# Octant Degeneracy and Quadrant of Leptonic CPV Phase at Long Baselineν Experiments and Baryogenesis

Authors: Kalpana Bora; Gayatri Ghosh; Debajyoti Dutta
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

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 angleθ23, 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 δCP can be predicted in this context. With respect to the latest global fit data of neutrino mixing angles, we predict the values of δCP for different cases. In this context we present favoured values of δCP (δCP range at ≥2σ) constrained by the latest updated BAU range and also confront our predictions of δCP with an up-to-date global analysis of neutrino oscillation data. We find that some region of the favoured δCP parameter space lies within the best fit values around δCP≃1.3π–1.4π. A detailed analytic and numerical study of baryogenesis through leptogenesis is performed in this framework within the nonsupersymmetric SO(10) models.


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## Body

## 1. Introduction

Today, physics is going through precision era; this is more so for neutrino physics. With the measurement of reactor mixing angleθ13 [1–7] precisely by reactor experiments, the unknown quantities left to be measured in neutrino sector are leptonic CP violating phase [8–13], octant of atmospheric angle θ23 [14–20], mass hierarchy, nature of neutrino, and so forth. Long baseline neutrino experiments (LBNE [21, 22], NOνA [23], T2K [24], MINOS [25], LBNO [26], etc.) may be very promising, in measuring many of these sensitive parameters.Measuring leptonic CP violation (CPV) is one of the most demanding tasks in future neutrino experiments [27]. The relatively large value of the reactor mixing angle θ13 measured with a high precision in neutrino experiments [1–5] 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 [28–30] 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 [31–34]. In this context, possible size of leptonic CP violation detectable through neutrino oscillations can be predicted. Recently, [8], we have explored possibilities 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(νμ→νe) 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 the 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 [35, 36]. But CPV phase measurement depends on value of reactor angle θ13, and hence precise measurement of θ13 plays crucial role in its CPV measurements. This fact was utilized recently by us [8], 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),(2)
reactor experiments.We considered both appearance (νμ→νe) and disappearance (νμ→νe) channels in both neutrino and antineutrino modes. Some of the observations made in [8] are as follows:(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 angleθ23, 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 improves its CPV sensitivity irrespective of octant.Aim of this work is to critically analyse the results presented in [8], in context of entanglement of quadrant of CPV phase and octant of θ23, and hence study the role of baryogenesis in resolving this entanglement. Though in [8] 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 [8]. Following the results of [8], 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 within the nonsupersymmetric SO(10) model [37], characterized by the presence of an intermediate mass scale where both the lepton number conservation and quark-lepton symmetry are broken. In the supersymmetric case, the two effects occur at the unification scale, and no intermediate scale is present.This is an era of precision measurements in neutrino physics. We therefore consider variation ofΔm312 range at its 3σ CL versus δCP range at ≥2σ over the corresponding distribution of χ2-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 θ13 range at its 3σ CL [6] from its central value versus δCP range at ≥2σ over χ2-minima distribution and find that for IH and LO case the allowed θ13 has a varied range altering within 8.7° to 9.4° in the upper quadrant (π to 2π) and 8.7° to 9.8° in the lower quadrant (0 to π). Similarly to IH and HO case the allowed θ13 has a varied range differing within 8.7° to 9.8°. As shown in our results in Section 4, in IH