# Implications ofδlCP~270° and θ23≳45∘ for Texture Specific Lepton Mass Matrices and 0νββ Decay

Authors: Rohit Verma
Journal: Advances in High Energy Physics
 (2016)

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## Abstract

We study the phenomenological consequences of recent results from atmospheric and accelerator neutrino experiments, favoring normal neutrino mass orderingm1<m2<m3, a near maximal lepton Dirac CP phase δl~270∘ along with θ23≳45∘, for possible realization of natural structure in the lepton mass matrices characterized by (Mij)~O(mimj) for i,j=1,2,3. It is observed that deviations from parallel texture structures for Ml and Mν are essential for realizing such structures. In particular, such hierarchical neutrino mass matrices are not supportive for a vanishing neutrino mass mν1→0 characterized by DetMν≠0 and predict mν1≃(0.1–8.0) meV, mν2≃(8.0–13.0)meV, mν3≃(47.0–52.0)meV, Σ≃(56.0–71.0)meV, and 〈mee〉≃(0.01–10.0)meV, respectively, indicating that the task of observing a 0νββ decay may be rather challenging for near future experiments.


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## 1. Introduction

One of the intriguing phenomena in particle physics is the origin of fermion masses which appear to span several orders of magnitudes starting with neutrinos to the top quark. The masses and flavor mixing schemes of quarks and leptons are significantly different with the quark sector exhibiting strong mass hierarchy, small mixing angles, and relatively heavier mass spectrum whereas the neutrinos are extremely light while two of their mixing angles are still large. In the current scenario, there is also a lack of consensus on the nature of neutrinos, that is, Dirac or Majorana along with doubts on the possible ordering of neutrino masses, namely, normal, that is,m1<m2<m3 (NO), or inverted, that is, m3<m1<m2 (IO). This nevertheless makes the task of constructing the fermion mass matrices nontrivial especially in the context of quark-lepton complementarity.The confirmation of Higgs Boson by the ATLAS and CMS collaborations [1] completes the Standard Model (SM) of particle physics. Within this model, the quark mass terms in the Lagrangian are expressible as(1)-Lmassquarks=q¯uLMuquR+q¯dLMdqdR+h.c.,where quL(R) and qdL(R) are the left (right) handed quark fields and Mq are the quark mass matrices with u, d for the “up” type and “down” type quarks. The resulting weak charged current quark interactions are given by(2)-Lccquarks=g2uct¯LγμVCKMdsbLWμ++h.c.,where VCKM=ULu†ULd is the Cabibbo-Kobayashi-Maskawa (CKM) matrix [2, 3] or the quark mixing matrix measuring the nontrivial mismatch between the flavor and mass eigenstates of quarks; for example,(3)ULu†MuMu†URu=Diagmu2,mc2,mt2,ULd†MdMd†URd=Diagmd2,ms2,mb2, where Uq are unitary matrices.Interestingly, the quark masses as well as the elements of CKM matrix observe a hierarchical pattern: namely,m1≪m2≪m3 and (Vub,Vtd)<(Vcb,Vts)<(Vus,Vcd)<(Vud,Vcs,Vtb). It is natural to expect this hierarchy to be embedded within the corresponding quark mass matrices: namely, (for q=u,d)(4)M11<M12,21≲M13,31<M22<M23,32<M33,with M22≪M33. Recent investigations [4] in this regard indicate that the current quark mixing data indeed permit the quark mass matrices to have such a natural and hierarchical structure provided (Mij)~O(mimj) for i,j=1,2,3, i≠j and (Mii)~O(mi). Such hierarchical mass matrices have been referred to asnatural mass matrices [5]. In particular, naturalness provides a rationale framework to correlate the observed fermion mass ratios, the corresponding mass matrices, and observed mixing angles. Specifically, for (M13)=M31≠0, the observed strong hierarchy among the quark masses and CKM elements gets naturally translated onto the structure of the corresponding mass matrices.A concomitant of such naturalness in mass matrices is the absence of parallel texture structure for the “up” and “down” type quark mass matrices [4]. Aparallel texture structure corresponds, for example, to the mass matrices Mu and Md with texture zeros at identical positions in both the mass matrices. Hierarchical structures have fetched greater importance in the literature as these predict certain very simple yet compelling relations among the CKM elements and the quark mass ratios [4, 6–14].However, the mass spectrum for leptons is quite distinguished from the quark sector, wherein the charged leptons masses are strongly hierarchical, that is,me≪mμ≪mτ, while at least two of the neutrinos are allowed to have the same order of mass. It should be interesting to investigate if naturalness can provide a unique explanation for the fermion mass matrices, corresponding observed mass spectra, and