Electroweak Baryogenesis

Third Latin American Symposium on High Energy Physics

PROCEEDINGS

Marta Losada Centro de Investigaciones, Universidad Antonio Nariño Cll. 59 No. 37-71, SantafédeBogotá, Colombia E-mail: [email protected]

Abstract: We present a summary of the physical framework necessary for electroweak baryogenesis. We discuss in detail the strong constraints on the physical parameters of the models from the preservation of the baryon asymmetry of the Universe for the Standard Model and the Minimal Supersymmetric Standard Model. For the Constrained MSSM we discuss the region in which also cosmological constraints from dark matter are simultaneously satisﬁed. We brieﬂy discuss alternative scenarios which can increase the available parameter space and in particular we comment on models with low reheating temperatures.

1. Introduction literature [3, 4].

Let us quickly summarize the current un- derstanding related to the other two requisites A compelling and consistent theory which can ex- for baryogenesis. Baryon number violation oc- plain the observed baryon asymmetry of the Uni- curs in the Standard Model through anomalous verse (BAU) is one of the most challenging the- processes. At low temperatures this anomalous oretical aspects of the interplay between particle baryon number violation only proceeds via a tun- physics and cosmology. Many mechanisms for nelling process which is exponentially suppressed, the production of the baryonic asymmetry have (− 4π ) at a rate Γ ∼ e αW . However, at ﬁnite temper- been discussed for diﬀerent periods of the evolu- ature these interactions are in equilibrium above tion of the early Universe which include GUT- the electroweak scale with a rate propotional to baryogenesis, leptogenesis, etc. For reviews on Γ ∼ α5 T 4 and may erase any previously pro- this subject see for instance [1]. The electroweak W duced B-asymmetry [5]1. scale is the last instance in the evolution of the

Universe in which the baryon asymmetry could At finite temperature T the rate Γs per unit have been produced. Moreover, the scenario of time and unit volume for fluctuations between electroweak baryogenesis places constraints on the neighboring minima with different baryon num- physical parameters of specific models which are ber is [6] testable at current and future accelerators. α 4 ζ7 Γ ∼ 105 T 4 W κ e−ζ, (1.1) In 1967, Sakharov [2] enunciated the neces- s 4π B7 sary ingredients for the production of the baryon asymmetry, which are: baryon number violation, wherewehaveusedζ(T)=Es(T)/T and Es(T )= 2 C and CP violation, and a departure from ther- [2mW (T )/αW ] B(λ/g ) is the sphaleron energy, 1 mal equilibrium. The Standard Model contains mW (T )= 2ghφ(T)i,B'1.9 is a function which all necessary physical aspects, and thus was con- depends weakly on the gauge and the Higgs quar- 2 sidered that solely within this framework baryo- tic couplings g and λ, αW = g /4π =0.033. genesis could be explained. As far as the required 1The loophole is that sphaleron transitions conserve CP-violation, present from the CKM matrix in B − L,soanetB−Lgenerated previously through in- the Standard Model, we refer the reader to the teractions in a specific model will not be erased. Third Latin American Symposium on High Energy Physics Marta Losada

Thus, below the electroweak phase transition the temperature expansion is sphaleron transition rate changes as the SU(2) 3 ∼ − 4πφ ≡ gauge field acquires a mass Γs/T e gT 2 2 3 4 2 e− Esph . m (φ)T m (φ)T m (φ) T T i − i + i (ln +C ), (1.2) 24 12π 64π2 µ2 i In 1985, Kuzmin, Rubakov and Shaposhni- kov[7] suggested that if the electroweak phase and from fermions transition was first order it provided a natural m2(φ)T 2 m4(φ) T 2 i − i (ln + C ). (1.3) way for the Universe to depart from equilibrium. 48 64π2 µ2 i Eventually bubbles of the broken phase nucle- ate and grow until they fill the Universe. Lo- We can easily see that the terms of the form cal departure of thermal equilibrium takes places m2(φ)T 2 ∼ φ2T 2, are responsible for symmetry near the walls of the expanding bubbles and if i restoration at very high temperature, and that asymmetries of some local charges are produced the terms −m3(φ)T ∼−φ3T produce a barrier which then diffuse into the unbroken phase where i in the potential as the temperature decreases and baryon number violation is active this converts thus allows a first order phase transition to oc- the asymmetries into a baryon asymmetry. Fi- cur. In many extensions of the Standard Model, nally, the baryon number flows into the broken finite temperature computations are complicated phase. Now it is necessary that the sphaleron by a hierarchy of mass scales, indeed many mass transitions are turned off after the phase transi- scales can appear for which the high-temperature tion (in the broken phase) so as to not wash out expansion is not adequate. In some cases, how- the asymmetry produced during the transition. ever a low-temperature expansion can be em- This is a crucial test for any model of electroweak ployed to a very good approximation. Recently baryogenesis. The condition which guarantees it was shown how a perturbative calculation to the non-erasure of the produced asymmetry oc- two-loops can be computed without any temper- curs when the transition rate of the sphaleron in- ature expansions [14]. teractions is small (out-of-equilibrium) compared to the Hubble expansion rate. In a radiation In general, we can parametrize the effective dominated Universe the Hubble parameter is H ∼ 2 potential at one-loop in the following way, T . MPl

