  We revisit the calculation of matter quantum effects on the graviton
self-energy on a flat Minkowski background, with the aim to acquire a deeper
understanding of the mechanism that renders the graviton massless. To this end,
we derive a low-energy theorem which directly relates the radiative corrections
of the cosmological constant to those of the graviton mass to all orders in
perturbation theory. As an illustrative example, we consider an Abelian Higgs
model with minimal coupling to gravity and show explicitly how a suitable
renormalization of the cosmological constant leads to the vanishing of the
graviton mass at the one-loop level. In the same Abelian Higgs model, we also
calculate the matter quantum corrections to the Newtonian potential and present
analytical formulae in terms of modified Bessel and Struve functions of the
particle masses in the loop. We show that the correction to the Newtonian
potential exhibits an exponential fall-off dependence on the distance $r$, once
the non-relativistic limit with respect to the non-zero loop mass is carefully
considered. For massless scalars, fermions and gauge bosons in the loops, we
recover the well-known results presented in the literature.
