# Phenomenology of Trimaximal Mixing with One Texture Equality

Authors: S. Dev; Desh Raj
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We study neutrino mass matrices with one texture equality and the neutrino mixing matrix having either its first (TM1) or second (TM2) column identical to that of the tribimaximal mixing matrix. We found that out of total fifteen possible neutrino mass matrices with one texture equality, only six textures are compatible with TM1 mixing and six textures are compatible with TM2 mixing in the light of the current neutrino oscillation data. These textures have interesting implications for the presently unknown parameters such as the neutrino mass scale, effective Majorana neutrino mass, effective neutrino mass, the atmospheric mixing, and the Dirac- and Majorana-type CP violating phases. We, also, present the S3 group motivation for some of these textures.


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## Body

## 1. Introduction

In the last two decades, significant advances have been made by various neutrino oscillation experiments in determining the neutrino masses and mixings. Various neutrino parameters like three mixing (solar, atmospheric, and reactor) angles and the two mass squared differences (Δm212 and Δm312) have been measured by various neutrino oscillation experiments with fairly good precision. In addition, the recent neutrino oscillation data hint towards a nonmaximal atmospheric mixing angle (θ23) [1] and Dirac-type CP-violating phase (δ) near 270° [2, 3]. However, many other attributes like leptonic CP-violation, neutrino mass ordering (normal mass ordering (NO) or inverted mass ordering (IO)), nature of neutrinos (Dirac or Majorana), and absolute neutrino mass scale are still unknown. Furthermore, the origin of the lepton flavor structure still remains an open issue. The neutrino mass matrix which encodes the neutrino properties contains several unknown physical parameters. The phenomenological approaches based on Abelian or non-Abelian flavor symmetries can play a significant role in determining the specific texture structure of the neutrino mass matrix with reduced number of independent parameters.Several predictive models such as texture zeros [4–25], vanishing cofactors [26–37], equalities among elements/cofactors [38, 39], and hybrid textures [40–46] amongst others can explain the presently available neutrino oscillation data, since the presence of texture equalities, just like texture zeros or vanishing cofactors, reduces the number of free parameters in the neutrino mass matrix and, hence, must have a similar predictability as that of texture zeros or vanishing cofactors. In the flavor basis, neutrino mass matrices with one texture equality and two texture equalities have been studied in the literature [38, 39]. The hybrid textures which combine a texture equality with a texture zero or a vanishing cofactor have been studied in the literature [40–46].In addition, discrete non-Abelian symmetries leading to the Tri-Bi-Maximal (TBM) [47, 48] neutrino mixing pattern have been widely studied in the literature. The TBM mixing matrix given by
(1)UTBM=23130−1613−12−161312predicts a vanishing reactor mixing angle (θ13=0) and maximal atmospheric mixing angle (θ23=π/4), and the solar mixing angle is predicted to be θ12 which issin−11/3. However, the nonzero value of θ13 confirmed by various neutrino oscillation experiments underlines the need for necessary modifications to the TBM mixing pattern to make it compatible with the present experimental data [49–53]. One of the simplest possibilities is to keep one of the columns of the TBM mixing matrix unchanged while modifying its remaining two columns to within the unitarity constraints. This gives rise to three mixing patterns, viz., TM1, TM2, and TM3 which have their first, second, and third columns identical to the TBM mixing matrix, respectively. The TM3 mixing scheme predicts θ13=0 and is, hence, phenomenologically unviable. The TM1 and TM2 mixing schemes have been successfully employed to explain the pattern of lepton mixing and have been extensively studied in the literature [54–71]. The TM1 mixing, in particular, gives a very good fit to the present neutrino oscillation data. Recently, neutrino mass matrices with texture zero(s) in combination with TM1 and TM2 mixing have been studied [72–74].In the present work, we study a class of neutrino mass matrices having one texture equality withTM1 or TM2 of the TBM in the neutrino mixing matrix. Neutrino mass matrices having one texture equality along with TM1 or TM2 of the TBM have a total of six free parameters and, hence, lead to very predictive textures for the neutrino mass matrices.There are a total of fifteen possible structures with one texture equality in the neutrino mass matrix, and they are listed in Table1.Table 1
Fifteen possible texture structures with one equality between two nonzero elements.
