Generalized F-theorem and the ϵ 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 a- 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, (Formula Presented) $$, where (Formula Presented). 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 S^4−ϵ, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the ϵ expansion of F˜ up to order ϵ⁵. Padé 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 (Formula Presented).