Majorana neutrino masses by D-brane instanton effects in magnetized orbifold models

We study Majorana neutrino masses induced by D-brane instanton effects in magnetized orbifold models. We classify the possible cases where neutrino masses can be induced. Three and four generations are favored in order to generate neutrino masses by D-brane instantons. Explicit mass matrices have specific features. Their diagonalizing matrices correspond to the bimaximal mixing matrix in the case with even magnetic fluxes, independently of the modulus value τ. On the other hand, for odd magnetic fluxes, diagonalizing matrices correspond nearly to the tri-bimaximal mixing matrix near τ = i, while they become the bimaximal mixing matrix for larger ${\rm Im} \, \tau$. For even fluxes, neutrino masses are modular forms of weight 1 on $T^2/\mathbb {Z}_2$, and they have symmetries such as $S_4^{\prime }$ and ${\Delta ^{\prime }}(96)\times \mathbb {Z}_3$.