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 Incoming links: Mass gap

For static sources, the spatial Fourier transform of a massive scalar propagator is

\int \frac{d ^{3} q}{( 2 π ) ^{3}} \frac{e ^{i q \cdot r}}{q ^{2} + M _{X}^{2}} = \frac{e ^{- M_{X} r}}{4 π r} .

Thus massive-vector exchange likewise produces a Yukawa potential,

V (r) \propto e^{- M_{X} r} / 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.

The range of a force is controlled by the lightest state that can carry it. Exchange of a particle of mass

m

produces a Yukawa potential proportional to

e^{- m r} / r

, with range

ℏ/ (m c)

. The electromagnetic interaction is long range because its carrier, the photon, is massless. The weak interaction is short range because the W bosons and Z boson are massive. Although the gluons of the strong interaction are massless in the Lagrangian, Quantum chromodynamics confines color and has a mass gap; only color-singlet hadrons propagate over macroscopic distances. The residual nuclear force is consequently controlled at long distance by massive pion exchange.