Predicting Leptonic CP phase by considering deviations in charged lepton and neutrino sectors

Recently, the reactor mixing angle $\theta_{13}$ has been measured precisely by Daya Bay, RENO and T2K experiments with a moderately large value. However, the standard form of neutrino mixing patterns such as bimaximal, tri-bimaximal, golden ratio of types A and B, hexagonal etc., which are based on certain flavor symmetries, predict vanishing $\theta_{13}$. Using the fact that the neutrino mixing matrix can be represented as $V_{\rm PMNS}=U_l^{\dagger} U_\nu P_\nu$, where $U_l$ and $U_\nu$ result from the diagonalization of the charged lepton and neutrino mass matrices and $P_\nu$ is a diagonal matrix containing Majorana phases, we explore the possibility of accounting for the large reactor mixing angle by considering deviations both in the charged lepton and neutrino sector. In the charged lepton sector we consider the deviation as an additional rotation in the (12) and (13) planes, whereas in neutrino sector we consider deviations to various neutrino mixing patterns through (13) and (23) rotations. We find that with the inclusion of these deviations it is possible to accommodate the observed large reactor mixing angle $\theta_{13}$, and one can also obtain limits on the CP violating Dirac phase $\delta_{CP}$ and Jarlskog invariant $J_{CP}$ for most of the cases. We then explore whether our findings can be tested in the currently running NO$\nu$A experiment with 3 years of data taking in neutrino mode followed by 3 years with anti-neutrino mode.


I. INTRODUCTION
The phenomenon of neutrino oscillations is found to be the first substantial evidence for physics beyond standard model. The results from various neutrino oscillation experiments established the fact that the three flavors of neutrinos mix with each other as they propagate and form the mass eigen states. The mixing is described by Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix V P M N S [1] analogous to the CKM mixing matrix in the quark sector, which can be parameterized in terms of three rotation angles θ 12 , θ 23 , θ 13 and three CPviolating phases, one Dirac phase δ CP and two Majorana phases ρ, σ as where c ij ≡ cos θ ij , s ij ≡ sin θ ij and P ν ≡ {e iρ , e iσ , 1} is a diagonal phase matrix, that is physically relevant if neutrinos are Majorana particles.
The solar and atmospheric neutrino oscillation parameters are precisely known from various neutrino oscillation experiments. Recently the reactor mixing angle has also been measured by the Daya Bay [2,3] RENO [4] and T2K [5,6] experiments. The global analysis of neutrino oscillation data has been performed by various groups [7][8][9][10]. After the discovery of sizable value of θ 13 , much attention has been paid to measure the last undetermined parameters of the standard model, namely the CP phases of the lepton sector. Of particular importance is the Dirac CP phase δ CP as it can be experimentally determined in neutrino oscillation experiments in the near future. Including the data from T2K and Daya Bay Forero et al. performed a global fit in Ref. [10] and found a hint for nonzero value of δ CP and a deviation of θ 23 from π/4 with the best fit values as δ CP ≃ 3π/2 and sin 2 θ 23 ≃ 0.57.
Understanding the origin of the patterns of neutrino masses and mixing, emerging from the neutrino oscillation data is one of the most challenging problem in neutrino physics. In fact, it is part of the more general fundamental problem in particle physics of understanding the origin of flavour i.e., the pattern of quark, charged lepton and neutrino masses and the mixing pattern in quark and lepton sector. As we know, the phenomenon of neutrino oscillation is characterized by two large mixing angles, the solar θ 12 and the atmospheric θ 23 mixing angles and one not so large mixing angle θ 13 . Initially it was believed that the reactor mixing angle would be vanishingly small and motivated by such anticipation many models  were proposed to explain the neutrino mixing pattern which are generally based on some kind of discrete flavor symmetries like S 3 , S 4 , A 4 etc. For an example, the tri-bimaximal (TBM) mixing pattern [11] is one such well motivated model having sin 2 θ 13 = 1 3 and sin 2 θ 23 = 1, which plays a substantive stimulus for model building. However, in TBM mixing pattern θ 13 is zero and the CP phase δ CP is consequently undefined. After the experimental discovery of moderately large θ 13 , various perturbation terms are added to the TBM mixing pattern and it was found that it can still be used to describe the neutrino mixing pattern or model building with suitable modifications [12].
