# Effects of Leptonic Nonunitarity on Lepton Flavor Violation, Neutrino Oscillation, Leptogenesis, and Lightest Neutrino Mass

Authors: Gayatri Ghosh; Kalpana Bora
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

Neutrino physics is a mature branch of science with all the three neutrino mixing angles and two mass squared differences determined with high precision. In spite of several experimental verifications of neutrino oscillations and precise measurements of two mass squared differences and the three mixing angles, the unitarity of the leptonic mixing matrix is not yet established, leaving room for the presence of small nonunitarity effects. Deriving the bounds on these nonunitarity parameters from existing experimental constraints, on cLFV decays such asμ→eγ, μ→τγ, and τ→eγ, we study their effects on the generation of baryon asymmetry through leptogenesis and neutrino oscillation probabilities. We consider a model where see-saw is extended by an additional singlet S which is very light but can give rise to nonunitarity effects without affecting the form on see-saw formula. We do a parameter scan of a minimal see-saw model in a type I see-saw framework satisfying the Planck data on baryon to photon ratio of the Universe, which lies in the interval 5.8×10-10<YB<6.6×10-10(BBN). We predict values of lightest neutrino mass and Dirac and Majorana CP-violating phases δCP, α, and β, for normal hierarchy and inverted hierarchy for one-flavor leptogenesis. It is worth mentioning that all these four quantities are unknown yet, and future experiments will be measuring them.


