Gauge Boson Masses in the 3-d, SU(2) Gauge-Higgs Model

TL;DR

The paper investigates magnetic screening masses in the high-temperature phase of the 3-d SU(2) gauge-Higgs model by computing gauge-field propagators in Landau gauge on the lattice and comparing with gauge-invariant spectra. It uses a 3-d effective theory derived from finite-temperature electroweak physics and performs finite-size analyses to extract W-boson masses from gauge-fixed correlators and from gauge-invariant operators. The main findings show a nonzero magnetic mass in the symmetric phase, m_w ≈ 0.158 in lattice units (≈ 0.35 g^2 T), and a rise above kappa_c consistent with a continuous transition; the gauge-invariant masses are larger in the symmetric phase, indicating decoupling between the Landau-gauge mass and the gauge-invariant vector mass. The results support a predominantly thermal origin for the high-temperature magnetic mass, with Higgs-induced contributions appearing in the broken phase and aligning with gap-equation analyses, while highlighting subtleties in comparing gauge-fixed and gauge-invariant spectra.

Abstract

We study gauge boson propagators in the symmetric and symmetry broken phases of the 3-d, $SU(2)$ gauge-Higgs model. Correlation functions for the gauge fields are calculated in Landau gauge. They are found to decay exponentially at large distances leading to a non-vanishing mass for the gauge bosons. We find that the W-boson screening mass drops in the symmetry broken phase when approaching the critical temperature. In the symmetric phase the screening mass stays small and is independent of the scalar--gauge coupling (the hopping parameter). Numerical results coincide with corresponding calculations performed for the pure gauge theory. We find $m_w = 0.35(1)g^2T $ in this phase which is consistent with analytic calculations based on gap equations. This is, however, significantly smaller than masses extracted from gauge invariant vector boson correlation functions. As internal consistency check we also have calculated correlation functions for gauge invariant operators leading to scalar and vector boson masses. Finite lattice size effects have been systematically analyzed on lattices of size $L^2\times L_z$ with $L=4-24$ and $L_z = 16 - 128$.

Figures (5)

• Figure 1: Gauge field correlation functions on $16^2 \times L_z$ lattices with $L_z = 32$ (squares), 40 (circles), 48 (upper triangles), 64 (lower triangles) and 128 (diamonds). Shown are correlation functions in the symmetric phase at $\kappa=0.1745$ (a) and the symmetry broken phase at $\kappa=0.17484$ (b). The curves give fits for $z\ge 8$. The fitting parameters are listed in Table 1.
• Figure 2: Local masses calculated at $\kappa = 0.1745$ from the correlation functions shown in Fig. 1a. In (a) we show local masses extracted according to Eq. (3.5) while (b) gives the result according to Eq.(3.6). The latter does assume $B=0$. The horizontal lines give the error band resulting from the fit on a $16^2\times 128$ lattice.
• Figure 3: Propagator masses obtained from fits to correlation functions calculated on lattices of size $16^2\times 32$. The curves show fits as explained in the text. The filled circle at $\kappa=0$ gives the result for the pure $SU(2)$ gauge theory.
• Figure 4: Vector ($a$) and scalar ($b$) masses at $\kappa = 0.1745$ (squares) and 0.17484 (circles) obtained from fits to gauge invariant correlation functions calculated on lattices of size $L^2\times 32$ with $L$ ranging from 4 to 24. Curves show exponential fits for the volume dependence of the masses.
• Figure 5: Scalar masses obtained from fits to the gauge invariant correlation function $G_s^a$ calculated on lattices of size $16^2\times 32$.