3 < Requiring Γs/T ∼ H at the bubble nucle- 1−loop 2 − 2 2 − 3 4 ation temperature Tb leads to the condition on Veff =(γT m )φ ETφ + λφ . (1.4) ζ(Tb). It can be shown that to avoid erasing any generated baryon asymmetry via sphaleron in- Thus, the phase transition will occur when ζ T )= Esph > 45. Given teractions we require ( c Tc ∼ that E is defined in terms of φ, this in turn 3 2 sph φ(Tc) ∼ E ∼ g ∼ mW implies that we must have a sufficiently strong g 2 , (1.5) Tc λ λ mh phase transition such that φ ∼> 1. Where φ is Tc where we have used the Standard Model lead- the order parameter of the transition. ing contributions to E. At zero-temperature, λ One of the main analytic tools for study- is related to the mass of the Higgs particle. At 2 2 ing the thermodynamics of the phase transition tree level in the SM, mh =4λφ ,sosmallλcorre- is the finite temperature effective potential. At sponds to small Higgs mass. When quantum cor- one-loop, the non-interacting bosonic or fermionic rections are included, the relationship becomes particles, whose mass depends on the background less simple, but the erasure condition still trans- field φ, contribute to the finite temperature effec- lates into an upper bound on the Higgs mass. ( ) φ Tc > 1 shows that we need a tive potential. Thus, requiring Tc ∼ small value of mh, which is not possible due to The contributions from bosons in the high- current experimental bounds.

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At two-loops there are new contributions which constrain a whole class of models which are de- could modify our previous conclusions, as the scribed by the same effective 3-d theory contain- 2−loop ∼ leading correction is of the form ∆Veff ing a single light scalar at the phase transition[12]. 2 2 2 mi(φ) ∼ 2 2 φ g T mi (φ)ln( T ) γBT φ ln( T ). Thus we can now parametrize 2. Electroweak Phase Transition in the MSSM 2 1/2 φ(Tc) ∼ E E 2γB + 2 + . (1.6) Tc λ λ λ We have shown that the relevant contributions For a given model, the parameters γ, E, λ and B to the strength of the first order phase transition are calculable functions of the zero temperature arise from bosonic particles. Extensions of the coupling constants. This shows that two-loop ef- Standard Model with new bosons could modify fects can strengthen the phase transition if B is the strength of the phase transition. In addition, large, and that we need a large E and a small we would also like to have new CP-violating in- quartic coupling λ. This does not happen in the teractions. Standard Model. The Minimal Supersymmetric Standard Model To summarize, although the ingredients for (MSSM) provides an interesting scenario in which the production of the baryon asymmetry exist in there are many new bosonic particles. In par- the Standard Model as far as having a mechanism ticular, the stops couple strongly to the Higgs which produces a departure of thermal equilib- field producing the largest modification to the rium and baryon number violating interactions, strength of the phase transition. Analytically, to we have shown using purely perturbative argu- determine the contribution from the stops we use ments that the simpler constraint on the preser- the finite temperature stop mass matrix which is vation of the produced asymmetry cannot be ful- given by filled. M11 M12 Mstops(φ, T )= , (2.1) A drawback to the perturbative evaluation of M21 M22 the effective potential is that it has infra-red di- where vergences as the Higgs vacuum expectation value 2 1 2 2 2 2 φ approaches zero. These are due to bosonic M11 = m + h sin βφ + a1T , Q 2 t modes whose only mass in the perturbative cal- 1 M12 = M21 = −√ ht sin βφ(A + µcot β), culation is proportional to φ. Resummation is 2 then employed to deal with these divergences. 2 1 2 2 2 2 M22 = m + h sin βφ + a2T , (2.2) The net effect in the case of the Standard Model U 2 t is to reduce the contribution from the gauge bosons where a1 and a2 are combinations of the relevant to the cubic term, thus weakening the strength of couplings constants in the theory. the phase transition. An alternative way to com- pute the ratio (1.6) is using Monte Carlo 3-d lat- In the previous expressions we have neglected tice simulations via dimensional reduction [11]. the contributions from the gauge couplings, this The important point is that this will include also will suffice to illustrate which are the most rel- all relevant non-perturbative effects. In dimen- evant contributions from the stop sector. Note sional reduction an effective 3-d theory is con- that for the case of zero stop-mixing, the contri- structed perturbatively integrating out all the bution from the right-handed stop to the cubic massive modes. The characteristics of the phase term of the effective potential is transition can then be determined by simulating on the lattice the remaining 3-dimensional theory of massless bosonic modes. One of the strong 2 3/2 2 1 2 2 2 4 2 (m˜ ) = m ++ h sin βφ +[ g points of the lattice simulations is that they can tR U 2 t 9 s

