Consistency of perturbed Tribimaximal, Bimaximal and Democratic mixing with Neutrino mixing data

We scrutinize corrections to tribimaximal (TBM), bimaximal (BM) and democratic (DC) mixing matrices for explaining recent global fit neutrino mixing data. These corrections are parameterized in terms of small orthogonal rotations (R) with corresponding modified PMNS matrices of the forms \big($R_{ij}^l\cdot U,~U\cdot R_{ij}^r,~U \cdot R_{ij}^r \cdot R_{kl}^r,~R_{ij}^l \cdot R_{kl}^l \cdot U$\big ) where $R_{ij}^{l, r}$ is rotation in ij sector and U is any one of these special matrices. We showed that for perturbative schemes dictated by single rotation, only \big($ R_{12}^l\cdot U_{BM},~R_{13}^l\cdot U_{BM},~U_{TBM}\cdot R_{13}^r$ \big ) can fit the mixing data at $3\sigma$ level. However for $R_{ij}^l\cdot R_{kl}^l\cdot U$ type rotations, only \big ($R_{23}^l\cdot R_{13}^l \cdot U_{DC} $\big ) is successful to fit all neutrino mixing angles within $1\sigma$ range. For $U\cdot R_{ij}^r\cdot R_{kl}^r$ perturbative scheme, only \big($U_{BM} \cdot R_{12}^r\cdot R_{13}^r$,~$U_{DC} \cdot R_{12}^r\cdot R_{23}^r$,~$U_{TBM} \cdot R_{12}^r\cdot R_{13}^r$\big ) are consistent at $1\sigma$ level. The remaining double rotation cases are either excluded at 3$\sigma$ level or successful in producing mixing angles only at $2\sigma-3\sigma$ level. We also updated our previous analysis on PMNS matrices of the form \big($R_{ij}\cdot U \cdot R_{kl}$\big ) with recent mixing data. We showed that the results modifies substantially with fitting accuracy level decreases for all of the permitted cases except \big($R_{12}\cdot U_{BM}\cdot R_{13}$, $R_{23}\cdot U_{TBM}\cdot R_{13}$ and $R_{13}\cdot U_{TBM} \cdot R_{13}$\big ) in this rotation scheme.