Thus, in a nutshell the experimental discovery that the reactor mixing angle is moderately large caused a profound change in the subject of flavor models which describe leptonic mixing. Many older models ceased to be valid and others are suitably modified by including appropriate perturbations to accommodate the observed θ 13 . In this paper we would like to consider few such well motivated models which are based on certain discrete flavour symmetries like A 4 , µ − τ symmetry etc, and study their consequences in explaining the observed data in the neutrino sector. These models include tri-bimaximal mixing (TBM), bi-maximal mixing (BM) [13], golden ratio type A (GRA), golden ratio type B (GRB) [14], hexagonal (HG) mixing patterns. However, as we know these models do not accommodate the recently observed moderately large reactor neutrino mixing angle θ 13 and hence to be modified suitably/generously to provide the leptonic mixing angles that are in agreement with the experimental data.
Another important issue in the study of neutrino physics is the determination of the CP violating phases and in particular the Dirac CP phase δ CP . Many dedicated long baseline experiments are planned to study CP violation in the neutrino sector. The theoretical prediction for the determination of CP phase in the neutrino mixing matrix depends on the approach and the type of symmetries one uses to understand the pattern of neutrino mixing.
Obviously a sufficiently precise measurement of δ CP will serve as a very useful constraint for identifying the approaches and symmetries, if any. In this work, we would also like to explore whehter it is possible to constrain the CP phase δ CP and if so whether it is possible to verify such predictions with the data from ongoing NOνA experiment.
The paper is organized as follows. In section II we present the basic framework of our analysis. The deviations to the mixing matrix due to neutrino and charged lepton sectors are discussed in Sections III and IV respectively. Section V contains summary and conclusion.

II. FRAMEWORK
It is well known that the lepton mixing matrix arises from the overlapping of the matrices that diagonalize charged lepton and neutrino mass matrices i.e., For the study of leptonic mixing it is generally assumed that the charged lepton mixing matrix U l as an identity matrix and the neutrino mixing matrix U ν has a specific form dictated by a symmetry which generally fixes the values of the three mixing angles in U ν .
The small deviations of the predicted values of the mixing angles from their corresponding measured values are considered, in general, as perturbative corrections arising from symmetry breaking effects. A variety symmetry forms of U ν have been explored in the literature e.g., tri-bimaximal (TBM), bi-maximal (BM), Golden ratio type-A (GRA), type-B (GRB), hexagonal (HG) and so on. In the case of TBM, BM, GRA, GRB and HG forms for U ν we have θ ν 23 = π/4, while θ ν 12 takes the values sin −1 (1/ √ 3), π/4, sin −1 (1/ √ 2 + r), sin −1 ( √ 3 − r/2) (r being the golden ratio i.e., r = (1 + √ 5)/2) and π/6 respectively. Thus, the neutrino matrix U ν corresponding to these cases has the form where θ ν 12 takes different values for the different symmetry forms of U ν as discussed above. In general if the charged lepton mixing matrix is considered to be a 3 × 3 unit matrix, i.e., U l =1, for which we have i) θ 13 = 0 in all cases.
iii) sin 2 θ 12 = 0.5 for BM forms; sin 2 θ 12 = 1/3 in the TBM case sin 2 θ 12 = 0.276 and 0.345 for GRA and GRB mixing and sin 2 θ 12 = 0.25 for HG mixing. Thus, one possible way to generate corrections for the mixing angles such that all the mixing angles θ 23 , θ 12 and θ 13 should be compatible with the observed data is to include suitable perturbative corrections to both the charged lepton and neutrino mixing matrices U l and U ν respectively. In this paper, we are interested to explore such a possibility. While considering the corrections to the neutrino mixing matrix we assume the charged lepton mass matrix to be unity and for correction to the charged lepton mass matrix we consider the neutrino mixing matrix to be either TBM/BM/GRA/GRB/HG form. Furthermore, we will neglect possible corrections to U ν from higher dimensional operators and from renormalization group effects.

