Abstract
If the coupling constants in QFT are promoted to functions of space-time, the dependence of the path integral on these couplings is highly constrained by conformal symmetry. We begin the present note by showing that this idea leads to a new proof of Zamolodchikov’s theorem. We then review how this simple observation also leads to a derivation of the a-theorem. We exemplify the general procedure in some interacting theories in four space-time dimensions. We concentrate on Banks-Zaks and weakly relevant flows, which can be controlled by ordinary and conformal perturbation theories, respectively. We compute explicitly the dependence of the path integral on the coupling constants and extract the change in the a-anomaly (this agrees with more conventional computations of the same quantity). We also discuss some general properties of the sum rule found in [1] and study it in several examples.
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References
Z. Komargodski and A. Schwimmer, On Renormalization Group Flows in Four Dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory to One Loop, Annals Phys. 158 (1984) 142 [INSPIRE].
J. Gasser and H. Leutwyler, Chiral Perturbation Theory: Expansions in the Mass of the Strange Quark, Nucl. Phys. B 250 (1985) 465 [INSPIRE].
V. Novikov, M.A. Shifman, A. Vainshtein and V.I. Zakharov, Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus, Nucl. Phys. B 229 (1983) 381 [INSPIRE].
N. Seiberg, Naturalness versus supersymmetric nonrenormalization theorems, Phys. Lett. B 318 (1993) 469 [hep-ph/9309335] [INSPIRE].
J. Polchinski, Scale and conformal invariance in quantum field theory, Nucl. Phys. B 303 (1988) 226 [INSPIRE].
Y. Nakayama, No Forbidden Landscape in String/M-theory, JHEP 01 (2010) 030 [arXiv:0909.4297] [INSPIRE].
D. Dorigoni and V.S. Rychkov, Scale Invariance + Unitarity → Conformal Invariance?, arXiv:0910.1087 [INSPIRE].
I. Antoniadis and M. Buican, On R-symmetric Fixed Points and Superconformality, Phys. Rev. D 83 (2011) 105011 [arXiv:1102.2294] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without Conformal Invariance: Theoretical Foundations, arXiv:1107.3840 [INSPIRE].
T.L. Curtright, X. Jin and C.K. Zachos, RG flows, cycles and c-theorem folklore, Phys. Rev. Lett. 108 (2012) 131601 [arXiv:1111.2649] [INSPIRE].
M. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].
S. Deser and A. Schwimmer, Geometric classification of conformal anomalies in arbitrary dimensions, Phys. Lett. B 309 (1993) 279 [hep-th/9302047] [INSPIRE].
L. Álvarez-Gaumé and E. Witten, Gravitational Anomalies, Nucl. Phys. B 234 (1984) 269 [INSPIRE].
A. Zamolodchikov, Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory, JETP Lett. 43 (1986) 730 [INSPIRE].
A. Cappelli, D. Friedan and J.I. Latorre, C theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].
A. Schwimmer and S. Theisen, Spontaneous Breaking of Conformal Invariance and Trace Anomaly Matching, Nucl. Phys. B 847 (2011) 590 [arXiv:1011.0696] [INSPIRE].
T. Pham and T.N. Truong, Evaluation of the derivative quartic terms of the meson chiral lagrangian from forward dispersion relation, Phys. Rev. D 31 (1985) 3027 [INSPIRE].
J. Distler, B. Grinstein, R.A. Porto and I.Z. Rothstein, Falsifying Models of New Physics via WW Scattering, Phys. Rev. Lett. 98 (2007) 041601 [hep-ph/0604255] [INSPIRE].
M. Dine, G. Festuccia and Z. Komargodski, A Bound on the Superpotential, JHEP 03 (2010) 011 [arXiv:0910.2527] [INSPIRE].
A. Adams, A. Jenkins and D. O’Connell, Signs of analyticity in fermion scattering, arXiv:0802.4081 [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
K.A. Intriligator and B. Wecht, The Exact superconformal R symmetry maximizes a, Nucl. Phys. B 667 (2003) 183 [hep-th/0304128] [INSPIRE].
I. Bah and B. Wecht, New N = 1 Superconformal Field Theories In Four Dimensions, arXiv:1111.3402 [INSPIRE].
M. Buican, A Conjectured Bound on Accidental Symmetries, Phys. Rev. D 85 (2012) 025020 [arXiv:1109.3279] [INSPIRE].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [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, A Finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [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].
T. Banks and A. Zaks, On the Phase Structure of Vector-Like Gauge Theories with Massless Fermions, Nucl. Phys. B 196 (1982) 189 [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
J.L. Cardy, Is There a c Theorem in Four-Dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
D. Green, Z. Komargodski, N. Seiberg, Y. Tachikawa and B. Wecht, Exactly Marginal Deformations and Global Symmetries, JHEP 06 (2010) 106 [arXiv:1005.3546] [INSPIRE].
Y. Nakayama, On ∈-conjecture in a-theorem, arXiv:1110.2586 [INSPIRE].
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ArXiv ePrint: 1112.4538
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Komargodski, Z. The constraints of conformal symmetry on RG flows. J. High Energ. Phys. 2012, 69 (2012). https://doi.org/10.1007/JHEP07(2012)069
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DOI: https://doi.org/10.1007/JHEP07(2012)069