  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,
$\tau$, associated with broken conformal symmetry, and the Goldstone mode,
$\beta$, 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 $\beta$ 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.
