Electroweak phase transitions in the secluded U(1)-prime-extended MSSM
aa r X i v : . [ h e p - ph ] J un Preprint typeset in JHEP style - HYPER VERSION
Electroweak phase transitions in the secluded U (1) ′ -extended MSSM Cheng-Wei Chiang , and Eibun Senaha , Department of Physics and Center for Mathematics and Theoretical Physics, NationalCentral University, 300 Jhongda Rd., Jhongli, Taiwan 320, R.O.C. Institute of Physics, Academia Sinica, 128 Sec. 2, Academia Rd., Nankang, Taipei,Taiwan 115, R.O.C. Physics Division, National Center for Theoretical Sciences, 101, Sec. 2 Kuang Fu Rd.,Hsinchu, Taiwan 300, R.O.C.E-mail: [email protected], [email protected]
Abstract:
The electroweak phase transition (EWPT) in the secluded- U (1) ′ -extendedMSSM (sMSSM) is studied. Using the effective potential at zero and finite temperatures,we search for the non-MSSM-like EWPT in which the light stop mass is larger than thetop quark mass. Scanning the parameters relevant to the EWPT, the upper limits of theHiggs boson masses, which are consistent with the strong first order EWPT, are derived.For the lightest CP -even and -odd Higgs bosons, we find m H < ∼
160 GeV and m A < ∼ CP violation is possible by the complexsoft supersymmetry breaking masses. It is observed that such a CP -violating effect doesnot spoil the strong first order EWPT for the typical parameter sets. Keywords: sMSSM, Electroweak phase transition, Higgs boson, CP violation. ontents
1. Introduction 12. The model 33. Higgs mass spectrum and vacuum conditions 6
4. Electroweak phase transition 9
5. Numerical evaluations 15 CP -conserving case 165.2 Scan analysis 185.3 CP -violating case 215.4 Comparisons 23
6. Conclusions and Discussions 24
1. Introduction
To explain the baryon asymmetry of the Universe (BAU) is one of the challenging problemsin particle physics and cosmology. From the latest cosmological observations, the BAU isfound to be [1]. n B n γ = (4 . − . × − (95% C . L . ) , (1.1)where n B is the difference between the number density of baryons and that of antibaryons,and n γ denotes the photon number density. If there is an inflation in the early Universe, anyprimordial BAU would be washed out. Therefore, the BAU must arise dynamically afterthe inflation. In order to generate the BAU from a baryon-symmetric universe, it is requiredthat [2] (1) baryon number ( B ) violation, (2) C and CP violation and (3) departure fromthermal equilibrium. The last condition is not mandatory if the CP T theorem does nothold. In principle the standard model (SM) can satisfy these three conditions. However,it turns out that the CP -violating phase in the Cabibbo-Kobayashi-Maswaka matrix [3] isway too small to generate the observed BAU [4] and a strong first order electroweak phase– 1 –ransition (EWPT) cannot be realized with the viable Higgs mass, m h > . S ) into the su-perpotential. Several versions of the singlet extended MSSMs have been proposed: theNext-to MSSM (NMSSM) [10, 11], the nearly MSSM (nMSSM) or the minimal non-MSSM(MNMSSM) [12], the U (1) ′ -extended MSSM (UMSSM) [13, 14, 15] and the secluded U (1) ′ -extended MSSM (sMSSM/S-model) [16, 17, 18, 19, 20] etc. In this class of the mod-els, there is no fundamental µ parameter in the superpotential, and it is generated after S develops its vacuum expectation value (VEV), i.e. , µ eff = λv S where λ denotes the di-mensionless coupling and v S is the VEV of S . It thus gives a solution to the so-called µ problem.It is known that the Higgs singlet which couples to the Higgs doublets can play a rolein strengthening the first order EWPT, and a light stop is not necessarily lighter than topquark in contrast to the MSSM case. A variety of patterns of the EWPT are possible in theNMSSM [21]. Detailed studies of the EWPT in the nMSSM can be found in Refs. [22, 23].For the analysis of the EWPT in the UMSSM, see, for example, Refs. [24]. The possibilityof electroweak baryogenesis in the sMSSM is outlined in the letter paper [18], and recentlyits full paper has come out [20]. It is demonstrated that the electroweak baryogenesisis successful in this model. Although a number of studies on the EWPT in the singlet-extended MSSM can be found in the literature, it is still not clear how heavy Higgs bosoncan be consistent with the strong first order EWPT in some models. Such a mass limitwould be indispensable for a test of EW baryogenesis scenario at colliders.In this paper, we examine the sMSSM EWPT with/without CP violation in the widerparameter space which has not been probed in Refs. [18, 20]. Since there are the singletcontributions in the Higgs potential, the stability of the Higgs potential is not manifest. Inour analysis, we require that the prescribed EW vacuum should be a global minimum atzero temperature. This vacuum condition can make the allowed region quite limited [19].To investigate the properties of the EWPT, we use the one-loop effective potential at zeroand nonzero temperatures taking the contributions from the Z and W bosons, the thirdgeneration of quarks and squarks into account. Under the theoretical and experimentalconstraints, we exclusively search for the non-MSSM-like EWPT by scanning the relevantparameters and work out the upper limits of the Higgs boson masses which are consistentwith the strong first order EWPT.In the sMSSM, owing to the soft supersymmetry (SUSY) breaking mass terms, it ispossible to realize both explicit and spontaneous CP violation at the tree level. The effectof such a CP -violating phase on the strength of the first order EWPT and the Higgs massspectrum are investigated. – 2 –n general, there are ten order parameters if spontaneous CP violation exists. It is anon-trivial task to investigate such an EWPT thoroughly, and the numerical calculation isextremely time consuming. We will focus exclusively on both the CP -conserving case andthe explicit CP -violating case in which the number of order parameters is reduced to six.The paper is organized as follows. In Sec. 2, we briefly explain the model and definethe notations. Section 3 is devoted to the tree-level analysis. The approximate formulaeof the Higgs spectrum are presented. We discuss the vacuum condition, which turns outto be the severest theoretical constraint in this study. In Sec. 4, we argue the EWPTqualitatively in some detail. To get some idea about the sMSSM EWPT, we outline thenMSSM EWPT by showing the analytical formulae. In Sec. 5, the numerical results arepresented. Sec. 6 contains the conclusions and discussions.