III. DEVIATION IN NEUTRINO SECTOR
In this section, we consider the corrections to the neutrino mixing matrix such that it can be written as where U 0 ν is one of the symmetry forms of the mixing matrix as described in Eq. (3) and U corr ν is the correction term. An important requirement is that the correction due to the matrix U corr ν should allow sizable deviation of the angle θ 13 from zero and also the required deviation of θ 23 and θ 12 so that they should be compatible with their measured values. These requirements imply that the simplest form of the correction matrix should be a rotation matrix either in 12, 23 or in 13 plane. There are several variants of this approach exist in the literature, generally for TBM mixing pattern. The main difference between the previous studies and our work is that apart from predicting the values of the mixing angles compatible with their experimental range, we have also looked into the possibility of constraining the CP phase δ CP .

A. Deviation due to 23 rotation
First we would like to consider additional rotation in the 23 plane. The PMNS mixing matrix in this case can be obtained by multiplying the neutrino mixing matrix U 0 ν with the 23 rotation matrix as follows where φ and α are arbitrary free parameters which can take any value in the range 0 ≤ φ ≤ π and 0 ≤ α ≤ 2π. The mixing angles sin 2 θ 12 , sin 2 θ 23 and sin θ 13 can be obtained using the relations Thus, with Eqs. (3), (5) and (6), one obtains the mixing angles as sin θ 13 = sin θ ν 12 sin φ , Thus, from Eq. (7), one can see that including the 23 rotation matrix as a perturbation, it is possible to have nonzero θ 13 , deviation of sin 2 θ 23 from 1/2 and sin 2 θ 12 from sin 2 θ ν which is sensitive to the Dirac CP violating phase. With Eq. (5), one can obtain the value of Jarlskog invariant as Comparing the two equations (8) and (9), one can obtain the expression for δ CP For numerical evaluation we vary the free parameters φ and α within their allowed range, i.e., 0 ≤ φ ≤ π and 0 ≤ α ≤ 2π. The value of θ ν 12 is kept fixed at the specified value, as predicted by the symmetry groups for different forms of neutrino mass matrices. With these input parameters, we present our results in Figure-  We next speculate on possible experimental indications which could support or rule out our findings. As we know neutrino physics is now entered the precision era as far as the measured parameters are concerned. The currently running experiments T2K and NOνA play a major role in this aspect. T2K will provide the accurate measurement of atmospheric neutrino mass square difference and the mixing angle while NOνA will provide complementary information about the neutrino mass ordering as well as possible indication of CP violation due to δ CP . We would like to see whether the constraints obtained on δ CP in our analysis could be probed in the NOνA experiment with either 3 years of data taking with neutrino mode and then followed by another 3 years with antineutrino mode. For our study, we do the simulations for NOνA off axis super beam experiment using GLoBES [15,16].
