  The two-particle finite-volume scattering formalism derived by L\"uscher and
generalized in many subsequent works does not hold for energies far enough
below the two-particle threshold to reach the nearest left-hand cut. The
breakdown of the formalism is signaled by the fact that a real scattering
amplitude is predicted in a regime where it should be complex. In this work, we
address this limitation by deriving an extended formalism that includes the
nearest branch cut, arising from single particle exchange. We focus on
two-nucleon ($NN \to NN$) scattering, for which the cut arises from pion
exchange, but give expressions for any system with a single channel of
identical particles. The new result takes the form of a modified quantization
condition that can be used to constrain an intermediate K-matrix in which the
cut is removed. In a second step, integral equations, also derived in this
work, must be used to convert the K-matrix to the physical scattering
amplitude. We also show how the new formalism reduces to the standard approach
when the $N \to N \pi$ coupling is set to zero.
