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

Abstract

The renormalization group functions are calculated in $D=4-ε$ dimensions for the $ϕ^4$-theory with two coupling constants associated with an ${O}(N)$-symmetric and a cubic interaction. Divergences are removed by minimal subtraction. The critical exponents $η$, $ν$, and $ω$ are expanded up to order $ε^5$ 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.