Abstract
We explore a formulation of the S-matrix in terms of the path integral with specified asymptotic data, as originally proposed by Arefeva, Faddeev, and Slavnov. In the tree approximation the S-matrix is equal to the exponential of the classical action evaluated on-shell. This formulation is well-suited to questions involving asymptotic symmetries, as it avoids reference to non-gauge/diffeomorphism invariant bulk correlators or sources at intermediate stages. We show that the soft photon theorem, originally derived by Weinberg and more recently connected to asymptotic symmetries by Strominger and collaborators, follows rather simply from invariance of the action under large gauge transformations applied to the asymptotic data. We also show that this formalism allows for efficient computation of the S-matrix in curved spacetime, including particle production due to a time dependent metric.
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S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
R. Balian and J. Zinn-Justin, Methods in Field Theory. Les Houches Summer School in Theoretical Physics, Session 28, July 28–September 6, 1975 (1976) [INSPIRE].
L.D. Faddeev and A.A. Slavnov, Gauge fields. Introduction to quantum theory, vol. 50, CRC Press (1993).
P. Deligne et al., Quantum fields and strings: A course for mathematicians. Vol. 1, 2, American Mathematical Society (1999) [INSPIRE].
J.E. Shrauner, C.L. Hammer and B. DeFacio, Path Integral Representation of S Matrix, Phys. Rev. D 18 (1978) 373 [INSPIRE].
A. Jevicki and C.-K. Lee, The S Matrix Generating Functional and Effective Action, Phys. Rev. D 37 (1988) 1485 [INSPIRE].
T. Adamo, S. Nakach and A.A. Tseytlin, Scattering of conformal higher spin fields, JHEP 07 (2018) 016 [arXiv:1805.00394] [INSPIRE].
T. Adamo, E. Casali, L. Mason and S. Nekovar, Scattering on plane waves and the double copy, Class. Quant. Grav. 35 (2018) 015004 [arXiv:1706.08925] [INSPIRE].
T. Adamo, A. Cristofoli and P. Tourkine, Eikonal amplitudes from curved backgrounds, SciPost Phys. 13 (2022) 032 [arXiv:2112.09113] [INSPIRE].
R. Gonzo, T. McLoughlin and A. Puhm, Celestial holography on Kerr-Schild backgrounds, JHEP 10 (2022) 073 [arXiv:2207.13719] [INSPIRE].
I.Y. Arefeva, L.D. Faddeev and A.A. Slavnov, Generating Functional for the s Matrix in Gauge Theories, Teor. Mat. Fiz. 21 (1974) 311 [INSPIRE].
T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New Symmetries of Massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of QED and Weinberg’s soft photon theorem, JHEP 07 (2015) 115 [arXiv:1505.05346] [INSPIRE].
D. Kapec, M. Pate and A. Strominger, New Symmetries of QED, Adv. Theor. Math. Phys. 21 (2017) 1769 [arXiv:1506.02906] [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. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP 11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
M. Campiglia and R. Eyheralde, Asymptotic U(1) charges at spatial infinity, JHEP 11 (2017) 168 [arXiv:1703.07884] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic charges in massless QED revisited: A view from Spatial Infinity, JHEP 05 (2019) 207 [arXiv:1810.04619] [INSPIRE].
N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space, Cambridge University Press, Cambridge, U.K. (1984) [https://doi.org/10.1017/CBO9780511622632] [INSPIRE].
B.S. DeWitt, Quantum Field Theory in Curved Space-Time, Phys. Rept. 19 (1975) 295 [INSPIRE].
S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations, Cambridge University Press (2005) [https://doi.org/10.1017/CBO9781139644167] [INSPIRE].
I. Papadimitriou, Multi-Trace Deformations in AdS/CFT: Exploring the Vacuum Structure of the Deformed CFT, JHEP 05 (2007) 075 [hep-th/0703152] [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].
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory, arXiv:1703.05448 [INSPIRE].
N. Miller, From Noether’s Theorem to Bremsstrahlung: a pedagogical introduction to large gauge transformations and classical soft theorems, arXiv:2112.05289 [INSPIRE].
S. Kim, P. Kraus and R.M. Myers, Systematics of boundary actions in gauge theory and gravity, JHEP 04 (2023) 121 [arXiv:2301.02964] [INSPIRE].
J. Preskill, Quantum field theory in curved spacetime, http://theory.caltech.edu/~preskill/notes.html (1990).
A. Strominger, Asymptotic Symmetries of Yang-Mills Theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
T. He and P. Mitra, Covariant Phase Space and Soft Factorization in Non-Abelian Gauge Theories, JHEP 03 (2021) 015 [arXiv:2009.14334] [INSPIRE].
A. Laddha and P. Mitra, Asymptotic Symmetries and Subleading Soft Photon Theorem in Effective Field Theories, JHEP 05 (2018) 132 [arXiv:1709.03850] [INSPIRE].
E. Himwich et al., The Soft \( \mathcal{S} \)-Matrix in Gravity, JHEP 09 (2020) 129 [arXiv:2005.13433] [INSPIRE].
K. Nguyen, A. Rios Fukelman and C.D. White, Celestial soft dressings from generalised Wilson lines, arXiv:2304.01250 [INSPIRE].
S. Choi and R. Akhoury, BMS Supertranslation Symmetry Implies Faddeev-Kulish Amplitudes, JHEP 02 (2018) 171 [arXiv:1712.04551] [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].
A.-M. Raclariu, Lectures on Celestial Holography, arXiv:2107.02075 [INSPIRE].
A.B. Prema et al., Celestial holography: Lectures on asymptotic symmetries, SciPost Phys. Lect. Notes 47 (2022) 1 [arXiv:2109.00997] [INSPIRE].
S. Pasterski, M. Pate and A.-M. Raclariu, Celestial Holography, in the proceedings of the Snowmass 2021, (2021) [arXiv:2111.11392] [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].
J. Schwinger, The Theory of Quantized Fields. III, Phys. Rev. 91 (1953) 728 [INSPIRE].
K. Prabhu, G. Satishchandran and R.M. Wald, Infrared finite scattering theory in quantum field theory and quantum gravity, Phys. Rev. D 106 (2022) 066005 [arXiv:2203.14334] [INSPIRE].
J.D. Bjorken and S.D. Drell, Relativistic quantum fields [INSPIRE].
S. Atul Bhatkar, Asymptotic conservation law with Feynman boundary condition, Phys. Rev. D 103 (2021) 125026 [arXiv:2101.09734] [INSPIRE].
Acknowledgments
We thank Thomas Dumitrescu, Temple He, Enrico Herrmann, Andrea Puhm, Trevor Scheopner, and Andy Strominger for discussions. P.K. and R.M. are supported in part by the National Science Foundation grant PHY-2209700.
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Kim, S., Kraus, P., Monten, R. et al. S-matrix path integral approach to symmetries and soft theorems. J. High Energ. Phys. 2023, 36 (2023). https://doi.org/10.1007/JHEP10(2023)036
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DOI: https://doi.org/10.1007/JHEP10(2023)036