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Nuclear Physics B 916 (2017) 430-462

www. elsevier. com/locate/nuclphysb

Fermion masses and mixing in SU(5) x D4 x U(1)

R. Ahl Laamaraab c, M.A. Loualidiac, M. Miskaouiac, E.H. Saidiac*

a LPHE-Modeling and Simulations, Faculty of Sciences, Mohammed V University, Rabat, Morocco b Centre Régional des Métiers de L'Education et de La Formation, Fès-Meknès, Morocco c Centre of Physics and Mathematics, CPM, Morocco

Received 17 October 2016; received in revised form 12 January 2017; accepted 13 January 2017

Editor: Tommy Ohlsson

Abstract

We propose a supersymmetric SU (5) x Gf GUT model with flavor symmetry Gf = D4 x U(1) providing a good description of fermion masses and mixing. The model has twenty eight free parameters, eighteen are fixed to produce approximative experimental values of the physical parameters in the quark and charged lepton sectors. In the neutrino sector, the TBM matrix is generated at leading order through type I seesaw mechanism, and the deviation from TBM studied to reconcile with the phenomenological values of the mixing angles. Other features in the charged sector such as Georgi-Jarlskog relations and CKM mixing matrix are also studied.

© 2017 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

Standard Model (SM) of elementary particle physics is a great achievement of modern quantum physics; but despite this success basic questions still remain without answer; one of them concerns the origin of the three generations of fermions, quark-lepton masses and mixing angles. Although the SM is sufficient to describe the masses of charged leptons and quarks, neutrinos (vi)i=1,2,3 are considered as massless particles in this model which is in conflict with observations. Indeed, neutrino oscillation experiments have shown that they have very tiny masses mi

* Corresponding author. http://dx.doi.org/10.10167j.nuclphysb.2017.01.011

0550-3213/© 2017 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.

Table 1

The global fit values for the squared-mass differences Am?, and mixing angles On as reported by

Ref. [6]. NH and IH stand for normal and inverted hierarchies respectively.

Parameters

Bestfit(+1:;-?:;+3:)(NH)

Bestfit(+i::-?::+3:)(iH)

Am?1 [ID-5 eV?] |Am?1 [10-3 eV2]|

sin? O1?

sin? O?3 sin? O31

7 60(+0.19; +0.39; +0.58) '•6o(-0.18;-0.34;-0.49)

( + 0.05;+0.H;+0.17) (-0.07;-0.13;-0.18)

(+0.016;+0.034;+0.052) (-0.016;-0.031;-0.045)

(+0.032;+0.056;+0.076)

(-0.124;-0.153;-0.174)

7.60(+0.19,+0.39,+0.58)

(-0.18;-0.34;-0.49)

(+0.05,+0.10,+0.16) (-0.06,-0.12,-0.18)

(+0.016,+0.034,+0.052) (-0.016,-0.031,-0.045)

(+0.025,+0.048,+0.067)

(-0.039,-0.138,-0.170)

0.0226

(+0.0012,+0.0024,+0.0036) (-0.0012,-0.0024,-0.0036)

0.0229

(+0.0012,+0.0023,+0.0036) (-0.0012,-0.0024,-0.0036)