2. The model
The sMSSM is one of the singlet-extended MSSM models, which can be regarded as aneffective theories of some unification theory such as string theory. The symmetry of themodel is SU (3) C × SU (2) L × U (1) Y × U (1) ′ Q ′ , where the extra U (1) is a remnant of thelarger symmetry of the UV theory. The Higgs sector comprises two Higgs doublets ( H d , H u )and four Higgs singlets ( S, S , S , S ). Among the singlets, the so-called secluded singlets( S i , i = 1 , , . ) play an essential role in ameliorating the severe experimental constraintson the Z ′ boson [16].It is desirable to have U (1) ′ charges ( Q ’s) chosen to make the model anomaly free.To this end, exotic chiral supermultiplets are generally required [14, 20, 25, 26]. For ourpurpose, we assume that they are heavy enough not to affect the phenomenology at theelectroweak scale. Neither will we address the gauge coupling unification issue here as itrequires the knowledge of full particle spectrum in the model. Instead, we focus exclusivelyon the Higgs sector to discuss the EWPT. Here, the so-called Model I sMSSM is considered,which can accommodate CP violation at the tree level. In Model I, the U (1) ′ charges ofthe Higgs fields must satisfy Q H d + Q H u + Q S = 0 , Q S = − Q S = − Q S = 12 Q S . (2.1)The superpotential ( W ) includes the following trilinear terms: W ∋ − ǫ ij λ b S b H id b H ju − λ S b S b S b S , (2.2)where b H d,u , b S, b S i , i = 1 , , , are the chiral superfields, λ and λ S are the dimensionlesscouplings. Due to the U (1) ′ symmetry, the S n ( n ∈ Z ) terms are forbidden. Therefore, weare not bothered by the domain wall problem, which can be induced by the spontaneousbreaking of the discrete symmetries. In general, the terms of the form b S b S , b S b S , b S b S and b S b S are allowed under the charge assignments (2.1). However, we will not consider suchquadratic terms since they may reintroduce the µ problem. Neither will we consider thecubic terms for simplicity. As we mentioned in the Introduction, once the Higgs singlet S develops its VEV, the effective µ term is generated by µ eff = λ h S i . Therefore, the scale of µ eff is not completely arbitrary but is fixed by the soft SUSY breaking parameters.– 3 –he Higgs potential at the tree level is given by V = V F + V D + V soft , (2.3)with V F = | λ | (cid:8) | ǫ ij Φ id Φ ju | + | S | (Φ † d Φ d + Φ † u Φ u ) (cid:9) + | λ S | ( | S S | + | S S | + | S S | ) , (2.4) V D = g + g † d Φ d − Φ † u Φ u ) + g | Φ † d Φ u | + g ′ (cid:16) Q H d Φ † d Φ d + Q H u Φ † u Φ u + Q S | S | + X i =1 Q S i | S i | (cid:17) , (2.5) V soft = m Φ † d Φ d + m Φ † u Φ u + m S | S | + X i =1 m S i | S i | − ( ǫ ij λA λ S Φ id Φ ju + λ S A λ S S S S + m SS SS + m SS SS + m S S S † S + h . c . ) , (2.6)where g , g and g ′ are the SU (2), U (1) and U (1) ′ gauge couplings, respectively. We willtake g ′ = p / g as motivated by the gauge unification in the simple GUTs. The softSUSY breaking masses m SS and m SS are introduced to break two unwanted global U (1)symmetries, and m S S is needed for explicit CP violation (ECPV).Here, we will follow the notation given in Ref. [19]. The Higgs VEVs and their fluctu-ation fields are parameterized asΦ d = e iθ √ ( v d + h d + ia d ) φ − d ! , Φ u = e iθ φ + u √ ( v u + h u + ia u ) ! , (2.7) S = e iθ S √ v S + h S + ia S ) , S i = e iθ Si √ v S i + h S i + ia S i ) , i = 1 − , (2.8)where v = q v d + v u ≃
246 GeV at the vacuum. The nonzero θ ’s can bring about spon-taneous CP violation (SCPV). However, not all θ ’s are independent. Due to the gaugeinvariance, the following four combinations are physical: ϕ = θ S + θ S , ϕ = θ S + θ S , ϕ = θ S + θ + θ , ϕ = θ S + θ S + θ S . (2.9)The analysis of SCPV at zero temperature can be found in Ref. [19]. To accommodateSCPV, the lightest Higgs boson mass should be less than about 125 GeV. Although it isinteresting to study SCPV at finite temperature, we will not pursue this possibility in thispaper. A comprehensive study of the SCPV at the finite temperature can be found inRef. [20]. – 4 –he tadpole conditions, which are defined by the first derivative of the Higgs potentialwith respect to the Higgs fields, are given by1 v d (cid:28) ∂V ∂h d (cid:29) = m + g + g v d − v u ) − R λ v u v S v d + | λ | v u + v S ) + g ′ Q H d ∆ = 0 , (2.10)1 v u (cid:28) ∂V ∂h u (cid:29) = m − g + g v d − v u ) − R λ v d v S v u + | λ | v d + v S ) + g ′ Q H u ∆ = 0 , (2.11)1 v S (cid:28) ∂V ∂h S (cid:29) = m S − ( R v S + R v S + R λ v d v u ) 1 v S + | λ | v d + v u ) + g ′ Q S ∆ = 0 , (2.12)1 v S (cid:28) ∂V ∂h S (cid:29) = m S − ( R v S + R v S + R λ S v S v S ) 1 v S + | λ S | v S + v S ) + g ′ Q S ∆ = 0 , (2.13)1 v S (cid:28) ∂V ∂h S (cid:29) = m S − ( R v S + R v S + R λ S v S v S ) 1 v S + | λ S | v S + v S ) + g ′ Q S ∆ = 0 , (2.14)1 v S (cid:28) ∂V ∂h S (cid:29) = m S − R λ S v S v S v S + | λ S | v S + v S ) + g ′ Q S ∆ = 0 , (2.15)1 v u (cid:28) ∂V ∂a d (cid:29) = 1 v d (cid:28) ∂V ∂a u (cid:29) = I λ v S = 0 , (2.16) (cid:28) ∂V ∂a S (cid:29) = I v S + I v S + I λ v d v u = 0 , (2.17) (cid:28) ∂V ∂a S (cid:29) = I v S − I v S + I λ S v S v S = 0 , (2.18) (cid:28) ∂V ∂a S (cid:29) = I v S + I v S + I λ S v S v S = 0 , (2.19) (cid:28) ∂V ∂a S (cid:29) = I λ S v S v S = 0 , (2.20)where ∆ = Q H d v d + Q H u v u + Q S v S + X i =1 Q S i v S i , (2.21) R i = Re( m SS i ) , I i = Im( m SS i ) , i = 1 , , (2.22) R = Re( m S S ) , I = Im( m S S ) , (2.23) R λ = Re( λA λ ) √ , I λ = Im( λA λ ) √ , (2.24) R λ S = Re( λ S A λ S ) √ , I λ S = Im( λ S A λ S ) √ , (2.25)– 5 –nd h X i is defined such that X is evaluated at the vacuum. In our analysis, the softSUSY breaking masses ( m , m , m S , m S i , i = 1 , , . ) are determined via the six tadpoleconditions (2.10)-(2.15). Here, all the Higgs VEVs are regarded as the input parametersand assumed to be nonzero.From the tadpole conditions with respect to the CP -odd Higgs fields, it follows that I λ = I λ S = 0 , I = I v S v S , I = − I v S v S . (2.26)Therefore, there is only one physical CP -violating phase at the tree level, and we takeArg( m S S ) ≡ θ S S as an input. After including the one-loop contributions, the relations(2.26) will be modified, and will be discussed in subsection 5.3.
