Abstract
We study the celestial description of the O(N) sigma model in the large N limit as introduced by Coleman, Jackiw and Politzer. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. We end with some topics for future study.
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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].
C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP 01 (2017) 112 [arXiv:1609.00732] [INSPIRE].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
S. Pasterski, Lectures on celestial amplitudes, Eur. Phys. J. C 81 (2021) 1062 [arXiv:2108.04801] [INSPIRE].
S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in 2022 Snowmass Summer Study, (2021) [arXiv:2111.11392] [INSPIRE].
H.T. Lam and S.-H. Shao, Conformal Basis, Optical Theorem, and the Bulk Point Singularity, Phys. Rev. D 98 (2018) 025020 [arXiv:1711.06138] [INSPIRE].
D. Nandan, A. Schreiber, A. Volovich and M. Zlotnikov, Celestial Amplitudes: Conformal Partial Waves and Soft Limits, JHEP 10 (2019) 018 [arXiv:1904.10940] [INSPIRE].
C.-M. Chang, Y.-t. Huang, Z.-X. Huang and W. Li, Bulk locality from the celestial amplitude, SciPost Phys. 12 (2022) 176 [arXiv:2106.11948] [INSPIRE].
W. Melton, Celestial Feynman Rules for Scalars, arXiv:2109.07462 [INSPIRE].
W. Fan, A. Fotopoulos, S. Stieberger, T.R. Taylor and B. Zhu, Conformal blocks from celestial gluon amplitudes, JHEP 05 (2021) 170 [arXiv:2103.04420] [INSPIRE].
W. Fan, A. Fotopoulos, S. Stieberger, T.R. Taylor and B. Zhu, Conformal blocks from celestial gluon amplitudes. Part II. Single-valued correlators, JHEP 11 (2021) 179 [arXiv:2108.10337] [INSPIRE].
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. Guevara, Celestial OPE blocks, arXiv:2108.12706 [INSPIRE].
S. Stieberger and T.R. Taylor, Symmetries of Celestial Amplitudes, Phys. Lett. B 793 (2019) 141 [arXiv:1812.01080] [INSPIRE].
Y.T.A. Law and M. Zlotnikov, Poincaré constraints on celestial amplitudes, JHEP 03 (2020) 085 [Erratum ibid. 04 (2020) 202] [arXiv:1910.04356] [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].
A. Fotopoulos and T.R. Taylor, Primary Fields in Celestial CFT, JHEP 10 (2019) 167 [arXiv:1906.10149] [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].
A. Fotopoulos, S. Stieberger, T.R. Taylor and B. Zhu, Extended Super BMS Algebra of Celestial CFT, JHEP 09 (2020) 198 [arXiv:2007.03785] [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].
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].
S.-H. Shao, Celestial Free Fields, https://www.youtube.com/watch?v=IlmiP8rsS2A&t=750s, from minute 12:45.
M.J.G. Veltman, Unitarity and causality in a renormalizable field theory with unstable particles, Physica 29 (1963) 186 [INSPIRE].
A. Denner and J.-N. Lang, The Complex-Mass Scheme and Unitarity in perturbative Quantum Field Theory, Eur. Phys. J. C 75 (2015) 377 [arXiv:1406.6280] [INSPIRE].
J.F. Donoghue and G. Menezes, Unitarity, stability and loops of unstable ghosts, Phys. Rev. D 100 (2019) 105006 [arXiv:1908.02416] [INSPIRE].
G. Menezes, Generalized unitarity method for unstable particles, arXiv:2111.11570 [INSPIRE].
H.S. Hannesdottir and S. Mizera, What is the iε for the S-matrix?, arXiv:2204.02988 [INSPIRE].
S.R. Coleman, R. Jackiw and H.D. Politzer, Spontaneous Symmetry Breaking in the O(N) Model for Large N*, Phys. Rev. D 10 (1974) 2491 [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].
D. Carmi, L. Di Pietro and S. Komatsu, A Study of Quantum Field Theories in AdS at Finite Coupling, JHEP 01 (2019) 200 [arXiv:1810.04185] [INSPIRE].
S.L. Adler, Consistency conditions on the strong interactions implied by a partially conserved axial vector current, Phys. Rev. 137 (1965) B1022 [INSPIRE].
