Deviation from tri-bimaximal neutrino mixing in A4 flavor symmetry

0.29 MB·

by  Mizue Honda

Deviation from tri-bimaximal neutrino mixing in A flavor symmetry
4
Mizue Honda
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
Morimitsu Tanimoto
Department of Physics, Niigata University, Niigata 950-2181, Japan
(Dated: February 2, 2008)
Thetri-bimaximalmixingisagoodapproximationforthepresentdataofneutrinomixingangles.
The deviation from the tri-bimaximal mixing is discussed numerically in the framework of the
A4 model. Values of tan2θ12, sin22θ23 and |Ue3| deviate from the tri-bimaximal mixing due to
the corrections of the vacuum alignment of flavon fields. It is remarked that sin22θ23 deviates
scarcely from 1 while sin2θ12 can deviate from 1/3 considerably and sinθ13 could be near the
8 present experimental upper bound. The CP violating measure JCP and the effective Majorana
0 neutrinomass hmeei are also discussed.
0
2 PACSnumbers: 11.30.Hv,14.60.Lm,14.60.Pq

n
a
J I. INTRODUCTION

9
Neutrino experimental data provide us an important clue to find an origin of the observed hierarchies in mass

]
matrices for quarks and leptons. Recent experiments of the neutrino oscillation go into the new phase of precise
h
p determinationofmixing anglesandmasssquareddifferences[1,2]. Thoseindicate the tri-bimaximalmixingforthree
- flavors in the lepton sector [3]. Therefore, it is very important to find a natural model that leads to this mixing
p
pattern with good accuracy.
e
Flavorsymmetryisexpectedtoexplainthemassspectrumandthemixingmatrixofquarksandleptons. Especially,
h
[ some predictive models with discrete flavor symmetries have been explored by many authors [4]-[15]. Among them,
the interesting models to give the tri-bimaximal mixing are based on the the non-Abelian finite group A4. Since
2
v the original papers [10] on the application of the non-Abelian discrete symmetry A4 to quark and lepton families,
muchprogresshasbeenmadeinunderstandingthe tri-bimaximalmixingforneutrinosinanumber ofspecificmodels
1
8 [11]-[15]. Therefore, it is important to test the A4 model experimentally.
1 We present the comprehensive analyses of the deviation from the tri-bimaximal mixing in the framework of the
0 A4 model, where the tri-bimaximal mixing is realized in the vacuum alignment of the flavon fields [14, 15]. Since
. the vacuum alignment is corrected by higher-dimensional operators, the deviation from the tri-bimaximal mixing is
1
0 predicted numerically.
8 Itis found thatsin22θ23 deviatesscarcelyfrom1 while sin2θ12 candeviate from1/3considerablyandsinθ13 could
0 be near the present experimental upper bound. The CP violating measure J and the effective Majorana neutrino
CP
: mass m are also predicted.
v ee
h i
i Thepaperisorganizedasfollows: wepresenttheframeworkofthemodelinSec. II,anddiscussthedeviationfrom
X
the tri-bimaximal mixing of neutrino flavors in Sec. III. In Sec.IV, the numerical results are presented. Section V is
r devoted to the summary. The useful relations among parameters are given in Appendix.
a
II. FRAMEWORK OF THE A4 MODEL
Thetri-bimaximalmixingpatternisagoodapproximationforthepresentdataofneutrinomixingangles. Therefore,
it is very important to find a natural model that leads to this mixing pattern with good accuracy. The interesting
models to give the tri-bimaximal mixing are based on the non-Abelian finite group A4, in which there are twelve
group elements and four irreducible representations: 1, 1′, 1′′ and 3. Under the A4 symmetry, the left-handed
lepton doublets ℓ are assumed to transform as 3, the right-handed charge lepton singlets ec, µc and τc as 1, 1, 1 ,
L ′ ′′
respectively. The flavor symmetry is spontaneously broken by two real 3s, ϕ, ϕ, and by three real singlets, ξ(1),
′ ′
ξ (1) and ξ (1 ), which are SU(2) gauge singlets.