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## Body

## 1. Introduction

Neutrinos have nonzero masses. There are 3 known flavors of neutrinos,νe, νμ, and ντ, each of which couples only to the charged lepton of the same flavor. νe, νμ, and ντ are superpositions of three mass eigenstates, |να>=∑Uαi∗|νi>, where α=e,μ,τ and νi is the neutrino of definite mass mi. The cosmological constraints of the sum of the ν masses bound are ∑im(νi)<0.23eV from CMB, Planck 2015 data (CMB15+ LRG+ lensing + H0) [1]. We note that the lepton mixing matrix U has a big mixing and we know almost nothing about the phases. The discoveries of neutrino mass and leptonic mixing have come from the observation of neutrino flavor change, να→νβ. CP violation interchanges every particle in a process by its antiparticle. This CP violation can be produced by the phase δCP in U. Neutrinos can have two types of mass term in the Lagrangian−Dirac and Majorana mass terms. To determine whether Majorana masses occur in nature, so that νi¯=νi, the favorable approach to seek is Neutrinoless Double Beta Decay (0νββ).In the conventional type I see-saw framework, there are Dirac and Majorana mass matricesmD and MR in the Lagrangian:(1)L=12NR¯MRNRc+NRmDνL+h.c. The low-energy mass matrix is given by(2)mν=-mDTMR-1mD. In the usual unitarity scenario, the three active neutrinos, the flavor eigenstates νe, νμ, and ντ are connected to the mass eigenstates ν1, ν2, and ν3 via να=Nαiνi, where N†N=1. Here N is the generalized neutrino mixing matrix which could be either unitary or nonunitary. In the diagonal charged lepton basis, mν is diagonalized by a unitary matrix as(3)UPmνP†U†=mνD. The Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix is UP, where U is(4)U=c12c13s12c13s13e-iδ-s12c23-c12s23s13eiδc12c23-s12s23s13eiδs23c13s12s23-c12c23s13eiδ-c12s23-s12c23s13eiδc23c13. Here θ12=33.56∘,θ23=41.6∘(50∘),θ13=8.46∘(8.49∘) [3] (see [3] for recent global fit values) are the solar, atmospheric, and reactor angles for Normal Ordering (Inverted Ordering), respectively. The Majorana phases reside in P, where(5)P=diag1eiαeiα+β. Cosmologists suggest that, just after the Big Bang, the Universe contained equal amounts of matter and antimatter. Today the Universe contains matter but almost no antimatter. This change needs that matter and antimatter act differently (CP violation). The CP-violating scenario to explain this change is leptogenesis. Leptogenesis is a natural outcome of the see-saw mechanism. In the see-saw picture, we assume that, just as there are 3 light neutrinos ν1,ν2,ν3, there are 3 heavy right-handed neutrinos M1,M2,M3, where MR~109-14 GeV, MR~M1,M2,M3 which were there in the Hot Big Bang. The MR decays modes are(6)M⟶l-+H+,M⟶l++H-,M⟶ν+H0,M⟶ν¯+H0¯,where l- are e-,μ-,τ- and H+,H-,H0 are SM Higgs. CP violation effects in the MR decays may result from phases in the decay coupling constants. This leads to unequal numbers of leptons (l- and ν) and antileptons (l+ and ν¯) in the Universe:(7)ΓM⟶l++H-≠ΓM⟶l-+H+. In leptogenesis, CP-violating decays of heavy Majorana neutrinos create a lepton−antilepton asymmetry [4] and then B+L violating sphaleron processes [5] at and above the electroweak symmetry breaking scale convert part of this asymmetry into the observed baryon-antibaryon asymmetry. The heavy neutrinos are see-saw partners of the observed light ones.Depending on mass of the lightest heavy RH Majorana neutrinos (whose decay causes leptogenesis) the leptogenesis can be of three types: unflavored (or one-flavor), two-flavored, and three-flavored leptogenesis. For unflavored leptogenesis, valid forM1≥1011 GeV, we have taken here M1~1012 GeV where the flavor of the final state leptons plays no role. It can be shown that for lower values of M1 it depends on the flavor of the final state leptons and hence is called flavored leptogenesis [6]. Here we consider unflavored leptogenesis. For unflavored leptogenesis, the decay asymmetry in the case of hierarchical heavy neutrinos is given by [7](8)ε1=18πυ21mDmD†11∑2,3ImmDmD†1j2fMj2M12, where f(x)=-3/2x for x> 1, i.e., for hierarchical heavy neutrinos. The baryon asymmetry of the Universe is proportional to the decay asymmetry ε1. The Dirac mass matrix mD in terms of a complex and orthogonal matrix R is [8](9)mD=iMRRmνdiagU†. The compatible quantity for leptogenesis is then(10)mDmD†=MRRmνdiagU†UmνdiagR†MR=MRRmνdiagR†MR If R is real then there is no leptogenesis at all. Here, we have taken R=U (see (4)). R consists of the low-energy mixing elements and the CP phases. Since unitarity of neutrino mixing matrix has not been proved yet, if it is nonunitary, then, for the neutrino mixing matrix N to be nonunitary, we have(11)να=Nαiνi, connecting the flavor and mass states. The nonunitary matrix N can be written as(12)N=1+ηU0, where U0=U∗P. If mν, which is diagonalized by a nonunitary mixing matrix, originates from the see-saw mechanism, we have(13)mD=iMRRmνdiagN†. And, thereupon, we have(14)mDmD†=MRRmνdiagN†NmνdiagR†MR. For nonunitary UPMNS