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3 2 1 / effective 3D theory constructed with dimensional + (1 + sin2 β)h2]T 2 , (2.3) 6 t reduction reproduces the perturbative 4D effective potential results. The 3-D theory naturally 2 < so, if mU ∼ 0 it can cancel the finite temperature incorporates the effects of resummation and some 3 3 contributions such that m˜ ∼ δEφ . This is tR higher order corrections. We use the results given called the light-stop mechanism [13]. The crucial in ref. [15] for the two-loop finite-temperature ef- effect of the parameters of the phase transition fective potential of the MSSM with a light stop. is that now at one-loop, It assumes that the b-quark Yukawa coupling is small (this restricts the value of tan β ∼< 12), and is calculated in the limit where all the supersym- φ(Tc) ∼ (E + δE) (2.4) ∼ Tc λ metric particles are heavy ( TeV) except for the ˜ stops. 3 3 At 3/2 ˜ where δE ∼ h sin β(1 − 2 ) ,andAt = t mQ 2 < A+µcot β. Something analogous would occur Allowing mU ∼ 0 is constrained at zero tem- ˜ for the left-handed stop mtL , which could also perature from the limits on the stop mass; for the strengthen the phase transition. However, this is current limits it is still possible for the light stop not possible for two reasons: the zero tempera- scenario to be available. We will return to this ture one-loop corrections to the Higgs mass from below when we discuss experimental constraints. 2 the stop sector are large enough to satisfy the Another constraint on the negative values of mU experimental constraints only when one of the appears as colour (and charge)-breaking minima stop fields is very heavy; secondly, precision elec- can develop in the stop direction both at zero ˜ troweak constraints require mtL to be heavy. It and finite-temperature. In fact, the strongest is also worth mentioning here that the two-loop constraint arises at finite-temperature, when a QCD effects from the stop sector are sizeable and two-stage phase transition can appear in certain they strengthen the phase transition [16, 18, 15]. regions of parameter space. The fact that the stop is light allows the possibility of tunneling The effects of non-zero doublet-singlet mix- into a color and charge breaking minimum [18, ing reduce the strength of the phase transition, 13, 15, 19], from which the Universe would subse- as shown in eq. (2.4), as it reduces the contri- quently undergo a transition to the SU(2) broken bution of the light stop to δE,italsomakesthe minimum. The analysis of ref. [19] shows that Higgs heavier and increases the value of the criti- the second phase transition may not take place, cal temperature. These undesirable effects of the thus giving stronger constraints on the allowed trilinears on the survival of the baryon asymme- parameter space. This gives a lower bound on try can be partially compensated by decreasing the stop mass for every set of values of the mix- 2 the value of mU . ing parameter and tan β. This lower bound is Simulations have also been done for the case larger than the direct experimental search limit on m˜ , as shown in fig. 1. where there is a colored SU(2) singlet among t2 the light/massless modes at φ = 0 [17]. This The conclusions of the two-loop perturbative corresponds to the light right-handed stop sce- analysis for the MSSM [18, 13, 15] are that the nario that could make the phase transition strong sphaleron transitions are suppressed when the enough in the MSSM. The area of MSSM param- stop and Higgs bosons are light enough. In the eter space where the electroweak phase transition MSSM at two loops, the light singlet stop contri- is strong enough was found in the lattice analy- E φ > bution to is sufficient to satisfy Tc ∼ 1. How- sis to be larger than the area found in pertur- ˜ ever, the maximum possible values of mt2 (the bation theory[17]. This means that calculating light stop mass), and mh for which the sphaleron φ(Tc)/Tc from the perturbative effective poten- transitions are suppressed depend also on the tri- tial is conservative, so this is why we employ the linear terms and on tan β, as discussed above. 2-loop effective potential. We construct the effective potential via dimensional reduction. The In fig. 1 we display the full allowed region