the mixing angles both for the quark and the lepton sectors.Since the neutrinos are massless within the SM, one has to explore beyond the realms of SM to comprehend the origin of neutrino masses and observe neutrino oscillations phenomenon. A simplistic way to achieve this is to extend the SM theory by assuming neutrinos as Dirac-like particles. In this case, the neutrinos acquire mass through the Higgs mechanism in the similar way as quarks and charged leptons do within the SM, through a Dirac mass term: for example,(5)-Lmassl=l¯LMllR+h.c.,-2LmassDirac=ν¯LMνDνR+h.c.,where Ml and MνD represent the charged-lepton and Dirac neutrino mass matrix, respectively. Indeed, the current experiments have not ruled out such a possibility. In this context, it is also observed that highly suppressed Yukawa couplings for Dirac neutrinos can naturally be achieved using models with extra spatial dimensions [15, 16] or through radiative mechanisms [17–22]. However, such a possibility is perceived to be highly unlikely due to several orders of magnitude difference among mα (α=e,μ,τ) and mνi (i=1,2,3).A rather convincing and natural explanation of neutrino masses can be obtained if neutrinos are assumed to be Majorana particles. This usually involves adding the lepton number (and flavor) violating Majorana mass terms for neutrinos in the Lagrangian; for example,(6)-2LmassMajorana=ν¯LMνLνRc+ν¯LcMνRνR,where MνL and MνR correspond, respectively, to the left and right-handed Majorana neutrino mass matrices and the latter usually has an extremely high mass scale. This facilitates generating the light neutrino masses through Type I or Type II seesaw mechanisms: namely,(7)Mν=-MνDTMνR-1MνD,Mν=MνL-MνDTMνR-1MνD,where Mν is usually a complex symmetric matrix: for example,(8)Mν=eνaνfνaνdνbνfνbνcν.This allows writing the corresponding charged weak current term for leptons as(9)-Lccleptons=g2νeνμντ¯LγμVeμτLWμ++h.c.,where V=VPMNS=UlL†UνL is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix [23] or the neutrino mixing matrix and emerges through the diagonalization of the matrices Ml and Mν: for example,(10)UlL†MlMl†UlR=Diagme2,mμ2,mτ2,UνL†MνMν†UνR=Diagmν12,mν22,mν32.This mixing matrix relates the neutrino flavor states to the neutrino mass eigenstates through(11)ναL=∑i=1,2,3VαiνiL.In the standard parametrization [24], the PMNS matrix is expressed as V=U·Po, where Po≡Diag{eiρ,eiσ,1} with ρ, σ being two Majorana CP violating phases and U can be parametrized in terms of three mixing angles θ12, θ13, and θ23 and one Dirac CP violating phase δl: namely,(12)U=c12c13s12c13s13e-iδl-s12c23-c12s13s23eiδlc12c23-s12s13s23eiδls23c13s12s23-c12c23s13eiδl-c12s23-s12s13c23eiδlc23c13with sij=sin⁡θij and cij=cos⁡θij for ij=12,13,23. The neutrino oscillation experiments provide constraints on the three mixing angles θ12, θ13, and θ23 along with the two mass square differences: namely, δm2=m22-m12 and Δm2=η[m32-(m12+m22)/2] with η=+1 for NO and η=-1 for IO cases.In the current scenario, the global picture of neutrino oscillation parameters for NO at3σ suggests [25](13)δm2=6.99–8.18×10-5eV2,Δm2=2.23–2.61×10-3eV2,s122=0.259–0.359,s132=0.0176–0.0295,s232=0.374–0.626,δl1σ=201∘–239∘.Moreover, the above data does not seem to forbid mν1=0 for NO or mν3=0 for IO cases, the signatures for which are obtained through DetMν=0. The Planck collaboration measurements of the cosmic microwave background [26] provide further insight into the sum of absolute neutrino masses: for example,(14)Σ=mν1+mν2+mν3<0.23eV.More recent results from long-baseline accelerator neutrino experiments T2K [27] and NOνA [28] are indicative of a near maximal Dirac CP phase: that is,(15)δl~270∘,θ23≳45∘along with preference for the normal ordering (NO) of neutrino masses. These results are also supported by the preliminary results from the atmospheric neutrino experiment at Super-Kamiokande [28]. In addition, a statistical analysis of the cosmological data [29] also indicates preference for NO providing maximum likelihood for Majorana effective mass: that is,(16)mee<16meV in neutrinoless double beta decay at 1σ where(17)mee=eiρUe12mν1+eiσUe22mν2+Ue32mν3.As the mixing angles are related to the corresponding mass matrices, it therefore becomes desirable to study the implications of a combination of NO,δCP~270∘ along with θ23≳45∘ for lepton mass matrices assuming quarks and lepton mass matrices have similar origins and investigate the conditions affecting the possibility of obtaining natural lepton mass matrices, synchronous with the quark sector. Nevertheless, from a top-down prospective, it should be more economical to have a common framework explaining the fermion masses and mixing for the quark and lepton sectors.


