The Effect of the Earth Matter on Three Neutrino Oscillations and Sensitivity to CP Phase Parameter

We find an analytical expression of neutrino evolution operator in the Earth matter using perturbative approach in the context of three neutrino oscillations. We find that our analytical expression is highly accurate by comparing its results with the numerical solutions of neutrino evolution equation at energy scales relevant for solar, reactor, atmospheric, and accelerator neutrinos. Using our analytical approach we study the accuracy of hypothesis of treating the Earth density piecewise constant. We also study how the Earth matter effect can change the sensitivity to CP phase parameter $\delta _{CP}$. Through nadir angle averaged conversion probabilities of neutrino and anti-neutrino, we find that the sensitivity to $\delta _{CP}$ is maximum in the energy range 0.2 to 1 GeV and through energy averaged conversion probabilities, we find that the sensitivity is maximum about nadir angle $72.5^{\text{o}}$ for neutrinos oscillating in the Earth matter.