and LO case the spectrum of δCP is mostly concentrated in the region 88° for θ13 around 9.09° to 9.2°, 9.25°, 9.35° to 9.5°, and 9.6° to 9.65°. Also δCP=276.5° exists for θ13 around 9.06° to 9.12°, 9.3° to 9.45°, 9.517°. δCP=290° survives for θ13 around, 9°, 9.15° to 9.3°, 9.35° to 9.5° in the higher quadrant (π to 2π). Similarly as shown in our results in Section 4, as allowed by the updated BAU limits in IH and HO case, the parameter space of δCP in the lower quadrant (0 to π) demands δCP to be around 69°, for θ13~8.86°–8.896°, 9.06°–9.22°, 9.32°, and 9.61°. There also exists δCP=95° which constrains θ13 to be around 8.99°, 9.24°, 9.352°, 9.5–9.64°, and 9.79°. In the upper quadrant (π to 2π) for the present updated BAU constraint, the allowed region of δCP parameter space becomes constrained with δCP=257.5°, for θ13 around 8.96°, 9.0974° to 9.22°, 9.5°, 9.6°, and 9.74°. Also the BAU constraint requires δCP to be equal to 288°, for θ13 around 9°, 9.06° to 9.35°, 9.5°, 9.69°, and 9.8°. Also δCP=295° survives for θ13 around 9.09°, 9.26°, 9.34°, 9.5°, 9.6°, and 9.69°. A part of the allowed δCP parameter space is found to lie within the best fit values of δCP≃1.3π–1.4π. As constrained by the current BAU bounds we present the 3D variation of the favoured range of parameters: Δm312 range within its 3σ C.L, θ13 range at its 3σ CL, and δCP range at ≥2σ varied within [0,2π] in Figure 2.As can be seen from the results presented in Section4, from Figures 3(a) and 3(b), we find that BAU can be explained most favourably for the following possible cases: δCP=1.536π, IH and LO of θ23; δCP=1.611π, IH and LO of θ23; δCP=0.488π, IH and LO of θ23; δCP=1.638π, IH and HO of θ23; δCP=1.6π, IH and HO of θ23; δCP=1.43π, IH and HO of θ23; δCP=0.777π, IH and HO of θ23; δCP=0.5277π, IH and HO of θ23; δCP=1.436π, NH and HO of θ23. It is worth mentioning that the value of δCP=1.43π and δCP=1.436π is close to the central value of δCP from the recent global fit result [31, 34, 38–42]. It is fascinating to notice that a nearly maximal CP-violating phase δCP=3/2π has been reported by the T2K [43], NOνA [44], and Super-Kamiokande experiments [45], even if the statistical significance of all these experimental results is below 3σ level. This accords with one of our calculated favoured solutions δCP=1.536π which exactly holds with the current BAU constraints. Moreover, such hints of a nonzero δCP were already present in global analyses of neutrino oscillation data, such as the one in [31–33]. 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 θ23 by using the current data of ν mixing parameters and identify the CPV phase and θ13 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 θ23 are yet not fixed. Also, they are significant in context of precision measurements on neutrino oscillation parameters.The paper is organized as follows. In Section2, we discuss entanglement of quadrant of CPV phase and octant of θ23. In Section 3, we present a review on leptogenesis and baryogenesis. In Section 4 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 Section 5, we present our conclusions.


## 2. CPV Phase and Octant ofθ23

As discussed above, from Figure  3 of [8], 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 Figure 1 we plot CP asymmetry,(1)ACP=Pνμ→νe-Pνμ¯→νe¯Pνμ→νe+Pνμ¯→νe¯as a function of leptonic CPV phase δCP, for 0≤δCP≤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 dependence of leptogenesis on octant of θ23 and quadrant of CPV phase. From Figure 1 we see that (2)ACPLO>ACPHO.Figure 1
CP asymmetry versusδCP at DUNE/LBNE, for both the hierarchies. Red and green solid (dotted) lines are for NH (IH) with types of curve to distinguish HO and LO as the true octant, respectively.Figure 2
In (a) and (b)δCP versus χ2 sensitivity corresponding to CP discovery potential at LBNEs, for both the hierarchies and octant shown.
(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 UPMNS. The regions are obtained by varying δCP range at ≥2σ over the corresponding χ2 minima distribution from Figure 2 and θ13 with its experimental values varied within 3σ. In Figure (a) (Figure (b)) we show the plot for the IH and LO case (IH and 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.
