundedness from below constraints on these couplings are generated for all directions. We then perform an extensive parameter scan in the 2HDMcT parameter space, delimited by the above derived theoretical constraints as well as by experimental limits. We find that an important triplet admixtures are still compatible with the Higgs data and investigate which observables will allow to restrict the triplet nature most effectively in the next runs of the LHC. Finally, we emphasize new production and decay channels and their phenomenological relevance and treatment at the LHC.


I. INTRODUCTION
After the discovery of a Standard-Model-like Higgs boson at the Large Hadron Collider (LHC) in 2012 [1, 2], the Standard Model (SM) of particle physics has been established as the most successful theory describing the elementary particles and their interactions. Despite its success, the SM has several drawbacks that have suggested theoretical investigations as well as experimental searches of physics beyond it. As an example, the observed neutrino oscillation cannot be explained within the SM [3]. Indeed, although the SM Higgs field is responsible for the generation of the masses of all known fundamental particles, it is unable to accommodate the tiny observed neutrino masses.
Within a renormalisable theory where new heavy fields are introduced, the neutrino masses are generated via the dimension-five Weinberg operator [4], this is the so called seesaw mechanism.
Different realisations of such a mechanism can be classified into three types: type I [5][6][7][8][9] in which only right-handed neutrinos coupling to the Higgs field, type II [10][11][12][13] where a new scalar field in the adjoint representation of SU (2) L and type III [14] which involves two extra fermionic fields.
In the above seesaw mechanisms heavy fields are supplemented to the SM spectrum in such a way the desired neutrino properties are reproduced once the electroweak symmetry is broken.
In the type II seesaw model, also dubbed Higgs triplet models (HTM), [15][16][17][18][19][20], the SM Lagrangian is augmented by a SU (2) L scalar triplet field ∆ with hypercharge Y ∆ = 2. In HTM, neutrino masses are proportional to the vacuum expectation value (vev) of the triplet field. Hence, the small values of neutrino masses is guaranteed by the smallness of the triplet vev assumed to be less than 1 GeV and the non-conservation of lepton number which is explicitly broken by a trilinear coupling µ term in the HTM scalar potential. The latter, protected by symmetry, is naturally small, thus ensuring small neutrino masses. The model spectrum contains several scalar particles, including a pair of singly charged Higgs boson (H ± ) and doubly charged Higgs boson (H ±± ). In addition, it also predicts a CP-odd neutral scalar (A 0 ) as well as two CP even neutral scalars, h and H. The lightest scalar h has essentially the same couplings to the fermions and vector bosons as the Higgs boson of the SM within a large region of the HTM parameter space.
By the new discovery of 125 scalar boson at the LHC, the phenomenology of two-Higgs-doublet mod (2HDM) has been investigated broadly in the literature. In the present work, due to the similarity in mass generation mechanism between type-II seesaw and the Higgs mechanism, we extend HTM and focus on the two Higgs double model extension to the type-II seesaw model, displaying phenomenological characteristics notably different from the scalar sector emerging from the HTM. In this context, we study several Higgs processes giving rise to the production times branching ratios of heavy Higgs bosons and focus on. Unlike most of the earlier studies, we consider here a framework where both type I and type II seesaw mechanisms are implemented and contribute to neutrino mass generation. We consider a high integrated luminosity of LHC collisions at a centre-of-mass energy of 13 TeV.
The content of the paper is laid out as follows. In Sec. II, we derive some crucial features of the 2HDMcT, with a focus on the particle content and the scalar potential of the model, followed by discussions on the minimization conditions of the scalar potential and the scalar mass spectra.
Besides, in this work we investigate the impact of two new terms introduced in the model scalar potential. Sec. III, is devoted to the study of the theoretical constraints on the scalar potential parameters from tree-level vacuum stability and perturbative unitarity of the scalar sector. In Sec.
IV, we further impose the LHC constraints associated with the 125 GeV Higgs boson and its signal strength to delimit the parameter space. Finally, we present some phenomenological aspects in Sec.??, benchmark points in Sec.VI and summarise our findings in Sec. VII.

