Abstract
In this paper, we analyze the loop corrections to celestial OPE for gluons and gravitons. Even at the loop level, the soft gluons and gravitons have conformal dimensions ∆ = 1 − \({\mathbb{Z}}_{\ge 0}\). The only novelty is the presence of higher poles. At one loop level, there are two types of conformal soft gluons with a single pole and a double pole in the ∆ plane. The celestial OPEs are obtained using the collinear splitting functions. In the case of gluons, the splitting functions receive loop corrections. After taking the holomorphic soft limit, we find the OPE of conformal soft gluons. We find a novel mixing of simple and double poles soft gluon operators in the OPE. In the case of gravitons, where splitting functions are known to be all loop exact, we still find a wedge algebra of w∞ which is in addition to the wedge algebra of w1+∞ already found by Strominger.
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References
A. Atanasov, W. Melton, A.-M. Raclariu and A. Strominger, Conformal block expansion in celestial CFT, Phys. Rev. D 104 (2021) 126033 [arXiv:2104.13432] [INSPIRE].
A. Ball, S.A. Narayanan, J. Salzer and A. Strominger, Perturbatively exact w1+∞ asymptotic symmetry of quantum self-dual gravity, JHEP 01 (2022) 114 [arXiv:2111.10392] [INSPIRE].
E. Crawley, N. Miller, S.A. Narayanan and A. Strominger, State-operator correspondence in celestial conformal field theory, JHEP 09 (2021) 132 [arXiv:2105.00331] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Asymptotic Symmetries and Celestial CFT, JHEP 09 (2020) 176 [arXiv:2005.08990] [INSPIRE].
A. Guevara, Notes on Conformal Soft Theorems and Recursion Relations in Gravity, arXiv:1906.07810 [INSPIRE].
H. Jiang, Holographic chiral algebra: supersymmetry, infinite Ward identities, and EFTs, JHEP 01 (2022) 113 [arXiv:2108.08799] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere, Phys. Rev. D 96 (2017) 065026 [arXiv:1701.00049] [INSPIRE].
S. Pasterski and S.-H. Shao, Conformal basis for flat space amplitudes, Phys. Rev. D 96 (2017) 065022 [arXiv:1705.01027] [INSPIRE].
S. Pasterski, S.-H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Correlators, Phys. Rev. D 96 (2017) 085006 [arXiv:1706.03917] [INSPIRE].
S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in the proceedings of the Snowmass 2021, Seattle, U.S.A., July 17–26 (2022) [arXiv:2111.11392] [INSPIRE].
S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [arXiv:2108.04801] [INSPIRE].
S. Banerjee and S. Pasterski, Revisiting the shadow stress tensor in celestial CFT, JHEP 04 (2023) 118 [arXiv:2212.00257] [INSPIRE].
S. Pasterski, A comment on loop corrections to the celestial stress tensor, JHEP 01 (2023) 025 [arXiv:2205.10901] [INSPIRE].
M. Pate, A.-M. Raclariu, A. Strominger and E.Y. Yuan, Celestial operator products of gluons and gravitons, Rev. Math. Phys. 33 (2021) 2140003 [arXiv:1910.07424] [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Conformally Soft Theorem in Gauge Theory, Phys. Rev. D 100 (2019) 085017 [arXiv:1904.10831] [INSPIRE].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
N. Arkani-Hamed, M. Pate, A.-M. Raclariu and A. Strominger, Celestial amplitudes from UV to IR, JHEP 08 (2021) 062 [arXiv:2012.04208] [INSPIRE].
E. Himwich et al., The Soft \(\mathcal{S}\)-Matrix in Gravity, JHEP 09 (2020) 129 [arXiv:2005.13433] [INSPIRE].
A. Ball et al., Uplifting AdS3/CFT2 to flat space holography, JHEP 08 (2019) 168 [arXiv:1905.09809] [INSPIRE].
Y. Pano, A. Puhm and E. Trevisani, Symmetries in Celestial CFTd, JHEP 07 (2023) 076 [arXiv:2302.10222] [INSPIRE].
T. McLoughlin, A. Puhm and A.-M. Raclariu, The SAGEX review on scattering amplitudes chapter 11: soft theorems and celestial amplitudes, J. Phys. A 55 (2022) 443012 [arXiv:2203.13022] [INSPIRE].
