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 arXiv:1409.1937 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, ${\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}}$, where $\tilde{F}=\sin (\pi d/2)\log Z_{S^d}$. 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 $\tilde{F}$ up to order ${ϵ}^5$. 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 ${\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}}$.

Comments: 41 pages, 7 figures. v3: minor improvements