(a) (b)For a given true hierarchy, there are eight degenerate solutions(3)δCP(first  quadrant)-θ23lower  octantδCP(second  quadrant)-θ23lower  octantδCP(third  quadrant)-θ23lower  octantδCP(fourth  quadrant)-θ23lower  octantδCP(first  quadrant)-θ23higher  octantδCP(second  quadrant)-θ23higher  octantδCP(third  quadrant)-θ23higher  octantδCP(fourth  quadrant)-θ23higher  octant.This eight-fold degeneracy can be viewed as(4)Quadrant  of  CPV  phase-Octant  ofθ23entanglement. 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 [8] 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 Figure  3 of [8]. Therefore 8-fold degeneracy of (3) gets reduced to 4-fold degeneracy, with our proposal [8]. Hence, following this 4-fold degeneracy still remains to be resolved: (5)δCP(first  quadrant)-θ23LOδCP(second  quadrant)-θ23LOδCP(third  quadrant)-θ23LOδCP(fourth  quadrant)-θ23LO.The possibility ofθ23>45°, that is, HO of θ23, is also considered in this work. In this context the degeneracy is (6)δCP(first  quadrant)-θ23HOδCP(second  quadrant)-θ23HOδCP(third  quadrant)-θ23HOδCP(fourth  quadrant)-θ23HO.In this work, we propose that leptogenesis can be used to break above-mentioned 4-fold degeneracy of (5) and (6). It is known that observed baryon asymmetry of the universe (BAU) can be explained via leptogenesis [46–50]. 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, δCP, in fermion mass matrices. If all the other parameters except leptonic δCP 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 δCP, and hence 4-fold entanglement of (5) and (6) can be broken. An experimental signature of CP violation associated with the Dirac phase δCP, in PMNS matrix [51] can in principle be obtained, by searching for CP asymmetry in ν flavor oscillation. To elucidate this proposal, we consider nonsupersymmetric SO(10) models, in which BAU arises due to leptogenesis, and this lepton-antilepton asymmetry [52] 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 mixing. We consider type I seesaw mechanism, just for simplicity.


## 3. Leptogenesis and Baryogenesis Using Type I Seesaw in SO(10) Models

In grand unified theories like SO(10), one right-handed heavy Majorana neutrino per generation is added to standard model and they couple with left-handed ν via Dirac mass matrix mD. When the neutrino mass matrix is diagonalised, we get two eigenvalues, light neutrino ~mD2/MR and a heavy neutrino state ~MR. This is called type I seesaw mechanism (it may be noted that type I seesaw may be obtained in other theories). Here, decay of the lightest of the three heavy RH Majorana neutrinos, M1, that is M3,M2≫M1, will contribute to l-l¯ asymmetry [53–58] (for leptogenesis), that is, ϵlCP. It may be noted that some results on leptogenesis in the context of SO(10) models have been discussed earlier in [59–65]. In the basis where RH ν mass matrix is diagonal, the type I contribution to ϵlCP is given by decay of M1(7)ϵlCP=ΓM1→lH-ΓM1→l¯H¯ΓM1→lH+ΓM1→l¯H¯,where Γ(M1→lH) means decay rate of heavy Majorana RH ν of mass M1 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, M2 and M3, decay. Any asymmetry produced by the out of equilibrium decay of M2 and M3 will be washed away by the lepton number violating interactions mediated by M1. Therefore, the final lepton-antilepton asymmetry is given only by the CP-violating decay of M1 to standard model leptons (l) and Higgs (H). This