Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 37D / e / Solution

For the static point source, take $A=0$ and $\alpha =0$. The scalar equation becomesIts spatial Fourier transform satisfiesThe stated inverse transform therefore giveswhere $\lambda$ absorbs the transform normalization and source strength. This is the Yukawa potential: a nonzero vector-field mass screens the interaction beyond the range $m^{-1}$. As $m\to 0$, it tends to the long-range Coulomb potential $\lambda /r$. The same limit removes the mass term that broke gauge invariance in part (b).