Flavour-symmetric type-II Dirac neutrino seesaw mechanism

Physics Letters B 779 (2018) 257–261

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Physics Letters B www.elsevier.com/locate/physletb

Flavour-symmetric type-II Dirac neutrino seesaw mechanism Cesar Bonilla a,∗ , J.M. Lamprea b , Eduardo Peinado b , Jose W.F. Valle a a AHEP Group, Institut de Física Corpuscular – C.S.I.C./Universitat de València, Parc Científic de Paterna, C/Catedrático José Beltrán, 2, E-46980 Paterna (Valencia), Spain b Instituto de Física, Universidad Nacional Autónoma de México (UNAM), A.P. 20-364, Ciudad de México 01000, Mexico

a r t i c l e

i n f o

Article history: Received 2 November 2017 Received in revised form 22 January 2018 Accepted 11 February 2018 Available online 13 February 2018 Editor: A. Ringwald

a b s t r a c t We propose a Standard Model extension with underlying A 4 flavour symmetry where small Dirac neutrino masses arise from a Type-II seesaw mechanism. The model predicts the “golden” flavourdependent bottom-tau mass relation, requires an inverted neutrino mass ordering and non-maximal atmospheric mixing angle. Using the latest neutrino oscillation global fit [1] we derive restrictions on the oscillation parameters, such as a correlation between δC P and mνlightest .

© 2018 The Authors. 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 .

Keywords: Neutrino masses and mixing Flavour physics

1. Introduction Full quark–lepton correspondence within the Standard Model would suggest neutrinos to be Dirac fermions, with the lepton mixing matrix completely analogous to the CKM matrix. However, if the ultimate description of particle physics is a four-dimensional quantum field theory, this situation is unlikely since, on general grounds, one expects neutrinos to be Majorana type [2]. In addition, the associated mechanisms to account for small neutrino mass as a consequence of their charge neutrality, such as the celebrated seesaw mechanism in its various realisations [2–6], all lead to Majorana neutrinos. From the experimental side, however, despite great efforts over several decades, neutrinoless double beta decay has not yet been detected [7]. According to the black-box theorem [8,9], such detection would provide the only robust way to establish the Majorana nature of neutrinos. Hence, for the time being, one must keep an open mind and re-examine the foundations of our ideological prejudices against having neutrinos as Dirac fermions. Till recently, one argument against Dirac neutrinos has been the absence of convincing realizations of the seesaw mechanism for this case. However this argument is fragile, and Dirac seesaw mechanisms have been shown to exist, both within the typeI [10–12], as well as type-II [13–15] realizations, for a brief classifi-

cation see [16,17].1 Moreover, the existence of right-handed states, required for Dirac masses, may be necessary in order to have a consistent high energy completion, or for realizing a higher symmetry, such as the gauged B–L symmetry present in the conventional SO(10) seesaw scenarios. Dirac neutrinos also arise in schemes with extra space–time dimensions such as string constructions, where right-handed states are needed to ensure anomaly cancellation through a generalized Green–Schwarz mechanism [21]. Alternatively, Dirac neutrinos arise from five-dimensional anti de Sitter space warped solutions to the hierarchy problem [22]. The truth of the matter is that the nature of neutrinos remains as mysterious as the mechanism providing their small masses, and hence we can not simply rule it out. In this letter, we consider the possibility of Dirac neutrinos resulting from a family symmetry construction in which the small neutrino mass arises à la seesaw, thus complementing the idea proposed in [23]. In addition to shedding light upon the pattern of neutrino oscillations, we also require the flavour structure to provide the generalised bottom-tau mass relation,

√

mτ

me m μ

=√

mb

md ms

,

(1)

proposed in [24,25]. This successful “golden” mass relation is derived without the need for invoking grand unification in a number

*

Corresponding author. E-mail addresses: cesar.bonilla@ific.uv.es (C. Bonilla), jmlamprea@estudiantes.fisica.unam.mx (J.M. Lamprea), epeinado@fisica.unam.mx (E. Peinado), valle@ific.uv.es (J.W.F. Valle).

