Charged lepton contributions to bimaximal and tri-bimaximal mixings for generating $\sin\theta_{13}\neq 0$ and $\tan^2\theta_{23}<1$

Bimaximal (BM) and tri-bimaximal (TB) mixings of neutrinos are two special cases of lepton mixing matrix, which predict the reactor angle $\theta_{13}=0$ and the atmospheric angle $\tan^2\theta_{23}=1$. Recent precision measurements and global analysis of oscillation parameters, have confirmed a non-vanishing value of $\theta_{13}$ as well as deviations of $\theta_{12}$ and $\theta_{23}$ from their maximal values predicted by BM or TB mixing. In this work we mainly concentrate on $\theta_{13}$ and $\theta_{23}$ to assign $\sin\theta_{13}\neq 0$ and $\tan^2\theta_{23}<1$ with the help of charged lepton corrections defined by $U_{PMNS}=U^{\dagger}_lU_{\nu}$. We first consider $U_{\nu}$ to be given separately by BM and TB mixing matrices and then find the possible forms of $U_l}$ such that the elements of PMNS matrix, finally yield $\sin\theta_{13}\neq 0$ and $\tan^2\theta_{23}<1$. To compute the values of mixing angles we assume the charged lepton correction to be of CKM-like. All the mixing matrices considered here satisfy the unitarity condition to leading order of expansion parameters. We also analyze both the mixing schemes in presence of Dirac CP phase and find expressions for the rephasing invariant quantity $J_{CP}$ which have been discussed in recent literature.


Introduction
Recent precision measurements [1][2][3][4] and latest global 3ν oscillation analysis [5] of neutrino mixing parameters, have confirmed non-vanishing value of θ 13 as well as deviation of atmospheric mixing angle from maximal value, θ 23 < π/4. One of the important aspects of neutrino physics is to understand such mixing patterns [6]. Charged lepton corrections [7] to neutrino mixing matrix is an attractive tool which can impart non-zero value of θ 13 as well as deviation of θ 23 from maximal value. We address the issue of charged lepton correction to both bimaximal(BM) and tri-bimaximal(TB) neutrino mixings to produce desired results.
To begin with we start with the lepton mixing matrix, known as Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [8], which is analogous to CKM matrix, V CKM = U † uL U dL for quark sector [9,10]. In relation (1), U l and U ν are the diagonalizing matrices for charged lepton and left-handed Majorana neutrino mass matrices respectively which are defined as : In the basis where charged lepton mass matrix is diagonal, m ν is expressible as [11] m ′ ν = U † lL m ν U lL .
In the standard Particle Data Group (PDG) parametrization [10], with three mixing angles and three CP phases-one Dirac CP phase (δ) and two Majorana CP phases (α, β), PMNS matrix has the form, where c ij = cos θ ij , s ij = sin θ ij with θ 12 being the solar angle, θ 23 being the atmospheric angle and θ 13 being the reactor angle and P = diag(1, e iα , e iβ ) contains the Majorana CP phases. In our present work we ignore all the CP phases. Then under µ − τ symmetry, with θ 13 = 0, PMNS matrix takes the form [ Table 1: Best fit, 1σ and 3σ ranges of parameters for NH obtained from global analysis [22] where we have ignored the CP violating phases. For our case we first consider the neutrino mixing pattern to be of bi-maximal nature. Then U ν = U BM is given by equation (5). We then take the following form of the lepton mixing matrix [15], wheres ij = sin θ l ij andc ij = cos θ l ij . This structure (8) had been studied earlier [15] but we study it again here in the light of latest observational data [5].
From equations (1), (5) and (8), we finally obtain the PMNS matrix U P M N S = U † l U BM as Let us now assume that the charged lepton corrections are Cabbibo-Kobayashi-Maskawa (CKM) like [10], which allows us to takẽ where the Wolfestein parameter λ is related to the Cabbibo angle (θ C ) by λ = sin θ C . Under this consideration, PMNS matrix in equation (9), can be approximated to the form,  Figure 1: Variation of tan 2 θ 12 with U 2 e3 for BM mixing after taking charged lepton correction. Dotted and Dashed lines represents 1σ and 3σ bounds respectively, obtained from the global analysis [22] And the expression in equation (8) becomes It can be emphasised here that both mixing matrices in equations (11) and (12) satisfy the unitarity condition as expected. Then equation (11) leads to With λ = 0.232 corresponding to |U e3 | 2 = 0.027, we get tan 2 θ 12 ≈ 0.50 and tan 2 θ 23 = 0.946. The variations of tan 2 θ 12 with |U e3 | 2 and tan 2 θ 23 with |U e3 | 2 are shown in Fig.1 and Fig.2 respectively for both 1σ and 3σ ranges ( Table 1) of latest global observational data [22]. As expected 3σ range of for BM mixing after taking charged lepton correction. Dotted and Dashed lines represents 1σ and 3σ bounds respectively, obtained from the global analysis [22] data can accomodate both tan 2 θ 12 and tan 2 θ 23 predictions. However, the 1σ range of data just marginally covers tan 2 θ 12 prediction at tan 2 θ 12 ≈ 0.5 (TB value) but not the tan 2 θ 23 prediction within the range. Certain theoretical refinements are needed in this front.

