Highly predictive and testable A4 flavor model within type-I and II seesaw framework and associated phenomenology

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics B 946 (2019) 114704 www.elsevier.com/locate/nuclphysb

Highly predictive and testable A4 flavor model within type-I and II seesaw framework and associated phenomenology Surender Verma ∗ , Monal Kashav, Shankita Bhardwaj Department of Physics and Astronomical Science, Central University of Himachal Pradesh, Dharamshala 176215, India Received 4 February 2019; received in revised form 18 July 2019; accepted 19 July 2019 Available online 25 July 2019 Editor: Tommy Ohlsson

Abstract We investigate neutrino mass model based on A4 discrete flavor symmetry in type-I+II seesaw framework. The model has imperative predictions for neutrino masses, mixing and CP violation testable in the current and upcoming neutrino oscillation experiments. The important predictions of the model are: normal hierarchy for neutrino masses, a higher octant for atmospheric angle (θ23 > 45◦ ) and near-maximal Diractype CP phase (δ ≈ π/2 or 3π/2) at 3σ C.L. These predictions are in consonance with the latest global-fit and results from Super-Kamiokande (SK), NOνA and T2K. Also, one of the important feature of the model is the existence of a lower bound on effective Majorana mass, |Mee | ≥ 0.047 eV (at 3σ ) which corresponds to the lower part of the degenerate spectrum and is within the sensitivity reach of the neutrinoless double beta decay (0νββ) experiments. © 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

1. Introduction The neutrino oscillation experiments have conclusively demonstrated that neutrinos have tiny mass and they do mix. Especially, with the observation of non-zero θ13 [1–6] the CP conserving * Corresponding author.

E-mail addresses: [email protected] (S. Verma), [email protected] (M. Kashav), [email protected] (S. Bhardwaj). https://doi.org/10.1016/j.nuclphysb.2019.114704 0550-3213/© 2019 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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+1.2 part of the neutrino mixing matrix is known to high precision: θ12 = 34.5+1.2 −1.0 , θ23 = 47.7−1.7 , 2 2 θ13 = 8.45+0.16 −0.14 [7]. Although, the two neutrino mass squared differences m12 and |m23 | have, also, been measured but there still exist two possibilities for neutrino masses to be either normal hierarchical (NH) or inverted hierarchical (IH). Understanding this emerged picture of neutrino masses and mixing, which is at odds with that characterizing the quark sector, is one of the biggest challenge in elementary particle physics. The Yukawa couplings are undetermined in the gauge theories. To understand the origin of neutrino mass and mixing one way is to employ phenomenological approaches such as texture zeros [8–23], hybrid textures [24–28], scaling [29–36], vanishing minor [37–39] etc. irrespective of details of the underlying theory. These different ansatze are quite predictive as they decrease the number of free parameters in neutrino mass matrix. The second way, which is more theoretically motivated, is to apply yet-to-be-determined non-Abelian flavor symmetry. In this approach a flavor symmetry group is employed in addition to the gauge group to restrict the Yukawa structure culminating in definitive predictions for values and/or correlations amongst low energy neutrino mixing parameters. Discrete symmetry groups have been successfully employed to explain non-zero tiny neutrino masses and large mixing angle in lepton sector [40–48]. There exist plethora of choices for flavor groups having similar predictions for neutrino masses and mixing patterns. In general, a flavor model results in proliferation of the Higgs sector making it sometime discouragingly complex. The group A4 [49–52] being the smallest group having 3-dimensional representation is widely employed as the possible underlying symmetry to understand neutrino masses and mixing with in the paradigm of seesaw mechanism [53–61]. It has been successfully employed to have texture zero(s) in the neutrino mass matrix which is found to be very predictive [15,20–22]. Another predictive ansatz is hybrid texture structure with one equality amongst elements and one texture zero in neutrino mass matrix. The hybrid texture of the neutrino mass matrix has been realized under S3 ⊗ Z3 symmetry with in type-II seesaw framework assuming five scalar triplets with different charge assignments under S3 and Z3 [27]. Also, some of these hybrid textures have been realized under Quaternion family symmetry Q8 [62]. In particular, the authors of Ref. [27] realized one of such hybrid texture with non-minimal extension in the scalar sector of the model which requires the imposition of an additional cyclic symmetry to write group-invariant Lagrangian. Also, the vacuum alignments have not been shown to be realizable. Keeping in view existing gaps, we are encouraged for realization of hybrid textures with minimal extension of scalar sector under group A4 (smallest group with 3-dimensional representation) assuming the vacuum alignments  0  = √ϑ (1, 1, 1)T . In fact, for A4 flavor model, it has 3 been shown in Ref. [63] that this VEV minimizes the scalar potential. In this work, for the first time, we have employed A4 flavor symmetry to realize hybrid texture structure of neutrino mass matrix. We present a simple minimal model based on group A4 with two right-handed neutrinos in type-I+II seesaw mechanism leading to hybrid texture structure for neutrino mass matrix. The same Higgs doublet is responsible for the masses of charged leptons and neutrinos [15]. In addition, one scalar singlet Higgs field χ and two scalar triplets i (i = 1, 2) are required to write A4 invariant Lagrangian. In Sec. 2, we systematically discuss the model based on group A4 and resulting effective Majorana neutrino mass matrix. Sec. 3 is devoted to study phenomenological consequences of the model. In this section we, also, study the implication to neutrinoless double beta decay (0νββ) process. Finally, in Sec. 4, we summarize the predictions of the model and their testability in current and upcoming neutrino oscillation/0νββ experiments.

