Quark-Lepton Complementarity Predictions for $\theta_{23}^{pmns}$ and CP Violation

In the light of recent experimental results on $\theta_{13}^{pmns}$, we re-investigate the complementarity between the quark and lepton mixing matrices and obtain predictions for most unsettled neutrino mixing parameters like $\theta_{23}^{pmns}$ and CP violating phase invariants $J$, $S_1$ and $S_2$. This paper is motivated by our previous work where in a QLC model we predicted the value for $\theta_{13}^{pmns}=(9{^{+1}_{-2}})^\circ$, which was found to be in strong agreement with the experimental results. In the QLC model the non-trivial correlation between CKM and PMNS mixing matrices is given by a correlation matrix ($V_{c}$). We do numerical simulation and estimate the texture of the $V_{c}$ and in our findings we get a small deviation from the Tri-Bi-Maximal (TBM) texture and a large from the Bi-Maximal one, which is consistent with the work already reported in literature. In the further investigation we obtain quite constrained limits for $sin^2{\theta_{23}^{pmns}}= 0.4235_{-0.0043}^{+0.0032}$ that is narrower to the existing ones. We also obtain the constrained limits for the three CP violating phase invariants $J$, $S_1$ and $S_2$: as $J<0.0315$, $S_{1}<0.12$ and $S_{2}<0.08$, respectively.


Introduction
In the past several years the results of neutrino oscillation experiments summarized in particle data group [1] have provided us a very robust evidence that the neutrinos are massive, the lepton flavours are mixed and they oscillate. The most recent result from Daya-Bay and other experiments [2,3] show relatively large value of θ pmns 13 , which was also suggested by a number of theoretical and phenomenological analyses [4]. So, the current experimental situation is such that we measured all the quark and charged lepton masses, and the value of the difference between the squares of the neutrino masses ∆m 2 12 = m 2 2 − m 2 1 and |∆m 2 23 | = |m 2 3 − m 2 2 |. We also know the value of the quark mixing angles and the mixing angles θ pmns 12 , θ pmns 23 and θ pmns 13 in the lepton sector. What still to be settled are mainly the absolute mass of neutrinos, mass hierarchy in the neutrino mass spectrum (i.e. to determine the sign of ∆m 2 23 ), and nature of neutrinos (Dirac or Majorana), etc. [5]. The precision on θ pmns 23 angle, in which quadrant it lies, is another challenge to settle. Finally, as θ pmns 13 is non-zero and not too small, there arises a hope to measure the CP violating phases.
In the line of Cabibbo-Kobayashi-Maskawa(U ckm ) mixing matrix in quark sector, the phenomenon of lepton flavour mixing is described by a 3×3 unitary matrix called Pontecorvo-Maki-Nakagawa-Sakata (U pmns ). Investigating global data fits of the experimental results, so far we have got a picture which suggests that the U pmns matrix contains two large and a small mixing angles; i.e. the θ pmns 23 ≈ 45 • , the θ pmns 12 ≈ 34 • and the θ pmns 13 ≈ 9 • . As these results are seen along with the quark flavour mixing matrix (U ckm ), where all the three mixing angles are small i.e. θ ckm 12 ≈ 13 • , θ ckm 23 ≈ 2.4 • and θ ckm 13 ≈ 0.2 • , a disparitycum-complementarity between quark and lepton mixing angles is noticed. Since, the quarks and leptons are fundamental constituents of matter and also that of particles' Standard Model(SM), the complementarity between the two families is seen as a consequence of a symmetry at some high energy scale. This complementarity popularly named 'Quark-Lepton Complementarity'(QLC) has been explored by several authors [6]- [11]. The relation is quite appealing to do the theory and phenomenology, however it is still an open question, what kind of symmetry could be there between these fundamental particles of two sectors.The possible consequences of QLC have been widely investigated in the literature. In particular a simple correspondence between the U pmns and U ckm matrices has been proposed and used by several authors [12]- [15] and analysed in terms of a correlation matrix [16]- [22].
The main motivation of the present work is to re-visit the QLC Model, which we proposed in 2007 [23]. The work came into light, when its predictions for the reactor mixing angle θ pmns 13 = (9 +1 −2 ) • was found to be in strong agreement with the experimental results [2,3]. In the light of recent reactor angle data and updated experimental statistics on the other mixing parameters it was necessary to work again on the model, investigate and update the predictions. Keeping in view the fact that the neutrino physics has entered in the precision era, we chose a more accurate Wolfenstein parametrization for U ckm i.e. preserving unitarity upto the sixth order -O(λ 6 )-so called Next-to-Leading order [24].
Using all these ingredients we re-investigate the probability density textures of the correlation matrix V c , numerically. We found that the most favoured pattern of V c slightly deviates from TriBi-Maximal (TBM) and largely from Bi-Maximal (BM) one. As per our investigations a clear non-trivial structure of V c and the strong indication of gauge coupling unification at high scale helped us to put constraints on the most un-settled angle θ pmns 23 and on unknown CP violation invariants J, S 1 and S 2 .
In the next section (2), we describe in brief the theory of the QLC model and the investigation of correlation matrix (V c ) using Monte Carlo simulation is done in section (3). The verification of our previous work with (V c ) 13 = 0 are checked and an updated version of V c matrix texture is obtained in the same section. As per the model procedure, using the most probable texture of the correlation matrix we derive the constraints on the θ pmns 23 mixing angle and lepton CP violating phase invariants J, S 1 , and S 2 in the section (4). Finally, the conclusions are summarized in the section (5).

