Exploring soft constraints on effective actions

TL;DR

This paper investigates effective actions with broken symmetries, revealing novel soft theorems and constraints in supersymmetric theories, and systematically determining amplitudes in conformal field theories with implications for non-renormalization.

Contribution

It introduces new soft theorems for broken symmetries, systematically constrains effective actions in supersymmetric and conformal theories, and derives amplitude relations up to high derivative orders.

Findings

Soft theorems with non-vanishing limits in broken symmetry scenarios.

Effective actions in pure dilaton sectors are determined by lower-order amplitudes.

Maximal supersymmetry constrains effective actions, leading to non-renormalization theorems.

Abstract

We study effective actions for simultaneous breaking of space-time and internal symmetries. Novel features arise due to the mixing of Goldstone modes under the broken symmetries which, in contrast to the usual Adler's zero, leads to non-vanishing soft limits. Such scenarios are common for spontaneously broken SCFT's. We explicitly test these soft theorems for $\mathcal{N}=4$ sYM in the Coulomb branch both perturbatively and non-perturbatively. We explore the soft constraints systematically utilizing recursion relations. In the pure dilaton sector of a general CFT, we show that all amplitudes up to order $s^{n} \sim \partial^{2n}$ are completely determined in terms of the $k$-point amplitudes at order $s^k$ with $k \leq n$. Terms with at most one derivative acting on each dilaton insertion are completely fixed and coincide with those appearing in the conformal DBI, i.e. DBI in AdS. With maximal supersymmetry, the effective actions are further constrained, leading to new non-renormalization theorems. In particular, the effective action is fixed up to eight derivatives in terms of just one unknown four-point coefficient and one more coefficient for ten-derivative terms. Finally, we also study the interplay between scale and conformal invariance in this context.