A. The Higgs sector
In a model with two Higgs doublets H 1,2 , the Two Higgs Doublet Type-II Seesaw Model (2HDMcT) contains an additional SU (2) L triplet Higgs field ∆ with hypercharge Y = 2 and lepton number L = −2, The most general renormalizable and gauge invariant Lagrangian of the 2HDMcT scalar sector is given by, where the scalar potential V (H, ∆), symmetric under a group SU (2) L × U (1) Y , reads as [21] V (H i , ∆) = V (H 1 , H 2 ) + V (∆) + V int (H 1 , H 2 , ∆) where : V (∆) = m 2 ∆ T r(∆ † ∆) +λ 8 (T r∆ † ∆) 2 +λ 9 T r(∆ † ∆) 2 (5) In the above, m 2 i , i=1,2,3 and m 2 ∆ are mass squared parameters, λ i , i=1,...,5 are dimensionless couplings not related to the triplet,λ i , i=8,9 are dimensionless couplings related to the triplet field, while µ i , i=1,2,3 with λ i , i=6,...,9, are dimensionless couplings that mixe all three Higgs fields. In Eq. 6, T r denotes the trace over 2 × 2 matrices, where for convenience we have used the 2×2 traceless matrix representation for the triplet. Also, the potential defined in Eq. 6 exhausts all possible gauge invariant renormalizable operators. For instance, a terms of the form λ 10 H † 1 ∆ † ∆H 1 and λ 11 H † 2 ∆ † ∆H 2 [21], which would be legitimate to add if ∆ contained a singlet component, can actually be projected on the λ 6,7 and λ 8,9 operators appearing in Eq. 6 thanks to the identity which is valid because ∆ is a traceless 2 × 2 matrix. Subsequently, we will assume that all these parameters are real valued. Indeed, apart from the µ i terms, all the other operators in V (H i , ∆) are self-conjugate so that, by hermicity of the potential, only the real parts of the λ's and the m 2 1 , m 2 2 , m 2 ∆ mass parameters are relevant. As for µ i , the only parameters that can pick up a would be CP-phases, these phases are unphysical and can always be absorbed in a redefinition of the fields H i and ∆. One thus concludes that the 2HDMcT Lagrangian is CP conserving. The electroweak symmetry is spontaneously broken when the neutral components of the Higgs fields acquire vacuum expectation values v 1 , v 2 and v t . Thus we can shift the Higgs fields in the following way, finding minimization conditions, or tree-level tadpole equations, given by where it is safe to take T i = 0 ( i=1,2,3) which leads to, with λ + ij = λ i + λ j ,λ + ij =λ i +λ j and λ 345 = λ 3 + λ 4 + λ 5 . If the terms associated to v 3 t are omitted in Eq. 11, then we can derive a new expression for v t as a function of the triplet scalar mass, Furthermore, for m ∆ sufficiently large compared to v 1,2 , we see that the above formula reduces to, v t ∼ µ 1 v 2 1 / √ 2m 2 ∆ , which is referred as type II seesaw mechanism. 1 The 2HDMcT model has altogether 24 degrees of freedom: 21 parameters originating from the scalar potential given by Eq. 3 and tree vacuum expectation values of the Higgs doublets and triplet fields. However, thanks to the three minimisation conditions, the W gauge boson mass and the correct electroweak scales, the parameters m 2 1 , m 2 2 , m 2 ∆ and v can be eliminated.

B. Higgs masses and mixing angles
In what follows, we will use Eqs. 9, 10 and 11 to trade the mass parameters m 2 11 , m 2 22 and m 2 ∆ for the rest of parameters given in the potential. Thus, the 14 × 14 squared mass matrix is given by, by denoting the corresponding VEV's Eq. 3 can be recast in a block diagonal form of one doubly degenerate eigenvalue m 2 H ±± and three 3 × 3 matrices denoted in the following by M 2 ± , M 2 CP odd and M 2 CPeven . The bilinear part of the Higgs potential is then given by: at tree-level. The elements of these mass matrices are explicitly presented below.
1 In the absence of µ1, µ2 and µ3 the m 2 ∆ becomes negative leading to a spontaneous violation of lepton number. The resulting Higgs spectrum contains a massless triplet scalar, called Majoron J. This model was excluded by LEP.

