  We consider the problem of soft scattering for the analogue of pion states in
gauge-fermion theories which approach a conformal fixed point in the infrared
limit. Introducing a fermion mass into such a theory will explicitly break both
scale invariance and chiral symmetry, leading to confinement and a spectrum of
bound states. We argue that in such a theory, the pion scattering length
diverges in the limit of zero fermion mass, in sharp contrast to QCD-like
theories where the chiral Lagrangian predicts a vanishing scattering length. We
demonstrate this effect both with a simple dimensional argument, and in a
generalized linear sigma model which we argue can be used to describe the
interactions of light scalar and pseudoscalar bound states in the soft limit of
a mass-deformed infrared-conformal theory. As a result, lattice calculations of
pion scattering lengths could be a sensitive probe for infrared scale
invariance in gauge-fermion theories.
