Relativistic partial waves for celestial amplitudes
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Springer
Received: August Revised: October Accepted: October Published: November
24, 10, 26, 27,
2020 2020 2020 2020
Y.T. Albert Law and Michael Zlotnikov Department of Physics, Center for Theoretical Physics, Columbia University, 538 West 120th Street, New York, NY 10027, U.S.A.
E-mail: [email protected], [email protected] Abstract: The formalism of relativistic partial wave expansion is developed for four-point celestial amplitudes of massless external particles. In particular, relativistic partial waves are found as eigenfunctions to the product representation of celestial Poincaré Casimir operators with appropriate eigenvalues. The requirement of hermiticity of Casimir operators is used to fix the corresponding integral inner product, and orthogonality of the obtained relativistic partial waves is verified explicitly. The completeness relation, as well as the relativistic partial wave expansion follow. Example celestial amplitudes of scalars, gluons, gravitons and open superstring gluons are expanded on the basis of relativistic partial waves for demonstration. A connection with the formulation of relativistic partial waves in the bulk of Minkowski space is made in appendices. Keywords: Scattering Amplitudes, Space-Time Symmetries ArXiv ePrint: 2008.02331
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)149
JHEP11(2020)149
Relativistic partial waves for celestial amplitudes
Contents 1
2 Preliminaries 2.1 Kinematic definitions and amplitude map to the celestial sphere 2.2 Poincaré algebra and generator representations on the celestial sphere
3 3 4
3 Choice of representation theory: ∆, J vs. m, s
4
4 Relativistic partial waves 4.1 Casimir operators and characteristic numbers 4.2 Relativistic partial waves as solutions to Casimir differential equations 4.3 Inner product 4.4 Orthogonality and completeness of relativistic partial waves 4.5 Relativistic partial wave expansion
6 6 7 11 12 14
5 Relativistic partial wave expansion examples 5.1 Scalar celestial amplitude 5.2 Gluon celestial amplitude 5.3 Graviton celestial amplitude 5.4 Open superstring gluon celestial amplitude
14 15 17 19 20
A Bulk Minkowski space representation of relativistic partial waves
22
B Relativistic partial waves from two three-point amplitudes
22
1
Introduction
Celestial amplitudes describe quantum field theoretic scattering on the celestial sphere at light-like infinity as opposed to the familiar scattering amplitudes in Minkowski space. The focus on the two-dimensional celestial sphere as the underlying space in this description of four-dimensional scattering leads to a notion of holography [1]. It is of interest to determine so called Celestial Conformal Field Theories (CCFT) living on the celestial sphere, which would give rise to celestial amplitudes without reference to their Minkowski space counterparts. Furthermore, this boundary parametrization is particularly convenient, since it equally well can be used to describe scattering in asymptotically flat space-time
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