contribution is [66](8)ϵl=-3M18πIm⁡Δm⊙2R122+ΔmA2R132υ2∑R1j2mj,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 RRT=1. R can be parameterized as [67](9)R=DM-1YνUDK-1,where Yν is the matrix of neutrino Yukawa couplings. υ is the vacuum expectation value of ν Dirac masses. In the flavour basis, where the charged lepton Yukawa matrix, Ye, and gauge interactions are flavour-diagonal, DK=UTKU, where K=YνTMR-1Yν. U is the PMNS matrix and MR is the RH neutrino Majorana scale. On the basis of right-handed neutrinos, DM=Diag(M1,M2,M3), where M3,M2≫M1. Equation (8) relates the lepton asymmetry to both the solar (Δm212) and atmospheric (ΔmA2) mass squared differences. Thus the magnitude of the matter-antimatter asymmetry can be predicted in terms of low energy oscillation parameters, Δm212, ΔmA2, and a CPV phase. In SO(10) models, the right-handed neutrino MR is generated from the Yukawa coupling of right-handed neutrinos with the Higgs field that breaks the unification or intermediate symmetry down to the standard model [68]. When such a Higgs field takes a VEV, the right-handed neutrinos get a Majorana mass. This happens because lepton number is broken at that scale. It has been discussed in [65] that in the supersymmetric case the mass scale of the right-handed neutrino is similar to the unification scale, MR~MU~1016 GeV, while in the nonsupersymmetric case the scale of MR is about the intermediate scale, MR~MI~1011 GeV [69, 70], MI being the scale of the quark-lepton symmetry [71].As has been discussed in [66], the lepton-antilepton asymmetry gets connected to both the solar and the atmospheric mass difference square. Following [72], in this work we choose a basis where the complex orthogonal matrix R takes the form, R=VCKM×UPMNS, where VCKM is the CKM matrix containing the quark mixing angles, and UPMNS is the PMNS matrix containing neutrino mixing parameters. In equations (9) and (15) of [72], if V is taken as quark mixing matrix (where, in SO(10) theories, as quarks and leptons appear in same representation, neutrino mixing matrix can be taken to be same as quark mixing matrix at high scales) and U is taken as PMNS matrix at low energies, then we get the relation R=VCKM×UPMNS. This assumption can also be justified, as it is well known that quark mixing CKM phase alone is not sufficient to explain the BAU, and leptonic CPV phase is needed to generate the observed BAU. Here, the matrix R is orthogonal since R†R=U†V†VU=1 (because U†U=V†V=1). Thus both the quark sector (quark mixing angles, phase) and the neutrino sector (ν mixing angles and the leptonic CPV phase) appear in the expression for R and in that case R can be parameterized as (10)Im⁡R132=-s2δqc223lc213ls213q-2sδqc13qc23lc213ls12qs13qs23l+2s-δl-δqc12qc13qc23lc13ls13qs13l-2sδlc12qc213qc13ls12qs23ls13l-s2δlc212qc213qs213l-2sδlc212qc213qs213lIm⁡R122=2sδqc13qc212lc23ls12qs13qs23l+2sδqc12qc13qc12lc13ls13qs12qs23l-s2δqc212ls213qs223l-2sδl-δqc13qc12lc223ls12qs13qs12ls13l-2sδl-δqc12qc13qc23lc13ls13qs212ls23l-2sδlc213qc12lc23ls212qs12ls23ls13l+2sδl-2δqc12lc23ls213qs12ls23ls13l-2sδlc12qc213qc13ls12qs212ls23ls13l+2sδl-δqc13qc12ls12qs12qs12ls223ls13l+2s2δl-2δqc223ls213qs212ls213l+2s2δl-δqc13qc23ls12qs13qs212ls23ls2⁡13l+s2δlc213qs212qs212ls223ls213lR11=c12qc13qc12lc13l+e-iδqs13qs12ls23l-e-iδle-iδqs13qc12lc23ls13l-c13qs12qc23ls12l-e-iδlc13qs12qc12ls23ls13lR12=c12qc13qc13ls12l-e-iδqs13qc12ls23l-e-iδle-iδqc23ls12ls13ls13q-c13qs12qc12lc23l-e-iδlc13qs12qs12ls23ls13lR13=e-iδqc23lc13ls13q-c13qc13ls12qs23l-e-iδlc12qc13qs13l.Here, c23l, s12l, c13l, and