′ ′ ′′ ′′ L
The relevant Yukawa couplings of leptons are given as follows:
y y y x
= eec(ϕℓ )h + µµc(ϕℓ ) h + ττc(ϕℓ )h + aξ(ℓ h ℓ h )+
LY Λ L d Λ L ′′ d Λ L ′ d Λ2 L u L u
x x x
b c
+ ξ (ℓ h ℓ h ) + ξ (ℓ h ℓ h ) + (ϕℓ h ℓ h )+h.c. , (1)
Λ2 ′′ L u L u ′ Λ2 ′ L u L u ′′ Λ2 ′ L u L u
2
where h and h are ordinary Higgs scalars, Λ is a cut-off scale, and y , x and x are dimensionless coefficients
d u α i
with order one. The effective Lagrangian without ξ (1) and ξ (1 ) was given by Altarelli and Feruglio [14]. The
′ ′ ′′ ′′
Lagrangian of Eq.(1) is a general one to give the A4 symmetric lepton mass matrices. The flavon fields ϕ, ϕ′, ξ, ξ′
and ξ develop the vacuum expectation values along the directions:
′′
ξ =ua, ξ′ =uc, ξ′′ =ub, ϕ =(v1,v2,v3), ϕ′ =(v1′,v2′,v3′) . (2)
h i h i h i h i h i
We take the three-dimensionalunitary representationmatrices of A4 in Ref.[10]. Then, if we put v1 =v2 =v3 =v,
the charged lepton mass matrix is given by
y 0 0
v e
ME =√3vd U0 0 yµ 0 , (3)
Λ
0 0 y !
τ
where v is the vacuum expectation value of h and
d d
1 1 1
1 1+i√3
U0 = 1 ω ω2 , ω = − . (4)
√3 1 ω2 ω ! 2
On the other hand, the left-handed Majorana mass matrix, which respects the A4 flavor symmetry, is written as
a+b+c f e
M = f a+ωb+ω2c d , (5)
ν
e d a+ω2b+ωc!
where
v2 v2 v2
a=x u u , b=x u u , c=x u u ,
a aΛ2 b bΛ2 c cΛ2
v2 v2 v2
d=xv u , e=xv u , f =xv u , (6)
1′ Λ2 2′ Λ2 3′ Λ2
and v is the vacuum expectation value of h . If c=b and e=f =0 are taken in Eq.(5), the neutrino mass matrix
u u
turns to
a+2b 0 0
M = 0 a b d . (7)
ν
0 −d a b!
−
In the flavor diagonal basis of the charged lepton mass matrix, the neutrino mass matrix is given as
a+ 2d b d b d
3 − 3 − 3
Mνf =U0†MνU0∗ = b− d3 b+ 23d a− d3  , (8)
b d a d b+ 2d
− 3 − 3 3
 
which leads to the tri-bimaximal mixing of flavors
2 1 0
3 √3
Utri bi = q1 1 1  , (9)
− −√6 √3 −√2
1 1 1
 
−√6 √3 √2 
 
with three mass eigenvalues
a b+d , a+2b , a+b+d . (10)
− −
Evenif a=0 or b=c=0 is taken, these mass eigenvalues give observedtwo neutrino mass scales ∆m2 and ∆m2
atm sol
although a moderate tuning of parameters is required. Actually, such models were presented in the previous works
[12, 14, 15].
It is important to discuss the origin of conditions b=c and e=f =0, which realizes the tri-bimaximal mixing. If
ξ and ξ are decoupled in the Yukawa couplings, b = c = 0 is obtained. On the other hand, if the field ϕ develops
′ ′′ ′
3
the vacuum expectation values ϕ = (v ,v ,v ) along the directions of v = v = 0, one can put e = f = 0. This
′ 1′ 2′ 3′ 2′ 3′
h i
vacuum alignment could be realized in the scalar potential with SUSY [14, 15].
The condition b = c is not expected unless ξ and ξ are decoupled. Then, the neutrino mixing deviates from the
′ ′′
tri-bimaximal mixing. On the other hand, the vacuum alignment v = v = 0 may be modified. Actually, higher
2′ 3′
dimensional operators contributing to the superpotential correct the vacuum alignment such as v = 0, v = 0.
2′ 3′
6 6
Moreover, the vacuum alignment ϕ = (v,v,v) could be also corrected through higher dimensional operators. This
h i
effect contributes to the charged lepton mass matrix. Therefore, the lepton flavor mixing deviates from the tri-
bimaximalonebyhigherdimensionaloperators. We discussthepatternofthisdeviationquantitativelyinthis paper.
III. DEVIATION FROM THE TRI-BIMAXIMAL MIXING
Let us discuss the deviation from the tri-bimaximal mixing of neutrinos. There are three sources of the deviation
as discussed in the previous section. As far as couplings of ξ and ξ are allowed in the Lagrangian of Eq.(1), b = c
′ ′′
is not expected unless other symmetry is imposed on these couplings. One cannot control the magnitude of the c/b
ratioin the frameworkofthe A4 flavorsymmetry. The caseofb=c has been discussedby Ma [13]. We willshow the
6
numerical result, which is consistent with the result in [13], in the next section.