matrix, N†N=1+2U0†ηU0≠1. Leptogenesis is no longer independent of the low-energy phases. It depends on the phases in U0 as well as the phases in η. Leptogenesis [9] is one of the exceedingly well-inspired frameworks which produces baryon asymmetry of the Universe through B + L violating electroweak sphaleron process [5, 10].In the conventional type I see-saw mechanism, due to the mixing of the left-handed and right-handed neutrinos the PMNS matrix is nonunitary. Nevertheless this nonunitarity is too small to have any observable effects in lepton flavor violation (LFV) or neutrino oscillation (see [7, 11, 12]). If we want to connect the lepton mixing matrix to leptogenesis via the Casas-Ibarra parametrization, then there should not be any significant sizeable contribution to mν and leptogenesis other than the usual see-saw terms. Hence we must decouple the origin of unitary violation from these terms. Mixing of the light neutrinos with new physics creates nonunitarity in the low-energy mixing matrix. We consider here a model as used in [7] where the see-saw mechanism is enlarged by an additional singlet sector, which leads to a 9 × 9 mass matrix. Here(15)L=12νLc¯NR¯X¯0mDTmTmDMR0m0MsνLNRcXc, where the upper left block is the one corresponding to usual type I see-saw mechanism. It can be diagonalized with a unitary matrix F, such that(16)FT0mDTmTmDMR0m0MsF=mνdiag000MR000Msdiag. The form of F is as given in [7]. As discussed in [7] with proper choice of various elements in F and of Ms, it can be shown that(17)mνdiag=-NT~mνDMR-1mνDN~, which shows that N=(N†~)-1 is the lepton mixing matrix, as there is no other significant contribution to the mass term of the light neutrinos. Thus in this model the usual see-saw mechanism remains unaltered with unmodified leptogenesis. Here MR does not couple to the new singlets but a sizeable nonunitary lepton mixing matrix N can be induced, thus providing us with a framework where we can apply (14).We consider here that lepton asymmetry is generated by out-of-equilibrium decay of heavy right-handed Majorana neutrinos into Higgs and lepton within the framework of type I see-saw mechanism. In a hierarchical case of three right-handed heavy Majorana neutrinosM2,3>M1, with the lepton asymmetry created by the decay of M1, the lightest one of three heavy right-handed neutrinos is [13](18)ε1α=18πυ21mD†mD11∑2,3ImmD∗α1mD†mD1jmDαjgxj+∑2,3ImmD∗α1mD†mDj1mDαj11-xj.Hereυ = 174 GeV is the vacuum expectation value of the SM Higgs and g(x)=x(1+1/1-x-1+xln⁡((1+x)/x)), where xj=Mj2/M12.At temperatures,T≥1012 GeV, all the charged leptons are in equilibrium because the direct and inverse decays are very frequent and wash out any asymmetry. At moderate temperatures T<1012 GeV (T<109 GeV), some particles decouple and thus flavor effects play an important role in the calculation of lepton asymmetry [6, 14–18]. The regions of temperatures belonging to 109<T/GeV<1012 and T/GeV<109 are, respectively, denoted as two and three flavor regimes of leptogenesis [19].The building blocks of matter are the quarks, the charged leptons, and the neutrinos. The discovery and study of the Higgs boson at the Large Hadron Collider (LHC) have provided strong evidence that the quarks and charged leptons derive their masses from a coupling to the Higgs field. Most theorists strongly believe that the origin of the neutrino masses is different from the origin of the quark and charged lepton masses. Neutrino oscillation has proved that neutrinos have nonzero masses. We and all matter may have descended from heavy neutrinos. We list the values ofmlightest for one-flavor for different hierarchies and unitarity and nonunitarity of UPMNS in Table 2 and check whether our values of mlightest are consistent with the constraints on the absolute scale of ν masses. The new results presented in this work are as follows:(i) We have calculated new values of nonunitarity parameters ofUPMNS matrix from the bounds on rare cLFV decays.(ii) Hence we predicted the absolute value of lightestν mass in this regard. The values of lightest ν mass lie in the ranges of 0.0018 eV to 0.0023 eV, 0.048 eV to 0.056 eV, 0.05 eV to 0.054 eV, and 0.053 eV to 0.062 eV in one-flavor leptogenesis regime.(iii) All these values satisfy the constraint∑im(νi)<0.23eV.(iv) The predicted values of CP-violating phases,δCP, and Majorana phases α and β are 36∘, 72∘, 108∘, 144∘, 180∘, 216∘, 252∘, 288∘, 324∘, and 360∘. Here the calculated values of each of them, i.e., the Dirac CP-violating phase, δCP, and the Majorana phases α and β, are found to be same.The paper is organized as follows. In Section2, we show the effect of low-energy phenomenology of nonunitarity on charged lepton flavor violating decays in type I see-saw theories and present the values of various parameters used in our analysis for the generation of baryon asymmetry of the Universe through the mechanism of leptogenesis. Section 3 contains our calculations and results. Section 4 contains analysis and discussions. Section 5 summarizes the work.


