Generalized \(F\)-theorem and the \(\varepsilon\) expansion

DOI10.1007/JHEP12(2015)155zbMATH Open1387.81274arXiv1507.01960MaRDI QIDQ2636339

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Publication date: 31 May 2018

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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, $ilde{F}_{mUV}>ilde{F}_{mIR}$, where $ildeF=sin(pid/2)log{Z}_{S^d}$. 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 $ildeF$ up to order $epsilo{n}^5$. Pade extrapolation of this series to $d=3$ gives results that are around $2-3\mathrm{\%}$ 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 $ilde{F}_{mUV}>ilde{F}_{mIR}$.

Full work available at URL: https://arxiv.org/abs/1507.01960

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