We have used the following input true values of neutrino oscillation parameters in our simulations: |∆m 2 ef f | = 2.4 × 10 −3 eV 2 , ∆m 2 21 = 7.6 × 10 −5 eV 2 , δ CP = 0, sin 2 θ 12 = 0.32, sin 2 2θ 13 = 0.1 and sin 2 θ 23 = 0.5. The relation between the atmospheric parameter |∆m 2 ef f | measured in MINOS and the standard oscillation parameter ∆m 2 31 in nature is given as [18] ∆m 2 31 = ∆m 2 ef f + ∆m 2 21 (cos 2 θ 12 − cos δ CP sin θ 13 sin 2θ 12 tan θ 23 ) , where ∆m 2 ef f is taken to be positive for Normal Hierarchy (NH) and negative for Inverted Hierarchy (IH). In order to obtain the allowed region for sin 2 2θ 13 and δ CP , we have generated true event spectrum by keeping above mentioned neutrino oscillation parameters as true values and generated test event spectrum by varying test values of sin 2 2θ 13 in the range [0.02:0.25] and that of δ CP in its full range[−π : π]. Finally, we calculated ∆χ 2 by comparing true event spectrum and test event spectrum. The obtained results in the sin 2 2θ 13 − δ CP plane are shown in Figure-2, which are overlayed with our predicted value of δ CP . The left panel shows the 1σ contours for the running of (3ν +0ν) years with NH as the true hierarchy.
The middle (right) panels represent (3ν + 3ν) years of data taking with NH (IH) as the true hierarchy. In these figures, the inner regions (bubbles) correspond to 1σ contours where as the outer curves represent 3σ contours. From these figures, one can see that our results are supported by NOνA data within 3σ C.L., however, with (3+3) years of data taking, NOνA could marginally exclude these results at 1σ C.L.
Now varying the free parameters in the range 0 ≤ φ ≤ π and 0 ≤ α ≤ 2π, we obtain the correlation plots between sin 2 θ 12 and sin 2 θ 13 as shown in the top left panel of Figure-2, where red, blue, green and cyan plots are for TBM, GRB, GRA and HG mixing patterns.
The correlation plot for HG (cyan) BM (not shown in the figure) lie outside the allowed 3σ regions. The δ CP phase is very loosely constrained in this case as presented in Figure-2.
We also overlayed the predicted value of δ CP for TBM over the NOνA simulated data. In this case also the predicted result is consistent with expected NOνA data. The relationship between the Jarlskog invariant J CP and sin 2 θ 13 as well as between J CP and δ CP are also shown in the figure. From the figure it can be seen that it could be possible to have large CP violation O(10 −2 ) in the lepton sector.
It should be noted that for deviation due to 12 rotation matrix does not accommodate the observed vale of θ 13 as U e3 = 0 for such case.

IV. DEVIATION IN THE CHARGED LEPTON SECTOR
In this section we will consider the deviation arising in the charged lepton sector. For the study of lepton mixing, it is generally assumed that the charged lepton mass matrix is diagonal and hence, the corresponding mixing matrix as an identity matrix. Now we first consider the simplest form of deviation for the charged lepton mixing matrix as a unitary rotation matrix either in (12), (23) or (13) plane. With this modification, one can write the PMNS matrix as where U ij is the rotation matrix in (ij) plane and U 0 ν is any one of the standard neutrino mixing matrix form TBM/BM/GRA/GRB/HG. However, correction arising due to U 23 rotation matrix is ruled out as it gives vanishing U e3 .
Now varying the free parameters in their allowed range we obtain the correlation plots between various mixing parameters, which are depicted in Figure-4. It should be noted that the correlation plots between sin 2 θ 13 and sin 2 θ 23 remain same for all the forms of neutrino mixing matrix U 0 ν as these mixing angles depend only on the free parameter φ and are independent of θ ν 12 (which takes different values for different mixing patterns). Furthermore, the CP violating phase is severely constrained in this scenario and the Jarlskog invariant is found to be significantly large as seen from the figure. Next we consider deviation due to additional rotation in 13 sector. In this case the PMNS matrix is given as The mixing angles obtained are sin θ 13 = sin φ √ 2 , sin 2 θ 23 = 1 1 + cos 2 φ , sin 2 θ 12 = 2 sin 2 θ ν 12 cos 2 φ + cos 2 θ ν 12 sin 2 φ − 1 √ 2 sin 2θ ν 12 sin 2φ cos α 1 + cos 2 φ .
The Jarlskog invariant and the CP phase are given as Since the results for this deviation pattern is almost similar to the correction due to 12 rotation case, one obtains the same constrains on δ CP as before.