Introduction
The Standard Model(SM) which governs the dynamics of interactions between fundamental particles contain massless neutrinos. But now its well established fact from the solar, atmospheric and reactor neutrino experiments [1][2][3][4][5] that neutrino switches flavor while traveling because of their tiny mass and flavor mixing. Thus its a clear evidence of Physics Beyond Standard Model. The understanding of extremely small neutrino mass(∼ meV ) and magnitude of mixing among different neutrino flavors are much deliberated issues in Particle Physics. The smallness of neutrino mass scale is well explained by seesaw mechanism [6] which links it to a new physical scale in Nature. However neutrino mixing indicate interesting pattern in which two mixing angles of a three flavor scenario seems to be maximal while third one remains small. Different mixing schemes like tribimaximal(TBM) [7], bimaximal(BM) [8] and democratic mixing(DC) [9] were explored to explain experimental neutrino mixing data. All these mixing scenarios have same predictions for the reactor mixing angle viz θ 13 = 0. The atmospheric mixing angle, θ 23 = 45 • for BM and TBM while for DC it takes the value 54.7 • . The solar mixing angle is maximal (i.e. 45 • ) for BM and DC while takes the value of 35.3 • for TBM mixing. As far origin of these special structures is concerned they can easily come from various discrete symmetries like A 4 [10], S 4 [11] etc. These issues are extensively discussed in literature. However, reactor based Daya Bay [1] experiment in China presented first conclusive result of non zero θ 13 usingν e beam with corresponding significance of 5.2σ. The value of 1-3 mixing angle consistent with data at 90% CL is reported to be in the range sin 2 2θ 13 = 0.092 ± 0.016(stat) ± 0.05(syst). Previously long baseline neutrino oscillation T2K experiment [2] observed the events corresponding to ν µ → ν e transition which is also consistent with non zero θ 13 in a three flavor scenario. The value of 1-3 mixing angle consistent with data at 90% CL is reported to be in the range 5 • (5.8 • ) < θ 13 < 16 • (17.8 • ) for Normal (Inverted) neutrino mass hierarchy. These results goes in well agreement with other oscillation experiments like Double Chooz [3], Minos [4] and RENO [5]. Moreover it is evident from recent global fit [12,13] for neutrino masses and mixing angles (given in Table 2) that these mixing scenarios can't be taken at their face value and thus should be investigated for possible perturbations. This issue has been taken up many times [14,15] in literature. In particular, ref. [16] and ref. [17] discussed the perturbations that are parametrized in terms of mixing angles to DC and TBM mixing respectively for obtaining large θ 13 . Here we looked into possible perturbations which are parameterized by one and two rotation matrices and are of the forms R ij · U, U · R ij , R ij · R kl · U, U · R ij · R kl . These corrections show strong correlations among neutrino mixing angles which are weakened with full perturbation matrix. Since the form of PMNS matrix is given by U P M N S = U † l U ν so these modifications may originate from charged lepton and neutrino sector respectively. Unlike previous studies [16][17][18], we used χ 2 approach [19] to investigate the situation of mixing angle fitting in parameter space and thus capture essential information about corresponding correction parameters. Also to study correlations among neutrino mixing angles, we allowed to vary all mixing angles in their permissible limits instead of fixing one of them at a particular value to study correlation between remaining two mixing angles. This in our view show a full picture and thus we present our results in terms of 2 dimensional scatter plots instead of line plots [18]. Moreover we confined ourselves to small rotation limit which in turn justify them to pronounce as perturbative corrections. We also looked into new cases apart from the ones that were presented in [18] and thus presenting a complete analysis on these rotation schemes. Here we also take opportunity to update our previous analysis [19] on R ij · U · R kl PMNS matrix for new mixing data. We find (χ 2 min , Best fit level) for different mixing cases [19] using previous mixing data [12] and compare it with the results that are obtained with new mixing data [13]. In this study, we works in CP conserving limit and thus all phases are taken to be zero. However we would like to emphasize that CP violation can have a deep impact on these studies. But including CP violation for the characterized perturbations is a elaborative task which in our opinion needs separate attention. So we leave the discussion of corrections coming from non zero CP phase for future discussions. These results can help in understanding the structure of corrections that these well known mixing scenarios require in order to be consistent with neutrino mixing data. Thus this study may turn out to be useful in restricting vast number of possible models which offers different corrections to these mixing schemes in neutrino model building physics. It would also be interesting to inspect the origin of these perturbations in a model dependent framework. However the investigation of all such issues is left for future consideration.
The main outline of the paper is as follows. In section 2, we will give general discussion about our work. In sections 3-7, we will present our numerical results for various possible perturbation cases. Finally in section 8, we will give the summary and conclusions of our study.

General Setup
A 3 × 3 Unitary matrix can be parametrized by 3 mixing angles and 6 phases. However 5 phases can be moved away leaving behind only 1 physical phase. Thus light neutrino mixing is given in standard form as [20] Table 2: Three-flavor oscillation neutrino mixing angles from fit to global data [13]. All mixing angles are in degrees(θ • ).
The resulting value of mixing angles from above matrices is given in Table 1. All of them predict vanishing value of θ 13 = 0 • . The atmospheric mixing angle(θ 23 ) is maximal in TBM and BM scenarios while it takes the larger value of 54.7 • for DC mixing. The value of solar mixing angle (θ 12 ) is maximal in BM and DC scenarios while its value is 35.3 • for TBM case. However these mixing angles are in conflict with recent experimental observations which provide best fit values at θ 13 ∼ 8 • , θ 12 ∼ 33 • and θ 23 ∼ 41 • . Thus these well studied structures need corrections [21][22][23] in order to be consistent with current neutrino mixing data. From theoretical point of view, neutrino mixing matrix U comes from the mismatch between diagonalization of charged lepton and neutrino mass matrix and is given as l U ν where U l and U ν are the unitary matrices that diagonalizes the charged lepton (M l ) and neutrino mass matrix (M ν ). Thus corrections that can modify the original predictions of above special structures can originate from following sources. Here U l crr and U ν crr are the correction matrices corresponding to charged lepton and neutrino sector respectively. For CP conserving case, U l crr and U ν crr are the real orthogonal matrices which can be parametrized in terms of 3 mixing angles as discussed previously.
In this study, we are considering first two cases i.e. possible modifications that comes from charged lepton and neutrino sector. The 3rd case is already investigated [19] in detail before. We tested different possibilities for fitting neutrino mixing data that are governed by either one or two mixing angles of perturbation matrix. These cases exhibit strong correlations among mixing angles which are weakened or hidden in situation where all angles are non zero. Here corresponding PMNS matrix is of the forms R X · U , U · R X , R X · R Y · U and U · R X · R Y where R X and R Y denote generic perturbation matrices and U is any of these special matrix. The perturbation matrices R X and R Y can be expressed in terms of mixing matrices as R X (R Y ) = {R 23 , R 13 , R 12 } in general, where R 23 , R 13 and R 12 represent the rotations in 23, 13 and 12 sector respectively and are given by where α, β, γ denote rotation angles. The corresponding PMNS matrix for single rotation case is given by: and where (αβ) = (12), (13), (23) respectively. The corresponding PMNS matrix for two rotation matrix thus becomes: and 13) where (ij), (kl) = (12), (13), (23) respectively. Now to get neutrino mixing angles the above matrices are compared with the standard PMNS matrix.
To investigate numerically the effect of these perturbations we define a χ 2 function which is a measure of deviation from the central value of mixing angles: with θ i (P ) are the theoretical value of mixing angles which are functions of perturbation parameters (α, β, γ). θ i are the experimental value of neutrino mixing angles with corresponding 1σ uncertainty δθ i . The corresponding value of χ 2 for different mixing schemes is given in Table 3.  We investigated the role of various possible perturbations for lowering the value of χ 2 from its original prediction in different mixing schemes. A good numerical fit should produce low χ 2 value in parameter space.