Charged lepton correction to TB Mixing
Tri-bimaximal neutrino mixing is a special case of mixing matrix with µ − τ symmetry. It can give a very close description of the experimental data except the case: θ 13 = 0. The TB neutrino mixing matrix (U ν = U T B ) is given in equation (5). In order to account for the charged lepton correction to the TB neutrino mixing, we start with the lepton mixing matrix which satisfies unitarity condition,  Figure 3: Variation of tan 2 θ 23 with sinθ 23 for TB mixing after taking charged lepton correction. Dotted and Dashed lines represents 1σ and 3σ bounds respectively, obtained from the global analysis [22] Using the form of U ν for TB, given by equation (5), we have U P M N S =Ũ † l U T B which reproduces the following PMNS matrix first proposed by King [16], This PMNS matrix has unique property of unitarity to leading order, and also predicts tan 2 θ 23 = 1. In order to have tan 2 θ 23 < 1 in the light of present experimental data [5], we now modify the charged lepton mixing matrix (16) by the relation whereR 23 has a structure similar to that of rotation matrix and is given bỹ withs 23 = sin θ l 23 andc 23 = cos θ l 23 .
which is lesser than maximal value for non-zero tanθ 23 . Assuming that the charged lepton corrections are Cabbibo-Kobayashi-Maskawa (CKM) like, we can have [10,17]s leading to tan 2 θ 23 = 0.85, where we have adopted λ = 0.2324 and A = 0.759. The variation of tan 2 θ 23 with sinθ 23 is shown in Fig.3. The prediction on tan 2 θ 12 is fixed at TB value while the change is confined to tan 2 θ 23 only and its variation with |U e3 | 2 along with 1σ and 3σ ranges of latest global observational data [22] is shown in Fig.4. At 3σ range the prediction on tan 2 θ 23 is in fair agreement with global data as like BM case. However, in TB case we notice an improvement of our prediction at 1σ range that it just passes through the 1σ region in the plot unlike the BM case.

Effects of Dirac CP phase
In this section we would like to discuss briefly the effects of CP violating phases in the proposed schemes. To observe the effects of the Dirac type CP phase in the BM scheme we follow two ways of introducing the phase. First case assumes a CP phase φ, coming from the charged lepton sector, with the unitary matrix [20] With this U l , U P M N S = U † l U BM yields In the second approach we introduce the CP phase δ, originating from neutrino sector, by the following relation [21], where U l is given by equation (8) and R 23 and R 12 are the 3 × 3 orthogonal rotation matrices with θ 23 = π 4 and θ 12 = π 4 respectively. Then equation (24) gives, Both the cases lead to a similar form of the rephasing invariant quantity defined as J CP = Im{U e2 U µ3 U * e3 U * µ2 }. For example, we get and from equations (23) and (25) respectively. We further calculate and from equations (23) and (25) respectively, which show the dependence of solar angle on the CP phase. For maximal CP violation (sin δ = ±1) we get |J BM CP | max ≈ 0.03989. From the relation sinθ 12 = √ 2 sin θ 13 along with the approximation cosθ 12 ≈ 1 (from eq.(10)) equation (27) gives which is consistent with the result of reference [7].
To incorporate the Dirac type CP effects in TB scheme we first adopt the Tri-bimaximal-Cabbibo mixing matrix U T BC proposed by King [16].
For δ = 0 equation (30) reproduces the mixing matrix given by equation (17). Then the relation U P M N S =R † 23 U T BC produces the following desired elements of the PMNS matrix, given in the set of equations (A), modified by the CP phase δ.
We also examine the structure of the PMNS matrix under the parameterization described in equation (24) where U † l is now given by equation (18) and R 23 and R 12 are respectively decribed by θ 23 = π 4 and θ 12 = arcsin 1 √ 3 .
We then obtain the following elements of the PMNS matrix: mixing matrices have basically been derived from rotation matrices and hence the conditions of unitarity of all diagonalising matrices including the final form of PMNS matrices discussed here, are satisfied at leading order. In such situation PMNS matrix proposed by King [16] is a pointer to the right direction. Asuming the charged lepton correction is CKM-like and taking λ = 0.232 we get sin 2 θ 13 = 0.027 for both BM and TB cases. For the same value of λ we calculate tan 2 θ 12 ≈ 0.50 and tan 2 θ 23 = 0.946 < 1 for BM case. After the introduction of Dirac CP phase we observe that tan 2 θ 12 is affected by the phase, but not tan 2 θ 23 . We also find that predictions on tan 2 θ 12 and tan 2 θ 23 in terms of |U e3 | 2 are consistent with the 3σ range of latest global observational data. However, at 1σ range the predictions are not comfortable. In case of TB mixing, the charged lepton correction only deviates the atmospheric angle. The solar angle remains fixed at its TB value (tan 2 θ 23 = 0.5). For λ = 0.232 and A = 0.759 we get tan 2 θ 23 = 0.85 < 1.
The variation of tan 2 θ 23 with |U e3 | 2 shows that at 3σ range the prediction on tan 2 θ 23 is smoothly consistent with global data. However, in TB case we get better agreement of our prediction with 1σ range of global data than that in BM case. Unlike the BM case, the inclusion of Dirac CP phase in TB mixing does not affect tan 2 θ 12 and tan 2 θ 23 . Finally we obtain two important expressions for the rephasing invariant quantity: J BM CP ≈ 1 4 sin θ 13 sin δ and J T B CP ≈ 1 3 √ 2 sin θ 13 sin δ which are consistent with the results of reference [7].
The deviation of solar mixing angle tan 2 θ 12 below the value of 0.50, can be introduced in realistic µ − τ symmetric neutrino mass matrices with specific choices of value of flavour twister term [15,18,19] present in the texture of the mass matrices, without affecting the good predictions on reactor and atmospheric mixing angles.