  The definition of the scattering volume for $p$-wave collisions needs to be
generalized in the presence of dipolar interactions for which the potential
decreases with the interparticle separation as $1/R^3$. Here, we propose a
generalized definition of the scattering volume characterizing the short-range
interactions in odd-parity waves, obtained from an analysis of the $p$-wave
component of the two-body threshold wave function. Our approach uses an
asymptotic model and introduces explicitly the anisotropic dipole-dipole
interaction, which governs the ultracold collision dynamics at long-range. The
short-range interactions, which are essential to describe threshold resonances,
are taken into account by a single parameter which is determined by the
field-free $s$-wave scattering length.
