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, \( {\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}} \), where \( \tilde{F}= \sin \left(\pi d/2\right) \log {Z}_{S^d} \). 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−ϵ, paying careful attention to the beta functions for the coefficients of curvature terms. This allows us to develop the ϵ expansion of \( \tilde{F} \) 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 \( {\tilde{F}}_{\mathrm{UV}}>{\tilde{F}}_{\mathrm{IR}} \).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S. Giombi and I.R. Klebanov, Interpolating between a and F, JHEP 03 (2015) 117 [arXiv:1409.1937] [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is There a c-Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
Z. Komargodski, The Constraints of Conformal Symmetry on RG Flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
C. Cordova, T.T. Dumitrescu and X. Yin, Higher Derivative Terms, Toroidal Compactification and Weyl Anomalies in Six-Dimensional (2,0) Theories, arXiv:1505.03850 [INSPIRE].
C. Cordova, T.T. Dumitrescu and K. Intriligator, Anomalies, Renormalization Group Flows and the a-Theorem in Six-Dimensional (1,0) Theories, arXiv:1506.03807 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c-Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
T. Maxfield and S. Sethi, The Conformal Anomaly of M5-Branes, JHEP 06 (2012) 075 [arXiv:1204.2002] [INSPIRE].
H. Elvang, D.Z. Freedman, L.-Y. Hung, M. Kiermaier, R.C. Myers and S. Theisen, On renormalization group flows and the a-theorem in 6d, JHEP 10 (2012) 011 [arXiv:1205.3994] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
T. Kawano, Y. Nakaguchi and T. Nishioka, Holographic Interpolation between a and F, JHEP 12 (2014) 161 [arXiv:1410.5973] [INSPIRE].
E. Witten, Multitrace operators, boundary conditions and AdS/CFT correspondence, hep-th/0112258 [INSPIRE].
S.S. Gubser and I.R. Klebanov, A universal result on central charges in the presence of double trace deformations, Nucl. Phys. B 656 (2003) 23 [hep-th/0212138] [INSPIRE].
D.E. Diaz and H. Dorn, Partition functions and double-trace deformations in AdS/CFT, JHEP 05 (2007) 046 [hep-th/0702163] [INSPIRE].
S. Giombi, I.R. Klebanov, S.S. Pufu, B.R. Safdi and G. Tarnopolsky, AdS Description of Induced Higher-Spin Gauge Theory, JHEP 10 (2013) 016 [arXiv:1306.5242] [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen, Nonperturbative formulas for central functions of supersymmetric gauge theories, Nucl. Phys. B 526 (1998) 543 [hep-th/9708042] [INSPIRE].
K.A. Intriligator and B. Wecht, The exact superconformal R-symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
D. Kutasov, A. Parnachev and D.A. Sahakyan, Central charges and U(1)(R) symmetries in N = 1 super Yang-Mills, JHEP 11 (2003) 013 [hep-th/0308071] [INSPIRE].
J.J. Heckman, D.R. Morrison, T. Rudelius and C. Vafa, Geometry of 6D RG Flows, JHEP 09 (2015) 052 [arXiv:1505.00009] [INSPIRE].
J.J. Heckman and T. Rudelius, Evidence for C-theorems in 6D SCFTs, JHEP 09 (2015) 218 [arXiv:1506.06753] [INSPIRE].
K.G. Wilson and M.E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28 (1972) 240 [INSPIRE].
A. Hasenfratz, P. Hasenfratz, K. Jansen, J. Kuti and Y. Shen, The equivalence of the top quark condensate and the elementary Higgs field, Nucl. Phys. B 365 (1991) 79 [INSPIRE].
J. Zinn-Justin, Four fermion interaction near four-dimensions, Nucl. Phys. B 367 (1991) 105 [INSPIRE].
B.i. Halperin, T.C. Lubensky and S.-k. Ma, First order phase transitions in superconductors and smectic - A liquid crystals, Phys. Rev. Lett. 32 (1974) 292 [INSPIRE].
L. Fei, S. Giombi and I.R. Klebanov, Critical O(N) models in 6 − ϵ dimensions, Phys. Rev. D 90 (2014) 025018 [arXiv:1404.1094] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Three loop analysis of the critical O(N) models in 6 − ϵ dimensions, Phys. Rev. D 91 (2015) 045011 [arXiv:1411.1099] [INSPIRE].
J.A. Gracey, Four loop renormalization of ϕ 3 theory in six dimensions, Phys. Rev. D 92 (2015) 025012 [arXiv:1506.03357] [INSPIRE].
K.G. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
I.T. Drummond and G.M. Shore, Conformal Anomalies for Interacting Scalar Fields in Curved Space-Time, Phys. Rev. D 19 (1979) 1134 [INSPIRE].
L.S. Brown and J.C. Collins, Dimensional Renormalization of Scalar Field Theory in Curved Space-time, Annals Phys. 130 (1980) 215 [INSPIRE].
S.J. Hathrell, Trace Anomalies and λϕ 4 Theory in Curved Space, Annals Phys. 139 (1982) 136 [INSPIRE].
I. Jack and H. Osborn, Background Field Calculations in Curved Space-time. 1. General Formalism and Application to Scalar Fields, Nucl. Phys. B 234 (1984) 331 [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
H. Kleinert and V. Schulte-Frohlinde, Critical properties of ϕ 4 -theories, World Scientific (2001).
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
M. Moshe and J. Zinn-Justin, Quantum field theory in the large-N limit: A review, Phys. Rept. 385 (2003) 69 [hep-th/0306133] [INSPIRE].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, to appear.
I.T. Drummond, Conformally Invariant Amplitudes and Field Theory in a Space-Time of Constant Curvature, Phys. Rev. D 19 (1979) 1123 [INSPIRE].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, On the Resolution of Singularities of Multiple Mellin-Barnes Integrals, Eur. Phys. J. C 62 (2009) 445 [arXiv:0901.0386] [INSPIRE].
V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1507.01960
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Fei, L., Giombi, S., Klebanov, I.R. et al. Generalized F-theorem and the ϵ expansion. J. High Energ. Phys. 2015, 1–37 (2015). https://doi.org/10.1007/JHEP12(2015)155
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/JHEP12(2015)155