Electromagnetic Form Factor of the Neutrino

Bernstein and Lee and, independently, Meyer and Schiff have recently published calculations of the neutrino electromagnetic form factor, obtaining results differing by a finite constant term. This difference can be traced back to how the $W$-meson contribution is regularized: The Bernstein-Lee calculation is gauge-invariant at every step, while Meyer and Schiff simply impose over-all neutrino charge neutrality at the end. The $\xi$-limiting process in addition sums a class of electromagnetic radiative corrections and assigns the value $\ln {\alpha }^{-1\mathrm{}}$ to the logarithmically divergent term in the $W$-meson contribution. Since the finite term, which is almost comparable to $\ln {\alpha }^{-1\mathrm{}}$ in magnitude, is not fixed by the $\xi$-limiting method, the neutrino form factor has actually been determined only to order of magnitude by this method. For this reason and because the $W$ mass is large (or infinite), we have determined the largest part of the neutrino form factor from the charged lepton contribution using a guage-invariant direct-interaction theory. This is obtained, without further calculation, from the photon vacuum polarization. The ${\nu }_e$ charge radius thus measures the same integral that appears in the perturbation-theory calculation of ${Z_{3e}}^{\frac{1}{2}}$, the charge renormalization in quantum electrodynamics.