3. Higgs mass spectrum and vacuum conditions
In this paper, we consider the one-loop corrections from the Z and W bosons, the thirdgeneration of quarks ( t, b ), and squarks (˜ t , , ˜ b , ). The one-loop effective potential at zerotemperature takes the form [27] V (Φ d , Φ u , S, S , , ) = X A c A ¯ m A π (cid:18) ln ¯ m A M − (cid:19) , (3.1)which is regularized in the DR scheme, ¯ m A is a field dependent mass, and M is therenormalization scale determined by the condition h V i = 0. The statistical factor of eachparticle is given respectively by c Z = 3 , c W = 6 , c t = c b = − N C , and c ˜ t , = c ˜ b , = 2 N C ,where N C is the color factor.In principle, the Z boson can mix with the Z ′ boson, and their mass matrix becomes2-by-2. However, the mass of Z ′ boson and the magnitude of the mixing angle are stronglyconstrained by experiments. A recent analysis of the constraints on the Z ′ boson mass andthe mixing angle can be found in Ref. [28]. As the mixing angle is constrained to be lessthan 10 − [28], we simply consider the case of no Z - Z ′ mixing, i.e. , tan β = p Q H d /Q H u .In addition, since the tan β ≃ O (1) is generic in the sMSSM [16, 17, 19], we will presentonly the case of tan β = 1. In this case, the secluded sector does not contribute to Eq. (3.1). Due to the additional contributions coming from the F -term and D -term of U (1) ′ , themass bound on the lightest Higgs boson is significantly relaxed compared to the MSSM.At the tree level, it is found that m H ≤ m Z cos β + | λ | v sin β + g ′ v ( Q H d cos β + Q H u sin β ) = (0 GeV) + (139 GeV) + (111 GeV) ≃ (178 GeV) , (3.2)where we have taken tan β = 1 , λ = 0 . , Q H d = Q H u = 1 in the second line. Thus, largeradiative corrections are not necessarily required for avoiding the LEP exclusion masslimits. However, we should note that because of the mixing terms between the doublets– 6 –nd singlets, m H can become smaller and even negative. For the above parameter set, weget m H = 12 " m S + | λ | v + 6 g ′ v S − rn m S + 2 g ′ (3 v S − v ) o + 4 v n R λ − ( | λ | − g ′ ) v S o , (3.3)where the mixing terms coming from the secluded sector are neglected, and CP is assumedto be conserved for simplicity. The parameter m S appearing in Eq. (3.3) is given by thetadpole condition for h S , i.e. , Eq. (2.12). Therefore, m H can become negative in the large R λ limit. It should be noted that R λ can be re-expressed in terms of m H ± . At the treelevel, m H ± = 1sin β cos β (cid:28) ∂ V ∂φ + d ∂φ − u (cid:29) = m W + 2 R λ sin 2 β v S − | λ | v . (3.4)Thus, m H can be unphysical in the large m H ± for a moderate value of v S . The one-loopformula of m H ± is explicitly given in Ref. [19]. As is done there, we take m H ± as an inputin place of | A λ | .To obtain more precise values in the Higgs mass spectrum, it is necessary to incorpo-rate the mixing terms between (Φ d , Φ u , S ) and ( S , S , S ) sectors, and also the one-loopcontributions (denoted by ∆ m H ). It is well-known that the dominant term of ∆ m H comesfrom the top/stop loops: ∆ m H = N C π v m t ln m ˜ t m ˜ t m t . (3.5)The explicit formulae of the mass matrix at the tree level are presented in Refs. [16, 19].For the one-loop expression, which is the same as the that of the NMSSM, see, for example,Ref. [11]. After the SU (2) Nambu-Goldstone bosons are rotated away, the mass matrix ofthe neutral Higgs bosons can be reduced to an 11-by-11 form. We diagonalize it numericallyto obtain m H i , where the subscript i is labeled in the ascending order of mass.Here, we comment on the possible range of m H ± . It is known that m H ± is constrainedfrom above by the vacuum condition. Namely, if we require that the energy level of theelectroweak vacuum v = q v d + v u ≃
246 GeV should be lower than the origin at whichthe energy level is normalized to zero, m H ± cannot exceed some critical value. For thetypical parameter sets, the maximal value of m H ± is found to be O (1 −
10) TeV [19].Actually, we can get a stronger constraint since the symmetric vacuum where v d = v u = 0is not necessarily located at the origin. We will discuss the vacuum condition in the nextsubsection. Before moving on to the analysis of the EWPT, we consider the structure of the zero-temperature effective potential in some detail. Although we restrict ourselves to the tree-level analysis for simplicity, it suffices to know the qualitative features of the Higgs potentialin the sMSSM. – 7 –he effective potential at the tree level takes the form V = 12 m v d + 12 m v u + 12 m S v S + X i m S i v S i − R v S v S − R v S v S − R v S v S − R λ v d v u v S − R λ S v S v S v S + g + g
32 ( v d − v u ) + | λ | v d v u + v d v S + v u v S )+ | λ S | v S v S + v S v S + v S v S ) + g ′ . (3.6)The EW vacuum defined by the tadpole conditions (in what follows we refer it as theprescribed EW vacuum) is not always the global minimum. In order to see this, we comparethe energy levels of the EW vacuum and the symmetric vacuum. The energy levels of thetwo vacua are given respectively by h V i vac = 12 R λ v d v u v S + 12 R λ S v S v S v S − g + g
32 ( v d − v u ) − | λ | v d v u + v d v S + v u v S ) − | λ S | v S v S + v S v S + v S v S ) − g ′ , (3.7) h V (sym)0 i vac = h V ( v d,u = 0) i vac = 12 R λ S ¯ v S ¯ v S ¯ v S − | λ S | v S ¯ v S + ¯ v S ¯ v S + ¯ v S ¯ v S ) − g ′ . (3.8)where ¯ v ’s are determined by the tadpole conditions using the potential V (sym)0 , and ¯∆ isgiven by ∆( v = ¯ v ). The energy level of the symmetric vacuum can in principle becomehigher than the origin in the limit of large positive R λ S .The difference between the energy levels of the two vacua is∆ h V i vac = h V (sym)0 i vac − h V i vac = − R λ v d v u v S + 12 R λ S (¯ v S ¯ v S ¯ v S − v S v S v S )+ g + g
32 ( v d − v u ) + | λ | v d v u + v d v S + v u v S ) − | λ S | h ¯ v S ¯ v S + ¯ v S ¯ v S + ¯ v S ¯ v S − v S v S − v S v S − v S v S i − g ′ − ∆ ) . (3.9)For a relatively large R λ , ∆ h V i vac < . As mentioned in subsection In the viable MSSM baryogenesis scenario, the EW vacuum is metastable and sufficiently long-lived,and the charge-color-breaking vacuum is the global minimum. On the other hand, the energy level of theEW symmetric vacuum is higher than that of the broken one as in the usual scenario. In such a case, thesuccessful EW symmetry restoration is possible [8]. – 8 –
450 470 490 510 530 550 570 590 610 630 650 -150-100-50050100150200250450 470 490 510 530 550 570 590 610 630 650
Figure 1:
Left: ∆ h V eff i vac = h V (sym)eff i vac − h V eff i vac as function of m H ± . Right: ∆ v S = ¯ v S − v S and ∆ v S i = ¯ v S i − v S i , i = 1 , ,
3, as a function of m H ± . We take tan β = 1, λ = 0 . λ S = 0 . v S = 500 GeV, v S = v S = v S = 1200 GeV, m SS = m SS = (50 GeV) , m S S = (200 GeV) , A λ S = A λ . R λ is nothing but a large m H ± . Therefore, the maximally allowed value of m H ± can be derived from this vacuum condition.The signs of the second, fifth and last terms of Eq. 3.9 depend on the magnitudesof ¯ v ’s. In Fig. 1, we plot various energy differences numerically. The left panel shows∆ h V eff i vac , where the one-loop corrections, V in Eq. (3.1), have been included in thenumerical calculation . As an illustration, we take tan β = 1, λ = 0 . λ S = 0 . v S = 500GeV, v S = v S = v S = 1200 GeV, m SS = m SS = (50 GeV) , m S S = (200 GeV) ,and A λ S = A λ . We call this parameter set Case 1, which we will discuss in greaterdetail in Sec. 5. In this case, a stable EW vacuum (∆ h V eff i vac >
0) exists for 496 GeV < ∼ m H ± < ∼
636 GeV. As m H ± decreases, the fifth term with a negative coefficient becomesdominant and then eventually results in ∆ h V eff i vac <
0. On the other hand, as m H ± increases, ∆ h V eff i vac < R λ term with a negativecoefficient as indicated by Eq. (3.9).In the right panel of Fig. 1, we plot ∆ v S and ∆ v S i , i = 1 , ,
3, where ∆ v S = ¯ v S − v S and∆ v S i = ¯ v S i − v S i . In the region where ∆ h V eff i vac is small, | ∆ v S | and | ∆ v S i | become large.As noticed in Ref. [21], the locations of the symmetric and broken vacua at zero temperature may yield some information about the EWPT. Namely, the sizable | ∆¯ v S | indicates thatthe singlet Higgs may be involved in realizing the non-MSSM-like EWPT. In such a case,since ∆ h V eff i vac is small, the EW symmetry is expected to be restored at a relatively lowtemperature.
