Octant Degeneracy, Quadrant of leptonic CPV phase at Long Baseline Neutrino Experiments and Baryogenesis

In a recent work by us, we have studied, how CP violation discovery potential can be improved at long baseline neutrino experiments (LBNE/DUNE), by combining with its ND (near detector) and reactor experiments. In this work, we discuss how this study can be further analysed to resolve entanglement of the quadrant of leptonic CPV phase and Octant of atmospheric mixing angle $ \theta_{23} $, at LBNEs. The study is done for both NH (Normal hierarchy) and IH (Inverted hierarchy), HO (Higher Octant) and LO (Lower Octant). We show how baryogenesis can enhance the effect of resolving this entanglement, and how possible values of the leptonic CP-violating phase $ \delta_{CP} $ can be predicted in this context. With respect to the latest global fit data of neutrino mixing angles, we predict the values of $ \delta_{CP} $ for different cases. In this context we present favoured values of $ \delta_{CP} $ ($ \delta_{CP} $ range at $ \geq $ 2$ \sigma $ ) constrained by the latest updated BAU range and also confront our predictions of $ \delta_{CP} $ with an up-to-date global analysis of neutrino oscillation data. We find that some region of the favoured $ \delta_{CP} $ parameter space lies within the best fit values around $ \delta_{CP} \simeq 1.3\pi-1.4 \pi $. A detailed analytic and numerical study of baryogenesis through leptogenesis is performed in this framework in a model independent way.

Measuring leptonic CP violation (CPV) is one of the most demanding tasks in future neutrino experiments [21]. The relatively large value of the reactor mixing angle θ 13 measured with a high precision in neutrino experiments [1] has opened up a wide range of possibilities to examine CP violation in the lepton sector. The leptonic CPV phase can be induced by the PMNS neutrino mixing matrix [22] which holds, in addition to the three mixing angles, a Dirac type CP violating phase in general as it exists in the quark sector, and two extra phases if neutrinos are Majorana particles. Even if we do not yet have significant evidence for leptonic CPV, the current global fit to available neutrino data manifests nontrivial values of the Dirac-type CP phase [23,24]. In this context, possible size of leptonic CP violation detectable through neutrino oscillations can be predicted. Recently, [4], we have explored possibiities of improving CP violation discovery potential of newly planned Long-Baseline Neutrino Experiments (earlier LBNE, now called DUNE) in USA. In neutrino oscillation probability expression P(ν µ → ν e ) relevant for LBNEs, the term due to significant matter effect, changes sign when oscillation is changed from neutrino to antineutrino mode, or vice-versa. Therefore in presence of matter effects, CPV effect is entangled and hence, one has two degenerate solutions -one due to CPV phase and another due to its entangled value. It has been suggested to resolve this issue by combining two experiments with different baselines [25,26]. But CPV phase measurements depends on value of reactor angle θ 13 , and hence precise measurement of θ 13 plays crucial role in its CPV measurements. This fact was utilised recently by us [4], where we have explored different possibilities of improving CPV sensitivity for LBNE, USA. We did so by considering LBNE with 1. Its ND (near detector).

And reactor experiments.
We considered both appearance (ν µ → ν e ) and disappearance (ν µ → ν e ) channels in both neutrino and antineutrino modes. Some of the observations made in [4] are 1. CPV discovery potential of LBNE increases significantly when combined with near detector and reactor experiments. 2. CPV violation sensitivity is more in LO (lower octant) of atmospheric angle θ 23 , for any assumed true hierarchy. 3. CPV sensitivity increases with mass of FD (far detector). 4. When NH is true hierarchy, adding data from reactors to LBNE improve its CPV sensitivity irrespective of octant.
