Universal effective couplings of the three-dimensional $n$-vector model and field theory

TL;DR

This paper computes universal ratios of coupling constants in the three-dimensional n-vector model using renormalization group methods, providing estimates for physical cases and analyzing asymptotic behaviors.

Contribution

It introduces four-loop and three-loop RG expansions for universal ratios and estimates their values for various n, advancing understanding of critical couplings in the n-vector model.

Findings

Universal ratio R_8* estimated for various n

Three-loop RG series for R_10 show large, rapidly growing coefficients

Numerical estimates for R_10 are hindered by series divergence

Abstract

We calculate the universal ratios $R_{2k}$ of renormalized coupling constants $g_{2k}$ entering the critical equation of state for the generalized Heisenberg (three-dimensional $n$-vector) model. Renormalization group (RG) expansions of $R_8$ and $R_{10}$ for arbitrary $n$ are found in the four-loop and three-loop approximations respectively. Universal octic coupling $R_8^*$ is estimated for physical values of spin dimensionality $n = 0, 1, 2, 3$ and for $n = 4,...64$ to get an idea about asymptotic behavior of $R_8^*$. Its numerical values are obtained by means of the resummation of the RG series and within the pseudo-$\varepsilon$ expansion approach. Regarding $R_{10}$ our calculations show that three-loop RG and pseudo-$\varepsilon$ expansions possess big and rapidly growing coefficients for physical values of $n$ what prevents getting fair numerical estimates.