Abstract
We establish a connection between the averaged null energy condition (ANEC) and the monotonicity of the renormalization group, by studying the light-ray operator ∫ duTuu in quantum field theories that flow between two conformal fixed points. In four dimensions, we derive an exact sum rule relating this operator to the Euler coefficient in the trace anomaly, and show that the ANEC implies the a-theorem. The argument is based on matching anomalies in the stress tensor 3-point function, and relies on special properties of contact terms involving light-ray operators. We also illustrate the sum rule for the example of a free massive scalar field. Averaged null energy appears in a variety of other applications to quantum field theory, including causality constraints, Lorentzian inversion, and quantum information. The quantum information perspective provides a new derivation of the a-theorem from the monotonicity of relative entropy. The equation relating our sum rule to the dilaton scattering amplitude in the forward limit suggests an inversion formula for non-conformal theories.
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References
D.M. Hofman and J. Maldacena, Conformal collider physics: Energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [INSPIRE].
L.J. Dixon, I. Moult and H.X. Zhu, Collinear limit of the energy-energy correlator, Phys. Rev. D 100 (2019) 014009 [arXiv:1905.01310] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, The light-ray OPE and conformal colliders, JHEP 01 (2021) 128 [arXiv:1905.01311] [INSPIRE].
K. Lee, B. Meçaj and I. Moult, Conformal Colliders Meet the LHC, arXiv:2205.03414 [INSPIRE].
T. Hartman, S. Kundu and A. Tajdini, Averaged Null Energy Condition from Causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
S. Caron-Huot, Analyticity in Spin in Conformal Theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
T. Faulkner, R.G. Leigh, O. Parrikar and H. Wang, Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition, JHEP 09 (2016) 038 [arXiv:1605.08072] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, Causality Constraints in Conformal Field Theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [INSPIRE].
S. Kundu, Subleading bounds on chaos, JHEP 04 (2022) 010 [arXiv:2109.03826] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
C. Córdova and S.-H. Shao, Light-ray Operators and the BMS Algebra, Phys. Rev. D 98 (2018) 125015 [arXiv:1810.05706] [INSPIRE].
Y. Hu and S. Pasterski, Celestial conformal colliders, JHEP 02 (2023) 243 [arXiv:2211.14287] [INSPIRE].
D.M. Hofman, Higher Derivative Gravity, Causality and Positivity of Energy in a UV complete QFT, Nucl. Phys. B 823 (2009) 174 [arXiv:0907.1625] [INSPIRE].
W.R. Kelly and A.C. Wall, Holographic proof of the averaged null energy condition, Phys. Rev. D 90 (2014) 106003 [Erratum ibid. 91 (2015) 069902] [arXiv:1408.3566] [INSPIRE].
G. Klinkhammer, Averaged energy conditions for free scalar fields in flat space-times, Phys. Rev. D 43 (1991) 2542 [INSPIRE].
R.M. Wald and U. Yurtsever, General proof of the averaged null energy condition for a massless scalar field in two-dimensional curved space-time, Phys. Rev. D 44 (1991) 403 [INSPIRE].
A. Folacci, Averaged null energy condition for electromagnetism in Minkowski space-time, Phys. Rev. D 46 (1992) 2726 [INSPIRE].
L.H. Ford and T.A. Roman, Averaged energy conditions and evaporating black holes, Phys. Rev. D 53 (1996) 1988 [gr-qc/9506052] [INSPIRE].
A. Borde, Geodesic focusing, energy conditions and singularities, Class. Quant. Grav. 4 (1987) 343 [INSPIRE].
S. Gao and R.M. Wald, Theorems on gravitational time delay and related issues, Class. Quant. Grav. 17 (2000) 4999 [gr-qc/0007021] [INSPIRE].
N. Graham and K.D. Olum, Achronal averaged null energy condition, Phys. Rev. D 76 (2007) 064001 [arXiv:0705.3193] [INSPIRE].
T. Hartman, S. Jain and S. Kundu, A New Spin on Causality Constraints, JHEP 10 (2016) 141 [arXiv:1601.07904] [INSPIRE].
D.M. Hofman et al., A Proof of the Conformal Collider Bounds, JHEP 06 (2016) 111 [arXiv:1603.03771] [INSPIRE].
C. Córdova, J. Maldacena and G.J. Turiaci, Bounds on OPE Coefficients from Interference Effects in the Conformal Collider, JHEP 11 (2017) 032 [arXiv:1710.03199] [INSPIRE].
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space, JHEP 01 (2020) 142 [arXiv:1908.04733] [INSPIRE].
D. Simmons-Duffin, The Lightcone Bootstrap and the Spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
N. Afkhami-Jeddi, T. Hartman, S. Kundu and A. Tajdini, Einstein gravity 3-point functions from conformal field theory, JHEP 12 (2017) 049 [arXiv:1610.09378] [INSPIRE].
