Possible Deviation from Exact Tribimaximal Mixing

A simple relation UMNS = V † CKM UTB between the lepton and quark mixing matrices (UMNS and VCKM ) is speculated under an ansatz that UMNS becomes an exact tribimaximal mixing UTB in a limit VCKM = 1. By using the observed CKM mixing parameters, possible values of neutrino oscillation parameters are estimated: sin θ13 = 0.024− 0.028, sin 2θ23 = 0.94− 0.95 and tan θ12 = 0.24− 1.00 depending on phase conventions of UTB. Recently, there has been considerable interest in the magnitude of the neutrino mixing angle θ13 (νe ↔ ντ mixing angle), because it is a key value not only for checking neutrino mass matrix models, but also for searching CP -violation effects in the lepton sector. Recent observed neutrino oscillation data are in favor of the so-called “tribimaximal mixing” [1] which predicts θ13 = 0, tan2 θ12 = 1/2 and sin 2 2θ23 = 1, since the present data yield the values tan 2 θ12 = 0.47 +0.06 −0.05 [2] and sin 2θ23 = 1.00−0.13 [3]. If the angle θ13 is exactly zero or negligibly small, the observation of the CP -violation effects in the lepton sector will be hopeless even in future. On the other hand, recently, Fogli et al. [4] have reported a sizable value sin θ13 = 0.016 ± 0.010 (1σ) from a global analysis of neutrino oscillation data. The tribimaximal lepton mixing is given by the form U TB =   2 √ 6 1 √ 3 0 − 1 √ 6 1 √ 3 − 1 √ 2 − 1 √ 6 1 √ 3 1 √ 2   . (1) Such a form with beautiful coefficients seems to be understood from a discrete symmetry of flavors [1]. In contrast to the lepton mixing matrix (Pontecorvo-Maki-Nakagawa-Sakata mixing matrix [5]) UMNS, the observed Cabibbo-Kobayashi-Maskawa [6] (CKM) quark mixing matrix VCKM seems to have no beautiful form with Clebsh-Gordan-like coefficients, and VCKM , rather, looks like nearly VCKM ≃ 1. It is unlikely that a theory which exactly leads to the tribimaximal mixing (1) simultaneously gives the CKM mixing matrix with small and complicated mixing values. Therefore, it is interesting to consider a specific case that a theory of flavor symmetry gives VCKM = 1 in the limit of UMNS = UTB . We consider that the observed form of the CKM matrix VCKM is due to some additional effects (e.g. symmetry breaking effects for the flavor symmetry). If it is true, then, the observed lepton mixing UMNS will also deviate from the exact tribimaximal mixing UMNS = UTB by additional effects which gives the deviation from VCKM = 1. For convenience of later discussions, we define the tribimaximal mixing by a form UTB = P ∗ LU 0 TBPR, (2)