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The light Higgs in supersymmetric models with Higgs triplets

In supersymmetric models the presence of Higgs triplets introduce new quartic interactions for the doublets that may raise the mass of the lightest CP-even field up to 205 GeV. We show that the complete effect of the triplets can be understood by decoupling them from the minimal sector and then analyzing the vacuum and the spectrum of the two-Higgs doublet model that results. We find that the maximum value of m_h is only achieved in a very definite region of the parameter space. In this region, however, radiative corrections decrease the bound to 190 GeV.

1. Introduction. In the minimal supersymmetric standard model (MSSM) the quartic couplings of the Higgs fields are not free parameters, they are related to the SU(2) L × U(1) Y gauge couplings [1]. This implies that the tree-level mass of the lightest CP -even scalar field is smaller than M Z : where tan β is the ratio H 2 / H 1 of vacuum expectation values (VEVs) of the two Higgs doublets.
There are two different mechanisms that can raise this upper bound on m 2 h . The first one is due to the fact that supersymmetry (SUSY) is broken. The SUSY values of the quartic couplings suffer radiative corrections [2] from top loops proportional to h 4 t ln(m 2 t /m 2 t ), where h t is the top Yukawa coupling and m t and mt are the fermion and the scalar masses, respectively. This translates into corrections to the Higgs mass of order ∆m 2 h ≈ (90 GeV) 2 . Also the soft SUSY-breaking trilinear V ⊃ A tt ct H 0 2 may have an impact on m 2 h . At one loop it gives nonlogarithmic corrections [3] proportional to h 2 The other mechanism to increase the bound on m 2 h requires the presence of extra Higgs fields and couplings, namely, gauge singlets [4] or triplets [5] and couplings of these fields with the doublets. Trilinears in the superpotential W involving two doublets and the extra field introduce new quartic interactions for the scalar doublets and then new contributions to m 2 h .
When introducing extra Higgs fields, however, there is another effect competing with the positive effect on m 2 h of the quartic couplings. It is due to the mixing between the lightest and heavier states. Such a mixing tends to decrease the smallest eigenvalue in the Higgs mass matrix. In a particular escenario one would expect that the new (arbitrary) parameters present, soft masses and trilinears, may be adjusted in order to cancel the mixing. In that case, for real VEVs the lightest neutral state would be obtained diagonalizing the 2 × 2 submatrix defined by the CP -even scalars in the doublets. Then the maximum value of m h would be obtained for heavy CP -odd states.
In particular, let us consider the presence of a pair of SU(2) L triplets (T, T ) of (−1, +1) hypercharge: These fields admit a term in the superpotential of type An analogous term can be obtained exchanging T ↔ T , where Although χ is in principle a free parameter, it is constrained by the following argument. Its evolution with the energy scale is given by The possibility to integrate the electroweak and the grand unification (GUT) scales is here a main motivation, as in any SUSY model. However, large initial values of χ would become nonperturbative before the GUT scale M X . Evolving the model down from M X one finds that at low energies χ is always smaller than ≈ 0.9. This value could be enough to put the bound in Eq. (4) around 205 GeV [5].
Our objective in this letter is to find out whether the bound in Eq. (4) can be saturated or not. We discuss a simple way to understand the effect of adding extra triplets (or any other vectorlike Higgs field) of mass m = M + O(m SU SY ). We decouple these scalar fields from the minimal sector keeping the terms of first order in 1/m. The effective model that results is a particular two-Higgs doublet model that depends on the physics of the triplets. The decoupling effects are equivalent to the mixing in the complete mass matrix, but much simpler to analyze. The method is justified when m 2 ≫ (v 2 , χ 2 4π m 2 SU SY ). In the triplet model this seems to be a necesary requirement because the VEVs grow as the inverse of their mass, and if T, T ≥ 10 GeV it predicts an unacceptable value of the ρ parameter (a different Weinberg angle measured from gauge boson masses and charged currents) [6]. We compare the value for m h obtained in the effective model with the exact numerical solution in the complete triplet model and find that the agreement is good even for m ≈ m SU SY . The method allows us to identify the only region in the parameter space that saturates the bound in Eq. (4). This region has a very definite pattern of SUSY breaking terms, with large radiative corrections that partially cancel top quark effects.
