Metastability of an Extended Higgs Model

We consider a singlet extended supersymmetric Higgs model. In the limit of it is possible to unravel the vacuum structure of this model analytically. We span the parameter space of the model. Specially we consider configurations in this space for a Higg’s mass of 125 GeV. We provide a detailed discussion of the issue of metastability of this model.


Introduction
The Standard Model SM of particle physics suffers from a hierarchy problem.Supersymmetric extensions of this model can solve the hierarchy problem 1-3 .But such extensions are encountered with a μ-problem 4, 5 .The bilinear supersymmetric Higgs mass term μH d H u in the superpotential, where H d and H u are one pair of Higgs doublets, does not violate supersymmetry susy , and gauge symmetry.Then the natural scale for μ is about Planck scale.However in order to get the weak scale correctly with unnatural cancelation we need μ to be about TeV scale.
One solution for this so-called μ problem is to substitute vacuum expectation value VEV of an extra gauge singlet field for the parameter μ.In the past thirty years various singlet extensions of the MSSM has been considered 6-9 .
A singlet extension of MSSM with mirror symmetry is defined by the superpotential 10 1.1

ISRN High Energy Physics
In this model Higgs singlet field S is coupled to a mirror world hidden sector indicated by tildes.And the dot product of the two Higgs doublets is defined by 1.2 As we do not consider the possibility that the charged Higgs fields could acquire vacuum expectation values we suppress occurrence of charged Higgs fields.So the product in 1.2 is taken to be equivalent to the product of the neutral fields.
In a recent paper 11 we showed that a true symmetry breaking minimum does not exist.And the model has two critical points, where at these points all first derivatives of the scalar potential with respect to the fields vanish.In our analysis we neglected the phases in the Higgs sector.We also did not include the soft susy breaking terms.
In this work we assume that the fields and the parameters in the Lagrangian of the Higgs sector are real.We attempt a phenomenological treatment of extended Higgs model and we give a partial analysis of soft susy breaking terms 12 .The motivation for the present work is as follows.
i The discovery of a nonzero vacuum energy density e 5.9 ± 0.2 meV 4 13 supports this point of view that we live in a metastable universe destined to ultimately undergo a phase transition to a susy world.So it is of interest to study field theoretical model that exhibit this property, such as an extended susy Higgs model.
ii To show that a true susy breaking minima is attainable by addition of soft Higgs masses.
iii Provide a detailed analysis of the parameter space of the model.
iv And finally to study the nature of phase transition to a future susy universe 13-16 .
Here a question may arise on the issue of metastability.In order to tunnel to a stable susy vacuum, the susy breaking in the metastable vacuum has to be spontaneous.But the susy breaking is parameterized in this paper with soft-breaking terms, which are an explicit breaking.How can this theory have a truly susy minimum?
The answer is that at present we do not have a good theory of susy breaking in our universe.But it is assumed that spontaneous breaking of supersymmetry occurs in a "hidden sector" of particles that have no or only very small coupling to the "visible sector" chiral supermultiplets of MSSM or its extensions.
Within this framework spontaneous symmetry breaking is communicated from the hidden sector where it originates to the observable sector by means of a third set of fields, the mediator, or the messenger fields.This mediation may take through gravity 17 or gauge interactions 18 .Supersymmetry breaking may be mediated by anomaly 19, 20 or extra dimension 21 as well.The result is the effective soft supersymmetry breaking in the visible sector.
Within this picture the appearance of explicit soft supersymmetry breaking terms are as the result of spontaneous supersymmetry breaking in a more fundamental theory.
It is evident when the susy is exact these soft mass terms will vanish.Hence during the transition to an exact susy phase Solution 1 these soft mass terms will disappear.
In 16 we studied this model for the case of tan β / 1.By utilizing numerical method we found configurations pertinent to an exothermic transition to a future susy universe.
In this work we consider the case of tan β 1.We find that in this case it is possible to give an analytical treatment of the subject matter.
This paper is organized in the following way.In section two we describe the model.We obtain the critical point condition on the parameters and vevs and we obtain some symmetric solutions.In section three we study the Higgs mass squared matrix of our symmetric solutions.We find the eigenvalues and we obtain the positivity constraints for these eigenvalues analytically.In section four we discuss the transition to a future susy universe.It is found that for the symmetric solution this transition is endothermic.We span over the soft squared masses.And finally in section five we present our conclusions.Technical details and special cases of soft squared masses are treated in the appendices.

The Model
The F term in the scalar potential in any supersymmetric model is derivable from the superpotential by

2.1
So for the neutral fields the F term of the scalar potential from the superpotential 1.1 is

2.2
The D term in the potential V D V D1 V D1 , where Here g 1 and g 2 are the U 1 and SU 2 gauge couplings.The structure of V D1 is similar to V D1 except that the Higgs doublet fields in this case belong to the mirror world.
The general structure of soft susy breaking term has a complicated form 13 .Here we only consider the soft mass squared Higgs term, namely,

4 ISRN High Energy Physics
Hence at the tree level the complete scalar potential of our model is

2.5
For simplicity we will ignore phases in the Higgs sector as well.The vacuum expectation values of the Higgs are given by Similarly for the Higgs in the mirror world we have

2.7
By minimizing the scalar potential we obtain with solutions.
2.9 which denotes a broken susy phase with no EWSB.

Higgs Mass Squared Matrices
In this section we compute the Higgs mass matrices for the solutions.