TexturesConstraints on elementsMν1Mee=MeμMν2Mee=MeτMν3Meμ=MμμMν4Mμμ=MμτMν5Meτ=MττMν6Mμτ=MττMν7Mee=MμτMν8Meτ=MμμMν9Meμ=MττMν10Mee=MμμMν11Mee=MττMν12Mμμ=MττMν13Meμ=MeτMν14Meμ=MμτMν15Meτ=MμτThere exists aμ-τ permutation symmetry between different structures of neutrino mass matrices, and the corresponding permutation matrix has the following form:
(2)P23=100001010.Neutrino mass matrices with one texture equality, therefore, are related to each other as(3)Mν′=P23MνP23T,leading to the following relations between the neutrino oscillation parameters:
(4)θ12′=θ12,θ13′=θ13,θ23′=π2−θ23,δ′=π−δ.Neutrino mass matrices with one texture equality related by theμ-τ permutation operation are
(5)Mν1↔Mν2,Mν3↔Mν5,Mν4↔Mν6,Mν7↔Mν7,Mν8↔Mν9,Mν10↔Mν11,Mν12↔Mν12,Mν13↔Mν13,Mν14↔Mν15.In the flavor basis, where the charged lepton mass matrixMl is diagonal, the complex symmetric Majorana neutrino mass matrix Mν can be diagonalized by a unitary matrixV′:
(6)Mν=V′MνdiagV′T,whereMνdiag=diagm1,m2,m3. The unitary matrix V′ can be parametrized as
(7)V′=PlVwithV=UPν,where(8)U=c12c13c13s12e−iδs13−c23s12−eiδc12s13s23c12c23−eiδs12s13s23c13s23s12s23−eiδc12c23s13−eiδc23s12s13−c12s23c13c23,Pν=1000eiα000eiβ,Pl=eiϕe000eiϕμ000eiϕτ,withcij=cosθij, sij=sinθij. Pν is the diagonal phase matrix containing the two Majorana-type CP-violating phases α and β. δ is the Dirac-type CP-violating phase. The phase matrix Pl is physically unobservable. The matrix V is called the neutrino mixing matrix or the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) [75–78] matrix. The effective Majorana neutrino mass matrix can be written as
(9)Mν=PlUPνMνdiagPνTUTPlT.The Dirac-type CP-violation in neutrino oscillation experiments can be described in terms of the Jarlskog rephasing invariant quantityJCP [79] with
(10)JCP=ImU11U22U12∗U21∗=sinθ12sinθ23sinθ13cosθ12cosθ23cos2θ13sinδ.The effective Majorana neutrino massMee, which determines the rate of neutrinoless double beta decay, is given by
(11)Mee=m1Ue12+m2Ue22+m3Ue32.There are many experiments such as CUORICINO [80], CUORE [81], MAJORANA [82], SuperNEMO [83], and EXO [84] which aim to achieve a sensitivity up to 0.01 eV for Mee. The KamLAND-Zen experiment [85] provides the upper limits on the effective Majorana neutrino mass which is given by
(12)Mee<0.36−0.156eV,at 90% confidence level (C.L.).The measurement of the absolute neutrino mass scale via the decay kinematics is usually described by the effective neutrino mass [86]
(13)mβ≡m12Ue12+m22Ue22+m32Ue32.Recently, the KATRIN [87] experiment has reported the upper limit of mβ<0.8eV at 90% C.L.Further, cosmological observations provide more stringent constraints on absolute neutrino mass scale by putting an upper bound on the sum of neutrino masses:(14)∑=∑i=13mi.Recent Planck data [88] in combination with baryon acoustic oscillation (BAO) measurements provide a tight bound on the sum of neutrino masses ∑mi≤0.12 eV at 95% C.L.


## 2.TM2 Mixing and One Texture Equality

A neutrino mass matrix withTM2 mixing can be written as
(15)MTM2=PlUTM2PνMνdiagPνTUTM2TPlT,where the mixing matrix TM2, also known as trimaximal mixing, can be parametrized [68–71] as
(16)UTM2=23cosθ1323sinθeiϕsinθ2−cosθ613−eiϕcosθ2−sinθ6−cosθ6−eiϕsinθ213eiϕcosθ2−sinθ6.The mass matrixMTM2 is invariant under the transformation G2TMTM2G2=MTM2 with G2=UTM2diag−1,1,−1UTM2†, as the generator of Z2 symmetry [90, 91]. Invariance of MTM2 under G2 when combined with one texture equality leads to the equality of three unphysical phases in MTM2, i.e., ϕe=ϕμ=ϕτ≡ϕl.The most general neutrino mass matrix withTM2 as the mixing matrix can be parametrized as
(17)MTM2=u+2x3v−x3w−x3v−x3w+2x3u−x3w−x3u−x3v+2x3≈abcbc+da−dca−db+d.The neutrino mass matrixMTM2 can be realized within the framework of an A4 model where the A4 flavor symmetry is spontaneously broken by two real A4 tripletsϕ,ϕ′, and three real A4 singlets, ξ,ξ′,ξ′′ which are SU2L gauge singlets [92–94]. Upon symmetry breaking, the VEVs of the flavone singlets and triplets take the alignments