Aim of this work is to critically analyse the results presented in [4], in context of entanglement of quadrant of CPV phase and octant of θ 23 , and hence study the role of baryogenesis in resolving this enganglement. Though in [4], we studied effect of both ND and reactor experiments on CPV sensitivity of the LBNEs, in this work we have considered only the effect of ND. But similar studies can also be done for the effect of Reactor experiments on LBNEs as well. The details of LBNE and ND are same as in [4]. Following the results of [4], either of the two octants is favoured, and the enhancement of CPV sensitivity with respect to its quadrant is utilized here to calculate the values of lepton-antilepton symmetry. This is done considering two cases of the rotation matrix for the fermions -CKM only, and CKM+PMNS. Then, this is used to calculate the value of BAU. This is an era of precision measurements in neutrino physics. We therefore consider variation of ∆m 2 31 range at its 3σ C.L Vs δ CP range at ≥ 2σ over the corresponding distribution of χ 2 -minima from figure 2. We calculate baryon to photon ratio, and compare with its experimentally known best fit value. As, constrained by the latest updated BAU limits, 5.7 × 10 −10 < BAU < 6.7 × 10 −10 , we plot θ 13 range at its 3 σ C.L [2] from its central value Vs δ CP range at ≥ 2σ over χ 2 -minima distribution and find that for IH, LO case the allowed θ 13 has a varied range altering within 8.7 0 to 9.4 0 in the upper quadrant (π to 2π) and 8.7 0 to 9.8 0 in the lower quadrant (0 to π). Similarly for IH, HO case the allowed θ 13 has a varied range differing within 8.7 0 to 9.8 0 . As shown in our results in Section IV, in IH, LO case the spectrum of δ CP is mostly concentrated in the region 88 0 for θ 13 around 9.09 0 to 9.2 0 , 9.25 0 , 9.35 0 to 9.5 0 , 9.6 0 to 9.65 0 . Also δ CP = 276.5 0 exists for θ 13 around 9.06 0 to 9.12 0 , 9.3 0 to 9.45 0 , 9.517 0 and δ CP = 290 0 for θ 13 around , 9 0 , 9.15 0 to 9.3 0 , 9.35 0 to 9.5 0 in the higher quadrant (π to 2π) for θ 13 around 9.09 0 to 9.5 0 . Similarly as shown in our results in Sec. 4, as allowed by the updated BAU limits in IH, HO case, the parameter space of δ CP in the lower quadrant (0 to π) demands δ CP to be around 69 0 , for θ 13 ∼ 8.86 0 − 8.896 0 , 9.06 0 − 9.22 0 , 9.32 0 , 9.61 0 . There also exists δ CP = 95 0 which constrains θ 13 to be around 8.99 0 , 9.24 0 , 9.352 0 , 9.5 − 9.64 0 and 9.79 0 . In the upper quadrant (π to 2π) for the present updated BAU constraint, the allowed region of δ CP parameter space becomes constrained with δ CP = 257.5 0 , for θ 13 around 8.96 0 , 9.0974 0 to 9.22 0 , 9.5 0 , 9.6 0 and 9.74 0 . Also the BAU constraint requires δ CP to be equal to 288 0 , for θ 13 around 9 0 , 9.06 0 to 9.35 0 , 9.5 0 , 9.69 0 , 9.8. Also δ CP = 295 0 survives for θ 13 around 9.09 0 , 9.26 0 , 9.34 0 , 9.5 0 , 9.6 0 , 9.69. A part of the allowed δ CP parameter space is found to lie within the best fit values of δ CP ≃ 1.3π − 1.4π. As constrained by the current BAU bounds we present the 3-D variation of the favoured range of parameters: ∆m 2 31 range within its 3σ C.L, θ 13 range at its 3σ C.L and δ CP range at ≥ 2σ varied within [0, 2π] in Fig. 2.
