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Towards the effective potential of the Littlest Higgs model

We compute the relevant parameters of the combined Higgs and \phi scalar effective potential in the Littlest Higgs (LH) model. These parameters are obtained as the sum of two kind of contributions. The first one is the one-loop radiative corrections coming from fermions and gauge bosons. The second one is obtained at the tree level from the higher order effective operators needed for the ultraviolet completion of the model. Finally we analyze the restrictions that the requirement of reproducing the standard electroweak symmetry breaking of the SM set on the LH model parameters.


Introduction
The discovery of a Higgs boson and the elucidation of the mechanism responsible for the electroweak symmetry breaking are some of the major goals of present and future searches in particle physics. The quadratically divergent contributions to the Higgs mass and the electroweak precision observables imply different scales for physics beyond the Standard Model (SM) ( 1 and 10 TeV respectively). This is the so called little hierarchy problem. As it is well known the mass of the Higgs boson receives loop corrections that are quadratic in the loop momenta. The largest contributions come from the top quark loop, with smaller corrections coming from loops of the electroweak gauge bosons and of the Higgs boson itself. Cancellations between the top sector and other sectors must occur in order to have the Higgs mass lighter than 200 GeV as expected from the electroweak precision test of the Standard Model (SM) which requires a fine-tuning of one part in 100. As this situation is quite unnatural various theories and models have been designed to solve this problem. For example, in Supersymmetric models the problem of quadratic Higgs mass divergences is resolved by the introduction of an opposite-statistic partner for each particle of the SM. The more recent idea of the Littlest Higgs model (LH) [1], inspired in an old suggestion by Georgi and Pais [2], tries to solve the little hierarchy problem by adding new particles with masses O(TeV) and symmetries which protect the Higgs mass from those dangerous quadratically divergent contributions (see [3] and [4] for reviews). These particles include the Goldstone bosons (GB) corresponding to a global spontaneous symmetry breaking (SSB) from the SU(5) to the SO(5) group, a new third generation vector quark called T and the gauge boson corresponding to an additional gauge group which contains at least a SU(2) R and eventually a new hypercharge U(1) . In this case cancellation occurs between same-statistics particles. However, LH models typically leave an uncanceled logarithmic mass contributions, which requires additional new contributions at some high scale to preserve a small Higgs boson mass. All of these new states could give rise to a very rich phenomenology, which could be probed at the CERN Large Hadron Collider (LHC) [5,6].
Nevertheless, it is clear that any viable model has to fulfill the basic requirement of reproducing the SM model at low energies. In particular, from the LH model it is, in principle, possible to compute the Higgs low-energy effective potential and then, by comparing with the SM potential, to obtain their phenomenological consequences including new restrictions on the parameter space of the LH model itself. For example, once obtained the one-loop corrections to the parameters of the standard Higgs potential, V = −µ 2 HH † + λ(HH † ) 2 ; where µ 2 and λ denote the well known Higgs mass and Higgs self-couplings parameters, restrictions over the LH parameters space can be obtained by imposing the condition µ 2 = λv 2 , where v is the SM vacuum expectation value ( H = (0, v)/ √ 2 ) with v ≃ 245 GeV. The µ 2 sign and value are well known [1,6], and effectively they are the right ones to produce the electroweak symmetry breaking, giving a Higgs mass m 2 H = 2µ 2 . However, the full expression for the radiative corrections to λ has not been analyzed in detail. In principle both µ 2 and λ receive contributions from fermion, gauge boson and scalar loops, besides others that could come from the ultraviolet completion of the LH model. We have previously computed the contributions to the Higgs effective potential in the LH model coming from the fermion and gauge boson sectors [7,8]. On the other hand, several relations for the threshold corrections to the λ parameter in the presence of a 10 TeV cut-off, depending of the UV-completion of the theory, has been reported before (see, for example [9]).
In this work we continue our program consisting in the computation of the relevant terms of the Higgs low-energy effective potential in the LH model and to analyze their phenomenological consequences. As has been mentioned before, we have started to developed this program in two previous articles [7,8]. First, we have computed and analyzed the fermion contributions to the low energy Higgs effective potential and we have illustrated the kind of constraints on the possible values of the LH parameters that can be set by requiring the complete LH Higgs effective potential to reproduce exactly the SM potential [7]. Second, the effects of virtual heavy and electroweak gauge bosons present in the LH model have been included in the analysis [8]. The radiative corrections to λ , at the one-loop level, have not been previously computed. First results are presented in the above two articles. We want to note that the computation of λ is important for several reasons: First, it must be positive, for the low energy effective action to make sense (otherwise the theory would not have any vacuum). In addition, from the effective potential above, one gets the simple formula m 2 H = 2λv 2 or, equivalently, µ 2 = λv 2 , being v set by experiment (for instance from the muon lifetime) to be v ≃ 245 GeV. Our phenomenological discussion in [7,8] have shown that the one-loop effective potential of the LH model cannot reproduce the SM potential with a low enough Higgs mass, m 2 H = 2λv 2 = 2µ 2 , to agree with the standard expectations. However, there are some indications suggesting that the effects of including interactions terms between Goldstone bosons (GB) and the other particles, i.e. fermions and gauge bosons and/or higher order GB loops could reduce the Higgs boson mass so that complete compatibility with the experimental constraints could be obtained.
The main objective of this work is to compute the effective potential for the doublet Higgs and the triplet φ , being both scalar fields of the LH model coming from some of the GB corresponding to the global SU(5) to SO(5) global symmetry breaking of the LH model. Its relevant terms can be read as [5], (1.1) This potential get contributions from radiative corrections and from effective operators coming from the ultraviolet completion of the LH model. With this potential we will study the regions of the LH parameter space giving rise to the SM electroweak symmetry breaking. Although radiative corrections from fermion and gauge boson loops are discussed in [7,8], the radiative contributions to λ φ 2 and λ H 2 φ have not been computed so far. A new constraints over the LH parameter space emerge once we impose the new relation between coefficients of the effective Higgs potential, imposed by the diagonalization of the Higgs mass matrix.
This work is organized as follows: In Section 2 we briefly explain the LH model and set the notation. Section 3 is devoted to the computation of the radiative corrections contributions to the effective potential at one-loop level. Next section is dedicated to the effective operator contribution. In Section 5 we analyze the constraints that our computation establishes on the LH parameters and, finally, in Section 6 we present the conclusions. The Goldstone bosons couplings to fermions and gauge bosons, needed for our computations, are listened in the Appendix.

