### Abstract

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 ϵ^{5}. 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).

Original language | English (US) |
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Article number | 155 |

Pages (from-to) | 1-37 |

Number of pages | 37 |

Journal | Journal of High Energy Physics |

Volume | 2015 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2015 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Keywords

- Field Theories in Lower Dimensions
- Renormalization Group

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## Cite this

*Journal of High Energy Physics*,

*2015*(12), 1-37. [155]. https://doi.org/10.1007/JHEP12(2015)155