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Baryogenesis Constraints on the Minimal

arXiv:hep-ph/9206266v1 30 Jun 1992BU-HEP 92-04January 1992Baryogenesis Constraints on the MinimalSupersymmetric Model1Stanley MyintDepartment of PhysicsBoston UniversityBoston, MA 02215, USAto appear in Phys. Lett.

BAbstractRequirement that the vacuum expectation values of Higgs fields immediately afterthe phase transition be large enough imposes constraints upon the parameters ofthe minimal supersymmetric model. In particular, one obtains the upper boundson the lighter CP-even Higgs mass and the soft supersymmetry breaking scale fordifferent values of the top quark mass.1This work was supported in part under NSF contract PHY–9057173, under DOE contracts DE–FG02–91ER40676 , DE–AC02–89ER40509 and by funds from the Texas National Research Laboratory Commissionunder grant RGFY91B6.

1IntroductionSeveral recent papers have studied the possibility of baryogenesis in the Standard Model andits minimal extensions. For example, references [1] - [6] impose constraints on the parametersof the Standard Model, minimal supersymmetric model (MSUSY), models with additionalbosons or singlet Majoron model.

In particular, authors of ref. [1] estimate the upper limiton the lighter CP-even Higgs in MSUSY to be equal to that in the Standard Model whichthey calculate to be 55 GeV.

However, they have restricted their analysis to the case of thetop quark lighter than about 115 GeV. The present paper includes the corrections due to theheavy top and its supersymmetric partners.The obvious question is: are these corrections important?

Contributions of the heavy topand stop to the effective potential have been calculated in many papers ( [7] - [13] ) and turnout to be a dominant part of the one loop effective potential . These results have been usedin ref.

[14] in order to impose limits on the parameters of MSUSY from existing data at LEPand in ref. [15] to analyze Higgs signals in the future hadron supercolliders.In this paper we consider the one loop effective potential including contributions from Wand Z gauge bosons, top and stop quarks and two CP-even Higgs bosons of the MSUSY inthe region of parameter space where they play a non-negligible role.Next we numerically calculate the zero and nonzero temperature VEV’s and impose therequirement that the latter be large enough immediately after the phase transition to suffi-ciently suppress baryon number violating processes.

This constraint gives upper bounds onthe lighter CP-even Higgs mass and on the soft SUSY breaking scale µ.In section 2, we briefly review how baryogenesis imposes constraints on parameters of theStandard Model and in section 3 we give a short description of the Higgs sector of MSUSY.Our calculation is presented in section 4 and the results are analyzed in section 5.2Constraints from Electroweak BaryogenesisMuch attention has been devoted recently to the possibility of creating the baryon asymmetryduring the electroweak phase transition. For this to be true one has to satisfy certain basicrequirements.

First, the amount of CP violation inherent in the model has to be large enoughto account for the observed baryon to photon ratio of about 10−8. Standard Model does notsatisfy this requirement but it is possible to extend it to have enough CP violation.

Secondly,the phase transition has to be first order to provide for departure from thermal equilibrium.Once these requirements are met it is possible to construct a mechanism which showshow the baryon asymmetry was created during the electroweak phase transition, ( see forexample ref. [3] ) instead of at much higher energies.

This lower energy scale will allow forpredictions of baryogenesis to be experimentally verifiable.Regardless of the details of the particular mechanism of creating the baryon asymmetry, itis important to make sure that baryon number violating processes after the phase transitiondo not erase any previously created symmetry. To be more specific: after the phase transitionat temperature Tc, the Higgs field acquires a vacuum expectation value v(Tc).

The rate ofbaryon number violation by thermal fluctuations is proportional to the Boltzmann factorexp(−Msph/Tc), where Msph is the mass of the sphaleron field configuration or equivalently the2

height of the barrier separating the gauge field configurations with different baryon numbers.Msph was calculated at zero T in ref. [16]:Msph(Tc) = 4πB( λg2) vgw(1)with: B( λg2) ∈[1.52, 2.70], forλg2 ∈[0, ∞), where the λ is Higgs self coupling.These baryon number violating processes have to proceed at a rate much smaller than theexpansion rate of the universe after the transition, therefore the exponent of the Boltzmannfactor has to be sufficiently large.