The Littlest Higgs model Lagrangian
The LH model is based on the assumption that there is a physical system with a global SU(5) symmetry that is spontaneously broken to a SO(5) symmetry at a high scale Λ through a vacuum expectation value of order f . Thus, 14 Goldstone bosons (GB) are obtained as a consequence of this breaking. In this work we will consider two different versions of the LH model. In the first one the global SU(5) symmetry is explicitly broken by a gauge group [SU(2) × U(1)] 2 . We refer to this version as Model I [7,8]. In the second one the gauge group is [SU(2) 2 × U(1)] (Model II ) [7,8]. In both cases some of the GB become pseudo-GB acquire their masses through radiative corrections coming from gauge bosons and t, b T fermions loops.
The LH low energy dynamics is then described by a non-linear sigma model lagrangian 3 plus the appropriate Yukawa terms. The corresponding lagrangian is given by [1,5,6], is the GB matrix field. Σ 0 is with 1 being the 2 × 2 unit matrix. The Π matrix can be parametrized as, Here H = (H 0 , H + ) is the SM Higgs doublet and φ is the triplet given by: The covariant derivative D µ is defined by: where g and g ′ are the gauge couplings, W a j (a = 1, 2, 3) and B j , B are the SU(2) and U(1) gauge fields, respectively, Q a 1ij = σ a ij /2 , for i, j = 1, 2 , Q a 2ij = σ a * ij /2 for i, j = 4, 5 and zero otherwise, The Yukawa Lagrangian in (2.1), L Y K , describes the interactions between GB and fermions, more exactly, the third generations of quarks plus the extra T quark appearing in the LH model. The indices in L Y K are defined such that m, n = 4, 5 , i, j = 1, 2, 3 , and with: and The SU(5) to SO(5) spontaneous breaking give rise to four massless gauge bosons (the SM gauge bosons) and four or three massive gauge bosons corresponding to Model I or Model II respectively. In the fermion sector, we obtain one massive T quark and two massless quarks, namely the top and the bottom quarks.
In order to compute the gauge bosons loops, the lagrangian L must be supplemented by the standard terms depending only on the gauge fields. For sake of simplicity we will work in the following in the Landau gauge. Then these terms can be written symbolically in the mass eigenstate basis as: where Ω stands for any of the gauge bosons: being the mass matrix eigenstates, Finally,Ĩ is the interaction matrix between the gauge bosons and the H and φ scalars as it is given in the Appendix.
For the quarks, the complete Lagrangian is, andÎ , being the scalar-quark interaction matrix, given also in the Appendix. For more details of the model, including Feynman rules and also some phenomenological results see, for example, [5].