(18)ξ=ua,ξ′=uc,ξ′′=ub,ϕ=v,v,v,ϕ′=v′,0,0.The neutrino mass matrix, in the flavor basis, is given by(19)u+2x3v−x3w−x3v−x3w+2x3u−x3w−x3u−x3v+2x3.The above mass matrix leads to theTM2 neutrino mixing matrix. For v=w, the above mass matrix leads to the TBM neutrino mixing matrix. The equality among the elements of mass matrix in Eq. (17) does not arise naturally, and hence, we assume additional constraints on the elements of mass matrix, e.g., u=v−x which leads to one equality between the 1,1 and 1,2 elements of MTM2 in Eq. (17). Therefore, all possible textures of neutrino mass matrices with TM2 mixing and one texture equality are given by
(20)MTM21=aabab+da−dba−da+d,MTM22=ababa+da−daa−db+d,(21)MTM23=b+daa−daaba−dba+d,MTM25=b+da−daa−da+dbaba,(22)MTM24=a+dba−dbaaa−dab+d,MTM26=a+da−dba−db+dabaa,(23)MTM27=MTM28=MTM29=abdbdadab,(24)MTM210=MTM215=abb−dbab−db−db−da+d,MTM211=MTM214=abb+dba+dbb+dba,(25)MTM212=MTM213=abbbb+da−dba−db+d,where the neutrino mass matrices in each equation are related byμ-τ symmetry. The neutrino mixing angles can be calculated by using the following relations:
(26)sin2θ13=U132,sin2θ12=U1221−U132,sin2θ23=U2321−U132.Substituting the elements ofU form Eq. (16) into Eq. (26), we get
(27)sin2θ13=23sin2θ,sin2θ12=13−2sin2θ,sin2θ23=121+3sin2θcosϕ3−2sin2θ.Using Eqs. (10) and (16), the Jarlskog rephasing invariant is given by
(28)JCP=163sin2θsinϕ,and the Dirac-type CP-violating phase can be calculated by using the equation [72–74]
(29)tanδ=cos2θ+22cos2θ+1tanϕ.From Eqs. (11) and (16), the effective Majorana mass for TM2 mixing is given by
(30)Mee=132m1cos2θ+m2e2iα+2m3sin2θe2iβ,and the effective neutrino mass for TM2 mixing can be calculated by using Eqs. (12) and (16) as
(31)mβ≡132m12cos2θ+m22+2m32sin2θ.The existence of one equality between the elementsa,b and c,d of the neutrino mass matrix MTM2 implies
(32)MTM2ab−MTM2cd=0,which yields the complex equation
(33)∑QVaiVbi−VciVdimi=0,where Q=eiϕa+ϕb−ϕc+ϕd and V is the PMNS matrix given in Eq. (7). The above equation can be rewritten as
(34)m1A1+m2A2e2iα+m3A3e2iβ=0,where
(35)Ai=QUaiUbi−UciUdi,with i=1,2,3 and a,b,c,d can take values e,μ, and τ. Since the TM2 mixing has equal elements in the second column, it leads to A2≡QUa2Ub2−Uc2Ud2=0. Therefore, using Eq. (16) in Eq. (34), we have
(36)m1A1+m3A3e2iβ=0.Simultaneous solution of the real and imaginary parts of Eq. (36) leads to
(37)ξ≡m3m1=ReA1ImA3sin2β−ReA3cos2β=A1A3,(38)β=12tan−1ReA3ImA1−ReA1ImA3ReA1ReA3+ImA1ImA3.Using experimentally available mass squared differencesΔm212 and Δm312Δm232 for NO (IO) with Eq. (37), the three neutrino mass eigenvalues are given by
(39)m1=Δm312ξ2−1,m2=Δm212+m12,m3=ξm1forNO,(40)m1=Δm232−Δm2121−ξ2,m2=Δm212+m12,m3=ξm1forIO,whereΔmij2=mi2−mj2, m1<m2<m3 for NO, and m3<m1<m2 for IO.For the numerical analysis, we generate∼107-109 points. The mass squared differences Δm212 and Δm312Δm232 for NO (IO) are varied randomly within their 3σ experimental ranges given in Table 2. Parameters θ,ϕ, and α are, also, varied randomly within their full ranges (0-90°), (0-360°), and (0-360°), respectively. Equation (38), (39), and (40) are used to calculate the Majorana-type CP-violating phase β and three mass eigenvalues (m1,m2, and m3) for both mass orderings. In addition, the mixing angles θ12,θ13, and θ23 are calculated by using Eq. (27) and must satisfy the experimental data given in Table 2. The Jarlskog invariant (JCP), Dirac-type CP-violating phase (δ), effective Majorana mass (Mee), effective neutrino mass (mβ), and the sum of neutrino masses (∑) are calculated by using Eqs. (28), (29), (30), (31), and (14), respectively.Table 2
Current neutrino oscillation parameters from global fits [89] with Δm3l2≡Δm312>0 for NO and Δm3l2≡Δm322=−Δm232<0 for IO.