S. Verma et al. / Nuclear Physics B 946 (2019) 114704

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Table 1 Field content of the model and charge assignments under SU (2)L and A4 . Symmetry

DiL

eR

μR

τR

ν1

ν2

χ



1

2

SU (2)L A4

2 3

1 1

1 1

1 1

1 1

1 1

1 1

2 3

3 1

3 1

2. The A4 model The group A4 is a non-Abelian discrete group of even permutations of four objects. It has four conjugacy classes, thus, have four irreducible representations (IRs), viz.: 1, 1 , 1

and 3. The multiplication rules of the IRs are: 1 ⊗ 1 = 1

, 1

⊗ 1

= 1 , 1 ⊗ 1

= 1, 3⊗ = 1 ⊕ 1 ⊕ 1

⊕ 3s ⊕ 3a where, (3⊗3)1 = a1 b1 + a2 b2 + a3 b3 , (3⊗3)1 = a1 b1 + ωa2 b2 + ω2 a3 b3 , (3⊗3)1

= a1 b1 + ω2 a2 b2 + ωa3 b3 , (3⊗3)3s = (a2 b3 + b2 a3 , a3 b1 + a1 b3 , a1 b2 + a2 b1 ) , (3⊗3)3a = (a2 b3 − b2 a3 , a3 b1 − a1 b3 , a1 b2 − a2 b1 ) , ω ≡ e2πi/3 and (a1 , a2 , a3 ), (b1 , b2 , b3 ) are basis vectors of the two triplets. Here, we present an A4 model within type-I+II seesaw framework of neutrino mass generation. In this model, we employed one SU (2)L Higgs doublet , one SU (2)L singlet Higgs χ and two SU (2)L triplet Higgs fields (1 , 2 ). The transformation properties of different fields under SU (2)L and A4 are given in Table 1. These field assignments under SU (2)L and A4 leads to the following Yukawa Lagrangian −L = ye (D¯ eL φ1 + D¯ μL φ2 + D¯ τ L φ3 )1 eR1 + yμ (D¯ eL φ1 + ω2 D¯ μL φ2 + ωD¯ τ L φ3 )1

μR

1

+ yτ (D¯ eL φ1 + ωD¯ μL φ2 + ω2 D¯ τ L φ3 )1 τR1

+ y1 (D¯ eL φ˜ 1 + D¯ μL φ˜ 2 + D¯ τ L φ˜ 3 )1 ν11 + y2 (D¯ eL φ˜ 1 + ωD¯ μL φ˜ 2 + ω2 D¯ τ L φ˜ 3 )1 ν21

T T − y1 (DeL C −1 DeL + DμL C −1 DμL + DτTL C −1 Dτ L )1 iτ2 11 T T − y2 (DeL C −1 DeL + ωDμL C −1 DμL + ω2 DτTL C −1 Dτ L )1 iτ2 21

− Mν1T C −1 ν1 − hχ χν2T C −1 ν2 + h.c.

