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Electroweak Baryogenesis

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Electroweak Baryogenesis Third Latin American Symposium on High Energy Physics PROCEEDINGS Electroweak Baryogenesis Marta Losada Centro de Investigaciones, Universidad Antonio Nari~no Cll. 59 No. 37-71, Santaf´edeBogot´a, Colombia E-mail: [email protected] Abstract: We present a summary of the physical framework necessary for electroweak baryoge- nesis. 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. 2 Third Latin American Symposium on High Energy Physics Marta Losada 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.
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