Numerical Results
In this section, we will discuss numerical findings of our study for Normal Hierarchy. We studied the role of various perturbative schemes in producing large θ 13 [14] and fitting other mixing angles. In Figs. 1-50, we present the results in terms of χ 2 over perturbation parameters and θ 13 over θ 23 − θ 12 plane for various studied cases. The numerical value of correction parameters α, β and γ are confined in the range [-0.5, 0.5] so as to keep them in perturbative limits. We enforced the condition χ 2 < χ 2 i (i = TBM, DC and BM) for plotting data points. In double rotation plots of χ 2 vs perturbation parameters (θ 1 , θ 2 ) red, blue and light green color regions corresponds to χ 2 value in the interval [0, 3], [3,10] and > 10 respectively. In figures of neutrino mixing angles, light green band corresponds to 1σ and full color band to 3σ values of θ 13 . Also '×' refers to the case which is unable to fit mixing angles even at 3σ level while '-' denotes the situation where θ 13 = 0.
In order to show the correlations between left and right figures we marked the χ 2 < 3, [3,10] regions in mixing angle plots with different color codings. The white region corresponds to 3 < χ 2 < 10 while yellow region belongs to χ 2 < 3. Horizontal and vertical dashed black, dashed pink and thick black lines corresponds to 1σ, 2σ and 3σ ranges of the other two mixing angles. Now we will take up the case of various possible forms of rotations one by one: Here we first consider the perturbations for which the form of modified PMNS matrix is given by U P M N S = R l ij .U . We will investigate their role in fitting the neutrino mixing angles.

12 Rotation
This case corresponds to rotation in 12 sector of these special matrices. Since for small rotation sin θ ≈ θ and cos θ ≈ 1 − θ 2 , so the neutrino mixing angles truncated at order O (θ 2 ) for this rotation are given by (iii) However same window of α for TBM (since U 23 is same in both cases) gives low(high) value of θ 12 for its negative(positive) range. These achieved values are well away from their allowed 3σ boundary. In DC case, |α| ∈ [0.169, 0.190] for setting θ 13 in its 3σ range which in turn fixes θ 12 away from its allowed 3σ range. (iv) The minimum value of χ 2 ∼ 24.0, 204.7 and 45.0 for BM, DC and TBM case respectively.
(v) Thus perturbed BM can be consistent at 3σ level while other two cases are not favored.

13 Rotation
This case pertains to rotation in 13 sector of these special matrices. The neutrino mixing angles for small perturbation parameter γ are given by Figs. 4-6 show the numerical results corresponding to TBM, BM and DC case. The main features of this perturbative scheme are given by (i) Here atmospheric mixing angle(θ 23 ) receives only minor corrections through sin θ 13 and thus its value doesn't change much from its original prediction.
(v) Thus BM case can be consistent at 3σ level while other two cases are not viable.