J. de Boer and S.N. Solodukhin, A Holographic reduction of Minkowski space-time, Nucl. Phys. B 665 (2003) 545 [hep-th/0303006] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429 [INSPIRE].
G. Mack and A. Salam, Finite component field representations of the conformal group, Annals Phys. 53 (1969) 174 [INSPIRE].
S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows, and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
Y.T.A. Law and M. Zlotnikov, Massive Spinning Bosons on the Celestial Sphere, JHEP 06 (2020) 079 [arXiv:2004.04309] [INSPIRE].
Y.T.A. Law and M. Zlotnikov, Relativistic partial waves for celestial amplitudes, JHEP 11 (2020) 149 [arXiv:2008.02331] [INSPIRE].
S. Mizera and S. Pasterski, Celestial geometry, JHEP 09 (2022) 045 [arXiv:2204.02505] [INSPIRE].
H. Bateman, Tables of integral transforms [volumes I & II], vol. 1, McGraw-Hill Book Company (1954), https://resolver.caltech.edu/CaltechAUTHORS:20140123-101456353.
D. Mazáč, A Crossing-Symmetric OPE Inversion Formula, JHEP 06 (2019) 082 [arXiv:1812.02254] [INSPIRE].
S. Pasterski, A. Puhm and E. Trevisani, Revisiting the conformally soft sector with celestial diamonds, JHEP 11 (2021) 143 [arXiv:2105.09792] [INSPIRE].
L. Donnay, S. Pasterski and A. Puhm, Goldilocks modes and the three scattering bases, JHEP 06 (2022) 124 [arXiv:2202.11127] [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes, arXiv:1308.1697 [INSPIRE].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
W. Bailey, Generalized Hypergeometric Series, in Cambridge tracts in mathematics and mathematical physics, The University Press (1935).
J. Maldacena and D. Stanford, Remarks on the Sachdev-Ye-Kitaev model, Phys. Rev. D 94 (2016) 106002 [arXiv:1604.07818] [INSPIRE].
D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].
S. Pasterski and H. Verlinde, Mapping SYK to the sky, JHEP 09 (2022) 047 [arXiv:2201.05054] [INSPIRE].
A. Kar, L. Lamprou, C. Marteau and F. Rosso, A Celestial Matrix Model, arXiv:2205.02240 [INSPIRE].
S. Jain, M. Mandlik, S. Minwalla, T. Takimi, S.R. Wadia and S. Yokoyama, Unitarity, Crossing Symmetry and Duality of the S-matrix in large N Chern-Simons theories with fundamental matter, JHEP 04 (2015) 129 [arXiv:1404.6373] [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].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D ≥ 3 Dimensions, SpringerBriefs in Physics, Springer (2016).
D. Simmons-Duffin, The Conformal Bootstrap, in Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, (2017), pp. 1–74, https://doi.org/10.1142/9789813149441_0001 [arXiv:1602.07982] [INSPIRE].
J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
A. Gadde, In search of conformal theories, arXiv:1702.07362 [INSPIRE].
M. Hogervorst, J. Penedones and K.S. Vaziri, Towards the non-perturbative cosmological bootstrap, arXiv:2107.13871 [INSPIRE].
K. Bulycheva, A note on the SYK model with complex fermions, JHEP 12 (2017) 069 [arXiv:1706.07411] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
P. Ferrero, K. Ghosh, A. Sinha and A. Zahed, Crossing symmetry, transcendentality and the Regge behaviour of 1d CFTs, JHEP 07 (2020) 170 [arXiv:1911.12388] [INSPIRE].
Y.-C. He, J. Rong and N. Su, Non-Wilson-Fisher kinks of O(N) numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs, SciPost Phys. 10 (2021) 115 [arXiv:2005.04250] [INSPIRE].
A. Bissi, A. Sinha and X. Zhou, Selected Topics in Analytic Conformal Bootstrap: A Guided Journey, arXiv:2202.08475 [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
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García-Sepúlveda, D., Guevara, A., Kulp, J. et al. Notes on resonances and unitarity from celestial amplitudes. J. High Energ. Phys. 2022, 245 (2022). https://doi.org/10.1007/JHEP09(2022)245
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DOI: https://doi.org/10.1007/JHEP09(2022)245