Subleading soft graviton theorem for loop amplitudes

Ashoke Sen

doi:10.1007/jhep11(2017)123

Abstract

Superstring field theory gives expressions for heterotic and type II string loop amplitudes that are free from ultraviolet and infrared divergences when the number of non-compact space-time dimensions is five or more. We prove the subleading soft graviton theorem in these theories to all orders in perturbation theory for S-matrix elements of arbitrary number of finite energy external states but only one external soft graviton. We also prove the leading soft graviton theorem for arbitrary number of finite energy external states and arbitrary number of soft gravitons. Since our analysis is based on general properties of one particle irreducible effective action, the results are valid in any theory of quantum gravity that gives finite result for the S-matrix order by order in perturbation theory without violating general coordinate invariance.

Highlights

We prove the subleading soft graviton theorem in these theories to all orders in perturbation theory for S-matrix elements of arbitrary number of finite energy external states but only one external soft graviton
We prove the leading soft graviton theorem for arbitrary number of finite energy external states and arbitrary number of soft gravitons
Since our analysis is based on general properties of one particle irreducible effective action, the results are valid in any theory of quantum gravity that gives finite result for the S-matrix order by order in perturbation theory without violating general coordinate invariance

Summary

Subleading soft theorem for one external soft graviton

We shall prove the subleading soft graviton theorem for amplitudes with one external soft graviton, but arbitrary number of finite energy external states. Since we are interested in computing the leading contribution from this graph, we can ignore terms involving ωμab since they involve derivatives of Sμν and have one or more powers of soft momentum For this amplitude the effect of coupling the soft graviton can be obtained by replacing the vielbeins as in (2.2) with Sμν given by the constant polarization tensor εμν. This is the standard form of the subleading soft graviton theorem given in [6] with the term inside the curly bracket representing orbital angular momentum operator and Jρν representing the spin angular momentum operator It was shown in [14] that in certain theories the soft graviton amplitude is determined by the form of the on-shell three point vertex of one soft graviton and a pair of finite energy particles. Our analysis shows that the subleading soft graviton amplitude is completely determined in terms of momentum and spin of the external finite energy particles, and no further information on the three point function is required

Leading soft theorem for multiple soft gravitons

A Justification for dropping Christoffel symbols from covariant derivatives