G. Travaglini et al., The SAGEX review on scattering amplitudes, J. Phys. A 55 (2022) 443001 [arXiv:2203.13011] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Goldilocks modes and the three scattering bases, JHEP 06 (2022) 124 [arXiv:2202.11127] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Celestial diamonds: conformal multiplets in celestial CFT, JHEP 11 (2021) 072 [arXiv:2105.03516] [INSPIRE].
S. Pasterski and A. Puhm, Shifting spin on the celestial sphere, Phys. Rev. D 104 (2021) 086020 [arXiv:2012.15694] [INSPIRE].
E. Casali and A. Puhm, Double Copy for Celestial Amplitudes, Phys. Rev. Lett. 126 (2021) 101602 [arXiv:2007.15027] [INSPIRE].
A. Guevara, E. Himwich, M. Pate and A. Strominger, Holographic symmetry algebras for gauge theory and gravity, JHEP 11 (2021) 152 [arXiv:2103.03961] [INSPIRE].
A. Strominger, w1+∞ Algebra and the Celestial Sphere: Infinite Towers of Soft Graviton, Photon, and Gluon Symmetries, Phys. Rev. Lett. 127 (2021) 221601 [arXiv:2105.14346] [INSPIRE].
A. Fotopoulos, S. Stieberger, T.R. Taylor and B. Zhu, Extended BMS Algebra of Celestial CFT, JHEP 03 (2020) 130 [arXiv:1912.10973] [INSPIRE].
W. Fan, A. Fotopoulos and T.R. Taylor, Soft Limits of Yang-Mills Amplitudes and Conformal Correlators, JHEP 05 (2019) 121 [arXiv:1903.01676] [INSPIRE].
T. Adamo, L. Mason and A. Sharma, Celestial w1+∞ Symmetries from Twistor Space, SIGMA 18 (2022) 016 [arXiv:2110.06066] [INSPIRE].
K. Costello, N.M. Paquette and A. Sharma, Top-Down Holography in an Asymptotically Flat Spacetime, Phys. Rev. Lett. 130 (2023) 061602 [arXiv:2208.14233] [INSPIRE].
K. Costello, N.M. Paquette and A. Sharma, Burns space and holography, JHEP 10 (2023) 174 [arXiv:2306.00940] [INSPIRE].
H. Krishna and Y.-P. Wang, Celestial holography from Chiral strings, arXiv:2308.11799 [INSPIRE].
A. Strominger, On BMS Invariance of Gravitational Scattering, JHEP 07 (2014) 152 [arXiv:1312.2229] [INSPIRE].
A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
V. Lysov, S. Pasterski and A. Strominger, Low’s Subleading Soft Theorem as a Symmetry of QED, Phys. Rev. Lett. 113 (2014) 111601 [arXiv:1407.3814] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Semiclassical Virasoro symmetry of the quantum gravity \(\mathcal{S}\)-matrix, JHEP 08 (2014) 058 [arXiv:1406.3312] [INSPIRE].
D. Kapec, P. Mitra, A.-M. Raclariu and A. Strominger, 2D Stress Tensor for 4D Gravity, Phys. Rev. Lett. 119 (2017) 121601 [arXiv:1609.00282] [INSPIRE].
S. Pasterski, A. Strominger and A. Zhiboedov, New Gravitational Memories, JHEP 12 (2016) 053 [arXiv:1502.06120] [INSPIRE].
D. Kapec, V. Lysov and A. Strominger, Asymptotic Symmetries of Massless QED in Even Dimensions, Adv. Theor. Math. Phys. 21 (2017) 1747 [arXiv:1412.2763] [INSPIRE].
D. Kapec, V. Lysov, S. Pasterski and A. Strominger, Higher-dimensional supertranslations and Weinberg’s soft graviton theorem, Ann. Math. Sci. Appl. 02 (2017) 69 [arXiv:1502.07644] [INSPIRE].
T. He, P. Mitra and A. Strominger, 2D Kac-Moody Symmetry of 4D Yang-Mills Theory, JHEP 10 (2016) 137 [arXiv:1503.02663] [INSPIRE].
D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, Adv. Theor. Math. Phys. 21 (2017) 1769 [arXiv:1506.02906] [INSPIRE].
A. Strominger, Magnetic Corrections to the Soft Photon Theorem, Phys. Rev. Lett. 116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
T.T. Dumitrescu, T. He, P. Mitra and A. Strominger, Infinite-dimensional fermionic symmetry in supersymmetric gauge theories, JHEP 08 (2021) 051 [arXiv:1511.07429] [INSPIRE].