Neutrino parameterNormal ordering (best fit)Inverted ordering (Δχ2=2.6)Bfp±1σ3σ rangeBfp±1σ3σ rangeθ12∘33.44−0.74+0.7731.27⟶35.8633.45−0.74+0.7731.27⟶35.87θ23∘49.2−1.3+1.039.5⟶52.049.5−1.2+1.039.8⟶52.1θ13∘8.57−0.12+0.138.20⟶8.978.60−0.12+0.128.24⟶8.98δCP∘194−25+52105⟶405287−32+27192⟶361Δm21210−5eV27.42−0.20+0.216.82⟶8.047.42−0.20+0.216.82⟶8.04Δm3l210−3eV2+2.515−0.028+0.028+2.431⟶+2.599−2.498−0.029+0.028−2.584⟶−2.413The numerical predictions for various neutrino parameters are given in Tables3 and 4. Table 3 provides numerical predictions for viable textures under the constrains from neutrino oscillation data, whereas Table 4 provides numerical predictions for viable textures under the constrains from cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data. The allowed range of parameter α is (0-360°) for all viable textures. JCP which lies in the range (−0.037-0.037) for all viable textures except MTM23MTM25 with NO (IO), and for these textures, the range of JCP is ±(0.011–0.037). The parameter θ is constrained to lie within the ranges (10.0°-11.1°) for all viable textures. The solar mixing angle (θ12) is constrained to lie in the ranges (35.68°-35.77°) for all allowed textures. Figures 1 and 2 show correlations among various neutrino oscillation parameters. The Dirac CP-violating phase δ and phase ϕ are linearly correlated as shown in Figure 2(e). Figure 1(h) depicts the correlation between θ12 and θ13. The Dirac-type CP-violating phase δ strongly depends on the Majorana-type CP-violating phase β as shown in Figure 1(a).Table 3
Numerical predictions for viable textures having one equality inMν with TM2 mixing at 3σ C.L. (only neutrino oscillation data are incorporated).
TextureOrderingmlowest (eV)Mee (eV)∑ (eV)mβθ23∘δ∘β∘MTM21NO0.003-0.00870.0-0.00830.062-0.0730.009-0.012839.5-51.410-155⊕205-36016-90⊕270-344MTM22NO0.003-0.00830.0-0.00820.061-0.0720.009-0.012439.5-51.40-160⊕202-3600-81⊕279-360MTM23NO0.025-0.50.006-0.50.10-1.510.026-0.539.5-4590-160⊕200-2700-68⊕292-360IO0.02-0.480.015-0.50.12-1.450.05-0.545-51.40-90⊕270-3600-90⊕270-360MTM24IO0.003-0.0080.014-0.050.1-0.110.048-0.05139.8-51.40-150⊕210-3600-77⊕283-360MTM25NO0.02-0.50.005-0.50.09-1.50.02-0.545-51.40-90⊕270-3600-90⊕270-360IO0.027-0.50.017-0.470.13-1.480.05-0.539.8-4590-150⊕209-2700-59⊕303-360MTM26IO0.003-0.0090.014-0.050.101-0.1110.048-0.05139.8-51.40-151⊕211-36019-90⊕270-340Table 4
Numerical predictions for viable textures having one equality inMν with TM2 mixing at 3σ C.L. (cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated).