4. Electroweak phase transition
Here, we give the necessary ingredients for calculating the EWPT and describe the EWPT The behaviors of ∆ h V eff i vac and ∆ h V i vac are almost the same. – 9 –ualitatively. The one-loop effective potential at finite temperature takes the form V (Φ d , Φ u , S, S , , ; T ) = T π X A c A I B,F (cid:18) ¯ m A T (cid:19) , (4.1)with I B,F ( a ) = Z ∞ dx x ln (cid:16) ∓ e −√ x + a (cid:17) , (4.2)where the subscripts of I B,F ( a ) denote boson (fermion). In order to reduce the com-putation time for the numerical integration in I B,F ( a ), we will use the fitting functionsemployed in Ref. [29] instead of Eq. (4.2). More explicitly,˜ I B,F ( a ) = e − a N X n =0 c b,fn a n , (4.3)are used, where c b,fn are determined by the least square method. For N = 40, | I B,F ( a ) − ˜ I B,F ( a ) | < − for any a , which suffices in our investigation. In the sMSSM, the structureof the tree-level potential is expected to be more important than the higher-order correc-tions unless a right-handed stop is lighter than top quark, which is the successful scenarioof the MSSM EW baryogenesis. In the following, we exclusively explore a non-MSSM-likeEWPT, i.e. , the heavy stop case. Hence, we will not include the two-loop contributions,and neither will we perform the ring-improvements in the effective potential for simplicity.For the EW baryogenesis to work, the sphaleron process, which is active in the sym-metric phase, must be decoupled when the EWPT completes. In other words, the sphaleronrate in the broken phase should be less than the Hubble parameter at that moment. Con-ventionally, the sphaleron decoupling condition is cast into the form ρ E T E > ζ, (4.4)where T E is the temperature at which the EWPT ends, ρ E is defined by the vacuumexpectation values of the two Higgs doublets at T E , i.e., ρ E = q ρ d ( T E ) + ρ u ( T E ), and ζ is an O (1) parameter which depends on the profile of the sphaleron, etc. It is, however,a non-trivial task to evaluate T E explicitly since a full knowledge of bubble dynamics isrequired. We thus use the condition ρ C /T C > ζ instead of Eq. (4.4), assuming that thesupercooling is not too large, where T C is defined by the temperature at which the effectivepotential has two degenerate minima and ρ C is the VEV at T C . To know the value of ζ within O (10%) accuracy, the sphaleron energy and zero-mode factors of the fluctuationsaround the sphaleron must be evaluated using the finite temperature effective potential.According to a recent study of the sphaleron decoupling condition with such an accuracy, ζ ≃ . ζ = 1 for simplicity. – 10 – AIAIIB EW vacuum
Figure 2:
The patterns of the EWPT.
As mentioned in Sec. 2, we will not consider the case of SCPV, which reduces the numberof order parameters relevant to the EWPT to six. Now let us introduce ρ = ( ρ d , ρ u , ρ S , ρ S , ρ S , ρ S ) (4.5)such that ρ ( T = 0) = v ≡ ( v d , v u , v S , v S , v S , v S ).As seen from Eq. (3.6), the effective potential in the ρ S direction has the form V ( ρ S ) ∋ − ( R ρ S + R ρ S ) ρ S + 12 m S ρ S + g ′ Q S ρ S + g ′ Q S (cid:16) X i Q S i ρ S i (cid:17) ρ S . (4.6)Since the linear term in ρ S can exist in principle, the point ( ρ d , ρ u , ρ S ) = (0 , ,
0) is notnecessarily a local minimum. The coefficient of ρ S term can vanish only when ρ S = ρ S =0 . Therefore, we may categorize the EWPT into the following two types:Type A: (0 , , ¯ ρ S , ¯ ρ S , ¯ ρ S , ¯ ρ S ) → ( ρ d , ρ u , ρ S , ρ S , ρ S , ρ S ),Type B: (0 , , , , , ¯ ρ S ) → ( ρ d , ρ u , ρ S , ρ S , ρ S , ρ S ),where the barred quantities denote the corresponding VEV’s in the symmetric phase. Thephase transition pattern of the U (1) ′ symmetry may be even more diverse, and a detailedanalysis of it is beyond the scope of the paper. In what follows, we will concentrate onthe EWPT of Type A and B. We call it a Type AI EWPT if ∆ v S > v S <
0. These types of EWPT are pictorially shown in Fig. 2 . In order tosee the qualitative features of the EWPT, we consider a rather simplified case. In the casewhere the temperature is high compared to the mass of the particle in the loop, I B,F ( a )in Eq. (4.2) can be expanded in powers of a = m /T as [31] I B ( a ) = − π
45 + π a − π a ) / − a (cid:18) ln a α B − (cid:19) + ζ (3)384 π a − · · · , (4.7) I F ( a ) = 7 π − π a − a (cid:18) ln a α F − (cid:19) + 7 ζ (3)384 π a − · · · , (4.8) In the parameter space explored in this paper, sgn( R , ) = +1 must be taken in order to be consistentwith the positivity of the Higgs squared mass. Of course there is additional axis of ρ S which is not shown here. – 11 –here ln α B = 2 ln 4 π − γ E ≃ . , ln α F = 2 ln π − γ E ≃ . , (4.9)and γ E ( ≃ . ζ (3)( ≃ . I B,F ( a ) are less than 5% if a < ∼ . a < ∼ . a ) / term with a negative coefficient inEq. (4.7) is crucial for the first order EWPT, as it gives rise to the potential barrierbetween the two degenerate minima. Such a cubic term originates from the zero frequencymodes in the bosonic thermal loop.In the other limit, where the temperature is low compared to the mass, I B,F ( a ) canbe expressed as [32] I B,F ( a ) ≃ ∓ a K ( a ) ≃ ∓ r π a / e − a (cid:20) a + · · · (cid:21) , (4.10)where K ( a ) is the modified Bessel function. The relative errors of I B,F ( a ) are less than5% if a > ∼ . a > ∼ . O (100) GeV. Therefore, thermal effects of the squarks considered hereinare exponentially suppressed, and do not play a significant role in realizing the first orderEWPT. We first begin with a simplified case. Suppose that g ′ = 0 and the VEVs of the secludedHiggs singlets are much larger than the others, which may correspond to the nMSSM-likelimit. Although the EWPT in the nMSSM has already been studied in Refs. [22, 23], wehere give a brief review of it to get a qualitative feature of the EWPT in the sMSSM. Sincethe nMSSM does not possess the cubic self-interacting term of the singlet Higgs boson inthe Higgs potential, the analysis of the EWPT becomes much simpler than that of anyother singlet-extended MSSMs.Let us consider the EWPT in the subspace ρ = ( ρ d , ρ u , ρ S ) assuming ρ S i = v S i . Herewe consider the tree-level potential and the dominant temperature-dependent contribu-tions, which are proportional to T . In the following, we show how the first order EWPTis realized without relying on the cubic term coming from the bosonic thermal loop asmentioned above.The effective potential here is reduced to V ( ρ , T ) = 12 M ( T ) ρ + 12 m S ρ S − ( c + ˜ R λ ρ ) ρ S + | λ | ρ ρ S + ˜ λ ρ , (4.11)where M ( T ) = m cos β + m sin β + G T ≡ M + G T , (4.12) c = R v S + R v S , ˜ R λ = R λ sin β cos β, (4.13)˜ λ = g + g β + | λ | β, (4.14)– 12 –nd we have subtracted ρ -independent terms from V ( ρ , T ). The value of G is given by thesum of the relevant couplings in the theory. For simplicity, the temperature dependence inthe mixing angle β is neglected. In order to have a stable vacuum in the symmetric phase, m S > ρ S , ρ S can be written in terms of ρ as ρ S = c + ˜ R λ ρ m S + | λ | ρ , (4.15)which gives the trajectory of the minimum values of ρ S as a function of ρ . Plugging thisback into the Higgs potential (4.11), we obtain V ( ρ, T ) = 12 M ( T ) ρ − ( c + ˜ R λ ρ ) m S + | λ | ρ ) + ˜ λ ρ . (4.16)The problem is now reduced to a one dimensional EWPT analysis. The parameters T C and ρ C for the first order EWPT are found to be T C = F ( ρ C ) − M a = F ( ρ C ) − F ( v ) a , (4.17) ρ C = 2 | λ | − m S + q m S ˜ λ (cid:12)(cid:12)(cid:12)(cid:12) ˜ R λ − | λ | c m S (cid:12)(cid:12)(cid:12)(cid:12) . (4.18)Here, F ( ρ ) is defined by F ( ρ ) = 2 ˜ R λ c + ˜ R λ ρ m S + | λ | ρ ! − | λ | c + ˜ R λ ρ m S + | λ | ρ ! − ˜ λ ρ . (4.19)From ρ C >
0, we find the condition for the first order EWPT [22] in terms of the modelparameters: ˜ λ < s m S (cid:12)(cid:12)(cid:12)(cid:12) ˜ R λ − | λ | c m S (cid:12)(cid:12)(cid:12)(cid:12) . (4.20)The physical implication of this condition becomes clearer if we expand the second termin Eq. (4.16) in powers of ρ /m S . To the sixth power of ρ , we obtain [23] V ( ρ, T ) = − c m S + 12 (cid:26) M ( T ) − cm S (cid:18) ˜ R λ − | λ | c m S (cid:19)(cid:27) ρ + 14 ( ˜ λ − m S (cid:18) ˜ R λ − | λ | c m S (cid:19) ) ρ + | λ | m S (cid:18) ˜ R λ − | λ | c m S (cid:19) ρ . (4.21)Condition (4.20) requires that the coefficient of the ρ -term be negative. In other words,the roles of the negative cubic term and the positive quartic term in the usual scenario arenow replaced by the negative quartic term and the positive sixth-power term. As already T C > – 13 –ointed out in Ref. [23], the form of the Higgs potential (4.21) is nothing but the SM Higgspotential with a dimension-six operator discussed in Refs. [33]. Hence the nMSSM can beregarded as one UV completion of such an effective theory.It should be stressed that in order to make the first order EWPT stronger, m S shouldnot be too large. This implies some constraints on the Higgs bosons whose masses comefrom m S . Another important implication is the following. From Eqs. (4.15) and (4.20), wefind that to have a first order EWPT the difference between the singlet VEV in the brokenphase and that in the symmetric phase must be larger than some critical value | ρ S − ¯ ρ S | = ρ m S (cid:12)(cid:12)(cid:12) ˜ R λ − | λ | c m S (cid:12)(cid:12)(cid:12) | λ | ρ m S > q m S ˜ λρ | λ | ρ m S , (4.22)where ¯ ρ S = c/m S . Conversely, if ρ S ≃ ¯ ρ S , the singlet Higgs field will not play a significantrole in realizing the first order EWPT, reducing to the MSSM-like EWPT.In the sMSSM case, because of the presence of the ρ S -term in the Higgs potential, theanalytic formula for the first order EWPT is not as simple as the nMSSM case. However,it is the same mechanism at work. That is, the first order EWPT is possible due to thenegative quartic term and the positive sixth-power term.Although Eq. (4.20) can provide a good approximation to the nMSSM [23], it fails to doso quantitatively in the sMSSM due to the presence of the secluded singlet Higgs bosons.The precise value of ¯ ρ S strongly depends on ¯ ρ S i through the tadpole conditions in thesymmetric phase. Indeed, it turns out that the above discussion is valid only qualitativelybut not quantitatively. We will present the numerical results in Sec. 5. As in the previous subsection, we focus only on the tree-level potential and the O ( T ) cor-rections coming from the finite-temperature effective potential. As noted in subsection 4.1, V can have an extremum at ( ρ d , ρ u , ρ S , ρ S , ρ S ) = (0 , , , ,
0) in Type B. It is thus usefulto prameterize the ρ -fields in terms of the five-dimensional polar coordinates ρ d = z cos δ cos γ cos α cos β, (4.23) ρ u = z cos δ cos γ cos α sin β, (4.24) ρ S = z cos δ cos γ sin α, (4.25) ρ S = z cos δ sin γ, (4.26) ρ S = z sin δ. (4.27)Using these variables, the effective potential at T C takes the form V ( z, T ) = c z − c z + c z = c z ( z − z C ) , (4.28)where we have subtracted z -independent terms in Eq. (4.28), and z C = c c , c = c c , (4.29)– 14 –ith c = 12 h c δ c γ c α ( m c β + m s β ) + m S c δ c γ s α + m S c δ s γ + m S s δ i − R c δ s γ c γ s α − R s δ c δ c γ s α − ( R + R λ S ρ S ) s δ c δ s γ + | λ S | c δ s γ + s δ ) ρ S + g ′ Q S ρ S h c δ c γ c α ( Q H d c β + Q H u s β ) + Q S c δ s γ s α + Q S c δ s γ + Q S s δ i + 12 G c δ c γ c α T C , (4.30) c = R λ c δ c γ s α c α s β c β , (4.31) c = g + g c δ c γ c α c β + | λ | c δ c γ c α ( c α s β c β + s α ) + | λ S | s δ c δ s γ + g ′ h c δ c γ c α + ( Q H d c β + Q H u s β ) + Q S c δ s γ s α + Q S c δ s γ + Q S s δ i , (4.32)where s α ≡ sin α, c α ≡ cos α , etc and the angles ( α, β, γ, δ ) are evaluated at T C . Let usdefine c = k + k T C . Then the critical temperature is given by T C = 1 k (cid:18) c c − k (cid:19) . (4.33)In Type B, the magnitude of R λ is very important for the strong first order EWPT.This is the same as the usual scenario in which the first order EWPT is induced by thenegative cubic term. Therefore, a large m H ± is favored. We also note that a smaller c ispreferred for a larger z C . This implies that a light Higgs boson is favored. We will quantifythe statements here in Sec. 5.