23 Rotation
In this case, θ 13 doesn't receive any corrections from perturbation matrix (i.e. θ 13 = 0) and the minimum value of χ 2 ∼ 1094.7, 1094.7 and 948.2 for BM, DC and TBM case respectively. Thus we left out this case for any further discussion.
Here we first consider the perturbations for which modified PMNS matrix is given by U P M N S = R r ij .U . We will investigate the role of these perturbations in fitting the neutrino mixing data.

12 Rotation
This case pertains to the situation where θ 13 = 0 and the minimum value of χ 2 ∼ 960.7, 1123.6 and 960.7 for BM, DC and TBM case respectively. Thus we haven't investigated it

13 Rotation
This case corresponds to rotations in 13 sector of these special matrices. The mixing angles for small perturbation parameter γ are given by Figs. 9-11 show the numerical results corresponding to TBM, BM and DC case. The main features of these corrections are given as: (i) Here solar mixing angle(θ 12 ) receives only small corrections through sin θ 13 and thus its value remain close to its original prediction. (iii)However for BM and DC, we require |γ| ∈ [0.196, 0.220] to fit θ 13 in its 3σ range. But said range drives θ 12 well away from its allowed 3σ range. (iii) The minimum value of χ 2 ∼ 183. 3, 196.5 and 9.6 for BM, DC and TBM case respectively.
(iv) Thus TBM case is consistent at 3σ level while other two cases are not viable.

23 Rotation
This case corresponds to rotation in 23 sector of these special matrices. The neutrino mixing angles for small rotation parameter β are given by  sin θ 13 ≈ |βU 12 |, (5.4) sin   Thus this case is ruled out at 3σ level. (iii)However for BM and DC, we require |β| ∈ [0.196, 0.220] to fit θ 13 in its 3σ range. But said range drives θ 12 well away from its allowed 3σ range. (iii) The minimum value of χ 2 ∼ 151.1, 162.9 and 52.2 for BM, DC and TBM case respectively.
(iv) Thus all cases are ruled out at 3σ level.
6 Rotations-R l ij .R l kl .U Here we first consider the perturbations for which modified PMNS matrix is given by U P M N S = R l ij .R l kl .U . We will investigate the role of these perturbations in fitting the neutrino mixing data.

12-23 Rotation
This case corresponds to rotations in 12 and 23 sector of these special matrices. The neutrino mixing angles for small perturbation parameters α and β are given by (iv) In BM and DC case its possible to fit all mixing angles at 2σ level while TBM case is completely excluded at 3σ level.
(ii) The minimum value of χ 2 ∼ 8.8, 140.0 and 18.2 for perturbative BM, DC and TBM case respectively. (iii) For TBM and BM case its possible to fit all mixing angles at 3σ level while DC case is not consistent even at 3σ level.

13-23 Rotation
This case corresponds to rotations in 13 and 23 sector of these special matrices. The neutrino mixing angles for small perturbation parameters γ and β are given by

23-12 Rotation
This case corresponds to rotations in 23 and 12 sector of these special matrices.
(ii) For this rotation scheme its possible to get χ 2 < 1 for DC in tiny viable parameter space while for BM and TBM cases χ 2 > 11 and χ 2 > 32 respectively. (iii) The minimum value of χ 2 ∼ 11.5, 0.02 and 32.3 for BM, DC and TBM case respectively.
(iv) The TBM case is completely excluded as it prefers much lower value of θ 12 while BM can only be consistent at 3σ level. Here DC case is most preferable as it can fit all mixing angles within 1σ range.  Now we comes to the perturbations for which modified PMNS matrix is given by U P M N S = U.R r ij .R r kl . We will investigate the role of these corrections in fitting the neutrino mixing data.

