Two simple textures of the magic neutrino mass matrix
Kanwaljeet S. Channey Sanjeev Kumar

Interferences between En and Mn (n=1,2...) optical transition moments
    A. I. Chichinin 

Institute of Chemical Kinetics and Combustion
 and
Novosibirsk State University,
 
630090 Novosibirsk, Russia

chichinin@kinetics.nsc.ru
    December 30, 2023
==============================================================================================================================================================


The Tri-Bimaximal (TBM) mixing predicts a vanishing θ_13. This can be
  attributed to the inherited μ-τ symmetry of TBM mixing. We break its
  μ-τ symmetry by adding a complex magic matrix with one variable to TBM neutrino mass matrix
  with one vanishing eigenvalue. We present two such textures and study their
  phenomenological implications.


In the Standard Model of electroweak and strong interactions, neutrino flavor
states ν_l (l=e,μ,τ) are the states which
form weak doublets with the corresponding charged lepton states l:

    j^μ = l̅γ^μ(1-γ_5) ν_l.

These neutrino states ν_l are the coherent combinations of the neutrino mass
states ν_i (i=1,2,3) which are also the eigenstates of the Hamiltonian in
vacuum. Neutrino flavor and mass states are related by the relation 

    ν_l = U_PMNSν_i,

where U_PMNS is the unitary matrix called
Pontecorvo-Maki-Nakagawa-Sakata mixing matrix. By convention, PMNS mixing matrix
is defined as


    U_PMNS = ([                      c_12 c_13                      s_12 c_13                     s_13 e^-ιδ;  -s_12c_23-c_12 s_23 s_13 e^ιδ  c_12 c_23-s_12 s_23 s_13 e^ιδ                      s_23 c_13;  s_12 s_23-c_12 c_23 s_13 e^ιδ -c_12 s_23-s_12 c_23 s_13 e^ιδ                      c_23 c_13;                                ])
         
             where s_ij=sinθ_ij, c_ij=cosθ_ij (i,j=1,2,3) and
             δ is the Dirac type CP violating phase.
             
In the flavor basis, where lepton mass matrix is diagonal, the neutrino mass
matrix M_ν is related with the unitary mixing matrix and the complex
neutrino masses m_d=diag(m_1,m_2 e^2 ια,m_3 e^2 ιβ) by the relation


    M_ν = U^*_PMNS m_d U^†_PMNS,

where α and β are the Majorana phases.
While analyzing the experimental results on neutrino mixing and mass matrices,
we can often look for some particular features like equalities <cit.>,
zeros <cit.>, hybrids of zeros
and equalities <cit.>, zero trace <cit.>, or some other pattern <cit.> in their elements.

Harrison, Perkins and Scott proposed one such type of mixing
matrix <cit.> which had ν_2 trimaximally mixed and ν_3
bimaximally mixed. Hence, they named it Tri-Bimaximal
(TBM) mixing. The TBM mixing matrix is 

    U_TBM = ([     √(2/3)     1/√(3)          0;    -√(1/6)     1/√(3)     1/√(2); -   √(1/6)     1/√(3)    -1/√(2) ]).

The mass matrix M_TBM corresponding to U_TBM
can be written by using 

    M_TBM = ( [    a    b    b;    b a+ d  b-d;    b  b-d  a+d;      ]).


TBM mixing matrix predicts sin^2 θ_13 = 0, sin^2
θ_12=1/3 and sin^2 θ_23 = 1/2. Mixing angles θ_12 and
θ_23 are in agreement at 3σ with their experimental values, sin^2 θ_12 =
0.306^+0.012_-0.012 and sin^2 θ_23 = 0.441^+0.027_-0.021,
provided by the latest global fit of the neutrino experimental
data <cit.>.

Another parametrization of the mixing matrix was given in Ref.
  <cit.>. In this parametrization, the mixing matrix was expressed
  as expansion in powers of the deviations of reactor, solar and atmospheric
mixing angles from their TBM value . Let r,s, and
a are the real parameters which give deviations to the reactor, solar and
atmospheric mixing angles from their TBM values:

    sinθ_13 = r/√(2), sinθ_12 = 1/√(3)(1+s), sinθ_23 = 1/√(2)(1+a).