As can be seen from the results presented in Sect. IV, we find that, BAU can be explained most favourably for the following possible cases: δ CP = 1.536π, IH, LO of θ 23 ; δ CP = 1.611π, IH, LO of θ 23 ; δ CP = 0.488π, IH, LO of θ 23 ; δ CP = 1.638π, IH, HO of θ 23 ; δ CP = 1.6π, IH, HO of θ 23 ; δ CP = 1.43π, IH, HO of θ 23 ; δ CP = .777π, IH, HO of θ 23 ; δ CP = .5277π, IH, HO of θ 23 ; δ CP = 1.436π, NH, HO of θ 23 . It is worth mentioning that the value of δ CP = 1.43π and δ CP = 1.436π is close to the central value of δ CP from the recent global fit result [24,27]. It is fascinating to notice that a nearly maximal CP-violating phase δ CP = 3 2 π has been reported by the T2K [28], NOνA [29] and Super-Kamiokande experiments [30], even if the statistical significance of all these experimental results is below 3σ level. This accords with one of our calculated favoured solution δ CP = 1.536π which exactly holds with the current BAU constraints. Moreover, such hints of a nonzero δ CP were already present in global analyses of neutrino oscillation data, such as the one in Ref. [23]. Our main aim in this work is to carry out a detailed analysis of the breaking of the entanglement of the quadrant of leptonic CPV phase and Octant of θ 23 by using the current data of ν mixing parameters and identify the CPV phase and θ 13 spectrum required to get the breaking favourable with the current BAU constraint. These results could be important keeping in view that the quadrant of leptonic CPV phase, and octant of atmospheric mixing angle θ 23 are yet not fixed. Also, they are significant in context of precision measurements on neutrino oscillation parameters.
The paper is organized as follows. In Section II, we discuss entanglement of quadrant of CPV phase and octant of θ 23 . In Section III, we present a review on leptogenesis and baryogenesis. In Sec. IV we show how the baryon asymmetry (BAU) within the SO(10) model, by using two distinct forms for the lepton CP asymmetry, can be used to break the entanglement. Finally in Sec. V, we present our conclusions.

II. CPV PHASE AND OCTANT OF θ23
As discussed above, from Fig. 3 of [4], we find that by combining with ND and reactor experiments, CPV sensitivity of LBNE improves more for LO (lower octant) than HO (higher octant), for any assumed true hierarchy. In Fig. 1 below we plot CP asymmetry, as a function of leptonic CPV phase δ CP , for 0 ≤ δ CP ≤ 2π. CP asymmetry also depends on the mass hierarchy. For NH, CP asymmetry is more in LO than in HO. For IH, CP asymmetry is more in LO than in HO. In this work we have used above information to calculate dependance of leptogenesis on octant of θ 23 and quadrant of CPV phase. From Fig. 1 we see that This eight-fold degeneracy can be viewed as entanglement. Out of these eight degenerate solutions, only one should be true solution. To pinpoint one true solution, this entanglement has to be broken. We have shown [4] that sensitivity to discovery potential of CPV at LBNEs in LO is improved more, if data from near detector of LBNEs, or from Reactor experiments is added to data from FD of LBNEs as shown in Fig. 3 of [4]. Therefore 8-fold degeneracy of (3) gets reduced to 4-fold degeneracy, with our proposal [4]. Hence, following this 4-fold degeneracy still remains to be resolved.
The possibility of θ 23 > 45 0 , ie HO of θ 23 is also considered in this work. In this context the degeneracy is In this work, we propose that leptogenesis can be used to break above mentioned 4-fold degeneracy of Eq. (5), (6). It is known that observed baryon asymmetry of the Universe (BAU) can be explained via leptogenesis [31][32][33][34][35]. In leptogenesis, the lepton-antilepton asymmetry can be explained, if there are complex Yukawa couplings or complex fermion mass matrices. This in turn arises due to complex leptonic CPV phases, δ CP , in fermion mass matrices. If all other parameters except leptonic δ CP phase in the formula for lepton -antilepton asymmetry are fixed, for example, then observed value of BAU from experimental observation can be used to constrain quadrant of δ CP , and hence 4-fold entanglement of (5),(6) can be broken. An experimental signature of CP violation associated to the Dirac phase δ CP , in PMNS matrix [36], can in principle be obtained, by searching for CP asymmetry in ν flavor oscillation. To elucidate this proposal, we consider model independent scenario, in which BAU arises due to leptogenesis, and this lepton-antilepton asymmetry [37] is generated by the out of equilibrium decay of the right handed, heavy Majorana neutrinos, which form an integral part of seesaw mechanism for neutrino masses and mixings. Since our proposal is model independent, we consider type I seesaw mechanism, just for simplicity.

