Abstract
We present an example of a limit cycle, i.e., a recurrent flow-line of the beta-function vector field, in a unitary four-dimensional gauge theory. We thus prove that beta functions of four-dimensional gauge theories do not produce gradient flows. The limit cycle is established in perturbation theory with a three-loop calculation which we describe in detail.
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J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance: an example, Phys. Lett. B 704 (2011) 74 [arXiv:1106.2540] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance: theoretical foundations, JHEP 07 (2012) 025 [arXiv:1107.3840] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Scale without conformal invariance at three loops, JHEP 08 (2012) 085 [arXiv:1202.4757] [INSPIRE].
K. Wilson and J.B. Kogut, The renormalization group and the ϵ-expansion, Phys. Rept. 12 (1974) 75 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c-theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Limit cycles and conformal invariance, arXiv:1208.3674 [INSPIRE].
E. Barnes, K.A. Intriligator, B. Wecht and J. Wright, Evidence for the strongest version of the 4D a-theorem, via a-maximization along RG flows, Nucl. Phys. B 702 (2004) 131 [hep-th/0408156] [INSPIRE].
A. 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].
H. Osborn, Derivation of a four-dimensional c-theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [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].
H. Elvang et al., On renormalization group flows and the a-theorem in 6D, JHEP 10 (2012) 011 [arXiv:1205.3994] [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].
S.L. Adler, Axial vector vertex in spinor electrodynamics, Phys. Rev. 177 (1969) 2426 [INSPIRE].
J. Bell and R. Jackiw, A PCAC puzzle: π0 → γγ in the σ-model, Nuovo Cim. A 60 (1969) 47 [INSPIRE].
D.B. Kaplan and A. Manohar, Strange matrix elements in the proton from neutral current experiments, Nucl. Phys. B 310 (1988) 527 [INSPIRE].
I. Jack and H. Osborn, General background field calculations with fermion fields, Nucl. Phys. B 249 (1985) 472 [INSPIRE].
A. Pickering, J. Gracey and D. Jones, Three loop gauge β-function for the most general single gauge coupling theory, Phys. Lett. B 510 (2001) 347 [Erratum ibid. B 535 (2002) 377] [hep-ph/0104247] [INSPIRE].
D. Wallace and R. Zia, Gradient properties of the renormalization group equations in multicomponent systems, Annals Phys. 92 (1975) 142 [INSPIRE].
M.E. Machacek and M.T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization, Nucl. Phys. B 222 (1983) 83 [INSPIRE].
K.G. Chetyrkin, M. Misiak and M. Münz, β-functions and anomalous dimensions up to three loops, Nucl. Phys. B 518 (1998) 473 [hep-ph/9711266] [INSPIRE].
H.K. Dreiner, H.E. Haber and S.P. Martin, Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry, Phys. Rept. 494 (2010) 1 [arXiv:0812.1594] [INSPIRE].
J.-F. Fortin, B. Grinstein and A. Stergiou, Cyclic unparticle physics, Phys. Lett. B 709 (2012) 408 [arXiv:1110.1634] [INSPIRE].
Y. Nakayama, Gravity dual for cyclic renormalization group flow without conformal invariance, Mod. Phys. Lett. A 26 (2011) 2469 [arXiv:1107.2928] [INSPIRE].
Y. Nakayama, Holographic renormalization of foliation preserving gravity and trace anomaly, Gen. Rel. Grav. 44 (2012) 2873 [arXiv:1203.1068] [INSPIRE].
J. Kuipers, T. Ueda, J. Vermaseren and J. Vollinga, FORM version 4.0, arXiv:1203.6543 [INSPIRE].
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ArXiv ePrint: 1206.2921
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Fortin, JF., Grinstein, B. & Stergiou, A. Limit cycles in four dimensions. J. High Energ. Phys. 2012, 112 (2012). https://doi.org/10.1007/JHEP12(2012)112
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DOI: https://doi.org/10.1007/JHEP12(2012)112