On the other hand, if the field ϕ develops the vacuum expectation values ϕ =(v ,v ,v ) along the directions of
′ ′ 1′ 2′ 3′
h i
v =v =0, one can put e=f =0. The vacuum alignment was discussed in the scalar potential with SUSY [15]. It
2′ 3′
is found that this vacuum alignment is spoiled by corrections of higher-dimensional operators, which are suppressed
by order 1/Λ [15]. The correction of the vacuum alignment ϕ =(v,v,v) is also investigated in the scalar potential
h i
with SUSY [15]. The correction of the order of 1/Λ may cause the deviation from the tri-bimaximal mixing through
the charged lepton mass matrix.
There are direct corrections to masses of charged leptons and neutrinos of order 1/Λ2 and 1/Λ3, respectively.
However,onecanassignZ3chargetopreventnewstructuresinEqs.(3)and(5)suchasℓL, ϕ′, ξ :ωandec, µc, τc :ω2
[15]. Forexample, the operatorec(ϕϕℓ )h does notcorrectthe chargedmassmatrix. Operators(ϕϕ)(ℓ ℓ ) h h ,
L d ′ ′ L L ′′ u u
(ϕϕ) (ℓ ℓ )h h , and ξ(ϕℓ ℓ ) h h contribute to the neutrino mass matrix. The first and second operators
′ ′′ L L ′ u u L L ′′ u u
give corrections in diagonal elements, which are absorbed in parameters b and c of Eq.(5), while the third one gives
corrections in off diagonal elements, which are absorbed in parameters d, e and f of Eq.(5).
Atfirststep,letusneglecttheeffectofthecorrectiononthe vacuumalignment ϕ =(v,v,v). Then,the deviation
h i
from the tri-bimaximal mixing comes from the neutrino sector. We take parameters
c=b (1+ǫ1) , e=ǫ2 d , f =ǫ3 d , (11)
where non-zero ǫ1, ǫ2 and ǫ3 lead to the deviation from the tri-bimaximal mixing.
Then, the neutrino mass matrix is given as:
m11 m12 m13
Mνf = m12 m22 m23 , (12)
m13 m23 m33!
where
2d
m11 a+ (1+ǫ2+ǫ3),
≃ 3
d
m12 b 2 (ǫ2+ǫ3) √3i(ǫ2 ǫ3) ,
≃ − 6{ − − − }
d
m13 b(1+ǫ1) 2 (ǫ2+ǫ3)+√3i(ǫ2 ǫ3) ,
≃ − 6{ − − }
d
m22 b(1+ǫ1)+ 2 (ǫ2+ǫ3)+√3i(ǫ2 ǫ3) ,
≃ 3{ − − }
d
m23 a (1+ǫ2+ǫ3),
≃ − 3
d
m33 b+ 2 (ǫ2+ǫ3) √3i(ǫ2 ǫ3) . (13)
≃ 3{ − − − }
In the first order of the perturbation, the neutrino masses m are
i
b d √3
m1 = a b+d ǫ1 (ǫ2+ǫ3)+i d(ǫ2 ǫ3) ,
− − 2 − 2 6 −
4
√3
m2 = a+2b+bǫ1 d(ǫ2+ǫ3) i d(ǫ2 ǫ3) ,
− − 3 −
3b 3d √3
m3 = a+b+d ǫ1+ (ǫ2+ǫ3) i d(ǫ2 ǫ3) , (14)
− − 2 2 − 2 −
and the MNS matrix elements [16] are
2 d
Ue1 = (ǫ2+ǫ3) ,
3 − √6(3b d)
r
−
1 d
Ue2 = + (ǫ2+ǫ3) ,
√3 √3(3b d)
−
b d
Ue3 = ǫ1 i(ǫ2 ǫ3) ,
− 2√2(a b) − √6(2a+b d) −
− −
1 3b d
Uµ1 = ǫ1 (ǫ2+ǫ3) ,
−√6 − 4√6(a b) − √6(3b d)
− −
1 d d
Uµ2 = (ǫ2+ǫ3) i(ǫ2 ǫ3) ,
√3 − 2√3(3b d) − √2(2a+b d) −
− −
1 b d
Uµ3 = + ǫ1+ i(ǫ2 ǫ3),
−√2 4√2(a b) √6(2a+b d) −
− −
1 3b d
Uτ1 = + ǫ1 (ǫ2+ǫ3) ,
−√6 4√6(a b) − √6(3b d)
− −
1 d d
Uτ2 = (ǫ2+ǫ3)+ i(ǫ2 ǫ3) ,
√3 − 2√3(3b d) √2(2a+b d) −
− −
1 b d
Uτ3 = + ǫ1+ i(ǫ2 ǫ3) , (15)
√2 4√2(a b) √6(2a+b d) −
− −
where all parameters are supposed to be real for simplicity. It is remarked that the A4 phase ω turns to the CP
violating phase if ǫ2 =ǫ3, that is , e=f is realized. The CP violationalso comes from phases in parametersa, b and
6 6
d. We will present numerical calculations by taking complex numbers for these parameters.
