Metastability of an Extended Higgs Model

Abstract

We consider a singlet extended supersymmetric Higgs model. In the limit of tan𝛽=1 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.

1. 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_𝑑𝐻_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] 𝑊=𝜆𝑆𝐻_𝑢⋅𝐻_𝑑−𝜐²+𝑆𝐻_𝑢⋅𝐻_𝑑−𝜐²+𝜇₀𝑆𝑆. (1.1) 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 𝐻_𝑢⋅𝐻_𝑑=𝐻⁰_𝑢⋅𝐻⁰_𝑑−𝐻⁺_𝑢𝐻⁻_𝑑. (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.

Solution 1. Exact susy with Electroweak Symmetry Breaking (EWSB).

Solution 2. Exact susy with no EWSB.

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) meV⁴ [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 tan𝛽≠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 tan𝛽=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.

2. The Model

The 𝐹 term in the scalar potential in any supersymmetric model is derivable from the superpotential by 𝑉_𝐹=_𝜙𝜕𝑊𝜕𝜙². (2.1)

So for the neutral fields the 𝐹 term of the scalar potential from the superpotential (1.1) is 𝑉_𝐹=𝜆²𝐻_𝑢𝐻_𝑑−𝜐²+𝜇₀𝑆²+|𝑆|²𝐻_𝑢²+𝐻_𝑑²+𝐻_𝑢𝐻_𝑑−𝜐²+𝜇₀𝑆²+𝑆²𝐻_𝑢²+𝐻_𝑑². (2.2)

The 𝐷 term in the potential 𝑉_𝐷=𝑉_𝐷1+𝑉_𝐷1, where 𝑉_𝐷1=𝑔²₁+𝑔²₂8𝐻_𝑑²−𝐻_𝑢²²+𝑔²₂2𝐻_𝑑²𝐻_𝑢²−𝐻_𝑢𝐻_𝑑². (2.3)

Here 𝑔₁ and 𝑔₂ 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, 𝑉_𝑆=𝜆²𝑚²_𝑆|𝑆|²+𝑚²_𝐻𝑢𝐻_𝑢²+𝑚²_𝐻𝑑𝐻_𝑑²+𝑚²_𝑆𝑆²+𝑚²𝐻_𝑢𝐻_𝑢²+𝑚²𝐻_𝑑𝐻_𝑑². (2.4)

Hence at the tree level the complete scalar potential of our model is 𝑉=𝑉_𝐹+𝑉_𝐷+𝑉_𝑆. (2.5)

For simplicity we will ignore phases in the Higgs sector as well. The vacuum expectation values of the Higgs are given by 𝐻_𝑢=𝜐₁,𝐻_𝑑=𝜐₂. (2.6)

Similarly for the Higgs in the mirror world we have 𝐻_𝑢=𝜐₁,𝐻_𝑑=𝜐₂,⟨𝑆⟩=𝑆₀,𝑆=𝑆₀. (2.7)

By minimizing the scalar potential we obtain 𝑆₀𝜐²₁+𝜐²₂+𝜇₀𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝑚²_𝑆𝑆₀𝑆=0,₀𝜐²₁+𝜐²₂+𝜇₀𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝑚²_𝑆𝑆₀𝜐=0,₂𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝜐₁𝑆²₀+𝜐₁4𝜆²𝑔²₁+𝑔²₂𝜐²₁−𝜐²₂+𝑚²_𝐻𝑢𝜐₁𝜐=0,₁𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝜐₂𝑆²₀+𝜐₂4𝜆²𝑔²₁+𝑔²₂𝜐²₂−𝜐²₁+𝑚²_𝐻𝑑𝜐₂𝜐=0,₂𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝜐₁𝑆²₀+𝜐₁4𝜆²𝑔²₁+𝑔²₂𝜐²₁−𝜐²₂+𝑚²𝐻_𝑢𝜐₁𝜐=0,₁𝜐₁𝜐₂−𝜐²+𝜇₀𝑆₀+𝜐₂𝑆²₀+𝜐₂4𝜆²𝑔²₁+𝑔²₂𝜐²₂−𝜐²₁+𝑚²𝐻_𝑑𝜐₂=0, (2.8) with solutions.

Solution 3. 𝜐₁=𝜐₂=𝜐₁=𝜐₂=0 and 𝑆₀=𝜇₀𝜐²𝜇²₀+𝑚²_𝑆,𝑆₀=𝜇₀𝜐²𝜇²₀+𝑚²_𝑆, (2.9) which denotes a broken susy phase with no EWSB.