One-loop Effective Potential
As it is well known the electroweak symmetry breaking in the LH model is triggered, in principle, by the Higgs potential generated by one-loop radiative corrections, including both, fermion and gauge boson loops. Obviously this potential is invariant under the electroweak gauge group SU(2) × U(1) . Its relevant terms are given in (1.1), being µ 2 and λ the Higgs mass and the Higgs self-couplings parameters respectively. Quartic terms involving φ 4 and H 2 φ 2 are not included since they are not relevant in our present computation. The coefficients λ , λ φ 2 and λ H 2 φ appearing in the potential (1.1) receive contributions from the tree-level higher order operators coming from the ultraviolet completion of the LH model (see Section 4) and also from the gauge boson and fermion radiative corrections as will be discussed in this section.
As described in detail in our previous articles [7,8], we first focused on the effective potential for the H doublet, obtaining the first two terms of the potential, where µ 2 f g and λ f g denote the sum of fermionic and the gauge boson contributions to µ 2 and λ . By imposing that these parameters should reproduce the SM relation m 2 where m H is the Higgs mass and v is the vacuum expectation value (vev), we found that this potential it is not sufficient to find a light Higgs mass and to satisfy the relation µ 2 f g = v 2 λ f g . Notice that v is set by the experiment (for instance from the muon lifetime) to be v ≃ 245 GeV and µ is forced by data to be at most of order 200 GeV. However, the inclusion of the Goldstone boson (GB) sector could channel the situation towards a complete compatibility with the SM and the experimental constraints. In this way, the next objective is to obtain the effective potential for the H and φ fields, including the radiative contributions from fermion and gauge boson loops and the ones coming from the effective higher order operators (treelevel contribution) [1,5,10]. In this work we concentrate on the computation of the fermionic and gauge bosons contributions to the remaining coefficients of the complete one-loop effective potential defined in (1.1), λ φ 2 and λ H 2 φ . For this purpose, we consider constant GB fields, i.e. ∂H = ∂φ = 0 . This assumption makes easier the computation since we have: On the other hand the action is quadratic in the fermionic fields. Therefore, this one-loop contribution is exactly computed.
We split the calculation in two parts: the first one is dedicated to the fermion sector, and the second one to the gauge boson sector. Details on how the effective action is computed, by using standard techniques (see for instance [11]), are given in [7,8]. In the following we summarize just the main steps needed for the calculation.

Fermionic contribution
Following the idea in [7], the fermionic part of the effective action can be expanded as: where we have neglected a constant, irrelevant for the computation of the effective action. The fermion propagator, G ab (x, y) , is given by: where dk ≡ d 4 k/(2π) 4 , and the interaction operators are: Here the subindex indicates the number of GB interacting with two fermions.
In order to obtain the fermionic contribution to the λ φ 2 and λ H 2 φ we only need consider the terms k = 1 and k = 2 in the expansion (3.3), respectively. The generic one-loop diagrams are shown in Fig.1. For k = 1 we get, For the case k = 2 one obtains,

S
(2) By using well known methods, and after some work in which the divergent integrals that emerge are regularized by using an ultraviolet cutoff Λ , we obtain the different contributions to the couplings. The Fermionic contributions are: and 9) where N c is the number of colors and, λ t and λ T are, respectively, the SM top Yukawa coupling and the heavy top Yukawa coupling, given by: (3.10) 8 For the purpose of illustration and the final discussion of this paper, we also summarize here the fermionic contribution to the other two parameters of the Higgs potential, µ 2 and λ , as have been obtained in [7]: and Observe that the λ 's parameters, λ f , λ φ 2 f and λ H 2 φ f , are quadratically divergent. This is due to the lack of any symmetry protecting them, unlike the µ parameter which is protected by a SU(3) global symmetry. This will be the case for the gauge sector too, as it will be seen in the following.