TextureOrderingmlowest (eV)Mee (eV)∑ (eV)mβθ23∘δ∘β∘MTM21NO0.003-0.00870.0-0.00830.062-0.0730.009-0.012839.5-51.410-155⊕205-36016-90⊕270-344MTM22NO0.003-0.00830.0-0.00820.061-0.0720.009-0.012439.5-51.40-160⊕202-3600-81⊕279-360MTM23NO0.025-0.0310.006-0.030.10-0.120.026-0.03239.5-40.34137-160⊕200-22245-68⊕292-316MTM24IO0.003-0.0080.014-0.050.1-0.1110.048-0.05139.8-51.40-150⊕210-3600-77⊕283-360MTM25NO0.021-0.0310.005-0.030.098-0.120.023-0.03249.7-51.40-42⊕317-36045-90⊕270-316MTM26IO0.003-0.0090.014-0.050.101-0.1110.048-0.05139.8-51.40-151⊕211-36019-90⊕270-340Figure 1
Correlation plots among various parameters for textures (a)MTM21, (b) MTM21, (c) MTM22, (d) MTM25, (e) MTM22, (f) MTM25, (g) MTM21, (h) MTM23, and (i) MTM25 with NO.Figure 2
Correlation plots among various parameters for textures (a)MTM24, (b) MTM24, (c) MTM24, (d) MTM26, (e) MTM26, and (f) MTM26 with IO.The main results for the neutrino mass matrices with one texture equality andTM2 mixing are listed in the following:
(i)
TexturesMTM27,MTM28, and MTM29 lead to two degenerate eigenvalues and are, hence, experimentally ruled out at 3σ C.L.(ii)
TexturesMTM210,MTM211,MTM212,MTM213,MTM214, and MTM215 lead to vanishing reactor mixing angle and, hence, are not viable at 3σ C.L.(iii)
TexturesMTM23 and MTM25 for IO are not consistent with the experimental data if cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated(iv)
TexturesMTM21 and MTM22 are consistent with NO only, whereas textures MTM24 and MTM26 are consistent with IO only(v)
For NO, texturesMTM24 and MTM26 are not consistent with the experimental data as the mixing angles θ12 and θ13 are not within the 3σ range(vi)
All viable textures cannot have zero lowest mass eigenvalue for both mass orderings(vii)
The atmospheric mixing angleθ23 is below (above) maximal for textures MTM23 (MTM25) and MTM25 (MTM23) with NO and IO, respectively(viii)
θ23 is maximal for δ∼π/2 or 3π/2 for textures MTM21 (MTM26) and MTM22 (MTM24) with NO (IO).(ix)
The parameterMee is found to be nonzero for all viable textures except MTM21 andMTM22. Mee gets its largest value when δ∼π/2 or 3π/2 for textures MTM23 and MTM25.(x)
For all viable textures, the effective neutrino mass (mβ) is well within the range provided by KATRIN experiment(xi)
The parametersm1m3, Mee, and ∑ get their largest value when θ23∼45∘ for textures MTM23 and MTM25 with NO (IO).


## 3.S3 Group Motivation

TheS3, permutation group of three objects, is the smallest discrete non-Abelian group. The permutation matrices in the three dimensional reducible representation are
(41)S1=100010001,S123=001100010,S132=010001100,S12=010100001,S13=001010100,S23=100001010,where matrices in each equation belong to the same class ofS3. The most general neutrino mass matrix invariant under the S3 group is proportional to the democratic matrix and is given by
(42)Mν=aDwithD=111111111,where a is a complex number and D is called the Democratic matrix. The exact S3 symmetry does not satisfy the current neutrino oscillation data, and hence, symmetry must be broken. Various models based on the S3 symmetry have been presented in Refs. [95–122]. In Ref. [123, 124], the S3 symmetry is broken by the linear combination of S3 group matrices and successfully generates the nonzero θ13.The mass matricesMTM2 in Eqs. (20)–(25) can be seen as the linear combination of a democratic part and a symmetry breaking part. The symmetry breaking matrix is the sum of two symmetric matrices out of which one is the S3 group matrix which can be any of the S12,S13,S23 matrices, and the other part is chosen in such a way that the resultant neutrino mass matrix still satisfies the magic symmetry [125–129] and remains invariant under Z2 symmetry. The mass matrix MTM21 can be rewritten as
(43)MTM21=aabab+da−dba−da+d≡aD+cS13+dΔ,where
(44)D=111111111,S13=001010100,Δ=00001−10−11,and a,c,d are arbitrary parameters with b=c+a.The S3 symmetry of the neutrino mass matrix is broken, and the resultant neutrino mass matrix still satisfies S3 invariant constraints
(45)Mνii−Mνjj=Mνkj−Mνkiwithi≠j≠k.This leads to a trimaximal eigenvector for the resultant neutrino mass matrix. For example, a typical form ofMTM21 neutrino mass matrix is given by
(46)MTM21=0.004406−0.005578i0.004406−0.005578i0.002587+0.006938i0.004406−0.005578i−0.003415+0.027066i0.010408−0.025705i0.002587+0.006938i0.010408−0.025705i−0.001597+0.014550i.In this analysis, we take the charged lepton mass matrix to be diagonal. If a horizontal symmetry exists, it must, simultaneously, be a symmetry of the neutrinos as well as the charged leptons before the gauge symmetry breaking. After the symmetry breaking when the fermions acquire nonzero masses, the neutrino sector and the charged lepton sector should be governed by different subgroups of the symmetry group in order to have nonzero mixing. Here, we considerS3 to be the residual symmetry in the neutrino sector and Z3 symmetry as the residual symmetry in the charged lepton sector which yields nondegenerate diagonal charged lepton mass matrix [130].Similarly, other viable textures in Eqs. (20)–(22) can, also, be decomposed into the democratic S3 invariant part and the symmetry breaking part. The phenomenologically viable mass matrices in Eqs. (20)–(22) are related as follows by S3 permutation symmetry:
(47)S123MTM21S123T=MTM24,S132MTM21S132T=MTM25,S12MTM21S12T=MTM23,S13MTM21S13T=MTM26,S23MTM21S23T=MTM22.


