Soft theorems from effective field theory

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.

A complete basis of helicity operators for subleading factorization

“…For the operators involving one ultrasoft gluon and two collinear quarks, we have the basis g us (q) n (q)n : The helicity selection rules act different for the two projections of the B us fields due to the different definition of helicity in the two cases. The Wilson coefficients of the operators that include B us(n)0 can be related to the Wilson coefficients of the leading power operators using RPI symmetry (see [66]). In particular, we have…”

“…In [66] it was shown that in the general case of N jets, certain subleading operators involving B a us(n)± are generated by the RPI expansion of the leading power operator. However, these particular operators vanish for the case of back to back jets that we consider here.…”

“…It is also interesting to mention the physical interpretation of the vanishing of these contributions. As was discussed in [66] these derivative terms can be interpreted as the orbital angular momentum contribution to the tree level Low-Burnett-Kroll (LBK) theorem [114,115], which vanishes for back to back jets.…”

“…9 For the corresponding tree level matching computation, the necessary SCET diagram calculations were carried out explicitly in ref. [66] for an arbitrary current. For completeness we give the full theory and SCET results needed for the matching calculation in the case studied here.…”

“…SCET allows for a systematic power expansion at the level of the Lagrangian, and simplifies many aspects of factorization proofs [65]. SCET has been used to study power corrections at the level of the amplitude [66] and to derive factorization theorems for B decays using subleading power operators (eg. [19, 21, 22, 24-26, 31, 67-69]), where many interesting processes only start at subleading power.…”