D. Meltzer and E. Perlmutter, Beyond a = c: gravitational couplings to matter and the stress tensor OPE, JHEP 07 (2018) 157 [arXiv:1712.04861] [INSPIRE].
A. Belin, D.M. Hofman and G. Mathys, Einstein gravity from ANEC correlators, JHEP 08 (2019) 032 [arXiv:1904.05892] [INSPIRE].
M. Kologlu, P. Kravchuk, D. Simmons-Duffin and A. Zhiboedov, Shocks, Superconvergence, and a Stringy Equivalence Principle, JHEP 11 (2020) 096 [arXiv:1904.05905] [INSPIRE].
A. Belin, D.M. Hofman, G. Mathys and M.T. Walters, On the stress tensor light-ray operator algebra, JHEP 05 (2021) 033 [arXiv:2011.13862] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Dispersive CFT Sum Rules, JHEP 05 (2021) 243 [arXiv:2008.04931] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, AdS bulk locality from sharp CFT bounds, JHEP 11 (2021) 164 [arXiv:2106.10274] [INSPIRE].
S. Caron-Huot, D. Mazac, L. Rastelli and D. Simmons-Duffin, Sharp boundaries for the swampland, JHEP 07 (2021) 110 [arXiv:2102.08951] [INSPIRE].
S. Caron-Huot, Y.-Z. Li, J. Parra-Martinez and D. Simmons-Duffin, Causality constraints on corrections to Einstein gravity, JHEP 05 (2023) 122 [arXiv:2201.06602] [INSPIRE].
T. Hartman and G. Mathys, Null energy constraints on two-dimensional RG flows, arXiv:2310.15217 [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].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
H. Casini and M. Huerta, A c-theorem for the entanglement entropy, J. Phys. A 40 (2007) 7031 [cond-mat/0610375] [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. Casini, E. Testé and G. Torroba, Markov Property of the Conformal Field Theory Vacuum and the a Theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, Irreversibility, QNEC, and defects, JHEP 07 (2023) 004 [arXiv:2303.16935] [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
T. Hartman, Y. Jiang, F. Sgarlata and A. Tajdini, Focusing bounds for CFT correlators and the S-matrix, arXiv:2212.01942 [INSPIRE].
S.L. Adler, Einstein Gravity as a Symmetry-Breaking Effect in Quantum Field Theory, Rev. Mod. Phys. 54 (1982) 729 [Erratum ibid. 55 (1983) 837] [INSPIRE].
A. Zee, Spontaneously Generated Gravity, Phys. Rev. D 23 (1981) 858 [INSPIRE].
D. Anselmi, Kinematic sum rules for trace anomalies, JHEP 11 (2001) 033 [hep-th/0107194] [INSPIRE].
D. Baumann, D. Green and T. Hartman, Dynamical Constraints on RG Flows and Cosmology, JHEP 12 (2019) 134 [arXiv:1906.10226] [INSPIRE].
J.J. Heckman and T. Rudelius, Evidence for C-theorems in 6D SCFTs, JHEP 09 (2015) 218 [arXiv:1506.06753] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, Anomalies, renormalization group flows, and the a-theorem in six-dimensional (1, 0) theories, JHEP 10 (2016) 080 [arXiv:1506.03807] [INSPIRE].
A. Stergiou, D. Stone and L.G. Vitale, Constraints on Perturbative RG Flows in Six Dimensions, JHEP 08 (2016) 010 [arXiv:1604.01782] [INSPIRE].
C. Córdova, T.T. Dumitrescu and K. Intriligator, 2-Group Global Symmetries and Anomalies in Six-Dimensional Quantum Field Theories, JHEP 04 (2021) 252 [arXiv:2009.00138] [INSPIRE].
J.J. Heckman, S. Kundu and H.Y. Zhang, Effective field theory of 6D SUSY RG Flows, Phys. Rev. D 104 (2021) 085017 [arXiv:2103.13395] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
G.P. Korchemsky, Energy correlations in the end-point region, JHEP 01 (2020) 008 [arXiv:1905.01444] [INSPIRE].
C.-H. Chang et al., Transverse spin in the light-ray OPE, JHEP 05 (2022) 059 [arXiv:2010.04726] [INSPIRE].
H. Chen, I. Moult, J. Sandor and H.X. Zhu, Celestial blocks and transverse spin in the three-point energy correlator, JHEP 09 (2022) 199 [arXiv:2202.04085] [INSPIRE].
C.-H. Chang and D. Simmons-Duffin, Three-point energy correlators and the celestial block expansion, JHEP 02 (2023) 126 [arXiv:2202.04090] [INSPIRE].
