Exploring soft constraints on effective actions

TL;DR

The paper develops a framework in which soft theorems for spontaneously broken space-time and internal symmetries, including their mixing, constrain the low-energy effective action. It combines perturbative (one-loop) and non-perturbative (instanton) tests in ${ m N}=4$ SYM on the Coulomb branch with on-shell recursion (soft-BCFW) to show that dilaton dynamics are highly constrained, and with maximal SUSY the action is fixed up to a small set of higher-derivative coefficients. The work reveals a deep link between scale and conformal invariance through soft theorems and provides a template for determining higher-point amplitudes from lower-point data, particularly within conformal DBI-like structures. It also extends the analysis to higher-dimensional SUSY theories, highlighting the universality of these soft constraints across dimensions and instanton sectors.

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.

Figures (1)

• Figure 1: Contributions to the soft limit