In above expressions, we have supposed the vacuum alignment ϕ = (v,v,v). The vacuum alignment may be
h i
sizeably corrected through higher dimensional operators. Then, the charged lepton mass matrix in Eq.(3) is not
preserved. The charged lepton mass matrix is modified in terms of correction parameters ǫch, ǫch, which are defined
1 2
as ϕ = v, (1+ǫch)v, (1+ǫch)v , as follows:
1 2
h i { }
1 0 0 1 1 1 y 0 0
v e
M = v 0 1+ǫch 0 1 ω ω2 0 y 0 . (16)
E dΛ 0 0 1 1+ǫch! 1 ω2 ω ! 0 0µ y !
2 τ
After a basis transformation of the left-handed charged lepton by the unitary matrix U0 in Eq.(4), we get ME′ as
Xy Yy Y y
v 1 e µ ∗ τ
ME′ =U0†ME =vdΛ√3 YY∗yye YXyyµ XYyyτ ! , (17)
e ∗ µ τ
where
X = 3+ǫch+ǫch 3 ,
1 2
≃
1
Y = ǫchω+ǫchω2 = (ǫch+ǫch)+√3i(ǫch ǫch) ,
1 2 −2{ 2 1 2 − 1 }
1
Y = ǫchω2+ǫchω = (ǫch+ǫch) √3i(ǫch ǫch) . (18)
∗ 1 2 2 1 2 1
−2{ − − }
Then, the left-handed mixing matrix of the charged lepton mass matrix is given by U0UE′ , where UE′ is given as
1 Y Y∗
X X
UE′ ≃−YXY∗ 1Y∗ XY1  , (19)
−X −X
 
5
which is obtained by diagonalizing ME′ ME′†. Therefore, the deviation from the tri-bimaximal mixing due to ǫc1h and
ǫch is given as
2
′
UMNS =UE† Utri bi , (20)
−
whose relevant mixing elements are given as
1 Y +Y 2 1
Ue1 ≃ √6 2+ X ∗ ≃ 3 1− 6(ǫc2h+ǫc1h) ,
(cid:18) (cid:19) r (cid:20) (cid:21)
1 Y +Y 1 1
Ue2 ≃ √3 1− X ∗ ≃ 3 1+ 3(ǫc2h+ǫc1h) ,
(cid:18) (cid:19) r (cid:20) (cid:21)
1 Y Y i
Ue3 ≃ √2 ∗X− ≃ √6(ǫc2h−ǫc1h) ,
1 Y 1 1
Uµ3 ≃ −√2 1+ X∗ ≃−√2 1− 6 (ǫc2h+ǫc1h)−i√3(ǫc2h−ǫc1h) . (21)
(cid:18) (cid:19) (cid:20) n o(cid:21)
It is noticed that a new CP violating phase appears due to corrections of the vacuum alignment.
IV. NUMERICAL RESULTS
Let us present numerical results as for the deviation from the tri-bimaximal mixing. At first, we discuss the case
a=0,b=c, e=f =0in the neutrino massmatrix ofEq.(5) to see the effect ofǫ1, whichdenotesthe deviationfrom
b=6 c. In6 this case, we neglect the effect of ǫch, ǫch in the charged lepton mass matrix, therefore, the charged lepton
1 2
mass matrix is given as in Eq.(3).
Next, we discuss the case of a = 0, b = c = 0, e = 0, f = 0, in which b = c is guaranteed by vanishing Yukawa
6 6 6
couplings as to ξ and ξ . This case leads to the normal mass hierarchy of neutrinos. We also discuss the case of the
′ ′′
inverted mass hierarchy of neutrinos, which could be realized in the case of a=0, b=c, e=0, f =0.
6 6
The effect of the correction in the charged lepton sector is also discussed in the last subsection.