12-13 Rotation
This case corresponds to rotations in 12 and 13 sector of these special matrices. The neutrino mixing angles truncated at order O (θ 2 ) for this mixing scheme is given by  sin θ 13 ≈ | − γU 11 + αγU 12 |, (7.1) sin In Figs. 33-35, we present the numerical results corresponding to BM, DC and TBM case. The main features of this perturbative scheme with θ 1 = γ and θ 2 = α are given by: (i) The perturbation parameters (α, γ) enters into these mixing angles at leading order and thus exhibit good correlations among themselves.
(ii) Here parameter space prefers two regions for mixing angles. In TBM and BM case for first allowed region θ 23 ∼ 40 • while for 2nd it lies around 50 • . In DC case higher θ 23 region remains outside 3σ band while lower θ 23 remains around 50 • .  (iii) The minimum value of χ 2 ∼ 1.1, 65.8 and 1.7 for BM, DC and TBM case respectively. (iv) It is possible to fit all mixing angles at 1σ level for TBM and BM while DC case can only be consistent at 3σ level.
(ii) This case is unfavorable for TBM and BM as χ 2 > 50 in all parameter space. However for DC case it is possible to get χ 2 < 2 in a tiny parameter region. (iii) The minimum value of χ 2 ∼ 50.8, 1.3 and 50.9 for BM, DC and TBM case respectively. (iv) TBM and BM is completely excluded while DC case can fit mixing angles at 1σ level.

13-12 Rotation
This case corresponds to rotations in 13 and 12 sector of these special matrices. The neutrino mixing angles for small perturbation parameters α and γ are given by  sin θ 13 ≈ |γU 11 |, (7.7) sin θ 23 ≈ | U 23 + γU 21 − γ 2 U 23 cos θ 13 |, (7.8) sin θ 12 ≈ | U 12 + αU 11 − α 2 U 12 cos θ 13 |. (7.9) Figs. 39-41 corresponds to BM, DC and TBM case respectively with θ 1 = γ and θ 2 = α. (i) The corrections to mixing angle θ 13 and θ 23 is only governed by perturbation parameter γ. Thus magnitude of parameter γ is tightly constrainted from fitting of θ 13 . This in turn permits very narrow ranges for θ 23 corresponding to negative and positive values of γ in parameter space. However θ 12 solely depends on α and thus can have wide range of possible values in parameter space.  (ii) The minimum value of χ 2 ∼ 15.5, 28.0 and 2.6 for BM, DC and TBM case respectively. (iii) DC and TBM are consistent at 3σ and 2σ level while BM is not consistent even at 3σ level.
(ii) The minimum value of χ 2 ∼ 146.6, 107.8 and 5.2 for BM, DC and TBM case respectively.
(iii) Thus BM and DC case is completely excluded as θ 12 ∼ 45 • which is well away from 3σ line. However TBM is still consistent at 2σ level.

