Generalized F-theorem and the epsilon expansion

Abstract

Some known constraints on Renormalization Group flow take the form of inequalities: in even dimensions they refer to the coefficient a of the Weyl anomaly, while in odd dimensions to the sphere free energy F. In recent work [ 1] it was suggested that the alpha-and F-theorems may be viewed as special cases of a Generalized F-Theorem valid in continuous dimension. This conjecture states that, for any RG flow from one conformal fixed point to another, (F) over tilde (UV)> (F) over tilde (IR),where (F) over tilde = sin(pi d/2) log Z(Sd). Here we provide additional evidence in favor of the Generalized F-Theorem. We show that it holds in conformal perturbation theory, i.e. for RG flows produced by weakly relevant operators. We also study a specific example of the Wilson-Fisher O(N) model and define this CFT on the sphere S4-epsilon, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the epsilon expansion of (F) over tilde up to order epsilon(5). Pade extrapolation of this series to d = 3 gives results that are around 2-3% below the free field values for small N. We also study RG flows which include an anisotropic perturbation breaking the O( N) symmetry; we again find that the results are consistent with (F) over tilde (UV)> (F) over tilde (IR).