and that the different flavors are mixed with some mixing angles Oij. The PMNS matrix which describe the mixing in the lepton sector contains two large angles O12 and O23 consistent with tribimaximal mixing matrix (TBM) [1], and a vanishing angle O13 which is in disagreement with the recent neutrino experiments1 [2-5]. The measurements of the mixing angles and the squared-mass differences was reported by several global fits of neutrino data [6-8]; see Table 1. This mixing together with the non-zero neutrino mass might be the best evidence of physics beyond the standard model; in this context, many models have been proposed in recent years, and Supersymmetric Grand Unified Theories (SUSY-GUTs) are one of the most appealing extension of the SM unifying three forces of nature in a single gauge symmetry group [9-11]. These quantum field theories contain naturally the right-handed neutrino needed to generate light masses for neutrinos through the seesaw mechanism. Moreover, particles are unified into different representations of the GUT groups; for instance, in S0(10) GUT model [11], all the fermions including the right-handed neutrino belong to the 16-dimensional spinor representation of S0(10), and in SU(5) GUT model, all the matter fits into two irreducible representations, the conjugate five F = 5 and the ten T = 10 [10]. In addition, extending GUT models with flavor symmetries might be the key to understand the flavor structure; indeed many flavor symmetries have been suggested in GUT models, in particular, the non-abelian discrete alternating A4 and symmetric S4 groups are widely studied in the literature. These discrete groups have been used in many papers to realize the TBM matrix [15], and used recently to accommodate a non-zero reactor angle [16-20], and lately, the models studied in Refs. [21,22] provided successfully the masses for all fermions and the mixing in the charged and chargeless sectors including spontaneous CP violation. In addition, there are many other non-abelian discrete groups proposed as family symmetry with the SU(5) GUT group; for example the SU(5) x T' model [23], and the SU(5) x A(96) model [24]. As for the flavor models based on S0(10) gauge group, we refer for instance to the S0(10) x A4 model [25], S£(10) x S4 model [26], S£(10) x PSL(2, 7) model [27], and S£(10) x A(27) model [28].

In this paper, we propose a supersymmetric SU (5) x Gf GUT model with flavor symmetry Gf = D4 x U (1) providing a good description of fermion masses; and leading as well to neutrino mixing properties agreeing with known results. The model has twenty eight free parameters in which we need to fix eighteen in order to produce the approximative experimental

1 In addition to the TBM matrix approximation, similar mixing matrices with vanishing O13 have been proposed such as Bimaximal (BM) [12], Golden-Ratio (GR) [13] and Democratic [14] mixing pattern.

values of the physical parameters in the quark and lepton sectors as given by Tables (5.2)-(5.3) and Tables (5.5)-(5.9). To fix ideas, let us comment rapidly some key points of this Gf based construction and some motivations behind the choice of the discrete D4 dihedral symmetry.

First, notice that the discrete flavor D4 symmetry is the finite dihedral group; and, like the alternating A4, it is also a non-abelian subgroup of the symmetric S4 with particular properties. It has 5 irreducible representations: four singlets 1p,q with indices p, q = ±1; and one doublet 20,0 offering therefore several pictures to engineer hierarchy among the three generations of matter; for example by accommodating one generation in a given 1p,q representation, while the two others in the 20,0 doublet. Another example is to treat the three generations in quite similar manner by accommodating them in 1-dimensional representations 1puqi but with different characters. Recall that the order of D4—which is 8—is linked to the sum of the squared dimensions of its five irreducible representations R1,..., R5 like 8 = 1+,+ + 1+ + 1-,+ + 1- + 22 0; the four representations Ri = 1p,q and the fifth R5 = 20,0 are indexed by the characters x(a), x (P) of the two non-commuting generators a and P of the dihedral D4; a remarkable feature of discrete group theory allowing to distinguish the four D4 singlets in a natural way.

Besides particularities of its singlet representations as well as its similarity with the popular alternating A4 group; our interest into a flavor invariance Gf d D4 has been also motivated from other reasons; in particular by the wish to complete partial results in supersymmetric GUTs which aren't embedded in brane picture of F-theory compactification along the line of [33]; and also by special features of the dihedral group. The discrete D4 symmetry has been considered as flavor symmetry in several models to study the mixing in the lepton sector, see for instance [29-31], and one of its interesting properties is that it predicts the x — t symmetry in a natural way as noticed by Grimus and Lavoura (GL) [29]. It was considered also in heterotic orbifold model building [32], as well as in constructing viable MSSM-like prototypes in F-theory [33]. But to our knowledge, the dihedral group D4 was never used as a flavor symmetry in GUT models which doesn't descend from string compactification; this lack will be completed in present study.

