= Solution
For static sources, the spatial Fourier transform of a massive scalar propagator is
$$
\int\frac{d^3\mathbf q}{(2\pi)^3}
\frac{e^{i\mathbf q\cdot\mathbf r}}{\mathbf q^2+M_X^2}
=\frac{e^{-M_Xr}}{4\pi r}.
$$
Thus massive-vector exchange likewise produces a <Yukawa potential>, $V(r)\propto e^{-M_Xr}/r$, with range $M_X^{-1}$ instead of the infinite range of the Coulomb potential.
<Quantum chromodynamics> supplies the requested massless-vector counterexample. Its <gluons> are massless, but <confinement> and the QCD <mass gap> prevent a long-range color force between color-singlet asymptotic states. The short range is generated by strong dynamics rather than by a vector-boson mass.
Back to article page
