A4 Model with Five Extra Scalars for Neutrino Masses and Mixing

A4 Model with Five Extra Scalars for Neutrino Masses and Mixing

A4 Model with Five Extra Scalars for Neutrino Masses and Mixing

Victoria Puyam Ng. Nimai Singh

Department of Physics, Manipur University,

Canchipur-795003, Manipur, India

Abstract

A modified neutrino mass model with five extra S U(2)L × U(1)Y singlet scalars is constructed using A4 discrete symmetry group. The resultant mass matrix is able to reproduce the current neutrino masses and mixing data with good accuracy. Our model predicts the relation between the neutrinoless double-beta decay parameter

|m| and oscillation parameters.

Keywords: A4 symmetry, scalars, Tribimax- imal(TBM), projection matrix, normal hierar- chy, inverted hierarchy

1 Introduction

21

31

The neutrino experiments have confirmed neutrino oscillations and mixing through the observation of solar and atmospheric neutri- nos indicating its mass thereby providing an important solid clue for a new physics beyond Standard Model(SM) of particle physics. At present, the neutrino oscillation experiments [1, 2, 3, 4] have measured the oscillation pa- rameters viz: mass squared difference(m2 and m2 ) and mixing angle(12 , 23 and 13) to a good accuracy. The bounds on the

absolute neutrino masses scale are also greatly reduced by direct and cosmological neutrino mass search experiments [5, 6]. However, the current data is still unable to explain several key issues such as the octant of 23, the neutrino mass ordering, CP violating phase etc.

The oscillation data reveals certain pattern of neutrino mixing matrix .Out of the several approaches to explain the observed pattern, the Tribimaximal(TBM) mixing[7, 8] used to be very favourable. And, it yields some interesting results like trivial value of 13 and CP violating phase and maximal 23 etc.

The A4 flavour symmetry model proposed by Altarelli and Feruglio[9] was the first at- tempt to accommodate TBM mixing scheme in a neutrino mass model. Following this example, many other neutrino mass models are constructed using different non-Abelian discrete symmetry groups[10]. However, the recently observed non-zero 13 disfavours Tribimaximal (TBM) and leads to the modi- fication of several mass models constructed with TBM [11, 12, 13]. As a result, the neutrino mixing pattern like TM1[14] and TM2 mixings[15] which are proposed with slight deviation from TBM gain momentum. Currently, they can predict the observed pat-

tern and mixing angles with good consistency.

In this present work, we proposed a model with five extra SM singlet scalars to explain neutrino mixing angles in their experimental range. The present model is constructed in T- diagonal basis. The non-zero value of 13 is obtained as a consequences of specific Dirac mass matrix which is constructed using anti-

symmetric part and its projection matrix arises

Fields Vacuum Expectation Value(VEV)

< S > (vs, vs, vs)

< T > (vT , 0, 0)

< Hu >,< Hd > vu, vd

< 1 >, < 2 >, < 3 > u1, u2, u3

Table 2: Vacuum Expectation Values of the scelar fields used in the model.

Ye c

Yµ c

Y c

y1 c

from the product of two A4 triplets. Here, we

LY = e (T l)1 Hd + µ (T l)1 Hd + (T l)1 Hd + 1(l )1

y2 y3

ya yb

present a detailed analysis on the neutrino os-

cillation parameters and its correlation among themselves and with neutrinoless double-beta

+ (2)1 (lc)1 + (3)1 (lc)1 + S (lHuc)A + S (lHuc)S +

1

M(cc) + h.c.

2

(1)

decay parameter |m|.