2. The triplet model. Let us consider the neutral Higgs sector of the model (from now on we drop the 0 superscript to indicate neutral fields). We include in W the terms Adding soft SUSY-breaking terms the relevant part of the scalar potential is whereg = (g 2 Y + g 2 L )/8 and all the fields are neutral. Field redefinitions can be used to set m 12 , A χ and χM real and positive. We asume for simplicity that A 2 M and χµ are real, but this does not guarantee that all the VEVs are real and positive (which would be the case if χµ ≥ 0 and A 2 M ≤ 0). The size of the mass parameters above is ( It is convenient to express the fields in terms of moduli and phases: The determination of the minimum of the potential in Eq. (7) and of m 2 h requires much algebra (in the general case with complex VEVs, the diagonalization of a 8 × 8 matrix). We propose, instead, to integrate the fields (T, T ) out, analize the effective model that results, and check numerically that for any particular choice of parameters the complete and the approximate models give the same spectrum for the four lightest fields.
3. The approximate model. We first rewrite (T, T ) in terms of mass eigenstates: where s α = sin α, c α = cos α and tan 2α = . Their masses are Integrating T 1 and T 2 out it results the two Higgs doublet model where the contributions of the triplets toχ 2 , λ 3 and λ 7 come from the diagrams (a), (b) and (c) in Fig. 1, respectively. We obtaiñ Higher dimensional operators would introduce corrections of order v 2 /M 2 1,2 , whereas one-loop corrections will be small if χ 2 4π m 2 SU SY < M 2 1,2 . To find the minimum and the spectrum of this model we express the fields in terms of moduli and phases: Then We can use an hypercharge transformation to set θ 1 = 0. The minimum conditions give then θ 2 = 0, v 1 and v 2 . In the 4 × 4 mass matrix the CP -odd sector, It is now straightforward to find the mass eigenvalues. In the CP -odd sector there is, in addition to the massless Goldstone, a field of mass m 2 A =m 2 12 /(sin β cos β). The lightest Higgs is in the CP -even sector, together with a field of mass m 2 H = O(m 2 12 ). m 2 h is bounded to be smaller than (M 33 M 44 − M 2 34 )/(M 33 + M 44 ), value that is saturated in the limitm 2 12 ≫ v 2 . In this limit we can obtain an approximate expression for m 2 h : where the couplings have the value specified in Eq. (12).
Before discussing how to tune the parameters in order to approach the bound in Eq. (4), let us check the efficiency of our approach. We have computed numerically the spectrum of light fields in the effective and in the complete triplet models for many different values of the parameters, changing the signs of A 2 M and χµ. We obtain that the results for m h in both models always agree within a 2% margin. For example, let us take a triplet model with M = 3 TeV, χ = 0.7,  4) is never approached. For M much larger than the SUSYbreaking massesχ 2 goes to zero, and the triplets decouple (as expected). If 2A χ c α is large and tends to cancel χMs α in one of the terms definingχ 2 , then in the other term χMc α + 2A χ s α will be large andχ 2 goes also to zero. The contribution to m 2 h proportional to λ 3 is always negative, whereas the one proportional to λ 7 can be positive if χµ < 0. However, in this case a sizeable contribution would require that all the couplings (χM, 2A χ and χµ) are of the same order, implying complex VEVs and mixing of the light Higgs with the (heavy) CP -odd sector. Such a mixing also would lower m 2 h . In any case, the λ 3,7 terms are not relevant in the region of large tan β, where the bound in Eq. (4), if saturated, allows a light Higgs of up to 205 GeV [5].
