Neutrino oscillations in the Sun probe long-range leptonic forces

Lepton number charges might be the source of long range forces. If one accepts that neutrinos produced in the Sun do indeed oscillate while crossing the interior of the Sun, then the shift in the phase of the neutrino wavefunction caused by an hypothetical potential associated to the leptonic charge of the electrons in the Sun could affect the oscillation pattern beyond what is actually observed. We show that a "fine structure" constant $\alpha_{L}$ in excess of $6.4 \times 10^{-54}$ is incompatible with present observational data. This bound is not valid for forces whose range is shorter than the size of the Sun.