Soft Theorems from Conformal Field Theory

TL;DR

This work investigates soft theorems for gauge and gravitational amplitudes through the lens of ambitwistor string theory, showing that soft limits act as Ward identities of a 2d CFT at null infinity. By expanding soft vertex operators on the ambitwistor string, the authors derive an infinite series of soft theorems that are valid to subleading order in Yang–Mills and to sub-subleading order in gravity, with the worldsheet charges organizing these terms. They reveal a braiding structure that encodes the algebra of soft limits and expose a simple mapping between soft gluon and graviton charges, hinting at a color–kinematics–like duality. The paper also extends to genus-one, computing the infrared-divergent 1-loop correction to the subleading soft graviton theorem, supporting the view that ambitwistor strings reproduce field-theory loop amplitudes in the infrared. Collectively, the results reinforce the interpretation of soft theorems as symmetry constraints of a 2d CFT at null infinity and demonstrate the power of ambitwistor strings as a calculational framework for both tree and loop amplitudes.

Abstract

Strominger and collaborators recently proposed that soft theorems for gauge and gravity amplitudes can be interpreted as Ward identities of a 2d CFT at null infinity. In this paper, we will consider a specific realization of this CFT known as ambitwistor string theory, which describes 4d Yang-Mills and gravity with any amount of supersymmetry. Using 4d ambitwistor string theory, we derive soft theorems in the form of an infinite series in the soft momentum which are valid to subleading order in gauge theory and sub-subleading order in gravity. Furthermore, we describe how the algebra of soft limits can be encoded in the braiding of soft vertex operators on the worldsheet and point out a simple relation between soft gluon and soft graviton vertex operators which suggests an interesting connection to color-kinematics duality. Finally, by considering ambitwistor string theory on a genus one worldsheet, we compute the 1-loop correction to the subleading soft graviton theorem due to infrared divergences.

Figures (5)

• Figure 1: If particle $i$ goes soft in a color-ordered Yang-Mills amplitude, the soft theorem follows from integrating its vertex operator around the vertex operators for particles $i-1$ and $i+1$, as depicted above for a genus zero worldsheet.
• Figure 2: The soft graviton theorem follows from integrating a soft graviton vertex operator around each of the hard vertex operators, as depicted above for a genus zero worldsheet.
• Figure 3: The commutator of two soft graviton limits from the point of view of CFT. In the left diagram, particle $n-1$ goes soft before particle $n$, and in the right diagram, particle $n$ goes soft before particle $n-1$. The bulk contributions correspond to points with two circles around them and the boundary contributions correspond to points with one circle around them.
• Figure 4: The commutator of two soft limits can be encoded by braiding one soft vertex operator around another.
• Figure 5: A non-separating degeneration of a toroidal worldsheet gives rise to a spherical worldsheet with two additional punctures.