2 Description of the model

In this work, we extended SM by adding five

In A4 symmetry model, the contraction of the two A4 triplets l and c into an- other triplet are specified by tensor product rules in two different ways. Each tensor product combination corresponds to certain

form of projection matrix. Therefore, the

extra S U(2)L × U(1)Y singlet scalars namely

mass terms ya

c b c

1, 2 and 3 which are singlet under A4 and the field T and S which are transformed as triplet. The standard model lepton doublets are assigned to triplet representation under A4, right handed charge lepton ec, µc, c and right

S (lHu )A and y S (lHu )S of Eq.(1) could have two independent new terms

with different coupling constant [17]. For our model, the projection matrix P in T-diagonal basis are as follows:

handed neutrino field c are assigned to 1,1 ,

1 and 3 representation respectively. The right

0 0 0

0 1 0

0 0 1

0 0 1

1 0 0 , 0 0 0

handed neutrino field c contributes to the ef-

PA =

,

fective neutrino mass matrix through type-I seesaw mechanism[16]. The transformation properties of the fields use in the model under

A4 × Z3 discrete group are given in the Table1

0 1 0

2 0 0

0 1 0

0 1 0

1 1 0

(2)

0 0 1

PS = 0 0 1 , 1 0 0 , 0 2 0 .

Fields l ec µc c c Hu,d S T 1 2 3

0 1 0

0 0 2

1 0 0

A4 3 1 1 1

3 1 3 3 1 1 1

Z3 2 1 1 1

SM (2, 1 ) (1,1) (1,1) (1,1) (1,0) (2,- 1 ) (1,0) (1,0) (1,0) (1,0) (1,0)

(3)

2 2

The Yukawa mass matrices can be derived

Table 1: Transformation properties of various

fields under A4 × Z3 group.

The Yukawa Lagrangian term for the lepton

by using the vacuum expectation value given in Table2 in Eq.(1). The obtained charged lep- ton mass matrix is diagonal and has the form

vdvT Ye 0 0

sector which are invariant under A4 transfor- mation are given in the equation:

Ml =

0 Yµ 0 . (4)

0 0 Y

The Majorana neutrino mass matrix has the structure

M 0 0

MR = 0 0 M . (5)

1. Numerical Analysis and Re- sults

   As the light neutrino mass matrix m obtained

   0 M 0

   For the Dirac neutrino mass matrix, the projection matrix given in Eq.(2) and (3)are utilised.Therefore, the resultant Dirac mass matrix is in the form

   in eq.(8) is in the basis where charge lepton mass matrix is diagonal. The Pontecorvo- Maki-Nakagawa-Sakata leptonic mixing matrix UPMNS which is necessary for the diagonalization of m becomes a unitary matrix U. Therefore, the light neutrino mass

   matrix m is diagonalized as:

   2a + c a + b + d a b + e

   MD = a b + d 2a + e a + b + c .

   a + b + e a b + c 2a + d

   ybvu.vs , b	= yavu .vs	yivu .ui
   	and c,

   where a =

   (6)

   where m

   m = UmdiagU (9)

   = diag(m , m , m ) is the

   d, e= ,

   diag

   1 2 3

   i=1,2 and 3.

   The effective neutrino mass matrix is ob- tained by using Type-I seesaw mechanism.

   light neutrino mass matrix in diagonal

   form. The three mass eigenvalues of the neutrino can be written as mdiag =

   m )

   diag(m1 , m2 + m2 m2 2

   1 21,

   +

   1 31

   m = (MT M1 MD) (7)

   in normal hierarchy(NH) and mdiag =

   D R

   diag( m2 + m2

   + m2 , m2 + m2 , m3)

   m11 m12 m13

   3 31

   21 3 23

   = m12 m22m23 . (8)

   in inverted hierarchy(IH). The upper

   bound on the sum of neutrino masses

   m13 m23 m33

   (I m

   = m1

   + m2

   + m3) obtained by the Planck

   where

   is 0.12 eV [6].

   m11 =

   1 ((2a + c)2 + 2(a + b d)(a b e)) M

   1

   The PMNS matrix U can be parametrised in terms of neutrino mixing angles and Dirac CP

   m12 = m21 =

   (3a2 + b2 + 2cd + e2

   M

   phase . Following PDG convention, U can

   takes the form

   + b(d + e) + a(6b 2c + d + e))

   1 i

   m13 = m31 =

   (3a2 + b2 + d2 + 2ce

   C12C13 S 12C13 S 13e

   M UPMNS = P S 12C23 C12S 23S 13ei C12C23 S 12S 23S 13ei S 23C13 . P,

   + b(d + e) + a(6b 2c + d + e)) 1

   S 12S 23 C12C23S 13ei C12S 23 S 12C23S 13ei C23C13

   (10)

   m22 =

   ((a + b + d)2 2(a + b c)(2a + e))

   M

   1

   m23 = m32 =

   (6a2 2b2 + c2 + 2de

   M

   where i j (for ij= 12,13,23) are the mixing angles (with Cij = cos i j and

   + b(d + e) + a(2c + d + e)))

   1

   S i j = sin i j) and is the Dirac CP phase. P = Diag(ei, ei, 1) contains

   m33 =

   2(a + b + c)(2a + d) + (a + b e)2).

