Abstract
QED with a large number N of massless fermionic degrees of freedom has a conformal phase in a range of space-time dimensions. We use a large N diagrammatic approach to calculate the leading corrections to C T , the coefficient of the two-point function of the stress-energy tensor, and C J , the coefficient of the two-point function of the global symmetry current. We present explicit formulae as a function of d and check them versus the expectations in 2 and 4 − ϵ dimensions. Using our results in higher even dimensions we find a concise formula for C T of the conformal Maxwell theory with higher derivative action \( {F}_{\mu \nu }{\left(-{\nabla}^2\right)}^{\frac{d}{2}-2}{F}^{\mu \nu } \). In d = 3, QED has a topological symmetry current, and we calculate the correction to its two-point function coefficient, C top J . We also show that some RG flows involving QED in d = 3 obey C UV T > C IR T and discuss possible implications of this inequality for the symmetry breaking at small values of N .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
A.C. Petkou, C(T) and C(J) up to next-to-leading order in 1/N in the conformally invariant 0(N) vector model for 2 < d < 4, Phys. Lett. B 359 (1995) 101 [hep-th/9506116] [INSPIRE].
K. Diab, L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, On C J and C T in the Gross-Neveu and O(N ) Models, arXiv:1601.07198 [INSPIRE].
A.N. Vasiliev and M.Yu. Nalimov, Analog of Dimensional Regularization for Calculation of the Renormalization Group Functions in the 1/n Expansion for Arbitrary Dimension of Space, Theor. Math. Phys. 55 (1983) 423 [INSPIRE].
A.N. Vasiliev, Yu.M. Pismak and Yu.R. Khonkonen, Simple Method of Calculating the Critical Indices in the 1/N Expansion, Theor. Math. Phys. 46 (1981) 104 [INSPIRE].
A.N. Vasiliev, Yu.M. Pismak and Yu.R. Khonkonen, 1/N Expansion: Calculation of the Exponents η and Nu in the Order 1/N 2 for Arbitrary Number of Dimensions, Theor. Math. Phys. 47 (1981) 465 [INSPIRE].
S.E. Derkachov and A.N. Manashov, The simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion, Nucl. Phys. B 522 (1998) 301 [hep-th/9710015] [INSPIRE].
M. Ciuchini, S.E. Derkachov, J.A. Gracey and A.N. Manashov, Quark mass anomalous dimension at O(1/N 2 f ) in QCD, Phys. Lett. B 458 (1999) 117 [hep-ph/9903410] [INSPIRE].
J.A. Gracey, Electron mass anomalous dimension at O(1/N 2 f ) in quantum electrodynamics, Phys. Lett. B 317 (1993) 415 [hep-th/9309092] [INSPIRE].
Y. Huh, P. Strack and S. Sachdev, Conserved current correlators of conformal field theories in 2+1 dimensions, Phys. Rev. B 88 (2013) 155109 [Erratum ibid. B 90 (2014) 199902] [arXiv:1307.6863] [INSPIRE].
Y. Huh and P. Strack, Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: Fermionic Dirac matter coupled to U(1) gauge field, JHEP 01 (2015) 147 [Erratum ibid. 03 (2016) 054] [arXiv:1410.1902] [INSPIRE].
S. Giombi, I.R. Klebanov and G. Tarnopolsky, Conformal QED d , F -Theorem and the ϵ Expansion, J. Phys. A 49 (2016) 135403 [arXiv:1508.06354] [INSPIRE].
S.M. Chester and S.S. Pufu, Towards bootstrapping QED 3, JHEP 08 (2016) 019 [arXiv:1601.03476] [INSPIRE].
R.D. Pisarski, Chiral Symmetry Breaking in Three-Dimensional Electrodynamics, Phys. Rev. D 29 (1984) 2423 [INSPIRE].
T. Appelquist, D. Nash and L.C.R. Wijewardhana, Critical Behavior in (2+1)-Dimensional QED, Phys. Rev. Lett. 60 (1988) 2575 [INSPIRE].
T. Nishioka and K. Yonekura, On RG Flow of τ RR for Supersymmetric Field Theories in Three-Dimensions, JHEP 05 (2013) 165 [arXiv:1303.1522] [INSPIRE].
K. Kaveh and I.F. Herbut, Chiral symmetry breaking in QED(3) in presence of irrelevant interactions: A renormalization group study, Phys. Rev. B 71 (2005) 184519 [cond-mat/0411594] [INSPIRE].
C.S. Fischer, R. Alkofer, T. Dahm and P. Maris, Dynamical chiral symmetry breaking in unquenched QED(3), Phys. Rev. D 70 (2004) 073007 [hep-ph/0407104] [INSPIRE].
J. Braun, H. Gies, L. Janssen and D. Roscher, Phase structure of many-flavor QED 3, Phys. Rev. D 90 (2014) 036002 [arXiv:1404.1362] [INSPIRE].
L. Di Pietro, Z. Komargodski, I. Shamir and E. Stamou, Quantum Electrodynamics in D = 3 from the ε Expansion, Phys. Rev. Lett. 116 (2016) 131601 [arXiv:1508.06278] [INSPIRE].
C. Strouthos and J.B. Kogut, The Phases of Non-Compact QED 3, PoS(LATTICE 2007)278 [arXiv:0804.0300] [INSPIRE].
O. Raviv, Y. Shamir and B. Svetitsky, Nonperturbative β-function in three-dimensional electrodynamics, Phys. Rev. D 90 (2014) 014512 [arXiv:1405.6916] [INSPIRE].
T. Appelquist and R.D. Pisarski, High-Temperature Yang-Mills Theories and Three-Dimensional Quantum Chromodynamics, Phys. Rev. D 23 (1981) 2305 [INSPIRE].
