The footprint of E7 in amplitudes of N=8 supergravity

TL;DR

The paper analyzes how the non-linearly realized $E_{7(7)}$ symmetry of ${\cal N}=8$ supergravity constrains low-energy amplitudes by studying one-soft-scalar emissions through the conserved Noether current. Using a general low-energy theorem for non-linear $G/H$ symmetries and focusing on the ${\rm Re}\Sigma$ portion of the $E_{7(7)}/SU(8)$ current to avoid dual-field issues, it computes the axial charge $g_A^a(0)$ for various external states. The central result is that $g_A^a(0)=0$ at tree level for all external configurations considered, implying that all one-soft-boson tree amplitudes vanish in ${\cal N}=8$ SG and supporting the unbroken symmetry at this order. The discussion then addresses extensions to higher loops, potential anomalies, and the broader implications for the role of $E_{7(7)}$ in the theory and its relation to string dualities and finiteness conjectures.

Abstract

We study the low energy theorems associated with the non-linearly realized continuous E7 symmetry of the on-shell N=8 supergravity. For Nambu-Goldstone bosons we evaluate the one-soft-scalar-bosonemission amplitudes by computing the E7 current matrix element on the one-particle external lines. We use the explicit form of the conserved E7 Noether current and prove that all such matrix elements vanish in the soft momentum limit,assuming the SU(8) symmetry of the S-matrix.This implies that all tree amplitudes vanish in the one-soft-boson limit. We also discuss the implications of unbroken E7 symmetry for higher-order amplitudes.

Figures (2)

• Figure 1: Three categories of diagrams contributing to $\langle\beta(P_f)\, {{\rm out}} | J_\mu^a(x) | \alpha(P_i)\, {{\rm in}} \rangle$. The small circle with cross stands for the current operator.
• Figure 2: The insertion of the non-linear part of the divergence of the Noether current into an external line of the on-shell amplitude.