Solution 4. Here we have 𝜐₁=𝜐₂=𝜐₁=𝜐₂=𝜐₀ moreover 𝑆₀=𝑆₀. This will correspond to a broken 𝑠𝑢𝑠𝑦 phase with EWSB. But as in this case 𝜐²₁+𝜐²₂=(246)², we have 𝜐₀=174 GeV. Moreover since tan𝛽=𝜐₁/𝜐₂, this solution then corresponds to the limit of tan𝛽=1.

3. Higgs Mass Squared Matrices

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

3.1. 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 𝑚²_𝐻𝑢=𝑚²_𝐻𝑑=𝑚²𝐻_𝑢=𝑚²𝐻_𝑑=𝑚²_𝐻,𝑚²_𝑆=𝑚²_𝑆. (3.1)

The Higgs mass matrix squared matrix for Solution 3 is given by 𝑀²=𝑎𝑏0000𝑏𝑎000000𝑒000000𝑎𝑏0000𝑏𝑎000000𝑒, (3.2) where the elements of the matrix are 𝑎=2𝜆²𝑆²₀+𝑚²_𝐻,𝑏=−2𝜆²𝑆²₀,𝑒=2𝜆²𝜇²₀+𝑚²_𝑆. (3.3)

We find that the degenerate physical Higgs masses are 𝜉₁=𝜉₂=𝜉₃=𝜉₄=2𝜆²2𝑆²₀+𝑚²_𝐻,𝜉₅=𝜉₆=2𝜆²𝜇²₀+𝑚²_𝑆. (3.4)

The Higgs contribution to the vacuum energy for this solution is 𝑉₃(0)=2𝜆²𝑚²_𝑆𝜇²₀𝜐⁴+𝑚²_𝑆𝜇²₀+𝑚²_𝑆². (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.

These conditions are satisfied if 𝑚²_𝑆>0,𝑚²_𝐻>−𝜇²₀𝜐⁴𝜇²₀+𝑚²_𝑆². (3.6)

3.2. 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 𝑀²=𝑎₁𝑏₁𝑐₁00𝑑₁𝑏₁𝑎₁𝑐₁00𝑑₁𝑐₁𝑐₁𝑒₁𝑑₁𝑑₁000𝑑₁𝑎₁𝑏₁𝑐₁00𝑑₁𝑏₁𝑎₁𝑐₁𝑑₁𝑑₁0𝑐₁𝑐₁𝑒₁, (3.7) where the elements of this matrix are given by 𝑎₁=2𝜆²𝜐²₀+𝑆²₀+𝑚²_𝐻+𝜐²₀𝑔²₁+𝑔²₂,𝑏₁=2𝜆²𝜐²₀−𝑆²₀−𝜐²₀𝑔²₁+𝑔²₂,𝑐₁=4𝜆²𝜐₀𝑆₀,𝑑₁=2𝜆²𝜇₀𝜐₀,𝑒₁=2𝜆²2𝜐²₀+𝜇²₀+𝑚²_𝑆. (3.8)

In a previous work we found the eigenvalues of this matrix [11]. The first two eigenvalues are 𝜂₁=𝜂₂=𝑎₁−𝑏₁=2𝜆²2𝑆²₀+𝑚²_𝐻+2𝜐²₀𝑔²₁+𝑔²₂. (3.9)

The third and the fourth eigenvalues satisfy 𝜂²−𝑎₁+𝑏₁+𝑒₁𝜂+𝑒₁𝑎₁+𝑏₁−2𝑐₁−𝑑₁²=0, (3.10) and finally the fifth and the sixth eigenvalues satisfy 𝜂²−𝑎₁+𝑏₁+𝑒₁𝜂+𝑒₁𝑎₁+𝑏₁−2𝑐₁+𝑑₁²=0. (3.11)

If we require that the quantity 𝑎₁+𝑏₁+𝑒₁=2𝜆²4𝜐²₀+𝜇²₀+𝑚²_𝐻+𝑚²_𝑆 (3.12) be positive, then the conditions for the last four eigenvalue to be positive are 𝑒₁𝑎₁+𝑏₁>2𝑐₁−𝑑₁²,𝑒₁𝑎₁+𝑏₁>2𝑐₁+𝑑₁². (3.13)

4. 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 2𝑆₀𝜐²₀+𝜇₀𝜐²₀−𝜐²+𝜇₀𝑆₀+𝑚²_𝑆𝑆₀𝜐=0,₂₀−𝜐²+𝜇₀𝑆₀+𝑆²₀+𝑚²_𝐻=0. (4.1)(4.2)