1

For the possibility of having radiative Dirac neutrino mass models see Refs. [18–

20].

https://doi.org/10.1016/j.physletb.2018.02.022 0370-2693/© 2018 The Authors. 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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C. Bonilla et al. / Physics Letters B 779 (2018) 257–261

   LY ⊃ Y i L¯ , H d  R + Y νi L¯ , φ 3 ν R + h.c., i 3

(3)

i

Fig. 1. Neutrino mass generation in the Type-II seesaw for Dirac neutrinos [13–15].

of flavour models [12,26–29], all of which lead to particular restrictions on the neutrino oscillation sector. The paper is organized as follows: In the next section we introduce the model, in section 3 we present the predictions and discuss the results, and finally we conclude in 4. 2. The model This model is constructed so that the smallness of the neutrino mass has a dynamical origin, given by the Type-II seesaw mechanism for Dirac neutrinos, illustrated in Fig. 1. In order to explain neutrino oscillation data in this scenario we introduce the flavour symmetry A 4 ⊗ Z3 ⊗ Z2 . After symmetry breaking there is an Abelian discrete symmetry Z3 which forbids the appearance of Majorana operators at the loop level in the broken phase, thus protecting the Diracness of neutrinos.

where the symbol [a, b]3i stands for the two ways of contracting two triplets of A 4 , a and b, into a triplet, as shown in the Appendix. Before proceeding we summarise our model structure by saying that, compared with the minimal Standard Model case, here one has three copies of the Higgs doublet H 1d,2,3 , three extra scalar iso-doublets φ1,2,3 and three right-handed neutrino states ν R 1 , R 2 , R 3 , all of them arranged into A 4 triplets. Moreover one has three iso-singlet scalars σ1,2,3 which could be assigned as a triplet or singlets of A 4 . After symmetry breaking, neutrinos get a small type-II seesaw mass, as a result of the small vacuum expectation value (vev) φ which is induced by the vevs of the σ1,2,3 , as proposed in [13–15]. 3. Flavour predictions and numerical results Two aspects of the flavour problem concern the explanation of mass hierarchies of quarks and leptons, as well as to account for the structure of mixing in each of these sectors, so disparate from each other. We will see how our model leads to a successful “golden” mass formula relating quark and lepton masses, despite the absence of grand unification. This is a flavour generalization of bottom-tau unification previously proposed in [25–29,12]. In addition we will derive the corresponding model-specific predictions for the lepton mixing matrix describing neutrino oscillations. While no predictions are made for the CKM quark mixing matrix, we show how it can be adequately fit in a simple way, see reference [26]. We now spell out the detailed flavour predictions of our model.

2.1. Lepton sector 3.1. Charged fermions and the generalised bottom-tau mass relation The particle assignments for the lepton sector and scalars are given in Table 1. The SU (2) scalar doublets H d = ( H 1d , H 2d , H 3d ) and φ = (φ1 , φ2 , φ3 ) transform as triplets under A 4 , where each component can be written as follows



H di

=

hdi + hdi 0



 φi0 φi = , φi− 

,

i = 1, 2, 3.

(2)

The vacuum expectation value (vev) alignments of these scalar   triplets are given by H d = ( v hd , v hd , v hd ) and φ = ( v φ1 , v φ2 , v φ3 ). 1

2

3

The complex scalars σi (with i = 1, 2, 3) responsible for inducing the small vev of φ could form a triplet or transform as singlet under A 4 .2 On the other hand all leptons, both left- and righthanded, including the right-handed neutrinos ν R = (ν R 1 , ν R 2 , ν R 3 ), transform as A 4 triplets. Given the charges under the cyclic groups Z3 ⊗ Z2 , one can easily see that the Z3 remains unbroken after symmetry breaking, because all scalars are blind under this symmetry. Such residual Z3 symmetry forbids the Majorana mass terms M R ν R ν R as well as the dimension-5 operators: L H d L H d , L φ˜ L φ˜ and L H d L φ˜ . This symmetry also forbids higher order operators derived from the product of the previous dimension-5 ˜ n as well as ν R ν R σ n . operators and ( H d † H d )n , (φ † φ)n and ( H d† φ) Finally, the Z2 charges are assigned as (−) to ν R , σ and φ and + to the other particles. As a result they act in a complementary way to the Z3 , forbidding the unwanted renormalisable Yukawa

˜ d ν R , where φ˜ = i σ2 φ ∗ and H˜d = i σ2 H d ∗ . couplings: L¯ φ˜  R and L¯ H In accordance with the previous discussion, the relevant part of Yukawa Lagrangian for the leptons is given as:

2 In the first case, the fields σi are arranged in an A 4 triplet, that is σ = (σ1 , σ2 , σ3 ) ∼ 3, while for the latter, they transform as a singlet under the flavour group, σi ∼ 1i .

