Abstract
The a-function is a proposed quantity defined for quantum field theories which has a monotonic behaviour along renormalisation group flows, being related to the β- functions via a gradient flow equation involving a positive definite metric. We demonstrate the existence of a candidate a-function for renormalisable Chern-Simons theories in three dimensions, involving scalar and fermion fields, in both non-supersymmetric and super-symmetric cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.L. Cardy, Is there a c-theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [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].
M.A. Luty, J. Polchinski and R. Rattazzi, The a-theorem and the asymptotics of 4D quantum field theory, JHEP 01 (2013) 152 [arXiv:1204.5221] [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].
H. Elvang and T.M. Olson, RG flows in d dimensions, the dilaton effective action and the a-theorem, JHEP 03 (2013) 034 [arXiv:1209.3424] [INSPIRE].
H. Osborn, Derivation of a four-dimensional c-theorem, Phys. Lett. B 222 (1989) 97 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c-theorem for four-dimensional renormalizable field theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
I. Jack and H. Osborn, Constraints on RG flow for four dimensional quantum field theories, Nucl. Phys. B 883 (2014) 425 [arXiv:1312.0428] [INSPIRE].
H. Osborn, Weyl consistency conditions and a local renormalization group equation for general renormalizable field theories, Nucl. Phys. B 363 (1991) 486 [INSPIRE].
B. Grinstein, A. Stergiou and D. Stone, Consequences of Weyl consistency conditions, JHEP 11 (2013) 195 [arXiv:1308.1096] [INSPIRE].
B. Grinstein, D. Stone, A. Stergiou and M. Zhong, Challenge to the a-theorem in six dimensions, Phys. Rev. Lett. 113 (2014) 231602 [arXiv:1406.3626] [INSPIRE].
B. Grinstein, A. Stergiou, D. Stone and M. Zhong, Two-loop renormalization of multiflavor ϕ 3 theory in six dimensions and the trace anomaly, Phys. Rev. D 92 (2015) 045013 [arXiv:1504.05959] [INSPIRE].
H. Osborn and A. Stergiou, Structures on the conformal manifold in six dimensional theories, JHEP 04 (2015) 157 [arXiv:1501.01308] [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].
Y. Nakayama, Consistency of local renormalization group in D = 3, Nucl. Phys. B 879 (2014) 37 [arXiv:1307.8048] [INSPIRE].
D.L. Jafferis, The exact superconformal R-symmetry extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [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].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
D.J. Wallace and R.K.P. Zia, Gradient properties of the renormalization group equations in multicomponent systems, Annals Phys. 92 (1975) 142 [INSPIRE].
L.V. Avdeev, G.V. Grigorev and D.I. Kazakov, Renormalizations in abelian Chern-Simons field theories with matter, Nucl. Phys. B 382 (1992) 561 [INSPIRE].
L.V. Avdeev, D.I. Kazakov and I.N. Kondrashuk, Renormalizations in supersymmetric and nonsupersymmetric non-Abelian Chern-Simons field theories with matter, Nucl. Phys. B 391 (1993) 333 [INSPIRE].
O. Antipin, M. Gillioz, E. Mølgaard and F. Sannino, The a theorem for gauge-Yukawa theories beyond Banks-Zaks fixed point, Phys. Rev. D 87 (2013) 125017 [arXiv:1303.1525] [INSPIRE].
O. Antipin, M. Gillioz, J. Krog, E. Mølgaard and F. Sannino, Standard model vacuum stability and Weyl consistency conditions, JHEP 08 (2013) 034 [arXiv:1306.3234] [INSPIRE].
W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193 [INSPIRE].
D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Regularization by dimensional reduction of supersymmetric and nonsupersymmetric gauge theories, Nucl. Phys. B 167 (1980) 479 [INSPIRE].
I. Jack, D.R.T. Jones, P. Kant and L. Mihaila, The four-loop DRED gauge β-function and fermion mass anomalous dimension for general gauge groups, JHEP 09 (2007) 058 [arXiv:0707.3055] [INSPIRE].
I. Jack and C. Poole, The a-function for gauge theories, JHEP 01 (2015) 138 [arXiv:1411.1301] [INSPIRE].
I. Jack, D.R.T. Jones and C. Poole, Three-dimensional gradient flows: beyond leading order, in preparation.
T. Morita and V. Niarchos, F-theorem, duality and SUSY breaking in one-adjoint Chern-Simons-Matter theories, Nucl. Phys. B 858 (2012) 84 [arXiv:1108.4963] [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: 1505.05400
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
Jack, I., Jones, D.R.T. & Poole, C. Gradient flows in three dimensions. J. High Energ. Phys. 2015, 61 (2015). https://doi.org/10.1007/JHEP09(2015)061
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2015)061