X iv : h ep - p h / 99 04 33 3 v 1 1 4 A pr 1 99 9 Ring diagrams and electroweak phase transition in a magnetic field

Electroweak phase transition in a magnetic field is investigated within the one-loop and ring diagram contributions to an effective potential. All fundamental fermions and bosons are included with their actual values of masses . The only free parameter remains the Higgs boson mass mH . It is shown that the ring diagrams affect the phase transition drastically . The effective potential is real at sufficiently high temperatures. For mH ≥ 75 Gev a metastable vacuum is not produced at all. It is also found that the magnetic field essentially influences the phase transition at high temperature. Upper limit on the possible field strengths in the phase transition epoch is derived: H < 0.1 · 1024G. 1. The concept of symmetry restoration at high temperature has been intensively used in studying the evolution of the Universe at its early stages. Nowadays it gives a possibility to investigate various problems of cosmology and particle physics [1],[2]. In particular, the type of the electroweak phase transition and hence the further evolution of the universe depends on the Higgs boson mass mH . Most investigations of the electroweak (EW) phase transition have included into consideration a high temperature as the main environment [2], [3]. But in recent years cogent arguments followed from different approaches in favour of the presence of strong magnetic fields at that stage have appeared [5], [6] (for recent review see [7]). So, the phase transition at high temperature and strong fields has to be of interest. Moreover, at present time when masses of all fundamental particles, except mH , are known there is a possibility to investigate in details the phase transition as a function of this parameter. One of the possibilities to have strong magnetic fields in the EW phase transition epoch was discussed by Vachaspati [5]. From his analysis it follows that under very general conditions the fields H ∼ T 2 i in the patches of sub-horizon scales can be generated during a large class of grand unified transitions [6],[7], where Ti is transition temperature. The second one is formation of the Savvidy vacuum magnetic state at high temperature (H ∼ gT , g is the gauge coupling constant) [10],[17],[18],[19]. In latter case only the abelian field configurations