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The Electroweak Phase Transition

arXiv:hep-ph/9207227v1 9 Jul 1992DFPD/92/TH/36The Electroweak Phase Transitionin a Nonminimal Supersymmetric ModelMassimo PietroniDipartimento di Fisica “Galileo Galilei”, Universit`adi Padova, Via F.Marzolo 8, 35131 Padova, ItaliaAbstractWe study the electroweak phase transition in a supersymmetricversion of the Standard Model, in which a gauge singlet superfield isadded to the Higgs sector. We show that the order of the transition isdetermined by the trilinear soft supersymmetry breaking terms ratherthan by the O(m3T) term in the 1-loop, T ̸= 0 corrections.Thisfact removes the Standard Model upper bound on the Higgs mass,mH < 55GeV , coming from the requirement that baryon asymmetryis not washed out by anomalous electroweak processes.

We perform anumerical analysis of parameter space including in the effective poten-tial top-stop contribution to 1-loop radiative corrections. We find thatthis model is compatible with the preservation of baryon asymmetryfor masses of the lightest scalar up to about 170 GeV.1

1. Introduction.Several years ago Sakharov [1] realized that the observed baryon asymme-try of the Universe (BAU) could have been generated at some early stage ofcosmic evolution at which three conditions were fulfilled: B-violating interac-tions, C and CP violation, and a departure from thermodynamic equilibrium.All these conditions are fulfilled in grand unification theories (GUT), in whichthe baryon asymmetry is generated in the out-of-equilibrium, B-violating,decay of some superheavy boson [2].

However this scenario presents severecomplications.First of all, as noted by Bocharev, Kuzmin and Shaposh-nikov[3], any nonzero fermion number (B+L) created at the GUT epoch isalmost completely erased [4] by the anomalous electroweak B+L-violatingprocesses [5], which are in equilibrium down to a temperature T ∼102 GeV.So, if no B-L is created at the GUT phase transition, then no baryon excesssurvives down to the Fermi scale.Moreover, if B-L-violating interactions(δL = 2 Majorana masses for neutrinos, R-parity violating interactions inSUSY models [5]...) are present and in equilibrium at high (T ≫100 GeV)temperature, then also an eventual B-L component of the GUT-generatedBAU vanishes before the onset of the electroweak era.The above considerations led many people to investigate the possibilitythat the BAU was generated at the electroweak phase transition,([6]-[13]).Here the difficulty is threefold; first, CP violation in the Standard Model istoo small [7], second, it is not obvious that the transition is of first orderstrongly enough to make the baryon number production effective and, third,we must avoid the wiping out of the baryon asymmetry by the anomalousprocesses. The efficiency of the latter is suppressed by the exponential of(minus) the ‘sphaleron’ mass [6] over the temperatureEsph(T)T= 4πv(T)gWBT ,(1)where B( λg2W ) is a slowly varying function of the couplings B(0) = 1.56,B(∞) = 2.72, and v(T) the value of the vacuum expectation value (VEV) ofthe Higgs field at the temperature T. The requirement that (1) is sufficientlylarge at the phase transition gives a lower bound [6] on the value of thevacuum expectation value (VEV) at the critical temperature (which we define2

here as the temperature at which the effective potential becomes flat at theorigin, V ′′(0) = 0)v(TC)TC>∼1.3. (2)In the Standard Model v(TC)/TC is given by the ratio of the coefficientof the term cubic in the Higgs field, δ, and the quartic self-coupling, λ.

So itis easy to see that (2) imposes an upper bound on the Higgs massMH246.2GeV2<∼2.3 δ. (3)The crucial δ coefficient is zero in the Standard Model at the tree level;it is generated by the one-loop finite temperature corrections [14,15] as asubleading term in the M/T expansion [15], and its evaluation requires acareful resummation of the leading infrared (IR) divergencies [14,16].

Despiteall the uncertainties related to this procedure, it seems by now establishedthat the Standard Model bound (3) has been ruled out by LEP results,MH > 57 GeV [17], or it will be in the near future.Since δ receives contribution by the bosonic degrees of freedom withmasses lighter than the temperatures in consideration 1, attempts have beenmade to relax the bound on MH by considering the Standard Model with anextended Higgs sector [6,8,9,18], or minimal SUSY Standard Model (MSSM),in which two higgs doublets are present [10]. In all these cases the cubic termis again a one-loop effect and its evaluation meets the same IR problems thatwe find in the Standard Model.

