  It has recently been reported [Phys. Rev. Lett. 117, 162502 (2016)] that (d,
p) cross sections can be very sensitive to the n-p interactions used in the
adiabatic treatment of deuteron breakup with nonlocal nucleon-target optical
potentials. To understand to what extent this sensitivity could originate in
the inaccuracy of the adiabatic approximation we have developed a leading-order
local- equivalent continuum-discretized coupled-channel model that accounts for
non-adiabatic effects in the presence of nonlocality of nucleon optical
potentials. We have applied our model to the astro- physically relevant
reaction $^{26m}$Al$(d, p) ^{27}$Al using two different n-p potentials
associated with the lowest and the highest n-p kinetic energy in the
short-range region of their interaction, respectively. Our calculations reveal
a significant reduction of the sensitivity to the high n-p momenta thus
confirming that it is mostly associated with theoretical uncertainties of the
adiabatic approximation itself. The non-adiabatic effects in the presence of
nonlocality were found to be stronger than those in the case of the local
optical potentials. These results argue for extending the analysis of the $(d,
p)$ reactions, measured for spectroscopic studies, beyond the adiabatic
approximation.