B. Deviation due to combined 23 and 13 rotation
Next we allow the deviation pattern as a product of two rotations in the 23 and 13 planes.
In this case the PMNS matrix can be given as This minimal form of U l = R 23 (θ l 23 )U 13 (θ l 13 , α) has been considered based on the assumption that U l has the similar structure as the CKM matrix connecting the weak eigen states of the down type quarks to the corresponding mass eigen states. For simplicity we have kept only two rotation matrices in U l as the the third mixing angle is assumed to be highly suppressed. This approximation is quite reasonable as we know that the CKM matrix is almost diagonal with the off diagonal elements strongly suppressed by the small expansion parameter λ = sin θ C (θ C , being the Cabibbo angle). Hence, such an assumption can naturally provide the small perturbations to the various leading order neutrino mixing matrix U 0 ν . Furthermore, this approximation is quite acceptable as the mass spectrum of charged leptons exhibits similar hierarchical structure as the down type quarks, i.e., (m e , m µ ) ≈ (λ 5 , λ 2 )m τ and (m d , m s ) ≈ (λ 4 , λ 2 )m b . This may imply that the charged lepton mixing matrix has a structure similar to the down type quark mixing and is governed by the CKM matrix.
Next we have to parameterize the mixing matrix (26) in a suitable way. Since, the mixing angle θ 13 receives more deviation than θ 23 angle for TBM pattern in particular and in general for all the forms that we have considered, we assume sin θ l 13 = sin θ C = λ, where, λ is a small expansion parameter analogous to the expansion parameter of Wolfenstein parametrization of the CKM matrix. The other mixing angle θ l 23 is assumed to be of the form where the parameter A = O(1). Using Eq. (6) and keeping terms of order (λ 3 ), we obtain the mixing angles as From this Eq. (28), one can see that, all the three the mixing angles receive significant corrections to their leading order values predicted by different patterns of neutrino mixing matrices U 0 ν . The Jarlskog invariant is found to be which is sensitive to the Dirac CP violating phase.
For numerical estimation we need to know the values of the three unknown parameters A, λ and α. In this analysis we assume the small expansion parameter λ to have the same value as that of the quark sector [20]: and treat A ∼ O(1) and the CP violating phase α as free parameters, i.e., we allow them to vary in their entire range −1 ≤ A ≤ 1 and 0 ≤ δ ≤ 2π. Now varying λ in its 3σ range, and using Eqs. (28) and (31), we present the correlation plots between different oscillation parameters for TBM mixing pattern in Figure-5. From the figure, it can be seen that in this formalism, the CP violating phase is severely constrained and the value of Jarlskog invariant is found to be O(10 −2 ). The predicted values of δ CP and J CP for other mixing patterns are presented in Table-2.

V. SUMMARY AND CONCLUSION
The recent observation of moderately large reactor mixing angle θ 13 has ignited a lot of interest to understand the mixing pattern in the lepton sector. It also opens promising perspectives for the observation of CP violation in the lepton sector. The precise determination of θ 13 in addition to providing a complete picture of neutrino mixing pattern, could be a signal of underlying physics responsible for lepton mixing and for the physics beyond standard model. In this paper we have considered a number of neutrino mixing patterns which are based on some discrete flavor symmetries like S 3 , A 4 , µ − τ , etc. However, these symmetry forms of the mixing matrices predict vanishing reactor and maximal atmospheric mixing angles. To accommodate the observed value of relatively large θ 13 , we consider the corrections due to both the charged lepton and neutrino sector and have shown that it is possible to explain the observed neutrino oscillation data with such corrections. The predicted values of δ CP are expected to be supported by the data from currently running NOνA experiment with (3ν +3ν) years of data taking. We have also shown that it is possible to predict the value of CP phase with such corrections. We have also found that sizable leptonic CP violation characterized by the Jarlskog invariant J CP , i.e., |J CP | ∼ 10 −2 could be possible in these scenarios.