Supersymmetric Electroweak Baryogenesis in the WKB Approximation

We calculate the baryon asymmetry generated at the electroweak phase transition in the minimal supersymmetric standard model, treating the particles in a WKB approximation in the bubble wall background. A set of diffusion equations for the particle species relevant to baryon generation, including source terms arising from the CP violation associated with the complex phase δ of the μ parameter, are derived from Boltzmann equations, and solved. The conclusion is that δ must be > ∼ 0.1 to generate a baryon asymmetry consistent with nucleosynthesis. We compare our results to several other recent computations of the effect, arguing that some are overestimates. Although theories to explain the baryon asymmetry of the Universe (BAU) abound, only electroweak baryogenesis has the hope of short-term testability. To calculate the asymmetry in detail one needs a specific model of electroweak physics beyond the standard model, and supersymmetry is the most popular idea for such new physics. Several groups have recently estimated the BAU coming from the minimal supersymmetric standard model (MSSM) at the electroweak phase transition (EWPT), assuming it is first order and so proceeds by bubble nucleation [1]-[6]. Despite the fact that the physics of the mechanism is essentially agreed upon—particles in the plasma interact with the bubble wall in a CP-violating manner, leading to a chiral asymmetry of quarks in front of the wall, which in turn biases sphalerons to produce the BAU—there is considerable disagreement on both methods and results. While it is straightforward to formulate diffusion equations for particle transport away from the wall, it is less obvious how to derive the CP-violating source terms (or equivalently the boundary conditions at the wall) for these equations, and it is here that most of the difficulty lies. Several methods have been proposed in which the source terms are first derived and then inserted into a set of diffusion equations, but without any first principles justification for the prescription used. In the present work we generalize a method which was introduced for the treatment of the non-supersymmetric two doublet model in [7], in which the diffusion equations and the source terms are derived together starting from a general set of classical Boltzmann equations. This provides a systematic and controlled approximation valid for bubble walls significantly thicker than the inverse temperature, which is the case in the MSSM. Here we will describe the results as concisely as possible; finer details will be presented in a longer publication [8]. The method is based on a WKB quasi-particle approximation to the particle dynamics and interactions in the background of the bubble wall, which is an expansion in derivatives of the Higgs field VEV’s that vary continuously in the wall. Such an expansion is appropriate because it well-describes the majority of particles in the plasma, whose wavelength (∼ 1/T ) is much shorter than the thickness of the wall (∼ 20/T ). When there is CP violation in the theory, such as is provided by the complex μ and At parameters of the MSSM, this approximation shows that particles and anti-particles experience a different force when interacting with the bubble wall, both dynamically as free WKB particles and in their