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The dashed line shown deﬁnes when the criti- 110.0 cal temperatures in the φ-andχ-directions are ˜ varying A equal,. For the same variations of tan β and At, 105.0 t according to the analysis of ref. [19] the region below this dashed line would be excluded. For v/Tc=1

H this value of m the maximum allowed masses 100.0 Q m < ˜ < are mh ∼ 105GeV and mt2 ∼ 170GeV.

− ˜ 95.0 In ﬁgure 2 we plot in the mh mt2 plane

At=0 φ contours of T =1formQ = 1 TeV. The param- ˜ ˜ eters that are varied are tan β,mtR and At.The 90.0 80.0 100.0 120.0 140.0 160.0 180.0 eﬀect of increasing tan β increases the value of

mt 2 mh, thus weakening the phase transition and so

a smaller value of mtR is necessary to maintain φ Figure 1: Available region in parameter space in T = 1. It is clear that the least stringent bound m − m m A˜ ˜ the h t˜2 plane for Q = 300 GeV, varying t on mt2 arises when At = 0, but the largest possi- β φ and tan defined by the contour of T =1.The ble value of the Higgs mass mh is determined by dashed line is defined when the critical temperatures the largest allowed value of the mixing param- φ− χ− in the and directions are equal for the same eter. The most important modification in the A˜ β variations of t and tan . analysis provided by the light-stop mechanism at 2-loops is that there is no longer an upper bound 120.0 on mh, from requiring the phase transition to be sufficiently strong, it is really a zero temperature 115.0 effect which gives the largest possible Higgs mass,

110.0 and as long as the stop is light enough there is a suﬃciently strong ﬁrst order phase transition

H 105.0 m which preserves the produced baryon asymmetry. 100.0 The current experimental bounds on the Higgs 95.0 masss, are for a Standard Model-like Higgs mh ∼>

90.0 106GeV; we emphasize that the results presented 100.0 110.0 120.0 130.0 140.0 150.0 160.0 170.0 m here are valid for large values of the pseudoscalar t2 mass mA, and a maximum value of tan β = 12,

φ as corrections from the bottom- sbottom sector Figure 2: Contours of =1in the mh − m˜ Tc t2 have not been included. The experimental bound plane for mQ =1TeV, varying A˜t. The contour for the Higgs for smaller values of m is about 10 which corresponds to no-mixing corresponds to the A GeV smaller for large tan β. From experiments ﬁrst curve on the right. For increasing values of the mixing parameter the curves shift to the left on the the stop mass is required to be from the Tevatron > o ≤ plane. mt˜ ∼ 90 GeV for mχ 50GeV. As can be seen from the results in ﬁgure 2 there is only a small window in which electroweak baryogenesis in the in parameter space coming from the requirement MSSM is possible. Before we discuss alternative of having a strong enough phase transition. The scenarios, which could increase the allowed re- solid vertical line is determined by varying tan β gion in parameter space, let us for the moment ˜ for At = 0 and the solid diagonal line corre- make a small digression. sponds to the variation of A˜t for tan β = 12. In the region to the left and below these solid lines sphaleron transitions are suppressed for values of 2 ∼< tan β ∼< 12 and 0 ∼< A˜t ∼< 280 GeV.