Mass of the neutral scalar field
In the basis (ρ 1 , ρ 2 , ρ 0 ) the neutral scalar mass matrix reads : Its diagonal terms are, while the off-diagonal terms are given by, The mass matrix can be diagonalised by an orthogonal matrix E which we parametrise as where the mixing angles α 1 , α 2 and α 3 can be chosen in the range the rotation between the two basis (ρ 1 , ρ 2 , ρ 0 ) and (h 1 , h 2 , h 3 ) diagonalises the mass matrix M 2 CPeven as, and leads to three mass eigenstates, ordered by ascending mass as: One choice of input parameters implemented in 2HDMcT consistes to use the following hybrid parameterisation, in which tan β = v 2 /v 1 = tan θ ± 1 = tan β 1 . In Appendix B, we discuss the second choice of input parameters in the physical basis 2HDMcT.
Using the Eqs. 32 and 16, one can easily express the reset of Lagrangian parameters in terms of those given by. 34. These are given by where v 0 = v 2 1 + v 2 2 and and β j (j=1,2,3) and 4 Higgs bosons masses, two of them correspond to the charged states H ± 1,2 masses, while the two others are matched to CP odd states A 1,2 as discussed previously.

C. Yukawa and gauge bosons textures
L Yukawa contains all the Yukawa sector of the SM plus one extra Yukawa term that leads after spontaneous symmetry breaking to (Majorana) mass terms for the neutrinos, without requiring right-handed neutrino states, where L denotes SU (2) L doublets of left-handed leptons, Y ν denotes neutrino Yukawa couplings, C the charge conjugation operator. The Z 2 symmetry is imposed in order to avoid tree-level FCNCs.
Furthermore, and in terms of the various α i which appear in the expressions of E ij matrix elements, we liste in table-I, all the CP even h i (i=1,2,3) Yukawa couplings for both type I and type II in the model. On the other hand, expanding the covariant derivative D µ , and performing the usual transformations on the gauge and scalar fields to obtain the physical fields, one can identify the Higgs couplings H i to the massive gauge bosons V = W, Z as given in table-II. Note that in our model, The triplet field ∆ does directly couple to the SM particles, so a new contribution will be appears, and the two couplings C h i V (V = W ± , Z) differs from one to another by a factor 2 associated to v t .

B. Boundedness From Below (BFB)
In order to derive the BFB constraints, we require that the vacuum is stable at tree level.
Generically, this means that the scalar potential has to be bounded from below at large scalar fields values in any directions of the field space. Thus, the potential is asymptotically dominated by the quartic terms, Therefore, to obtain in this model the full set of BFB conditions valid for all directions, it is suitable to only consider V (4) (H 1 , H 2 , ∆) than to study the full scalar potential. As an illustration, consider for instance the case where there is no coupling between doublets H i and triplet ∆ Higgs bosons, i.e. λ 3 = λ 4 = λ 5 = λ 6 = λ 7 = λ 8 = λ 9 = 0. Obviously, one can see that, Given this, all the other necessary and sufficient conditions for stability are listed in Appendix D and can be read as, with the corresponding 2HDM BFB constraints well known in the literature, whereas the Ω i (i = 1, .., 5) stand for the new contraints added as follows, Ω 3 = λ 9 2λ 2 ≥ |λ 9 | λ 8 +λ 9 or 2λ 7 + λ 9 + 4λ 2λ9 − λ 2 9 1 + 2λ 8 λ 9 besides other are mentioned in Appendix D.

C. Bounds from theoretical constraints
In order to validate our rough analytical understanding and to further explore the impact of the unitarity and BFB constraints we use the numerical machinery. There are many possibilities what to use as input parameters. Naively using the initial Lagrangian parameters will hardly produce points which are in agreement with the Higgs measurements. Therefore, we trade hybrid parameterization. With that choice, the full set of parameters is given by Eq. 34. As first step, we show in Fig. 1, all generated points in the planes λ 6 vs λ 8 , λ 7 vs λ 9 andλ 8 vsλ 9 .
In the 2HDMcT, the presence of the triplet field implies a new scalar couplings λ 6,7,8,9 etλ 8,9 and the vacuum stability condition requires that not only λ 1,2 ≥ 0 but also λ 7,8 ≥ 0 with the conditions in Eq. (47). By varying λ 1,...5 in [−8π : 8π], we show in Fig. 1, the allowed domains on λ 6,7,8,9 etλ 8,9 plans without conflicting with the theoretical constraints. We assume that the seesaw mechanism at the TeV scale that we consider here, is a "low-energy" effective phenomenological manifestation at high energy scale. We therefore assume that the couplings remain perturbative up to GUT scale.