III. LEPTOGENESIS AND BARYOGENESIS IN TYPE I SEESAW SO(10) MODELS
In Grand Unified theories like SO(10), one right handed heavy Majorana neutrino per generation is added to Standard Model [38][39][40][41], and they couple to left handed ν via Dirac mass matrix m D . When the neutrino mass matrix is diagonalised, we get two eigen values − light neutrino ∼ m 2 D MR and a heavy neutrino state ∼ M R . This is called type I See Saw mechanism. Here, decay of the lightest of the three heavy RH Majorana neutrinos, M 1 , i.e M 3 , M 2 ≫ M 1 will contribute to l −l asymmetry (for leptogenesis), i.e ǫ CP l . In the basis where RH ν mass matrix is diagonal, the type I contribution to ǫ CP l is given by decay of M 1 where Γ(M 1 → lH) means decay rate of heavy Majorana RH ν of mass M 1 to a lepton and Higgs. We assume a normal mass hierarchy for heavy Majorana neutrinos. In this scenario the lightest of heavy Majorana neutrinos is in thermal equilibrium while the heavier neutrinos, M 2 and M 3 , decay. Any asymmetry produced by the out of equilibrium decay of M 2 and M 3 will be washed away by the lepton number violating interactions mediated by M 1 . Therefore, the final lepton-antilepton asymmetry is given only by the CP-violating decay of M 1 to standard model leptons (l) and Higgs (H). This contribution is [42]: where υ is the vev of the SM Higgs doublet that breaks the SM gauge group to U (1) em . R is a complex orthogonal matrix with the property that RR T = 1. R can be parameterized as [43]: where Y ν is the matrix of neutrino Yukawa couplings.  Figure 2: In Fig. 2a and 2b δCP Vs χ 2 sensitivity corresponding to CP discovery potential at LBNEs, for both the hierarchies and octant is shown.  (8) relates the lepton asymmetry to both the solar (∆m 2 21 ) and atmospheric (∆m 2 A ) mass squared differences. Thus the magnitude of the matter-antimatter asymmetry can be predicted in terms of low energy oscillation parameters, ∆m 2 21 , ∆m 2 A and a CPV phase. Here matrix R is dependent on both U P MN S and V CKM , and it can be shown that, ImR 2 13 = −s(2δ q )c 2 23 l c 2 13 l s 2 13 q − 2s(δ q )c13 q c23 l c 2 13 l s12 q s13 q s23 l +2s(−δ l − δ q )c12 q c13 q c23 l c13 l s13 q s13 l − 2s(δ l )c12 q c 2 13 q c13 l s12 q s23 l s13 l − s(2δ l )c 2 12 q c 2 13 q s 2 13 l − 2s(δ l )c 2 12 q c 2 13 q s 2 13 l ImR 2 12 = 2s(δ q )c13 q c 2 12 l c23 l s12 q s13 q s23 l + 2s(δ q )c12 q c13 q c12 l c13 l s13 q s12 q s23 l − s(2δ q )c 2 12 l s 2 13 q s 2 23 l − 2s(δ l − δ q )c13 q c12 l c 2 23 l s12 q s13 q s12 l s13 l − 2s(δ l − δ q )c12 q c13 q c23 l c13 l s13 q s 2 12 l s23 l − 2s(δ l )c 2 13 q c12 l c23 l s 2 12 q s12 l s23 l s13 l + 2s(δ l − 2δ q )c12 l c23 l s 2 13 q s12 l s23 l s13 l −2s(δ l )c12 q c 2 13 q c13 l s12 q s 2 12 l s23 l s13 l + 2s(δ l − δ q )c13 q c12 l s12 q s12 q s12 l s 2 23 l s13 l + 2s(2δ l − 2δ q )c 2 23 l s 2 13 q s 2 12 l s 2 13 l + 2s(2δ l − δ q ) c13 q c23 l s12 q s13 q s 2 