## 2. Lepton Mass Matrices

Phenomenologically, the problem of constructing the fermion mass matrices has always been a difficult task within the framework of Standard Model (SM) and its possible extensions, wherein the flavor structure of these matrices is usually not constrained by the gauge symmetry. As a result, the matricesMl and Mν remain arbitrary 3×3 complex matrices, thereby involving several free parameters as compared to the number of physical observables, namely, six lepton masses, three mixing angles, and one Dirac-like CP phase δl along with two Majorana phases ρ and σ.In this regard, the “texture zero” ansatz initiated by Weinberg [30] and Fritzsch [31, 32] has been quite successful in explaining the fermion masses and mixing patterns [33–52]. However, one requires handling all possible texture structures on a case to case basis. In this context, a common framework allowing for the study of such possibilities is more desirable. This is addressed in the following section.


## 3. Constructing the PMNS Matrix

In order to reconstruct the PMNS matrix, one requires obtaining the diagonalizing transformations for the corresponding mass matrices. To start with, forq=l,ν, we consider the following two possibilities of texture one-zero mass matrices, as(18)Mq=eqeiψqaqeiαq0aqeiαqdqeiωqbqeiβq0bqeiβqcqeiγq,Mq′=0aq′eiαqfq′eiΔqaq′eiαqdq′eiωqbq′eiβqfq′eiΔqbq′eiβqcq′eiγq, referred to as Type I and Type II structures, respectively, in the following text.One may also consider these matrices to be Hermitian for Dirac neutrinos. Using standard procedures, it is not possible to obtain the exact diagonalizing transformations for the latter case. In order to avoid a large number of free parameters in these matrices, we assume that the phases are factorizable in these, requiring(19)ψq=2αq,ωq=0,Δq=αq+βq,γq=2βq for symmetric Mq and Mq′ and(20)ψq=0,ωq=0,Δq=αq+βq,γq=0 for Hermitian Mq and Mq′.The diagonalization ofMq above is realized using(21)MqDiag=OqTM~qOq=Diagm1,-m2,m3,with 1,2,3=e,μ,τ for q=l and 1,2,3=ν1,ν2,ν3 for q=ν. Here(22)Pq=Diage-iαq,1,e-iκβq,M~q=PqMqQq=eqaq0aqdqbq0bqcq,and Q=P (symmetric case) and Q=P† (Hermitian case). Considering eq and dq as free parameters, one can write [45](23)
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1such that(24)cq=m1-m2+m3-dq-eq,aq=m1-eqm2+eqm3-eqcq-eq,bq=cq-m1m3-cqcq+m2cq-eq,m1>eq>-m2,m3-m2-eq>dq>m1-m2-eq. The above constraints on the parameters eq and dq nevertheless allow hierarchical mass matrices: that is, eq<aq<dq<bq<cq. Texture rotation from (13) and (31) positions in Mq to (11) position in Mq′ is realized by rotating (11) element in Mq to (13) and (31) positions in Mq′ through a unitary transformation Rq on Mq using(25)Mq⟶Mq′=RqTMqRq,for symmetric mass matrices and(26)Mq⟶Mq′=Rq†MqRq,for Hermitian case, where Rq is a complex rotation matrix in the 1–3-generation plane: for example,(27)Rq=cos⁡η13q0-e-iαq-κβqsin⁡η13q010eiαq-κβqsin⁡η13q0cos⁡η13q,where κ=+1 for symmetric matrices and κ=-1 for Hermitian matrices.The condition of a texture zero rotation from(13,31) positions in Mq to (11) position in Mq′ requires(28)0=eqcos2⁡η13q+cqsin2⁡η13q,which can be translated to(29)tan2⁡η13q=-eqcq⟹tan⁡η13q=τq-eqcq,where τq=±1 and eq is always negative. Note that the rotation angle η13q is not a free parameter and is completely fixed through eq and cq due to repositioning of texture zeros as a result of the rotation Rq. One can now relate the matrix elements in Mq′ with the corresponding elements in Mq: for example,(30)aq′=aqcos⁡η13q+τqbqsin⁡η13q,bq′=bqcos⁡η13q-τqaqsin⁡η13q,cq′=cqcos2⁡η13q+eqsin2⁡η13q,dq′=dq,fq′=-eqcq.The texture rotation in 1–3-generation plane allowsdq′=dq. Note that fq′∝-eq, while the other off-diagonal elements essentially get rescaled due to texture rotation. Furthermore, for eq~-m1, one expects fq′~O(m1m3) allowing hierarchical structures in Type II possibility: namely, al′<fl′<dl′<bl′<cl′ along with aν′~fν′~dν′≲bν′≲cν′ since O(mν1mν2)~O(mν1mν3)~O(mν2) are allowed by oscillation data. Henceforth, it is trivial to obtain the orthogonal transformation Oq′ for Mq′ (symmetric case) as(31)Oq′=PqRqTPq†Oq=R~qTOq and (Hermitian case) as(32)Oq′=Pq†Rq†PqOq=R~qTOqwith(33)Mq′Diag=Oq′TM~q′Oq′=MqDiag.with M~q′=PqM′Qq. Note that in the absence of texture rotation, R~q=I (unit matrix) for Mq while(34)R~q=cos⁡η13q0-sin⁡η13q010sin⁡η13q0cos⁡η13qfor Mq′ signifying the corresponding effect of such rotation on real diagonalizing transformation Oq′. The resulting mixing matrix for Mq and/or Mq′ may be constructed as(35)V=OlTR~lPlPν†R~νTOν.Also PlPν†=Diag(e-iϕ1,1,eiϕ2), ϕ1=αl-αν, and ϕ2=βν-βl (symmetrics case) or ϕ2=βl-βν (Hermitian case). Note that a change in sign for aq′ and fq′ can always be accommodated in the redefinition of phases αq and βq which only appear implicitly in the PMNS matrix through ϕ1 and ϕ2. Considering the six lepton masses, ϕ1, ϕ2, dq, and eq as free parameters, one can reconstruct the unitary mixing matrix V using the above procedure and confront it with the current oscillation data. In lieu of this, we restrict our investigation to only texture four-zero mass matrices involving ten free parameters. Furthermore, the condition of naturalness forbids a texture zero at (33) matrix elements.Recent works [53–56] in this regard suggest that there exist several viable texture structures of lepton mass matrices. Most of these investigations work in the flavor basis with diagonal charged-lepton mass matrix or enforce parallel texture structures for lepton mass matrices Ml and Mν. In this letter, we investigate all possible structures for four-zero lepton mass matrices, both symmetric and/or Hermitian, assuming factorizable phases (for simplicity) in these. The resulting structures are summarized in Tables 1 and 2 wherein we enlist all texture five and four zeros in agreement with current data at 3σ. Xl and Xν in the tables represent the position of texture zeros in the corresponding mass matrices. It is observed that the constraints of naturalness, near maximal δl, s232≳0.50, and normal ordering for neutrino masses, taken together, greatly reduce the number of possible viable structures and only a few possibilities seem to survive the test. The possibility of a vanishing neutrino mass is also studied for these texture structures.Table 1
Viable texture five zeros in relation tos232≳0.5, δl~270∘, naturalness, and mν1=0.
Sr. X l X ν (a)s232≳0.5 (b)δl~270∘ (c) natural (a + c) (b + c) D e t  M ν = 0 1 11,22,13,31 11,13,31 √ × √ √ × × 2 11,13,31 11,22,13,31 √ × × × × × 3 11,13,31,23,32 11,13,31 √ × × × × × 4 12,21,22,13,31 11,13,31 √ × × × × × 5 11,13,31,23,32 11,12,21 √ × × × × × 6 11,22,13,31 11,12,21 √ × √ √ × × 7 12,21,22,13,31 11,12,21 √ × √ √ × × 8 11,12,21,23,32 11,13,31 √ × × × × ×Table 2
Viable texture four zeros in relation tos232≳0.5, δl~270∘, naturalness, and DetMν=0.
Sr. X l X ν (a)s232≳0.5 (b)δl~270∘ (c) natural (a + c) (b + c) D e t  M ν = 0 1 11,13,31 11,13,31 √ √ √ √ × × 2 13,31,23,32 11,13,31 √ √ × × × × 3 11 , 22 11 , 13 , 31 √ √ √ √ √ × 4 11,13,31 11,22 √ √ √ × × √ 5 13,31 11,22,13,31 √ √ √ × √ × 6 11,22,13,31 13,31 √ × √ √ × √ 7 13,31 11,13,31,23,32 √ √ × × × × 8 11,13,31,23,32 13,31 √ × × × × √ 9 11 11,22,13,31 √ √ √ × √ × 10 11,22,13,31 11 √ × √ √ × √ 11 11 11,13,31,23,32 √ √ × × × × 12 11,13,31,23,32 11 √ × × × × × 13 22 , 13 , 31 11 , 13 , 31 √ √ √ √ √ × 14 11 , 12 , 21 11 , 13 , 31 √ √ √ √ √ × 15 11,13,31 11,12,21 √ √ √ √ × × 16 13,31,23,32 11,12,21 √ √ × × × × 17 22,13,31 11,12,21 √ √ √ √ × × 18 11,12,21 11,22 √ √ × × × √ 19 11 , 22 11 , 12 , 21 √ √ √ √ √ × 20 12,21,13,31 11,13,31 √ √ × × × × 21 12,21,13,31 11,22 √ √ × × × × 22 22,12,21,13,31 13,31 √ × × × × √ 23 12,21,22,13,31 11 √ × √ √ × × 24 11,23,32 11,13,31 √ × × × × × 25 11,12,21,23,32 11 √ × × × × × 26 11,12,21,23,32 13,31 √ × √ √ × × 27 13,31 11,12,21,23,32 √ √ × × × × 28 11 11,12,21,23,32 √ √ × × × ×