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Dark Matter Abundance and Electroweak baryo- by genesis in the CMSSM

As we have shown above, the preservation of 2 2 the baryon asymmetry is still possible in a small dmU 16 αs 2 ht 2 2 2 2 = M3 −2 (m +m +m +A ) window of parameter space in the context of the dt 3 4π 16π2 Q U h t (2.5) MSSM. Another salient feature of this supersym- 2 mGUT metric model is that it has a dark matter can- where M3 is the gluino mass and t =ln Q2 . 2 ∼ didate, namely the lightest supersymmetric par- It can be shown that mU 0 is obtained for | |∼ | | ticle (LSP). If R-parity is conserved the LSP is Ao 11 m1/2 . The important point to notice stable and its mass density is large enough, 0.1 < is that for fixed mo,m1/2 we can independently Ωh2 < 0.3, to be cosmologically interesting for a vary Ao to produce a light stop. There is a sim- mostly gaugino-like neutralino [21]. In ref. [20] it ilar term proportional to the tau-Yukawa cou- was determined if there exists a region in which plings and for some ranges of medium to large combined experimental, theoretical and cosmo- tan β the light stau mass can be driven negative. logical constraints could be satisfied. The com- Another important constraint from baryogenesis bined constraints that were used included: LEP is the requirement of relatively small stop mix- bounds on particle masses, the branching ratio ing, thus we choose the sign of µ to be opposite for b → sγ, precision electroweak constraints to to A so as to cancel the off-diagonal elements of the ρ-parameter, electroweak phase transition boundsthe stop mass matrix. and the relic abundance constraint. In figure 3 we plot the combined set of exper- − In the constrained MSSM (CMSSM), there imental and theoretical constraints in the m1/2 −Ao plane. This figure is for values of tan β =12 are only four parameters mo,m1/2,Ao,tan β and the sign(µ) which define parameter space. Over and mo = 145 GeV. The presence of a quasi-fixed much of parameter space of the CMSSM the LSP point for At at low tan β tends to drive At to pos- is a bino. For a gaugino-type neutralino annihi- itive values in the vicinity of 2m1/2, which tends | | | | lation occurs via sfermion exchange into fermion to be much larger than µ cot β. Smaller At can be generated at moderate tan β by taking pairs (leptons). As mo increases the slepton becomes more massive thus decreasing the neutralino A0 large and negative. Accordingly, to obtain annihilation rate. Thus, the neutralino relic abun- essentially zero-mixing in the stop sector we con- dance increases, until an upper bound for the sider negative A0 and positive µ in fig. 3. The light shaded region is excluded because it con- scalar mass parameter mo is obtained. A lot of work has been done constraining SUSY models tains either a tachyonic stop or stau. The dashed using combined bounds from accelerator experi- contour combines the experimental bounds from ments and the cosmological relic abundance[21]. chargino, stop and Higgs searches with the condition that the LSP be the neutralino, to avoid This defines a specific allowed region in the mo − an excess of charged dark matter. In Fig. 3, −m1/2 plane. We are particularly interested in the overlap with the area of parameter space where the vertical left side of the dashed contour is the singlet stop soft SUSY-breaking mass is neg- due to the chargino bound, the diagonal piece ative for the preservation of the baryon asym- which parallels the light shaded region is due metry. In order to do this the low energy pa- to the stop bound, and the horizontal piece is rameters are obtained by running the couplings the line mχ = mτ˜R The experimental Higgs con- and gaugino masses with 2-loop renormalization straint, form LEP189 run, does not provide a group equations, and the 1-loop renormalization useful bound for this value of tan β. The solid group equation for other masses. In fact, the line gives the BAU constraint: above the solid light stop mass constraint from the electroweak line, sphaleron processes wash out the baryon phase transition is the hardest to satisfy. The asymmetry, while below the solid line the baryon one-loop renormalization group equation is given asymmetry is preserved. In the figures, the BAU boundary is typically set by the condition that

6 Third Latin American Symposium on High Energy Physics Marta Losada the stop mass be suﬃciently light. Lastly, the we would like to mention some of the possible dark hatching marks the region where the neu- ways of enlarging parameter space. tralino relic density violates the cosmological up- 2 ≤ per bound Ωχe h 0.3. The region which is • A straightforward procedure is to redo the allowed by all of the experimental and cosmolog- electroweak phase transition analysis with ical constraints is then highlighted by diagonal values of the pseudoscalar mass mA ∼ 100 shading. GeV with the light stop mechanism, as the

-1000 LEP bound is weaker. Also we can explore a) larger values of tan β by including the contributions from the bottom-sbottom sector.

