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Gauge fields in the presence of the electroweak bubble wall

Takahiro Kubota

Abstract

The gauge field theory of the standard electroweak model in the presence of the electroweak bubble wall is investigated with an eye toward its applications to microscopic phenomena, which are supposed 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 the so-called 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.

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 an eye toward its applications to microscopic phenomena, which are supposed 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 and boson masses. A novel method of massive gauge field quantization in the -gauge with 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 the so-called 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 and boson waves on the bubble wall during the phase transition in the early universe.

Paper Structure

This paper contains 33 sections, 166 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The electroweak bubble wall
  3. Wave propagation modes
  4. Asymptotic fields
  5. The Klein-Gordon type real scalar field
  6. Scalar field quantization in the presence of the electroweak bubble wall
  7. The retarded and advanced Green's functions
  8. The special case of $z$-independent mass (no bubble wall)
  9. The BRST analyses in the electroweak theory
  10. The gauge fixing and auxiliary fields
  11. The Faddeev-Popov ghost Lagrangian
  12. The BRST transformation
  13. The conserved current and the conserved charge
  14. Field quantization in the presence of the electroweak bubble wall
  15. Asymptotic gauge fields
  16. ...and 18 more sections

Figures (2)

  • Figure 1: A typical profile of the Higgs condensate $v(z)$, that connects the symmetry-restored region ($z \to -\infty$) and the symmetry-broken region ($z \to + \infty$).
  • Figure 2: The relations among the four members of the BRST quartet. The arrows with $Q_{B}$ means that the fields and operators are transformed by the BRST transformation along the arrows. The fields and operators connected by left-right arrows have non-vanishing commutation or anti-commutaion relations.