A. a6=0, b6=c, e=f =0
The couplings ξ′ and ξ′′ generally lead to b=c in the A4 symmetry. Then, the neutrino mixing deviates from the
6
tri-bimaximal mixing due to ǫ1. We suppose the vacuum alignment v2′ =v3′ =0, that is e=f =0 in order to see the
effect of b=c clearly. This case has been studied analytically by Ma at first [13]. Our numerical result is completely
6
consistent with the results in Ref. [13] .
As seenin Eq.(15), Ue1 andUe2 areindependent ofǫ1 in the first
order approximation. The effect of (ǫ21) increases tan2θ12 toward
O
thelargervaluethanthetri-maximalmixing1/2. Takingtherecent
tan2Θ12
experimental data as input [17]: 0.6
∆m232 =(1.9 3.0) 10−3eV2 , sin22θ23 >0.92
∼ ×
∆m221 =(8.0+−00..43)×10−5eV2 , sin22θ12 =0.86+−00..0034 , (22) 0.5
at 90%C.L., we plot the allowed region on Ue3 tan2θ12 plane in
| |−
Figure 1, where we take a, b, d and ǫ1 as complex parameters. We
also take the experimental upper bound
0.05 0.1 0.15 ÈUe3È
sin22θ13 <0.19 . (23)
The prediction of tan2θ12 is almost at the high end of the ex- pFlIaGn.e1in: thTehceasaelloowfendonr-ezgeiroonǫo1.n |Ue3|−tan2θ12
perimental allowed region, while the experimental central value is
tan2θ12 =0.45. If we take tan2θ12 =0.502as the upper bound, we
get the bound of Ue3 0.05, where ǫ1 is smaller than 0.35. Therefore, the relation b=c should be satisfied at the
| |≤ | |
35% level. Therefore, it is preferred to decouple ξ and ξ in the Lagrangianof Eq.(1), that is b=c=0.
′ ′′
It may be noticed that direct corrections to neutrino masses induce small values of b and c even if ξ and ξ are
′ ′′
decoupled in the Yukawa couplings. Since these corrections on neutrino masses and mixing angles are not leading
ones, we neglect them hereafter.
6
ÈUe3È ÈUe3È
0.2 0.2
HaL HbL
0.15 0.15
0.1 0.1
0.05 0.05
0.42 0.44 0.46 0.48 0.5 tan2Θ12 0.96 0.98 1 sin22Θ23
Jcp <mee>@eVD
0.03 HcL 0.02 HdL
0.02
0.015
0.01
ÈU È 0.01
e3
0.05 0.1 0.15 0.2
-0.01
0.005
-0.02
-0.03 0.005 0.01 0.015 0.02m1@eVD
FIG.2: Theallowed regionon(a)tan2θ12−|Ue3|,(b)sin22θ23−|Ue3|,(c)|Ue3|−JCP,and(d)m1−hmeeiplanesinthecase
of the normal mass hierarchy of neutrinos with non-zero ǫ2 and ǫ3.
B. a6=0, b=c=0, e6=0, f 6=0
Let us consider the simple case of b = c = 0. Since the tri-bimaximal mixing with the relevant neutrino mass
spectrum is realized at the limit of ǫ2 = ǫ3 = 0 [12], evolutions of the non-zero ǫ2 and ǫ3 lead to the deviation from
the tri-bimaximal mixing.
The ǫ2 and ǫ3 are expected to be real if we suppose the vacuum expectation values (v1′,v2′,v3′) to be real. Since a
is taken to be realin general,we have a phase φd in addition to the A4 phase ω. Then we take following parameters:
a , d= d eiφd , ǫ2 , ǫ3 , (24)
| |
wherea,ǫ2 andǫ3 denoterealones[19]. Inthecaseofǫ2 =ǫ3 =0,parametersa, d, φd aregivenintermsof∆m2atm
and ∆m2 as shown in Appendix. | |
sol
As seen in Eq.(15), we have an approximate relation
1
Uµ3 Ue3 . (25)
≃−√2 −
It is not easily to test this relation in the future experiments unless the phase of Ue3 is known.
It is noticed in Eq.(14) with b= 0 that one gets the neutrino mass spectrum with the normal mass hierarchy, but
cannot get the inverted one [12]. We show the numerical results in Figure 2 and Table I. As seen in Fig.2(a), plots
on tan2θ12 Ue3 plane cover all experimental allowed region. Therefore, there is no prediction in this case. On
the other ha−nd|, pl|ots on sin22θ23 Ue3 plane in Fig.2(b) indicate the correlation between sin22θ23 and Ue3 . One
expects sin22θ23 0.98 if Ue3 −0.|05 w|ill be confirmed in the future experiments. In the case of Ue3 | 0.0|1, θ23
≥ | | ≤ | | ≤
deviates scarcely from the maximal mixing while θ12 could deviate considerably from the tri-maximal mixing.
