Electromagnetic Form Factor of the Neutrino

Abstract

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 $\ensuremath{\xi}$-limiting process in addition sums a class of electromagnetic radiative corrections and assigns the value $\mathrm{ln}{\ensuremath{\alpha}}^{\ensuremath{-}1}$ to the logarithmically divergent term in the $W$-meson contribution. Since the finite term, which is almost comparable to $\mathrm{ln}{\ensuremath{\alpha}}^{\ensuremath{-}1}$ in magnitude, is not fixed by the $\ensuremath{\xi}$-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 ${\ensuremath{\nu}}_{e}$ 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.