The maximum value of m h would be obtained for a SUSY mass M and a SUSYbreaking trilinear A χ both much smaller than the SUSY-breaking masses of the triplets: . In this limit the scalar triplets decouple but the quartic coupling that they introduce in the Higgs doublet sector remains. In consequence, the tree-level bound in Eq. (4) would be approached.
This very definite pattern of SUSY breaking parameters, however, has obvious implications at the quantum level. We have here a large splitting between the fermion and the scalar components of the triplet superfields and also a large Yukawa coupling (see Eq. (6)), very much like in the top quark sector. To estimate the radiative effects let us focus on the region of large tan β, where the light neutral Higgs φ is basically H 2 . We simplify and assume that M ≈ m t and all the SUSY-breaking masses coincide, m SU SY ≈ 1 TeV, with a generic suppression of scalar trilinears that makes negligible nonlogarithmic corrections. Below m SU SY we have the standard model, with V = m 2 φ φ † φ + λ 4 (φ † φ) 2 , plus the fermion components of the triplets. At m SU SY the quartic scalar coupling is and its running down to the electroweak scale is given by In the MSSM one obtains λ(m 2 Here there is a partial cancellation in the β function, with and where we have neglected the electroweak gauge couplings and the evolution of h t and λ with the scale (both with decreasing effect on the estimated size of radiative corrections).

Conclusions.
The mass of the Higgs in the MSSM is constrained to be smaller than M Z at the tree level and smaller than around 130 GeV once radiative (SUSY-breaking) corrections are included. In more general models there are new fields and new quartic Higgs interactions raising m h . However, the new fields also introduce mixing with the Higgs doublets, which decreases m h . In models with gauge singlets it is easy to see that this mixing can be fine tuned to zero. However, in models with triplets the larger number of fields and parameters makes the analysis too complicated.
We have presented a method that allows to understand the complete effect of the triplets on m h . It is based on an effective model that results integrating the triplets out and keeping only their effect on the quartic couplings of the doublets. In this effective model the bound on m h is very simple to estimate (see Eq. (16)). The procedure gives an excellent approximation for triplet masses larger than the electroweak scale v ≈ 174 GeV, even if the triplets and the doublets have similar masses.
Using this method we show that, in order to modify substantially the MSSM values of m h , the scalar triplets must have a small SUSY mass (their mass must be basically a SUSYbreaking term) and their trilinears must be suppressed respect the masses. The large splitting in the triplet supermultiplet, with heavy scalars and light fermions, together with the large Yukawa coupling χ and the absence of scalar trilinears, define a very clear pattern of SUSYbreaking parameters with implications on the size of radiative corrections. We have estimated these corrections and obtained that they are significantly smaller than in the MSSM.
We conclude that the triplet model is able to provide values of m h of up to 190 GeV. The value 205 GeV seems an overestimate, since it would require M ≈ m SU SY and large SUSY-breaking trilinears (it is based on a cancellation between trilinears of order m SU SY that we show cannot take place). The value m h = 190 GeV is still larger than m h = 155 GeV [7] of the singlet model with intermediate vectorlike matter. Probably, the triplet model has ingredients that make it a less appealing framework from a model building point of view: the need to avoid triplet VEVs, the need for four pairs of colour triplets at low energy to obtain gauge unification, the need to avoid extra matter with electroweak charges at intermediate scales (g Y and g L are near their perturbative fixed point values), or the need to incorporate a generalized R-parity to be realistic (the usual Z 2 matter parity of the MSSM cannot avoid here, for example, unacceptable neutrino masses). However, the pattern of SUSY-breaking terms (large scalar masses versus trilinears) and the presence of double charged leptons at ≈ 200 GeV required to saturate the bound define an interesting region of its parameter space. For example, there the triplet VEVs are naturally small: easily within the experimental limit of 10 GeV.