We plot allowed region J , which is the Jarlskog invariant [18] in Fig.2(c). Since the CP violation comes from
CP
phases of ω and φd, the predicted region is rather wide even if Ue3 is fixed. The predicted absolute value reaches
| |
0.03.
In Fig.2(d), we show the predicted effective Majorana neutrino mass m , which is related with the neutrinoless
ee
h i
double beta decay rate,
hmeei=|Mνf[1,1]|= m1c212c213eiρ+m2s212c213eiσ+m3s213e−2iδD , (26)
(cid:12) (cid:12)
(cid:12) (cid:12)
7
Ε
3 ÈU È
0.3 HaL 0.2e3
HbL
0.2
0.15
0.1
ÈΕ È 0.1
0.05 0.1 0.15 0.2 0.25 0.3 2
-0.1 0.05
-0.2
ÈΕ -Ε È
0.1 0.2 0.3 0.4 0.5 0.6 2 3
-0.3
FIG. 3: The allowed region on (a) |ǫ2|−ǫ3 and (b) |ǫ2−ǫ3|−|Ue3| planes.
TABLEI: Predictions in subsections B and C
b=c=0 , Normal. a=0 , Inverted.
tan2θ12 0.404∼0.502 0.443∼0.502
sin22θ23 0.95∼1 0.99∼1
|Jcp| ≤0.031 ≤0.023
hmeei ≥3.5 meV 14∼22 meV
|Ue3| ≤0.16 ≤0.10
where c and s denote cosθ and sinθ , respectively, δ is a so called the Dirac phase, and ρ,σ are the Majorana
ij ij ij ij D
phases. As seen in Fig.2(d), m 3.5 meV is predicted. The magnitude increases proportional to the value of
ee
h i ≥
m1 in the case of the normal mass hierarchy of neutrinos. The large value 20 meV is expected for rather degenerate
neutrino masses.
It may be useful to comment on the allowed values of ǫ2 and ǫ3. We
show the allowed region on ǫ2 ǫ3 plane and ǫ2 ǫ3 Ue3 plane in
| |− | − |−| |
Figure 3. Both ǫ2 and ǫ3 are varied in the restricted region 0.3 0.3
− ∼
as seen in Fig.3(a). It is found in Fig.3(b) that Ue3 is approximately ArgHUe3L@°D
| |
proportional to the magnitude of ǫ2 ǫ3. Since ǫ2 and ǫ3 are cut at 150 HcL
− | | | |
0.3 by hand, Ue3 is bounded by 0.16. 100
| |
We also present the phase of Ue3, which is shown in Figure 4. The 50
phase of Ue3 is predicted around 90◦, which is almost independent of 0.05 0.1 0.15 0.2 ÈUe3È
|Ue3|. Therefore, sin22θ23 deviates at most in a few percent from the -1-0500
maximal mixing as seen in Eq.(25).
-150
C. Inverted mass hierarchy
FIG. 4: The predicted phase of Ue3 versus
|Ue3| in the case of non-zero ǫ2 and ǫ3. The
In this subsection, we discuss the case of the inverted mass hierarchy, phaseis predicted around 90◦.
which was presented in the case of a = 0, b = c = 0 [12]. In this case,
6
b=cmaybeaccidentalbecausetheA4symmetrydoesnotguaranteethis
relation. Therefore,wediscussthiscaseonlyinphenomenologicalinterest
of the inverted neutrino mass hierarchy. We take following parameters:
b=c , d= d eiφd , ǫ2 , ǫ3 , (27)
| |
where b, c, ǫ2 and ǫ3 denote real ones. In the case of ǫ2 =ǫ3 =0, parameters b, d, φd are given in terms of ∆m2atm
and ∆m2 as shown in Appendix. | |
sol
We show the numerical results in Figure 5 and Table I. As Ue3 increases, the deviation from tan2θ12 = 1/2 can
be larger, while sin22θ23 0.99 is obtained. As seen in Fig.5(|b), |the allowed region on sin22θ23 tan2θ12 plane is
≥ −
restricted close to the tri-bimaximal mixing. It is noticed that Ue3 is smaller than 0.10.
| |
We alsoplotallowedregionJCP inFig.5(c). Itis remarkablethat JCP is almostdetermined if Ue3 is fixed. This
| | | |
fact means that the inverted mass hierarchy is realized in the restricted value of φ .