so forth represent the cosine of atmospheric mixing angle, sine of solar mixing angle, and cosine of reactor mixing angle, respectively. Similarly 23q, 12q, and 13q are the quark mixing angles. δl and δq are the leptonic CPV phase and quark CPV phase, respectively. Here the R equation presumably holds at the GUT scale but weak scale values can be used in the calculations since it is well known that quark mixing parameters do not change much under renormalization group (RG) evolution in hierarchical mass spectrum of SO(10) theories. Hence CKM matrix at high scales can be used at low energies also and U is taken as PMNS matrix at low energies. We also consider R=UPMNS, where R matrix consists of ν mixing angles and the leptonic CPV phase. Thus when left-right symmetry is broken at high intermediate mass scale MR in SO(10) theory, CP asymmetry in this case is given by (11)ϵl=-3M18πIm⁡ΔmA2R132υ2∑Rij2mj,where (12)R112=cos2⁡θ12lcos2⁡θ13l,R122=sin2⁡θ12lcos2⁡θ13l,R132=cos2⁡δlsin2⁡θ13l+sin2⁡δlsin2⁡θ13l,Im⁡R132=-sin2⁡2δlsin2⁡θ13l.The neutrino oscillation data used in our numerical calculations are summarised as follows [31, 38–42]: (13)Δm21210-5eV2=7.60-0.18+0.19Δm31210-3eV2=2.48-0.07+0.052.38-0.06+0.05sin2⁡θ12=0.323±0.016sin2⁡θ23=0.567-0.124+0.0320.573-0.039+0.025sin2⁡θ13=0.0226±0.00120.0229±0.0012.For Δm312, sin2⁡θ23, and sin2⁡θ13, the quantities inside the bracket correspond to inverted neutrino mass hierarchy and those outside the bracket correspond 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 mixing using updated values of running quark and lepton masses in SUSY SO(10) have also been presented in [73]. Though we consider 3-flavour neutrino scenario, 4-flavour neutrinos with sterile neutrinos as fourth flavour are also possible [74]. It is worth mentioning that ν masses and mixing can lead to charged lepton flavor violation in grand unified theories like SO(10) [75].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 right-handed Majorana neutrinos produce a lepton asymmetry which is transformed into a baryon asymmetry [72] by electroweak sphaleron processes [76–79]. Lepton asymmetry is partially converted to baryon asymmetry through B + L violating sphaleron interactions [80]. As proposed in [65], a baryon asymmetry can be generated from a lepton asymmetry. The baryon asymmetry is defined as(14)YB=nB-nB¯s=nB-nB¯7nγ=ηB7,where nB, nB¯, and nγ are number densities of baryons, antibaryons, and photons, respectively, s is the entropy density, η is the baryon to photon ratio, and 5.7×10-10≤ηB≤6.7×10-10 (95% CL) [81]. The lepton number is converted into the baryon number through electroweak sphaleron process [76–79]: (15)YB=aa-1YL,a=8NF+4NH22NF+13NH,where NF is the number of families and NH is the number of light Higgs doublets. In case of SM, NF=3 and NH=1. The lepton asymmetry is as follows:(16)YL=dϵlg∗.d is a dilution factor and g∗=106.75 in the standard case [65] is the effective number of light degrees of freedom in the theory. The dilution factor d [65] is d=0.24/k(ln⁡k)0.6 for k≥10 and d=1/2k,d=1 for 1≤k≤10, and 0≤k≤1, respectively, where the parameter k [65] is k=MP/1.7υ232πg∗(MD†MD)11/M1; here MP is the Planck mass. We have used the form of Dirac neutrino mass matrix MD from [37].