Since the parameters r,s, and a are very small, we can expand the mixing
matrix about U_TBM in the powers of r,s, and a. We present here the mixing
matrix to the first order in r,s, and a

    U_p ≈(
    [          √(2/3)( 1- 1/2s )                1/√(3)(1+s)                1/√(2)re^ιδ;      - 1/√(6)(1+s-a+re^ιδ)  1/√(3)(1-1/2s-a-1/2re^ιδ)                1/√(2)(1+a);        1/√(6)(1+s+a-re^ιδ) -1/√(3)(1-1/2s+a-1/2re^ιδ)                1/√(2)(1-a) ]).


The mixing angle θ_13 is non zero as measured by the
recent experiments: T2K<cit.>, Daya Bay<cit.>, RENO<cit.>
and DOUBLE CHOOZ<cit.>. This leads to the realization that although
TBM ansatz is ruled out by the experiments, it can still be used
as leading order contribution to the neutrino mass matrix. We can add
perturbations to M_TBM so as to generate the non-zero θ_13. TBM mass
matrix obeys both the magic symmetry and the μ-τ exchange symmetry. Magic symmetry
means sum of the elements of each row and column of mass matrix remains the
same, whereas μ-τ exchange symmetry means that the neutrino mass matrix is invariant under the simultaneous
interchange of its second and third (μ-τ) indices.

A neutrino mass matrix that is invariant under magic symmetry and μ-τ
exchange symmetry predicts maximal θ_23 and vanishing θ_13. These predictions
are very close to the present neutrino oscillation data. This indicates that
we can satisfy the present experimental data by introducing small
perturbations to the magic mass matrix. Magic symmetry also provides sum rules
between the mixing angles due to trimaximal structure of ν_2 <cit.> which in return reduces the number of free parameters. These sum rules can be tested at the future neutrino oscillation experiments.

In the present paper, we propose two simple textures of M_magic that break the
μ-τ symmetry of TBM neutrino mass matrix but preserve its magic
symmetry. These textures can be written as

    M^i_magic = M_TBM +M'_i , (i=a,b)

The μ-τ breaking term M_i in these textures is function of only one complex variable η = z
e^ιχ. To reduce the number of independent variables, in our study we
have considered the M_TBM with
vanishing lowest eigenvalue. This assumption will lead to the condition that
b=a in the Eq. <ref>. Forms of M_TBM and M'_i (i=a,b)
studied in the present work for normal hierarchy are:

    M_TBM = (
    [   a   a   a;   a a+d a-d;   a a-d a+d;     ]), M'_a = (
    [  0  0  η;  0  0  η;  η  η -η;    ]), M'_b = (
    [ 0 η 0; η 0 0; 0 0 η;   ]).

We then study the phenomenological implications for these textures of neutrino mass matrix.

While perturbing the TBM mass matrix by adding an extra matrix, we can
break M_TBM in such a way that out of the two symmetries that it possesses, we break only one.
Since μ-τ symmetry predicts vanishing θ_13, preserving magic
symmetry is a feasible choice. 

If a transformation G_j of the neutrino fields leaves the neutrino
mass matrix unchanged such that

    G^T_j M_ν G_j = M_ν,

 the transformation G_j is called a symmetry of mass matrix M_ν. The transformation matrix
can be calculated using the relation, G_j = 1-u_j u^T_j (j=1,2,3) where u_j is the column
of matrix corresponding to the symmetry G_j. The transformation matrix corresponding to the magic symmetry
is given below

    G_2 = (
    [  1/3 -2/3 -2/3; -2/3  1/3 -2/3; -2/3 -2/3  1/3;      ]).

Therefore, a mass matrix M_magic that preserves the magic symmetry will obey the relation
G^T_2 M_magic G_2 = M_magic. Mixing matrix corresponding to such mass matrices will have
their middle column same as that of U_TBM (trimaximal) and can be described in
terms of two independent variables θ and ϕ

    U_TM=(
        [                                                                                                                           √(2/3)cosθ                                                                                                                               1/√(3)                                                                                                                           √(2/3)sinθ;                                                                                                           e^i ϕsinθ  -cosθ/√(3)/√(2)                                                                                                                               1/√(3) -e^i
                                                                                                             ϕcosθ  -sinθ/√(3)/√(2);                                                                                                            -cosθ/√(3)-e^i ϕsinθ/√(2)                                                                                                                               1/√(3)                                         e^i
                                                                      ϕcosθ  -sinθ/√(3)/√(2);                                                                                                                                      ]).

