Exact relation of lattice and continuum parameters in three-dimensional SU(2)+Higgs theories

TL;DR

This work derives exact two-loop relations between lattice and continuum parameters for three-dimensional SU(2)+Higgs theories arising from dimensional reduction of high-temperature electroweak physics. By matching a gauge-invariant observable, the minimum of the effective potential $V(\min)$, the authors express the lattice mass parameter $m_L^2$ in terms of continuum quantities $m_3^2(\mu)$, $g_3$, and $\lambda_3$, including a precise analytic form for the two-loop counterterm $\delta m_L^2(\hbar^2)$ and the vacuum counterterm $\delta V_L(\hbar^2)$; logarithmic, linear, and constant $g_3^4$ and $\lambda_3 g_3^2$ contributions are all accounted for. The calculation confirms cancellations of ultraviolet divergences and matches well with previous lattice Monte Carlo results, providing a robust tool for translating lattice simulations of high-temperature gauge theories into continuum physical observables relevant to the electroweak phase transition. The results enhance the accuracy of nonperturbative studies of dimensional-reduced effective theories and their implications for cosmology.

Abstract

The essential features of the high-temperature electroweak phase transition are contained in a three-dimensional super-renormalizable effective field theory. We calculate the exact counterterms needed for lattice simulations of the SU(2)-part of this theory. Scalar fields in both fundamental and adjoint representations are included. The three-dimensional U(1)+Higgs theory is also discussed.

Figures (1)

• Figure 1: The irreducible two-loop graphs contributing to the two-loop effective potential in the L-scheme. Wiggly line is the vector propagator, dashed line is the scalar propagator, and double line is the ghost propagator.