No BAU • The explicit inclusion of CP-violation into the analysis of the production and decay

0 rate of the Higgs boson seem to hint that -1500 m 2 A ∼ < t 0 a relatively light Higgs boson could have escaped detection at LEP2[22, 23]. This m2 clearly would enlarge parameter space by τ∼ < 0 allowing a smaller value of mh, which would also imply that a heavier stop would also be -2000 100 150 200 allowed. m 1/2 • We could slightly modify the CMSSM by Figure 3: The region of mSUGRA parameter space splitting mQ (or mU ) from the other boson satisfying the BAU, experimental and cosmologi- masses related to mo. The effects of CP- cal constraints described in the text, for tan β = violation mentioned above are also relevant 12,m0 = 145 GeV and µ>0, is marked by diago- here. nal shading. Above the solid line, sphaleron processes • One can consider other models such as the wipe out the baryon asymmetry. The light shaded region contains tachyons. The dark hatched region has Two Higgs Doublet model, the Next to Min- h2 > . imal Supersymmetric Standard Model, or Ωeχ 0 3. The dotted lines are contours of con- h2 . the MSSM with R-parity violation in which stant Ωeχ =01. either the Higgs mass bounds are modified or we can have different couplings (not fixed by supersymmetry) which contribute E For a more complete discussion of this sce- to the ratio of λ . nario see ref. [20]. We have presented these re- • Alternative cosmologies can modify the con- sults because although at present LEP2 bounds straints on the parameters related to the seem to exclude the combined region of allowed production and preservation of the baryon parameter space, a caveat may exist which can asymmetry of the Universe. We will discuss open up room for this region once more. The a particular scenario next. caveat is related to the inclusion of CP-violation in the MSSM which we will discuss a bit further in the next section. Electroweak Baryogenesis in Low Treheat Models It is well known that the flatness and hori- 3. Alternative Scenarios zon problems of the standard big bang cosmology are solved if during the early Universe the energy density was dominated by some form of Our results show that the allowed regions for vacuum energy. The observed large scale density baryogenesis in the (C)MSSM are quite small and temperature fluctuations can also be gener- given the strong experimental constraints. Here ated within this framework.

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12 8 strained by laboratory bounds, also disequilib- 95 b) 16 100 rium is hard to come by as the Universe is ex- 20 panding very slowly, so it is close to equilibrium.

This would seem to rule out the possibility

100 of baryogenesis within the context of these models. Nevertheless, let us inspect more carefully what are the implications for electroweak baryo- 90 95 genesis. For other baryogenesis models see refs. [27, 28]. The crucial point is to realize that reheating is not an instantaneous process and that the maximum temperature Tm during reheating can be much larger that Treheat[26]. For a radia- 100150 200 tion dominated Universe the temperature scales ∼ −1 m as T a ; during reheating the temperature 1/2 Treheat is given by T = Tmf(a), where Tm = 1/2 , αφ Figure 4: Contours of constant Higgs mass (dot- and αφ is defined by the decay rate of the in- δρ × 4 dashed), chargino mass (dotted) and 10 (solid) flaton field, Γφ = αφMφ. The function f(a)= are displayed. −3/2 −4 1/2 κ(a −a ) ,isuchthatf(ao)=1hasits 8 2/5 maximum value for ao =(3) . The important thing to note is that the Hubble parameter is now The inflationary stage can be parametrized modified to by the evolution of some scalar field φ, the inflaton, which is initially displaced from the minimum of its potential. Inflation ends when the T 2 H ∼ H , (3.1) potential energy associated with the inflaton field RD 2 Treheat becomes smaller that the kinetic energy of the thus for smaller values of T the Hubble pa- field. The process which converts a low-entropy reheat rameter increases with respect to its value in a cold Universe dominated by the energy of the co- radiation dominated Universe. Let us now sup- herent motion of the φ-fieldintoahighentropy pose that the reheating temperature T hot Universe dominated by radiation is called re- reheat T . We have seen that during the reheating pro- heating. c cess the thermal bath may reach temperatures When the Universe becomes radiation dom- which are much larger than electroweak phase ∼ 4 ∼ inated the energy density scales like ρ Treheat, transition critical temperature Tc TEW . This this defines Treheat. Recall that for a radiation means that the Universo can undergo a phase dominated Universe the Hubble parameter is H ∼ transition in which the electroweak symmetry is 2 T . A common assumption is that the post- broken although the Universe is not yet radia- MPl inflationary Universe contained a plasma in ther- tion dominated. The main difference is that the mal equilibrium at a temperature Treheat TEW phase transition occurs in a matter dominated ∼ 102 GeV. However, experimentally, all we know Universe with an expansion rate given by eq. is that Treheat ≥ few MeV, which is required (3.1). This implies that electroweak baryogen- from nucleosynthesis. Low reheat scenarios are esis may occur even when Treheat Tc. This particularly appealing as they avoid the over- is a nontrivial result and is the crucial fact that production of dangerous relics[24, 25]. A par- we want to point out. As discussed above, the ticularly interesting question is what happens to generation of the baryon asymmetry is mediated Sakharov’s conditions of baryon number viola- by sphaleron transitions in the unbroken phase, < tion and disequilibrium in models with low reheat which are in equilibrium at temperatures T ∼ 4 2 1/3 ∼ 4 2/3 temperatures. In fact, baryon number violating αW MpTreheat 10 (Treheat/1GeV) GeV. interactions at low temperature are strongly con- We can now show that the requirement that the

