  Two-dimensional coherent spectroscopy (2DCS) is a nonlinear spectroscopy
technique capable of identifying whether apparent continua in linear response
are made out of multiplets of sharp deconfined quasiparticles. This makes it a
potent tool for experimental identification of fractionalized phases. Previous
discussions have focused on limits where the quasiparticles in question are
infinitely long lived. In this manuscript we discuss 2DCS in the regime where
the fractionalized quasiparticles can themselves decay. We introduce a powerful
path integral based approach, whereby the computation of nonlinear
susceptibilities reduces to an efficient exercise in diagrammatic perturbation
theory. We apply this method to compute the 2DCS response of the
one-dimensional transverse field Ising model, in the presence of
integrability-breaking perturbations. We discuss aspects of the self-energy of
the fractionalized quasiparticles that may be extracted via 2DCS, such as the
momentum-dependent decay rate.
