Journal of Modern Physics, 2013, 4, 495-500
http://dx.doi.org/10.4236/jmp.2013.44070 Published Online April 2013 (http://www.scirp.org/journal/jmp)
M Expansion of UPMNS and Neutrino Mass Matrix

in
Terms of sinθ13 for Inverted Hierarchical Case
Subhankar Roy, Nangkham Nimai Singh
Department of Physics, Gauhati University, Guwahati, India
Received January 1, 2013; revised January 31, 2013; accepted February 8, 2013
Copyright © 2013 Subhankar Roy, Nangkham Nimai Singh. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
The recent observational data supports the deviation from Tri-bimaximal (TBM) mixing. Different theories suggest the
interdependency among the observational parameters involving the mixing angles. On phenomenological ground, we
try to construct the PMNS matrix, UPMNS with certain analytic structure satisfying spontaneously the unitary condition,
in terms of a single observational parameter 13 sin .

We hypothesise the three neutrino masses, mi as functions of
M
13
sin

and then construct the neutrino mass matrix

with certain exact and expandable form. We assume the con-
vergence of the model to TBM mixing when 13

is taken 0. The mass matrix so far obtained can be employed for va-
rious applications including the estimation of matter-antimatter asymmetry of the universe.
Keywords: Inverted Hierarchy; Tri-Bimaximal Mixing; Neutrino Masses; -

 Symmetry
1. Introduction
Recent results published by Double Chooz [1], Daya Bay
[2], RENO [3], T2K [4] and MINOS [5] collaborations
assure relatively large reactor angle 

13

. Also recent
global neutrino data analysis [6] insists on 23 π4


.
Tri-bimaximal mixing [7] is associated with 13 0


, and
23 π4

. This symmetry has a strong theoretical support
because of its relation with so called

  symmetry of
neutrino mass matrix.

  symmetry, in turn is asso-
ciated with 4
A
[8-12], one of the candidates of discrete
flavour symmetry groups. But in order to comply with the
recent experimental results, some perturbations have to be
introduced in this mixing pattern. Whether the corrections
[13,14] are needed or a new mixing scheme is to be in-
troduced, is still an open question [15].
Literature [16,17] shows the dependency of the mixing
angles on one another. If this is true, then we are allowed
to choose a single parameter capable of describing all the
three mixing angles.
We move a step ahead and express the three masses
under this parameter. This helps us to define a simplified
neutrino mass model with a single parameter only.
Out of all the three observational parameters concern-
ing the mixing angles 13
sin

is the smallest one. So, we
choose 13
sin

as the guiding parameter. We consider tri-
bimaximal mixing pattern and -


sin 0.
symmetry as the
first approximation. Hence the model is supposed to
produce T. B. M mixing when we put 13

 We try
to keep the structure of the three rotation matrices


U13


23
U


U12

,

and

in analytical form so that
they can satisfy the unitary condition
 
†
ij ij
UU I 



13
s
,
without any prior approximation.
We start with the following ansatz,
, (1)


12
1,
5
3
s

(2)
23
1
2 2
s




sins
. (3)
where, ij ij


, and then construct the PMNS mixing
matrix and then the neutrino mass matrix in the usual way.
2. Construction of the PMNS Matrix
We consider the charged lepton mass matrix to be diago-
nal. Hence we can choose PMNS UU

. We propose the
three rotation matrices as:
C
opyright © 2013 SciRes. JMP

S. ROY, N. N. SINGH 496



1
2
13
10
01
e0
U





2i
1
i2
2
e
0
1
 










, (4)


1
22
23
10
024
11
022
U 

 




1
22
0
11
2
22
4
22



















, (5)

1
22
12
22
325
53
12
53
35
00
U 




 
 

1
22
10
5
3
2
0
25
3
1



















123
2 3
123
ee e
uuu
uuu
uuu
 
 







. (6)
We have,
 
PMNS 23 13 121
UU
UU
  (7)
where,

1
22
1,ua
1e

1
22
11,
3





i
3e, e
u
25 e
u





 
i
11
e,
2a
153 3
30 2
ub


 