K.-W. Huang, Stress-tensor commutators in conformal field theories near the lightcone, Phys. Rev. D 100 (2019) 061701 [arXiv:1907.00599] [INSPIRE].
K.-W. Huang, Lightcone Commutator and Stress-Tensor Exchange in d > 2 CFTs, Phys. Rev. D 102 (2020) 021701 [arXiv:2002.00110] [INSPIRE].
M. Beşken, J. De Boer and G. Mathys, On local and integrated stress-tensor commutators, JHEP 21 (2020) 148 [arXiv:2012.15724] [INSPIRE].
G.P. Korchemsky and A. Zhiboedov, On the light-ray algebra in conformal field theories, JHEP 02 (2022) 140 [arXiv:2109.13269] [INSPIRE].
K.-W. Huang, d > 2 stress-tensor operator product expansion near a line, Phys. Rev. D 103 (2021) 121702 [arXiv:2103.09930] [INSPIRE].
K.-W. Huang, Approximate symmetries in d = 4 CFTs with an Einstein gravity dual, JHEP 09 (2022) 053 [arXiv:2202.09998] [INSPIRE].
S. De, Y. Hu, A. Yelleshpur Srikant and A. Volovich, Correlators of four light-ray operators in CCFT, JHEP 10 (2022) 170 [arXiv:2206.08875] [INSPIRE].
A.L. Fitzpatrick and K.-W. Huang, Universal Lowest-Twist in CFTs from Holography, JHEP 08 (2019) 138 [arXiv:1903.05306] [INSPIRE].
A.L. Fitzpatrick, K.-W. Huang and D. Li, Probing universalities in d > 2 CFTs: from black holes to shockwaves, JHEP 11 (2019) 139 [arXiv:1907.10810] [INSPIRE].
A.L. Fitzpatrick et al., Model-dependence of minimal-twist OPEs in d > 2 holographic CFTs, JHEP 11 (2020) 060 [arXiv:2007.07382] [INSPIRE].
K.-W. Huang, R. Karlsson, A. Parnachev and S. Valach, Freedom near lightcone and ANEC saturation, JHEP 05 (2023) 065 [arXiv:2210.16274] [INSPIRE].
T. Bautista, L. Casarin and H. Godazgar, ANEC in λϕ4 theory, JHEP 01 (2021) 132 [arXiv:2010.02136] [INSPIRE].
T. Bautista and L. Casarin, ANEC on stress-tensor states in perturbative λϕ4 theory, JHEP 01 (2023) 097 [arXiv:2210.11365] [INSPIRE].
S. Caron-Huot et al., Detectors in weakly-coupled field theories, JHEP 04 (2023) 014 [arXiv:2209.00008] [INSPIRE].
Z. Komargodski, M. Kulaxizi, A. Parnachev and A. Zhiboedov, Conformal Field Theories and Deep Inelastic Scattering, Phys. Rev. D 95 (2017) 065011 [arXiv:1601.05453] [INSPIRE].
D. Meltzer, Higher Spin ANEC and the Space of CFTs, JHEP 07 (2019) 001 [arXiv:1811.01913] [INSPIRE].
G.P. Korchemsky, G. Oderda and G.F. Sterman, Power corrections and nonlocal operators, AIP Conf. Proc. 407 (1997) 988 [hep-ph/9708346] [INSPIRE].
N.A. Sveshnikov and F.V. Tkachov, Jets and quantum field theory, Phys. Lett. B 382 (1996) 403 [hep-ph/9512370] [INSPIRE].
A.V. Belitsky et al., Energy-Energy Correlations in N = 4 Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 071601 [arXiv:1311.6800] [INSPIRE].
A.V. Belitsky et al., Event shapes in \( \mathcal{N} \) = 4 super-Yang-Mills theory, Nucl. Phys. B 884 (2014) 206 [arXiv:1309.1424] [INSPIRE].
A.V. Belitsky et al., From correlation functions to event shapes, Nucl. Phys. B 884 (2014) 305 [arXiv:1309.0769] [INSPIRE].
J.M. Henn, E. Sokatchev, K. Yan and A. Zhiboedov, Energy-energy correlation in N = 4 super Yang-Mills theory at next-to-next-to-leading order, Phys. Rev. D 100 (2019) 036010 [arXiv:1903.05314] [INSPIRE].
R. Gonzo and A. Pokraka, Light-ray operators, detectors and gravitational event shapes, JHEP 05 (2021) 015 [arXiv:2012.01406] [INSPIRE].
G.P. Korchemsky, E. Sokatchev and A. Zhiboedov, Generalizing event shapes: in search of lost collider time, JHEP 08 (2022) 188 [arXiv:2106.14899] [INSPIRE].