Mass Squared Matrix of Solution 3
In the space of H u , H d , S, H u , H d , and S this mass squared matrix is obtained from the second derivative of the scalar potential.For simplicity we impose the following conditions on the soft squared masses 3.1 The Higgs mass matrix squared matrix for Solution 3 is given by where the elements of the matrix are

3.3
We find that the degenerate physical Higgs masses are

3.4
The Higgs contribution to the vacuum energy for this solution is

3.5
The conditions for this solution to be a true supersymmetry breaking vacuum is i the physical Higgs masses be all positive, ii the vacuum energy be also positive.

ISRN High Energy Physics
These conditions are satisfied if 3.6

Mass Squared Matrix of Solution 4
The condition of 3.1 for the soft squared masses is implied by this solution.And the Higgs mass squared matrix for this case is where the elements of this matrix are given by

3.8
In a previous work we found the eigenvalues of this matrix 11 .The first two eigenvalues are 3.9 The third and the fourth eigenvalues satisfy and finally the fifth and the sixth eigenvalues satisfy

3.11
If we require that the quantity 3.12 ISRN High Energy Physics 7 be positive, then the conditions for the last four eigenvalue to be positive are 3.13

Metastable Aspects of the Model
To discuss transition from Solution 4 to Solution 1, we consider the equations of motions, for the symmetric solution they are The Higgs contribution to the vacuum energy for a symmetric solution is Upon substituting the values of S 0 and m 2 H the vacuum energy of Solution 4 is where But due to the positivity constraints of the Higgs mass matrix an exothermic phase transition υ < υ 0 does not occur see Appendix A for the proof .
For the case of an endothermic transition υ > υ 0 , the vacuum energy is positive if the values of the m 2 S satisfy where ς 1 and ς 2 are the roots of Bς 2 Cς D 0. 4.9 We note that ς 1 < 0 and ς 2 < 0. In Appendix B we show that the region m 2 S < ς 2 is ruled out by the positivity constraint of the Higgs mass squared matrix, there we also show that m 2 H > 0.
As noted earlier the experimental value of EWSB requires υ 0 174 GeV.
For the Higgs mass we consider m H 125 GeV 22 .
In units of υ 0 a typical solution with positive soft squared masses is Choosing a larger value for m 2 S will result in a smaller value for m 2 H . Another type of solution when m 2 S < 0 is The special cases where one of the soft squared masses is zero is treated in Appendix C.

Conclusions
For simplicity we did not included phases in the Higgs sector.However by the inclusion of soft squared masses we showed that the model has a rich vacuum structure.For the symmetric solution we discussed the phase transition and we showed it was an endothermic transition.We also provided bounds on the soft squared masses.It will be interesting to consider other solutions of the model.Or consider the case where all the fields in the Higgs sector are complex.We plan to report on these issues in the future.

A. The Case of an Exothermic Phase Transition
Here we show that an exothermic transition from the symmetric solution does not exist.
For the case of an exothermic transition υ < υ 0 , the vacuum energy is positive if the values of the m 2 S satisfy and from 4.9 where The physical Higgs masses are positive if But utilizing 4.5 So the positivity requirement of the physical Higgs masses becomes From the values of ς 1 and ς 2 we find that 2υ 2 0 m 2 S < 0. Therefore the condition stated in A.4 cannot be satisfied for an exothermic transition.

B. The Case of an Endothermic Phase Transition
In this appendix first we show that for an endothermic transition υ > υ 0 the region m 2 S < ς 2 is ruled out by the positivity constraints of the Higgs squared mass matrix.The form of the positivity condition is identical to that of A.4 .But in this region again 2υ 2 0 m 2 S < 0. Therefore the first term in the left hand side of this equation is negative.By using 4.5 it is easy to see that for an endothermic transition the second term in the left hand side is negative as well.But depending on the choice of the parameters of the model the right hand is always positive either for plus sign or minus sign .Hence the values m 2 S < ς 2 are not acceptable.Here we prove that the negative values of m 2 H are not permitted in an endothermic transition.First we assume m 2 H < 0 and then from the equations of motion we calculate m 2

S
and S 0 and finally we show that the positivity constraints for the Higgs mass squared matrix is violated.From 4.5 we obtain So for an endothermic transition and assuming a negative value for m 2 H we have From this expression the allowed range of m 2 S is Combining this result with similar constraint from the positivity of Higgs vacuum energy we obtain as É > E. For an endothermic transition the values of E and É are greater than unity and in the domain of B.5 the quantity m 2 S 2υ 2 0 has positive value.Next we consider the case of μ 0 > 0. From 4.4 we see that the value of S 0 is positive.So for negative value of m 2 H and from the equation of motion we find Now if we substitute the upper bound of the quantity m 2 S 2υ 2 0 in the physical Higgs mass constraint we get but the second term in the left hand side of B.7 is negative, and upon substituting the value of S 0 in the right hand side of this expression we find that the physical Higgs mass constraint is violated.Similar result holds for μ 0 < 0 case.So negative values of m 2 H are not allowed.

C. Special Cases of Soft Squared Masses
In this appendix we study the model with vanishing The Higgs vacuum energy in this situation is We note that when S 0 0 we have V 4 0 0. The condition for the positivity of Higgs mass squared matrix is This energy is positive if υ 0 < υ.Therefore this case corresponds to an endothermic transition.
Again we consider m H 125 GeV.An acceptable configuration for in this case is

2 0 m 2 S
have shown that for an endothermic transition this condition is satisfied.Hence this solution corresponds to an exact susy with broken EWSB.However we see that the solution S 0 2υ /μ 0 cannot satisfy the positivity of Higgs mass squared matrix for this case which is2υ 2 0 m 2 S > 4 S 2 0 ± S 0 μ 0 , C.5therefore this value of S 0 does not correspond to a true minimum.When m S 0, the Higgs vacuum energy will be Solution 1. Exact susy with Electroweak Symmetry Breaking EWSB .