Theory of the QLC Model
The Standard Model of particle physics (SM) provides the best theoretical description of physical world at energies so far probed by experiments. The neutrinos are the only fermions in SM without right-handed partners and any electrical charge. Since they do not have any right handed partners, they are massless. However, the recent experimental results show that they mix and do oscillate, such that they are massive and have relatively small masses. This suggests a need to learn physics beyond the SM framework i.e. any new phenomena must appear at some scale associated with the existence of neutrino masses and mixing. Any realistic model, such as see-saw model, must produce such tiny non-zero masses and large mixing when reduced to an effective low energy theory. Thus it is quite possible that SM of particle interactions is a low-energy limit of some underlying theory whose true structure will emerge only when higher energy scales are probed. Certainly a deep understanding of the algebraic relationship between quarks and leptons at high energy scale will be interesting. At the same time the stability check of the RGE effects on the model equation is also essential.
The flavour mixing stems from the mismatch between the left handed rotations of the uptype and down-type quarks, and the charged leptons and neutrinos. Such mixing of quarks and leptons has always been of great interest and remains a mystery in particle physics. In the SM the mixing of quark and lepton sectors is described by the matrices U ckm and U pmns , respectively, which show up in the charged current interaction described by the Lagrangian as: where This relation when viewed with the observed pattern of mixing angles of quarks and leptons and combined with the pursuit for unification i.e. symmetry at some high energy leads the concept of quark lepton complementarity i.e. QLC. To look for such unification, it is useful to work in a basis where the quark and lepton Yukawa matrices are related. In general, the Yukawa matrices for quarks are taken as Y u and Y d i.e. for up and down quark sectors, respectively. The diagonalizing matrices are given by where the Y ∆ are diagonal and the U and V are unitary matrices. Such that the observable quark mixing matrix U ckm is given by However, for the charged lepton sector, the Yukawa matrix is given by For neutrino sector, we introduce one right-handed singlet neutrino per family; i.e. M R as the Majorana mass matrix for the right handed neutrino and M D the Dirac mass matrix. This leads to light neutrino masses given by the Type-I see-saw mechanism after breakdown of electroweak symmetry where U l , V l and U 0 , V 0 diagonalize the charged lepton and M D , respectively.
The neutrino mass matrix can be rewritten as where V c represents the rotation of M ∆ The mixing matrix V c is here defined to verify the equality U ν ≡ U 0 V c and is such that in lines of V ckm the lepton mixing matrix is given by In grand unification, the down-type quarks and the charged leptons are in general assigned into a multiplet, we assume that the following simple relations hold In the same way, if we call Y ν the Yukawa coupling that will generate the Dirac neutrino mass matrix M D , we have also the relation Such that If we further assume Y ν ≈ Y u , which can be realized in some larger gauge group such as generic SO (10). In addition, for symmetric form of the down-type quark Yukawa matrix, there arises an interesting relationship amongst the quark mixing, lepton mixing and the correlation matrices V c . Combining the relations derived for quark and lepton we get and ,Using the equations we get The form of V c can be obtained under some assumptions about the flavor structure of the theory where the quantity Ψ is a diagonal matrix Ψ = diag(e ιψ i ) and the three phases ψ i are free parameters as they are not restricted by present experimental evidences. This is more appropriate to do because in Grand Unified Theories (GUTs), once quarks and leptons are inserted in the same representation of the underlying gauge group, in order to counter the phase mismatch one has to include arbitrary but non-trivial phases between the quark and lepton mixing matrices.
The simplest possiblity for the correlation matrix, which has been widely explored in literature is V c = U ckm · U pmns . Here the correlation matrix V c is taken bi-maximal in nature such that two angles of 45 • and one 0 • . Such that, the original form proposed for QLC relation was where θ l,q ij are the respective lepton and quark mixing angles. However, in our model we adopt a more general and democratic approach for the correlation matrix i.e. it may take any form of texture as suggested by the input form theory and experimental data from quark and lepton sectors.