Shaposhnikov [17] showed that:Msph(Tc)Tc≥45(2)This equation was derived under the assumption [22] that at T = Tc (1) can be approxi-mated by the same expression with v, λ, g2 evaluated at T = Tc .How does this constrain parameters of the model? For instance, in the Standard Model,the one loop effective potential V T(φ) of the Higgs field at a temperature T much higherthan the masses of the particles is given by:V T(φ) = γ(T 2 −T 22 )φ2 −ETφ3 + λT4 φ4(3)where the coefficients γ, E, λT are positive and are determined by the parameters of theStandard Model.

In particular, λT is the temperature dependent effective quartic self couplingof the Higgs field.The phase transition occurs near the point where the temperature dependent effectivemass of the Higgs field vanishes:γ(T 2 −T 22 ) ≈0(4)Then, the VEV is given by:v(Tc) ≈3ETλT(5)If we want to have v(Tc) sufficiently large to satisfy (1) and (2), we must have λT suffi-ciently small. This in turn imposes the upper limit on the Higgs self coupling λ and thereforeon the Higgs mass.The similar thing will happen in MSUSY as our results will show.3Higgs Sector of MSUSYHiggs sector of MSUSY (see for instance ref.

[18]) contains two complex Higgs doublets withthe following SU(3) × SU(2) × U(1) quantum numbers:H1 = H01H−1!∈(1, 2, −1/2) , H2 = H+2H02!∈(1, 2, +1/2)(6)3

From these eight real fields spontaneous symmetry breaking decouples three unphysicalGoldstone bosons and one is left with five physical Higgs bosons, namely: two CP-evenscalars, one CP-odd scalar and a pair of charged scalars. The tree level Higgs potential:V=m21|H1|2 + m22|H2|2 −m23(H1H2 + h.c.) + g218 (H+1 ⃗σH1 + H+2 ⃗σH2)2+g228 (|H1|2 −|H2|2)2(7)can be restricted to the real components of the neutral Higgs fields, φ1=Re H01,φ2 = Re H02:Vtree = m21|φ1|2 + m22|φ2|2 −m23φ1φ2 + g21 + g228(φ21 −φ22)2(8)One can always choose such a field basis that m23, v1, v2 are real and positive.

Constantsm1, m2 and m3 have to satisfy certain conditions. Requiring that the potential be boundedfrom below gives:m21 + m222≥m23(9)and the spontaneous symmetry breaking condition is:m43 ≥m21m22(10)Here v1 and v2 are proportional to vacuum expectation values of φ1 and φ2:< φ1 >≡v1√2, < φ2 >≡v2√2(11)and tanβ is defined to be their ratio: tanβ ≡v2/v1.

Here:qv21 + v22 = 246 GeV(12)to reproduce the measured values of gauge boson masses:m2w = g214 (v21 + v22) , m2z = g21 + g224(v21 + v22)(13)Fields φ1 and φ2 couple to down and up type quarks respectively. For example, after thespontaneous symmetry breaking top quark gets the mass:m2t = h2t < φ2 >2(14)However, if the supersymmetry is softly broken, its scalar superpartner “stop” will havethe different mass:m2˜t = h2t < φ2 >2 +µ2(15)where we consider only the case of common soft supersymmetry-breaking mass µ for ˜tL and˜tR and vanishing off-diagonal elements of the 2×2 stop mass matrix.

(The following analysiscan be easily generalized to include all of these terms. )4

Finally CP-even physical eigenstates h and H, with masses mh < mH are obtained bydiagonalizing the mass matrix:M ≡12 ∂2Vtree∂φi∂φj!min= ACCB! (16)and their masses are given by:m2h,H = 12(A + B ∓q(A −B)2 + 4C2)(17)All this was at the tree level, but, as was already mentioned, in order to obtain limits onthe Higgs mass it is paramount to include one -loop corrections.

