TY - JOUR

T1 - Generalized F-theorem and the ϵ expansion

AU - Fei, Lin

AU - Giombi, Simone

AU - Klebanov, Igor R.

AU - Tarnopolsky, Grigory

N1 - Publisher Copyright:
© 2015, The Author(s).
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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 S4−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 ϵ5. 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).

AB - 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 S4−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 ϵ5. 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).

KW - Field Theories in Lower Dimensions

KW - Renormalization Group

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U2 - 10.1007/JHEP12(2015)155

DO - 10.1007/JHEP12(2015)155

M3 - Article

AN - SCOPUS:84952888667

VL - 2015

SP - 1

EP - 37

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 12

M1 - 155

ER -