

 19 Apr 1994

 NEXT-TO-LEADING ORDER DEBYE-SCREENING IN SPONTANEOUSLY BROKEN GAUGE THEORIES*

Anton K. Rebhan Theory Group, DESY Notkestr. 85 D-22603 Hamburg, Germany

1. SUMMARY

The main r^ole of the Debye screening mass in the perturbative treatment of the electro-weak phase transition is the reduction of the cubic term that determines the strength of a first-order transition. In this note I point out that the standard definition of the Debye mass is unphysical. Its next-to-leading order corrections in resummed perturbation theory are gauge dependent generally in nonabelian gauge theories, and even in Abelian theories when in the Higgs phase. A gauge independent definition can be obtained from a gap equation for the propagator rather than the self-energy, which turns out to be perturbatively under control in the Higgs phase, but sensitive to the nonperturbative magnetic mass scale in the symmetric phase of nonabelian theories.

2. ABELIAN HIGGS MODEL

In the Abelian Higgs model with Lagrangian

L = \Gamma 14 F 2 + jD_\Phi j2 + *v2j\Phi j2 \Gamma *j\Phi j4; p2\Phi = ' + iO/ (1)

and R,-gauge fixing term Lg:f: = \Gamma 12, (@A \Gamma ,2e'O/)2, the leading-order results for the various masses at high temperature read

m2L = 13 e2T 2 + m2; m2T = m2 j e2'2; (2)

m2' = 3*'2 \Gamma *v2 + ( e

2

4 + *3 )T 2; (3)

m2O/ = *'2 \Gamma *v2 + ( e

2

4 + *3 )T 2 + ,m2; (4)

where mL and mT are the longitudinal (Debye) and the transverse (magnetic) mass of the photon propagator.1

* Contributed talk at the NATO Advanced Research Workshop "Electroweak Physics

and the Early Universe", 23 - 25 March 1994, Sintra, Portugal

+ 1/2 + 1/2 Figure 1. Dressed one-loop diagrams for \Pi 00 in the Abelian Higgs model.

1/2 - +1/2 +1/2 Figure 2. Additional dressed one-loop corrections to the longitudinal gauge boson propagator. Wavy, dotted, full, and dashed lines correspond to gauge bosons, Faddeev-Popov ghosts, Higgs and Goldstone particles, resp.; a blob on these lines marks one-loop dressed propagators.

The next-to-leading order result for \Pi 00(0), which is usually taken as the definition of the Debye mass squared, is given by the dressed one-loop diagrams of Fig. 1,1

\Pi 00(k0 = 0; k ! 0) = 13 e2T 2 + m2 \Gamma e

2T

4ss i 4m

2 mL+m' + m' + mO/j : (5)

Because all higher-order calculation to date have been performed in the Landau gauge, it seems to have gone unnoticed that this definition of the Debye mass is gauge dependent through its dependence on the Goldstone boson mass (4), so that it cannot be the correct one.

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