To build the supersymmetric model SU(5) x D4 x U (1)f, we need building blocks of the construction and their couplings; in particular the chiral superfields of the prototype; their quantum numbers under flavor symmetry and their superpotential W ($). After identifying the SU(5) superfield spectrum with appropriate D4 quantum numbers, we introduce an additional global U(1)f symmetry which will make our model quasi-realistic—U(1)f = U (1). As we will show; this extra continuous symmetry is needed to control the superpotential in the quark-and lepton-sectors, and also to prevent dangerous operators that mediate rapid proton decay. Our SU(5) x D4 x U (1) model involves, in addition to the usual SU(5) superfield spectrum collected in Tables (2.7)-(2.8), eleven flavon superfields carrying quantum numbers under the flavor symmetry D4 x U(1) as given by (2.13)-(2.14); these flavon superfields will play an important role in obtaining the appropriate masses for the quarks and leptons. Moreover, we have twenty eight free parameters—fifteen Yukawa coupling constants, eleven flavon VEVs, the 45-dimensional Higgs VEV and the cutoff scale A—where we fix eighteen of them; eight in the quark and charged lepton sectors and ten in the neutrino sector. We end this study by performing a numerical study, where we use the experimental values of sin 0ij and Amij to make predictions concerning numerical estimations of the parameters obtained in the neutrino sector.

The paper is organized as follows. In section 2, we present the superfield content of the SU(5) model as well as a superfield spectrum containing flavons superfields in D4 representations. Then, we assign U(1) charges to all the superfields of the model. In section 3, we first study the neutrino mass matrix and its diagonalization with the TBM matrix; then we study the deviation of the TBM matrix by introducing extra flavon superfields, and we make a numerical study

to fix the parameters of the neutrino sector. In section 4, we study the mass matrix of the up quark sector and we make a comment concerning the scale of the flavon VEVs derived from the experimental values of the quark up masses; then, we analyse the down quarks-charged leptons sector by calculating their mass matrices as well as the mixing matrix of the quarks. In section 5, we give our conclusion and numerical results. In Appendix A, we give all the higher dimensional operators yielding to the rapid proton decay which are forbidden by the U(1) symmetry. In Appendix B, we give useful tools and details on D4 tensor products.

2. SU(5) model with D4 x U(1) flavor symmetry

In this section, we first describe the chiral superfields content of the supersymmetric SU (5) GUT model; then we extend this model by implementing the D4 flavor symmetry accompanied with extra flavon superfields which are gauge singlets. This extension is further stretched with a flavor symmetry U(1) needed to exclude unwanted couplings.

2.1. Superfields in SU(5) model

In this subsection, we review briefly the building blocks of the usual supersymmetric SU(5)-GUT model that contain the minimal supersymmetric model (MSSM) quarks and lep-tons as well as the right-handed neutrino; we also use this description to fix some notations and conventions. We will focus mainly on the chiral superfields of the model and the invariant superpotential; the Kahler sector of the model involving as well gauge superfields is understood the presentation. The chiral sector of SU(5) model has two kinds of building blocks: matter and Higgs; they are as follows

• Matter superfields

In supersymmetric SU(5) -GUT, each family F of quarks Q (with colors r, b, g) and leptons L fits nicely into a reducible SU(5) representation involving the leading irreducible 1, 5, 10. In superspace language, left-handed fermions are described by chiral superfields Fi = 5i and Ti = 10i; the right-handed neutrinos are also described by chiral superfields but living in SU(5) singlets Ni = 1i. The index i = 1 2, 3 refers to the three possible generations of matter Fi = {Fi ,Ti,Ni}; for example the first family F1, the constituents of F1 and T1 are explicitly as follows [35]

I dcr \

f 0 ug -ub ur dr

-ug 0 uC ub db

4 -uC 0r ug dg

-ur -ub -ug 0 ec

- dr -db -dg -ec 0

• Higgs superfields

We distinguish several kinds of SU(5)-GUT Higgs superfields; in particular the H5, H5, H24 and the H45. The chiral superfields H5 = 5Hu and H5 = 5Hd are respectively the analogue of two light Higgs doublet superfields Hu and Hd of the MSSM; in general the MSSM Higgs doublet Hd is a combination of the H5 Higgs with the 45-dimensional Higgs denoted by H45. This extra Higgs superfield will also used later on in order to distinguish the down quarks masses from the leptons masses.