Summary and Conclusions
Tribimaximal(TBM), bimaximal(BM) and Democratic(DC) mixing matrices offers to explain the neutrino mixing data with a common prediction of vanishing reactor mixing angle. The atmospheric mixing angle(θ 23 ) is maximal in TBM and BM scenarios while it takes  the larger value of 54.7 • for DC mixing. The value of solar mixing angle (θ 12 ) is maximal in BM and DC scenarios while its value is 35.3 • for TBM case. However experimental observation of non zero reactor mixing angle (θ 13 ≈ 8 • ) and departure of other two mixing angles from maximality is asking for corrections in these mixing schemes.
In this study, we investigated the perturbations around TBM, BM and DC mixing scenarios. These modifications are expressed in terms of three orthogonal rotation matrices R 12 , R 13 and R 23 which acts on 12, 13 and 23 sector of unperturbed PMNS matrix respectively. We looked into various possible cases that are governed by one and two rotation matrices with corresponding modified PMNS matrices of the forms R ij · U, R ij · R kl · U, U · R ij , U · R ij · R kl where U is any one of these special matrices. As the form of PMNS matrix is given by U P M N S = U † l U ν so these corrections may originate from charged lepton and neutrino sector respectively. In this study, we restricted ourselves to CP conserving case by setting phases to be zero. The effects of CP violation on mixing angles will be reported somewhere else. For our analysis we constructed χ 2 function which is a measure of deviation from experimental best fit values of mixing angles. The numerical findings are presented in terms of χ 2 vs perturbation parameters and as correlations among different neutrino mixing angles. The final results are summarized in Table 4 and Table 5 for single and double rotation cases respectively. For the sake of completion, we also included the results of R ij · U · R kl perturbative scheme [19] and updated them for new mixing data in Table 6.
The rotation R 12 · U , imparts negligible corrections to θ 23 and thus its value remain close to its unperturbed prediction. Here BM case can achieve χ 2 ∼ 24 and thus allowed at 3σ level while other two cases are not consistent. In rotation R 13 · U , θ 23 receives very minor corrections through θ 13 and thus stays close to its original prediction. The BM case is consistent at 3σ level while other two cases are not allowed. For R 23 · U case, θ 13 is still zero and thus it was not investigated any further. The rotation U · R 12 , doesn't impart corrections to 13 mixing angle and thus we left its further discussion. For U · R 13 , θ 12 receives very minor corrections only through θ 13 and thus its range remains quite close to its unperturbed value. The only perturbed TBM is allowed at 3σ level while other two mixing schemes are not consistent with mixing data. The U · R 23 rotation scheme is not consistent with mixing data for all three mixing schemes.
For rotation R 12 · R 13 · U , θ 23 remain close to its unperturbed value. Thus TBM and BM cases can be consistent at 3σ level while DC is not allowed. The rotation R 12 · R 23 · U for BM and DC can fit all mixing angles within 2σ level. However TBM case is completely excluded for this rotation. The perturbation scheme R 13 · R 12 · U scheme is much similar to R 12 · R 13 · U . Here also TBM and BM cases can be consistent at 3σ level while DC is not consistent. For rotation R 13 · R 23 · U , TBM case is excluded while BM and DC is consistent at 3σ level. In R 23 · R 12 · U , TBM and DC is completely excluded while BM can only be viable at 3σ level. Thus this case is not much preferable. The perturbation scheme R 23 · R 13 · U is much favorable for DC case as it successfully fits all mixing angles at 1σ level with lowest value of χ 2 among all mixing schemes. However TBM case is excluded for this rotation while BM can only be viable at 3σ level.
The perturbation scheme U · R 12 · R 13 is much preferable since it is possible to fit all mixing angles within 1σ range for TBM and BM case. However DC case can only be allowed at 3σ level. The rotation scheme U · R 12 · R 23 is only preferable for DC while the other two cases in scheme are excluded at 3σ level. The mixing angles can be fitted at 1σ level in this rotation for DC case. In rotation U · R 13 · R 12 , it is possible to fit all mixing angles within 2σ range for perturbed TBM matrix and 3σ level for perturbed DC case. However BM is not viable as it failed to fit mixing angles even at 3σ level. The perturbed rotation U · R 13 · R 23 imparts corrections to θ 12 only at O(θ 2 ). Thus its value remain quite close to its original value and hence BM and DC cases are excluded while TBM can be consistent at 2σ level. The rotation scheme U · R 23 · R 12 is only little preferable for DC case as it is possible to fit mixing angles at 3σ level while other two cases are not consistent. Like U · R 13 · R 23 case, rotation scheme U · R 23 · R 13 also imparts negligible corrections to θ 12 . Thus BM and DC case is excluded while perturbed TBM case is still consistent at 2σ level.
We also updated our previous analysis [19] on R ij · U · R kl PMNS matrices for new mixing data. To draw the comparison, we found (χ 2 min , Best fit level) for different mixing cases [19] using old mixing data [12] and compared it with new obtained values. The corresponding results are given in Table 6. In all allowed cases, except R 12 · U BM · R 13 , R 23 · U T BM · R 13 and R 13 · U T BM · R 13 , minimum χ 2 value increases which imply fitting gets tougher for those cases. With further improvement of accuracy on mixing angles, many cases might be ruled out completely.
This completes our discussion on checking the consistency of various perturbative rotations with neutrino mixing data. This study might turn out to be useful in restricting vast number of possible models which offers different corrections to these mixing schemes in neutrino model building physics. It thus can be a guideline for neutrino model building. However all such issues including the origin of these perturbations are left for future investigations.

Acknowledgements
The author acknowledges the support provided by CERN ROOT group for data plotting.

A Results: Summary
In this appendix, we summarized all our results for easy reference on discussed rotation schemes. We also updated our previous analysis [19] with new mixing data which was done for R ij · U · R kl rotation scheme.
In Table 4, we presented our result corresponding to R ij · U, U · R ij PMNS matrices. Table 5 and Table 6 contains the results corresponding to R ij · R kl · U, U · R ij · R kl and R ij · U · R kl PMNS matrices respectively.   Table 6: A comparison of (χ 2 min , Best fit level) for perturbation scheme that are dictated by R ij · U · R kl PMNS matrix with previous results [19]. Since previous mixing data [12] have two regions of θ 23 so here we mentioned best fit among those two possibilities.