8 Third Latin American Symposium on High Energy Physics Marta Losada sphalerons are out-of-equilibrium in the broken asymmetry in this context. The continous decays phase is easier to fulfill if Treheat Tc than in of the scalar field φ add entropy into the thermal the standard cosmology. By once again compar- bath when the temperature decreases from Tc to 3 ing the rate Γs/T , given in eq. (1.1), with the Treheat.IfwetakeBcto be the baryon asymme- new expansion rate of eq. (3.1) we find try to entropy density ratio nB/s generated at the electroweak phase transition, one finds that ζ(T ) ∼> 7logζ(T ) + 9 log 10 + log κ b b the final baryon asymmetry is now +2log(T /T ) reheat b 5 > S.C. nB Treheat ∼ ζ(Tb) +2log(Treheat/Tb) .(3.2) ∼ Bc . (3.4) s Tc This inequality is the standard one [1, 5], with This means that, for T ∼ 10 GeV, the mech- one crucial difference: the presence of the last reheat anism of baryogenesis at the electroweak scale term. The presence of this term implies that, if has to be more efficient by a factor ∼ 105 than the reheating temperature is much smaller than in the standard case. Although this is can be Tc (or equivalently the Universe is expanding very difficult it is not impossible to achieve. quickly) sphalerons are out-of-equilibrium in the broken phase easier than in the standard cosmology. Numerically, we can see that if we take −1 4. Conclusions ζ(Tb) ' 1.2ζ(Tc)[12],thenforκ=10 and > Treheat ∼ 1(10) GeV, we obtain that ζ(Tc) ∼ 28(33), which translates into We have presented an overview of the status of electroweak baryogenesis in the Standard Model hφ(Tc)i ∼> 0.77(0.92). (3.3) and the (C)MSSM. We have shown that the avail- T c able region in parameter space for the MSSM is This bound shows how the constraint on the or- highly constrained by experimental results. We der parameter changes compared to the stan- have also presented alternative scenarios which > dard result, hφ(Tc)i/Tc ∼ 1, obtained for the can increase the allowed region of parameter space. same value of κ. This finding clearly enlarges In particular, we have discussed the scenario of the available region in parameter space where non-standard cosmology with low reheat temper- the sphaleron bound is satisfied and relaxes the atures in which baryogenesis could still occur and upper bound on the stop mass in the MSSM the implications for electroweak baryogenesis. and on the Higgs mass in other extensions of the SM. The implication for the SM is that al- Acknowledgements though current LEP bounds on the Higgs mass still rule out electroweak baryogenesis, for small I would like to thank my collaborators S. David- values of the Higgs mass the phase transition is son, T. Falk and A. Riotto for enjoyable collabo- now strong enough for sphaleron transitions to rations of different aspects of the work presented be suppressed. From the lattice results of Ref. here. I would also like to thank the organizers [12] we can determine that Eq. (3.3) implies that of SILAFAE-III for an interesting and enjoyable the electroweak phase transition would be strong < meeting. enough for baryogenesis for mh ∼ 50 GeV. More interesting, for the MSSM in the region of allowed Higgs masses the new bound could increase References the upper bound on the stop mass by about 10 < GeV to mt˜ ∼ 180 GeV for all other parameters [1] For recent reviews, see W. Buchmuller, fixed. S. Fredenhagen, hep-ph/0001098; A. Riotto, hep-ph/9807454 ; A. Riotto and We have seen that preserving a baryon asym- M. Trodden, Annual Reviews of Nuclear metry is easier if Treheat Tc, however one must and Particle Science 49, 35 (1999). V.A. also check the effects on the production of the Rubakov, M.E. Shaposhnikov, Usp. Fiz.

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