12 l s23 l Sin 2 13 l + s(2δ l )c 2 13 q s 2 12 q s 2 12 l s 2 23 l s 2 13 l Here, c23 l , s12 l , c13 l , etc represents the cosine of atmospheric mixing angle, sine of solar mixing angle and cosine of reactor mixing angle repectively. Similarly 23 q , 12 q , 13 q are the quark mixing angles. δ l and δ q are the leptonic CPV phase and quark CPV phase respectively. When left-right symmetry is broken at high intermediate mass scale M R in SO(10) theory, CP asymmetry is given by where |R 11 | 2 = cos 2 (θ l 12 )cos 2 (θ l 13 ), |R 12 | 2 = sin 2 (θ l 12 )cos 2 (θ l 13 ), |R 13 | 2 = cos 2 (δ l )sin 2 (θ l 13 ) + sin 2 (δ l )sin 2 (θ l 13 ) and ImR 2 13 = −sin 2 (2δ l )sin 2 (θ l 13 ) The neutrino oscillation data used in our numerical calculations are summarised as follows [27].
For ∆m 2 31 , sin 2 θ 23 , sin 2 θ 13 , the quantities inside the bracket corresponds to inverted neutrino mass hierarchy and those outside the bracket corresponds to normal mass hierarchy. The errors are within the 1σ range of the ν oscillation parameters. It may be noted that some results on neutrino masses and mixings using updated values of running quark and lepton masses in SUSY SO(10) have also been presented in [44]. Though we consider 3-flavour neutrino scenario, 4-flavour neutrinos with sterile neutrinos as fourth flavour, are also possible [45]. It is worth mentionng that ν masses and mixings can lead to charged lepton flavor violation in grand unified theories like SO(10) [46].
The origin of the baryon asymmetry in the universe (baryogenesis) is a very interesting topic of current research. A well known mechanism is the baryogenesis via leptogenesis, where the out-of-equilibrium decays of heavy righthanded Majorana neutrinos produce a lepton asymmetry which is transformed into a baryon asymmetry by electroweak sphaleron processes [47][48][49]. Lepton asymmetry is partially converted to baryon asymmetry through B+L violating sphaleron interactions [50]. As proposed in [51], a baryon asymmetry can be generated from a lepton asymmetry. The baryon asymmetry is defined as: where n B , nB, n γ are number densities of baryons, antibaryons and photons respectively, s is the entropy density, η is the baryon-to-photon ratio, 5.7 × 10 −10 ≤ η B ≤ 6.7 × 10 −10 (95 % C.L) [52]. The lepton number is converted into the baryon number through electroweak sphaleron process [47][48][49].
where N f is the number of families and N H is the number of light Higgs doublets. In case of SM, N f = 3 and N H = 1. The lepton asymmetry is as follows: d is a dilution factor and g * = 106.75 in the standard case [51], is the effective number of light degrees of freedom in the theory. The dilution factor d [51] is, d = 0.24 k(lnk) 0.6 for k ≥ 10 and d = 1 2k , d = 1 for 1 ≤ k ≤ 10 and 0 ≤ k ≤ 1 respectively, where the parameter k [51] , here M P is the Planck mass. We have used the form of Dirac neutrino mass matrix M D from [53].

