Journal of Modern Physics, 2012, 3, 1556-1561
http://dx.doi.org/10.4236/jmp.2012.310192 Published Online October 2012 (http://www.SciRP.org/journal/jmp)
Different Mixing Scenerio of Qu asi-Degenerate
Neutrino with Charged Lepton
Correction
Mrinal Kumar Das
Department of Physics, Tezpur University, Tezpur, India
Email: mkdas@tezu.ernet.in
Received August 16, 2012; revised September 15, 2012; accepted September 22, 2012
ABSTRACT
Theoretically, in order to achieve non-zero θ13 a little deviation from Tribimaximal Mixing (TBM) pattern is needed,
especially on θ13 without perturbing the atmospheric and solar mixing angles. In this work we computed the neutrino
mixing angles by disturbing the θ13 as well as θ12 in Bimaximal (BM) and Hexagonal mixing (HM) using non-diagonal
charged lepton mass. Considering the standard form of mass texture which satisfies TBM we have shown the quasi-
degenerate nature of neutrino. This quasi degenerate type of mass matrix for BM and HM is then used to calculate the
deviated mixing pattern which are consistent with recent neutrino oscillation data.
Keywords: Neutrino Mixing, Neutrino Masses, Charged Lepton Correction
1. Introduction
Recent solar, atmospheric, reactor, and accelerator neu-
trino experiments have provided concrete evidence that
neutrinos are massive and they change their flavors dur-
ing propagation. In the standard neutrino oscillation pic-
ture three active neutrinos are involved, with mass-
squared differences of order 10 and 10 eV2. The
deficit in the neutrino flux from solar and atmospheric
neutrinos have confirmed that at least two neutrinos
should have non-zero masses. The mixing pattern and the
tiny neutrino masses makes the explanation of the origin
of neutrino masses and leptonic flavor mixing one of the
most prominent problems in the particle physics. The
mixing of lepton flavors is described by a 3 × 3 unitary
matrix, whose nine elements are commonly parametrized
in terms of three rotation angles and three CP-violating
phases. Defining three unitary rotation matrices in the
complex planes one can express neutrino mixing in terms
of three rotations:
53
i23
23
23
23
0
e ,
0e
s
sc

i13
13
13
0e
10,
cs
sc
i12
12 12
i12
12 1212
e0
=e 0,
001
cs
Us c






=cosc
23 23
i
10
=0Uc
23
13
13
i13
=0
e0
U
(1)
13
(2)
(3)
where ij ij
and ij ij
=sins1, 3ij
with
23 13 12
=,UUUUP
PMNS
13 1213 1213
ii
23121213 2312231213 2313 23
i
12 231213 2312 2323131213 23
ee,
e
i
UU
cccs se
cs cssccssscsP
ss csccs csscc



 




.
Using these three rotations, expression of neutrino mix-
ing with standard parameterizations become
(4)
where U is known as Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) matrix and can be written as
(5)
where
ii
12
Diage, e,1P

is a diagonal phase matrix
which contains two non-trivial Majorana phases of CP
violation. This also involves just three irremoveable
physical phases ij
. In this parameterizations the Dirac
phase
which enters the CP odd part of neutrino os-
cillation probabilities is given by 1323 12


13
. The
recent global fit [1] to the various neutrino experimental
data has given the following mixing angle values and
non-zero of
.
C
opyright © 2012 SciRes. JMP
M. K. DAS 1557
2
23
sin
0.073
0.058
= 0.466
0.019
0.018
= 0.312
0.016 0.010.
(6)
2
12
sin
. (7)
and
2
13
sin =
(8)
In view of above mixing angles it is clear that the
Tribimaximal mixing (TBM) [2-4],
PMNS
3
U
21 0
3
11 1
=,
632
11 1
63 2