Moreover, as discussed in ref. [13], there isa large suppression (O(α8W)) of the baryon asymmetry in models contain-ing only scalar doublets, without extra CP violation sources than the Higgspotential and the Yukawa couplings.In this paper we study the electroweak phase transition in the simplestpossible extension of the MSSM with soft supersymmetry breaking, in whicha gauge singlet superfield is added to the superpotential [19,20].

From thepoint of view of the phase transition the main feature of this model is the oc-currence of trilinear soft SUSY breaking terms in the Higgs sector of the treelevel effective potential. These terms behave as effective cubic terms in theradial direction from the origin of the field space to the vacuum, so that the1in fact it may be shown numerically that the M/T expansion is a good approximationup to MT ∼2.2 [9]3

potential has a barrier along this direction already at the tree-level. More-over, including the leading (no IR problem) finite temperature corrections, atone-loop, we get also positive linear terms which enhance the barrier betweenthe false and the true vacuum.

As a result, this model leads very naturallyto a strongly first order phase transition and, as we will see, the bound (2)may be fulfilled in a large portion of the parameter space corresponding tovalues of the mass of the lighter scalar up to about 170 GeV .2. The model.We assume [20] that the squark and slepton fields have vanishing vacuumexpectation values so that we can restrict our attention to the Higgs sectorof the superpotential involving the superfields ˜H1, ˜H2 and ˜N onlyWHiggs = λ ˜H1 ˜H2 ˜N −13k ˜N3.The tree level scalar potential is given byV=VF + VD + VSoft;(4)VF=|λ|2 h|N|2(|H1|2 + |H2|2) + |H1H2|2i+ k2|N|4 −(λk∗H1H2N2∗+ h.c);VD=g2 + g′28(|H2|2 −|H1|2)2 + g22 |H†1H2|2;VSoft=m2H1|H1|2 + m2H2|H2|2 + m2N|N|2−(λAλH1H2N + h.c.) −(13kAkN3 + h.c.),where H1 ≡ H01H−!, H2 ≡ H+H02!.Note in particular the presence of trilinear terms in VSoft, which are absentin the Higgs sector of the MSSM.Redefining the global phases of H1 and N we can always take λAλ andkAk real and positive, while by an SU(2)XU(1) global rotation we putv+ ≡< H+ >= 0, v2 ≡< H02 >∈R+.4

We will not discuss the CP violation aspect of the BAU generation prob-lem, so we assume CP conservation. Choosing λk > 0 we avoid explicit CPviolation and find three degenerate minima (note that the problem has adiscrete Z3 symmetry[21]) corresponding to different values for φ1 and φx,which are respectively the phases of v1 ≡< H1 > and x ≡< N >φ1=0,φx=0;φ1=23π,φx=43π;φ1=43π,φx=23π.

(5)Due to our assumption of CP conservation, the true vacuum of the theoryis the one with both phases equal to zero.We will work in the unitary gaugeH1 = (Re H01 +i√2 sin β A0)sin β C+ ∗!, H2 = cos β C+(Re H02 +i√2 cos β A0)!,where, as usual, tgβ = v2/v1, andA0≡√2(sin β Im H01 + cos β Im H02);C+≡cos β H+ + sin βH−∗.The combinations orthogonal to C+ and A0 are the would-be Goldstonebosons giving masses to the W ± and Z0. The physical degrees of freedomof the model are given by three scalars, two pseudoscalars and one chargedscalar.Before proceeding in the calculation of the finite temperature corrections,we wish to comment on some aspects of the tree level potential which will beused in what follows.

