Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 305 / 1 / c

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- Solution c

Solution

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.

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