13
(9)
can give the close description to the neutrino oscillation
as well with a minor correction in
. The predictions
of Equation (9) viz. 2
sin
23
1
2
13
sin 0 and
2
12
sin3
1, are consistent with atmospheric and solar
neutrino oscillation with minor correction in 13
. As the
global analysis of neutrino data has provided hints for
non-zero 13
[5-7]. The first observational hint for non-
zero 13
has come from the T2K experiment [8]. After
T2K experiment MINOS experiment also disfavor the
13 0
[9]. To achieve non-zero 13
theoretically is an
interesting topic in neutrino physics. Now, recent analy-
sis on neutrino mixing it is proposed by many papers
[10-12] that apart from TBM there are some other mixing
pattern like Bimaximal mixing (BM), Hexagonal mixing
(HM) and Tetragonal mixing can also give the alternative
description of neutrino mixing with correction. Among
these, tribimaximal mixing gives very close description
of the experimentally found mixing angles when the best
fit values are presumed.
In the proposed work we give a description of neutrino
mixing which is achieved in the framework of bimaximal,
tribimaximal and hexagonal mixing with the help of
charged lepton correction. In the present paper, we take
non-diagonal charged lepton mass in expression of
L
L
given by Equation (11). This correction can be realised in
the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix
in the form of PMNSl
UU
m
UU
 , where l
U diago-
nalise the charged lepton mass matrix
L
R. The matrix
l is considered as Cabibbo like mixing matrix as al-
ready been discussed in a recent work [13] for TBM case.
This correction will deviate the solar mixing 12
m
U
and
CHOOZ mixing 13
to the experimental range in case
BM and HM without disturbing the 23
.
The seesaw mechanism is used to construct neutrino
mass models e.g.: Quasi-degenerate, Normal Hierarchi-
cal (NH) and Inverted Hierarchical (IH), are discussed in
our earlier work [14,15]. Out of these three models which
model can give good prediction to the neutrino oscilla-
tion is also a topical question in recent neutrino physics.
In this work we analyze on Quasi-degenerate neutrino
mass model (NH and IH) with different mixing pattern as
discussed above. In Section 2, we shown neutrinos are
quasi-degenerate (NH and IH) in nature considering neu-
trinos are mixed tribimaximally. Next in Section 3, dif-
ferent neutrino mass models with BM and HM consider-
ing neutrinos are quasi-degenerate in nature are discussed.
Then with the help of charged lepton correction how
these mixing can predict new mixing which are consis-
tent with recent experimental data along with non-zero
13
is also discussed. And in the final section i.e. in Sec-
tion 4, there are some concluding remarks.
2. Quasi-Degenerate Neutrino
As mentioned in the introduction neutrino mass eigen-
values can have three kinds of pattern normal, inverted
hierarchical and quasi-degenerate. We know, from neu-
trino oscillations that the neutrino mass pattern is non-
degenerate. The pattern is hierarchical, if 1
mm
2
m2
m
2
1
mm
m
,
1 is smallest neutrino mass and is solar mass
square difference. When a
tm
, the pattern is in-
verted hierarchical, where 3 is the smallest mass. In
the quasi-degenerate case both the ordering may be pos-
sible.
The most popular neutrino mixing which gives the
TBM form is given by Equation (9). This can be gener-
ated with two generators S and T of 4
A
symmetry, one
of which gives charged lepton mass matrix diagonal and
other gives the invariant neutrino mass matrix
[16], where is given by
TBM
m
TBM
m


TBM
11
,
22
11
22
AB B
mBABDABD
BABD ABD








 


=mAB
(10)
gives 1
, 2 and 3. The
mass matrix given by Equation (10) is constructed on the
basis of the type I see-saw mechanism,
=2mAB=mD
1
=,
T
L
LLRRRLR
mmMm
(11)
where
L
R is diagonal. The present neutrino oscillation
data gives the following information for the mass square
difference.
m
52 252
21
7.05 10eV8.34 10eVm


32 232
31
2.0710 eV2.7510 eVm


,
,
Copyright © 2012 SciRes. JMP
M. K. DAS
JMP
1558
252
232
6510eV ,
2.4010eV .
m m
<0.61eV
m m
0.105217
0.05632
0.10905

0.0001196
0.005786
0.100396





=,
l
UUU
tive mixing pattern can also arrive in the experimental
range with corrections. In this section we try to produce a
mixing pattern, from Bimaximal, Hexagonal as well as
Tribimaximal mixing with charged lepton correction,
which is consistent with recent neutrino oscillation data. In
general, the Pontecorvo-Maki-Nakagawa-Sakata (PMNS)
matrix with charged lepton correction can be expressed
as product of two unitary matrices as
with the following best fit values
21
31
7.m
m