Numerical Analysis
In this section we investigate the texture of V c correlation matrix taking into account the experimental updates and the Wolfenstein parametrization for U ckm i.e. unitary up to O(λ 6 ) also called as Next-to-Leading order [24] as The Wolfenstein parameters λ, A, ρ, η are where On the other hand the lepton mixing matrix U pmns is parametrized as [23] Here φ ∼ diag(1, 1, e ιφ ) and φ m ∼ diag(e ιφ 1 , e ιφ 2 , 1) are diagonal matrices containing the Dirac and Majorana CP violating phases, respectively. Such that However, the values of the U ckm parameters [1] and U pmns angles [25] are taken as under at 1-σ level For the unknown phases φ 1 , φ 2 and the three ψ i (14), as they are not constrained by any experimental data, we vary their values between the interval [0, 2π] in a flat distribution.
We perform the investigation by using the numerical method of Monte Carlo simulation, which generated one billion i.e. 10 8 values for each variable with two-sided Gaussian distributions around the mean values of the observables including the quark parameters A, λ,ρ,η. Such that we obtained fine, smooth histogram for probability density of each elements of the V c .
In our previous work we have seen that the 13-element of the correlation matrix is strongly weighted to zero, namely The corresponding histograms of probability density distribution for all the 9-elements of V c matrix are shown in figure 1 as the respective nine panels.
In these histograms we have compared the V c matrix with the BM and TBM structure of matrices. The solid lines correspond to the TBM matrix and the dashed line in all the panels represents the corresponding BM matrix element. It can be seen from the figure(s) that the V c generated is deviated slightly from TBM and largely from BM. This has also been reported earlier in the literature [26]. There are panels in the figure corresponding to 13, 23, 33 elements in which the dashed and solid lines are overlapped, as their values are exactly same.
On comparing the weighted value of each element taken from the histograms, in figure 1, of V c matrix with corresponding elements of BM and TBM matrices, we can see as  .
We can see that this is exactly the equation 21 of the work [23]. Furthermore for the CKM matrix up to the order O(λ 6 ) the above equation becomes

Results and Discussions
Unlike our previous work, we see a probabilistic departure from the BM and TBM pattern, i.e. specifically V 13 c = 0. The reason behind such a situation is that the estimation of this element V 13 c is weighted heavily by the value θ pmns 13 as compared to the other PMNS angles and CKM parameters. So, keeping θ pmns 13 free to vary flatly between 0.00 to 0.32 as we did in [23] the nontrivial combination PMNS angles, λ's and other non-restricted phases etc. leads to the probability density of the element V 13 c to peak at zero. However, for a very restricted Gaussian range for θ pmns 13 the peak of probability distribution of the element V 13 c gets shifted to 0.18, as in the present study.
It may be noted that there exists some flavor models, which imply a correlation V c with (V c ) 13 = 0. In some discrete flavor symmetries such as A 4 dynamically broken into Z 3 , as in [27] and [28], or S 3 softly broken into S 2 , as in [29], the TBM structure appears naturally. However, in the present case where V c deviates from BM and TBM structure i.e. V 13 c = 0, we need to look for some relevant flavour models. As per our model procedure, in order to constrain the lepton mixing parameter namely the less constrained ones θ pmns 23 and the three lepton CP violating phase invariants (J, S 1 , S 2 ) we use the inverse equation [23]  In the figure 4 we compare our results with the values of Global data analysis given by the various groups [4]. This figure can be further compared with results of [21] where the similar comparison is done for QLC Leading and Next-to-Leading order. As such, two things seen are quite common to discuss that by considering the higher order approximation in the CKM matrix the 1-σ ranges of sin 2 θ pmns 23 values is constricted significantly and are consistent with all the data fits available, but quite constrained.