A. Oblic parameters
Strong indirect probe of physics beyond SM is provided by the oblique parameters S, T and U.
More than that, the calculations of several observables check their dependences on those oblique parameters, for example, and not as a limitation, the ρ parameter [24], i.e. ρ = m 2 W /m 2 Z c 2 W . In the SM, this parameter is equal to 1 at tree level. In the THDMcT, the new triplet contributions to W and Z masses readily form Eq. 14 and the kinetic terms in Eq. 2 leads to write, the modified form of the ρ parameter reads The impact of a 2HDMcT in the so-called electroweak precision requires that ρ to be close to its SM value: ρ = 1.0004 +0.0003 −0.0004 [25]. Then, one gets an upper bound for v t < 5 GeV. Furthermore, the major contribution to the T-parameter comes from the loops involving the scalar triplet when v t equal to zero or less. Also, since deviations from the Standard Model expectations in U are negligible [26], then we will assume the latter to be zero and consider only S and T . We compute their χ 2 ST contribution through, For U = 0,the electroweak fit gives the values.
The contribution of the scalar triplet to S and T reads as [27] and m 0 = m h 2 , m h 3 , while s w stands for the sinus of the Weinberg angle θ w . The function ξ(x, y) and F (x, y) are defined by, We include both the measured signal rates from the ATLAS and CMS Run I and Run II and their combinaisons in our study via the public code HiggsSignals-2.2.1beta [30].

V. LIGHT AND HEAVY HIGGS PHENOMENOLOGY
In this section we study the influence of the constraints presented in the previous section (indirect, LEP, Tevatron and LHC constraints) on the free parameters. For this purpose we generate a set of 10 9 points randomly for each of the two different types of model defined in Tables I and II with random values for each of the free parameters. The available ranges we use in the simulation are given: λ 1 , λ 3 and λ 4 are set respectively to the values 0.15, 1.6 and 1.6 for the sake of simplification.
The ranges for λ 6 , λ 7 , λ 8 , λ 9 ,λ 8 andλ 9 resulted from the unitarity and boundedness constraints as can be seen from Fig. 2. For simplicity, let us classify the dimensionless parameters in the scalar potential into two different sets according to the following two types of constraints: • First set of constraints includes the unitarity, vacuum stability and BFB constraints as well as non-tachyonic masses. We refer to this set as C 1 .
• The second set of constraints contains C 1 and constraints from Higgs data. We refer to this set as C 2 .
Looking now at the planeλ 8 vsλ 9 are not very much restricted by C 2 constraint due to the fact thatλ 8 andλ 9 are always dependent of the vev of scalar triplet. In Fig. 3, we present the allowed points in the (sign(C h 1 V ) sin(α 1 − π/2) , tan β) plane, that passes all constraints in type II (left) and type I (right) at 1σ, 2σ and 3σ. In type-II, one can appreciate that the mixing angle α 1 seems more constrained than in type-I. Results are shown by imposing the conditions C 1 and C 2 . The later has a strong impact on how the mixing angles are constrained. Fig.3(left) displays wrong-sing Yukawa couplings scenario at 2σ. In Fig. 4, the allowed ranges are plotted in the planes m A 1 , m A 2 , m h 3 and m H ±± vs m H ± 2 . The left-columns panels corresponds to type-II, the right-column to the type-I, all points passed the constraints mentioned above at 1σ (yellow), 2σ (blue) and 3σ (red). As can be seeing most of the masses can be light in type-I less than 300 GeV. Looking only at the yellow points, those which pass the C 2 constraint, we can see that the m A 2 , m H ± 2 , m h 3 and m H ±± masses are bounded in type II where we find that most of the yellow points lie in the ranges m H ± The sensitivity of the Higgs couplings of h 1 at the LHC is not appreciably better than 20%, leaving thus a significant window of opportunity for new physics. Here we investigate the correlations among relevant couplings within the C1 and C 2 constraints. In Fig 5, we show these correlations which are consistent within the BRs of m h 1 = 125 GeV within 1σ (yellow) and 2σ(blue). This is related to the fact that the central values of some Higgs couplings deviate from the SM, which strongly restrict the range of deviations from the SM. While the decay h 1 → Zγ is not yet observed at the LHC, the correlation between the h 1 → W W, γγ and h → bb couplings can now be measured by the experiments. The ratio of the BR of W W to that γγ can be measured with an accuracy better than 5% and an integrated luminosity that is expected to be accumulated by the High Luminosity LHC. In the following, we will scrutinize the impact of the searches for heavy Higgs particles in 2HDMcT ordered by their decay products. First we will address the bosonic decays to V V (V = γ, Z, W ± ) and fermionic mode τ − τ + branching ratios of the neutral Higgs (h 2 , h 3 , A 1 and A 2 ) using both CMS and ATLAS Higgs data for 8 TeV [31][32][33] and 13 TeV [34][35][36][37][38]. After that, we will turn towards the pair production of h i h j . The narrow width approximation will be applied throughout this section, we will comment on its validity at the end of the text section. We define the cross section as, where σ SM (gg → φ) is the cross section for Higgs production in gluon fusion in the SM, and are the partial decay rates in 2HDMcT and SM respectively. We use Sushi v1.6.0 public code [39,40] at NNLO QCD to perform the calculation of the cross sections for Higgs production in gluon fusion (ggF ) and bottom-quark annihilation in the SM at 13 TeV. The relative coupling of h 2,3 to the two vector bosons ZZ and W W is universal and type independent. However, the production of the h 2,3 differs between the types. In Fig. (6)   For the W W searches, Run-I data dictate the limit until 350 GeV and the high mass range is dominated by Run-II data. Direct searches for a heavy Higgs decaying to two photons constrain σ × BR by roughly one magnitude compared the ATLAS limit in CP-even decay modes (see Fig. 7). The searches in the di-photon decay channel of a pseudoscalar Higgs yield a suppression of σ × BR one to three orders of magnitude compared to the ATLAS limit for m A 1,2 ≤ 400 GeV certain intermediate σ × BR regions for low m A 1,2 are disfavoured by the prior.
In Fig. 8 we plot the production cross section in proton proton fusion times τ τ branching ration after imposing C 3 constrains. The errors for χ-square fit are 99.7% CL (red), 95.5% CL (blue) and 68% CL (yellow). The red(green) solid line in Fig. 8 is the upper limit on the cross-section times branching ration from the AT LAS 13 TeV results [45] (CM S 13 TeV results [46]). The colored lines in Fig. 7 denote the observed limits from 13 TeV ATLAS data [47].
Finally, in Fig. (9) we present a sample of points generated for Higgs pair production in the scenario where the observed Higgs boson is the lightest scalar h 1 (top panels) and the next-tolighest scalar (bottom panels). We present cross sections as a function of the masses of the two new scalars that can be involved in the chain decay contributions. We overlay three layers of point for which the total cross section is within 3σ (red), 2σ(blue) and 1σ respectively at the LHC Run-2. In the upper plots, where h 1 chain decays become possible above 250 GeV. In Fig.(

