For an anisotropic Euclidean ${\phi }^4$ theory with two interactions $\left[u\right({\sum }_{i=1\mathrm{}}^M{\phi }_i^2{)}^2+v{\sum }_{i=1\mathrm{}}^M{\phi }_i^4]$ the $\beta$ functions are calculated from five-loop perturbation expansions in $d=4\mathrm{}-\varepsilon$ dimensions, using the knowledge of the large-order behavior and Borel transformations. For $\varepsilon =1\mathrm{}$, an infrared-stable cubic fixed point for $M>~3$ 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 Padé approximations, but only the evidence presented in this work seems to be sufficiently convincing to draw this conclusion.

• Received 21 February 1997

DOI:https://doi.org/10.1103/PhysRevB.56.14428