 

 
i
2
12353e,ucb



30


1
22
3
121
2
u

  ,
 
i
1
111 e
2510
23
ubd


 
 
 
 

 ,


i
2
11
353 e
230
2
ua b



 


 ,


2
3
11,
2
ub



and,
1
22
12 ,
325
53
a  






1
22
222,b


1
22
150 309,c 

1
2
2
50 10
33
d 




 

††
PMNS PMNS
100
010.
001
ij ij
UU UU 


 
  

.
It can be checked that,
(8)
And we get,
22
22
12 2
1
2510 33
tan 5010 33
e
e
u
u






(9)
22
22
12 2
1
2510 33
tan 5010 33
e
e
u
u






(10)
After interpreting the above two relations in terms of
sinθ13, we have,
223
1213 1313
11 12
tansinsinsin ,
22 425
    (11)

223
231313 13
13
tan1 sin2sin2sin
5

 
PMNS
U0
. (12)
for

 and 0.156

 are shown below,
TBM
21
0
33
11 1
63 2
11 1
63 2
U












i
ii
ii
0.82740.5395 0.156e
0.42450.0822e0.6511 0.0536e0.6214 .
435 0.1015e0.5270 0.0662e0.7678
U







 




,
0.3

S. ROY, N. N. SINGH 497
This is clear from the above analysis that 12
and , if 13
2
tan 0.5  
ta sin
2
23
n 1  0


2
23
tan  
(TBM mixing). At
(N. H), 0.156 (I. H) (the best-fit value of sinθ13)
[6], we get and ,
which are very close to the best fit results [6]: 12
0.1
0.657,0.654
2
tan
55
2
12
tan  

0.425

=
0.443 (N. H or I. H) and (N. H) and
0.644 (I. H). This is shown in Figure 1, where the varia-
2
23
tan 0.628  
tions of 2
12
tan

and ta 2
23
n

are plotted against 13 sin

.
3. Jarkslog Parameter (Jcp)
We introduce the CP phase

in U13 as shown in Equa-
tion (8). The inclusion of cp

does not affect 2
12
tan

or
2
tan 23

(Equations (9) and (10)).
We obtain the as,
cp
J

2
1131
1
22
22
Im 1
122
243 2
253
cpe e
Juuuu







 



 
1
11
25
23
sin
5

 

 
 




,
(13)
Maximum cp i.e., max J
J
is obtained for π2


.
For, max 0.156,
J
is obtained as 0.0341. The varia-
tion of max
J
with respect to sin 13

is shown in Figure
2. Also the variation of cp with
J

(with  or 13 sin

fixed at 0.156), is plotted in Figure 3.
4. Generation of Neutrino Mass Matrix with
Inverted Hierarchy
We now apply the PMNS mixing matrix from Equation
(7), to construct the neutrino mass matrix with inverted
hierarchy (I. H). We try to interpret the masses in terms of
the same parameter 13 sin

.
We choose on phenomenological ground the absolute
values of three neutrino masses in units of eV as,
4
607 ,
1250 8


1
m (14)
5
4
7,
8 3



2
61
1250
m (15)
4
3
7.
8
m (16)
leading to,
4
25
21
121 761
Δ
15625005000 1875
m
910 71,
12 9
 (17)
4
910
71 .
12 9

2
Δm2 m
0,2 Δ7.74 eV
2252
7.53 10eVm 
0.156.
25
23
3721 42761
Δ
15625005000 1875
m
 (18)
The variations of 21 and 23 Δ with  are shown
in Figures 4 and 5. At TBM mixing condition, i.e., at
we get, ,
and 21 Δ,
are obtained at
25
10
21
m
23
eV ,
3
eV
23
Δ10m
23 10m

2.38
2
2.43 2
Figure 1. Variation of tan2θ12 and tan2θ23 with . 
Figure 2. Variation of Jmax with . 
Figure 3. Variation of Jcp with δ.
Figure 4. Variation of with . m2
21
Δ