The SU(5) GUT symmetry is broken down to the standard model symmetry SU(3)C x SU(2)L x U(\)Y by the VEV of the adjoint Higgs H24. This is done by choosing (H24> along the following particular Cartan direction in the Lie algebra of SU (5)

(H24> =

so the SU(5) fields are given in standard model terms as 10m ^ (3, 2) 1 + (3, 1) -4 + (1,1)2

5M ^ (1, 2)_1 + (3, 1)2 5hu ^ (1, 2)1 + (3,1)-2 5nd ^ (1, 2)_1 + (3, 1)2

24 ^ (8, 1)o + (1, 3)o + (1,1)o + (3, 2)-5 + (3,2) 5

as well as

45 ^ (8, 2)! + (6,1)_2 + (3, 3)_2 + (3, 2) _7 + (3,1)_1 + (3,1) 8 + (1,2)1 (2.5)

In what follows we describe our extension of supersymmetric SU (5) -GUT by a global flavor symmetry G f which is given D4 x U (1)f, the product of the finite discrete Dihedral group and the U (1)f global continuous phase.

2.2. Implementing D4 flavor symmetry

Here, we present our extension of the supersymmetric SU(5) GUT model by the flavor symmetry D4, details of the Dihedral group D4 are provided in Appendix B. First, we give the D4-quantum numbers of the superfields of usual SUSY SU(5) matter; then we describe the needed extra matter required by dihedral flavor symmetry.

In the usual SU(5) model reviewed in previous subsection, the matter and Higgs superfields are as collected in first line of Tables (2.7)-(2.8); they are unified in the SU(5) representations with link to MSSM as

10m = (uc,ec ,Ql)

5m = (dC,L)

5hu = (Au,Hu) 5Hd = (Ad,Hd)

The three generations of 10^ and 5m are denoted as T and F respectively, the three right-handed neutrinos denoted as N are singlets under SU(5); and the two GUT Higgses denoted as H5 and H5 like 5hu and 5Hd.

In our extension with a D4 flavor symmetry, we have a larger set of chiral superfields that can be organized into two basic subsets: (a) the usual SU(5) matter and Higgs superfields; but carrying as well quantum numbers under D4; and (b) an extra subset of chiral superfields required by D4 flavor invariance; they are as described below.

a) Matter and Higgs sectors in SU (5) x D4 The superfield content of this sector is same as the SU(5) matter and Higgs superfields; but with extra quantum numbers under D4 flavor invariance as given here below

Matter Ti T2 T3 F1 F2,3 N1 N2,3

SU(5) loi 10i 10m 51 —2 3 5 11

D4 1+,- 1+,- 1+,+ 1+,- 2o,o 1+,+ 2o,o

Higgs H5 H5 H45

SU(5) 5ru 5Hd 45h

D4 1+,- 1+,+ 1+,-

The matter superfields 10'm of the three generations i = 1, 2, 3 are assigned into the D4 representations 1+,-, 1+,- and 1+,+ respectively; while the 5m matter superfields are assigned into the D4 singlet 1+,- and the D4 doublet 20,0. The right-handed neutrino N sits in the D4 trivial singlet 1+,+, and the two N2,3 sit together in the D4 doublet 20,0. The GUT Higgses H5, Hj and H455 are put in different D4 singlets; 1+,-, 1+,+ and 1+,- respectively.

b) Flavon sector

In addition to the SU(5) superfields of (2.7)-(2.8), the SU(5) x D4 model has eleven flavon chiral superfields namely four doublets and seven singlets; they transform as singlets under gauge group SU (5), but carry charges under D4 flavor symmetry as follows

Flavons r Î2 F 0 <P n X a p p ' Z

SU(5) 1 1 1 1 1 1 1 1 1 1 1

D4 1+,- 1+,- 1+,- 2o,o 2o,o 1+,+ 2o,o 2o,o 1+,- 1-,- 1+,+

These flavon superfields couple to the matter and Higgs superfields of the model. The above quantum numbers are required by the building of the chiral superpotential WSU5 xD4 of the super-symmetric model. This complex superpotential is a superspace density which, after performing superspace integration, leads to a space time lagrangian density LSU5xD4 describing matter couplings through Higgs and flavons. The typical form of LSU5xD4 is given by

LSU5 XD4 = y d2eWsu5x.D4 + hc (2.10)

where the generic 's stand for the chiral superfields of Tables (2.7)-(2.9). This superpotential involves several free coupling parameters to be studied in forthcoming sections. The flavons in Table (2.9) have been required by D4 invariance; they are briefly commented below:

(i) Neutrinos couplings

Invariant neutrinos superpotential WSU5xD4 (N,..) under D4 flavor symmetry requires in turns the flavons n, X, P, p', Z, a:

• the flavon n and x are needed to produce the TBM matrix in the neutrino mass matrix.