In the effective potentialapproach used in ref. [7] and [8] , masses of the Higgs bosons are approximated with theeigenvalues of the matrix of second derivatives of the one-loop effective potential evaluatedat its minimum.The one loop effective potential at zero temperature is given by the expression:V 0 = Vtree(Q) +164π2Str(M4(φ)"ln M2(φ)Q2−32#)(18)Here M2(φ)is the field dependent squared mass matrix, Q is the renormalization scaleand the supertrace is given by:Str fM2=Xi(−1)2Jigifm2i(19)The sum runs over all the physical particles i of spin Ji, field dependent mass eigenvaluemi and multiplicity gi that couple to fields φ1 and φ2.

In our case these are W and Z bosons,top and stop quarks and h and H bosons with multiplicities:gw = 6 , gz = 3 , gt = g˜t = 12 , gh = gH = 1(20)We have neglected the contributions due to other quark-squark flavors. This is justifiableinsofar as their masses are small.

As was pointed out in ref. [7], also the bottom-sbottomcontributions can be non negligible for very large values of tanβ.This case will not berelevant for us.4Nonzero Temperature Effective PotentialWhen the temperature is nonzero, effective potential gets a contribution [19]:△VT = T 42π2XigiI±"mi(φ)T#(21)where gi are multiplicities as before, mi(φ) are field dependent masses and I−(I+) which areto be used for bosons (fermions) are given by:I±(y) = ∓Z ∞0x2 ln1 ± e−√x2+y2dx(22)5

This contribution to the effective potential describes the interactions of the Higgs bosons withthe thermal bath surrounding them. Expressions (22) are rather difficult to operate with,especially when the masses depend on fields in a complicated way.

Fortunately, as shown inref. [2], one can always use either high temperature (small y) or low temperature expansion(high y), so that the mistake in determining △VT is never bigger than 10 percent.High temperature expansions of (4.2) are given by:h−(y) = −π445 + π212y2−π6y3 −y432 ln y2cb!, h+(y) = −7π4360 + π224y2 + y432 ln y2cf!,ln cb ≈5.41, ln cf ≈2.64,(23)Whereas the low temperature expansion is:l(y) = −rπ2 y3/2e−y 1 + 158y!

(24)In this calculation we will always use one of these expansions or linear interpolationbetween them. By substituting field dependent values of mw, mz, mt, m˜t, mh and mH fromequations (13) - (17) into the expressions for the effective potential (18) and (21) one obtainsthe full zero and nonzero temperature one-loop effective potentials.

The critical temperaturein this system is close to the point where the temperature dependent effective mass matrixhas a zero eigenvalue.What can we get out of this? If we take mt and µ to be our input parameters we have:3 SUSY parameters m1, m2 and m3; 2 zero-temperature VEV’s < φ1 > and < φ2 >, 2nonzero-temperature VEV’s < φ1 >T and < φ2 >T and the critical temperature – altogethereight unknowns.How many conditions do we have?

First the fixed magnitude of the zero T VEV (11) and(12), then 2 zero-T minima:∂V 0∂φ1= ∂V 0∂φ2= 0(25)next 2 nonzero-T minima:∂V T∂φ1= ∂V T∂φ2= 0(26)the critical temperature condition:det ∂2V T∂φi∂φj!φ1=φ2=0,Tc≈0(27)and, finally, from (1) and (2), the condition that nonzero-T VEV be sufficiently large:v(Tc) =qv21(Tc) + v22(Tc) ≥vcrit(Tc) =45gwTc4πB(λ/g2w)(28)If we take the equality in (28) (which corresponds to the upper limit on the mh ), we haveseven equations in eight unknowns, therefore we can impose one relation between them. This6

was done numerically in the form: mh = mh(tanβ) for different values of parameters mt andµ. Here, mh is the upper limit on the mass of the lighter CP-even Higgs field.This program was realized in MATHEMATICA and maxima of curves mh(tanβ) are givenin figures 1,2 and 3 for mt = 115 GeV, 150 GeV and 200 GeV respectively.The argument λeff/g2 in the function B(λeff/g2) was determined in ref.

