Neutrino Mass Spectrum for Co-Bimaximal Mixings from Quantum Gravity

We consider non-renormalizable interaction term as a perturbation of the conventional neutrino mass matrix. Quantum gravitational (Planck scale )effects lead to an effective $SU(2{)}_L\times U(1)$ invariant dimension-5 Lagrangian involving neutrino and Higgs fields,On symmetry breaking, this operator gives rise to correction to the neutrino masses and mixing. The gravitational interaction $M_X=M_{pl}$ which gives rise to additional terms in neutrino mass matrix. We also assume that, just above the electroweak breaking scale, neutrino masses are nearly degenerate and their mixing is Co-bimaximal mixing by assuming mixing angle ${\theta }_{13}\mathrm{\ne }0={10}^o$, ${\theta }_{23}=\frac{\pi }{4}$, $\tan\theta_{12}^ 2 = \frac{1-3sin\theta_{13}^2}{2} = 34^{o}$andDiracphase$\delta=\pm\frac{\pi}{2}$.There additional term can be considered to be perturbation of the GUT scale Co-bimaximal neutrino mass matrix. The relation consider with solar and atmospheric neutrino oscillation data predicted above GUT scale $m_1^{\mathrm{\prime }}\simeq 0.00001eV-0.00003eV$, $m_2^{\mathrm{\prime }}\simeq 0.00008eV-0.00012$, and $m_3^{\mathrm{\prime }}\simeq 0.000207eV-000320eV$.