Abstract
We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov c-theorem, and derive further independent constraints along the flow. In particular, we identify a natural C-function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1)nC(n)(μ2) ≥ 0. The completely monotonic C-function is identical to the Zamolodchikov C-function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov c-theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the c-theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.
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References
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].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
S. Giombi and I.R. Klebanov, Interpolating between a and F , JHEP 03 (2015) 117 [arXiv:1409.1937] [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].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
J.J. Heckman and T. Rudelius, Evidence for c-theorems in 6D SCFTs, JHEP 09 (2015) 218 [arXiv:1506.06753] [INSPIRE].
C. Cordova, T.T. Dumitrescu and X. Yin, Higher derivative terms, toroidal compactification, and Weyl anomalies in six-dimensional (2, 0) theories, JHEP 10 (2019) 128 [arXiv:1505.03850] [INSPIRE].
C. Cordova, 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].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [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].
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].
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. Kundu and A. Tajdini, Averaged null energy condition from causality, JHEP 07 (2017) 066 [arXiv:1610.05308] [INSPIRE].
D.M. Hofman and J. Maldacena, Conformal collider physics: energy and charge correlations, JHEP 05 (2008) 012 [arXiv:0803.1467] [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].
T. Hartman, S. Jain and S. Kundu, Causality constraints in conformal field theory, JHEP 05 (2016) 099 [arXiv:1509.00014] [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. Cordova, 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].
C. Córdova and S.-H. Shao, Light-ray operators and the BMS algebra, Phys. Rev. D 98 (2018) 125015 [arXiv:1810.05706] [INSPIRE].
T. Bautista and H. Godazgar, Lorentzian CFT 3-point functions in momentum space, JHEP 01 (2020) 142 [arXiv:1908.04733] [INSPIRE].
M. Beşken, J. De Boer and G. Mathys, On local and integrated stress-tensor commutators, JHEP 21 (2020) 148 [arXiv:2012.15724] [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].
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].
D. Baumann, D. Green and T. Hartman, Dynamical constraints on RG flows and cosmology, JHEP 12 (2019) 134 [arXiv:1906.10226] [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].
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].
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. Hartman and G. Mathys, Averaged null energy and the renormalization group, JHEP 12 (2023) 139 [arXiv:2309.14409] [INSPIRE].
J.L. Cardy, The central charge and universal combinations of amplitudes in two-dimensional theories away from criticality, Phys. Rev. Lett. 60 (1988) 2709 [INSPIRE].
S. Bernstein, Sur les fonctions absolument monotones (in French), Acta Math. 52 (1929) 1.
J.D. Tamarkin, On a theorem of S. Bernstein-Widder, Trans. Amer. Math. Soc. 33 (1931) 893.
A. Cappelli, D. Friedan and J.I. Latorre, c-theorem and spectral representation, Nucl. Phys. B 352 (1991) 616 [INSPIRE].
M. Bander and C. Itzykson, Quantum field theory calculation of two-dimensional Ising model correlation function, Phys. Rev. D 15 (1977) 463 [INSPIRE].
R. Haag, Local quantum physics: fields, particles, algebras, Springer, Berlin, Heidelberg, Germany (1992) [https://doi.org/10.1007/978-3-642-97306-2] [INSPIRE].
R. Verch, The averaged null energy condition for general quantum field theories in two-dimensions, J. Math. Phys. 41 (2000) 206 [math-ph/9904036] [INSPIRE].
Z. Komargodski, The constraints of conformal symmetry on RG flows, JHEP 07 (2012) 069 [arXiv:1112.4538] [INSPIRE].
D. Meltzer, Dispersion formulas in QFTs, CFTs, and holography, JHEP 05 (2021) 098 [arXiv:2103.15839] [INSPIRE].
H. Epstein, V. Glaser and A. Jaffe, Nonpositivity of energy density in quantized field theories, Nuovo Cim. 36 (1965) 1016 [INSPIRE].
Acknowledgments
We thank Simon Caron-Huot, Horacio Casini, Jeevan Chandra, Clay Cordova, Diego Hofman, Austin Joyce, Denis Karateev, Murat Kologlu, Zohar Komargodski, Juan Maldacena, David Meltzer, 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. GM is supported by the Simons Foundation grant 488649 (Simons Collaboration on the Nonperturbative Bootstrap) and the Swiss National Science Foundation through the project 200020_197160 and through the National Centre of Competence in Research SwissMAP. 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. Null energy constraints on two-dimensional RG flows. J. High Energ. Phys. 2024, 102 (2024). https://doi.org/10.1007/JHEP01(2024)102
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DOI: https://doi.org/10.1007/JHEP01(2024)102