Abstract
The singular limits of massless gauge theory amplitudes are described by an effective theory, called soft-collinear effective theory (SCET), which has been applied most successfully to make all-orders predictions for observables in collider physics and weak decays. At tree-level, the emission of a soft gauge boson at subleading order in its energy is given by the Low-Burnett-Kroll theorem, with the angular momentum operator acting on a lower-point amplitude. For well separated particles at tree-level, we prove the Low-Burnett-Kroll theorem using matrix elements of subleading SCET Lagrangian and operator insertions which are individually gauge invariant. These contributions are uniquely determined by gauge invariance and the reparametrization invariance (RPI) symmetry of SCET. RPI in SCET is connected to the infinite-dimensional asymptotic symmetries of the S-matrix. The Low-Burnett-Kroll theorem is generically spoiled by on-shell corrections, including collinear loops and collinear emissions. We demonstrate this explicitly both at tree-level and at one-loop. The effective theory correctly describes these configurations, and we generalize the Low-Burnett-Kroll theorem into a new one-loop subleading soft theorem for amplitudes. Our analysis is presented in a manner that illustrates the wider utility of using effective theory techniques to understand the perturbative S-matrix.
Highlights
The modern study of the perturbative S-matrix is a mature field which traces its roots to the Parke-Taylor formula for maximally-helicity-violating (MHV) amplitudes [1] in the 1980s, the unitarity methods of refs. [2, 3] in the early 1990s and the identification of perturbative gauge theory as a string theory in twistor space [4] in the early 2000s
At tree level we show that the subleading LBK soft factor S(sub)(s) in gauge theory is reproduced by matrix elements involving the subleading gauge invariant soft-collinear effective theory (SCET) Lagrangian and operators
We present the calculation that shows how these operators arise in appendix C, including the demonstration that they all are uniquely determined by the reparametrization invariance (RPI) symmetry at this order
Summary
The modern study of the perturbative S-matrix is a mature field which traces its roots to the Parke-Taylor formula for maximally-helicity-violating (MHV) amplitudes [1] in the 1980s, the unitarity methods of refs. [2, 3] in the early 1990s and the identification of perturbative gauge theory as a string theory in twistor space [4] in the early 2000s. The corrections encoded in this soft theorem come from the region of the loop integral in which the loop momenta is collinear to external particles, and this situation violates the assumptions required in deriving the simpler Low-Burnett-Kroll result. We derive a soft theorem for real emission graphs containing two collinear particles that are not well-separated in phase space The result includes both a direct emission contribution and an amplitude coupling to the soft limit of the 1 → 3 splitting amplitude.
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