More on Soft Theorems: Trees, Loops and Strings

TL;DR

This paper investigates the universality and validity of soft theorems across various theories, including gauge theories, gravity, and string theory, at tree and loop levels, revealing conditions under which they hold or require modifications.

Contribution

It demonstrates that planar integrands in N=4 SYM satisfy soft theorems to all orders, and explores the modifications needed for gravity and string theories, highlighting the role of anomalies and interactions.

Findings

Planar integrand for N=4 SYM satisfies tree-level soft theorems at all orders.

Soft theorems hold for non-supersymmetric Yang-Mills and gravity at the integrand level.

Superstring amplitudes obey the same soft theorems as field theory, unlike bosonic strings.

Abstract

We study soft theorems in a broader context, addressing their fate at loop level and their universality in effective field theories and string theory. We argue that for gauge theories in the planar limit, loop-level soft gluon theorems can be made manifest already at the integrand level. In particular, we show that the planar integrand for N=4 SYM satisfies the tree-level soft theorem to all orders in perturbation theory and provide strong evidence to this effect for integrands in N<4 SYM. We consider soft theorems for non-supersymmetric Yang-Mills theories and gravity, and show the validity of integrand soft theorem, while loop corrections to the integrated soft theorems are intimately tied to the presence of conformal anomalies. We then address the question of universality of the soft theorems for various theories. In effective field theories with F^3 and R^3 interactions, the soft theorems are not modified. However for gravity theories with R^2 phi interactions, the sub-sub-leading order soft graviton theorem, which is beyond what is implied by the extended BMS symmetry, requires modifications at tree level for non-supersymmetric theories, and at loop level for N<5 supergravity due to anomalies. Finally, for superstring amplitudes at finite alpha', via explicit calculation for lower-point examples as well as world-sheet OPE analysis for arbitrary multiplicity, we show that the superstring amplitudes satisfy the same soft theorem as its field-theory counterpart. This is no longer true for bosonic closed strings due to the presence of R^2 phi interactions.