Abstract
We analyze the low-energy behavior of scattering amplitudes involving gravitons at loop level in four dimensions. The single-graviton soft limit is controlled by soft operators which have been argued to separate into a factorized piece and a non-factorizing infrared divergent contribution. In this note we show that the soft operators responsible for the factorized contributions are strongly constrained by gauge and Poincaré invariance under the assumption of a local structure. We show that the leading and subleading orders in the soft-momentum expansion cannot receive radiative corrections. The first radiative correction occurs for the sub-subleading soft graviton operator and is one-loop exact. It depends on only two undetermined coefficients which should reflect the field content of the theory under consideration.
Highlights
The low energy behavior of graviton scattering amplitudes with a single soft graviton momentum has been related [1] to Ward identities of the extended Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [2]
The single-graviton soft limit is controlled by soft operators which have been argued to separate into a factorized piece and a non-factorizing infrared divergent contribution
In distinction to the leading Weinberg pole, which is a function of the soft and hard momenta and the soft graviton polarization, the suband sub-subleading soft behavior of the (n + 1)-point graviton amplitude can be expressed in terms of differential operators in the
Summary
The low energy behavior of graviton scattering amplitudes with a single soft graviton momentum has been related [1] to Ward identities of the extended Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [2]. Starting again from on-shell gauge invariance, factorization and Poincaré symmetry, we adapt the formalism to the appropriate mass dimensions at different loop levels and identify the restricted class of operators which can appear at leading, subleading and sub-subleading order. The undetermined coefficient depends on the matter content of the gauge theory in question, whereas the tensorial form of the subleading soft-gluon operator is universal It is precisely at this order where one first encounters the non-local parts mentioned above: As was analyzed carefully in Section 3.2 of [16] the single-minus one-loop (n + 1)-gluon amplitude with a soft leg of positive helicity develops a factorized non-local subleading soft pole, whose corresponding operator depends on next-to-nearest neighboring hard legs of the soft leg. Our formalism is unable to capture this behavior at present
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