We study the Blume-Capel universality class in $d=\frac{10}{3}-\epsilon$ dimensions. The renormalization group flow is extracted by looking at poles in fractional dimension of three loop diagrams using $\overline{\mathrm{MS}}$. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $\epsilon =\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the operator product expansion coefficients to the leading order. Our findings agree with and complement conformal field theory results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as special cases.

• Received 11 July 2017

DOI:https://doi.org/10.1103/PhysRevD.96.081701