(1)

where, φ˜ = iτ2 φ ∗ and yi (i = e, μ, τ, 1, 2, 1 , 2 ) are Yukawa coupling constants. The above Lagrangian leads to charged lepton mass matrix ml , right handed Majorana mass matrix mR and Dirac mass matrix mD given by ml = UL diag(ye , yμ , yτ )ϑ,   M 0 mR = , 0 N

(2) (3)

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⎛

x mD = ⎝ x x

⎞ y ωy ⎠, ω2 y

(4)

after spontaneous symmetry breaking with vacuum expectation values (VEVs) as  0  = ϑ √ (1, 1, 1)T and χ0  = ε for Higgs doublet and scalar singlet, respectively. It has been thor3

oughly studied in literature [63–65] that vacuum expectation value scalar potential. Here UL is ⎞ ⎛ 1 1 1 1 ⎝ 1 ω2 ω ⎠ , √ 3 1 ω ω2 which diagonalizes ml , N = hχ ε, x = √ϑ y1 and y = 3 effective Majorana neutrino mass matrix is

ϑ √ y . 3 2

ϑ √ (1, 1, 1)T 3

(6)

Using Eqns. (3) and (4) we get ⎛ 2 y2 ωy 2 x2 x M + N M + N ⎜ 2 ωy 2 ω2 y 2 x2 x mν1 = ⎜ ⎝ M+ N M + N M

+

(5)

The type-I seesaw contribution to

T mν1 = mD m−1 R mD .

x2

minimizes A4

ω2 y 2

x2

N

M

+

y2 N

ω2 y 2 x2 M + N y2 x2 M + N ωy 2 x2 N + N

⎞ ⎟ ⎟. ⎠

(7)

Assuming VEVs υj (j = 1, 2) for scalar triplets 1 , 2 , respectively, the type-II seesaw contribution to effective Majorana mass matrix is ⎞ ⎛ c+d 0 0 ⎠, c + ωd (8) mν2 = ⎝ 0 0 0 c + ω2 d where c = y1 υ1 and d = y2 υ2 . So, effective Majorana mass matrix is given as mν = mν1 + mν2 . The charge lepton mass matrix ml can be diagonalized by the transformation Ml = UL† ml UR , where UR is unit matrix corresponding to right handed charged lepton singlet fields. In charged lepton basis the effective Majorana mass matrix is given by ⎛ ⎞ d c x2 0 9 + 3M 9 ⎜ y2 c⎟ (9) Mν = ⎝ d ⎠ 9 3N 9 d c 0 9 9 which symbolically can be written as ⎛ ⎞ X  0 Mν = ⎝  X X ⎠ , 0 X 

(10)

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where  denotes the equality between elements and X denotes arbitrary non-zero elements. In literature, such type of neutrino mass matrix structure is referred as hybrid textures [24–26]. On changing the assignments of the fields we can have two more hybrid textures. For example, if we assign ν2 ∼ 1 , χ ∼ 1 and 1 ∼ 1 we end up with ⎛ ⎞ X X  ⎝X  0 ⎠. (11)  0 X Similarly, the field assignments ν2 ∼ 1 , χ ∼ 1 and 2 ∼ 1 result in effective neutrino mass matrix ⎛ ⎞ X 0  ⎝ 0  X⎠. (12)  X X In the next section, we study the phenomenological consequences of these neutrino mass matrices. 3. Phenomenological consequences of the model In charged lepton basis, the effective Majorana neutrino mass matrix, Mν is given by Mν = V Mνdiag V T , where V = U.P and ⎛ m1 Mνdiag = ⎝ 0 0