T. He, D. Kapec, A.-M. Raclariu and A. Strominger, Loop-Corrected Virasoro Symmetry of 4D Quantum Gravity, JHEP 08 (2017) 050 [arXiv:1701.00496] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A. Nande, M. Pate and A. Strominger, Soft Factorization in QED from 2D Kac-Moody Symmetry, JHEP 02 (2018) 079 [arXiv:1705.00608] [INSPIRE].
D. Kapec, M. Perry, A.-M. Raclariu and A. Strominger, Infrared Divergences in QED, Revisited, Phys. Rev. D 96 (2017) 085002 [arXiv:1705.04311] [INSPIRE].
M. Pate, A.-M. Raclariu and A. Strominger, Color Memory: A Yang-Mills Analog of Gravitational Wave Memory, Phys. Rev. Lett. 119 (2017) 261602 [arXiv:1707.08016] [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
F. Cachazo and A. Strominger, Evidence for a New Soft Graviton Theorem, arXiv:1404.4091 [INSPIRE].
T. He, V. Lysov, P. Mitra and A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem, JHEP 05 (2015) 151 [arXiv:1401.7026] [INSPIRE].
S. Atul Bhatkar and B. Sahoo, Subleading Soft Theorem for arbitrary number of external soft photons and gravitons, JHEP 01 (2019) 153 [arXiv:1809.01675] [INSPIRE].
S. Chakrabarti et al., Subleading Soft Theorem for Multiple Soft Gravitons, JHEP 12 (2017) 150 [arXiv:1707.06803] [INSPIRE].
S. Chakrabarti et al., Testing Subleading Multiple Soft Graviton Theorem for CHY Prescription, JHEP 01 (2018) 090 [arXiv:1709.07883] [INSPIRE].
A. Laddha and A. Sen, Sub-subleading Soft Graviton Theorem in Generic Theories of Quantum Gravity, JHEP 10 (2017) 065 [arXiv:1706.00759] [INSPIRE].
A. Laddha and A. Sen, Gravity Waves from Soft Theorem in General Dimensions, JHEP 09 (2018) 105 [arXiv:1801.07719] [INSPIRE].
B. Sahoo and A. Sen, Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in Four Dimensions, JHEP 02 (2019) 086 [arXiv:1808.03288] [INSPIRE].
A. Sen, Soft Theorems in Superstring Theory, JHEP 06 (2017) 113 [arXiv:1702.03934] [INSPIRE].
A. Sen, Subleading Soft Graviton Theorem for Loop Amplitudes, JHEP 11 (2017) 123 [arXiv:1703.00024] [INSPIRE].
A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev. D 24 (1981) 3281 [INSPIRE].
B. Sahoo and A. Sen, Classical soft graviton theorem rewritten, JHEP 01 (2022) 077 [arXiv:2105.08739] [INSPIRE].
B. Sahoo, Classical Sub-subleading Soft Photon and Soft Graviton Theorems in Four Spacetime Dimensions, JHEP 12 (2020) 070 [arXiv:2008.04376] [INSPIRE].
L. Donnay, A. Puhm and A. Strominger, Conformally Soft Photons and Gravitons, JHEP 01 (2019) 184 [arXiv:1810.05219] [INSPIRE].
E. Himwich and A. Strominger, Celestial current algebra from Low’s subleading soft theorem, Phys. Rev. D 100 (2019) 065001 [arXiv:1901.01622] [INSPIRE].
A. Puhm, Conformally Soft Theorem in Gravity, JHEP 09 (2020) 130 [arXiv:1905.09799] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Revisiting the conformally soft sector with celestial diamonds, JHEP 11 (2021) 143 [arXiv:2105.09792] [INSPIRE].
Y. Pano, S. Pasterski and A. Puhm, Conformally soft fermions, JHEP 12 (2021) 166 [arXiv:2108.11422] [INSPIRE].
J. Cotler, N. Miller and A. Strominger, An integer basis for celestial amplitudes, JHEP 08 (2023) 192 [arXiv:2302.04905] [INSPIRE].
L. Freidel, D. Pranzetti and A.-M. Raclariu, A discrete basis for celestial holography, JHEP 02 (2024) 176 [arXiv:2212.12469] [INSPIRE].
L. Donnay et al., Celestial open strings at one-loop, JHEP 10 (2023) 047 [arXiv:2307.03551] [INSPIRE].
H.A. González, A. Puhm and F. Rojas, Loop corrections to celestial amplitudes, Phys. Rev. D 102 (2020) 126027 [arXiv:2009.07290] [INSPIRE].