N.K. Nielsen, The Energy Momentum Tensor in a Nonabelian Quark Gluon Theory, Nucl. Phys. B 120 (1977) 212 [INSPIRE].
M.F. Zoller and K.G. Chetyrkin, OPE of the energy-momentum tensor correlator in massless QCD, JHEP 12 (2012) 119 [arXiv:1209.1516] [INSPIRE].
D. Gepner, Nonabelian Bosonization and Multiflavor QED and QCD in Two-dimensions, Nucl. Phys. B 252 (1985) 481 [INSPIRE].
I. Affleck, On the Realization of Chiral Symmetry in (1+1)-dimensions, Nucl. Phys. B 265 (1986) 448 [INSPIRE].
E.A. Ivanov, A.V. Smilga and B.M. Zupnik, Renormalizable supersymmetric gauge theory in six dimensions, Nucl. Phys. B 726 (2005) 131 [hep-th/0505082] [INSPIRE].
E.A. Ivanov and A.V. Smilga, Conformal properties of hypermultiplet actions in six dimensions, Phys. Lett. B 637 (2006) 374 [hep-th/0510273] [INSPIRE].
A.V. Smilga, 6D superconformal theory as the theory of everything, hep-th/0509022 [INSPIRE].
A.V. Smilga, Chiral anomalies in higher-derivative supersymmetric 6D theories, Phys. Lett. B 647 (2007) 298 [hep-th/0606139] [INSPIRE].
M. Beccaria and A.A. Tseytlin, Conformal a-anomaly of some non-unitary 6d superconformal theories, JHEP 09 (2015) 017 [arXiv:1506.08727] [INSPIRE].
J.A. Gracey, Six dimensional QCD at two loops, Phys. Rev. D 93 (2016) 025025 [arXiv:1512.04443] [INSPIRE].
M. Ciuchini, S.E. Derkachov, J.A. Gracey and A.N. Manashov, Computation of quark mass anomalous dimension at O(1/N 2 f ) in quantum chromodynamics, Nucl. Phys. B 579 (2000) 56 [hep-ph/9912221] [INSPIRE].
E. Witten, \( \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [INSPIRE].
R.G. Leigh and A.C. Petkou, \( \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) \) action on three-dimensional CFTs and holography, JHEP 12 (2003) 020 [hep-th/0309177] [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].
A. Cappelli, D. Friedan and J.I. Latorre, C theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].
A. Petkou, Conserved currents, consistency relations and operator product expansions in the conformally invariant O(N) vector model, Annals Phys. 249 (1996) 180 [hep-th/9410093] [INSPIRE].
S.J. Hathrell, Trace Anomalies and QED in Curved Space, Annals Phys. 142 (1982) 34 [INSPIRE].
I. Jack and H. Osborn, Analogs for the c Theorem for Four-dimensional Renormalizable Field Theories, Nucl. Phys. B 343 (1990) 647 [INSPIRE].
S. El-Showk, Y. Nakayama and S. Rychkov, What Maxwell Theory in D ≠ 4 teaches us about scale and conformal invariance, Nucl. Phys. B 848 (2011) 578 [arXiv:1101.5385] [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].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-Theorem without Supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [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].
N. Karthik and R. Narayanan, No evidence for bilinear condensate in parity-invariant three-dimensional QED with massless fermions, Phys. Rev. D 93 (2016) 045020 [arXiv:1512.02993] [INSPIRE].
T. Appelquist, A.G. Cohen and M. Schmaltz, A new constraint on strongly coupled gauge theories, Phys. Rev. D 60 (1999) 045003 [hep-th/9901109] [INSPIRE].
A. Hasenfratz and P. Hasenfratz, The equivalence of the SU(N ) Yang-Mills theory with a purely fermionic model, Phys. Lett. B 297 (1992) 166 [hep-lat/9207017] [INSPIRE].
J.A. Gracey, Quark, gluon and ghost anomalous dimensions at O(1/N f ) in quantum chromodynamics, Phys. Lett. B 318 (1993) 177 [hep-th/9310063] [INSPIRE].
D.I. Kazakov and G.S. Vartanov, Renormalizable 1/N f Expansion for Field Theories in Extra Dimensions, JHEP 06 (2007) 081 [arXiv:0707.2564] [INSPIRE].
D.B. Ali and J.A. Gracey, Anomalous dimension of nonsinglet quark currents at O(1/N 2 f ) in QCD, Phys. Lett. B 518 (2001) 188 [hep-ph/0105038] [INSPIRE].
D. Dudal, J.A. Gracey, V.E.R. Lemes, R.F. Sobreiro, S.P. Sorella and H. Verschelde, Renormalization properties of the mass operator A α μ A α μ in three dimensional Yang-Mills theories in the Landau gauge, Annals Phys. 317 (2005) 203 [hep-th/0409254] [INSPIRE].
J.F. Bennett and J.A. Gracey, Three loop renormalization of the SU(N c ) nonAbelian Thirring model, Nucl. Phys. B 563 (1999) 390 [hep-th/9909046] [INSPIRE].
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
G. Bhanot, K. Demeterfi and I.R. Klebanov, (1+1)-dimensional large-N QCD coupled to adjoint fermions, Phys. Rev. D 48 (1993) 4980 [hep-th/9307111] [INSPIRE].
H. Osborn and A. Stergiou, C T for non-unitary CFTs in higher dimensions, JHEP 06 (2016) 079 [arXiv:1603.07307] [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: 1602.01076
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
Giombi, S., Tarnopolsky, G. & Klebanov, I.R. On C J and C T in conformal QED. J. High Energ. Phys. 2016, 156 (2016). https://doi.org/10.1007/JHEP08(2016)156
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)156