d
8
ÈUe3È tan2Θ12
0.12
HaL 0.5 HbL
0.1
0.48
0.08
0.46
0.06
0.44
0.04
0.42
0.02
0.42 0.44 0.46 0.48 0.5 tan2Θ12 0.992 0.996 1sin22Θ23
Jcp <mee>@eVD
0.03 HcL 0.025 HdL
0.02 0.02
0.01 0.015
ÈU È
0.02 0.04 0.06 0.08 0.1 e3 0.01
-0.01
0.005
-0.02
m3@eVD
-0.03 0.002 0.004 0.006
FIG. 5: Theallowed region on (a) tan2θ12−|Ue3|, (b) sin22θ23−tan2θ12,(c) |Ue3|−JCP,and (d) m3−hmeei planes in the
case of theinverted mass hierarchy of neutrinos.
The effective Majorana neutrino mass is predicted to be a rather narrow range m = 14 22 meV as seen in
ee
h i ∼
Fig.5(d). Therefore,onemayexpectthattheneutrinolessdoublebetadecaywillbeobservedinthefutureexperiments.
D. The effect of the charged lepton
Inthissubsection,wediscusstheeffectofthechargedleptonmassmatrix. Ifthevacuumalignment ϕ =(v,v,v)is
h i
sizeably correctedthrough higher dimentional operators,we cannot neglect the contribution from the chargedlepton
mass matrix to predict the deviation from the tri-bimaximal mixing. In order to examine the effect of correction
parameters ǫch, ǫch in Eq.(16) clearly, we take b=c=0 and e=f =0 with the normal mass hierarchy of neutrinos
1 2
at first step.
We show numerical results in Figure 6 and Table II. As seen in Fig.6(a), plots on tan2θ12 Ue3 plane cover all
experimentalallowedregion. Ontheotherhand,plotsonsin22θ23 tan2θ12planeinFig.6(b)in−di|cate|sin22θ23 0.99
− ≥
while θ12 can deviate from the tri-maximal mixing considerably.
We also plot allowedregionJ in Fig.6(c). Since the CP violationis only due to ω, J is clearlydetermined if
CP CP
| |
Ue3 is fixed. The predicted effective Majorana neutrino mass is mee 2.3 meV as seen in Fig.6(d).
| In|order to see values of ǫc1h and ǫc2h, we plot allowed regions onhǫc1hi≥ǫc2h and Ue3 ǫc1h ǫc2h planes. As seen in
Fig.7(a), the relative sign of ǫch and ǫch is almost opossite, and these−magnitude|s ar|e−a|t mo−st 0.|3. Furtheremore, it
1 2
is found that Ue3 is proportional to the magnitude of ǫc1h ǫc2h as seen in Fig.7(b).
| | −
We also examine the case of a = 0, b = c = 0 and e = f = 0 with the inverted hierarchy of neutrino masses. The
6
numerical result is not so changed in the above case as summarized in Table II.
At the next step, we present the numerical result in the case that both charged lepton and the neutrino mass
matricescontributeto the deviationfromthe tri-bimaximalmixing. Taking ǫc1h 0.05, ǫc2h 0.05, ǫ2 0.05,and
| |≤ | |≤ | |≤
ǫ3 0.05, which guarantees that corrections of higher-dimensional operators do not spoil the leading order picture
| |≤
as discussed in the previous work [15], we predict the following values for the normal neutrino mass hierarchy:
tan2θ12 =0.404 0.502 , sin22θ23 =0.997 1 , Ue3 0.047 ,
∼ ∼ | |≤
J 0.011 , m 4.2 meV . (28)
CP ee
| |≤ h i≥
9
ÈUe3È tan2Θ12
0.2
0.5
0.15 HaL 0.48 HbL
0.46
0.1
0.44
0.05 0.42
0.42 0.44 0.46 0.48 0.5 tan2Θ12 0.992 0.996 1sin22Θ23
Jcp <mee>@eVD
0.02
0.04 HdL
HcL 0.015
0.02
0.01
0.05 0.1 0.15 0.2 ÈUe3È
-0.02 0.005
-0.04 0.005 0.01 0.015 0.02 m1@eVD
FIG. 6: The allowed region on (a) tan2θ12−|Ue3|, (b) sin22θ23−tan2θ12, (c) |Ue3|−JCP, and (d) m1−hmeei planes with
non-zero ǫc1h,ǫc2h in thecharged lepton sector.
