Phase Transition Couplings in U(1) and SU(N) Regularized Gauge Theories

TL;DR

This paper calculates phase transition couplings in U(1) and SU(N) gauge theories using a 2-loop approximation, aligning theoretical predictions with lattice results and estimating SU(N) couplings relevant for the Planck scale.

Contribution

It provides a 2-loop approximation for critical couplings in gauge theories and estimates SU(N) couplings based on abelianization ideas, improving upon previous one-loop results.

Findings

Critical electric coupling in U(1) gauge theory is approximately 0.208.

The SU(N) critical coupling relates to U(1) coupling via a specific N-dependent formula.

The estimated critical couplings agree with lattice results and inform Planck scale physics.

Abstract

Using a 2-loop approximation for $\beta$-functions, we have considered the corresponding renormalization group improved effective potential in the Dual Abelian Higgs Model (DAHM) of scalar monopoles and calculated the phase transition (critical) couplings in U(1) and SU(N) regularized gauge theories. In contrast to our previous result $\alpha_{crit} \approx 0.17$, obtained in the one-loop approximation with the DAHM effective potential (see Ref.[20]), the critical value of the electric fine structure constant in the 2-loop approximation, calculated in the present paper, is equal to $\alpha_{crit}\approx 0.208$ and coincides with the lattice result for compact QED [10]: $\alpha_{crit}^{lat} \approx 0.20\pm 0.015$. Following the 't Hooft's idea of the "abelization" of monopole vacuum in the Yang--Mills theories, we have obtained an estimation of the SU(N) triple point coupling constants, which is $\alpha_{N,crit}^{-1}=\frac{N}{2}\sqrt{\frac{N+1}{N-1}} \alpha_{U(1),crit}^{-1}$. This relation was used for the description of the Planck scale values of the inverse running constants $\alpha_i^{-1}(\mu)$ (i=1,2,3 correspond to U(1), SU(2) and SU(3) groups), according to the ideas of the Multiple Point Model [16].