## 4. Fritzsch-Like Four Zeros

It has been observed [36, 47] that, in the absence of δl~270∘ constraint, the Fritzsch-like texture four-zero mass matrices are physically equivalent to the generic lepton mass matrices. Interestingly, these matrices can be obtained from the above structures using the assumption of eq=0 and fq′=0. In particular, Rq=R~q=I, where I is a unit matrix, for this case. The predictions from these matrices and their experimental tests can be found in previous works. To start with, using (13), (35) and allowing free variations to the parameters mν1, dl, dν, ϕ1, and ϕ2, we first reconstruct the viable structures for M~l (in units of GeV) and M~ν (in units of eV) for dν~mν2 using the available oscillation data and obtain the following best-fits:(36)M~l=00.007–0.01000.007–0.0100–0.8220.423–0.92400.423–0.9240.822–1.644GeV,M~ν=00.0066–0.010400.0066–0.01040.0076–0.01150.0223–0.026000.0223–0.02600.0302–0.0383eV,along with ϕ1=0∘–50∘,267∘–360∘ and ϕ2=180∘–285∘. The corresponding predictions for the absolute neutrino masses Σ and mee read mν1=(2.96–6.70)meV, mν2=(9.05–11.50)meV, mν3=(47.7–51.9)meV, Σ=60.2–69.6meV, and mee=0.008–9.00meV, respectively. In the context of agreement with δl~270∘ along with θ23≳45∘, it is observed that naturalness is allowed in Mν independent of s23 octant. This is depicted in Figure 1 where one observes that dν≲mν2 is still consistent with s232≳0.5. However, one finds that the near maximal constraint of δl≃270∘ requires large deviation of Ml from a possible natural structure. In particular, we identify three vital sources for CP violation in these matrices: namely, the two nontrivial phases ϕ1, ϕ2 along with the free parameter dl as elaborated in Figure 2, indicating dl>0.6 GeV ~mτ/3≫mμ is required to obtain δl≃270∘. This also implies that Fritzsch-like texture five-zero matrices (dl=0) should be ruled out by δl≃270∘. Our study reveals this conclusion to hold true for all possible texture five-zero structures, all of which seem to be ruled out by a near maximal δl; see Table 1. This calls upon investigating alternate texture structures, which on the one hand account for near maximal δl and at the same time allow possible natural structures for Ml and Mν (i.e., Mjk~O(mjmk)).Figure 1
s 23 2 versus dν for Fritzsch-like four zeros.Figure 2
δ l versus dl for Fritzsch-like four zeros.


