Massive spinning bosons on the celestial sphere
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Springer
Received: May 8, 2020 Accepted: May 22, 2020 Published: June 11, 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: A natural extension of the Pasterski-Shao-Strominger (PSS) prescription is described, enabling the map of Minkowski space amplitudes with massive spinning external legs to the celestial sphere to be performed. An integral representation for the conformal primary wave function (CPW) of massive spinning bosons on the celestial sphere is derived explicitly for spin-one and -two. By analogy with the spin-zero case, the spinning bulk-to-boundary propagator on Euclidean AdS is employed to extend the massive CPW integral representation to arbitrary integer spin, and to describe the appropriate inverse transform of massive spinning CPWs back to the plane wave basis in Minkowski space. Subsequently, a massive spin-s momentum operator representation on the celestial sphere is determined, and used in conjunction with known Lorentz generators to derive Poincar´e symmetry constraints on generic massive spinning two-, three- and four-point celestial amplitude structures. Finally, as a consistency check, three-point Minkowski space amplitudes of two massless scalars and a spin-one or -two massive boson are explicitly mapped to the celestial sphere, and the resulting three-point function coefficients are confirmed to be in exact agreement with the results obtained from Poincar´e symmetry constraints. Keywords: Scattering Amplitudes, Space-Time Symmetries ArXiv ePrint: 2004.04309
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)079
JHEP06(2020)079
Massive spinning bosons on the celestial sphere
Contents 1
2 Spinning massive conformal primary wave functions 2.1 On-shell massive and massless momentum parametrization in the bulk 2.2 Map of spinning massive amplitudes to the celestial sphere 2.3 Spin-zero, -one and -two massive conformal primary wave functions 2.4 Arbitrary integer spin massive conformal primary wave functions 2.5 Completeness of massive spinning conformal primary wave function basis
4 4 5 6 8 9
3 Spin-s massive momentum representation and Poincar´ e algebra
10
4 Poincar´ e constraints on massive spinning celestial amplitudes 4.1 Two-point structure 4.2 Three-point structure 4.3 Four-point structure
12 12 14 16
A Spin-one and -two massive polarization tensors
18
B Polynomial encoding of symmetric traceless transverse tensors review B.1 Tensors living on pˆ2 + 1 = 0 in R1,3 B.2 Tensors living on q 2 = 0 in R1,3 and projection onto the boundary
18 18 19
C Example spinning massive amplitudes mapped to the celestial sphere C.1 Celestial amplitude of two massless scalars and one massive spin-1 boson C.2 Celestial amplitude of two massless scalars and one massive spin-2 boson
22 23 24
1
Introduction
Minkowski space scattering amplitudes can be equivalently mapped
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