Bosonic contribution
Here we concentrate in the gauge boson contributions at the one-loop level. We use the Landau gauge so that we do not have to consider any ghost field at this level. In this case the effective action can be expand as: where G is the gauge boson propagator given by (Landau gauge): 14) and the interaction operators are: The generic diagrams for this computation are given in Fig.2. In this case, we only need to consider the term k = 1 to obtain the two parameters λ φ 2 and λ H 2 φ . Then, we get: As it was mentioned above we consider in our analysis two different models: The original LH with two U(1) groups (Model I ) and the other one with just one U(1) group (Model II ). As there is no mixing between the SU(2) and U(1) groups, the only difference among these two models occurs in the U(1) sector.

Model I
The contributions from gauge boson sector to the λ φ 2 and λ H 2 φ parameters are given by: Note that this last parameter only receives contributions from the heavy gauge boson sector.
For the sake of completeness and our phenomenological discussion we also list here the results for the gauge boson contributions to µ 2 and λ , as obtained in [8]: (3.20) where g ≡ = g 1 g 2 g 2 1 + g 2 2 , s ψ = sin ψ = g 1

Model II
The corresponding results for this model are: As it was expected, the U(1) sector does not have any influence on λ II H 2 φg .
In addition the µ 2 II g and λ II g parameters are given by [8]: and With these results, the radiative contributions at one-loop level to the Higgs potential parameters are completed.

Effective Operators
As discussed above, the Higgs potential gets gauge boson and fermion one-loop contributions in the LH model. In addition, the potential coefficients also receive contributions from additional operators coming from the ultraviolet completion of the LH model. Thus these operators must be consistent with the symmetry of the theory [1,5,10]. At the lowest order they can be parameterized by two unknown coefficients a and a ′ ∼ O(1) . The form of these effective operators are, for the fermion sector [5], where i, j, k, m, n run over 1,2,3 and w, x, y, z run over 4,5 and for the gauge sector (Model I ), with j = 1, 2 and Q a j and Y j being the generators of the SU(2) j and U(1) j groups, respectively.
In the case of the Model II : where j = 1, 2 and Y is the generator of the unique U(1) group.
By expanding the GB field matrix Σ in these effective operators, we obtain their different contributions to the coefficients of the effective potential (1.1): To summarize, the complete results for these parameters is the sum of the contributions coming from the effective operators, as given above, and the radiative contributions coming from the fermion and gauge boson sector, which were given in Section 3.

