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 -