First, it contains seven free parameters: λ, k, Aλ, Ak,m2H1, m2H2 and m2N. Imposing the stationarity conditions in (H01, H02, N) =(v1, v2, x) with the constraintv21 + v22 ≡v2 = (246.2 GeV )2,we express the soft masses in terms of the six parameters λ, k, Aλ, Ak, tgβ5

and xm2H1=λ(Aλ + kx)xtgβ −λ2(x2 + v2 sin2 β)−g2+g′24v2 cos 2βm2H2=λ(Aλ + kx) x cotgβ −λ2(x2 + v2 cos2 β)+g2+g′24v2 cos 2βm2N=λAλv22x sin 2β + kAkx −λ2v2 −2k2x2+λkv2 sin 2β(6)Note that (6) does not guarantee that (v1, v2, x) is the global minimum ofthe effective potential (4); for each choice of the parameters we must verifythat this is indeed the case.In addition, we have to be sure that v−vanishes in the vacuum; lookingat the potential along the charged direction,Vch=|H−|2"m2H1 + λ2|N|2 + g24|H1|2 + |H2|2+g′24 (|H1|2 −|H2|2)#+ g2 + g′28|H−|4,we see that a sufficient condition in order to have v−= 0 as a global minimumism2H1 + λ2x2 + g24 v2 + g′24 v2 cos 2β > 0(7)Now we introduce the finite temperature corrections at the one-loop level.As usual [15] they can be split into T = 0 and T ̸= 0 contributions.The former are given by ([22])∆V 1−loopT=0=164π2Str(M4(φ)"lnM2(φ)Q2−32#),(8)where M4(φ), with φ = (H1, H2, N), is the field dependent mass matrix andQ the renormalization point. The Q2 dependence in (8) is compensated bythat of the renormalized parameters, so that the full effective potential isindependent of Q2.We have approximated expression (8) by considering only the top and stopcontributions: in addition, we take the two stops’ masses to be degeneratem2˜tL,R ≈˜m2Q + h2t|H2|2 , m2t = h2t|H2|2,(9)6

where ˜m2Q is the squarks soft mass (we assume ˜m2U = ˜m2Q) and ht is theYukawa coupling. We fix the renormalization scale Q at a value Q such thatthe tree level relations (6), expressed in terms of the renormalized parameters,do not change, i.e.∂∆V 1−loopT=0∂H2 Q2 = Q2H2 = v2= 0.

(10)Since we have neglected the ˜tL −˜tR mixing terms in the stop mass matrix,the correction (8) does not affect the pseudoscalar mass matrix [22], while itmodifies the scalar spectrum.The T ̸= 0 part of one-loop corrections [15] is given by the expression∆V 1−loopT̸=0(φ, T)=XFgFT 42π2Z ∞0dx x2ln1 + e−x2+m2F (φ)T 2 12 (11)+XBgBT 42π2Z ∞0dx x2ln1 −e−x2+m2B(φ)T 2 12 (12)where m2F (B)(φ) is the tree-level mass of a fermion (boson) in presence ofthe background fields φ(= H1, H2, N), gF (B) is the corresponding number ofdegrees of freedom and the sum runs over all fermions (F) and bosons (B)of the theory. For values of the fields such that m(φ)/T < 1 we can expand(11, 12) as ([15])∆V 1−loopT̸=0(φ, T)≈PF gFT 4−7π2720 +m2F (φ)48T 2 + Om4F (φ)T 4 lnmF (φ)T+PB gBT 4−π290 +m2B(φ)24T 2 + Om3B(φ)T 3,(13)where we have neglected the subleading m3B/T 3 term in the expansion forthe bosons.

As we said, this term is essential in the Standard Model as in theMSSM in order to obtain a first order phase transition, while in the presentmodel the barrier between the true and the false vacua is given mainly bythe tree-level trilinear terms. We have checked numerically that the cubicterm coming from one-loop corrections changes the values of TC and V (TC)only by a few percent.7

The degrees of freedom corresponding to masses m > T are Boltzmann-suppressed and in this limit the correction (15) reduces to ([9])∆V 1−loopT̸=0(φ, T) ∼XF,BgiT 2(2π)32 m2isTmie−miT"1 + 158Tmi+ O T 2m2i!#.Since we are considering temperatures of the order of MW the exponentialfactor in the previous expression allows us, with good approximation, toneglect SUSY particles with masses above the SUSY threshold which weassume to be˜m ≈1TeVLighter particles will contribute to (13) even if it may happen that M(φ)/T >∼1 for some of them, at large values of the field. However (see the footnoten.

1) this does not change matters essentially with respect to the situationin which we use (11,12). Moreover the use of (13) allows us to perform ananalytic study of the critical temperature and of the phase transition whichcan help us to understand the main properties of the model, otherwise hiddenby a numerical evaluation of the integrals in (11,12).In the following we will derive the mass matrix for the relevant degreesof freedom of the theory on the background of all the scalar fields.