• the flavons p, p', Z and a are added to generate the deviation from TBM matrix. (ii) Quarks and charged leptons superpotentials

Flavor symmetry invariant superpotentials WSU5xD4 (T,F,..) involving quarks and charged leptons require the flavon superfields T, F, $, v with quantum numbers as listed in (2.9) for the following purposes:

• the three flavons T and F contribute to the up-, charm- and top-quark masses respectively.

• the two flavons T and ^ are also needed by down quarks/charged leptons in order to generate masses for the first two families.

• the flavon $ is required by down quarks/charged leptons in order to produce the mass of the third family.

• the flavon v is needed for two goals: first to contribute to the mass of the first two generations of down quarks/charged leptons together with the flavon singlets T and and second to couple to the 45-dimensional Higgs H45 in order to distinguish between the down quarks and charged leptons mass matrices.

2.3. Need of U(1)f symmetry

In order to engineer a semi-realistic model, we need additional flavor symmetries; in our D4 based proposal, we found that we have to add an abelian U(1) symmetry to fully control the couplings of SU(5) x D4 model for reasons such as the ones given below:

(i) Eliminate unwanted couplings

The global U(1) symmetry is necessary to eliminate unwanted couplings and to produce the observed mass hierarchies, it makes the model quasi-realistic for the two following things:

• first to control the superpotential of the quark and lepton sectors in the SU(5) x D4 model; for example the flavon F, transforming as 1+,-, is used to generate a heavy mass for the top quark; but the two other flavons T and ^ share the same D4 representation 1+,_ and so can couple quark and lepton superfields in a D4 invariant manner. These coupling cannot be dropped out without imposing an extra constraint; moreover, the three flavons could be mixed in the operators of each family of the Yukawa up type; so they could affect the top quark mass, and consequently risking to lose the mass hierarchy between the top and the up, charm quarks. This issue is handled by accommodating the flavons which possess the same D4 representation in different U(1) representations as in Table (2.13).

• second, the U(1) charge assignments are chosen to produce the TBM as well as its deviation to get a non-zero reactor angle in the neutrino sector which will be discussed in section 3.

(ii) Avoid rapid proton decay

The U(1) flavor symmetry is also needed to forbid the operators yielding to rapid proton decay such as the couplings of type 10m.5m.5m. The SU(5) x D4 model have several invariant operators of this type and of other types which will be discussed in Appendix A; they are prevented by the extra global U(1) symmetry with charge assignments as in the following tables:

* families

matter T1 T2 T3 F1 F2,3 N1 N2,3

SU(5) 10m 10m 10m 51 —2 3 52,3 11 V

D4 1+,- 1+,- 1+,+ 1+,- 2o,o 1+,+ 2o,o

U(1) 12 7 -27 14 14 -6 -6

(2.11)

* Higgs

Higgs H5 H5 H45

SU(5) 5ru 5Hd 45h

D4 1+,- 1+,+ 1+,-

U(1) -8 11 10

(2.12)

* flavons

3. Neutrino sector in SU(5) x D4 x U(1) model

flavons r F 0 V

SU(5) 1 1 1 1 1

D4 1+,- 1+,- 1+,- 20,0 20,0

U(1) -6 -16 62 2 -31

flavons n X a P P' z

SU(5) 1 1 1 1 1 1

D4 1+,+ 20,0 20,0 1+,- 1-,- 1+,+

U(1) 12 12 -24 -24 -24 36

(2.13)

(2.14)

In this section, we first study the mass matrices of Dirac and Majorana neutrinos; then we use the seesaw type I to get a neutrino mass matrix compatible with TBM as a leading approximation. Next, we study the deviation from TBM by adding new flavons. Notice that the right-handed neutrinos are SU (5) singlets, thus the light neutrino masses are only generated through type-I seesaw mechanism [34]

mv = mDMR

where the mD and MR are the Dirac and the Majorana mass matrices respectively.

