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For static sources, the spatial Fourier transform of a massive scalar propagator isThus 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 $e^{-mr}/r$, with range $\hslash /(mc)$. 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.

After electroweak symmetry breaking, the Higgs field has vacuum expectation value $⟨H⟩=(0,v/\sqrt{2}{)}^T$. A Yukawa interaction $-y_f{\bar{f}}_{L}H{f}_R+h.c.$ then becomes a fermion mass with $m_f=y_{f}v/\sqrt{2}$. The Higgs gauge-covariant kinetic term $|D_{\mu }H{|}^2$ giveswhile the radial fluctuation is the massive Higgs boson. The photon and all eight gluons correspond to unbroken gauge generators and remain massless. In the minimal renormalizable Standard Model, which has no right-handed neutrinos, the neutrinos also remain massless; adding right-handed neutrinos permits Higgs-generated Dirac masses.