We willthen discuss bounds on the parameter space coming from the requirement ofa correct symmetry breaking pattern (i.e. no VEV’s for A0, ImN and forC+), and on the LEP limits on chargino masses.

Finally we will scan theallowed parameter space, find the values of the critical temperatures, and theminimum of the potential at those temperatures.3. The mass matrix.Let us consider the matrix of the second derivatives of the tree-level ef-fective potential: due to C and CP conservation it is a block-diagonal 6x6matrix containing the 3x3 neutral scalars matrix, the 2x2 pseudoscalar ma-trix and the charged scalar mass.

The elements of these matrices are givenby:a)neutral scalars:8

in the basis (ReH1, ReH2, ReN) we haveM2S 11=2m2H1 + 2λ2(|N|2 + |H02|2)+g2+g′22[Re(H012) + 2|H01|2 −|H02|2 −|H+|2 + |H−|2] + g2|H+|2;M2S 22=2m2H2 + 2λ2(|N|2 + |H01|2)+g2+g′22[Re(H022) + 2|H02|2 −|H01|2 −|H−|2 + |H+|2] + g2|H−|2;M2S 33=2m2N + 2λ2(|H01|2 + |H02|2 + |H+|2 + |H−|2) + 4k2 [3(ReN)2 + (ImN)2]−4kAkReN −4λkRe(H1H2);M2S 12=[4λ2 −2(g2 + g′2)]ReH01ReH02 −4λAλReN−4λk [(ReN)2 −(ImN)2] −4λ2 −g22Re(H+H−);MS 13 = 4λ2ReH01ReN −4λkRe(H02N∗) −2λAλReH02;MS 23 = 4λ2ReH02ReN −4λkRe(H01N∗) −2λAλReH01;b)pseudoscalars:in the basis (A0, ImN)MP 11=sin2 β"m2H1 + λ2(|N|2 + |H02|2) + g22 |H+|2+g2 + g′24(2|H01|2 −|H02|2 −Re(H012) + |H−|2 −|H+|2#;(14)9

MP 22=m2N + +2λ2(|H1|2 + |H2|2)+4k2[2|N|2 −Re(N2)] + 4λkRe(H1H2) + 4kAkReN;(15)MP 12=1√2 sin βn4λ2ImNImH01 −4λkRe(N∗H02) + 2λAλReH02o+1√2 cos βn4λ2ImNImH02 −4λkRe(N∗H01) + 2λAλReH01o;c)charged scalar:m2C=cos2 β"m2H2 + λ2(|N|2 + |H−|2) + g2 −g′24|H01|2+g2 + g′24(|H02|2 + 2|H+|2 −|H−|2)#+sin β cos β"−λ2(H01H02 −H+H−) −g2 + g′24(H+H−)+g22 (H01H02) + λkN2 + λAλN∗+ h.c.#+sin2 β"m2H1 + λ2(|N|2 + |H+|2) + g2 −g′24|H02|2+g2 + g′24(|H01|2 + 2|H−|2 −|H+|2)#.Next we come to the SUSY particles. Among the squarks and leptonswe should consider only the stop-quarks ˜tL,R, the others having negligibleYukawa couplings.

However ˜tL,R masses (9) are dominated by ˜mQ, which atthe electroweak scale is much greater than MW (its renormalization groupequation is dominated by the strong coupling [19]) and then their contribu-tion is Boltzmann-suppressed.Charginos have a 2x2 mass matrix (in the basis ( ˜H−, ˜W −))M2ch = λNg√2H02g√2H01M2!,(16)10

where M2 is the gaugino direct mass term. AssumingM2 ≈˜m ≫MW ≈λxwe getmch12 ≈λ2|N|2 ; mch22 ≈M22 + 2g2(|H01|2 + |H02|2),which shows that the lightest chargino may be in thermodynamic equilibriumand contributes to (13).