IV. CALCULATIONS, RESULTS AND DISCUSSION
For the purpose of calculations, we use the current experimental data for three neutrino mixing angles as inputs, which are given at 1σ − 3σ C.L, as presented in [27]. Here, we perform numerical analysis for both the hierarchies and octants. We explore the baryon asymmetry of the universe using Eq. (7)-Eq. (16) of the two hierarchies (NH and IH), two octants− LO and HO, w ND, w/o ND (with and without near detector) and δ CP range at ≥ 2σ over the corresponding distribution of χ 2 -minima (for maximum sensitivity from Fig. 2(a), 2(b), for which the CP discovery potential of the DUNE is maximum). For our purpose, we shall carry out a general scanning of the parameters: δ CP range at ≥ 2σ (from Fig. 2(a), 2(b)), θ 13 at its 3σ C.L and ∆m 2 31 at its 3σ C.L using the data given by the oscillation experiments [1,2,27]. We scan the parameter space for IH, HO/LO in in the light of recent ratio of the baryon to photon density bounds, 5.7 × 10 −10 ≤ η B ≤ 6.7 × 10 −10 (CMB) [52] in the following ranges: Similarly constrained by the present BAU bounds we perform random scans for the following range of parameters in NH, HO/LO case: We find that the updated BAU limit [52] together with a large θ 13 [1][2][3] puts significant constraints on the δ CP -θ 13 parameter space in the IH, LO case. As can be seen from Fig. 3a, a part of the paramater space survives for δ CP ≃ 1.3 − 1.4π in the IH, LO case as allowed by the current BAU constraint 5.7 × 10 −10 ≤ η B ≤ 6.7 × 10 −10 (CMB), corresponding to θ 13 around 8.9 0 − 9.5 0 . This leads to the conclusion that the parameter space for the best fit values of δ CP ≃ 1.3 − 1.4π is allowed by the present BAU constraint. The allowed regions in Fig. 3a for the lower quadrant (0 to π) requires δ CP spectra, i.e. δ CP to be equal to 88 0 for θ 13 around 9.09 0 to 9.2 0 , 9.25 0 , 9.35 0 to 9.5 0 , 9.6 0 to 9.65 0 . Almost continuous values of δ CP ranging from 50 0 to 140 0 are allowed for θ 13 , 9.3 0 to 9.5 0 . For, θ 13 around 9.7 0 , the values of δ CP mostly favoured are 50 0 , 69 0 , 88 0 , 90 0 , 95 0 , 99 0 , 120 0 , 130 0 .. The allowed region in the upper quadrant (π to 2π) necessitates δ CP to be around 276.5 0 , for θ 13 around 9.06 0 to 9.12 0 , 9.3 0 to 9.45 0 , 9.517 0 as allowed by the current BAU bounds. Also δ CP = 290 0 exists for θ 13 around , 9 0 , 9.15 0 to 9.3 0 , 9.35 0 to 9.5 0 . Almost continuous δ CP ranging from 230 0 to 340 0 are allowed for θ 13 around 9.4 0 .
(a) (b) Figure 3: Allowed region constrained by the present BAU bounds, 5.7 × 10 −10 < ηB < 6.7 × 10 −10 for δCP , θ13 for the case when R matrix consists of both VCKM and UP M NS . The regions are obtained by varying δCP range at ≥ 2σ over the corresponding χ 2 minima distribution from fig. 2 and θ13 with its experimental values varied within 3σ. In Fig. 3a (3b) we show the plot for the IH, LO case (IH, HO case). The blue (cyan) horizontal line represents δCP = 1.3π(1.4π) around which the best fit values of CPV phase δCP are assumed to lie.