Numerical Results and Phenomenological Discussion
In this section we discuss about the constraints on the possible values of the LH parameters. In our previous works we focused on the analysis of the constraints on the LH parameters by considering the effective potential only for the LH doublet (3.1) [7,8]. Our computation included the effect of virtual heavy quarks t, b and T , together with the heavy and electroweak gauge bosons W ′ , W , B ′ and B , present in the LH model. By imposing that the potential (3.1) has a minimum whenever µ 2 = λv 2 ( v = 246 GeV), we found that the obtained values for the µ parameter were too high to be compatible with the expected Higgs mass, which should not be larger than about 200 GeV according to the electroweak precision data.
It is clear that a similar analysis should be done if we consider the complete effective Higgs potential as given in (1.1). In this case, by diagonalizing the Higgs mass matrix, the Higgs mass is given to the leading order by m 2 [5]. Therefore, the LH parameters must satisfy the condition: .
In the following we will discuss about the constraints that the condition (5.1) imposes on the LH parameters space. In this way, we should also take into account other constraints imposed by requiring the consistency of the LH models with the electroweak precision data. There exist several studies of the corrections to electroweak precision observables in the Little Higgs models, exploring whether there are regions of the parameter space in which the model is consistent with data [3][4][5][6][12][13][14][15][16]. In the Model I with a gauge group SU(2) × SU(2) × U(1) × U(1) one has a multiplet of heavy SU(2) gauge bosons and a heavy U(1) gauge boson. The last one leads to large electroweak corrections and some problems with the direct observational bounds on the Z ′ boson from Tevatron [12,13]. Then, a very strong bound on the symmetry breaking scale f , f > 4 TeV at 95% C.L, is found [12]. However, it is known that this bound is lowered to 1 − 2 TeV for some region of the parameter space [13] by gauging only SU(2) × SU(2) × U(1) (Model II ). Thus in the following, we focus in the LH version called Model II.
In order to avoid small values of the W ′ mass and a very strong coupling constant we will set M W ′ > 0.6 TeV and g R ≤ 3 in our numerical discussion. We have found that for very small or very large values of the gauge group mixing angles, µ 2 f g is not positive and the SSB does not occur. However, due to the dependence of heavy gauge coupling constants on the mixing angles, it is found that c ψ < 0.1 or c ψ ∼ 1 imply a very strong gauge coupling. Accordingly we will work with 0.1 < c ψ < 0.9 , and then we will ensure that µ 2 has the right sing to generate a SSB. Besides, taking into account the restrictions on the parameters given in [8], we also set the following ranges: 0.5 < λ T < 2 , 0.8 TeV < f < 1 TeV and 10 TeV < Λ < 12 TeV. The condition λ T > ∼ 0.5 is established setting the bounds on the couplings λ 1 , λ 2 ≥ m t /v or λ 1 λ 2 ≥ 2(m t /v) 2 from the top mass [5]. The condition m T < ∼ 2.5 TeV is required in order to avoid a large amount of fine-tuning in the Higgs potential [1,6]. Then, since m T grows linearly with f , f should be lesser than about one TeV [7]. Finally, Λ is restricted by the standard condition Λ ∼ 4πf [17]. On the other hand the a and a ′ parameters are expected to be O(1) [1,5,10]. Values of the symmetry breaking scale f around 4 TeV are also allowed by the electroweak precision observables [12,13]. However, this value of f implies that m T is always greater than 5.7 TeV, when λ T > 0.5 . A fine-tuning of 0.8% is estimated for a Higgs mass of 200 GeV [12]. Besides, one gets M W ′ > 2.6 TeV. In addition, we have found that for f = 4 TeV, the allowed region for the LH parameters space, satisfying the condition imposed by the minimum of the Higgs potential, is smaller [7,8]. In fact, values around 1 − 3 TeV are the preferred ones for our selected choices of the LH parameters.
By considering the LH parameters bounded as described above and imposing the condition (5.1), we analyze the constraints on the different LH parameters with the inclusion of the effects of both contributions, the radiative ones (for the Model II ) and the effective operators. We found that the minimum µ value is µ = 0.39 TeV; for f = 0.8 TeV, Λ = 10 TeV, λ T = 0.72 , c ψ = 0.34 and a = a ′ = 0 , and the maximum is µ = 0.761 TeV; for f = 0.95 TeV, Λ = 11.5 TeV, λ T = 1.8 , c ψ = 0.71 , a = 1 and a ′ = 0.3 . Clearly, the minimum value corresponds with the case of considering the radiative corrections alone. In this case, small values of cos ψ are preferred. This is due to the fact that the fermionic contribution to the µ parameter is very large for higher energies, while λ does not change strongly with f and Λ . In this way, to satisfy the condition (5.1), small values of cos ψ are needed in order to reduce µ . As it was expected, large φ mass, M φ = 4.1 TeV, is found for values of the parameters corresponding to the minimum value of µ .
In Fig.3  respectively. Thus, the columns represent different a and a ′ values. Starting from left to right the values are the following: a = 0.3 and a ′ = 0.2 , a = 0.3 and a ′ = 0.7 , a = 0.9 and a ′ = 0.2 , and finally, a = 0.9 and a ′ = 0.7 . One can see that small a ′ values are preferred in order to satisfy the SSB condition. Since in this case we are taking into account both, the radiative corrections and the effective operators, the fermion sector contributions become even more important than in the cases of considering some of these contributions alone. Therefore, the strongly dependence on a ′ caused by the top's Yukawa couplings, gives large values of the λ 's parameters for high a ′ values making difficult to satisfy the eq.(5.1). However, the case of a is quite different. From Fig.3 it is clear that the results do not depend strongly on this parameter since its contribution is suppressed by the gauge couplings g and g ′ and by the not so small values of c ψ .
It is interesting to note that if one consider only the contributions to the potential coming from the effective operators (except for the µ parameter which still will be dominated by the radiative corrections), the SSB condition is not easily satisfied. The reason is very simple; the λ 's parameters at tree-level does not depend neither on f nor on Λ , while µ does. For example, the dependence of µ on f appears trough the heavy particle masses and then µ is directly proportional to f . In this way, µ rises very quickly with f meanwhile the λ 's parameters do not. Therefore high values of the parameters a and a ′ are needed in order to satisfy the condition (5.1). We find that only for f = 0.8 TeV, Λ = 10 TeV, and a ≥ 0.75 y a ′ ≥ 0.95 the equality (5.1) is satisfied. A more detailed analysis of the allowed values of the constants a and a ′ , in agreement with the electroweak precision fits, is given in [12]. The lowest value we found for µ is µ = 0.49 TeV, corresponding to f = 0.8 TeV, Λ = 10 TeV, c ψ = 0.2 λ = 2.59 , a = 0.75 and a ′ = 0.95 , being the mass of the φ scalar 2.86 TeV.
From the discussion above we see that in all cases the µ values are higher than 350 GeV. This is far away from the expected bound of the order of 200 GeV predicted by the SM precision tests. Therefore, it is clear that the inclusion of the interactions terms between GB and the other particles is not enough to reduce the Higgs boson mass so that the complete compatibility with the experimental constraints can be obtained. There are some indications suggesting that contributions coming from the scalar sector must reduce the absolute value of µ 2 and thus the Higgs mass. Although the scalar loops contributions have not been analyzed before (except the case of the radiatively generated scalar operators that have been discussed in [18]), the expression for the leading correction to the Higgs mass parameter, µ 2 φ , is presented in several articles. In particular, this correction is given by [1], Let us now estimate the size of the above contribution for the case in which we have obtained the minimum value for µ , µ = 0.39 TeV; corresponding to f = 0.8 TeV, Λ = 10 TeV, λ T = 0.72 , c ψ = 0.34 and a = a ′ = 0 , with M φ = 4.1 TeV. Thus, by taking Λ = 10 TeV, M φ = 4.1 TeV and assuming that λ ≃ 1 3 (at tree level) for having a Higgs mass of order 200 GeV [12], we get µ φ = −0.14 TeV. This implies that the µ value is reduced to be µ = 250 GeV. Note, however, that the quartic coupling λ is fixed to a particular value in the above analysis. Since small changes in the input parameters of the model produce large changes in the value of λ (and therefore the value of µ could vary), the radiative corrections to this coupling coming from the scalar sector must be also taken into account in a full analysis [19].