[5] for the generalcase of a two doublet model:λeff = λ1 cos4 βT + λ2 sin4 βT + 2h cos2 βT sin2 βT(29)Here βT is the nonzero-T “mixing angle of VEV’s”:tan βT = v2(Tc)v1(Tc)(30)In the MSUSY case:λ1 = λ2 = g21 + g224, h = −g21 + g224(31)therefore:λeff = g21 + g224cos2(2βT)(32)There are several causes of uncertainty in this calculation. First, we have assumed thatthe phase transition happens at point Tc where (27) is satisfied.

This is not true. It wasalready noticed in ref.

[1] that since the phase transition happens earlier, when the vacuumexpectation value is smaller than at Tc, the actual bound is stronger than the one we take.In other words, if we were able to calculate the phase transition temperature exactly theupper limit on the Higgs mass would be lower than the one we impose. Unfortunately, at thephase transition the one loop effective potential is not accurate at the origin due to infrareddivergences and therefore we can just estimate the critical temperature.Secondly, when using eq.

(28) for the MSUSY we used the fact derived in [5] that theupper bound on the sphaleron mass in MSUSY is the sphaleron mass of the Standard Model.Therefore, our bound is again weaker than the actual one but still it will turn out to be verystrong. However, the more precise calculation would require calculating the sphaleron massin MSUSY at the critical temperature.Finally, as is usual in the study of phase transitions in early universe, we are using effectivepotential which is a static quantity for a system which not only evolves but evolves out ofequilibrium.One should keep all of these caveats in mind when interpreting the results which are givenin the next section.5Results and DiscussionFigures 1, 2 and 3 show upper limits on the Higgs mass for mt=115, 150 and 200 GeVrespectively.

Points are obtained as maxima of curves mh(tanβ) for different values of µ. Foreasier visibility they have been connected by straight lines. One can draw two conclusionsfrom these results.7

First, for values of µ = 150GeV (which is the asymptotic lower experimental mass limitat 90% c.l. for a gluino mass lower than 400 GeV - see ref.

[20] ), we get the upper limit onthe Higgs mass to be 51 GeV, 54.5 GeV and 63 GeV for 3 different values of the top mass.One should compare this with experimental lower limit of 41 GeV 2 (see [14] and [21] ).Second, for the considered region of tanβ > 1 , there was always a maximal value of µabove which v(Tc) was never big enough to satisfy requirement (28). This gives the upperlimit on the soft-SUSY breaking scale of about 750 GeV, 250 GeV and 170 GeV for threetop masses considered.

One should compare this with the previously mentioned asymptoticlower mass limit of 150 GeV.As a conclusion one can establish the following “no loose theorem” from these results:either the top quark is heavy (fig 3.) in which case Higgs can be as heavy as 63 GeV butSUSY breaking scale is very close to its experimental lower limit or top is lighter than 150GeV (fig 1. and 2.) but then the Higgs mass is lighter than 55 Gev and thus close to itsexperimental lower limit.As with all other calculations in supersymmetric models this one has the trouble thatthere are simply too many unknown parameters.

We have considered that region in param-eter space which has been searched by experiments [14] ,[20], [21] ( i.e. for a common softsupersymmetry-breaking mass for ˜tL and ˜tR and vanishing off-diagonal mass elements forthe stop mass matrix ).

The limits that we obtain complement the experimental results andseverely limit the parameter space of MSUSY.Acknowledgments This research was supported by grants from NSF, DOE and TexasNational Research Laboratory Commission.I would like to thank Mitchell Golden and Stephen Selipsky for useful discussions, Chris-tian Mannes for help with MATHEMATICA and particularly my advisor Andrew Cohen forproposing this project to me and for discussing it at all its stages.References[1] M. Dine, P. Huet, R. Singleton; Santa Cruz preprint SCIPP 91/08[2] G. Anderson, L. Hall; Berkeley preprint UCB-PTH-9141[3] A. Nelson, D. Kaplan, A. Cohen; Boston Un. preprint BUHEP-91-15[4] N. Turok, J. Zadrozny; Princeton preprint PUPT-91-1225[5] A. Bochkarev, S. Kuzmin, M. Shaposhnikov; Phys.

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