The constraints imposed on the δ CP , θ 13 parameter in NH, HO/LO space are found to be more severe as compared to IH, HO/LO space. For, NH, LO only a particular value of CP violating phase, δ CP = 258.5 0 corresponding to θ 13 = 9.02375 is consistent with the BAU constraint. From our analysis we find that for NH, HO case we are unable to resolve the entanglement of the quadrant of δ CP and octant of θ 23 since no point in the parameter space (δ CP , θ 13 ) is in consistent with the recent ratio of baryon to photon density bounds, 5.7 × 10 −10 < η B < 6.7 × 10 −10 . Therefore, this indicates that IH is the most favoured hierarchy for breaking the 4-fold degeneracy of Eq. (5), (6). All the analysis presented above is for the case when R matrix consists of both V CKM and U P MN S .
No points in the δ CP − θ 13 parameter space, consistent with the BAU constraint, is able to break the entanglement of the quadrant of δ CP and octant of θ 23 , when R matrix consists of U P MN S only.

V. CONCLUSION
Measuring CP violation in the lepton sector is one of the most challenging tasks today. A systematic study of the CP sensitivity of the current and upcoming LBNE/DUNE is done in our earlier work [4] which may help a precision measurement of leptonic δ CP phase. In this work, we studied how the entanglement of the quadrant of leptonic CPV phase and octant of atmospheric mixing angle θ 23 at LBNE/DUNE, can be broken via leptogenesis and baryogenesis. Here, we have considered the effect of ND only in LBNE, on sensitivity of CPV phase measurement, but similar conclusions would hold for the effect of reactor experiments as well. This study is done for both the octants and hierarchies. We considered two cases of fermion rotation matrix -PMNS only, and CKM+PMNS. Following the results of [4], the enhancement of CPV sensitivity with respect to its quadrant is utilized here to calculate the values of lepton-antilepton symmetry. Then, this is used to calculate the value of BAU. This is an era of precision measurements in neutrino physics. We therefore considered variation of ∆m 2 31 and θ 13 within its 3σ range from their central values. We calculated baryon to photon ratio, and compared with its experimentally known best fit value.
We have made a complete numerical analysis of the 3 dimensional parameters, δ CP , θ 13 and ∆m 2 31 that encode the breaking of the entanglement of the quadrant of CPV phase and Octant of θ 23 in presence of the latest constraints on |η B |, 5.7 × 10 −10 < η B < 6.7 × 10 −10 , by taking neutrino oscillation mixings and mass scales as indicated by the experiments. By allowing δ CP range to vary within [0 − 2π] interval at ≥ 2σ over the χ 2 − minima distribution from Fig. 2, we have studied the absolute values of both θ 13 , δ CP parameters in order to break the 4-fold degeneracy of Eq. (5), (6).
The main results of this work are presented in Table I and II and Eq. (19) which show that leptonic CPV phase in all the four quadrants are allowed which lie within the constraints of present BAU. These values also contain the best fit values of leptonic CPV phase as discussed earlier.
These results could be important, as the quadrant of leptonic CPV phase, and octant of atmospheric mixing angle θ 23 are yet not fixed experimentally. Also, they are significant in context of precision measurements of neutrino oscillation parameters, specially the leptonic CPV phase, ∆m 2 31 and the reactor angle θ 13 . Future experiments like DUNE/LBNEs and Hyper-Kamionande [54] looking for the leptonic CPV phase δ CP together with an improvement in the precision determination on the mixing angles would certainly provide worthy informations to support or rule out the scenario presented in this work for breaking the entanglement of quadrant of CPV phase and Octant of θ 23 . Figure 5: Allowed 3-D region constrained by the present BAU bounds, 5.7 × 10 −10 < ηB < 6.7 × 10 −10 for δCP , θ13 and ∆m 2 31 for the case when R matrix consists of both VCKM and UP M NS . The regions are obtained by varying δCP range at ≥ 2σ over the corresponding χ 2 minima distribution from fig. 2, θ13 with its experimental values varied within 3σ from its central values and ∆m 2 31 at its 3σ C.L. The results of our calculation are presented for IH, HO case.