The complete particle assignment of our model is inspired in the one in [26], and is shown in Table 2, including both gauge as well as flavour transformation properties. From Eq. (3) one sees that the charged lepton mass matrix, following [25–27], can be parametrised as:

⎛

0

m = ⎝ b α a e i θ

⎞

a α e i θ 0 b ρ

b e i θ a ρ ⎠ , 0

(4)

where a = v hd (Y 1 + Y 3 ) and b = v hd (Y 2 + Y 4 ) are real Yukawa 2 2 couplings, θ is a unremovable complex phase3 and the H d vev  d alignment is parameterised as H = ( v hd , v hd , v hd ) = v hd (ρ , 1, α ), with

α = v hd / v hd and ρ = v hd / v hd . 3

1

2

1

2

3

2

2

The bi-unitary invariants of the squared mass matrix M 2 = † m m are determined as:

TrM 2 = m21 + m22 + m23 ,

(5)

2

det M  = m21m22 m23 , 2 2

2 2

(TrM  ) − Tr( M  ) =

2m21m22

+ 2m22m23

(6)

+ 2m21m23 .

(7)

We work under the assumptions ρ α , ρ 1, b > a and ρ ba which, at leading order, ensure adequate family mass hierarchy as well as mixing patterns. One can show from Eqs. (5)–(7) that:

3 We assume that the non-conservation of CP symmetry comes entirely from the neutrino sector. Thus in our analysis we fixed θ = 0.

C. Bonilla et al. / Physics Letters B 779 (2018) 257–261

Table 1 Charge assignments for the particles involved in the neutrino mass generation mechanism, where

SU (2) L ⊗ U (1)Y A4

Z3 Z2

259

ω3 = 1.

L¯

R

νR

Hd

φ

σ or σi

(2, −1/2)

(1, −1)

(1, 0)

(2, 1/2)

(2, −1/2)

(1, 0)

3

3

3

ω

ω

3 1

3 1

3 or 1i 1

+

−

+

−

−

ω

2

+

Table 2 Particle content and quantum numbers for the model.

SU (2) L ⊗ U (1)Y A4 Z3 Z2 Z 2d

Q¯

L¯

u Ri

dR

R

νR

Hu

Hd

φ

σ /σi

(2, 1/6)

(2, −1/2)

(1, 2/3)

(1, −1/3)

(1, −1)

(1, 0)

(2, −1/2)

(2, 1/2)

(2, −1/2)

(1, 0)

3 1

3

1i 1

3 1

3

3

ω2

ω

ω

3 1

3 1

3 1

3/1i 1

+ +

+ +

+ +

+ −

+ +

− +

+ +

+ −

− +

− +

(b ρ )2 ≈ m23 ,

(8)

3

(9)

(b   ) ≈ m21m22m23 , (a b 2 )2 ≈ m22m23 .

ρα

2

ρ

(10)

Solving the system in Eqs. (8)–(10), we can find the approximate expressions 4 :

a ≈

m2 m3



m1 m2

α

 b ≈

,

m1 m2

and

α

ρ m3 ≈√ . α m1 m2

√

(11) d

Notice that also the down-type quarks couple to the H and hence have the same flavour structure. This implies that the parameters ρ and α in Eq. (11) are common to the charged leptons and the down-type quarks, i.e. ρ = ρd and α = αd . From this we derive the generalised bottom-tau mass relation in Eq. (1):

√

mτ

me m μ

=√

mb

md ms

,

in a straightforward way. This generalised down quark-charged lepton mass relation, Eq. (1), follows from our flavour group assignments. Although it has been obtained also in other realizations of A 4 family symmetry [24,26,27], these are not equivalent. 3.2. Fermion masses and mixing In this section we focus on the lepton mixing matrix, because it is in this sector that our model makes non-trivial predictions. However, the CKM matrix describing quark mixing will be adequately described, providing, in addition, an input for the lepton mixing matrix. In analogy with the previous subsection, Eq. (3) gives the neutrino mass matrix, which can also be parametrised as:

⎛

0

mν = ⎝ bν e i θν αν aν

aν αν 0 bν e i θν ρν

⎞

bν e i θν aν ρν ⎠ , 0

(12)

where aν = v φ2 (Y ν1 + Y ν3 ) and bν = v φ2 (Y ν2 + Y ν4 ) are real Yukawa couplings, θν is the complex phase that cannot be rotated away under SU (2) transformations, and characterizes the strength of CP violation in the lepton sector. The vev-alignment of φ can be written as φ = ( v φ1 , v φ2 , v φ3 ) = v φ2 (ρν , 1, αν ), with αν = v φ 3 / v φ 2 and ρν = v φ 1 / v φ 2 .