## 2. Low-Energy Phenomenology of Nonunitarity and Leptogenesis

One interesting feature of nonunitarity of the PMNS matrix can be studied in rare charged lepton flavor violation (LFV) decay processes. In the light of unitarity violation in decays such asα→βγ,(α,β)=(τ,μ),(τ,e)or(μ,e), the branching ratio is [7](19)BRα→β+γBRα→β+νν¯=100α96πNN†αβ2. Also(20)BRτ→μ+γBRτ→μ+νν¯≃25α6πημτ2;BRτ→μ+γ4.2×10-10≃ημτ225×10-8.Using the latest updated constraint on BR(τ→μ+γ)=4.4×10-8 [2], one can derive bounds on |ημτ| from (23). It can be shown that(21)BRτ→μ+γBRμ→e+γ=BRτ→μ+ντνμ¯BRμ→e+νμνe¯×ημτ2ημe2. Now, we calculate the ratio (22)BRτ→μ+ντνμ¯BRμ→e+νμνe¯=0.176745.Thus we find constraints on |ημe| from (24), using the latest constraint on BR(μ→e+γ), where BR(μ→e+γ)=4.2×10-13 [2]. Again we have(23)BRτ→μ+γBRτ→e+γ=BRτ→μ+ντνμ¯BRτ→e+νμνe¯×ημτ2ητe2. From our calculation, the ratio BR(τ→μ+ντνμ¯)/BR(τ→e+νμνe¯) is(24)BRτ→μ+ντνμ¯BRτ→e+νμνe¯=2.509. And then we calculate the latest updated bounds on |ητe|. The calculations are summarized in Table 1. The study of the effects of leptonic nonunitarity on two- and three-flavored leptogenesis is presented in [20]. We note that interesting results on cLFV μ→eγ in NUSM, NUHM, NUGM, and mSUGRA models are presented in [21] in which we have predicted some values of new SUSY particles that may be detected at next run of LHC.Table 1
Our calculated constraints on nonunitarity parametersητe, ητμ, and ημe using branching ratios of latest cLFV decays taken from [2].
Serial No. Latest updated Branching Ratios Calculated bounds on cLFV Decays on|η|αβ 1 B R ( μ → e + γ ) = 4.2 × 1 0 - 13 η μ e = 6.64733013 × 1 0 - 6 2 B R ( τ → μ + γ ) = 4.4 × 1 0 - 8 η τ μ = 5.11766 × 1 0 - 3 3 B R ( τ → e + γ ) = 3.3 × 1 0 - 8 η τ e = 7.021 × 1 0 - 3Table 2
Our calculated results formlightest with inverted hierarchy, normal hierarchy, and one-flavor leptogenesis. The symbol ✓ (×) is used when YB is within (not within) updated BAU range.
Case m l i g h t e s t One Flavor NH, non-unitarity 0.0018 eV to 0.0023 eV ✓ NH, unitarity 0.048 eV to 0.056 eV ✓ IH, non-unitarity 0.05 eV to 0.054 eV ✓ IH, unitarity 0.053 eV to 0.062 eV ✓Returning to leptogenesis, the baryon asymmetry should lie in the interval5.8×10-10<YB<6.6×10-10 [22]. In general, we have taken complex and orthogonal matrix R=UPMNS, but R can be taken as R=VCKM×UPMNS in non-SUSY SO(10) models [18] (see [18] for detailed discussion) as studied in the context of breaking entanglement of octant of θ23 and δCP in the light of baryon asymmetry of the Universe through the mechanism of leptogenesis.For the normally ordered light neutrino masses, we have(25)MRdiag=diagM1,M2,M3=M1diag1,M2M1,M3M1=M1diag1,m1m2,m1m3, with m1∈[10-6eV,10-1eV], and m22-m12=7.60×10-5eV2,m32-m12=2.48×10-3eV2 as is evident from the neutrino oscillation data [3], with m1 being the lightest one of three neutrino masses. For the inverted ordered light neutrino masses, we have(26)MRdiag=diagM1,M2,M3=M1diag1,M2M1,M3M1=M1diag1,m1∗m3m22,m1m2, with m3 being the lightest one of the three neutrino masses. Next, we do the parameter scan for one-flavored leptogenesis of a minimal see-saw model satisfying the Planck data on baryon to photon ratio of the Universe for four cases:(i) Normal hierarchical structure neutrino masses, nonunitarity of PMNS matrix(ii) Normal hierarchical structure neutrino masses, unitarity of PMNS matrix(iii) Inverted hierarchical structure of neutrino masses, nonunitarity of PMNS matrix(iv) Inverted hierarchical structure neutrino masses, unitarity of PMNS matrix.We perform random scan of the parameter space for NH and IH in the light of recent ratio of the baryon to photon density bounds5.8×10-10<ηB<6.6×10-10 in the following ranges:(27)m1m3∈10-6eV,0.1eV10-6eV,0.1eV,δCP∈0,2π,α∈0,2π,β∈0,2π.While doing parameter scan, we find values of lightest neutrino mass, Majorana phasesα, β, and Dirac CPV phase δCP, for which baryon to photon ratio YB lies in the given range, for above four cases. This is done for one-flavor/unflavored leptogenesis regime.


