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 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 $S^{4-\epsilon }$, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the $\epsilon$ expansion of $\tilde{F}$ 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 ${\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}}$.

41 pages, 7 figures. v3: minor improvements

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• 3d Abelian Gauge Theories at the Boundary
• Accidental Symmetries and the Conformal Bootstrap
• Entanglement entropy of the large $N$ Wilson-Fisher conformal field theory
• Scalar CFTs and Their Large N Limits
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• Excitation spectra of quantum matter without quasiparticles I: Sachdev-Ye-Kitaev models
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• Scalar Multiplet Recombination at Large N and Holography
• Regularizing 3D conformal field theories via anyons on the fuzzy sphere
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• RG flows on $S^d$ and Hamiltonian truncation
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