4

Fig. 2. The regions in the atmospheric mixing angle θ23 and the lightest neutrino mass m3 plane allowed by current oscillation data are the shaded (green) areas, see text. The horizontal dashed line represents the best-fit value for sin2 θ23 , whereas the horizontal shaded region corresponds to the 1σ allowed region from Ref. [1].

These approximations are in 1% agreement with the exact numerical solution.

From the invariants Eqs. (5)–(7) of the squared neutrino mass † matrix M ν2 = mν mν :

Tr( M ν2 ) = (a2ν + b2ν )(1 + αν2 + ρν2 ), 2

6

6

(13)

3 3

2

2

det( M ν ) = (aν + bν + 2aν bν cos(3θν ))αν ρν ,  1 (TrM ν2 )2 − Tr( M ν4 ) 2

(14)

= a2ν b2ν (1 + αν4 + ρν4 ) + (a4ν + b4ν )(ρν2 + αν2 (1 + ρν2 )),

(15)

we performed a numerical scan over the parameter regions5 for the solutions of Eqs. (13)–(15) that reproduce the measured ele†

ments of the leptonic mixing matrix V = U l U ν . To do this we use as inputs the 3σ values for the three neutrino mixing angles and the two mass squared differences from the global fit [1]. The U ν matrix comes from the bi-unitary transformation for the neutri†





nos U ν mν V ν = D ν , where D = diag mν1 , mν2 , mν3 , while the U l comes from the equivalent transformation for the charged leptons, † e.g. U  m V  = D  . We now turn to the CKM matrix describing quark mixing. Although we have no family symmetry prediction for the CKM matrix, we notice that it can be accommodated in the same way as described in [26]. This fixes the value for the αd parameter

5 The number of free parameters in the leptonic sector is 9, two of them are fixed from the quark sector. The other 7 are used to fit 3 masses plus two splittings, 3 mixing angles and one neutrino CP phase.

260

C. Bonilla et al. / Physics Letters B 779 (2018) 257–261

Fig. 3. Correlation between the CP violation and the lightest neutrino mass. Left: correlation between the Jarlskog invariant and the lightest neutrino mass m3 allowed by the current oscillation data from Ref. [1]. Right: We plot also the allowed region for the correlation between the Dirac CP phase δC P and the lightest neutrino mass m3 .

which enters also in the leptonic sector. In order to adequately fit the CKM matrix we need αd = αl ≈ 1.58 [26]. Taking into account the numerical values for the charged lepton masses me = 0.511006 MeV, mμ = 105.656 MeV and mτ = 1776.96 MeV and assuming that the complex phase θ = 0, as in Refs. [25–27], we find that the resulting contribution from the charged lepton sector to the neutrino mixing matrix is fixed and close to be diagonal, see for instance [25]. 3.3. Neutrino oscillation predictions In order to determine the neutrino oscillation predictions of the model, we have performed a numerical scan in which parameters are varied randomly in the ranges

αν ∈ [−10, 10] ,

ρν ∈ [−10, 10] and θν ∈ [0, 2π ] .

(16)

Only those choices for which the undisplayed and wellmeasured oscillation parameters are within 3σ of the values obtained in the latest neutrino oscillation global fit of Ref. [1] are kept.6 This way we have obtained the model-allowed regions in terms of the “interesting” and poorly determined oscillation parameters θ23 , δC P as well as the lightest neutrino mass eigenvalue. These are displayed in shaded (green) regions in the Figs. 2, 3 and 4. In contrast, the band in Fig. 2 and the unshaded regions are the 90 and 99%CL regions in Fig. 4 are obtained directly from the unconstrained three-neutrino oscillation global fit [1], see Figures 4 and 5 in reference [1]. We find that the model is only compatible with the inverted ordering for the neutrino mass eigenvalues. The consistent parameter regions for the atmospheric mixing angle sin θ23 vs. the lightest neutrino mass m3 are given in Fig. 2, while the CP violation parameters δC P and J C P vs. m3 are displayed in Fig. 3. From the plot given in Fig. 3 one sees that the allowed region for the lightest neutrino mass m3 is within the range [6.4 × 10−4 eV, 2.7 × 10−3 eV]. We can see that only masses above ∼ 0.002 eV allow J C P = 0, signifying no CP violation, while for lower masses such value is always non-zero. The shaded areas in Fig. 4 are obtained from a numerical scan that filters those parameter choices for which the well-measured undisplayed oscillation parameters lie within 3σ of the best fit values obtained in the latest neutrino oscillation global fit in Ref. [1]. These

6 This phenomenological requirement implies that the allowed regions for the free parameters are: 0.75  |αν , ρν |  1.25, 0.03  |aν , bν |  0.04 and −π /3  θν  π /3.