0 m2 0

(13) ⎞ 0 0 ⎠. m3

U is Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix and in standard PDG representation is given by ⎞ ⎛ c12 c13 s12 c13 s13 e−iδ ⎟ ⎜ (14) U = ⎝ −s12 c23 − c12 s13 s23 eiδ c12 c23 − s12 s13 s23 eiδ c13 s23 ⎠ , s12 s23 − c12 s13 c23 eiδ where sij = sin θij and cij ⎛ 1 0 0 ⎜ 2iα 0 ⎝0 e 0

0

−c12 s23 − s12 s13 c23 eiδ

c13 c23

= cos θij . The phase matrix, P is ⎞ ⎟ ⎠,

e2i(β+δ)

where α, β are Majorana phases and δ is Dirac-type CP violating phase. The neutrino mass model described by Eqn. (10) imposes two conditions on the neutrino mass matrix Mν , viz.: (Mν )ab = 0, (Mν )uv = (Mν )mn , where a = u = 1, b = m = n = 3 and v = 2 for neutrino mass matrix in Eqn. (9).

(15)

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It leads to two complex equations amongst nine parameters, viz.: three neutrino masses (m1 , m2 , m3 ), three mixing angles (θ12 , θ23 , θ13 ) and three CP violating phases (δ, α, β) m1 Ua1 Ub1 + m2 Ua2 Ub2 e2iα + m3 Ua3 Ub3 e2i(β+δ) = 0,

(16)

and m1 (Uu1 Uv1 − Um1 Un1 ) + m2 (Uu2 Uv2 − Um2 Un2 )e2iα + m3 (Uu3 Uv3 − Um3 Un3 )e2i(β+δ) = 0. m1 m3

(17) m2 m3

We solve Eqn. (16) and (17) for mass ratios and





m1 R13 ≡

e−2i(β+δ)

m

3

Ua3 Ub3 Uu2 Uv2 − Ua2 Ub2 Uu3 Uv3 + Ua2 Ub2 Um3 Un3 − Ua3 Ub3 Um2 Un2

,

=

Ua2 Ub2 Uu1 Uv1 − Ua1 Ub1 Uu2 Uv2 + Ua1 Ub1 Um2 Un2 − Ua2 Ub2 Um1 Un1



m2 2i(α−β−δ)



R23 ≡ e

m3



Ua1 Ub1 Uu3 Uv3 − Ua3 Ub3 Uu1 Uv1 + Ua3 Ub3 Um1 Un1 − Ua1 Ub1 Um3 Un3

,

=

Ua2 Ub2 Uu1 Uv1 − Ua1 Ub1 Uu2 Uv2 + Ua1 Ub1 Um2 Un2 − Ua2 Ub2 Um1 Un1

(18)

(19)

m2 1 where the ratios R13 ≡ m m3 and R23 ≡ m3 . The ratio R23 can be obtained from R13 using the transformation θ12 → π2 − θ12 . The mass ratios R13 and R23 along with measured neutrino masssquared differences provide two values of m3 , viz.: ma3 and mb3 , respectively and is given by  m212 + m223 ma3 = , (20) 2 1 − R13  m223 b . (21) m3 = 2 1 − R23

These two values of m3 must be consistent with each other, which results in Rν = The ratios

2 − R2 R23 13 2 | |1 − R23

m1 −2i(β+δ) m3 e

≡

m221 |m223 |

and

.

(22)

m2 2i(α−β−δ) , m3 e

to first order in s13 , is given by 2



2 e2iδ s − c c s 2 2 + c23 e−iδ s13 s23 c23 m1 −2i(β+δ) 12 12 23 23 e ≈− 2 + , 5 m3 s23 s12 s23 2



2 e2iδ c + s c s 2 e−iδ s13 s23 + c23 c2 m2 2i(α−β−δ) 12 12 23 23 e ≈ − 23 + . 2 5 m3 s23 c12 s23

Using these approximated mass ratios we find

2  4 s13 s12 − c12 c23 s23 c23 2 2 R13 ≈ 4 − cos δ 2c23 7 s12 s23 s23   s13  2 4 4 2 2 s12 − c12 c23 s23 c23 + s23 + 2c23 s23 cos 2δ , − 3 s12 s23