3.1. Neutrino mass matrix and tribimaximal mixing

We begin by considering Dirac mass matrix involving left- and right-handed neutrinos; and turn after to calculate the Majorana masses.

3.1.1. Dirac neutrinos

The Dirac mass matrix couples the left-handed neutrinos in the (Fi)i=1,2,3 to the right-handed ones (Ni )i=i,2,3 living in different representations of SU (5) x Gf with flavor symmetry Gf =

D4 x U (I). As described in section 2, the F1 lives in the non-trivial D4 singlet 1+,— while F2 and F3 live together in the D4 doublet 20,0; they have the same U(1) charge qFi = 14. The right-handed neutrinos have different quantum numbers under D4; the N1 lives in the D4 representation 1+,+ while N2 and N3 live together in the D4 doublet 20,0; they have the same U (1) charge qNi = -6. The chiral superpotential WD (F, N, H) for neutrino Yukawa couplings respecting gauge invariance and flavor D4 x U(1) symmetry is given by

Wd = X1N1F1H5 + ^2N2,3F2,3HS (3.2)

where A^ and X2 are Yukawa coupling constants. Using the tensor product of D4 irreducible representations given in Eqs. (B.4)-(B.5) and denoting the Higgs by Hu, the superpotential (3.2) become

Wd = k1Hu (VeLe) + k2Hu (v,L, + VtLt) (3.3)

When the Higgs doublet develop its VEV as usual (Hu) = vu, we get the Dirac mass matrix of neutrinos

% 0 0 \

mD = vu\ 0 X2 0 (3.4)

0 0 X2

3.1.2. Majorana neutrinos

A Majorana mass matrix couples the three right-handed neutrinos Ni to themselves; this mass matrix is obtained from the superpotential WM(N,...) respecting gauge invariance and flavor symmetry of the model. Using Tables (2.11)-(2.14), one can check that this chiral superpotential is given by

Wm = A3N1 Nin + X4N2,3N2,3V + ¿5NIN2,3X (3.5)

In this expression, we have added the third term involving the flavon x to satisfy the TBM conditions and to generate appropriate masses for the neutrinos. This term—which is at the renor-malizable level—will contribute to the entries (12) and (13) in the Majorana mass matrix. By using the multiplication rule of D4 representations, the superpotential WM develops into

Wm = A3 (V1V1) n + A4(V2V3 + V3V2)n + A5V1 (V2X2 + V3X1) (3.6)

and by taking the VEVs of the flavons x and n as

<X1> = (X2> = Ux, (n> = Un we find the Majorana neutrino mass matrix MR as follows

(k3Vv k5Vx

^5VX 0 Uvv \ (3.7)

X5Vx X4VV

The light neutrino mass matrix is obtained using type I seesaw mechanism formula mv = mDM —1mD, and we find

^5UX A4 Un 0

X^X4 Un

X3X4V:2 -2X2 uX X1X2X5Ux

X1X2X5Ux X3X4 ^-2x2^

X1 ^2^5Ux

À3À4U2-2À2UX

X3X4 U2 -2^2 u'x

À1À2^4À5UnUx

\ À3À2U3-2À4 X^UX

X2X2Ux XiX^U 3-2X4 X^nUx

5 Ux -

X2(x2U2 -X3X4Un^

X3X^U^-2X4X^ UnU2

X3 X2 u3 -2X4 X2 Un U2

X2X2U2 X2X5Ux

X3X^U3-2X4X^UnUx y

mv = U

this form of mv can realize the TBM matrix by adopting the following A.1 =

X4vv = X3VV+X5Vx

so the above mass matrix mv is diagonalized as Mv = UT mvU = diag(m1, m2, m3) with the TBM matrix U given by