CP violating invariants in the lepton sector
We also investigate the consequences of the non-trivial structure of the V c correlation matrix upon the undetermined CP invariants in the lepton sector. As we know that there are two kinds of CP invariants parametrizing CP violating effect in the leptonic sector. Analogous to the quark sector Jarlskog invariant J that parametrizes the effects related to the Dirac phase, and two additional CP invariants S 1 and S 2 that parametrize the effects related to the Majorana phases, which arise if neutrinos are Majorana particles. The J invariant describes all CP breaking observables in neutrino oscillations. The most general form of this is given as Using the parametrization 20 the equation becomes J = 1 8 sin 2θ 12 sin 2θ 13 sin 2θ 23 cos θ 13 sin φ; Similarly, the other two CP invariants S 1 and S 2 are related to Majorana phases, as under Here the mixing angles and phases involved on the RHS of the CP invariants are obviously corresponding to the PMNS mixing matrix. From the above equations we can see that the two Majorana phases appear in S 1 and S 2 but not in J. Using simulation for all the three equations of CP invariants, the effect of non-zero but sizeable value of θ 13   The above results can compared to our previous work [22] where J assumes any value between -0.04 & +0.04 and |S 1 | < 0.14, |S 2 | < 0.11 at 1 − σ level for BM and TBM structure of the V c .    Thus it is clear that our present ranges for the CP invariants are narrower i.e. J by 21%, S 1 by 14% and S 2 by 27%.

Conclusions
The recent non-zero value ( (9 • )) of reactor mixing angle (θ pmns 13 ) from Daya Bay, RENO and other experiments [2,3] was found to be in strong agreement with the one predicted by our group [23] in 2007. This drew quite a large attention of scientific community towards the model we used. We took benefit of recent experimental developments and re-investigated our model i.e. mainly the correlation between the U ckm quark and U pmns lepton mixing matrices. In fact, the advantages of the present work over the similar other ones existing in literature e.g. the most recent one of Junpei in [24] are quite remarkable. In fact, we had the following motivation points to continue: 1) Model prediction already tested by the experiments; 2) Experimental updates and global data fits available, specifically on the value of θ pmns

13
; 3) Wolfenstein parametrization up to the order O(λ 6 ) to obtain better precision; 4) Unlike other work it was a numerical study supplemented by the analytical one; 5) Quite general and natural in procedure.
As such, it was imperative to re-visit the model, update the things and get some more insight for further predictions. Being most general and natural in procedure, i.e. not restricting the phase mismatch matrix, the disadvantage of this model is that it gives too broad range of parameters even if the experimental data in hand have had quite a large precision. Despite of the fact, after performing rigorous numerical computations and analytical study, we have got interesting results to share with the scientific community.
In a detailed analysis we estimated and found the texture of the correlation matrix V c , which is slightly deviated from a TriBi-Maximal pattern, and largely with a Bi-Maximal pattern. This observation that V c is much closer to TBM than BM has been also reported in literature [26]. This conclusion endorsed the results of the previous studies [16] and is in agreement with other qualitative arguments that favor the CKM matrix to measure the deviation of the PMNS matrix from exact Bimaximal mixing [30].
The most interesting result and conclusion we could draw from this work is the quadrant of θ pmns 23 , which is significantly shifted to < 45 • , i.e. about 5 σ below the maximal mixing.
We obtained a notable improvement in precision of the predictions of angle as as we moved from 4 th order to the 6 th order approximation i.e. the 1-σ region of θ pmns 23 is constrained by about 90%.
Another consequences of the model are the predictions for CP violating phase invariants J, S 1 and S 2 ; i.e. |J |< 0.0315 |S 1 |< 0.12 |S 2 |< 0.08, which are constrained by 21%, 14%, and 27%, respectively as compared to the previous results.
The observed deviation from TBM and BM values can lead to violation of the TBM conditions and that is to deviating significantly from the TMB form of neutrino mass matrix. It can be a manifestation of some other symmetry existing at high energy scale. In other words, we obtain a texture, which suggests a new kind of flavour symmetry at high scale or TBM broken at low energy scale or could be a weakly broken TBM at the high scale. This opens up several new question to be addressed.
As far as the stability of the QLC relation is concerned, the kind of model relation obtained from the GUT models with some flavor symmetry has been checked by the several works and has been concluded that they are stable under the RGE effects and the radiative corrections are small for supersymmetric parameter tan β ≤ 40 [31]- [34]. It is quite possible that the QLC relation holds good at the unification scale after counting RGE effects such as for tan β ≤ 40 the deviation in the V c from TBM structure is introduced. If supersymmetry is discovered with tan β ≤ 40 and that would be a strong hint for some relevant flavor symmetry models and their specific Higgs pattern. The future test of the model will be the results from neutrino experiments, in particular, regarding determination of the quadrant of θ 23 and the observance of CP violation, and relevant flavour symmetry found at the high energy scale.