Using 1, 2 and 3 from Equation (10), mass
square differences and the sum of the three mass eigen-
values i
i can be expressed in
terms of A, B and D. By solving these three equations in
terms of A, B and D values of 1, 2 and 3 are
calculated with MATHEMATICA and listed in the Ta-
ble 1, which are quasi-degenerate in nature.
m
cosmo
m
m (14)
U
diagonalizes the neutrino mass matrix as where
T
diagonal =,
LL LL
mUmU
These values of A, B and D are now used to construct
the neutrino mass matrices with the help of Equation (10)
for NH and IN case are:
(15)
l
U diagonalizes the charged lepton mass matrix as
Normal hierarchical:
T
diagonal =.
L
RlLRl
mUmU (16)
0.052487 0.105217
0.105217 0.10905
0.105217 0.05632
LL
m
(12) In our earlier work [14,15], in construction of
L
L
m,
using seasaw I we were considered the dirac mass
L
R
M
is diagonal, which can be considered as either charged
lepton or up quark type. However, a general form of the
Dirac neutrino mass matrix is given by
Inverted hierarchical:
0.1063024 0.0001196
0.0001197 0.100396
0.00011966 0.005786
LL
m


(13) 00
00
001
m
n
LR
m





(17)
After Diagonalising the above mass matrices, calcu-
lated mass square differences and mixing angles are
listed in the Table 2, which shows TBM property.

where
f
mtanm
corresponds to
for
,6,2mn
m

in the case of charged lepton and t for
,8,4mn
in the case of up-quarks. Here
can pick value be-
tween 0.104 and 0.247 for the Dirac neutrino mass ma-
trix. In this pattern the PMNS matrix given by Equation
(14) doesnot involve l. In the present analysis we use
non-diagonal
3. Deviation from Original Mixing Pattern
with Charged Lepton Correction U
U
L
R in seesaw I and due to this reason
one has to use the contribution l in the PMNS matrix.
We construct a deviated neutrino mixing matrix U i.e.
PMNS matrix using Equation (14), where charged lep-
tons mass matrices are considered to be non-diagonal.
This predicts mixing angles, are consistent with recent
oscillation data. To construct U we consider
From analysis of recent neutrino oscillation parameter it
is observed that neutrino mixing is very close to TBM
pattern. Deviation from TBM is recently reported [13,
17], where important corrections are incorporated to
make the mixing angle match with the experimental data.
Different possible alternatives to this TBM are, e.g. Bi-
maximal, Trimaximal, Hexagonal mixing as well golden
ration angles, discussed in recent work [10]. This alterna-
m
U
U
in three
different forms e.g. BM, HM as well as TBM and the
Table 1. Values of A, B, D, m1, m2, m3.
Type A B D m1 m2 m3
NH 0.052487 0.105217 0.16537 0.157704 0.157947 0.16537
IH 0.106302 0.0001197 0.09461 0.106182 0.106541 0.09461
Table 2. Mass square differences and mixing angles.
252
21 10 eVm

 232
31 10 eVm
Copyright © 2012 SciRes.
Type
2
12
sin
2
23
sin
13
sin
NH 7.67 2.39 0.37 0.5 0
IH 7.68 2.40 0.33 0.5 0
M. K. DAS 1559
elements of matrix l are calculated considering it is a
Cabibbo-Kobayashi-Maskawa like mixing i.e. [13]:
U
23
13
sin ,
l
ab
12 23
sin, sin,
ll



(18)
where
varies between 0.104 to 0.247 and a, b varies
between 0.2 to 5. From Equation (18) calculated l
are
used to construct the elements of matrix .
l
U
3.1. Hexagonal Mixing
The standard form of hexagonal mixing matrix is
HM
2
=22
22
U
310
2
13
1
22 2
131
22 2










(19)
which can predict 2
12
1
sin= 4
, 2
23
1
sin=2
2
sin=0
and
13 . Using Equation (19) one can also construct
L
L
m for HM as
1
2
3
00
00
00
mm
m





HM
310
2
2
131
22 222
131
22 222
310
2
2
131
,
22 222
131
22 222
T
LL
m





 









 





(20)
and has a texture
HM 8
=
LL
AB
mBABD
BABD A










8
,
33
88
33
B
ABD
BD














(21)
where mass eigenvalues are
12
2
=,=
3
mABm A
3.2. Bimaximal Mixing
The standard form of bimaximal mixing is written as:
BM
11 0
22
11 1
=22 2
11 1
22 2
U










(23)
3
6,=.
BmD
(22)
which predicts 2
12
1
sin 2
, 2
23
1
sin 2
2
13
and sin
0
.
Using Equation (23) for the quasi-degenerate case one
can construct neutrino mass model for BM case as:
1
HM
2
3
11 0
22 00
11100
22 200
11 1
22 2
11 0
22
11 1
,
22 2
11 1
22 2
T
LL
m
mm
m