## 4. 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 at1σ–3σ C.L, as presented in [31, 38–42]. Here, we perform numerical analysis for both the hierarchies and octants. We explore the baryon asymmetry of the universe within nonsupersymmetric SO(10) models [37] using (7)–(16) of the two hierarchies (NH and IH), two octants LO and HO, w ND, w/o ND (with and without near detector), and δCP range at ≥2σ over the corresponding distribution of χ2-minima (for maximum sensitivity from Figures 2(a) and 2(b), for which the CP discovery potential of the DUNE is maximum). For our purpose, we shall carry out general scanning of the parameters: δCP range at ≥2σ (from Figures 2(a) and 2(b)), θ13 at its 3σ CL, and Δm312 at its 3σ CL using the data given by the oscillation experiments [1–6, 31, 38–42]. In this calculation, we have chosen M1~2.154×1011 GeV. Also we note that, for values of M1<2.154×1011 GeV, the baryon asymmetry becomes lower than the observed value. We scan the parameter space for IH and HO/LO in the light of recent ratio of the baryon to photon density bounds, 5.7×10-10≤ηB≤6.7×10-10 (CMB) [81] in the following ranges: (17)δCP∈0,2πθ13∈7.8°,9.9°Δm312∈-2.54∗10-3,-2.20∗10-3eV2.Similarly constrained by the present BAU bounds we perform random scans for the following range of parameters in NH, HO/LO case: (18)δCP∈0,2πθ13∈7.7°,9.9°Δm312∈2.30∗10-3,2.65∗10-3eV2.We find that the updated BAU limit [81] together with a large θ13 [1–7] puts significant constraints on the δCP-θ13 parameter space in the IH and LO case. As can be seen from Figure 3(a), a part of the parameter space survives for δCP≃1.3–1.4π in the IH and LO case as allowed by the current BAU constraint 5.7×10-10≤ηB≤6.7×10-10 (CMB), corresponding to θ13 around 8.9°–9.5°. This leads to the conclusion that the parameter space for the best fit values of δCP≃1.3–1.4π is allowed by the present BAU constraint. The allowed regions in Figure 3(a) for the lower quadrant (0 to π) require δCP spectra, that is, δCP to be equal to 88° for θ13 around 9.09° to 9.2°, 9.25°, 9.35° to 9.5°, and 9.6° to 9.65°. Almost continuous values of δCP ranging from 50° to 140° are allowed for θ13, 9.3° to 9.5°. For θ13 around 9.7°, the values of δCP mostly favoured are 50°, 69°, 88°, 90°, 95°, 99°, 120°, and 130°. The allowed region in the upper quadrant (π to 2π) necessitates δCP to be around 276.5°, for θ13 around 9.06° to 9.12°, 9.3° to 9.45°, and 9.517° as allowed by the current BAU bounds. Also δCP=290° exists for θ13 around 9°, 9.15° to 9.3°, and 9.35° to 9.5°. Almost continuous δCP ranging from 230° to 340° are allowed for θ13 around 9.4°.For IH and HO case in Figure3(b), a part of the δCP-θ13 parameter space exists for δCP≃1.3–1.4π constrained by the current BAU limits for θ13 around 9°, 9.097°, 9.1°, 9.2°, 9.3°, 9.5°, 9.55°, 9.6°, and 9.7°. Thus the parameter space for the best fit values of δCP≃1.3–1.4π is allowed by the present BAU constraint [81]. Figure 3(b) reveals that the allowed regions for the lower quadrant (0 to π) require δCP to be around 69°, for θ13~8.86°–8.896°, 9.06°–9.22°, 9.32°, and 9.61°. There also exists δCP=88° which constrains θ13 to be around 9.0° to 9.1°. Also δCP=95° survives for θ13 around 8.99°, 9.24°, 9.352°, 9.5°–9.64°, and 9.79°. Almost continuous values of δCP ranging from 50° to 140° are allowed for θ13 varying from 8.75° to 9.75°. In the upper quadrant (π to 2π) for the present updated BAU constraint, the allowed region of δCP parameter space becomes constrained with δCP=257.5°, for θ13 around 8.96°, 9.0974° to 9.22°, 9.5°, 9.6°, and 9.74°. Also the BAU constraints necessitate δCP to be around 288°, for θ13 around 9°, 9.06° to 