Appendices Appendix A: Diagonalization Matrices
The rotation between the physical and non-physical states are given by: The matrices elements for each sector are given below.

charged sector
In terms of the mixing angles θ where in the hybrid parameterization P I , these elements are given by,

CP odd sector
For the terms of odd sector, the rotation matrix O can be expressed in terms of the mixing angles βi as,

Appendix B: Setting the model parameters
There are currently two choices of input parameters implemented in 2HDMcT. To ease export formalities for second set, a simplified approximations should be adopted for such purpose. Indeed, the Cij(i, j = 1, 2, 3) and Oij(i, j = 1, 2, 3) are nearly the same as vt << v0, which involves Using the Eqs. B1, B2 and B3, the matrices C and O become, One can then provide the following expressions for the non-physical parameters, with t x = tan x, ct x = 1/ tan x, c x = cos x, s x = sin x, s 2x = sin 2x, cs x = 1/ cos x, se x = 1/ sin x

Appendix C: Unitarity Constraints Matrices
The unitarity constraints are derived in the basis of unrotated states, corresponding to the fields before electroweak symmetry breaking. The quartic scalar vertices have in this case a much simpler form than the complicated functions of λ i , α 1 , O ij and C ij obtained in the physical basis (H ±± , H ± 1 , H ± 2 , G ± , H 1 , H 2 , H 3 , A 1 , A 2 and G 0 ) of mass eigenstate fields. The S-matrix for the physical fields is related by a unitary transformation to the S-matrix for the unrotated fields.
At the points given in D41, the eigenvalues of H can be expressed in terms of λ s by solving the following equation: Hence, at this stage, and for G III to be convex, the eigenvalues Λ 1 and Λ 2 for each point should be positive quantities. Within sight of their long and complicated expressions, we will not show them here, though they would be taken into account in our calculations. Furthermore, one must also make sure that for each point, both 0 ≤ η 0 , ζ 0 ≤ 1 for which we request that, G III (η 0 , ζ 0 ) > 0 ∀ ξ ∈ [0, 1] and z ∈ [−1, 1].