Infinite Set of Soft Theorems in Gauge-Gravity Theories as Ward-Takahashi Identities

TL;DR

It is shown that the soft photon, gluon, and graviton theorems can be understood as the Ward-Takahashi identities of large gauge transformation, i.e., diffeomorphism that does not fall off at spatial infinity.

Abstract

We show that the soft photon, gluon and graviton theorems can be understood as the Ward-Takahashi identities of large gauge transformation, i.e., diffeomorphism that does not fall off at spatial infinity. We found infinitely many new identities which constrain the higher order soft behavior of the gauge bosons and gravitons in scattering amplitudes of gauge and gravity theories. Diagrammatic representations of these soft theorems are presented.

Figures (5)

• Figure 1: The Feynman diagrams which contribute to the LHS of the WT identity. The black dot represents the insertion of the current $J_0$. There are infinitely many diagrams where $n$-legs attach with $J_0$, and we show the first three diagrams here. Since the intermediate state should be a Nambu-Goldstone (NG) one particle state, only the first diagram gives contribution at tree level.
• Figure 2: Diagrammatic representation of Eq. \ref{['Eq:soft photon two-point']}.
• Figure 3: Schematic diagrammatic representation of the soft photon theorem, Eq. \ref{['Eq:soft photon result']}.
• Figure 4: Schematic diagrammatic representation of the soft gluon theorem, Eq. \ref{['Eq:soft gluon result']}.
• Figure 5: Schematic diagrammatic representation of the soft graviton theorem, Eq. \ref{['Eq:soft graviton result']}. The wavy line represents the graviton.