 


















(24)
the texture has form [15]
121 1
BM
122
122
12 2
=1
1
L
Lo
mm



 







(25)
where mass eigen values are

1112
2112
3
132,
132,
o
o
o
mm
mm
mm




m
0.104669
(26)
The values can be calculated for HM as well as
1,2, 3
BM case, which are quasi-degenerate in nature using
similar procedure adopted in Section 2 for TBM case.
For HM A
, , 0.128304B 0.210183D
and for BM 1
5
7.2 10
 3
3.9 10

m
, 2 are calcu-
lated using MATHEMATICA. Values of mass square
differences and mixing angles for BM & HM are given
in Table 3. The expression for o is defined as [15]
2
mmv12
6.6 10
o
v
U
ofo
Now elements of
, where .
are taken from Equation (19)
and (23) for HM and BM respectively. Then using Equa-
tion (18) choosing the suitable value for
, a and b ele-
ments of l have been calculated. Finally deviated
atrix U is constructed using the following equatio
U
n: m
Copyright © 2012 SciRes. JMP
M. K. DAS
1560
Table 3. Values of θ12, θ23, θ13 and mass scales.
Type 252
21 10eVm


22 2
31 23
mm 3
10 eV
2
12
sin
2
23
sin
13
sin
Bimaximal Mixing 7.91 2.50 0.50 0.50 0
Hexagonal Mixing 7.67 2.40 0.25 0.50 0
Table 4. Values of θ12, θ23 and θ13.
Mixing type a b
2
12
sin
2
23
sin
13
sin
Bimaximal Mixing 0.25 0.25 0.18 0.26 0.44 0.047
Hexagonal Mixing 0.3 0.3 0.16 0.31 0.42 0.061
Tri-Bimaximal Mixing0.3 0.3 0.2 0.29 0.49 0.064
13 1213121313 121312
PMNS23121213 2312231213 2313 2323121213 2312231213
12 2312132312 2323131213 23
llll l
lllllll lllll
ll lllll lll ll
cc csscc cs
UUcscsscc ssscscs cssccss
ss csccs css cc
 
  


 




12 2312132312 232313
ss csccs cs
  

13
s
2313 23
1213 23
.scs
s cc
 
 





