Generalized $F$-Theorem and the $ε$ Expansion

TL;DR

The paper provides a perturbative validation of a Generalized $F$-Theorem in continuous dimension by studying RG flows between conformal fixed points. It develops the sphere-based formalism for renormalization in curved space, computes the $O(N)$ Wilson-Fisher fixed point on $S^{4-\epsilon}$ up to $O(\epsilon^5)$, and analyzes conformal perturbations with one or more weakly relevant operators across dimensions, including the Gross-Neveu model, using Padé resummation to extract $d=3$ behavior. The results show monotonic decreases of $\tilde F = -\sin(\frac{\pi d}{2})F$ along RG flows and provide explicit coefficients for several models, reinforcing the universality and applicability of the generalized theorem. The work also clarifies the role of curvature counterterms in preserving conformal invariance on curved backgrounds and offers quantitative predictions for $F$-theorem constraints in non-integer dimensions with practical extrapolations to three dimensions. Overall, the paper extends the reach of $F$-theorem-type constraints to a continuum of dimensions and a broad class of interacting CFTs.

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, $\tilde F_{\rm UV} > \tilde F_{\rm IR}$, where $\tilde F=\sin (πd/2)\log Z_{S^d}$. 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 $\tilde F_{\rm UV} > \tilde F_{\rm IR}$.

Figures (7)

• Figure 1: Diagrams contributing to the free energy in $\phi^{4}$-theory up to fifth order. Each line represents the sphere propogator $\langle \phi(x)\phi(y)\rangle = C_{\phi}/s(x,y)^{d-2}$. Symmetry factors are not included in these graphs.
• Figure 2: $\tilde{F}/\tilde{F}_s$ for Ising model in $2\le d\le 4$. The solid line is the result obtained by averaging Padé approximants $\textrm{Pad\'e}_{[m,n]}$ with $m+n=6$ and fixed $d=2$ boundary condition. The dashed line corresponds to the $\epsilon$ expansion result (\ref{['F-Is-ep5']}) without resummation.
• Figure 3: Structure of the one-loop RG flow trajectories for $N=2,4,6,8$. The black dot indicates the free UV fixed point, the blue one the $O(N)$ fixed point, the green one the $N$ decoupled Ising models, and the red one the anisotropic fixed point. For $N<4$, the $O(N)$ fixed point is IR stable, while for $N>4$ the cubic point is. For $N=4$, the $O(N)$ and cubic fixed point coincide.
• Figure 4: External line and propogator for the integrals on a sphere.
• Figure 5: Basic relations for rewriting Feynman integrals on a sphere in the Mellin-Barnes form.