9.35°, 9.5°, 9.69°, and 9.8° as allowed by the current BAU bounds. Also δCP=295° survives for θ13 around 9.09°, 9.26°, 9.34°, 9.5°, 9.6°, and 9.69°. Almost continuous δCP ranging from 232° to 340° are allowed for θ13 around 8.9° and 9.65°.The constraints imposed on theδCP and θ13 parameter in NH, HO/LO space are found to be more severe as compared to IH and HO/LO space. For, NH and LO only a particular value of CP violating phase, δCP=258.5° corresponding to θ13=9.02375°, is consistent with the BAU constraint. From our analysis we find that for NH and HO case we are unable to resolve the entanglement of the quadrant of δCP and octant of θ23 since no point in the parameter space (δCP,θ13) is consistent with the recent ratio of baryon to photon density bounds, 5.7×10-10<ηB<6.7×10-10. Therefore, this indicates that IH is the most favoured hierarchy for breaking the 4-fold degeneracy of (5) and (6). All of the analysis presented above is for the case when R matrix consists of both VCKM and UPMNS.No points in theδCP-θ13 parameter space, consistent with the BAU constraint, are able to break the entanglement of the quadrant of δCP and octant of θ23, when R matrix consists of UPMNS only.In Figures4 and 5 we display the allowed 3D space (δCP, θ13, and Δm312) for breaking the 4-fold degeneracy of (5) and (6) by varying the leptonic δCP phase in all of its possible range [0–2π] at ≥2σ (from Figure 2), the mixing parameter θ13 within its 3σ level, and Δm312 at its 3σ C.L [6].Figure 4
Allowed region constrained by the present BAU bounds,5.7×10-10<ηB<6.7×10-10 for δCP, θ13, and Δm312 for the case when R matrix consists of both VCKM and UPMNS. The regions are obtained by varying δCP range at ≥2σ over the corresponding χ2 minima distribution from Figure 2 and θ13 with its experimental values varied within 3σ and Δm312 at its 3σ C.L. The results of our calculation are presented for IH and LO case.Figure 5
Allowed 3D region constrained by the present BAU bounds,5.7×10-10<ηB<6.7×10-10 for δCP, θ13, and Δm312 for the case when R matrix consists of both VCKM and UPMNS. The regions are obtained by varying δCP range at ≥2σ over the corresponding χ2 minima distribution from Figure 2, θ13 with its experimental values varied within 3σ from its central values, and Δm312 at its 3σ CL. The results of our calculation are presented for IH and HO case.From Figure4 one can easily see the favoured values of δCP, θ13, and Δm312 for IH and LO case, allowed by the updated recent ratio of photon density to baryon density bounds, 5.7×10-10<ηB<6.7×10-10 (shown in Table 1).Table 1
The summary of our calculated values ofδCP, θ13, and Δm312 in case of IH and LO for R1j elements of R matrix comprising UPMNS and VCKM.
Serial number Leptonic CPV phaseδCP [ Δ m 31 2 ]  e V 2 θ 13 Quadrant ofδCP 1 δ C P = 8 8 ° [ - 2.54 ∗ 1 0 - 3, -2.21∗10-3] 9.041 7 °, 9.0697°, 9.0974°, 9.3917°, 9.4417°, 9.6167°, 9.6417° I 2 δ C P = 276 . 5 ° [ - 2.45 ∗ 1 0 - 3, -2.21∗10-3] 9.066 7 °, 9.0974°, 9.3167°–9.4417°, 9.5167° IV 3 δ C P = 29 0 ° [ - 2.42 ∗ 1 0 - 3, -2.54∗10-3] 9 °, 9.1817°, 9.2667°, 9.3667°–9.4417° IVFor IH and HO case, the results of our numerical analysis are shown in Figure5 which shows allowed (δCP, θ13, Δm312) space as allowed by the current BAU bounds. The values of δCP, θ13, and Δm312 which are favoured simultaneously consistent with ηB constraints, 5.7×10-10<ηB<6.7×10-10 [81], are shown in Table 2. For NH case we get only one point as shown in (19)NH,HO,III  quadrant  of  LeptonicδCPphaseδCP=258.4°or1.436π.Table 2
The summary of our calculated values ofδCP, θ13, and Δm312 in case of IH and HO for R1j elements of R matrix comprising UPMNS and VCKM.