(27)
This new mixing pattern for suitable value of
,
and is the new mixing pattern deviated from BM and
HM and meet with the recent oscillation data. Values of
mixing angles for the pattern along with the suitable
parameters are listed in the Table 4 in case of HM, BM
as well as TBM.
a
b
4. Conclusion
Tribimaximal mixing provides a very close description of
neutrino mixing angles. However the present hint of
nonzero 13
coming from analysis of global neutrino
oscillation data may indicate that it is broken. Here in
this work, we try to use the charge lepton correction in
PMNS matrix to get which can predicts recent neutrino
oscillation parameters. Different neutrino mixings e.g.
TBM, BM, HM as well as tetragonal mixing are well
established and can explain the different pattern of neu-
trino in context of their mass. Analysis on these mixing
are very mass important to give comments on two im-
portant aspects e.g. neutrino mass hierarchy and non-zero
13
. In this work we have tried to show quasi-degenerate
(NH, IH) property of neutrino by parameterized standard
form of neutrino mass matrix using 21 , 31 and
i
i considering neutrino is tribimaximaly mix. Then
the deviated mixing pattern from TBM, BM and HM
have been constructed using charged lepton corrections.
From this analysis it can be conclude that neutrino can
mix tribimaximaly, hexagonally, and bimaximaly which
can predicts the neutrino oscillation parameters accu-
rately in
2
m2
m
m
3
level with a small correction in charged
lepton part. There is a good scope for extension of this
work with CP violating phases. By using 21 , 31
and it is also possible to get non-zero
2
m2
m
i
im
13
and
solar and atmospheric mixing angle directly in the range
of experimental values.
5. Acknowledgements
Author would like to thank Werner Rodejohann of Max-
Planck Institute, Germany for useful discussion during
this work. MKD is also supported by start-up grant of
Tezpur University, Tezpur.
REFERENCES
[1] K. Nakamura, et al., “Review of Particle Physics,” Jour-
nal of Physics G: Nuclear and Particle Physics, Vol. 37,
No. 7A, 2010, Article ID: 075021.
doi:10.1088/0954-3899/37/7A/075021
[2] P. F. Harrison, D. H. Perkins and W. G. Scott, “Tri-Bi-
maximal Mixing and the Neutrino Oscillation Data,”
Physics Letters B, Vol. 530, No. 1, 2002, pp. 167-173.
[3] Z. Z. Xing, “A Full Determination of the Neutrino Mass
Spectrum from Two-Zero Textures of the Neutrino Mass
Matrix,” Physics Letters B, Vol. 533, No. 1-2, 2002, pp.
85-90. doi:10.1016/S0370-2693(02)02062-2
[4] X. G. He and A. Zee, “Some Simple Mixing and Mass
Matrices for Neutrinos,” Physics Letters B, Vol. 560, No.
1-2, 2003, pp. 87-90. doi:10.1016/S0370-2693(03)00390-3
[5] G. L. Fogli, et al., “Hints of θ13 > 0 from Global Neutrino
Data Analysis,” Physics Letters B, Vol. 101, No. 14, 2008,
Article ID: 141801.
doi:10.1103/PhysRevLett.101.141801
[6] M. C. Gonzalez-Garcia, M. Maltoni and J. Salvodo,
“Updated Global Fit to Three Neutrino Mixing: Status of
the Hints of θ13 > 0,” Journal of High Energy Physics, No.
4, 2010, pp. 56-76.
[7] T. Schwetz, M. Tortola and J. W. F. Valle, “Where We
Copyright © 2012 SciRes. JMP
M. K. DAS 1561
Are on θ13: Addendum to ‘Global Neutrino Data and Re-
cent Reactor Fluxes: Status of Three-Flavor Oscillation
Parameters’,” New Journal of Physics, Vol. 13, 2011, Ar-
ticle ID: 063004. doi:10.1088/1367-2630/13/10/109401
[8] K. Abe, et al., “Indication of Electron Neutrino Appear-
ance from an Accelerator-Produced Off-Axis Muon Neu-
trino Beam,” Physical Review Letters, Vol. 107, 2011,
Article ID: 041801.
[9] L. Whitehead, et al., “Recent Results from MINOS, Joint
Experimental-Theoretical Seminar,” Joint Experimental-
Theoretical Seminar, Fermilab, 24 June 2011.
[10] C. H. Albright, A. Dueck and W. Rodejohann, “Possible
Alternatives to Tri-Bimaximal Mixing,” The European
Physical Journal C, Vol. 70, No. 4, 2010, pp. 1099-1110.
doi:10.1140/epjc/s10052-010-1492-2
[11] Z. Xing, “Tetramaximal Neutrino Mixing and Its Implica-
tions on Neutrino Oscillations and Collider Signatures,”
Physical Review D, Vol. 78, No. 1, 2008, Article ID:
011301. doi:10.1103/PhysRevD.78.011301
[12] W. Grimus and L. Lavoura, “A Model for Trimaximal
Lepton Mixing,” Journal of High Energy Physics, No. 9,
2008, p. 106. doi:10.1088/1126-6708/2008/09/106
[13] S. Goswami, S. T. Petcov, S. Ray and W. Rodejohann,
“Large |Ue3| and Tribimaximal Mixing,” Physical Review
D, Vol. 80, No. 5, 2009, Article ID: 053013.
doi:10.1103/PhysRevD.80.053013
[14] M. K. Das, M. Patgiri and N. N. Singh, “Numerical Con-
sistency Check between Two Approaches to Radiative
Corrections for Neutrino Masses and Mixings,” Pra-
mana—Journal of Physics, Vol. 65, No. 6, 2006, pp.
995-1013.
[15] N. N. Singh, M. Patgiri and M. K. Das, “Discriminating
Neutrino Mass Models Using Type-II See-Saw Formula,”
Pramana—Journal of Physics, Vol. 66, No. 2, 2006, pp.
361-375.
[16] G. Altarelli and F. Feruglio, “Discrete Flavor Symmetries
and Models of Neutrino Mixing,” Reviews of Modern
Physics, Vol. 82, 2010, pp. 2701-2729.
[17] S. Boudjemaa and S. F. King, “Deviations from Tribi-
maximal Mixing: Charged Lepton Corrections and Re-
normalization Group Running,” Physical Review D, Vol.
79, No. 3, 2009, Article ID: 033001.
doi:10.1103/PhysRevD.79.033001
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