   M

   two Majorana CP phases and , while

   P = Diag(ei1 , ei2 , ei3 ) consists of three unphysical phases 1,2,3 that can be removed via the charged-lepton field rephasing[18]. The neutrino mixing angles 12, 23 and 13 in terms of the elements of U are given below:

   in Fig. 1a-1b. While the predicted value of 13 and 23 are just outside 1 range but well within 3 range. Fig.2 shows that the prediction of , 23 and 13 for IH are well within 3 range. Therefore, the present model slightly prefers normal hierarchy(NH).

   S 2 = |U12 |2

   2 = |U23 |2

   2 = |U

   |2 .

   12 1|U13 |2 S 23

   1|U13 |2 S 13

   13

   (11)

   The effective neutrino mass |m| is charac- terized by

   In order to show that the present model is in consistent with the present neutrino oscillation data. We solve the free parameters by gen- erating sufficiently large number of random points. The points in parameters space need

   to satisfy Eqs.(8) and the accuracy level is de-

   1 j

   m = |U2 mj|, (12)

   where mj are the Majorana masses of three light neutrinos. The upper limits of the ef- fective neutrino mass are obtained by: Gerda [5] as |m| < 104 228 meV corresponds

   76 0 25

   termined by the experimental error on the neu- to

   Ge(T1/2 > 9 × 10

   ), CUORE [20]

   trino oscillation parameters. The experimental

   as |m| < 75 350 meV corresponds to

   130Te(T 0 25

   values of the observables are given in Table3.

   1/2 > 3.2 × 10

   ) and KamLAND-Zen

   The model predictions of the neutrino oscil-

   [21] as |m| < 61 165 meV corresponds

   136 0 25

   lations parameters in 3 confidence level are to

   Xe(T1/2 > 1.07 × 10

   ). The model

   shown in Fig. 1 and Fig. 2. One of the impor- tant prediction of model is that solar neutrino mixing angle 12 lies around 35.7 in both NH and IH cases.

   Parameters Best fit±1 2 3 12/ 34.3 ± 1.0 32.3-36.4 31.4-37.4

   prediction of the effective Majorana mass

   |m| vs Jarlskog invariant J in 3 range for both NH and IH are shown in Fig.3a and Fig.[4b] respectively. The correlation between and |m| are depicted in Fig. 3b for NH and Fig. 4a for IH.

   0.12

   13/(NO) 8.53+0.13

   0.14

   13/(IO) 8.58+0.12

   8.27-8.79 8.20-8.97

   8.30-8.83 8.17-8.96

   23/(NO) 49.26 ± 0.79 47.35-50.67 41.20-51.33

2. Conclusion

   0.97

   23/(IO) 49.46+0.60

   m2 [105 eV2] 7.50+0.22

   47.35-50.67 41.16-51.25

   7.12-7.93 6.94-8.14

   In conclusion, we have presented a neutrino

   21 0.20

   |m2 |[103 eV2](NO) 2.55+0.02

   2.49-2.60 2.47-2.63

   mass model that can accommodate current

   31 0.03

   |m2 |[103 eV2](IO) 2.45+0.02

   31 0.03

   22

   /(NO) 194+24

   28

   /(IO) 284+26

   2.39-2.50 2.37-2.53

   152-255 128-359

   226-332 200-353

   neutrino oscillation data. The model has some

   characteristic prediction for neutrino oscilla- tion parameters. The solar neutrino angle 12 centred around 35.7 for both mass ordering.

   Table 3: The global-fit result for neutrino os- cillation parameters [19].