(23)

(24)

S. Verma et al. / Nuclear Physics B 946 (2019) 114704

2 R23



2  s13 c12 + s12 c23 s23 2 ≈ 4 − cos δ 2c23 7 c12 s23 s23   s13  2 4 4 2 2 − c12 + s12 c23 s23 c23 + s23 + 2c23 s23 cos 2δ , 3 c12 s23

7

4 c23

(25)

and 2 2 R23 − R13 ≈

s13

4 4 + 2c2 s 2 cos 2δ s − 4 sin 2θ sin3 2θ cos δ + s23 (4 sin 2θ12 − sin 2θ23 ) c23 13 12 23 23 23 8 8 sin 2θ12 s23

. (26)

2 − R 2 > 0 which is possible if m2 must be greater than m1 , or equivalently, R23 13 4 4 2 2 (4 sin 2θ12 − sin 2θ23 )(c23 + s23 + 2c23 s23 cos 2δ)s13 > 4 sin 2θ12 sin3 2θ23 cos δ),

which translates to constraint on δ given by     4 sin 2θ12 sin3 2θ23 sin2 2θ23

cos 2δ < 1. 4 4 cos δ − 4 s 4 + s23 2 c23 + s23 (4 sin 2θ12 − sin 2θ23 ) c23 13

(27)

(28)

Using the experimental data shown in Table 2, we find that δ can take values only near δ ≈ 90◦ or δ ≈ 270◦ for the model to be consistent with solar mass hierarchy. However, it will, further, get constrained by the requirement of Rν to be within its experimental range. For normal hierarchy 2 − R 2 > 0 and 1 − R 2 > 0 (or R13 < 1 and R < 1). Substituting δ ≈ 90◦ or 270◦ in (NH), R23 23 13 23 R23 2 > 0 (to leading order in s ) yields cot4 θ < 1 i.e. θ is above Eqn. (24), the condition 1 − R23 13 23 23 2 − R 2 > 0 and maximality (θ23 > 45◦ ). Similarly, for inverted hierarchy (IH), the condition R23 13 2 < 0 (or R13 < 1 and R > 1) predicts θ below maximality (θ < 45◦ ). 1 − R13 13 23 23 R23 With the help of constraints derived for δ, θ23 for NH as well as IH, it is straightforward to show that Rν (Eqn. (22)) is O(10−2 ) for NH and O(10−1 ) for IH. For example, if δ = 87.6◦ , θ23 = 48◦ , θ12 = 34.5◦ and θ13 = 8.5◦ , Rν is 0.025 for NH. Similarly, if δ = 87.6◦ , θ23 = 44◦ , θ12 = 34.5◦ and θ13 = 8.5◦ , Rν is 0.849 for IH. Thus, the requirement that model-prediction for Rν must lie within its experimentally allowed range hints towards normal mass hierarchy. These approximated analytical results will be extremely helpful to comprehend the phenomenological predictions obtained from the numerical analysis which is based on the exact constraining Eqns. (18) and (19). In numerical analysis, we have used Eqn. (22) as our constraining equation to obtain the allowed parameter space of the model i.e. Rν must lie within its 3σ experimental range. We have used latest global-fit data shown in Table 2. The experimentally known parameters such as mass-squared differences and mixing angles are randomly generated with Gaussian distribution whereas CP violating phase δ is allowed to vary in full range (0◦ –360◦ ) with uniform distribution (≈ 107 points). The mass ratios R13 and R23 depend on θ12 , θ23 , θ13 and δ. Using experimental data shown in Table 2, we first calculate the prediction of the model for Rν with normal as well as inverted hierarchy. It is evident from Fig. 1 that Rν is O(10−1 ) for IH i.e. outside the experimental 3σ range of Rν which is, also, in consonance with above analytical discussion. Hence, inverted hierarchy (IH) is ruled out at more than 3σ . In Fig. 2(a), we have depicted correlation between δ and θ23 at 3σ . θ23 = 45◦ is not allowed 2 must be less than 1. Also, the point (θ = 45◦ , δ = 90◦ or 270◦ ) is not allowed because 1 − R23 23 otherwise Rν < 0. θ23 is found to be above maximality and Dirac-type CP violating phase δ is

