Expansion of $U_{PMNS}$ and Neutrino mass matrix $M_{\nu}$ in terms of $sin\theta_{13}$ for Inverted Hierarchical case

The recent observational data supports the deviation from Tri-bimaximal (TBM) mixings. Different neutrino mass models suggest the interdependency among the observational parameters involving the mixing angles. On phenomenological ground we try to construct the PMNS matrix $U_{PMNS}$ with certain analytic structure satisfying the unitary condition, in terms of a single observational parameter $sin\theta_{13}$. We hypothesise the three neutrino masses $m_{i}$ as functions of $sin\theta_{13}$ and then construct the neutrino mass matrix $M_{\nu}$. We assume the convergence of the model to TBM mixing when $\theta_{13}$ is taken zero. This mass matrix so far obtained can be employed for various applications including the estimation of matter-antimatter asymmetry of the Universe.


Introduction
Recent results published by Double Chooz [1], Daya Bay [2], RENO [3], T2K [4] and MINOS [5] collaborations assure relatively large reactor angle (θ13). Also the recent global neutrino oscillation data analysis [ 6] insists on θ23 < π/4. Tri-bimaximal Mixing [7] is associated with θ13 = 0, and θ23 = π/4. This symmetry has a strong theoretical support because of its relation with so called µ−τ symmetry of neutrino mass matrix. µ−τ symmetry which is associated with A4 discrete flavour symmetry group [8][9][10][11][12]. But in order to comply with the recent experimental results, some perturbations have to be introduced in this mixing pattern. An open question is whether the corrections [13,14] are needed or a new mixing scheme is to be introduced [15].
In the present literature [16,17] we find the dependency of the mixing angles on one another. If this is true, then we are allowed to choose a single parameter capable of describing all the three mixing angles. We move a step ahead and express the three masses under this parameter. This helps us to define a simplified neutrino mass model with a single parameter only.
Out of all the three observational parameters concerning the mixing angles, sin θ13 is the smallest one. So, we choose sin θ13 as the guiding parameter. We consider tri-bimaximal mixing pattern and µ−τ symmetry as the first approximation. Hence the model is supposed to produce T.B.M mixing when we put sin θ13 = 0. We try to keep the structure of the three rotation matrices U (θ13), U (θ12) and U (θ23) in analytical form so that they can satisfy the unitary condition [U (θij )] † U (θij ) = I. We start with the following ansatz, where, sij = sin θij , and then construct the PMNS mixing matrix and then the neutrino mass matrix in the usual way.

Construction of the PMNS matrix
We consider the charged lepton mass matrix to be diagonal. Hence we can choose UP M NS = Uν . We propose the three rotation matrices as: We have, where, and, It can be checked that, and we get, After interpreting the above two relations in terms of sin θ13, we have,  3 Jarkslog parameter ( J cp ) We introduce the CP phase δ in U13 as shown in eq (8). The inclusion of δcp does not affect tan 2 θ12 or tan 2 θ23 [eq(9), eq(10)]. We obtain the Jcp as, Maximum Jcp, i.e., Jmax is obtained for δ = π 2 . For, ǫ = 0.156, Jmax is obtained as 0.0341. The variation of Jmax with respect to sin θ13 is shown in Fig.2 . Also the variation of Jcp with δ ( with ǫ or sin θ13 fixed at 0.156 ), is plotted in Fig.3.

Summary
We have started with a parameter ǫ equating this to sin θ13 and construct the PMNS matrix, UP M NS . Then we represent the neutrino masses (mi=1,2,3) in terms of the same parameter sin θ13, i.e ǫ. We verify our hopothesis by comparing the ranges of the mass squared differences as a result of our ansatz with the 1σ range, experimentally obtained. We take the range of ǫ as the experimental 1σ range of sin θ13 [6]. We obtain the range of ∆m 2 21 and ∆m 2 23 as (7.46−7.58)×10 −5 eV 2 and (2.42−2.44)×10 −3 eV 2 respectively. The respective ranges obtained, lie within the experimental 1σ boundary [6]. This provides a support to our hypothesis mi as mi(ǫ). This is to be emphasised that the UP M NS matrix as proposed in eq.(7) satisfy the unitary condition and is not dependent on the choice of the order of ǫ. The introduction of δcp does not affect tan 2 θ12 and tan 2 θ23 in our calculation. The maximum Jcp obtained is 0.034 ( with respect to ǫ = sin θ13 = 0.156 ). Finally we concentrate on the construction of Mν, the neutrino mass matrix. The present investigation though phenomenological, gives a complete picture of the texture of the neutrino mass matrix which can be employed in other applications regarding baryon asymmetry of the Universe [18]. Although we have constructed the mass matrix for inverted hierarchical model, yet we can extend our technique to Normal as well as Quasidegenerate mass models.