## 5. Natural Lepton Mass Matrices

In this context,eq≠0 and/or fq′≠0 in the corresponding mass matrices provide greater possibility of realizing naturalness in corresponding mass matrices as compared to the Fritzsch-like structures wherein interactions between the first and third generations of leptons are suppressed due to texture zeros invoked at (11) and (13,31) matrix elements. At least, for the quark sector, nonvanishing (13,31) elements are observed to be crucial in effectuating the natural structures of corresponding mass matrices. A careful analysis of all possible texture four-zero structures reveals that only four possibilities for natural structures are allowed by recent data; see Table 2. We categorize these as Type I and Type II, based on the texture structure of Mν.


### 5.1. Type IMν(11)=Mν(13,31)=0

Case A (Ml(22)=Ml(13,31)=0). The viable best-fit structures of the lepton mass matrices are summarized below:(37)M~l=-0.003–00.007–0.01900.007–0.01900.416–0.42600.416–0.4261.644–1.647GeV,M~ν=00.0053–0.010600.0053–0.01060.0057–0.01230.0221–0.027200.0221–0.02720.0285–0.0394eV,with ϕ1=0∘–340∘, ϕ2=98∘–265∘ and mν1=(1.99–7.01) meV, mν2=(8.65–11.3)meV, mν3=(47.7–51.9)meV, Σ=(58.7–70.0)meV, and mee=(0.01–9.23)meV, respectively. Like the Fritzsch-like texture four zeros, s122∝mν1 [36, 38, 47, 57] as depicted in Figure 3. However, the other two mixing angles are fixed by the free parameter dν illustrated in Figures 4 and 5. The latter also indicates that natural structure for Mν is allowed independent of s23 octant, with dν≲mν2 also accounting for s232>0.5. Finally, the parameter el≪mμ accounts for near maximal δl as shown in Figure 6. In particular, a small deviation of δl→270∘±30∘ provides greater agreement of el~5MeV with the notion of naturalness in the corresponding mass matrix.Figure 3
s 12 2 versus mν1 for Case A.Figure 4
s 13 2 versus dν for Case A.Figure 5
s 23 2 versus dν for Case A.Figure 6
δ l versus el for Case A.Case B (Ml(11)=Ml(22)=0). We obtain the following viable best-fit structures for these lepton mass matrices: namely,(38)M~l′=00.001–0.0070.0003–0.0890.001–0.00700.413–0.4230.0003–0.0890.413–0.4231.644GeV,M~ν=00.0056–0.011100.0056–0.01110.0065–0.01160.0223–0.026600.0223–0.02660.0294–0.0390eV,wherein ϕ1=0∘–11∘,251∘–360∘, ϕ2=89∘–268∘ and mν1=(2.26–7.53)meV, mν2=(8.73–11.6)meV, mν3=(47.7–52.0)meV, Σ=(59.0–71.0)meV, and mee=(0.01–10.0)meV, respectively. mν1 dependence for s122 remains the same as before while the other two mixing angles are fixed by the parameter dν. Furthermore, apart from phases ϕ1 and ϕ2, δl is now fixed by the parameter fl=-elcl as shown in Figure 7. Naturalness in Ml′ and Mν seems to be in good agreement with δl~270∘ and s232≳0.5 compatible with fl′~0.075 GeV ~O(memμ)<mμ and dν≲mν2, respectively. A greater agreement with naturalness in Ml is achieved for δl→270∘±30∘ up to fl′~0.05 GeV.Figure 7
δ l versus fl′ for Case B.Case C (Ml(11)=Ml(12,21)=0). The viable best-fit structures so obtained for these lepton mass matrices are shown below:(39)M~l′=000.029–0.16700.003–0.1030.395–0.5800.029–0.1670.395–0.5801.54–1.64GeV,M~ν=00.0022–0.011300.0022–0.01130.0027–0.01170.0223–0.027700.0223–0.02770.0283–0.0399eV,wherein ϕ1=0∘–36∘,175∘–360∘, ϕ2=97∘–265∘ and mν1=(0.4–7.8)meV, mν2=(8.4–11.7)meV, mν3=(47.6–52.0)meV, Σ=(56.9–71.2)meV, and mee=(0.02–9.6)meV, respectively. It is noteworthy that the condition of texture zero at Ml′(12,21), that is, al′=0, fixes the parameter el and hence(40)fl′=-elclthrough (30) with (41)el=-memμmτdlcl.This results in only one free parameter dl=dl′ in Ml′. This is depicted in Figure 8. This parameter also determines the Dirac-like CP phase as shown in Figure 9. Other observations pertaining to the dependence of mixing angles remain the same as in previous cases. It is clear that naturalness in Ml′ and Mν is in good agreement with δl~270∘ and s232≳0.5 compatible with fl′~0.064GeV~O(memμ)<mμ and dν≲mν2, respectively.Figure 8
f l ′ versus dl′ for Case C.Figure 9
δ l versus dl′ for Case C.