Conclusions
In this work we have computed the Higgs and φ bosons effective potential of the LH model (1.1). We have considered two kind of contributions to the parameters of this effective potential. Firstly, we have concentrated on the fermions and gauge bosons one-loop radiative corrections. In this case, we observe that the λ 's parameters are O(Λ 2 ) , since they are not protected as the Higgs mass is in the LH model. Secondly, we compute the contributions to the parameters from the effective operators coming from the ultraviolet completion of the LH model. Here, the obtained parameters depend exclusively on the two new unknown coefficients a and a ′ , as well as the mixing angle c ψ and the T Yukawa coupling. This is an important difference between the results obtained at one-loop which also depend on the cutoff Λ .
The resulting potential has the right form to produce a spontaneously symmetry breaking, since µ 2 > 0 for some regions of the LH model parameter space. Thus, if the condition (5.1) is satisfied, the electroweak symmetry is broken. By using the obtained effective potential we have analyzed the constraints imposed on the LH parameters in order to reproduce the SM electroweak symmetry breaking. We observe that if one only consider the effective operator contribution, the SSB condition is not easily satisfied. However, if the radiative corrections or both contributions are taken into account the allowed ranges of the parameters are much wider. The explanation comes from the way in which the coefficients of the potential, µ and λ 's, depend on f and Λ , as it was discussed above.
Finally, we numerically analize the LH parameter space that can be set by requiring the LH Higgs effective potential to reproduce exactly the SM potential, and its compatibility with the present phenomenological constraints on the Higgs boson mass. The lowest value found for the µ parameter is 390 GeV, which implies a Higgs boson mass of m H ≃ 550 GeV which is far from the current bound of about 200 GeV. As a consequence, we conclude that radiative corrections, coming from the Higgs itself and φ fields, could also provide relevant contributions to the effective potential if the LH model is able to reproduce the SM at low energies. An estimation of the scalar contribution to the µ parameter leads to a value of µ = 250 GeV, and thus m H ≃ 350 GeV. Nevertheless, the full contribution from the triplet, and thus the triplet mass M φ , is required to correct the Higgs mass in improved computations. The value of the Higgs quartic coupling, λ , receives several contributions which have a non-trivial dependence on the various parameters of the model and have no being computed so far. Work is in progress in order to compute these contributions and to check if then the value of Higgs mass will get closer to the current bound [19].