Abstract
In this paper, we study loop corrections to the recently proposed new soft theorem of Cachazo-Strominger, for both gravity and gauge theory amplitudes. We first review the proof of its tree-level validity based on BCFW recursion relations, which also establishes an infinite series of universals soft functions for MHV amplitudes, and a generalization to supersymmetric cases. For loop corrections, we focus on infrared finite, rational amplitudes at one loop, and apply recursion relations with boundary or double-pole contributions. For all-plus amplitudes, we prove that the subleading soft-theorems are exact to all multiplicities for both gauge and gravity amplitudes. For single-minus amplitudes, while the subleading soft-theorems are again exact for the minus-helicity soft leg, for plus-helicity loop corrections are required. Using recursion relations, we identify the source of such mismatch as stemming from the special contribution containing double poles, and obtain the all-multiplicity one-loop corrections to the subleading soft behavior in Yang-Mills theory. We also comment on the derivation of soft theorems using BCFW recursion in arbitrary dimensions.
Highlights
The soft behavior in eq (1.1) has recently been understood as a consequence of Bondi, van der Burg, Metzner, and Sachs (BMS) symmetry [13,14,15,16,17], which is a large diffeomorphism transformations that translate between asymptotic flat solutions
In this paper, we study loop corrections to the recently proposed new soft theorems of Cachazo and Strominger [1], for both gravity and gauge theory amplitudes
We first review the proof of its tree-level validity based on BCFW recursion relations, which establishes an infinite series of universals soft functions for MHV amplitudes, and a generalization to supersymmetric cases
Summary
The Cachazo-Strominger tree-level soft theorem can be most derived via BCFW representation of tree amplitudes, where the singular part of the amplitude in the soft limit is given solely by two-particle factorization channels. The holomorphic soft limit is achieved by scaling λs → ǫλs, and using momentum conservation to solve two anti-holomorphic spinors (λa, λb) in terms of other λ’s. Color-ordered Yang-Mills amplitude has milder soft-divergence compared to gravity in the holomorphic soft limit, due to the milder little-group rescaling. The soft gluon theorem can be derived in a parallel fashion with gravity by using the BCFW representation of tree-level amplitudes, and the divergent term under the holomorphic soft limit is again isolated to the two particle channel indicated in figure 1(only one term i = 1 contributes because of the color ordering): An+1(1 , . The gravity soft factors can be expressed as a double copy of that of Yang-Mills theory, Ks2i. Mills amplitudes [36], it would be very interesting to see whether there is a deeper physical reason behind this double copy relation.
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