### 5.2. Type IIMν(11)=Mν(12,21)=0

Case D (Ml(11)=Mν(22)=0). The best-fit values obtained for this possibility are summarized below:(42)M~l′=00.007–0.0940.0004–0.2200.007–0.09500.348–0.4230.0004–0.2200.348–0.4231.644GeV,M~ν=000.006–0.01900.0041–0.01200.0178–0.02690.006–0.0190.0178–0.02690.0282–0.0393eV,wherein ϕ1=0∘–23∘,256∘–360∘, ϕ2=98∘–261∘ and mν1=(1.1–7.9)meV, mν2=(8.4–12.5)meV, mν3=(47.6–51.9)meV, Σ=(57.5–70.8)meV, and mee=(0.01–9.56)meV, respectively. It is observed that naturalness is in good agreement with δl~270∘ and s232≳0.5 compatible with |fl′|~0.088GeV~O(memμ)<mμ; see Figure 10 and dν≲mν2, respectively. Again a greater agreement with naturalness in Ml can be achieved for δl→270∘±30∘ up to |fl′|~0.05GeV.Figure 10
δ l versus fl′ for Case D.


## 5.1. Type IMν(11)=Mν(13,31)=0

Case A (Ml(22)=Ml(13,31)=0). The viable best-fit structures of the lepton mass matrices are summarized below:(37)M~l=-0.003–00.007–0.01900.007–0.01900.416–0.42600.416–0.4261.644–1.647GeV,M~ν=00.0053–0.010600.0053–0.01060.0057–0.01230.0221–0.027200.0221–0.02720.0285–0.0394eV,with ϕ1=0∘–340∘, ϕ2=98∘–265∘ and mν1=(1.99–7.01) meV, mν2=(8.65–11.3)meV, mν3=(47.7–51.9)meV, Σ=(58.7–70.0)meV, and mee=(0.01–9.23)meV, respectively. Like the Fritzsch-like texture four zeros, s122∝mν1 [36, 38, 47, 57] as depicted in Figure 3. However, the other two mixing angles are fixed by the free parameter dν illustrated in Figures 4 and 5. The latter also indicates that natural structure for Mν is allowed independent of s23 octant, with dν≲mν2 also accounting for s232>0.5. Finally, the parameter el≪mμ accounts for near maximal δl as shown in Figure 6. In particular, a small deviation of δl→270∘±30∘ provides greater agreement of el~5MeV with the notion of naturalness in the corresponding mass matrix.Figure 3
s 12 2 versus mν1 for Case A.Figure 4
s 13 2 versus dν for Case A.Figure 5
s 23 2 versus dν for Case A.Figure 6
δ l versus el for Case A.Case B (Ml(11)=Ml(22)=0). We obtain the following viable best-fit structures for these lepton mass matrices: namely,(38)M~l′=00.001–0.0070.0003–0.0890.001–0.00700.413–0.4230.0003–0.0890.413–0.4231.644GeV,M~ν=00.0056–0.011100.0056–0.01110.0065–0.01160.0223–0.026600.0223–0.02660.0294–0.0390eV,wherein ϕ1=0∘–11∘,251∘–360∘, ϕ2=89∘–268∘ and mν1=(2.26–7.53)meV, mν2=(8.73–11.6)meV, mν3=(47.7–52.0)meV, Σ=(59.0–71.0)meV, and mee=(0.01–10.0)meV, respectively. mν1 dependence for s122 remains the same as before while the other two mixing angles are fixed by the parameter dν. Furthermore, apart from phases ϕ1 and ϕ2, δl is now fixed by the parameter fl=-elcl as shown in Figure 7. Naturalness in Ml′ and Mν seems to be in good agreement with δl~270∘ and s232≳0.5 compatible with fl′~0.075 GeV ~O(memμ)<mμ and dν≲mν2, respectively. A greater agreement with naturalness in Ml is achieved for δl→270∘±30∘ up to fl′~0.05 GeV.Figure 7
δ l versus fl′ for Case B.Case C (Ml(11)=Ml(12,21)=0). The viable best-fit structures so obtained for these lepton mass matrices are shown below:(39)M~l′=000.029–0.16700.003–0.1030.395–0.5800.029–0.1670.395–0.5801.54–1.64GeV,M~ν=00.0022–0.011300.0022–0.01130.0027–0.01170.0223–0.027700.0223–0.02770.0283–0.0399eV,wherein ϕ1=0∘–36∘,175∘–360∘, ϕ2=97∘–265∘ and mν1=(0.4–7.8)meV, mν2=(8.4–11.7)meV, mν3=(47.6–52.0)meV, Σ=(56.9–71.2)meV, and mee=(0.02–9.6)meV, respectively. It is noteworthy that the condition of texture zero at Ml′(12,21), that is, al′=0, fixes the parameter el and hence(40)fl′=-elclthrough (30) with (41)el=-memμmτdlcl.This results in only one free parameter dl=dl′ in Ml′. This is depicted in Figure 8. This parameter also determines the Dirac-like CP phase as shown in Figure 9. Other observations pertaining to the dependence of mixing angles remain the same as in previous cases. It is clear that naturalness in Ml′ and Mν is in good agreement with δl~270∘ and s232≳0.5 compatible with fl′~0.064GeV~O(memμ)<mμ and dν≲mν2, respectively.Figure 8
f l ′ versus dl′ for Case C.Figure 9
δ l versus dl′ for Case C.