Serial number δ C P [ Δ m 31 2 ]  e V 2 θ 13 Quadrant ofδCP 1 9 5 ° [ - 2.54 ∗ 1 0 - 3, -2.21∗10-3] 8.991 7 °, 9.2417°, 9.352°, 9.5167°, 9.567°, 9.6417°, 9.7917° II 2 6 9 ° [ - 2.54 ∗ 1 0 - 3, -2.38∗10-3] 8.866 7 °, 8.8917°, 8.9667°, 9.0667°, 9.0917°, 9.1667°, 9.2167°, 9.3167°, 9.6167° I 3 14 0 ° [ - 2.54 ∗ 1 0 - 3, -2.21∗10-3] 8.766 7 °, 8.8667°, 8.9167°, 8.9667°, 9.1167°, 9.1417°, 9.1917°–9.2417°, 9.3167°, 9.3417°, 9.5667°, 9.6917° II 4 257 . 5 ° [ - 2.54 ∗ 1 0 - 3, -2.21∗10-3] 8.966 7 °, 9.0974°, 9.1417°, 9.1667°, 9.2167°, 9.4917°, 9.5917°, 9.6167°, 9.7417° III 5 29 5 ° [ - 2.3 ∗ 1 0 - 3, -2.21∗10-3] 8.491 7 °, 8.8167°, 9.0917°, 9.2667°, 9.4917°, 9.6167° IV 6 28 8 ° [ - 2.44 ∗ 1 0 - 3, -2.28∗10-3] 8.781 7 °, 8.9917°, 9.0667°, 9.1167°, 9.1417°, 9.1917°, 9.2417°, 9.3167°, 9.3417°, 9.5167°, 9.6917°, 9.7917° IV


## 5. 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 [8] which may help a precision measurement of leptonic δCP phase. In this work, we studied how the entanglement of the quadrant of leptonic CPV phase and octant of atmospheric mixing angle θ23 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 [8], 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 Δm312 and θ13 within its 3σ range from their central values. We calculated baryon to photon ratio and compared with its experimentally known best fit value.It has been found [65] that, in SUSY theories, YB does not lie in the allowed range of the order of 10-10. Nonsupersymmetric SO(10) model is favoured for baryogenesis with respect to the supersymmetric model. In the nonsupersymmetric model [37], we find that the baryon asymmetry is around the required order of magnitude YB~10-10. We have made a complete numerical analysis of the 3-dimensional parameters, δCP, θ13, and Δm312, that encode the breaking of the entanglement of the quadrant of CPV phase and octant of θ23 in the presence of the latest constraints on ηB, 5.7×10-10<ηB<6.7×10-10, by taking neutrino oscillation mixing and mass scales as indicated by the experiments. By allowing δCP range to vary within [0–2π] interval at ≥2σ over the χ2-minima distribution from Figure 2, we have studied the absolute values of both θ13 and δCP parameters in order to break the 4-fold degeneracy of (5) and (6).The current data shows a preference ofδCP towards 1.5π. From our analysis, one of the leptonic CPV phases determined in IH and LO of θ23 case is δCP=276.5° or 1.536π corresponding to θ13~9.0667°,9.0974°,9.3167°–9.4417°,9.5167°, and Δm312∈[-2.45∗10-3,-2.21∗10-3]eV2 which is near the preferred data δCP=3/2π reported in [43–45]. The current analysis reveals that our calculated value of δCP=1.43π of IH and HO of θ23 case is close to the best fit value of δCP=1.48π for inverted ordering from global fit results [31, 38–42] and also δCP=1.436π of NH and HO of θ23 case is favoured with the recent hint of δCP=1.41π for normal hierarchy [31, 38–42].The main results of this work are presented in Tables1 and 2 and (19) which show that leptonic CPV phases 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 angleθ23 are yet not fixed experimentally. Also, they are significant in context of precision measurements of neutrino oscillation parameters, specially the leptonic CPV phase, Δm312, and the reactor angle θ13.Future experiments like DUNE/LBNEs and Hyper-Kamiokande [82] looking for the leptonic CPV phase δCP together with an improvement in the precision determination on the mixing angles would certainly provide worthy information to support or rule out the scenario presented in this work for breaking the entanglement of quadrant of CPV phase and octant of θ23.



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