R. Bhardwaj et al., Loop-level gluon OPEs in celestial holography, JHEP 11 (2022) 171 [arXiv:2208.14416] [INSPIRE].
Z. Bern, S. Davies and J. Nohle, On Loop Corrections to Subleading Soft Behavior of Gluons and Gravitons, Phys. Rev. D 90 (2014) 085015 [arXiv:1405.1015] [INSPIRE].
A. Strominger and A. Zhiboedov, Gravitational Memory, BMS Supertranslations and Soft Theorems, JHEP 01 (2016) 086 [arXiv:1411.5745] [INSPIRE].
G. Grammer Jr. and D.R. Yennie, Improved treatment for the infrared divergence problem in quantum electrodynamics, Phys. Rev. D 8 (1973) 4332 [INSPIRE].
H. Krishna and B. Sahoo, Universality of loop corrected soft theorems in 4d, JHEP 11 (2023) 233 [arXiv:2308.16807] [INSPIRE].
D. Ghosh and B. Sahoo, Spin-dependent gravitational tail memory in D = 4, Phys. Rev. D 105 (2022) 025024 [arXiv:2106.10741] [INSPIRE].
S. Atul Bhatkar, Ward identity for loop level soft photon theorem for massless QED coupled to gravity, JHEP 10 (2020) 110 [arXiv:1912.10229] [INSPIRE].
S. Atul Bhatkar, New asymptotic conservation laws forelectromagnetism, JHEP 02 (2021) 082 [arXiv:2007.03627] [INSPIRE].
G.F. Sterman, Infrared divergences in perturbative QCD, AIP Conf. Proc. 74 (1981) 22 [INSPIRE].
J.C. Collins and G.F. Sterman, Soft Partons in QCD, Nucl. Phys. B 185 (1981) 172 [INSPIRE].
S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multileg one loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140] [INSPIRE].
R. Akhoury, R. Saotome and G. Sterman, Collinear and Soft Divergences in Perturbative Quantum Gravity, Phys. Rev. D 84 (2011) 104040 [arXiv:1109.0270] [INSPIRE].
L. Magnea, Non-abelian infrared divergences on the celestial sphere, JHEP 05 (2021) 282 [arXiv:2104.10254] [INSPIRE].
F. Herzog, Y. Ma, B. Mistlberger and A. Suresh, Single-soft emissions for amplitudes with two colored particles at three loops, JHEP 12 (2023) 023 [arXiv:2309.07884] [INSPIRE].
I. Feige and M.D. Schwartz, Hard-Soft-Collinear Factorization to All Orders, Phys. Rev. D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
C.D. White, Factorization Properties of Soft Graviton Amplitudes, JHEP 05 (2011) 060 [arXiv:1103.2981] [INSPIRE].
L. Donnay and R. Ruzziconi, BMS flux algebra in celestial holography, JHEP 11 (2021) 040 [arXiv:2108.11969] [INSPIRE].
L. Donnay, K. Nguyen and R. Ruzziconi, Loop-corrected subleading soft theorem and the celestial stress tensor, JHEP 09 (2022) 063 [arXiv:2205.11477] [INSPIRE].
S. Agrawal, L. Donnay, K. Nguyen and R. Ruzziconi, Logarithmic soft graviton theorems from superrotation Ward identities, JHEP 02 (2024) 120 [arXiv:2309.11220] [INSPIRE].
Y. Hu and S. Pasterski, Celestial conformal colliders, JHEP 02 (2023) 243 [arXiv:2211.14287] [INSPIRE].
Y. Hu and S. Pasterski, Detector operators for celestial symmetries, JHEP 12 (2023) 035 [arXiv:2307.16801] [INSPIRE].
Acknowledgments
We are indebted to J. Bhambure, R. Bhardwaj, Y. Hu, B. Sahoo, G. Stermann, Y.P. Wang, for helpful discussions. We would like to especially thank R. Bhardwaj for going through the previous draft and pointing out errors. We would also like to thank G. Stermann for important comments on the draft. The work of HK is supported by NSF grant PHY 2210533.
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Krishna, H. Celestial gluon and graviton OPE at loop level. J. High Energ. Phys. 2024, 176 (2024). https://doi.org/10.1007/JHEP03(2024)176
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DOI: https://doi.org/10.1007/JHEP03(2024)176