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S. Verma et al. / Nuclear Physics B 946 (2019) 114704

Table 2 The latest global-fit results of neutrino mixing angles and neutrino masssquared differences used in this analysis [7]. Parameters

Best-fit ±1σ

3σ range

m221 [10−5 eV2 ] m231 [10−3 eV2 ] (NH) m231 [10−3 eV2 ] (IH) Sin2 θ12 /10−1 Sin2 θ23 /10−1 (NH) Sin2 θ23 /10−1 (IH) Sin2 θ13 /10−2 (NH) Sin2 θ13 /10−2 (IH)

7.55+0.20 −0.16

7.05–8.14

2.50 ± 0.03

2.41–2.60

2.42+0.03 −0.04

2.31–2.51

3.20+0.20 −0.16

5.47+0.20 −0.30 5.51+0.18 −0.30 2.160+0.083 −0.069 2.220+0.074 −0.076

2.73–3.79 4.45–5.99 4.53–5.98 1.96–2.41 1.99–2.44

Fig. 1. Rν as a function of m1 /m3 for normal hierarchy (NH) and inverted hierarchy (IH).

constrained to a very narrow region in Ist and IVth quadrant. In Fig. 2(b), 2(c) and 2(d), we have shown the normalized probability distributions of θ23 and δ. The 3σ ranges of these parameters are given in Table 3. One of the desirable feature of a neutrino mass model is its prediction of the observable(s) which can be probed outside the neutrino sector. One such process is 0νββ decay, the amplitude of which is proportional to effective Majorana neutrino mass |Mee | given by 2 2 2 2 2iα 2 2iβ |Mee | = |m1 c12 c13 + m2 s12 c13 e + m3 s13 e |.

(29)

In Fig. 3, we have shown 23 − |Mee | correlation plot at 3σ . The important feature of the present model is the existence of lower bound |Mee | > 0.047 eV (at 3σ ) which is within the sensitivity reach of 0νββ decay experiments like SuperNEMO [66], KamLAND-Zen [67], NEXT [68,69], nEXO [70]. The |Mee |-sin2 θ23 parameter spaceis, further, constrained with inclusion of the cosmological bound on sum of neutrino masses ( i mi < 0.24 eV at 95% C.L., TT, TE, EE+lowE+lensing) [71] in our numerical analysis. In particular, there exist an upper (lower) bound on Mee ≤ 0.070 eV (θ23 ≥ 48◦ ) at 3σ . A similar analysis of neutrino mass matrices shown in Eqns. (11) and (12) reveals that these textures are not compatible with present global-fit data on neutrino masses and mixing including latest hints of normal hierarchical neutrino masses, higher octant of θ23 and near maximal Diracsin2 θ

S. Verma et al. / Nuclear Physics B 946 (2019) 114704

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Fig. 2. δ − θ23 correlation plot at 3σ (Fig. 2(a)) and probability distribution plots for θ23 (Fig. 2(b)) and δ (Fig. 2(c) and 2(d)).

type CP violating phase δ [7,20,72,73]. It has been already shown in reference [27] that hybrid textures are stable against the one loop RG effects. Consequent to the small renormalization group (RG) effects, the phenomenological consequences of hybrid texture structure predicted at higher energy scale can be studied with same structure at electroweak scale. Furthermore, being a minimal model, it will have interesting implications for leptogenesis which will be discussed elsewhere. The extension of the scalar field sector with scalar singlet (χ ) and scalar triplets (i ) in addition to Higgs doublet ( i ) may provide interesting phenomenology in collider experiments. The important feature of Higgs triplet model is presence of doubly charged scalar boson (±± ) in addition to H ± W ∓ Z vertex at tree level [74]. The decay of doubly charged bosons (±± ) to

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S. Verma et al. / Nuclear Physics B 946 (2019) 114704