## 4.TM1 Mixing and One Texture Equality

The neutrino mixing matrix with first column identical to TBM can be parametrized [60–67] as
(48)UTM1=23cosθ3sinθ3−16cosθ3+eiϕsinθ2sinθ3−eiϕcosθ2−16cosθ3−eiϕsinθ2eiϕcosθ2+sinθ3,and the corresponding neutrino mass matrix is given by
(49)MTM1=PlUTM1PνMνdiagPνTUTM1TPlT.The most general neutrino mass matrix withTM1 mixing can be written as
(50)MTM1=a2b2c2b4b+da−b−c−d2ca−b−c−d4c+d.The mass matrixMTM1 is invariant under the transformation G1TMTM1G1=MTM1 where G1=UTM1diag1,−1,−1UTM1† is the generator of Z2 symmetry. This along with equality condition restricts the three unphysical phase angles to ϕe=ϕμ=ϕτ≡ϕl.All possible textures of neutrino mass matrices withTM1 mixing and one texture equality are given by
(51)MTM11=2b2b2c2b4b+db−c−d2cb−c−d4c+d,MTM12=2c2b2c2b4b+d−b+c−d2c−b+c−d4c+d,MTM13=a2b2c2b2ba+b−c2ca+b−c4c−2b,MTM15=a2b2c2b4b−2ca−b+c2ca−b+c2c,MTM14=a2b2c2b12a+3b−c12a+3b−c2c12a+3b−c12a−5b+7c,MTM16=a2b2c2b12a+7b−5c12a−b+3c2c12a−b+3c12a−b+3c,MTM17=a2b2c2b3b−ca2ca3c−b,MTM18=a2b2c2b2ca+3b−3c2ca+3b−3c6c−4b,MTM19=a2b2c2b6b−4ca−3b+3c2ca−3b+3c2b,MTM110=a2b2c2ba3b−c2c3b−ca−4b+4c,MTM111=a2b2c2ba+4b−4c3c−b2c3c−ba,MTM112=MTM113=a2b2b2b4b+da−2b−d2ba−2b−d4b+d,MTM114=a2b2c2ba+b−c2b2c2ba−3b+3c,MTM115=a2b2c2ba+3b−3c2c2c2ca−b+c,where textures represented in each equation are related byμ-τ permutation symmetry. The neutrino mixing angles for TM1 mixing in terms of parameters θ and ϕ are [72–74] given by
(52)sin2θ13=13sin2θ,sin2θ12=1−23−sin2θ,sin2θ23=121+6sin2θcosϕ3−sin2θ.ForTM1 mixing, the Jarlskog rephasing invariant [72–74] is
(53)JCP=166sin2θsinϕ,and the Dirac-type CP-violating phase [72–74] is given by
(54)tanδ=cos2θ+55cos2θ+1tanϕ.The effective Majorana mass forTM1 mixing can be calculated by using Eqs. (11) and (48) as
(55)Mee=132m1+m2cos2θe2iα+m3sin2θe2iβ,and the effective neutrino mass for TM1 by using Eqs.(12) and (48) is given by
(56)mβ=132m12+m22cos2θ+m32sin2θ.The existence of one equality between the elements of the neutrino mass matrix implies(57)MTM1ab−MTM1cd=0,which yields the following complex equation:
(58)∑QVaiVbi−VciVdimi=0,where Q=eiϕa+ϕb−ϕc+ϕd. The above equation can be rewritten as
(59)m1A1+m2A2e2iα+m3A3e2iβ=0,where
(60)Ai=QUaiUbi−UciUdi,with i=1,2,3 and a,b can take values e,μ, and τ. Solving the real and imaginary parts of Eq. (59) simultaneously, we obtain the following two mass ratios:
(61)ζ≡m2m1=ReA1ImA3−ReA3ImA1cos2β+ReA1ReA3+ImA1ImA3sin2βReA3ImA2−ReA2ImA3cos2α−β+ReA2ReA3+ImA2ImA3sin2α−β,ξ≡m3m1=ReA2ImA1−ReA1ImA2cos2α−ReA1ReA2+ImA1ImA2sin2αReA3ImA2−ReA2ImA3cos2α−β+ReA2ReA3+ImA2ImA3sin2α−β.These mass ratios can be used to calculate the ratio of mass squared differences (Rν) which is given by
(62)Rν≡Δm212Δm312=ζ2−1ξ2−1Rν≡Δm212Δm232=ζ2−1ζ2−ξ2,for NO and IO, respectively. Since, Δm212 and Δm312Δm232 for NO (IO) are experimentally known, the parameter Rν should lie within its experimentally allowed range for a texture equality to be compatible with the current neutrino oscillation data. The neutrino mass eigenvalues can be calculated by using the relations
