  We consider the low-energy scattering of two ultracold polarized dipoles with
both a short-range interaction (SRI) and a weak dipole-dipole interaction (DDI)
which is far away from shape-resonances. In previous analytical studies, the
scattering amplitude in this system was often calculated via the first-order
Born approximation (FBA). Our results show that significant derivations from
this approximation can arise in some cases. In these cases, the SRI can
significantly modify the inter-dipole scattering amplitudes even if the
scattering amplitudes for the SRI alone are much smaller than the dipolar
length of the DDI. We further obtain approximate analytical expressions for
these inter-dipole scattering amplitudes.