It predicts the mixing angles as follows

Sin2 012 =

sin2 023 =

sin2 0,3 = 0

the eigen-masses are

j2„ 2 k1vu

X3V„ - X^Vx

X2„ 2 X1uu

X3VV + 2X5 vx '

m3 = -

X2„ 2

X3Un + X5UX

(3.10)

(3.11)

(3.12)

which yield to a non-vanishing solar and the atmospheric mass-squared differences Am21 and

3.2. Deviation of mixing angles 013 and 023

In this subsection we study the deviation from TBM matrix which consists of breaking the fi-r symmetry in the neutrino mass matrix in order to reconcile the reactor angle 013 with the global fit data in Table 1. Recently, the deviation from TBM using additional flavons has been extensively studied in the literature and there are two matrix perturbations that allow for a suitable deviation of the mixing angles (for deviation by using non-trivial singlets, see for example Ref. [36]), they are:

8MH = e

M3 = e

(3.13)

where the indices (12), (33), (13) and (22) are the elements that should be perturbed in the neutrino matrix to deviate from TBM and e is the deviation parameter.

Using the flavon superfields a, Z, P and p' of Table (2.14), we see that we can perform a symmetric perturbation of the superpotential (3.5) that induces a deviation of the Majorana neutrino mass matrix MR of Eq. (3.7). Thus, the additional higher dimensional operators that respect the symmetries of the model are as follows:

SWm = — (X6N1N2,3VZ + X7N2'3N2'3PZ + X&N2'3N2'3P 'Z)

(3.14)

The invariance of SWM may be explicitly exhibited by using the D4 representation language, N1N2'3^Z ~ 1+'+ ® 20'0 ® 20'0 ® 1+'+

N2'3N2'3PZ ~ 20'0 ® 20'0 ® 1+'- ® 1+'+ (3.15)

N2,3N2,3P'Z ~ 20'0 ® 20'0 ® 1-'- ® 1+'+

Hence, to obtain the desired D4 invariant, the tensor product between the D4 doublets should be 1+,+ for the first term, 1+,— for the second term and 1—,— for the last term. Thus, we obtain

SWm = A (k6(v1V3)aZ + A7 (V2V2 + V3V3)pZ + As (V2V2 - V3V3) p'z)

Assuming that

A7 = As, A6 = 2A7 and if we choose the VEVs of the flavons as

(P) = [p') = (a), with (a) = (va, 0)T we get the second matrix perturbation in Eq. (3.13)

'0 0 e SM = A | 0 e 0

, e 0 0,

e = A6

(Z )(a) A2

(3.16)

(3.17)

(3.18)

(3.19)

With this correction, the previous Majorana neutrino mass matrix MR gets deformed as

MR = A

/ A3U,

A5UX A

a5ux A

A3Un +A5UX

a5ux A

A3Vy +A5UX A

(3.20)

V A +e A

In order to extract the mixing matrix and the neutrino masses, we will parameterize MR in the following way

A3V„

a5ux A

which leads to

MR = A

c + e a + c

(3.21)

(3.22)

Notice that since the Dirac mass matrix mD is diagonal (see Eq. (3.4)), it does not affect the correction induced in the Majorana matrix MR, and by using type I seesaw mechanism formula mVff = mDM'—lmTD, we obtain the new neutrino mass matrix with elements given explicitly as

m0 / 2 2\

11 = — (^a + 2ac + c J

m22 = — (c2 + 2ce + e2) k

m0 2 m33 = - — (as - c ) k

m0 i 2\

m12 = m21 = —— ^ae + ce + ac + c j

(3.23)

m13 = m31 =

ce - ac - c2 + e2

m0 2 2

m23 = m32 =--(—a — ac + c + ec)

where k = a3 + la2c — ac2 — lace — 2c3 — c2e + Ice2 + e3 and m0 = This is a symmetric matrix that can be diagonalized by a similarity transformation like mdviag = U TmevffU. The system of eigenvectors and eigenvalues can be computed perturbatively; we find up to order

O(e2), the unitary matrix U which diagonalize the neutrino mass matrix m, given in terms of its eigenvectors as

sil 4a\[6 V3 1 3s 1

-3 -- -- +

V2 + 4aV22

+ °(s2)

(3.24)

