Gauge fields in the presence of the electroweak bubble wall

Abstract

The gauge field theory of the standard electroweak model in the presence of the electroweak bubble wall is investigated with a view to its applications to microscopic phenomena, which are believed to have occurred during the phase transition in the early universe. The asymptotic fields are defined anew so that the effects of the position-dependent Higgs condensate are taken into account through the position-dependent $W$ and $Z$ boson masses. A novel method of massive gauge field quantization in the $R_ξ$-gauge with $ξ=1$ is proposed for the case of the position-dependent masses. Our procedure is based on the eigenfunction expansion method associated with second-order differential operators, i.e., a sort of generalized Fourier expansion. The commutation relations of creation and annihilation operators of various wave propagation modes are given in terms of what is known as the spectral function. The decoupling of unphysical states from the physical S-matrix is also investigated along the line of Kugo-Ojima's quartet mechanism on the basis of the BRST symmetry. It is pointed out that one of the quartet fields is not merely the unphysical scalar field but should be a linear combination of the unphysical scalar and the gauge fields. The physical and unphysical polarizations of the gauge field waves are unambiguously distinguished and this will help us evaluate the friction caused by the physical polarization states of $W$ and $Z$ boson waves on the bubble wall during the phase transition in the early universe.