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Register a new userCorrelating lepton mixing angles and mixing matrix with Wolfenstein parameters
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Inspired by a new relation $theta_{13}^{rm PMNS}={theta_C}/{sqrt{2}}$ observed from the relatively large $theta_{13}^{rm PMNS}$, we find that the combination of this relation with the quark-lepton complementarity and the self-complementarity results in correlations of the lepton mixing angles with the quark mixing angles. We find that the three mixing angles in the PMNS matrix are all related to the Wolfenstein parameter $lambda$ in the quark mixing, so they are also correlated. Consequently, the PMNS matrix can be parameterized by $lambda$, A, and a Dirac CP-violating phase $delta$. Such parametrizations for the PMNS matrix have the same explicitly hierarchical structure as the Wolfenstein parametrization for the CKM matrix in the quark mixing, and the bimaximal mixing pattern is deduced at the leading order. We also discuss implications of these phenomenological relations in parametrizations.
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We propose a new parametrization for the quark and lepton mixing matrices: the two 12-mixing angles (the Cabibbo angle and the angle responsible for solar neutrino oscillations) are at zeroth order pi/12 and pi/5, respectively. The resulting 12-elements in the CKM and PMNS matrices, V_{us} and U_{e2}, are in this order irrational but simple algebraic numbers. We note that the cosine of pi/5 is the golden ratio divided by two. The difference between pi/5 and the observed best-fit value of solar neutrino mixing is of the same order as the difference between the observed value and the one for tri-bimaximal mixing. In order to reproduce the central values of current fits, corrections to the zeroth order expressions are necessary. They are small and of the same order and sign for quarks and leptons. We parametrize the perturbations to the CKM and PMNS matrices in a triminimal way, i.e., with three small rotations in an order corresponding to the order of the rotations in the PDG-description of mixing matrices.
Starting with the Wolfenstein form for the leptonic mixing matrix we show that renormaliztion group evolution brings that to the observed large mixing at low energies.
Many unified models predict two large neutrino mixing angles, with the charged lepton mixing angles being small and quark-like, and the neutrino masses being hierarchical. Assuming this, we present simple approximate analytic formulae giving the lepton mixing angles in terms of the underlying high energy neutrino mixing angles together with small perturbations due to both charged lepton corrections and renormalisation group (RG) effects, including also the effects of third family canonical normalization (CN). We apply the perturbative formulae to the ubiquitous case of tri-bimaximal neutrino mixing at the unification scale, in order to predict the theoretical corrections to mixing angle predictions and sum rule relations, and give a general discussion of all limiting cases.
We discuss the constraints of lepton mixing angles from lepton number violating processes such as neutrinoless double beta decay, (mu^-)-(e^+) conversion and K decay, $K^- to pi^+mu^-mu^-$ which are allowed only if neutrinos are Majorana particles. The rates of these processes are proportional to the averaged neutrino mass defined by $<m_{ u} >_{a b}equiv |sum_{j=1}^{3}U_{a j} U_{b j}m_j|$ in the absence of right-handed weak coupling. Here $a, b (j)$ are flavour(mass) eigen states and $U_{a j}$ is the left-handed lepton mixing matrix. We obtain the consistency conditions which are satisfied irrelevant to the concrete values of CP violation phases (three phases in Majorana neutrinos). These conditions constrain the lepton mixing angles, neutrino masses $m_i$ and (< m_{ u} >_{a b}). By using these constraints we obtain the limits on the averaged neutrino masses for (mu^-)-(e^+) conversion and K decay, $K^- to pi^+mu^-mu^-$.
Advantages of the original symmetrical form of the parametrization of the lepton mixing matrix are discussed. It provides a conceptually more transparent description of neutrino oscillations and lepton number violating processes like neutrinoless double beta decay, clarifying the significance of Dirac and Majorana phases. It is also ideal for parametrizing scenarios with light sterile neutrinos.
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