## 5.2. Type IIMν(11)=Mν(12,21)=0

Case D (Ml(11)=Mν(22)=0). The best-fit values obtained for this possibility are summarized below:(42)M~l′=00.007–0.0940.0004–0.2200.007–0.09500.348–0.4230.0004–0.2200.348–0.4231.644GeV,M~ν=000.006–0.01900.0041–0.01200.0178–0.02690.006–0.0190.0178–0.02690.0282–0.0393eV,wherein ϕ1=0∘–23∘,256∘–360∘, ϕ2=98∘–261∘ and mν1=(1.1–7.9)meV, mν2=(8.4–12.5)meV, mν3=(47.6–51.9)meV, Σ=(57.5–70.8)meV, and mee=(0.01–9.56)meV, respectively. It is observed that naturalness is in good agreement with δl~270∘ and s232≳0.5 compatible with |fl′|~0.088GeV~O(memμ)<mμ; see Figure 10 and dν≲mν2, respectively. Again a greater agreement with naturalness in Ml can be achieved for δl→270∘±30∘ up to |fl′|~0.05GeV.Figure 10
δ l versus fl′ for Case D.


## 6. Conclusions

Assuming factorizable phases in lepton mass matrices, we show that natural mass matrices characterized by(Mij)~O(mimj) for i,j=1,2,3, i≠j, and (Mii)~O(mi) provide a reasonable explanation for the observed fermion masses and flavor mixing patterns in the quark as well as the lepton sectors. It is also observed that deviations from parallel texture structures for Ml,d and Mν,u are essential for establishing such natural structures. Such phenomenological textures have also been observed to be stable under the renormalization group running from the lightest right-handed neutrino mass scale to the electroweak scale [39, 40, 53, 58, 59].Interestingly, naturalness in the lepton sector impliess12∝O(mν1/mν2) and s232∝dν/cν or s23∝O(mν2/mν3) such that the observed large values of these mixing angles are perhaps indicative of the possible realization of the neutrino mass ratios as obtained above: that is, mν1≃(0.1–8.0)meV, mν2≃(8.0–13.0)meV, mν3≃(47.0–52.0)meV, Σ≃(56.0–71.0)meV, and mee≃(0.01–10.0)meV, respectively. In particular, the possibility of a vanishing neutrino mass, that is, mν1=0, is not supported by natural lepton matrices. From the point of view of 0νββ decays, these results seem to indicate that multiton scale detectors may be required to possibly observe signals for such processes.



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*Source: 2094323-2016-11-22.xml*