5. Numerical evaluations
We give numerical results in this section. To this end, T C and ρ C are determined using theone-loop effective potential at zero temperature, together with the fitting functions (4.3) of I B,F ( a ). First we discuss the CP -conserving case. The case of CP violation is argued insubsection 5.3. Before showing the numerical results, we list the experimental constraintsimposed in our numerical calculations.For a Higgs boson lighter than 114 . ξ < k ( m H i ) , (5.1)where ξ = g H i ZZ /g SM H i ZZ and k is the 95% C.L. upper limit derived from the LEP exper-iments as a function of the Higgs boson mass. For the ρ parameter corrections, ∆ ρ < . × − must be satisfied. As mentioned briefly below Eq. (3.1), the constraints ofthe Z ′ boson must be taken into account. The Z ′ boson mass ( m Z ′ ) and the mixing be-tween Z and Z ′ bosons ( α ZZ ′ ) are constrained by the direct searches of the Z ′ boson andthe EW precision measurements. The typical values are found to be m Z ′ > α ZZ ′ < O (10 − ) [28]. Since we here consider the α ZZ ′ = 0 case and at least one of– 15 –he VEV’s of the secluded singlets is taken to be O (1) TeV, the Z ′ constraints are easilysatisfied. For the relevant SUSY particles, we impose m ˜ χ ± >
104 GeV and m ˜ χ >
46 GeV.In order to extract information about the doublet-singlet mixing effects, we define theMSSM fractions by [17] ξ H i = (cid:16) O ( H )1 i (cid:17) + (cid:16) O ( H )2 i (cid:17) , ξ A i = (cid:16) O ( A )1 i (cid:17) , (5.2)where O ( H ) and O ( A ) are the orthogonal matrices which diagonalize the mass-squaredmatrices of the CP -even and CP -odd Higgs bosons, respectively. The parameter ξ char-acterizes to what extent φ (= H, A ) comprises the doublet components. For ξ = 1, φ arepurely composed of the doublets while ξ = 0 means that φ is purely the singlet components. CP -conserving case We turn off CP violation in this and the next subsection. As an example we take Q H d = Q H u = 1 , tan β = 1 , | λ S | = 0 . , A t = A b = 2 m ˜ q + µ eff / tan β, sgn( R λ S ) = sgn( R , ) = sgn( R ) = +1 , | m S S | = (200 GeV) ,m ˜ q = m ˜ t R = m ˜ b R = 1000 GeV , M = 200 GeV , M = M ′ = 300 GeV , (5.3)where m ˜ q , m ˜ t R and m ˜ t R are soft SUSY breaking masses of the stop and sbottom, M , M and M ′ are the gaugino masses associated with U (1) Y , SU (2) L and U (1) ′ , respectively.The quantity sgn( R λ ) is fixed bysgn( R λ ) = sgn (cid:18) m H ± − m W + | λ | v − ∆ m H ± (cid:19) , (5.4)where ∆ m H ± is the one-loop contribution to the charged Higgs boson mass.Since the mechanisms of the strong first order EWPT in Type A and Type B are dif-ferent from each other qualitatively, we will explore both parameter spaces. The followingtwo cases are representative points for Types A and B, respectively.Case 1 : v S = 500 GeV , v S = v S = v S = 1200 GeV , | A λ S | = | A λ | , (5.5)Case 2 : v S = 500 GeV , v S = v S = 100 GeV , v S = 1500 GeV , | A λ S | = 1000 GeV . (5.6)As mentioned above, | A λ | is determined via the mass formula of m H ± . The remaininginput parameters are λ , m H ± , | m SS i | , i = 1 ,
2, by varying which we search for the strongfirst order EWPT.In the left panel of Fig. 3, we plot ρ C , T C , m H , and m A as a function of m H ± in Case 1. As shown in Fig. 1, the allowed region, where the prescribed EW vacuum isthe global minimum, corresponds to the range, 496 GeV < ∼ m H ± < ∼
636 GeV. It is foundthat there are two ranges in which ρ C /T C > < ∼ m H ± < ∼
516 GeV(green-colored region) and 614 GeV < ∼ m H ± < ∼
636 GeV (blue-colored region). The strongfirst order EWPT is possible only for the region with sizable | ∆ v S | ; | ∆ v S | > ∼
110 GeV is– 16 – .10.20.30.40.50.60.70.80.91450 470 490 510 530 550 570 590 610 630 650 M SS M f r a c t i o n s Figure 3:
Dependence of ρ C , T C , m H , , m A and the MSSM fractions of H , H and A on m H ± in Case 1 with λ = 0 . | m SS | = | m SS | = (50 GeV) . obtained in Case 1. In such a region, a low T C is enough for the EW symmetry restorationas expected. We also plot the masses of H , H and A . To be consistent with the strongfirst order EWPT, the mass of the lightest CP -even Higgs boson m H should be less thanabout 130 GeV.In the right panel of Fig. 3, the MSSM fractions of H , H and A are plotted. Devia-tions of ξ H , from unity in the regions where the EWPT is strongly first order imply thatthe singlet Higgs bosons play a crucial role here. Since ξ A < . A is predominantly thesinglet-like CP -odd Higgs boson.Next we examine the dependences of | m SS | and | m SS | on ρ C /T C . In Fig. 4, weshow ρ C , T C , m H and m A as a function of q | m SS i | , where | m SS i | ≡ | m SS | = | m SS | is assumed. As | m SS i | increases, ρ C /T C decreases. Since a larger | m SS i | gives a larger m S via the tadpole condition, this tendency is understandable from the discussions givenin subsection. 4.2. Correspondingly, the Higgs boson masses which are mainly originatedfrom m S are constrained, leading to the mass bounds: m H < ∼
135 GeV and m A < ∼ ρ C , T C , m H , and m A (left panel) and ξ H , and ξ A (right panel) as a function of m H ± . Similar to Case 1, due to the vacuum condition the allowed region is limited tothe relatively small range 539 GeV < ∼ m H ± < ∼
592 GeV. Within the interval, there aretwo regions where the strong first order EWPT is possible: 539 GeV < ∼ m H ± < ∼
548 GeV(green-colored region) and 572 GeV < ∼ m H ± < ∼
592 GeV (magenta-colored region). Theformer corresponds to Type AI EWPT and the latter to Type B EWPT. For Type B,as discussed in subsection 4.3, the magnitude of R λ , and thus that of m H ± is crucial forthe strength of the first order EWPT. As m H ± increases, ρ C /T C becomes larger. Fromthe right panel of Fig. 5, we find that the doublet-singlet mixing plays an essential role– 17 – igure 4: Dependence of ρ C , T C , m H and m A on q | m SS i | , where | m SS i | = | m SS | = | m SS | isassumed, in Case 1 with λ = 0 . m H ± = 500 GeV. .10.20.30.40.50.60.70.80.91530 540 550 560 570 580 590 600 M SS M f r a c t i o n s Figure 5:
Dependence of ρ C , T C , m H , and m A (left panel) and ξ H , and ξ A (right panel) on m H ± in Case 2 with λ = 0 . | m SS | = | m SS | = (200 GeV) . in realizing the strong first order EWPT as it should be. The mass limits of H and A consistent with ρ C /T C > m H < ∼