S. ROY, N. N. SINGH 498
Figure 5. Variation of with .
For simplicity in the texture of the neutrino mass matrix,
we avoid the inclusion of cp

. Using Equation (7) for
PMNS (with cp
U0


) and Equations (14)-(16) for i,
we construct the neutrino mass matrix
m
M

as follows,
1
T
PMNS2 PMNS
3
11 1213
21 2223
31 3233
00
00
00
.
m
MUm U
m
mmm
mmm
mmm



 








(19)
where,
m2
23
Δ




2
62 242
11
7671
11,
81258 5
3
mAB 

 
 
 

 



 


 
11
2 4
22
671 1
11
82125852
232
111
1,
525
323
mACA
BCA

52
12
1
22
71 
  




   
  



  
 

  
  




 






 


 
11
52 2
22
13
14
22
7111
11
8552
332
67 11
1,
5
mCBC A
AAC



 

125 8 2
23


 

 
 
 
  



 

 




 
 
2
82 12585
23
11 ,
25
23
BAC
2
24
42
22
7167121
12
2
mCA




 
 



  

 
 
 

 





 
 


 


   
  
4
2
23
71
1
82
2
1,
5
3
mC
A
CA

 








 

 


 

 
4
11 1
(
5225
32 3
671 11
125 8522
322
BC AC
AC

  


  

 

 
 
 
 


 



 

 


 
 
4
2
33
2
4
71
1
85
32
611
7.
125 825
23
mC BCA
CA


 





 
 
 

 



 

 

 
2
1
2


S. ROY, N. N. SINGH 499
and,

1
22
22 ,
325
53





A

1
45
2
7,
8 3




61
1250
B

1
22
24
2
C  




0
-
1.
At, (T. B. M mixing), Equation (22) reduces to


22 0
22
11
symmetric mass matrix form,
1
1
1
M
m







1,2



1,
, (20)
with

 for inverted hierarchy. Equation (19) leads
to
1
2
121 1
3750
1
0.0157 0.0323
0.0323 0.0083
0.0323 0.0083
MM








11
1 1
4 4
11
44
0.0323
0.0083 ,
0.0083


















12
,,
1250 1250
mm
3
6061 0in eVm
0.156
0.0255
0.0118 .
0.0142
MM M







0.0068
0.0035
5 0.0059







3
0005

sin
(21)
At , Equation (22) leads to
0.0189 0.0362
0.0362 0.0049
0.0255 0.0118
 






(22)
where,
0.0032 0.0039
Δ0.0039 0.0034
0.0068 0.003
M








12
0.0485,0.0493, 0.mm m in eV.
5. Summary
We have started with a parameter equating this to
13
thesis by comparing the ranges of the mass squared dif-
ferences as a result of our ansatz with the 1

range,
experimentally obtained. We take the range of as the
experimental

1

range of 13 sin

[6]. We obtain the
range of and as
and
2
21
m2
23
m

52
7.46 - 7.5810eV




32
2.42 - 2.4410eV

 respectively. The respec-
tive ranges obtained, lie within the experimental 1

boundary [6]. This provides a support to our hypothesis
i as


and construct the PMNS matrix, PMNS . Then
we represent the neutrino masses U


1,2,3i in terms of
the same parameter
m
13
nsi ,

i.e. We verify our hypo-
.
m

i. This is to be emphasised that the PMNS
U
matrix as proposed in Equation (7) satisfies the unitary
condition and is not dependent on the choice of the order
of . The introduction of cp
m


does not affect 12
2
tan

and 23
2
tan

in our calculation. The maximum ob-
tained is 0.034 (with respect to 13
cp
J
sin 0.156


,
). Fi-
nally we concentrate on the construction of
M

the
neutrino mass matrix. The present investigation though
phenomenological, gives a complete picture of the texture
of the neutrino mass matrix which can be employed in
other applications regarding baryon asymmetry of the
universe [18]. Although we have constructed the mass
matrix for inverted hierarchical model, yet we can extend
our technique to Normal as well as Quasidegenerate mass
models.
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