Stability of 3D Cubic Fixed Point in Two-Coupling-Constant ϕ^4-Theory

Abstract

For an anisotropic euclidean $ϕ^4$-theory with two interactions $[u (\sum_{i=1^M ϕ_i^2)^2+v \sum_{i=1}^M ϕ_i^4]$ the $β$-functions are calculated from five-loop perturbation expansions in $d=4-\varepsilon$ dimensions, using the knowledge of the large-order behavior and Borel transformations. For $\varepsilon=1$, an infrared stable cubic fixed point for $M \geq 3$ is found, implying that the critical exponents in the magnetic phase transition of real crystals are of the cubic universality class. There were previous indications of the stability based either on lower-loop expansions or on less reliable Pad\'{e approximations, but only the evidence presented in this work seems to be sufficently convincing to draw this conclusion.