   In normal hierarchy, the Dirac CP violating phase is obtained in 1 range as shown

   The predicted range of and atmospheric neu- trino mixing angle 23 for NH are in good agreement with their respective experimental range. The present model also predicted the correlation between and m. The presented model slightly preferred NH data.

   195

   190

   185

   180

   49.0 49.2 49.4 49.6 49.8 50.0 50.2

   23

   (a)

   195

   °

   190

   °

   216

   42.2 42.4 42.6 42.8 43.0

   °

   23

   (a)

   °

   185

   180

   8.3 8.4 8.5 8.6 8.7 8.8

   °

   13

   (b)

   216

   8.3 8.4 8.5 8.6 8.7 8.8

   °

   13

   (b)

   50.2

   50.0

   49.8

   °

   23

   49.6

   49.4

   49.2

   49.0

   °

   23

   42.8

   42.6

   42.4

   42.2

   48.8

   35.70 35.71

   °

   12

   (c)

   35.70 35.71

   °

   12

   (c)

   50.2

   50.0

   49.8

   23/°

   49.6

   49.4

   49.2

   /°

   43.0

   23

   42.8

   49.0

   48.8

   0.058 0.059 0.060 0.061

   m3(eV)

   (d)

   42.6

   42.4

   42.2

   0.056 0.057 0.058 0.059 0.060 0.061 0.062

   m3(eV)

   Figure 1: Correlation plot for different neu- trino oscillation parameters for normal mass ordering(NH).

   (d)

   Figure 2: Correlation plot for different neu- trino oscillation parameters for inverted mass ordering(IH).

   –

   –

   –

   J

   –

   -0.024

   -0.025

   -0.026

   22 24 26 28 30

   m(meV)

   (a)

   –

   –

   J

   –

   –

   –

   –

   22 24 26 28 30

   m(meV)

   (a)

   195

   °

   °

   190

   185

   180

   22 24 26 28 30

   m(meV)

   (b)

   216

   22 24 26 28 30

   m(meV)

   (b)

   Figure 3: Model predictions for Jarlskog in- variant versus effective Majorana mass and Dirac CP violating phase versus effective Ma- jorana mass for NH.

   Figure 4: Model predictions for Jarlskog in- variant versus effective Majorana mass and Dirac CP violating phase versus effective Ma- jorana mass for IH.

   A APPENDIX: A4 GROUP

   2

   A4 is the even permutation group of 4 objects with 4! elements. It has four irreducible rep- resentations, namely 1, 1, 1 and 3. All the elements of the group can be generated by two elements S and T. The generators S and T sat-

   3 × 3 = 1

   : P =

   1

   0	1	0
   1	0	0
   0	0	1

   (A.5)

   isfy the relation

   =

   3 × 3 1

   0 0 1

   : P1 = 0 1 0

   S 2 = (S T )3 = T 3 = 1. (A.1)

   The multiplication rules of any two irre- ducible representations under A4 are given by

   1 0 0

   (A.6)

   2 0 0 0 1 0 0 0 1

   3 × 3 = 3S : PS = 0 0 1 , 1 0 0 , 0 2 0

   3 3 = 1 1 1

   3S 3A

   1 1 = 1 1 1 = 1

   A1 1 1 = 1 1 1 = 1 .

   3 1/1/1 = 3 1/1 /1 3 = 3

   0 1 0 0 0 2 1 0 0

   0 0 0 0 1 0 0 0 1

   (A.2)

   3 × 3 = 3A : PA = 0 0 1 , 1 0 0 , 0 0 0

   The contractions of any two particle fields

   0 1 0 0 1 0 1 1 0

   (A.8)

   and

   under A4 are specified by the tensor

   product rules. In our case we considered the contraction of Majorana fields:

   The detailed studied on the contractions of

   fields under A4 symmetry can be found in the appendix of Ref.[17].

   T P

   = (A.3)

   where P is the projection matrix and is result of the contraction. P is specified for each tensor product combination and it is also different in S-diagonal or T-diagonal basis. Then, the form of P for all triplet contractions in T diagonal basis for Eq.(A.3) are given below:

   1 0 0

   3 × 3 = 1 : P1 = 0 0 1

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