The LEP bound on the lowest chargino mass [23]mχ± > 45GeVwill be imposed to constrain the parameter space for λ and x.The 5x5neutralinos mass matrix, M2N has two heavy (of order M1,2) Boltzmann sup-pressed eigenvalues. With the choice of the parameters that will be illustratedin the following, the off-diagonal terms MN 122, MN 132 and MN 232, are negli-gible in comparison with the three diagonal ones, MN 112, MN 222 and MN 332,so that the approximate masses for the three lightest neutralinos are givenby the following expressionsmN12≈λ2(|N|2 + |H02|2) + (g2 + g′2)|H01|2;mN22≈λ2(|N|2 + |H01|2) + (g2 + g′2)|H02|2;mN32≈λ2(|H01|2 + |H02|2) + 4k2|N|2.Now we are ready to write the one-loop, T ̸= 0, correction in the hightemperature limit.

It is given by∆V 1−loopT̸=0(φ, T)=T 224 [TrM2S + TrM2P + 2m2C + 6M2W + 3M2Z+6m2t + 2m2ch1 + m2N1 + m2N2 + m2N3]′,where [· · ·]′ means that the sum runs on the light masses only. As the field-dependent mass of a given particle becomes much greater than the tempera-ture, the contribution of that particle to the previous expression is switchedoff.

Numerical evaluations [9] of the integrals in (11, 12) show that we canapproximate them at better than ten percent using expression (13) for massesup to m/T ∼2.11

4. Determination of the critical temperature.We defined the critical temperature TC as that value of T at which theorigin of the field space becomes a saddle point for the effective potential.In fact, it is well known [9,24] that the transition proceeds by quantum tun-nelling and is completed at a temperature higher than TC, however we arenow interested in determining the order of the transition, which is parame-terized by the vacuum expectation value at TC (v(TC) = 0 means a secondorder phase transition, v(TC) ̸= 0 a first order one).It is easy to check that at high temperature symmetry is restored, in thesense that the origin is a minimum of the effective potential.The critical temperature is then defined by the conditiondethMTS2(TC)iφi=0 = 0,(17)where the effective mass matrix MTS is given by the second derivatives of thefull one-loop potential at finite temperature with respect to the scalar fields.In the origin of the field space the effective mass matrix is approximativelydiagonal in the basis (ReH1, ReH2, ReN)MTS2(TC)φi=0 = 2m2H1 + T 224 CH1T 224 C120T 224 C12m2H2 + T 224 CH2 + ∆rad000m2N + T 224 CN,(18)whereCH1=λ2(5 + cos2 β) + g24 (22 −cos 2β) + g′24 (8 −3 cos 2β);CH2=6h2t + λ2(5 + sin2 β) + g24 (22 + cos 2β) + g′24 (8 + 3 cos 2β);C12=sin 2β(g22 −λ2);CN=9λ2 + 12k2 + 3λk sin 2β,and∆rad = 3h2t8π2 ˜m2log˜m2˜m2 + m2t.12

Neglecting the off-diagonal term, the critical temperature is given by thehighest among T SH1, T SH2 and T SN whereT Si2 = −24m2i + δi, H2∆radCi(i = H1, H2, N). (19)Obviously the effective potential can become flat only along those directionscorresponding to negative soft masses.

Negative values of m2N may howeverbe dangerous for CP conservation, which we have assumed as a ‘boundarycondition’ for the vacuum at T=0; if we look at the effective mass matrix forpseudoscalars in the origin of field space, and in the basis (A0, ImN), we getMTP2 = 2 m2H1 sin2 β + m2H2 cos2 β + T 224 PA000m2N + T 224 PN!wherePA0=λ2(3 + 52 sin2 2β) + 516(g2 + g′2) cos2 2βPN=9λ2 + 12k2 −3λk sin 2β.Then, at the temperatureT PN2 = −24m2NPN≥T SN2the potential becomes flat, in the origin, along the direction ImN, whileit remains convex in the direction ReN.So, if at that temperature thetransition in the scalar fields has not yet taken place 2 then the imaginary partof the fields acquires a vacuum expectation value, and, when the temperaturegoes to zero, the vacuum is no more φ1 = φx = 0 (see eq. (5)).Due to the heavy top [19], m2H2 can run to negative values at low energy,whereas, if λ is not too large, m2H1 remains positive.