This mixing matrix, has a trimaximally mixed column leading to its nomenclature
as trimaximal mixing. One of the general form for M_magic can be written as

    M_magic =   (
    [       a       b       c;       b     a+d     c-d;       c     c-d a+b-c+d;         ]).

This mass matrix can be diagonalized by using the equation

    M_d = U^T_TMM_magicU_TM.

The rationale of choosing the form of M_magic as given in Eq.
(<ref>) is that it reduces to M_TBM (Eq.
(<ref>)) for c=b. It is the difference of b and c that breaks the
μ-τ symmetry of M_TBM. Therefore, to break μ-τ exchange
symmetry and to generate non-zero θ_13, we can allow b and c to
differ by a small 

The diagonal elements of M_d will give us neutrino masses and Majorana phases, whereas the off-diagonal
elements, when equated to zero, will give the variables θ and ϕ of
U_TM in terms of the parameters of M_magic. We can
calculate the mixing angles in terms of θ and ϕ from the elements of
U=U_TM using the relations

    sin^2 θ_12 = |U_12|^2/1-|U_13|^2, sin^2 θ_23 = |U_23|^2/1-|U_13|^2,sin^2 _13 = |U_13|^2.

  We can calculate the Dirac phase δ from the Jarlskog rephasing
  invariant measure of CP violation,

    J = Im[U_12 U_23 U^*_13 U^*_22],

using the relation,

    J = c_12 s_12 c_23 s_23 c^2_13 s_13sinδ.

Neutrino masses and Majorana phases can be calculated from M_d using the following relations
  
    |m_1| =|[M_d]_11|, |m_2| = |[M_d]_22|,|m_3|=|[M_d]_33|,

  
    α = 1/2([M_d]_22/[M_d]_11)  , β = 1/2([M_d]_33/[M_d]_11).

We can write M_magic as the sum of M_TBM and a μ-τ
symmetry breaking term M':

    M_magic = M_TBM + M'

where M' is also invariant under G_2.
Looking at Eq. (<ref>), we observe that we can write M_magic as sum of four matrices:


    M_magic=(
    [ a 0 0; 0 0 a; 0 a 0;   ])+(
    [  0  0  0;  0  d -d;  0 -d  d;    ])+(
    [ 0 b 0; b 0 0; 0 0 b;   ])+(
    [  0  0  c;  0  0  c;  c  c -c;    ]).


There are two ways to write M_magic as combination of
M_TBM and M' as shown in Eq.
(<ref>). First is by considering b=a, c=a+η

    M^a_magic = (
    [   a   a   a;   a a+d a-d;   a a-d a+d;     ]) + (
    [  0  0  η;  0  0  η;  η  η -η;    ])

and second is by
considering b=a+η, c=a

    M^b_magic = (
    [   a   a   a;   a a+d a-d;   a a-d a+d;     ]) +(
    [ 0 η 0; η 0 0; 0 0 η;   ]),

where η = z e^ιχ is responsible for the breaking of μ-τ symmetry. The assumption a=b in M_TBM results in vanishing lowest
eigen value of M_TBM. We made this assumption to reduce the number of
free parameters in M_TBM. This gives us the two textures studied in
this paper given in Eq. (<ref>) (M^a_magic and M^b_magic).

We can diagonalize these mass matrices by using Eq. (<ref>) and obtain
our predictions from Eqs. (<ref>-<ref>). Equating the
nondiagonal entry [m_d]_13 with zero for these textures will give us
predictions for the variables
θ and ϕ in terms of a, d, z and χ.


For texture M^a_magic, the nondiagonal entry [m_d]_13, θ
and ϕ are as follows:

    [m_d]_13   = 1/4 e^-2 i ϕ(2 √(3) z cos 2 θ  e^i
           (χ +ϕ )) 
       - 1/4 e^-2 i ϕ( sin 2 θ(4 d +e^i χ z (-3+e^2 i ϕ))), 
    tanϕ   = -d tanχ/d-z χ, 
    tan 2 θ   = -2 √(3) z cos (ϕ -χ )/-4 d cos 2 ϕ -z (cosχ -3 cos (2 ϕ -χ )).

 For texture M^b_magic we obtain the following relations for [m_d]_13, θ
 and ϕ:

    [m_d]_13   = 1/4 e^-2 i ϕ(-sin 2 θ(4 d+e^i
                    χ z (1+e^2 i ϕ))) 
       - 1/4 e^-2 i ϕ(2 √(3) z 
      cos 2 θ  e^i (χ +ϕ )), 
    ϕ   = -χ, 
    tan 2 θ   = √(3) z cos (ϕ -χ )/-2 d cos 2 ϕ -z cosϕcos (ϕ -χ ).


