So where are the expressions? I can't seem to find them.
To first order the interaction between, say, a proton and a neutron depends
on the exchange of a virtual pion, and the potential is given by the Yukawa
formula:
V(r) = (1/r) exp(-r/R) * (some coupling constant)
where R is the range of the interaction, and is the inverse of the mass (in
the appropriate units) of the particle that carries the interaction. If
you want to see the units explicitly, it's
1/R = m c / h
Meanwhile, to first order the interaction between, say, a proton and
electron depends on the exchange of a virtual photon. The potential is
given by the same type of formula, but since the photon is massless it
simplifies to
V(r) = (1/r) * (some other coupling constant)
which should be familiar.
The weak force is (surprise!) the same story, involving the exchange of
"intermediate vector bosons" i.e. W and Z particles.
BTW, there is a plague of "derivations" of the Yukawa potential that
revolve around a completely bogus alleged invocation of the uncertainty
principle. AFAICT they are little better than guessing the right answer by
dimensional analysis. That has its place, but please don't pretend it's
something it's not! There is no delta E * delta t uncertainty principle!
You can get the Yukawa result just as easily using a non-bogus argument
that depends on barrier penetration and evanescent waves.