Fig. 4. The allowed regions of the atmospheric mixing angle and δC P are indicated in shaded (green). They result from a numerical scan keeping only those choices that lie within 3σ of their preferred best fit values Ref. [1]. The unshaded regions are 90 and 99%CL regions obtained directly in the unconstrained three-neutrino oscillation global fit [1].

should be compared with the unshaded 90 and 99%CL regions obtained directly in the unconstrained three-neutrino oscillation global fit [1]. 4. Conclusions We have proposed a A 4 ⊗ Z3 ⊗ Z2 flavour extension of the Standard Model where the small neutrino masses are generated from a type II Dirac see-saw mechanism. The model addresses both aspects of the flavour problem: the explanation of mass hierarchies of quark and leptons, as well as restricting the structure of the lepton mixing matrix. Concerning the first point, our model leads to a successful “golden” mass relation between quark and lepton masses, proposed previously. In addition, the model provides flavour predictions for the lepton mixing matrix relevant for neutrino oscillations. First of all, inverted neutrino mass ordering and non-maximal atmospheric mixing angle are predicted. While this is at odds with the results of the latest neutrino oscillation global fit in [1], we stress that the neither the preference for normal ordering nor the indication for a given octant are currently statistically significant, since the general three-neutrino fit gives four possible closely separated local minima. In any case, taken at face value, the model at hand would suggest a slight preference for the higher octant, since it predicts inverted neutrino mass ordering. We have also found a positive hint for CP violation, δC P = 0, if mνlightest  0.002 eV, while bigger masses are consistent with CP

C. Bonilla et al. / Physics Letters B 779 (2018) 257–261

conserving solutions. The model indicates regions for the CP phase δC P and the atmospheric angle that nicely agree with the currently preferred ones, though these global fit determinations are not yet fully robust. Concerning the CKM quark mixing matrix, we also saw that, although no definite predictions are made, the required CKM matrix elements can be adequately described, and they also fix the contribution to the neutrino mixing matrix that comes from the charged lepton sector. Finally, we note that the residual flavour symmetry forbids the Majorana mass terms at any order and provides, by construction, a natural realization of a type II Dirac see-saw mechanism for small neutrino masses. Acknowledgements This research is supported by the Spanish grants FPA201458183-P, SEV-2014-0398 (MINECO) and PROMETEOII/2014/084 (Generalitat Valenciana). E.P. is supported by DGAPA-PAPIIT RA101516 and IN100217. J.M.L. would like to thanks AHEP group at IFIC for its kind hospitality while part of this work was carried out, and DGAPA-PAPIIT RA101516 and CONACYT (México) for financial support. Appendix A. The A 4 product representation The non abelian discrete group A 4 , or the group of the even permutation of four elements, has four irreducible representations [30–32]: three singlets 11 , 12 , and 13 and one triplet 3 and two generators: S and T following the relations S 2 = T 3 = ( S T )3 = I . The one-dimensional unitary representations are

11 : S = 1, T = 1, 12 : S = 1, T = ω, 13 : S = 1, T = ω2 , where

⎛

(A.1)

ω3 = 1. In the basis where S is real diagonal, ⎞ ⎞ ⎛

1 0 0 S = ⎝ 0 −1 0 ⎠ and 0 0 −1

0 T =⎝0 1

1 0 0

0 1⎠ . 0

(A.2)

The product rule for the singlets are

11 ⊗ 11 = 12 ⊗ 13 = 1, 12 ⊗ 12 = 13 , 13 ⊗ 13 = 12 ,

(A.3)

and triplet multiplication rules are

(ab)11 (ab)12 (ab)13 (ab)31 (ab)32

= = = = =

a1 b 1 + a2 b 2 + a3 b 3 , a1 b1 + ωa2 b2 + ω2 a3 b3 , a1 b1 + ω2 a2 b2 + ωa3 b3 , (a2 b3 , a3 b1 , a1 b2 ) , (a3 b2 , a1 b3 , a2 b1 ) ,

(A.4)

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