Total cross sections for the production of single $W$ mesons in electron-positron colliding beams and in the electron scattering experiments have been calculated. If the first type of cross sections are estimated by calculating the simplest lowest-order dominant Feynman graph, one finds that they are remarkably large (∼${10}^{-30\mathrm{}}$ ${\mathrm{cm}}^2$). Indeed, a calculation in the literature which includes all lowest-order contributions apparently confirms this result. These estimates are in striking disagreement with calculations for the related process of neutrino-induced production of single $W$ mesons. These suggest a cross section of ∼${10}^{-37\mathrm{}}$ ${\mathrm{cm}}^2$, a result which is to be expected from a naive power-counting argument. We have calculated the cross sections and have found that they are in fact quite small and of the same order of magnitude as the neutrino-induced cross sections. We present a general gauge-invariance argument which illustrates how almost any trivial error in the calculation can easily lead to errors in the result of at least ${\mathrm{}\left[\right(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{o}\mathrm{n})/(\mathrm{mass}\mathrm{of}\mathrm{the}\mathrm{positron}\left)\right]}^2$∼${10}^6$. Basically, gauge invariance demands a critical cancellation in the differential cross section when the mass of the exchanged photon ($K^2$) becomes very small. If such a cancellation does not take place (because of some small error), the region around $K^2=0\mathrm{}$ is enormously enhanced by the smallness of the mass of the positron, and large errors are induced in the final result. An examination of the calculations that lead to a large total cross section does indeed show that each violates the constraint of gauge invariance.

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