It seems also plausiblethat, for small λ and k also m2N remains positive, although a detailed reso-lution of the RGE’s for the soft parameters would be necessary in order toclarify this point.2Note [20] that in (ReH1, ReH2, ReN) = (v1, v2, x) the point (A0, ImN) = (0, 0) isalways a minimum, as the pseudoscalar mass matrix has only positive eigenvalues.13

However, in what follows, we will constrain the parameter space by meansof eq. (6), requiring that the only negative soft mass is m2H2.The critical temperature will then be given, in any case, byTC ≈T SH2 =vuut−24m2H2 + ∆radCH2,where the ≈is due to the fact that we have neglected the off-diagonal termin (16).

Anyway, numerical results will be obtained by solving the condition(17) with the complete matrix.5. Minimization of the potentialOnce we have ensured the correct pattern of symmetry breaking, i.e.

noVEV’s for ImN, A0 and C+ (this last condition is guaranteed by eq. (7)) wecan study the minimization of the potential, at the critical temperature, inthe scalar directions only.Before illustrating the numerical results of this minimization let us givean intuitive view of what actually happens, namely, of the crucial role playedby the trilinear soft SUSY breaking parameters Aλ and Ak in the formationof a minimum of the effective potential for large values of the fields (∼TC).We define the polar coordinatesReH01=Y cos α cos βT;ReH02=Y cos α sin βT;ReN=Y sin α.The full effective potential may then be written as the sum of terms of dif-ferent order in ˜Y ≡Y/T˜V ≡VT 4 = a˜Y + b˜Y 2 + c ˜Y 3 + d ˜Y 4 + · · · ,(20)where the ellipses indicate terms coming from ∆V 1−loopT=0, which, in our ap-proximation, are different from zero in the direction H02 only and, due to the14

renormalization condition (10), will be small as long as we search a minimumat TC with H02 of the same order of v2.The coefficients in (20) are given by:i) linear terms:a = 18 sin 2β sin αλAλT ;ii) quadratic terms:b = 1T 2"cos2 α(m2H1 cos2 βT + m2H2 sin2 βT) + m2N sin2 α + T 224 (· · ·)#;here we do not write explicitly the terms coming from finite temperature cor-rections; for the present purposes what matters is that the complete quadraticterm is positive or zero at T = TC in any direction;iii) cubic terms:c = −1T sin αλAλ cos2 α sin 2βT + 23kAk sin2 α;iv) quartic terms:d=λ2 cos2 αsin2 α + 14 cos2 α sin2 2βT+ k2 sin4 α−λk4 sin2 2α sin 2βT + g2 + g′28cos4 α cos2 2βT.At large values of the fields, ˜Y 2 ≥a/c(∼O(1)) (that is Y ≥TC) the potentialhas a minimum at˜Y ≈−3c4d;the condition for this to be a global minimum is (neglecting the quadraticterm)c3ad2 ∼sin2 αλAλ cos2 α sin 2βT + 23kAk sin2 α3Aλ sin 2βT 2Cλ5≫1(21)Assuming that eq. (21) is satisfied, the question is now to determine thedirection of the vacuum in the field space.Since the only negative term15

appearing in (20) is the cubic one, and it is proportional to sin α , there willbe always a ReN-component in the direction of the vacuum.However, for large Aλ the l.h.s.of (21) decreases for sin α = 1 (it isthe effect of the linear term that raises the potential in that direction), sothat the sin α ̸= 0 directions become favorite, and an electroweak first-order(nonzero VEV’s for ReH1, ReH2) phase transition takes place.6. Numerical results.Contrary to the case of the Standard Model, the constraint (2) on thevacuum expectation value of the fields at the critical temperature cannot bewritten as a simple bound on the masses of the scalars in this model.

Infact, approximate analytic expressions for these masses can be obtained [20]in the limiting cases x ≫v1, v2 or x ≪v1, v2, but we are here interestedin values of x of the same order of the electroweak VEV’s v1, v2 3. So weare induced to perform a numerical investigation on the parameter space inorder to find the region in which the bound (2) is satisfied, and then look atthe corresponding values for the mass of the lightest scalar.In addition to the six tree-level parameters, we have to consider also thoseappearing in the one-loop corrections, namely, the top Yukawa coupling ht,the stop soft SUSY breaking mass ˜m2Q, and the gauginos direct masses M1and M2 (14).

