  We consider the metric-affine formulation of bumblebee gravity, derive the
field equations, and show that the connection can be written as Levi-Civita of
a disformally related metric in which the bumblebee field determines the
disformal part. As a consequence, the bumblebee field gets coupled to all the
other matter fields present in the theory, potentially leading to nontrivial
phenomenological effects. To explore this issue we compute the weak-field limit
and study the resulting effective theory. In this scenario, we couple scalar
and spinorial matter to the effective metric which, besides the zeroth-order
Minkowskian contribution, also has the vector field contributions of the
bumblebee, and show that it is renormalizable at one-loop level. From our
analysis it also follows that the non-metricity of this theory is determined by
the gradient of the bumblebee field, and that it can acquire a vacuum
expectation value due to the contribution of the bumblebee field.