Corresponding mixing angles and Dirac type CP violating phase then can be
calculated from these θ and ϕ by using the following relations

    sin^2 θ_12   =   1/3-2 sin ^2 θ,   
    sin^2 θ_23   =   1/2(1+ √(3)sin 2 θcosϕ/3-2 sin ^2 θ),   
    sin^2 θ_13   =   2 sin ^2 θ/3,
    ^2 δ   =    ^2 ϕ -3 sin ^2 2 θ ^2 ϕ/(3-2 sin ^2 θ)^2.




  Substituting the values of θ and ϕ in terms of a, b, z and
  χ in [m_d]_11, [m_d]_22 and [m_d]_33, we can obtain the three
  neutrino masses (m_1, m_2 and m_3) and the Majorana phases (α and
  β) using Eqs. (<ref>, <ref>), where


    [ m_d ]_11   = -1/2 e^-2 i ϕ(sin^2
                   θ(-4 d+3 e^i χ z) ) 
       - 1/2 e^-2 i ϕ( ( 2 √(3) z cosθ  e^i (χ +ϕ ))
                                 + z cos ^2 θ  e^i (χ +2 ϕ )),
    [ m_d ]_22   = 3 a + e^i χ z,

and

    [ m_d ]_33   = 1/2 e^-2 i ϕ(cos ^2 θ(4 d-3 e^i χ z) )
       + 1/2 e^-2 i ϕ( z sinθ 
                                   e^i (χ +ϕ )(2 √(3)cosθ
                                   -e^i ϕsinθ)).

Similarly, for texture M^b_magic, we have

    [ m_d ]_11   = 1/2 e^-2 i ϕ(sin ^2 θ(4 d+e^i χ z)+√(3) z sin 2
                                   θ  e^i (χ +ϕ ))
       + 1/2 e^-2 i ϕ( z cos ^2 θ(-e^i (χ +2 ϕ ))),
    [ m_d ]_22   = 3 a+e^i χ z ,

and

    [ m_d ]_33   = 1/2 e^-2 i ϕ(cos ^2 θ(4 d+e^i χ z) ) 
       - 1/2 e^-2 i ϕ( z sinθ
                                     e^i (χ +ϕ )(2 √(3)cosθ +e^i ϕsinθ)).

These relations give us the three neutrino masses and two Majorana phases from
 Eqs. (<ref>, <ref>).

 Current neutrino experiments cannot observe the three neutrino masses directly. β-
 decay experiments <cit.> are sensitive to the effective neutrino mass m_β given
 as


    m^2_β = m^2_1 |U_e1|^2 + m^2_2 |U_e2|^2 + m^2_3 |U_e3|^2.

The effective neutrino mass m_ββ given as

    m_ββ = |m_1 U^2_e1 + m_2 U^2_e2 + m_3 U^2_e3|.

can be measured in the neutrino-less double β-decay experiments <cit.>.
 
 We can obtain the parameters r,s, and a defined in
   Eq. (<ref>) in terms of θ and ϕ by comparing the
   values of mixing angles given in Eq. (<ref>-<ref>) with Eq.
   (<ref>). These relations are given as follows

    r     =    2/√(3)sinθ,
    
    s    =   1/√(1-2/3sin^2θ) - 1,
    
      a    =   √(1+ (√(3)sin 2θcosϕ/3-2 sin ^2θ))-1.

 
We perform a Monte Carlo analysis for these two textures by generating the variables a,
d, z and χ using uniform random distributions. The parameter space of these
variables is restricted by imposing the experimental constraints <cit.> on Δ m^2_12=m^2_2-m^2_1, Δ m^2_23=m^2_3-m^2_2,
θ_12 and θ_23 at 3σ C. L. These ranges are shown in Tab.
<ref>.
The allowed parameter space for a, d, z and χ is displayed in Fig. <ref> and the allowed ranges can
be read from Tab. <ref>. The correlation plots between
θ_23 and δ are presented in Fig. <ref> for both the textures. Here, θ_23 is well within the experimental range
at 3σ
([38.4 , 52.8], shown as horizontal dashed lines) for
both the textures. However, for 1 σ range of
  θ_23, we find that only allowed ranges for δ
  are [215-251] and [287-312] ruling out δ =[250-286].
At 3σ, for both the textures, δ should be either 90^o or
270^o for a maximal θ_23. The value of δ shifts towards 0^o
or 180^o when θ_23 takes its extreme values
around 38^o or 51^o. This feature is testable at the experiments like
NOνA <cit.>
and T2K <cit.>. The degeneracy in
the Fig. <ref> is because of the contributions of positive and negative values of the parameters a and d. The allowed values of these parameters can be positive or negative and contribute to the different branches of this figure.