H. Chen, I. Moult, X.Y. Zhang and H.X. Zhu, Rethinking jets with energy correlators: Tracks, resummation, and analytic continuation, Phys. Rev. D 102 (2020) 054012 [arXiv:2004.11381] [INSPIRE].
H. Epstein, V. Glaser and A. Jaffe, Nonpositivity of energy density in Quantized field theories, Nuovo Cim. 36 (1965) 1016 [INSPIRE].
M. Visser, Scale anomalies imply violation of the averaged null energy condition, Phys. Lett. B 349 (1995) 443 [gr-qc/9409043] [INSPIRE].
D. Urban and K.D. Olum, Averaged null energy condition violation in a conformally flat spacetime, Phys. Rev. D 81 (2010) 024039 [arXiv:0910.5925] [INSPIRE].
D. Meltzer, Dispersion Formulas in QFTs, CFTs, and Holography, JHEP 05 (2021) 098 [arXiv:2103.15839] [INSPIRE].
R. Haag, Local quantum physics: Fields, particles, algebras, Springer Berlin Heidelberg (1992).
P. Kravchuk, J. Qiao and S. Rychkov, Distributions in CFT. Part II. Minkowski space, JHEP 08 (2021) 094 [arXiv:2104.02090] [INSPIRE].
S. Deser, M.J. Duff and C.J. Isham, Nonlocal Conformal Anomalies, Nucl. Phys. B 111 (1976) 45 [INSPIRE].
M.J. Duff, Observations on Conformal Anomalies, Nucl. Phys. B 125 (1977) 334 [INSPIRE].
M.J. Duff, Twenty years of the Weyl anomaly, Class. Quant. Grav. 11 (1994) 1387 [hep-th/9308075] [INSPIRE].
Y. Nakayama, Scale invariance vs conformal invariance, Phys. Rept. 569 (2015) 1 [arXiv:1302.0884] [INSPIRE].
J. Wess and B. Zumino, Consequences of anomalous Ward identities, Phys. Lett. B 37 (1971) 95 [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].
D. Karateev, J. Marucha, J. Penedones and B. Sahoo, Bootstrapping the a-anomaly in 4d QFTs, JHEP 12 (2022) 136 [arXiv:2204.01786] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Relative entropy and the RG flow, JHEP 03 (2017) 089 [arXiv:1611.00016] [INSPIRE].
V. Balasubramanian, J.J. Heckman and A. Maloney, Relative Entropy and Proximity of Quantum Field Theories, JHEP 05 (2015) 104 [arXiv:1410.6809] [INSPIRE].
J. Stout, Infinite Distance Limits and Information Theory, arXiv:2106.11313 [INSPIRE].
J. Erdmenger, K.T. Grosvenor and R. Jefferson, Towards quantifying information flows: relative entropy in deep neural networks and the renormalization group, SciPost Phys. 12 (2022) 041 [arXiv:2107.06898] [INSPIRE].
J. Stout, Infinite Distances and Factorization, arXiv:2208.08444 [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].
F. Baume, B. Keren-Zur, R. Rattazzi and L. Vitale, The local Callan-Symanzik equation: structure and applications, JHEP 08 (2014) 152 [arXiv:1401.5983] [INSPIRE].
G.M. Shore, The c and a-theorems and the Local Renormalisation Group, Springer, Cham (2017) [https://doi.org/10.1007/978-3-319-54000-9] [INSPIRE].
J. Maldacena, S.H. Shenker and D. Stanford, A bound on chaos, JHEP 08 (2016) 106 [arXiv:1503.01409] [INSPIRE].
T. Hartman, Holography and Energy Conditions, Lectures at the 2018 Bootstrap School, held at Caltech, (July 2018) https://www.youtube.com/watch?v=AuPuwWj83KU.
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].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York, U.S.A. (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
Acknowledgments
We are grateful to Simon Caron-Huot, Horacio Casini, Jeevan Chandra, Clay Cordova, Diego Hofman, Austin Joyce, Denis Karateev, Murat Kologlu, Zohar Komargodski, Juan Maldacena, David Meltzer, Ian Moult, Joao Penedones, Shu-Heng Shao, Nathan Seiberg, David Simmons-Duffin, and John Stout for helpful discussions. This work is funded by NSF grant PHY-2014071. We also acknowledge support from NSF grant PHY-1748958 for participation in a KITP workshop. Part of this work was performed in part at Aspen Center for Physics, which is supported by NSF grant PHY-2210452.
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Hartman, T., Mathys, G. Averaged null energy and the renormalization group. J. High Energ. Phys. 2023, 139 (2023). https://doi.org/10.1007/JHEP12(2023)139
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DOI: https://doi.org/10.1007/JHEP12(2023)139