## 3. Calculations and Results

Results of our analysis have been presented in Figures1–9. It can be seen from Figure 1 that in the one-flavor regime NH structure of neutrino masses and nonunitarity textures of PMNS matrix can give rise to correct baryon asymmetry of the Universe, 5.8×10-10<YB<6.6×10-10, if the lightest ν mass lies around 0.0018 eV to 0.0023 eV. A close numerical inspection of the situation reveals that for a lightest neutrino mass of 0.0024 eV-0.005 eV one can exceed the upper bound on YB. The dependence of lightest neutrino mass m1 on Majorana phases α, β is shown in (a), (b) of Figure 2, respectively, YB being constrained in the order 10-10. We find that M1=1012 GeV is favored in the light of baryon asymmetry of the Universe for one-flavor regime.Figure 1
Scatter plot of the lightest neutrino massm1 against the baryon asymmetry of the Universe with normal hierarchy, nonunitarity case in one-flavor leptogenesis regime. The values of m1 [eV] along the x-axis are multiplied by 10-2.Figure 2
Variation of lightest neutrino massm1 with Majorana phases α and β in case of NH, nonunitarity case in one-flavor leptogenesis regime. m1 is in eV. The values of m1 [eV] in (a) and (b) along the x-axis are multiplied by 10-2.
(a) (b)Figure 3
Scatter plot of the lightest neutrino massm1 against the baryon asymmetry of the Universe with normal hierarchy, unitarity case in one-flavor leptogenesis regime.Figure 4
Variation of lightest neutrino massm1 on Dirac CP phase δCP and Majorana phases α in case of NH unitarity case in one-flavor leptogenesis regime.
(a) (b)Figure 5
Scatter plot of the lightest neutrino massm3 against the baryon asymmetry of the Universe with inverted hierarchy, nonunitarity case in one-flavor leptogenesis regime.Figure 6
Variation of lightest neutrino massm3 against Dirac CP phase δCP and Majorana phase α in case of IH, nonunitarity case in one-flavor leptogenesis regime. The values of m3 [eV] in (b) along the x-axis are multiplied by 10-2.
(a) (b)Figure 7
Scatter plot of the lightest neutrino mass against the baryon asymmetry of the Universe and Dirac CP phaseδCP with inverted hierarchy, unitarity case in one-flavor leptogenesis regime. For YB in the order 10-10, the lightest ν mass m3 is concentrated in the region 0.053eV to 0.062 eV. The values of m3 [eV] in (b) along the x-axis are multiplied by 10-2.
(a) (b)Figure 8
Variation of lightest neutrino massm3 with Majorana phases α and β for IH, unitarity texture of UPMNS in one-flavor leptogenesis regime. The values of m3 [eV] in (a) along the x-axis are multiplied by 10-2.
(a) (b)Figure 9
Variation ofP(νμ→νe) against energy E in Long Baseline Neutrino Experiments with value of Dirac CP phase, δCP=1.45π, equal to the best fit value [3]. The yellow (red) curve in the figure corresponds to unitarity (nonunitarity) of PMNS matrix.Figure3 shows the scatter plot of the lightest neutrino mass m1 against the baryon asymmetry of the Universe with Normal hierarchy and unitary UPMNS in one-flavor regime. For YB to be in the range, 5.8×10-10<ηB<6.6×10-10, m1 lies between 0.048 eV and 0.056 eV. For YB in the order 10-10, the lightest neutrino mass m1 is mostly concentrated in the region 0.043 eV to 0.06 eV.In Figure4 we have shown the variation of lightest neutrino mass m1 with Dirac CP phase δCP and Majorana phase α for NH (normal hierarchy) and unitarity texture. For YB to be in the consistent BAU range 5.8×10-10<YB<6.6×10-10, one of the values of δCP predicted by us, i.e., δCP=252.9∘ is also favored in the recent global fit values, δCP=253.8∘ for normal hierarchy. It can be seen from Figure 5 that, in the one-flavor regime, IH structure of neutrino masses and nonunitarity textures of PMNS matrix can give rise to baryon asymmetry of the Universe, of the order of 10-10, if the lightest neutrino mass lies around 0.05 eV to 0.054 eV. Few points lie in the region, 5.8×10-10<YB<6.6×10-10.Figure6 shows the scattered plot of lightest neutrino mass m3 against Dirac CPV phase δCP and Majorana phase α in IH of neutrino mass and nonunitarity case of one-flavor leptogenesis regime. Figure 7 shows the variation of the lightest neutrino mass against the baryon asymmetry of the Universe and Dirac CP phase δCP with inverted hierarchy and unitarity case in one-flavor leptogenesis regime. For YB in the order 10-10, the lightest ν mass m3 is concentrated in the region 0.053 eV to 0.062 eV. In Figure 8 we have shown the scatter plot of m3 with Majorana phases α and β for IH and unitarity texture of UPMNS in one-flavor leptogenesis regime.In Figure9, we have shown the effect of nonunitarity on probability P(νμ→νe) for a particular case of Long Baseline Neutrino Experiments (DUNE, FNAL, USA). We have used the value of baseline to be L = 1300 Km and values of other oscillation parameters are taken from latest global fit [3]. It can be seen that nonunitarity affects the probability, which means that its effect could be studied in neutrino oscillation experiments provided we reach the required precision level.


