  The metric-affine bumblebee model in the presence of fermionic matter
minimally coupled to the connection is studied. We show that the model admits
an Einstein frame representation in which the matter sector is described by a
non-minimal Dirac action without any analogy in the literature. Such
non-minimal terms involve unconventional couplings between the bumblebee and
the fermion field. We then rewrite the quadratic fermion action in the Einstein
frame in the basis of 16 Dirac matrices in order to identify the coefficients
for Lorentz/CPT violation in all orders of the non-minimal coupling $\xi$. The
exact result for the fermionic determinant in the Einstein frame, including all
orders in $\xi$, is also provided. We demonstrate that the axial contributions
are at least of second order in the perturbative expansion of $\xi$.
Furthermore, we compute the one-loop effective potential within the weak field
approximation.
