On Sugawara construction on celestial sphere
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Springer
Received: May 25, 2020 Accepted: August 10, 2020 Published: September 22, 2020
On Sugawara construction on celestial sphere
a
Department of Physics, College of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China b Department of Physics, Northeastern University, Boston, MA 02115, U.S.A. c Department of Sciences, Wentworth Institute of Technology, Boston, MA 02115, U.S.A. d Max-Planck-Institut f¨ ur Physik, Werner-Heisenberg-Institut, 80805 M¨ unchen, Germany
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Conformally soft gluons are conserved currents of the Celestial Conformal Field Theory (CCFT) and generate a Kac-Moody algebra. We study celestial amplitudes of Yang-Mills theory, which are Mellin transforms of gluon amplitudes and take the double soft limit of a pair of gluons. In this manner we construct the Sugawara energy-momentum tensor of the CCFT. We verify that conformally soft gauge bosons are Virasoro primaries of the CCFT under the Sugawara energy-momentum tensor. The Sugawara tensor though does not generate the correct conformal transformations for hard states. In Einstein-YangMills theory, we consider an alternative construction of the energy-momentum tensor, similar to the double copy construction which relates gauge theory amplitudes with gravity ones. This energy momentum tensor has the correct properties to generate conformal transformations for both soft and hard states. We extend this construction to supertranslations. Keywords: Conformal and W Symmetry, Scattering Amplitudes ArXiv ePrint: 2005.10666
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP09(2020)139
JHEP09(2020)139
Wei Fan,a Angelos Fotopoulos,b,c Stephan Stiebergerd and Tomasz R. Taylorb
Contents 1
2 Remarks on gluon operator products and Sugawara construction
3
3 Gluon amplitudes, gauge current insertion and operator products 3.1 Gluon amplitudes and operator products for SU(N ) 3.2 Gluon color sums and operator products for general gauge group
8 9 12
4 Mellin transform and the Sugawara energy-momentum tensor 4.1 Sugawara energy-momentum tensor and conformally soft gluons 4.2 Comments on the OPE of the Sugawara tensor for hard operators
17 17 21
5 The general gauge group using OPE and conformal soft limits of currents 23 6 Energy-momentum tensor from shadow transform and double copy 6.1 A double copy construction of the energy momentum tensor 6.2 A double copy construction for supertranslations
24 24 28
7 Conclusions
29
A Solution of the n-particle momentum-conservating delta function
30
B Seven-gluon NMHV amplitude and T S ¯ j OPE
31
C Sugawara OPE with soft shadow operators
33
1
Introduction
The null infinity of D = 4 asymptotically flat spacetime is the product of a conformal two-sphere (celestial sphere) CS 2 with a null line. It was realized some time ago, that for asymptotically flat space-times the Poincare group can be extended to the local BMS group. The loca
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