  Conformal field theories that exhibit spontaneous breaking of conformal
symmetry (a moduli space of vacua) must satisfy a set of bootstrap constraints,
involving the usual data (scaling dimensions and OPE coefficients) as well as
new data such as the spectrum of asymptotic states in the broken vacuum and
form factors. The simplest bootstrap equation arises by expanding a two-point
function of local operators in two channels, at short distance using the OPE
and at large distance using the EFT in the broken vacuum. We illustrate this
equation in what is arguably the simplest perturbative model that exhibits
conformal symmetry breaking, namely the real $ABC$ model in $d = 4 -\epsilon$
dimensions. We investigate the convergence properties of the bootstrap equation
and check explicitly many of the non-trivial relations that it imposes on
theory data.