We will assume a common value for ˜m2Q, M21 and M22˜m2Q = M21 = M22 = ˜m2 = 1TeV.Then we constrain ht, λ, and k by means of the renormalization group anal-ysis performed by the authors of ref [25].Requiring that the coupling λremains perturbative up to a large (say 1016 GeV) scale, they find, at MZλ2(MZ) < 2M2Zv2= 0.274,(22)and3One of the original reasons for the introduction of this model was the so called µ-problem, whose solution requires λx ∼O(MW ) [20].16

ht(MZ) ≈0.97(23)Moreover, RGE’s for the couplings have the fixed ratio point (neglecting g, g′with respect to ht)k2λ2 = 12. (24)We will fix λ at its upper value as given by (22) and ht and k according to(23), (24).For fixed M2 and λ the experimental limit [23] on the chargino mass givesa lower bound on x, approximativelyx ≥xmin ≈45GeVλ.

(25)We then fix tanβ to some typical value (we will use tanβ = 2, 10) and require(see the discussion on the critical temperature)m2H1 > 0, m2H2 < 0.This, in turn, impliesAminλ< Aλ < Amaxλ,(26)whereAminλ=1x"λcotgβ x2 + v24!+ λv24 sin 2β −kx2#;Amaxλ=1x"λtgβ x2 + v24!+ λv24 sin 2β −kx2#,or, equivalentlyλsin2 β x2 + v24!< m2c <λcos2 β x2 + v24!,where m2c is the mass of the charged scalar. Eq.

(26) automatically ensuresthat eq. (8) is satisfied, i.e., that the charged fields do not acquire any VEV.Finally, the requirement m2N > 0 implies a lower bound on AkAk > 2kx + λv2kxλ −sin 2βAλ2x + k.(27)17

For any triplet (x, Aλ, Ak) satisfying (25), (26), and (27), we minimizenumerically the effective potential at zero temperature and verify that thevacuum is at (ReH01, ReH02, ReN) = (v1, v2, x). We then calculate the criticaltemperature TC according to the definition (17), and, finally, minimize theeffective potential at TC.In Fig.

1 we report v(Tc)/Tc against the lightest scalar (mS1) mass for allthe points that passed the above mentioned selections. As we see there are alot of points corresponding to v(Tc)/Tc > 1.3 and mS1 > 60GeV .

Note thatthe maximum allowed values for mS1 reach 170GeV .In Figs. 2a, 2b we plot the allowed region (i.e.

the points correspondingto v(Tc)/Tc > 1.3) in the mS1 −mc plane, where mc is the charged scalarmass; x is fixed at 300 GeV, while the upper and lower limits for mc aregiven by (26).Finally, in Fig. 3a, 3b we recover the result we got from the naive discus-sion on the minimization of the potential, namely, the crucial role played byAλ in the determination of the direction of the transition.

The dashed linesrepresent the values of v(TC), while the continuos ones those of TC.In conclusion, we have showed explicitly that minimal extensions of theMSSM allow to easily circumvent the Higgs mass problem arising from thestudy of the electroweak phase transition in the Standard Model and in theMSSM. The main role is played by the trilinear soft SUSY breaking terms,which are present in any supersymmetric model with extra SU(2)-singlet su-perfields.

Radiative effects are also important as, due to the heavy top, theyforce the direction of the transition (i.e. the flat direction of the potential atTC) to be always ReH2.AcknowledgmentsThe author would like to thank A. Masiero, who inspired this work andfollowed each stage of its preparation, and G.F. Giudice, who read themanuscript and provided helpful comments and suggestions.

Useful discus-sions with D. Comelli, F. Illuminati and F. Zwirner are also acknowledged.18

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Figure Captions.Fig. 1 v(TC)/TC ratio vs. the lightest scalar mass, for all the allowed regionfor the parameters λ, k, x, Aλ, and Ak (see text), and tgβ = 2.Fig.

2a Upper and lower bounds for the lightest scalar mass coming fromthe various conditions imposed on parameter space (see text) plus the re-quirement that v(TC)/TC) > 1/3. x has been fixed to 300 GeV and tgβ = 2.Fig.

2b The same as in fig 2a with tgβ = 10.Fig. 3a v(TC) and TC dependence from Aλ, for x = 300GeV , Ak = 190GeVand tgβ = 2.Fig.

3b Same as in Fig. 3a for x = 400GeV , Ak = 300GeV and tgβ = 10.23

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