The Higgs contribution to the vacuum energy for a symmetric solution is 𝑉(0)=2𝜆²𝜐²₀−𝜐²+𝜇₀𝑆₀²+2𝜐²₀𝑆²₀+2𝑚²_𝐻𝜐²₀+𝑚²_𝑆𝑆²₀. (4.3)

From (4.1) the value of 𝑆₀ is 𝑆₀𝜇=−⁰𝜐²₀−𝜐²𝐴,where𝐴=𝜇²₀+2𝜐²₀+𝑚²_𝑆. (4.4)

From (4.2) the value of 𝑚²_𝐻 is 𝑚²_𝐻=−𝜐²₀−𝜐²+𝜇²₀𝜐²₀−𝜐²𝐴−𝜇²₀𝜐²₀−𝜐²²𝐴². (4.5) Upon substituting the values of 𝑆₀ and 𝑚²_𝐻 the vacuum energy of Solution 4 is 𝑉₄(0)=−2𝜆²𝜐²₀−𝜐²𝐴²𝐵𝑚⁴_𝑆+𝐶𝑚²_𝑆+𝐷, (4.6) where 𝐵=𝜐²₀+𝜐²,𝐶=4𝜐²₀+𝜐²𝐵,𝐷=4𝜐²₀𝐵+𝜇²₀. (4.7)

But due to the positivity constraints of the Higgs mass matrix an exothermic phase transition (𝜐<𝜐₀) does not occur (see Appendix A for the proof).

For the case of an endothermic transition (𝜐>𝜐₀), the vacuum energy is positive if the values of the 𝑚²_𝑆 satisfy 𝑚²_𝑆>𝜍₁or𝑚²_𝑆<𝜍₂, (4.8) where 𝜍₁ and 𝜍₂ are the roots of 𝐵𝜍²+𝐶𝜍+𝐷=0. (4.9)

We note that 𝜍₁<0 and 𝜍₂<0. In Appendix B we show that the region 𝑚²_𝑆<𝜍₂ is ruled out by the positivity constraint of the Higgs mass squared matrix, there we also show that 𝑚²_𝐻>0.

As noted earlier the experimental value of EWSB requires 𝜐₀=174 GeV.

For the Higgs mass we consider 𝑚_𝐻=125 GeV [22].

In units of 𝜐₀ a typical solution with positive soft squared masses is 𝜐²₀=1,𝜐²=2,𝜇₀=2,𝑚²_𝑆=3.06,𝑚²_𝐻=0.51. (4.10)

Choosing a larger value for 𝑚²_𝑆 will result in a smaller value for 𝑚²_𝐻.

Another type of solution when 𝑚²_𝑆<0 is 𝜐²₀=1,𝜐²=2,𝜇₀=1,𝑚²_𝑆=−0.22,𝑚²_𝐻=0.51. (4.11)

The special cases where one of the soft squared masses is zero is treated in Appendix C.

5. 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.

Appendices

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<𝜐₀), the vacuum energy is positive if the values of the 𝑚²_𝑆 satisfy 𝜍₂<𝑚²_𝑆<𝜍₁, (A.1) and from (4.9) 𝜍₁=−2𝜐²₀−𝜇₀21−√𝐸,𝜍₂=−2𝜐²₀−𝜇₀21+√𝐸, (A.2) where 𝐸=1−8𝜐²₀𝜐²₀−𝜐²𝜇²₀𝜐²₀+𝜐². (A.3)

The physical Higgs masses are positive if 2𝜐²₀2𝜐²₀+𝑚²_𝑆+𝑚²_𝐻2𝜐²₀+𝑚²_𝑆+𝜇²₀>8𝜐²₀𝑆²₀±𝜇₀𝑆₀. (A.4) But utilizing (4.5) 𝑚²_𝐻2𝜐²₀+𝑚²_𝑆+𝜇²₀=−𝜐²₀−𝜐²2𝜐²₀+𝑚²_𝑆+𝜇₀𝑆₀𝜐²₀𝜐1−²𝜐²₀. (A.5)

So the positivity requirement of the physical Higgs masses becomes 𝜐²₀+𝜐²2𝜐²₀+𝑚²_𝑆+𝜇₀𝑆₀𝜐²₀𝜐1−²𝜐²₀>8𝜐²₀𝑆²₀±𝜇₀𝑆₀. (A.6)

From the values of 𝜍₁ and 𝜍₂ we find that 2𝜐²₀+𝑚²_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>𝜐₀) the region 𝑚²_𝑆<𝜍₂ 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 2𝜐²₀+𝑚²_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 𝑚²_𝑆<𝜍₂ are not acceptable.