(63)m2=m12+Δm212,m3=m12+Δm312,m2=m12+Δm212,m3=m12+Δm212−Δm232,for NO and IO, respectively.For numerical analysis, we follow the same procedure as inTM2 mixing except that the parameters β and m1 are generated randomly within their allowed ranges. The mass eigenvalues are calculated by using Eq. (63), and texture equality is imposed by requiring the parameter Rν in Eq. (62) to lie within its 3σ experimental range.The numerical predictions for unknown parameters are summarized in Table5 (where constrains only from neutrino oscillation data are used) and Table 6 (where the constrains from cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are used). The allowed ranges of the parameter θ12 are (34.24∘-34.42∘) for all viable textures. The parameter θ is constrained to lie in the ranges (14.3∘-15.7∘), whereas JCP lies in the ranges ±0.026−0.036 for all viable textures. The Majorana phase α varies in the range (0-360∘) for all viable textures with NO only. Correlation plots among various neutrino oscillating parameters are shown in Figures 3 and 4 for NO and IO, respectively. Mee strongly depends on the Majorana phase α as shown in Figure 3(a) for NO and Figure 4(b) for IO. As shown in Figure 3(d), θ12 is inversely proportional to θ13 which is the classical prediction of TM1 mixing. The Dirac-type CP-violating phase is constrained to lie in the regions around 90° and 270° which is consistent with the recent observations in the long-baseline neutrino oscillation experiments such as T2K and NOvA [2, 3] which shows a preference for the Dirac-type CP-violating phase δ to lie around δ∼270∘. The main implications for textures having TM1 mixing with one texture equality are summarized in the following:
(i)
For IO, texturesMTM11, MTM12, and MTM17 are not consistent with the neutrino oscillation data at 3σ C.L.(ii)
For NO, texturesMTM14 and MTM16 predict large θ13 and small θ12 and are, hence, experimentally ruled out at 3σ C.L.(iii)
TexturesMTM112 and MTM113 predict a vanishing reactor mixing angle and degenerate mass eigenvalues and are, hence, not viable for both mass orderings(iv)
TexturesMTM13, MTM15, MTM110, MTM114, MTM115, and MTM111for both mass orderings and textures MTM17, MTM18 for NO predict large ∑mi and are, hence, not viable with experimental data when cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated(v)
All viable textures cannot have zero lowest mass eigenvalue for both mass orderings(vi)
The atmospheric mixing angleθ23 is maximal for δ∼π/2 or 3π/2 for all viable textures(vii)
The parameterMee is found to be bounded from below for all viable textures except MTM11 and MTM12 with NO(viii)
The parametermβ is found to lie within the current experimental range for all viable textures(ix)
The Dirac-type CP-violating phaseδ is directly proportional to the parameter ϕ for all viable texturesTable 5
Numerical predictions for viable textures having one equality inMν with TM1 mixing at 3σ C.L. (only neutrino oscillation data are incorporated).
TextureOrderingmlowest (eV)Mee (eV)∑ (eV)mβδ∘β∘MTM11NO0.0039-0.00660.0-0.00720.062-0.0690.009-0.01263-125⊕235-29721-65⊕116-159⊕202-244⊕296-339MTM12NO0.0039-0.00660.0-0.00720.062-0.0690.009-0.01264-125⊕235-29626-68⊕112-154⊕206-249⊕291-334MTM13NO0.046-0.670.027-0.550.16-2.00.002-0.00364-122⊕235-2970-43⊕137-222⊕318-360IO0.05-0.420.06-0.410.21-1.250.077-0.4267-125⊕235-2940-44⊕136-227⊕314-360MTM14IO0.007-0.0160.017-0.0510.107-0.1210.048-0.05265-124⊕235-2950-324MTM15NO0.04-0.920.02-0.880.15-2.80.045-0.9264-124⊕235-2960-36⊕143-218⊕322-360IO0.062-0.360.064-0.290.22-1.10.079-0.3766-125⊕236-2930-35⊕144-213⊕324-360MTM16IO0.008-0.0170.017-0.0520.108-0.1230.049-0.52465-125⊕234-2940-324MTM17NO0.085-0.720.032-0.70.26-2.20.085-0.7263-125⊕236-2970-37⊕143-218⊕323-360MTM18NO0.03-0.320.02-0.310.118-0.980.031-0.3263-124⊕236-2970-53⊕128-235⊕308-360IO0.014-0.230.03-0.220.116-0.70.05-0.2466-125⊕235-2940-65⊕115-244⊕296-360MTM19NO0.025-0.310.018-0.190.107-0.930.025-0.3267-125⊕235-2970-50⊕130-224⊕312-360IO0.0148-0.330.028-0.260.117-1.00.05-0.3366-125⊕235-2940-56⊕124-236⊕306-360MTM110