Fig. 3. sin2 θ23 − |Mee | correlation plot at 3σ . The current sensitivity of SuperNEMO, KamLAND-Zen, NEXT, nEXO (5 yr) for |Mee | is, also, shown. The shaded region depicts the |Mee |-sin2 θ23 parameter space disallowed by incorporating the cosmological bound on sum of neutrino masses i.e. i mi < 0.24 eV (at 95% C.L.) [71] in our numerical analysis. Table 3 Prediction of the model for θ23 , δ and |Mee |. θ23 (bfp, 3σ range)

δ (bfp(s), 3σ range(s))

|Mee | (3σ lower bound)

48.68◦ 45.17◦ –50.70◦

88.58◦ 87.50◦ –89.70◦ 271.42◦ 270.20◦ –272.50◦

≥0.047 eV

charged leptons connects the neutrino sector and collider physics of the model. Due to involvement of the same couplings in neutrino sector and decay of ±± , collider phenomenology can be studied independently. With two scalar triplets (1,2 ), the physical Higgs are related to doubly charged scalar bosons (++ ) as  ++     ++  H1 cos θ sin θ 1 = − sin θ cos θ H2++ ++ 2 where θ is mixing angle. These physical Higgs can decay through different decay channels such as H ++ → l + l + , H ++ → W + W + , H ++ → H + W + , H ++ → H + H + . The last two decay channels might be kinematically suppressed as they depend on the mass difference of H + and H ++ . In particular, the decay mode H ++ → l + l + is dominant, however, it will depend on the VEV acquired by the scalar triplets [75]. For this decay mode, branching ratio depends on Yukawa couplings i.e. structure of neutrino mass matrix. The observation of dilepton decay mode, in collider experiment may, in general, shed light on the neutrino mass hierarchy. 4. Conclusions In conclusion, we have presented a neutrino mass model based on A4 flavor symmetry for leptons within type-I+II seesaw framework. The model is economical in terms of extended scalar sector and is highly predictive. The field content assumed in this work predicts three textures for Mν based on the charge assignments under SU (2)L and A4 . However, only one (Eqn. (10)) is found to be compatible with experimental data on neutrino masses and mixing angles. We have

S. Verma et al. / Nuclear Physics B 946 (2019) 114704

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studied the phenomenological implications of this texture in detail. The solar mass hierarchy 2 − R 2 > 0 constrains Dirac-type CP violating δ to narrow ranges 87.50◦ –89.70◦ and i.e. R23 13 270.20◦ –272.50◦ at 3σ . The sharp correlation between Dirac-type CP violating phase δ and atmospheric mixing angle θ23 demonstrates the true predictive power of the model (Fig. 2(a)). The predictions for these less precisely known oscillation parameters (δ and θ23 ) are remarkable which can be tested in neutrino oscillation experiments like T2K, NOνA, SK and DUNE to name a few. We have, also, calculated effective Majorana neutrino mass |Mee |. The important feature of the model is existence of lower bound on |Mee | which can be probed in 0νββ decay experiments like SuperNEMO, KamLAND-Zen, NEXT and nEXO. The main predictions of the model are: i. ii. iii. iv.

normal hierarchical neutrino masses. θ23 above maximality (θ23 > 45◦ ). near-maximal Dirac-type CP violating phase δ = 88.58◦ or 271.42◦ . 3σ range of effective Majorana neutrino mass 0.047 ≤ |Mee | ≤ 0.070 eV.

A precise measurements of Dirac-type CP violating phase δ, neutrino mass hierarchy and θ23 is important to confirm the viability of the model presented in this work. Acknowledgements The authors thank R.R. Gautam for useful discussions. S.V. acknowledges the financial support provided by UGC-BSR and DST, Government of India vide Grant Nos. F.20-2(03)/ 2013(BSR) and MTR/2019/000799/MS, respectively. M.K. acknowledges the financial support provided by Department of Science and Technology, Government of India vide Grant No. DST/INSPIRE Fellowship/2018/IF180327. The authors, also, acknowledge Department of Physics and Astronomical Science for providing necessary facility to carry out this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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