Dilaton Effective Action with $\mathcal{N}=1$ Supersymmetry

TL;DR

This work constructs the four-dimensional dilaton-axion effective action for $ S=1$ SCFTs, incorporating both the conformal and $U(1)_R$ anomalies through a Weyl–gauge invariant action $S_ ext{inv}$ and a Wess–Zumino piece $S_ ext{WZ}$. Supersymmetry fixes a unique completion of the action so that the four-derivative sector reproduces the anomaly structure via $ riangle a$ and $ riangle c$, while the axion $eta$ (Goldstone of broken $U(1)_R$) does not modify the four-dilaton scattering that underpins the $a$-theorem; SUSY Ward identities equate dilaton and axion amplitudes, ensuring the $2 o2$ dilaton amplitude remains the relevant probe. Matching to the Schwimmer–Theisen superspace result constrains the coefficients so that $ abla$-invariants do not affect the four-point dilaton amplitude, and a flat-space analysis shows $ abla$-invariant contributions cancel or are constrained to avoid spoiling $ riangle a>0$. The analysis extends the Komargodski–Schwimmer program to $ obreak N=1$ SUSY with axions and demonstrates that the $a$-theorem is robust against additional massless modes, both in SUSY and non-SUSY settings, with potential pathways to broader global-symmetry and higher-supersymmetry extensions.

Abstract

We clarify the structure of the four-dimensional low-energy effective action that encodes the conformal and $U(1)$ R-symmetry anomalies in an $\mathcal{N}=1$ supersymmetric field theory. The action depends on the dilaton, $τ$, associated with broken conformal symmetry, and the Goldstone mode, $β$, of the broken $U(1)$ R-symmetry. We present the action for general curved spacetime and background gauge field up to and including all possible four-derivative terms. The result, constructed from basic principles, extends and clarifies the structure found by Schwimmer and Theisen in arXiv:1011.0696 using superfield methods. We show that the Goldstone mode $β$ does not interfere with the proof of the four-dimensional $a$-theorem based on $2 \to 2$ dilaton scattering. In fact, supersymmetry Ward identities ensure that a proof of the $a$-theorem can also be based on $2 \to 2$ Goldstone mode scattering when the low-energy theory preserves $\mathcal{N}=1$ supersymmetry. We find that even without supersymmetry, a Goldstone mode for any broken global $U(1)$ symmetry cannot interfere with the proof of the four-dimensional $a$-theorem.