Subleading Soft Graviton Theorem for Loop Amplitudes

TL;DR

This work proves the subleading soft graviton theorem to all orders for loop amplitudes with any number of finite-energy external states and a single soft graviton, and the leading soft graviton theorem for any number of soft gravitons, within heterotic and type II string field theories in backgrounds with $D\ge 5$. The key technique is a covariantization of the Lorentz-gauge fixed 1PI effective action with respect to a soft graviton, using $e_\mu^{\ a}=\delta_\mu^{\ a}+S_\mu^{\ a}$ and flat tangent-space indices, so that soft emissions can be captured by tree-level diagrams. The subleading theorem features a leading $1/(p_i\cdot k)$ pole proportional to $\varepsilon_{\mu\nu} p_i^\mu p_i^\nu$ acting on the hard amplitude, plus subleading terms involving derivatives with respect to external momenta and angular-momentum generators $J^{ab}$; the leading multi-soft result generalizes to $m$ soft gravitons as a product of single-soft factors acting on the hard amplitude. The findings provide a universal, coordinate-invariant description of soft graviton emissions in perturbative quantum gravity, with potential applicability beyond string theory to any finite S-matrix quantum gravity framework.

Abstract

Superstring field theory gives expressions for heterotic and type II string loop amplitudes that are free from ultraviolet and infrared divergences when the number of non-compact space-time dimensions is five or more. We prove the subleading soft graviton theorem in these theories to all orders in perturbation theory for S-matrix elements of arbitrary number of finite energy external states but only one external soft graviton. We also prove the leading soft graviton theorem for arbitrary number of finite energy external states and arbitrary number of soft gravitons. Since our analysis is based on general properties of one particle irreducible effective action, the results are valid in any theory of quantum gravity that gives finite result for the S-matrix order by order in perturbation theory without violating general coordinate invariance.

Figures (4)

• Figure 1: Source of the leading contribution to the string loop amplitude with one external soft graviton. $\epsilon_i$ and $p_i$ denote the polarization and momentum of the $i$-th external finite energy particles, while $\varepsilon_{\mu\nu}$ and $k$ denote the polarization and momentum of the external soft graviton.
• Figure 2: Source of the subleading contribution to the string loop amplitude with one external soft graviton.
• Figure 3: Two external soft gravitons attached to different external lines carrying finite momenta.
• Figure 4: Two external soft gravitons attached to the same external line carrying finite momenta.