## 4. Analysis and Discussion

We present here some comments that reflect the main inference of this work. From the results presented in Table2 we see that in all the four cases there exist values of lightest neutrino mass, satisfying the constraint ∑im(νi)<0.23eV and those of present day baryon asymmetry of the Universe. The analysis of our results can be summarized as follows:(i) One-flavor leptogenesis: The value of lightest neutrino mass shifts to higher value in IH case, as compared to NH. Nonunitarity effects decrease the value of lightest neutrino mass in both NH and IH and the diminishing effect is more severe in NH.(ii) We also found that nonunitarity affects the probability ofν oscillation P(νμ→νe).(iii) The predicted values of CP-violating phases,δCP, and Majorana phases α and β are 36∘, 72∘, 108∘, 144∘, 180∘, 216∘, 252∘, 288∘, 324∘, and 360∘. Here the calculated values of each of the Dirac CP-violating phase, δCP, and the Majorana phases α and β are found to have the same set of values, i.e., 36∘, 72∘, 108∘, 144∘, 180∘, 216∘, 252∘, 288∘, 324∘, and 360∘.


## 5. Conclusion

In conclusion, in this work, we have considered the possibility that the neutrino mixing matrix (considering charged lepton mass matrix to be diagonal),UPMNS, could be nonunitary and then calculated the limits on nonunitary parameters ημe, ητe, and ητμ (see Table 1) from latest constraints on branching ratios of cLFV decays. It is well known that in usual type I see-saw mechanism, mixing of left- and right-handed neutrinos may lead to nonunitarity but it has been found that [11] its effect is not significant for processes like lepton flavor violation and neutrino oscillation. Therefore we consider here a model (see [7]) where see-saw is extended by an additional singlet (very light) which, although inducing nonunitarity of the UPMNS matrix, leaves formula for see-saw mechanism unmodified. This nonunitarity however may affect leptogenesis. Baryogenesis through leptogenesis is believed to be responsible for producing the matter-antimatter asymmetry present in the present day Universe, which can be expressed through parameter YB (baryon to photon ratio). We then analysed how the nonunitarity of UPMNS can affect leptogenesis and hence calculated the values of lightest ν mass, Dirac CPV phase δCP, and Majorana phases α and β, such that YB lies in the present day constraints (5.8×10-10<YB<6.6×10-10). This was done using type I see-saw mechanisms for producing light neutrino masses.Above analysis was done for different cases: NH neutrino masses, unitaryUPMNS; NH neutrino masses, nonunitary UPMNS; IH neutrino masses, unitary UPMNS; IH neutrino masses, nonunitary UPMNS. We discussed these issues for unflavored leptogenesis regimes, for which M1≥1012 GeV, where M1 is the lightest one of the three heavy right-handed Majorana neutrinos, whose out-of-equilibrium decay produces lepton asymmetry (which in turn can be converted to BAU).In this work, we have calculated new limits on nonunitarity parameters using latest bounds on cLFV decays and thus predicted values of lightest neutrino mass (Table2) for both the hierarchies, which is still unknown experimentally. We also have predicted values of CPV phases -δCP (Dirac phase) and α and β (Majorana phases), which are also unknown so far (Table 3). Though Majorana phases do not affect neutrino oscillation probability, they may affect neutrino mass measurements in 0νββ experiments. Hence the results in this work are important, keeping in view that in future experiments will be endeavoring to measure the values of absolute value of neutrino mass and CP-violating phase δCP and α,β (Majorana phases). Future measurements related to Dirac CPV phase in neutrino experiments will validate or contradict some of the results presented here. Our analysis in this work only provides a benchmark for consistent works affiliated to model building.Table 3
The results for Dirac CPV phaseδCP and two Majorana phases α, β of all the four cases mentioned above in one flavor leptogenesis regime in this work. Here the calculated values of each of them (the Dirac CP-violating phase, δCP, and the Majorana phases α and β) are found to have the same set of values, i.e., 36∘, 72∘, 108∘, 144∘, 180∘, 216∘, 252∘, 288∘, 324∘, and 360∘.
δ C P, α, β 3 6 ∘, 72∘, 108∘, 14 4 ∘, 180∘, 216∘, 25 2 ∘, 288∘, 324∘, 360∘



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