Bubble Wall Velocity in the MSSM

We compute the wall velocity in the MSSM with W , tops and stops contributing to the friction. In a wide range of parameters including those which fulfil the requirements of baryogenesis we find a wall velocity of order vw ≈ 10 −2 much below the SM value. 1 Bubble Wall Equation of Motion in the MSSM Energy conservation leads to the equations of motion of an electroweak bubble wall interacting with a hot plasma of particles: h1 + ∂VT (h1, h2) ∂h1 + ∑ i ∂mi ∂h1 ∫ d3p (2π)2Ei δfi(p, x) = 0, (1) h2 + ∂VT (h1, h2) ∂h2 + ∑ j ∂mj ∂h2 ∫ d3p (2π)2Ej δfj(p, x) = 0. (2) where fi = f0,i+δfi is the distribution function for a particle species in the heat bath [1,2]. We have to sum over all particle species i. The distribution function is divided up into equilibrium part f0,i and out-of-equilibrium part δfi. The equilibrium part has been absorbed into the equilibrium temperature dependent effective potential VT (h1, h2). In the following we will restrict ourselves to late times leading to a stationarily moving domain wall where the friction stops the bubble wall acceleration. This is a reasonable assumption for the late stage of the bubble expansion where baryogenesis takes place. To appear in the Proceedings of SEWM2000, Marseille, June 14-17, 2000 2 Fluid Equations The deviations δfi from the equilibrium population density are originating by a moving wall. They are derived by Boltzmann equations in the fluid frame: dtfi ≡ ∂tfi + ẋ ∂ ∂x fi + ṗx ∂ ∂px fi = −C[fi], (3) with the population density fi and energy E = √ px +m 2(x). C[fi] represents the scattering integral. The classical (WKB) approximation is valid for p ≫ 1/Lw (“thick wall”). For particles with E, p>∼ gT this should be fulfilled. Infrared particles are supposed not to contribute to the friction. This is a crude approximation and there are additional contributions [3] which further lower the wall velocity. In the MSSM the wall thickness Lw is of order 15/T–40/T , as found in [4, 5], and Lw ≫ 1/T is fulfilled. Those particles which couple very weakly to the Higgs are denoted as “light particles”. Particles coupling strongly to the Higgs are heavy in the Higgs phase and therefore called “heavy”. “superheavy” particles as the “left handed” stops do not appear in our calculation besides their remnants in the effective potential. We treat as “heavy” particles top quarks, (right handed) stops, and W bosons. The Higgses are left out. Further contributions produce an even smaller velocity. We assume now that the interaction between wall and particle plasma is the origin of small perturbations from equilibrium. We will treat perturbations in the temperature δT , velocity δv and chemical potential δμ and linearize the resulting fluid equations. Then the full population density fi of a particle species i in the fluid frame is given by