Effect of the CP violation on neutrino mixing

After the short review of neutrino physics, we show the constraints on a neutrino mass matrix M in order to explain the maximal atmospheric neutrino mixing and maximal CP violation. Using these constraints, we show five types of mass orderings among the three neutrino masses m(1), m(2) and m(3) that explain the observed mass hierarchy of Delta m(atm) >> Delta m(circle dot). Our matrix M is described by the flavor neutrino masses, M-e mu, M-e tau, and M-mu tau, where M-ij stands for the ij matrix element of M (i, j = e, mu, tau), which satisfy |M-mu mu|(2) - |M-tau tau|(2) = |M-e tau|(2) - |M-e mu|(2). If |M-e mu| = |M-e tau|, giving either M-e tau = -sigma e(i theta)M(e mu), or M-e tau = -sigma e(i theta)M(e mu)*, is imposed, we find that |M-mu mu| = |M-tau tau| I is necessarily satisfied and this relation can be called "complex" mu-tau symmetry. It turns out that the real and imaginary parts of our neutrino mass matrix for theta = 0, respectively, describe the mu-tau symmetric one and its breaking one.