Abstract
The soft factorization theorem for 4D abelian gauge theory states that the \( \mathcal{S} \)-matrix factorizes into soft and hard parts, with the universal soft part containing all soft and collinear poles. Similarly, correlation functions on the sphere in a 2D CFT with a U(1) Kac-Moody current algebra factorize into current algebra and non-current algebra factors, with the current algebra factor fully determined by its pole structure. In this paper, we show that these 4D and 2D factorizations are mathematically the same phenomena. The soft ‘t Hooft-Wilson lines and soft photons are realized as a complexified 2D current algebra on the celestial sphere at null infinity. The current algebra level is determined by the cusp anomalous dimension. The associated complex U(1) boson lives on a torus whose modular parameter is \( \tau =\frac{2\pi i}{e^2}+\frac{\theta }{2\pi } \). The correlators of this 2D current algebra fully reproduce the known soft part of the 4D \( \mathcal{S} \)-matrix, as well as a conjectured generalization involving magnetic charges.
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References
T. He, P. Mitra and A. Strominger, 2D Kac-Moody symmetry of 4D Yang-Mills theory, JHEP 10 (2016) 137 [arXiv:1503.02663] [INSPIRE].
M. Campiglia and A. Laddha, Asymptotic symmetries of QED and Weinberg’s soft photon theorem, JHEP 07 (2015) 115 [arXiv:1505.05346] [INSPIRE].
M. Campiglia, Null to time-like infinity Green’s functions for asymptotic symmetries in Minkowski spacetime, JHEP 11 (2015) 160 [arXiv:1509.01408] [INSPIRE].
D. Kapec, M. Pate and A. Strominger, New symmetries of QED, arXiv:1506.02906 [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].
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].
A. Strominger, Asymptotic symmetries of Yang-Mills theory, JHEP 07 (2014) 151 [arXiv:1308.0589] [INSPIRE].
A. Bassetto, M. Ciafaloni and G. Marchesini, Jet structure and infrared sensitive quantities in perturbative QCD, Phys. Rept. 100 (1983) 201 [INSPIRE].
I. Feige and M.D. Schwartz, Hard-soft-collinear factorization to all orders, Phys. Rev. D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
M. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press, Cambridge U.K., (2014) [ISBN:9781107034730] [INSPIRE].
A. Strominger, Magnetic corrections to the soft photon theorem, Phys. Rev. Lett. 116 (2016) 031602 [arXiv:1509.00543] [INSPIRE].
A. Strominger, Lectures on the infrared structure of gravity and gauge theory, arXiv:1703.05448 [INSPIRE].
T. He, P. Mitra, A.P. Porfyriadis and A. Strominger, New symmetries of massless QED, JHEP 10 (2014) 112 [arXiv:1407.3789] [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].
Y.-T. Chien, M.D. Schwartz, D. Simmons-Duffin and I.W. Stewart, Jet physics from static charges in AdS, Phys. Rev. D 85 (2012) 045010 [arXiv:1109.6010] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [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].
F.E. Low, Scattering of light of very low frequency by systems of spin 1/2, Phys. Rev. 96 (1954) 1428 [INSPIRE].
F.E. Low, Bremsstrahlung of very low-energy quanta in elementary particle collisions, Phys. Rev. 110 (1958) 974 [INSPIRE].
S. Weinberg, Infrared photons and gravitons, Phys. Rev. 140 (1965) B516 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
A.V. Manohar, Deep inelastic scattering as x → 1 using soft collinear effective theory, Phys. Rev. D 68 (2003) 114019 [hep-ph/0309176] [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 and A. Strominger, Asymptotic symmetries of massless QED in even dimensions, arXiv:1412.2763 [INSPIRE].
A. Mohd, A note on asymptotic symmetries and soft-photon theorem, JHEP 02 (2015) 060 [arXiv:1412.5365] [INSPIRE].
S.G. Avery and B.U.W. Schwab, Noether’s second theorem and Ward identities for gauge symmetries, JHEP 02 (2016) 031 [arXiv:1510.07038] [INSPIRE].
K. Colwell and J. Terning, S-duality and helicity amplitudes, JHEP 03 (2016) 068 [arXiv:1510.07627] [INSPIRE].
M. Campiglia and A. Laddha, Subleading soft photons and large gauge transformations, JHEP 11 (2016) 012 [arXiv:1605.09677] [INSPIRE].
E. Conde and P. Mao, Remarks on asymptotic symmetries and the subleading soft photon theorem, Phys. Rev. D 95 (2017) 021701 [arXiv:1605.09731] [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].
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Nande, A., Pate, M. & Strominger, A. Soft factorization in QED from 2D Kac-Moody symmetry. J. High Energ. Phys. 2018, 79 (2018). https://doi.org/10.1007/JHEP02(2018)079
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DOI: https://doi.org/10.1007/JHEP02(2018)079