ar X iv : h ep - p h / 96 04 24 9 v 1 4 A pr 1 99 6 ELECTROWEAK BARYOGENESIS IN A SUPERSYMMETRIC MODEL To appear in HEP 95

It is well known that there is difficulty in sustaining the hypothesis 1,2 of baryogenesis at the electroweak phase transition in the minimal standard model. To overcome these difficulties attention has been given to extensions of the minimal standard model, involving the addition of extra scalars . Prominent among these is the minimal supersymmetric standard model, MSSM, where the Higgs sector is just two doublets . Here we shall discuss, with a perturbative treatment, the next-to-minimal model, NMSSM, which has additionally one singlet Higgs scalar .In the absence of hard information we have to adopt a hypothesis on the SUSY breaking scale and on the spectrum of the particles, and ours is the simplest possible. We follow a number of papers of recent years in taking the SUSY breaking scale,MS, to be of the order of 1 TeV; we take perfect supersymmetry above that scale. Then at MS the quartic scalar couplings are fixed by the gauge couplings and two more parameters. We then use the renormalization group equations to run down the quartic couplings to the electroweak scale, where we investigate the nature of the phase change. There are also cubic and quadratic supersymmetry breaking couplings,and there results a space of variable parameters in which we investigate what proportion leads to a first order electroweak phase change, and so is compatible with electroweak baryogenesis. There has been quite considerable previous work on the electroweak phase change in the MSSM. We are not aware of so much on the NMSSM . The work of Pietroni has pointed up that the NMSSM, in contrast to the MSSM, has cubic terms in the scalar field potential at tree level leading to the possibility of a potential barrier in radial directions even at tree level. That work uses a unitary gauge which we consider to be not so secure a basis for the consideration of phase changes as the Landau gauge which we use . We start with the tree level potential,which is 5