Two-loop neutrino masses and the solar neutrino problem

The addition of m singlet right-handed neutrinos to the standard model leads to radiatively generated mass corrections for the SU(2${)}_{\mathit{L}}$ doublet neutrinos. For those neutrinos which are massless at the tree level after this addition, this implies a small mass generated at the two-loop level via ${\mathit{W}}^{\pm }$ exchange. We calculate these mass corrections exactly by obtaining an analytic form for the general case of n doublets and m singlets. As a phenomenological application, we study the m=1, n=3 case in detail, for which there are two massive and two massless neutrinos at the tree level. After calculating the loop-induced masses, the full mass spectrum is examined in the light of experimental data from the four solar neutrino experiments and the COBE satellite. Requiring compatibility with these experiments enables us to constrain the parameter space and to obtain a handle on the scale of new physics, as typified by the mass of the heavy singlet. We also obtain the interesting result that the final mass eigenvalues, corrected for loop effects, may all be significantly different from the seesaw value of the lighter of the two neutrinos which have tree level masses, thus demonstrating the importance of the loop effects considered here.