Soft Theorems from Compactification

TL;DR

This paper derives soft theorems for gravitons, vectors, and scalars in lower dimensions via compactification, confirming their consistency through explicit amplitude calculations involving Kaluza-Klein modes.

Contribution

It provides a unified derivation of soft theorems in compactified theories and verifies them with explicit amplitude computations involving massive and massless states.

Findings

Soft theorems for graviton, vector, and scalar fields are derived in lower dimensions.

Explicit amplitude calculations confirm the soft theorem results for Kaluza-Klein modes.

Agreement is found when the gyromagnetic ratio is set to g=1 in the Fierz-Pauli Lagrangian.

Abstract

We analyze the single subleading soft graviton theorem in $(d+1)$ dimensions under compactification on $S^1$. This produces the single soft theorems for the graviton, vector and scalar fields in $d$ dimension. For the compactification of $11$-dimensional supergravity theory, this gives the soft factorization properties of the single graviton, dilaton and RR 1-form fields in type IIA string theory in ten dimensions. For the case of the soft vector field, we also explicitly check the result obtained from compactification by computing the amplitudes with external massive spin two and massless finite energy states interacting with soft vector field. The former are the Kaluza-Klein excitations of the $d+1$ dimensional metric. Describing the interaction of the KK-modes with the vector field at each level by the minimally coupled Fierz-Pauli Lagrangian, we find agreement with the results obtained from the compactification if the gyromagnetic ratio in the minimally coupled Fierz-Pauli Lagrangian is taken to be $g=1$.