NO0.034-0.70.014-0.650.13-2.10.034-0.763-124⊕237-29743-141⊕221-314IO0.028-0.40.021-0.270.14-1.20.05-0.4166-125⊕235-29534-145⊕215-325MTM111NO0.031-0.30.011-0.270.122-0.860.032-0.2965-125⊕235−29547-128⊕229-313IO0.03-0.510.023-0.480.14-1.520.05-0.5165-124⊕236-29544-136⊕224-316MTM114NO0.04-0.480.02-0.360.16-1.430.04-0.4863-123⊕237-29747-132⊕230-314IO0.06-0.40.028-0.380.21-1.20.07-0.4165-126⊕234-29443-135⊕225-316MTM115NO0.04-0.550.01-0.50.15-1.70.04-0.5564-123⊕236-29553-125⊕237-307IO0.06-0.520.03-0.50.22-1.60.07-0.5265-125⊕235-29554-126⊕234-305Table 6
Numerical predictions for viable textures having one equality inMν with TM1 mixing at 3σ C.L. (cosmological and neutrinoless double beta decay bounds along with neutrino oscillation data are incorporated).
TextureOrderingmlowest (eV)Mee (eV)∑ (eV)mβδ∘β∘MTM11NO0.0039-0.00660.0-0.00720.062-0.0690.009-0.01264-125⊕235-29621-65⊕116-158⊕201-245⊕295-339MTM12NO0.0039-0.00660.0-0.00720.062-0.0690.009-0.01264-125⊕235-29626-68⊕112-154⊕206-249⊕291-334MTM14IO0.007-0.0160.017-0.0510.107-0.120.048-0.05266-125⊕235-2950-322MTM16IO0.008-0.0170.017-0.0510.108-0.120.049-0.05265-125⊕235-2950-320MTM18IO0.014-0.0170.038-0.0490.116-0.120.05-0.05284-124⊕236-27311-53⊕125-170⊕193-235⊕307-347MTM19NO0.025-0.0310.018-0.0310.107-0.120.026-0.32112-125⊕235-2471-29⊕152-208⊕332-359IO0.0148-0.01670.037-0.0470.117-0.120.05-0.05267-95⊕269-29316-44⊕135-168⊕191-225⊕316-348Figure 3
Correlation plots among various parameters for textures (a)MTM11, (b)MTM11, (c) MTM12, (d) MTM12, (e) MTM12, and (f) MTM19 with NO.Figure 4
Correlation plots among various parameters for textures (a)MTM14, (b) MTM14, (c) MTM16, (d) MTM18, (e) MTM19, and (f) MTM16 with IO.


## 5. Summary

We studied the phenomenological implications of one texture equality in the neutrino mass matrix withTM1 or TM2 mixing. The presence of one texture equality in Mν with TM1 or TM2 as the mixing matrix reduces the number of free parameters significantly and, hence, leads to very predictive neutrino mass matrices. Out of total fifteen possible textures of Mν, thirteen textures are phenomenologically allowed with TM1 mixing, and only six textures are allowed with TM2 mixing in the light of current neutrino oscillation data at 3σ C.L. However, the number of viable textures reduced to six for TM1 mixing if the constrains from cosmology and neutrinoless double beta decay experiments along with neutrino oscillation data are used. Since, the TM2 mixing predicts a value of θ12 away from its best fit value, TM1 mixing is phenomenologically more appealing. In this analysis, we have obtained interesting predictions for unknown parameters such as the Dirac- and Majorana-type CP-violating phases, effective Majorana neutrino mass, effective neutrino mass, Jarlskog rephasing invariant, neutrino mass scale, and the sum of neutrino masses. For TM1 mixing, the Dirac-type CP-violating phase δ is restricted to the regions around π/2 and 3π/2, the atmospheric mixing angle θ23 is maximal for δ∼π/2 or 3π/2, and the lowest neutrino mass eigenvalue cannot be zero for all viable textures. For TM2 mixing, the CP-violating phases δ and β are strongly correlated, θ23 is below (above) maximal for textures MTM23MTM25 and MTM25MTM23 with NO and IO, respectively, and the lowest mass eigenvalue cannot be zero for all viable textures. For MTM2 mass matrices with one texture equality, the residual S3 symmetry is broken, and the resulting neutrino mass matrix is invariant under Z2 64.



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*Source: 4952562-2022-09-30.xml*