Exact Five-Loop Renormalization Group Functions of $\phi^4$-Theory with O(N)-Symmetric and Cubic Interactions. Critical Exponents up to $\ep^5$

TL;DR

This paper computes five-loop renormalization group functions for a $$-theory with $O(N)$ and cubic interactions, providing high-order critical exponents that clarify the stability of fixed points and universality classes.

Contribution

It presents the first five-loop calculations of RG functions and critical exponents for $$-theory with $O(N)$ and cubic interactions, extending previous three-loop results.

Findings

Cubic fixed point is stable for N

Critical exponents are very close across universality classes

Numerical differences are less than 1%, challenging experimental distinction

Abstract

The renormalization group functions are calculated in $D=4-\epsilon$ dimensions for the $\phi^4$-theory with two coupling constants associated with an ${O}(N)$-symmetric and a cubic interaction. Divergences are removed by minimal subtraction. The critical exponents $\eta$, $\nu$, and $\omega$ are expanded up to order $\epsilon^5$ for the three nontrivial fixed points O(N)-symmetric, Ising, and cubic. The results suggest the stability of the cubic fixed point for $N\geq3$, implying that the critical exponents seen in the magnetic transition of three-dimensional cubic crystals are of the cubic universality class. This is in contrast to earlier three-loop results which gave $N > 3$, and thus Heisenberg exponents. The numerical differences, however, are less than a percent making an experimental distinction of the universality classes very difficult.