137 GeV and m A < ∼ In order to search for strong first order EWPT in the wider parameter space, we performscans in the λ - m H ± and q | m SS i | - m H ± planes, respectively.– 18 – igure 6: Left: Case 1 with | m SS | = | m SS | = (50 GeV) in the λ - m H ± plane. Right: Case 1with λ = 0 . | m SS | = | m SS | in the q | m SS i | - m H ± plane. We show the results for Case 1 in Fig. 6. In the left plot, we take | m SS | = | m SS | =(50 GeV) and show the regions with sufficiently strong first order EWPT in the λ - m H ± plane. In the area surrounded by the solid black curves, the prescribed EW vacuum isthe global minimum. The region below the magenta dashed line is excluded by the LEP95% C.L. exclusion limit, Eq. (5.1). The green (blue) region corresponds to Type AI (AII)EWPT. We also overlay the contours of the mass of the lightest Higgs boson, m H = 140GeV (red), 160 GeV (green) and 180 GeV (orange). Around the λ = 1 region, m H = 140to 160 GeV is consistent with the strong first order EWPT. Here, we do not impose theperturbativity of λ in the investigation. However, the maximally allowed m H satisfying ρ C /T C > λ = 0 . q | m SS i | - m H ± plane and | m SS | = | m SS | is assumed. The curves have the same meanings as the left plot except that theHiggs boson mass contours are drawn for A . Here, we plot for m A = 100 GeV (red),150 GeV (green), 200 GeV (orange) and 250 GeV (gray). It should be emphasized that m A <
250 GeV in order to be consistent with the strong first order EWPT. This featurediffers from the MSSM.The numerical results of Case 2 is shown in Fig. 7. In the left panel, we take | m SS | = | m SS | = (200 GeV) and show the regions with sufficiently strong first order EWPTin the λ - m H ± plane. The magenta region represents Type B EWPT. The meanings ofthe remaining curves are the same as in Case 1. The λ dependence of Type B EWPT isexpected from the analysis done in subsection 4.3. Since c ∝ λ and c ∝ λ in the potential(4.28), ρ C /T C becomes smaller as λ increases. In both Type A and Type B EWPT, it isobserved that m H = 160 GeV supports the strong first order EWPT if λ ≃ igure 7: Left: Case 2 with | m SS | = | m SS | = (200 GeV) in the λ - m H ± plane. Right: Case 2with λ = 0 . | m SS | = | m SS | in the q | m SS i | - m H ± plane. In the right panel, we take λ = 0 . q | m SS i | - m H ± plane, and | m SS | = | m SS | ≡| m SS i | is assumed. In Case 2, the dependence of | m SS i | on the strength of the first orderEWPT is smaller. This is because both v S and v S are taken to be small (100 GeV) sothat the variation of m S as a function of | m SS i | is much milder than that in Case 1. The m A contours are drawn in the red curve (100 GeV), the green curve (150 GeV), and theorange curve (200 GeV), respectively. Asymptotically, m A approaches around 214 GeV as | m SS i | increases. Therefore, there is no significant difference in the Higgs mass spectrumbetween Case 1 and Case 2.Here, we comment on the other cases. Although the magnitude of v S is relevant to therealization of the non-MSSM-like EWPT, so that it should be taken around O (100) GeV,the maximal value of ρ C /T C is not sensitive to its exact value. Similarly, the strength of thefirst order EWPT does not depend sensitively on the other unvaried parameters appearingin the tree-level Higgs potential. Therefore, the upper limits of the Higgs boson massesallowed for the strong first order EWPT would not change substantially.Even in the case of the non-MSSM-like EWPT, we could also argue that the light stopis lighter than the top quark. In Case 1 with the light stop, no significant enhancement onthe strength of the first order EWPT is observed. It implies that the singlet Higgs bosonsand the light stop do not contribute to the first order EWPT constructively. This mayfollow from the fact that the light stop gives a − ρ -like term in the effective potential whilethe singlet Higgs bosons produces a − ρ -like term. In Case 2 with the light stop, however,the stop contribution is constructive, leading to about 60% enhancement on ρ C /T C . Sincethe right-handed soft SUSY breaking mass and the off-diagonal term in the stop massmatrix are taken to be small in the light stop scenario, the upper bound of m H obtainedabove is virtually unchanged. – 20 – .3 CP -violating case We now discuss CP violation in this subsection. The relations between the CP -violatingphases at the one-loop level are given by the one-loop tadpole conditions [19]: I λ = − N C π v (cid:20) m t I t sin β f ( m t , m t ) + m b I b cos β f ( m b , m b ) (cid:21) , (5.7) I λ S = 0 , I = I v S v S , I = − I v S v S , (5.8)where I q = Im( λA q ) / √ , q = t, b . Here A q is the soft SUSY-breaking trilinear coupling,and f ( m , m ) is defined by f ( m , m ) = 1 m − m (cid:20) m (cid:18) ln m M − (cid:19) − m (cid:18) ln m M − (cid:19)(cid:21) . (5.9)To see the CP -violating effect that the MSSM does not possess, we take I t = I b = 0 sothat I λ = 0. It follows from Eq. (5.8) thatsin θ SS = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m S S m SS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v S v S sin θ S S , (5.10)sin θ SS = − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m S S m SS (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v S v S sin θ S S , (5.11)where θ SS , = Arg( m SS , ). In the CP -violating case, the prefactors in Eqs. (5.10) and(5.11) cannot be chosen arbitrarily. For instance, the following inequalities must hold forsin θ S S = 1: | m S S | v S v S ≤ | m SS | , | m S S | v S v S ≤ | m SS | . (5.12)As we have discussed so far, small | m SS , | are favored for the strong first order EWPTin Case 1, which in turn imposes constraints on the magnitude of | m S S | via Eq. (5.12).Since | m S S | appears in some of the diagonal elements of the neutral Higgs bosons, i.e. ,the ( h S , , h S , ) and ( a S , , a S , ) elements [19], the Higgs boson masses mainly comingfrom those elements are lowered in accordance with | m S S | .In Fig. 8, we plot ρ C , T C , and the three Higgs boson masses m H i , i = 1 , , , for Case1 with | λ | = 0 . m H ± = 630 GeV, | m SS | = | m SS | = (50 GeV) and | m S S | = (20 GeV) (left panel); and for Case 2 with | λ | = 0 . m H ± = 590 GeV, and | m SS | = | m SS | =(200 GeV) (right panel). In both cases, the CP -violating effect on the strength of thefirst EWPT is relatively mild. In particular, no significant dependence is observed in Case2. This is because the mass matrix elements of the CP -even and -odd mixing part are notsufficiently large to alter the Higgs boson masses significantly.Since | m S S | assumes a smaller value to be consistent with Eq. (5.12) in Case 1, someof the Higgs masses are lowered as seen in the left panel of Fig. 8. In this case, H and H are purely singlet-like Higgs bosons and may be challenging to detect at colliders.– 21 –
250 0 2 0 40 60 8 0 100 120 140 160 180 050100150200250 0 20 40 60 8 0 100 120 140 160 180
Figure 8:
Left: Case 1 with | λ | = 0 . m H ± = 630 GeV, | m SS | = | m SS | = (50 GeV) , and | m S S | = (20 GeV) . Right: Case 2 with | λ | = 0 . m H ± = 590 GeV, and | m SS | = | m SS | =(200 GeV) . Figure 9:
Dependence of various quantities on the CP-violating phase θ S S . Here we take λ =0 . , m H ± = 675 GeV, v S = 560 GeV, v S = v S = 100 GeV, v S = 1000 GeV, | m SS | = (334 GeV ), | m SS | = (106 GeV ), and | m S S | = (200 GeV ). In contrast to Case 1, such a small | m S S | is not mandatory in Case 2 since v S and v S are small enough to satisfy Eq. (5.12). Therefore, there is no significant differencebetween the Higgs mass spectrum of the CP -conserving case and that of the CP -violatingcase.Depending on the choices of the Higgs VEVs, one can possibly find cases that havea significant dependence on θ S S . One example is shown in Fig. 9. Here we take | λ | =0 . , m H ± = 675 GeV, v S = 560 GeV, v S = v S = 100 GeV, v S = 1000 GeV, | m SS | =(334 GeV) , | m SS | = (106 GeV) , and | m S S | = (200 GeV) . In this case, a stable EW– 22 –acuum exists only for θ S S < ∼ ◦ , beyond which it shifts to a metastable one. It is foundthat ρ C /T C < θ S S = 0. As θ S S increases, however, ρ C /T C gets enhanced andeventually ρ C /T C > θ S S > ∼ . ◦ . This example clearly illustrates thatCP violation in the model can enhance the first order EWPT. It should be noted thatthis behavior may not be observed in the MSSM, for the strong first order EWPT favorsvanishing off-diagonal elements of the stop mass matrix, where the CP -violating phaseresides. On the contrary, such a CP -violating phase can make the EWPT weaker in theMSSM [9].Finally, we briefly comment on the constraints from the electric dipole moment (EDM).Here, we only consider CP violation originating from the soft SUSY breaking masses m SS , m SS and m S S . In such a case, the contributions to the EDM’s of electron and neutronand so on are small enough to satisfy the current experimental limits [20]. In this subsection, we compare our results with other models. The mass spectra havingthe strong first order EWPT are summarized in Table 1. In the sMSSM, m H < ∼
160 GeVand m A < ∼
250 GeV are required. However, there is no constraint on the stop mass. It iswell-known that in the MSSM, in addition to the requirement of a light Higgs boson, themass of the lighter stop must be smaller than that of top quark. According to Ref. [8], m H < ∼
127 GeV is required. An even severer bound m ˜ t <
120 GeV is found to be requiredby the strong first order EWPT.In the NMSSM, it is claimed that m H ≃
170 GeV is compatible with the strong firstorder EWPT [34]. In Ref [34], m ˜ q = m ˜ t = 1000 GeV are taken, the two stop masses areassumed to be degenerate, and the perturbativity of λ is taken into account. The sphalerondecoupling condition is ρ C /T C > ∼ .
3. Here, T C is defined as the temperature at which theeffective potential at the origin is destabilized in some direction. Such a temperature isalways lower than the temperature that we define in this paper, rendering a larger ρ C /T C .In the nMSSM, the CP -odd Higgs boson must have an upper bound on its mass inorder to realize the strong first order EWPT as discussed in subsection 4.2. In Ref. [22],such an upper limit is found to be m A < ∼
250 GeV, which is approximately the same asthe sMSSM.In the columns of NMSSM, nMSSM and UMSSM in Table 1, we leave a question markfor those cases where we are not aware of any literature that gives an upper bound onthe Higgs boson mass consistent with the requirement of strong first order EWPT. As areference, we cite the value of the CP -even Higgs boson in the nMSSM from Ref. [22]. Itis found that the lightest CP -even Higgs boson with a mass of 130 GeV is consistent withthe strong first order EWPT. The main differences between our work and Ref. [22] arethe following: (1) The soft SUSY-breaking parameters are taken as m ˜ q = m ˜ t = 500 GeVand A t = 100 GeV. (2) The perturbativity of λ ( < ∼ .
8) is imposed. (3) The sphalerondecoupling condition is ρ C /T C > ∼ .
3. When we na¨ıvely adopt those three conditions in thesMSSM, the upper bound on m H in the sMSSM approaches around 130 GeV.In the NMSSM [35] and the UMSSM [24], it is found that the strength of the firstorder EWPT can be enhanced for a larger mass of the lightest Higgs boson. In contrast,– 23 –MSSM MSSM NMSSM nMSSM UMSSM m H < ∼
160 GeV < ∼
127 GeV [8] < ∼
170 GeV [34] ? ? m A < ∼
250 GeV — ? < ∼
250 GeV [22] ? m ˜ t — < ∼
120 GeV [8] < m t — — — Table 1:
The mass spectra for the strong first order EWPT in the various models. we do not observe such an effect in the sMSSM.
6. Conclusions and Discussions
We have investigated the possible parameter space where first order EWPT is possible andstrong enough for successful EW baryogenesis in the sMSSM. We demonstrate two typicalexamples. In Case 1 where all of the secluded singlet Higgs VEV’s are taken to be ofthe order of TeV, Type A EWPT is realized. This pattern of the EWPT is the same asin the nMSSM in the sense that the EWPT is possible by the non-cubic coupling with anegative coefficient. However, the dependences of the strength of the EWPT on m H ± or A λ in both models are different from each other because of the U (1) ′ contributions and thesecluded singlet sector. In Case 2 where two of the secluded singlet Higgs VEV’s are takento be O (100) GeV, Type B EWPT is realized. In Type B, the parameters most relevantto the strong first order EWPT are m H ± and λ . As m H ± increases, ρ C /T C is enhanced.However, as λ increases, the magnitude of ρ C /T C reduces. The mechanism of strong firstorder EWPT in Type B is quite similar to the usual case where one has negative cubic andpositive quartic terms.By scanning the parameters most relevant to the EWPT, we have obtained the typicalHiggs mass spectrum that is consistent with the strong first order EWPT. In the sMSSM,it is found that m H < ∼
160 GeV and m A < ∼
250 GeV must be satisfied. Unlike the MSSM,the light stop mass is not necessarily smaller than the top quark mass.We have also worked out the impact of the CP -violating phase on the strength ofthe first order EWPT. In most of the parameter space, the dependence of ρ C /T C on θ S S is quite mild. However, according to the tadpole conditions for the CP -odd Higgsbosons, | m SS | , | m SS | and | m S S | , v S and v S , cannot be freely chosen. This leads to theconstraints on the Higgs mass spectrum, especially for Case 1.Our numerical study suggests that to have non-MSSM-like EWPT, the singlet HiggsVEV’s, particularly v S , in the broken phase and the symmetric phase must be significantlydifferent from each other. For the typical parameter sets, | ∆ v S | >
100 GeV must besatisfied. This condition is almost temperature-independent in our analysis. In principle,∆ v S can be derived provided that the soft SUSY-breaking masses are known, or moreprecisely, once the global structure of the Higgs potential is completely determined. Thedetermination of a sizable | ∆ v S | at zero temperature from collider experiments may beevidence of strong first order EWPT in the singlet-extended MSSM.– 24 – cknowledgments E. S. would like to thank Koichi Funakubo for useful discussions. This work was partlycarried out while E. S. visited KEK under the KEK-NCTS Exchange Program. This workis supported in part by the National Science Council of Taiwan, R. O. C. under GrantNo. NSC 97-2112-M-008-002-MY3 and in part by the NCTS.
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