Correlation between Leptonic CP Violation and mu-tau Symmetry Breaking
Abstract
Considering the
μ
-
τ
symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic CP violation by determining three eigenvectors associated with
M=
M
†
ν
M
ν
for a complex flavor neutrino mass matrix
M
ν
in the flavor basis. Since the Dirac CP violation is absent in the
μ
-
τ
symmetric limit, leptonic CP violation is sensitive to the
μ
-
τ
symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase (
δ
) arises from the
μ
-
τ
symmetry breaking part of
M
eμ,eτ
and an additional phase (
ρ
) is associated with the
μ
-
τ
symmetric part of
M
eμ,eτ
, where
M
ij
stands for an
ij
matrix element (
i,j
=
e,μ,τ
). The phase
ρ
is redundant and can be removed but leaves its effect in the Dirac CP violation characterized by
sin(δ+ρ)
. The perturbative results suggest the exact formula of mixing parameters including that of
δ
and
ρ
, which turns out to be free from the effects of the redundant phases. As a result, it is generally shown that the maximal atmospheric neutrino mixing necessarily accompanies either
sin
θ
13
=0
or
cos(δ+ρ)=0
, the latter of which indicates maximal CP violation, where
θ
13
is the
ν
e
-
ν
τ
mixing angle.