The correlation between the phases δ and β is displayed in Fig. <ref> and
that between α and β is shown in Fig. <ref>. Variation of m_β and
m_ββ with CP violating phase δ is presented in Fig.
<ref> and <ref> respectively. 





From Fig. <ref>, <ref> and <ref>, it is
  clear that our predictions for m_β and m_ββ are very small as compared to
  the sensitivities of the near future β-decay experiments like KATRIN
  <cit.>, Project 8 <cit.> and double
  β-decay experiment EXO-200 <cit.>, KamLAND-Zen <cit.>. If any of these experiments will be successful in measuring the
  m_β or m_ββ, our textures will be ruled out. Correlations between the mixing angles for
  fixed values of δ (0,45,90,135,180,225,270,315,360) are
  given in the Fig. <ref>. Here the plots for (δ,δ+180)
  are identical. Similarly, the plots for
  (δ+45,δ+135,δ+225,δ+315) and (δ+90,δ+270)
  are same. The solid lines represent the
1σ experimental range and dashed lines represents the 3σ
experimental range.

We have constructed the textures M^a_magic and M^b_magic
for normal hierarchy with vanishing lowest eigenvalue of M_TBM. We can
obtain similar textures for neutrino masses with inverted hierarchy which can
also be written as M^i_magic = M_TBM+M'_i. Assuming that
M_TBM has lowest vanishing eigen value, the
forms of M_TBM and M'_i for the inverted hierarchy case are as
follows:

    M_TBM = (
    [     a a+2 d a+2 d; a+2 d   a+d   a+d; a+2 d   a+d   a+d;       ]), M'_a = (
    [  0  0  η;  0  0  η;  η  η -η;    ]), M'_b = (
    [ 0 η 0; η 0 0; 0 0 η;   ]).

However, as shown in the Fig. <ref>, we find that the experimental ranges for the mixing angle θ_13 and the
ratio Δ m^2_12 / |Δ m^2_23| cannot be satisfied simultaneously
for the inverted hierarchy case. Thus, the corresponding textures with inverted hierarchy are ruled out.

In one of our previous study <cit.>, we had demonstrated the idea that viable textures TM_1 and TM_2 of the neutrino mass matrix can be created which are modifications of the mass matrix corresponding to TBM mixing. We did not provide any rationale to these textures. However in the present work, we propose a systematic method to modify the neutrino mass matrix corresponding to the TBM mixing M_TBM. We modify the M_TBM by breaking μ - τ symmetry but preserving the magic symmetry. We generate two such textures M^a_magic and M^b_magic. The texture M_2 of our previous study and the texture M^b_magic of our present study are related by μ - τ exchange symmetry resulting in identical predictions for these textures. The another texture M_1 of the previous study did not preserve either μ - τ symmetry or the magic symmetry. Similarly the texture M^a_magic of our current study is different from any of the previously proposed textures as can be seen by comparing the Fig. <ref>, <ref>, <ref> and <ref> of the present manuscript.

In conclusion, we have investigated two simple textures of the neutrino
mass matrix with magic symmetry. These textures can be written as combination of
TBM mass matrix with a vanishing eigenvalue and a simple perturbation matrix with
one complex parameter preserving the magic symmetry. These textures have four
real parameters: a, d,  z, and χ. We find the allowed ranges for
these parameters and present the resulting correlations between θ_23
and δ. We find that δ should be around 90^o or 270^o for
maximal θ_23 mixing for both textures. When θ_23 is around
40^o or 50^o, δ can take values near 0^o or 180^o. Such
correlations are generic features of magic symmetry and are testable at future
neutrino experiments like NOνA and T2K. Our textures have definite
predictions for m_β
<cit.> and
m_ββ <cit.> which can be tested at β-decay and neutrino-less double β-decay experiments.




                                                           
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