Εch
2
0.3 HaL ÈUe3È
0.2 0.2
0.1 0.15 HbL
Εch 0.1
-0.2 -0.1 0.1 0.2 0.3 1
-0.1 0.05
-0.2 0.1 0.2 0.3 0.4 0.5 ÈΕch1-Εch2È
-0.3
FIG. 7: Theallowed region on (a) ǫc1h−ǫc2h and (b) |ǫc1h−ǫc2h|−|Ue3| planes.
These predictedmixing anglesshouldbe takenasa typicaldeviationfromthe tri-bimaximalmixing inthe A4 model.
TABLE II:Predictions in subsection D
b=0 , Normal. a=0 , Inverted.
tan2θ12 0.404∼0.502 0.404∼0.502
sin22θ23 0.994∼1 0.995∼1
|Jcp| ≤0.046 ≤0.046
hmeei ≥2.3 meV 14∼22 meV
|Ue3| ≤0.22 ≤0.22
10
V. SUMMARY
We have examined the deviation from the tri-bimaximal mixing of the neutrino flavorsin the framework of the A4
model. Taking account corrections of the vacuum alignment of flavon fields, the deviation from the tri-bimaximal
mixing are estimated quantitatively.
In the case of the normal mass hierarchy of neutrinos, there is the correlation between sin22θ23 and Ue3 . We
expect sin22θ23 0.98if Ue3 0.05 will be confirmedin the future experiments. If the strongerbound U|e3 | 0.01
≥ | |≤ | |≤
will be obtained in the future, θ23 is expected to be almost maximal mixing while θ12 could be deviated considerably
from the tri-maximal mixing.
In the case of the inverted mass hierarchy of neutrinos, the deviation from tan2θ12 = 1/2 becomes larger as
Ue3 increases. The allowed region on sin22θ23 tan2θ12 plane is restricted close to the tri-bimaximal mixing.
| | −
It is remarkable that JCP is almost determined if Ue3 is fixed. Moreover, the neutrinoless double beta decay
| | | |
is expected to be observed in the future experiments because the predicted effective Majorana neutrino mass is
m =14 22 meV.
ee
h Thiedevia∼tionthroughthechargedleptonsectorisalsoexamined. Ifparametersareconstrainedsuchas ǫch 0.05,
1
ǫc2h 0.05, ǫ2 0.05, and ǫ3 0.05, tan2θ12 can deviate considerably from the tri-maximal mixin|g 1/|2≤while
s|in2|2≤θ23 0.|997| ≤and Ue3 |0.0|4≤7 are obtained. These values indicate a typical deviation from the tri-bimaximal
≥ | | ≤
mixing in the A4 model.
Theprecisionmeasurementsofthedeviationfromthetri-bimaximalmixingoftheneutrinoflavorsprovideacrucial
test of the A4 flavor symmetry with the vacuum alignment of flavon fields.
Acknowledgments
The work of M.T. has been supported by the Grant-in-Aid for Science Research of the Ministry of Education,
Science, and Culture of Japan Nos. 17540243and 19034002.
Appendix
In this appendix, we show relations among mass eiganvalues and parameters at the limit of the tri-bimaximal
mixing.
In the case of b=c=0 with the normal hierarchy, mass eigenvalues m0 are given as
i
m0 =a+ deiφd , m0 =a , m0 = a+ deiφd , (29)
1 2 3
| | − | |
where a is taken to be real. Then, we have
1
∆m2 m0 2 m0 2 = 4ad cosφ , ∆m2 m0 2 m0 2 = ∆m2 d2 , (30)
atm ≡| 3| −| 1| − | | d sol ≡| 2| −| 1| 2 atm−| |
which give
∆m2 1
a= atm , d = ∆m2 ∆m2 . (31)
−4d cosφ | | 2 atm− sol
| | d r
In the case of a=0 and b=c=0 with the inverted hierarchy, mass eigenvalues m0 are given as
6 i
m0 = b+ deiφd , m0 =2b , m0 =b+ deiφd , (32)
1 2 3
− | | | |
where b is taken to be real. Then, we have
∆m2 m0 2 m0 2 = 4bd cosφ , ∆m2 m0 2 m0 2 =3b2+2bd cosφ d2 , (33)
atm 1 3 d sol 2 1 d
≡| | −| | − | | ≡| | −| | | | −| |
which gives
∆m2
b = atm , (34)
−4d cosφ
d
| |
∆m2 ∆m2 1 3
d2 = atm sol + (1+ )∆m4 +4∆m2 ∆m2 +4∆m4 . (35)
| | − 4 − 2 4 cos2φ atm atm sol sol
r d