Here we prove that the negative values of 𝑚²_𝐻 are not permitted in an endothermic transition. First we assume 𝑚²_𝐻<0 and then from the equations of motion we calculate 𝑚²_𝑆 and 𝑆₀ and finally we show that the positivity constraints for the Higgs mass squared matrix is violated. From (4.5) we obtain 𝑚²_𝐻=𝜐²−𝜐²₀𝐴²2𝜐²₀+𝑚²_𝑆²+𝜇²₀2𝜐²₀+𝑚²_𝑆−𝜇²₀𝜐²−𝜐²₀. (B.1)

So for an endothermic transition and assuming a negative value for 𝑚²_𝐻 we have 2𝜐²₀+𝑚²_𝑆²+𝜇²₀2𝜐²₀+𝑚²_𝑆−𝜇²₀𝜐²−𝜐²₀<0. (B.2)

From this expression the allowed range of 𝑚²_𝑆 is −𝜇²₀21+𝐸́<𝑚²_𝑆+2𝜐²₀𝜇<−²⁰21−𝐸́, (B.3) where ́4𝜐𝐸=1+²−𝜐²₀𝜇²₀. (B.4)

Combining this result with similar constraint from the positivity of Higgs vacuum energy we obtain −𝜇²₀21−√𝐸<𝑚²_𝑆+2𝜐²₀𝜇<−²⁰21−𝐸́, (B.5) 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 𝑚²_𝑆+2𝜐²₀ has positive value.

Next we consider the case of 𝜇₀>0. From (4.4) we see that the value of 𝑆₀ is positive. So for negative value of 𝑚²_𝐻 and from the equation of motion we find 𝑆₀>𝜇₀2𝐸́−1. (B.6)

Now if we substitute the upper bound of the quantity 𝑚²_𝑆+2𝜐²₀ in the physical Higgs mass constraint we get 𝜇²₀𝜐²₀́+𝜇𝐸−1²⁰2𝑚²_𝐻𝐸́+1>8𝜐²₀𝑆²₀+𝜇₀𝑆₀, (B.7) but the second term in the left hand side of (B.7) is negative, and upon substituting the value of 𝑆₀ in the right hand side of this expression we find that the physical Higgs mass constraint is violated. Similar result holds for 𝜇₀<0 case. So negative values of 𝑚²_𝐻 are not allowed.

C. Special Cases of Soft Squared Masses

In this appendix we study the model with vanishing 𝑚²_𝐻=0 or vanishing 𝑚²_𝑆=0.

When 𝑚²_𝐻=0 from equations of motions we find 2𝑆₀𝜐²₀−𝜇₀𝑆²₀+𝑚²_𝑆𝑆₀=0, (C.1) with solutions 𝑆₀=0,𝑆₀=2𝜐²₀+𝑚²_𝑆𝜇₀. (C.2)

The Higgs vacuum energy in this situation is 𝑉₄(0)=2𝜆²𝑆²₀𝑆²₀+2𝜐²₀+𝑚²_𝑆. (C.3)

We note that when 𝑆₀=0 we have 𝑉₄(0)=0. The condition for the positivity of Higgs mass squared matrix is 2𝜐²₀+𝑚²_𝑆>0, (C.4) in previous we 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 𝑆₀=(2𝜐²₀+𝑚²_𝑆)/𝜇₀ cannot satisfy the positivity of Higgs mass squared matrix for this case which is 2𝜐²₀+𝑚²_𝑆>4𝑆²₀±𝑆₀𝜇₀, (C.5) therefore this value of 𝑆₀ does not correspond to a true minimum.

When 𝑚_𝑆=0, the Higgs vacuum energy will be 𝑉₄(0)=−8𝜆²𝜐²₀−𝜐²𝜐⁴₀2𝜐²₀+𝜇²₀²𝜐²₀+𝜐²+𝜇²₀. (C.6)

This energy is positive if 𝜐₀<𝜐. Therefore this case corresponds to an endothermic transition.

Again we consider 𝑚_𝐻=125 GeV. An acceptable configuration for in this case is 𝜐²₀=1,𝜐²=2,𝜇₀=1.13,𝑚²_𝑆=0.0,𝑚²_𝐻=0.51. (C.7)