Solar neutrinos and the principle of equivalence.

We study the proposed solution of the solar neutrino problem which requires a flavor nondiagonal coupling of neutrinos to gravity. We adopt a phenomenological point of view and investigate the consequences of the hypothesis that the neutrino weak interaction eigenstates are linear combinations of the gravitational eigenstates which have slightly different couplings to gravity, ${\mathit{f}}_{1}$G and ${\mathit{f}}_{2}$G, \ensuremath{\Vert}${\mathit{f}}_{1}$-${\mathit{f}}_{2}$\ensuremath{\Vert}\ensuremath{\ll}1, corresponding to a difference in redshift between electron and muon neutrinos, \ensuremath{\Delta}z/(1+z)\ensuremath{\sim}\ensuremath{\Vert}${\mathit{f}}_{1}$-${\mathit{f}}_{2}$\ensuremath{\Vert}. Our ${\mathrm{\ensuremath{\chi}}}^{2}$ analysis of the available solar neutrino data on observed event rates rules out most of the relevant parameter space, allowing only \ensuremath{\Vert}${\mathit{f}}_{1}$-${\mathit{f}}_{2}$\ensuremath{\Vert}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}14}$ for small values of the mixing angle [2\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$\ensuremath{\lesssim}${\mathrm{sin}}^{2}$(2${\mathrm{\ensuremath{\theta}}}_{\mathit{G}}$)\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}2}$] and ${10}^{\mathrm{\ensuremath{-}}16}$\ensuremath{\lesssim}\ensuremath{\Vert}${\mathit{f}}_{1}$-${\mathit{f}}_{2}$\ensuremath{\Vert}\ensuremath{\lesssim}${10}^{\mathrm{\ensuremath{-}}15}$ for large mixing [0.6\ensuremath{\lesssim}${\mathrm{sin}}^{2}$(2${\mathrm{\ensuremath{\theta}}}_{\mathit{G}}$)\ensuremath{\lesssim}0.9]. We show that the recoil-electron spectrum measured by the Kamiokande II Collaboration can be used to exclude part of the allowed regions obtained above. We analyze the prospects of using future spectral measurements of solar neutrinos to distinguish the oscillation mechanism due to the violation of the equivalence principle from more conventional mechanisms which require neutrinos to have nondegenerate masses. We find that, for small mixing angles, the flavor nondiagonal coupling to gravity leads to predictions regarding the shape of the $^{8}\mathrm{neutrino}$ spectrum which will be distinguishable in the upcoming SNO and Super-Kamiokande experiments and which are independent of solar models.