Infinite soft theorems from gauge symmetry

Zhi-Zhong Li,Hung-Hwa Lin,Shun-Qing Zhang

doi:10.1103/physrevd.98.045004

Zhi-Zhong Li, Hung-Hwa Lin + Show 1 more

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https://doi.org/10.1103/physrevd.98.045004

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Abstract

In this letter we show that the soft behaviour of photons and graviton amplitudes, after projection, can be determined to infinite order in soft expansion via ordinary on-shell gauge invariance. In particular, as one of the particle's momenta becomes soft, gauge invariance relates the non-singular diagrams of an n-point amplitude to that of the singular ones up to possible homogeneous terms. We demonstrate that with a particular projection of the soft-limit, the homogeneous terms do not contribute, and one arrives at an infinite soft theorem. This reproduces the result recently derived from the Ward identity of large gauge transformations. We also discuss the modification of these soft theorems due to the presence of higher-dimensional operators.

Highlights

It has long been known that on-shell gauge invariance can be utilized to obtain universal soft behaviours of scattering amplitudes for photons and gravitons
Taking one of the momenta of an n-point amplitude (Mn) to be soft, the gauge invariance of the soft leg relates the finite part of the amplitude to the singular diagrams, which is given by the product of a threepoint vertex and the n−1-point amplitude
We demonstrate how on-shell gauge invariance can fix higher order soft limit of photons and gravitons up to an undetermined homogeneous term Rμ

Summary

INTRODUCTION

It has long been known that on-shell gauge invariance can be utilized to obtain universal soft behaviours of scattering amplitudes for photons and gravitons. A few years ago, Strominger and collaborators [5, 6] demonstrated that the soft-theorems for gravitons can alternatively be interpreted as a consequence of extended Bondi, van der Burg, Metzner and Sachs (BMS) symmetry [7, 8] This generated new interest in soft-theorems of amplitudes and its relationship with underlying symmetry. The infinite order soft-theorem derived in [16] is precisely the part of the amplitude that are completely determined by ordinary gauge symmetry. We will demonstrate this for photon and gravitons.

SOFT THEOREM FROM WARD IDENTITY

Photon Soft Theorem

Graviton

EXAMPLE OF HOMOGENEOUS TERMS

EFFECT OF HIGHER DIMENSIONAL OPERATORS

CONCLUSION AND DISCUSSION