V V6 V3 V2 '

consequently, the reactor and atmospheric angles develops into

Sin 013 =

sin 023 =

(3.25)

while the solar angle 6\2 maintain its TBM value; sin 6\2 = . It is easy to check that the matrix

U coincides with the TBM matrix in the limit e ^ 0. As for the eigenvalues of m, , they read up to order O(e2),

c+V a2 -as+s2

s+a+2c

(3.2l)

c+^j a2-as+s2

Using these masses, we calculate the solar and the atmospheric mass-squared differences

Am2 _ \m2 - 4 m0(3ae+3ce+6ac+3c^ "msol Am21 * 4(a—c)(a+2c)(ae—4ce+ac+a2—2c2)

c(s-2a)

Am^m = Am21 = 2m0 (a2-c2)(-2as+a2-c2)

(3.27)

Since the parameters a and c contribute to the tiny mass of neutrinos (see Eq. (3.26)), the VEVs vv and vx should be small and close to the cutoff un, vx < A which means that

|a| < 1, |c| < 1

(3.28)

3.2.1. Fixing a for allowed sin 0ij

Focusing on relations in Eq. (3.25), we fix the parameter of deviation e in the range of O(i0), and we use the experimental values of sin0ij given in Table 1; then, we plot in Fig. 1 sin 023 as a function of sin 013 in terms of the ratio a induced by the VEV of the singlet n. The values of the ratio | that are compatible with both sin 013 and sin 023 are shown in the left panel (right panel) of Fig. 1 within their 3a allowed range for the normal hierarchy (inverted hierarchy) case; see Table 1. We observe that for the left panel, the mixing angles 013 and 023 vary within the acceptable 3a ranges

0.138 < sin 013 < 0.161

(3.29)

0.626 < sin 023 < 0.638

Fig. 1. Left: sin $23 as a function of sin $13 with the relative parameter a shown in the palette. Right: The same variation as in the left panel but for inverted hierarchy. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

for the orange line which corresponds to

0-38 < - < 0.45

(3.30)

0.138 < sin 013 < 0.162 0.776 < sin 623 < 0.788

for the blue line which corresponds to

-0.45 < - < -0.38 a

As for the right panel of Fig. 1, the mixing angles 913 and 023 vary within the acceptable 3a ranges

0.139 < sin 013 < 0.144 0.634 < sin 023 < 0.637

for the orange line which corresponds to -

(3.31)

(3.32)

(3.33)

0-39 < - < 0.41

0.139 < sin 013 < 0.163 0.776 < sin 023 < 0.788

for the blue line which corresponds to -

-0.46 < - < -0.39 a

(3.34)

(3.35)

(3.36)

In order to get estimations of the parameter a, we plot in the left panel in Fig. 2 sin013 as a function of £ with the parameter a shown in the palette on the right while sin 023 is considered as an input parameter to get the value of the parameter a compatible with both mixing angles. We observe that the values of sin013 in the interval [0.138, 0.162] for e of ) corresponds to

-0.25 < a < -0.0007

(3.37)

Fig. 2. Left: sin $13 as a function of e with the relative parameter a shown in the palette. Right: The same as in the left panel but for sin $23 instead of sin $13.

while for values of sin013 in the interval [0.138, 0.161], we have

0.0003 < a < 0.25 (3.38)

Normally, the left panel in Fig. 2 is sufficient to obtain the allowed ranges of the parameter a because the intervals obtained in Eqs. (3.37)-(3.38) are compatible with both mixing angles 013 and 023, but the allowed range of the parameter a in the left panel provide us only the allowed values of sin 013. To extract the allowed ranges of sin 023 that are compatible with the ranges of the parameter a obtained in Eqs. (3.37)-(3.38), we plot in the right panel of Fig. 2 sin 023 as a function of e with the parameter a shown in the palette on the right while sin 013 is considered as an input parameter. We observe that the values of sin 023 in the interval [0.776, 0.788] corresponds to the range of the parameter a given in Eq. (3.37)

-0.25 < a < -0.0002 (3.39)

while for values of sin023 in the interval [0.626, 0.638], we have the range of a given in Eq. (3.38).

0.0004 < a < 0.25 (3.40)

3.2.2. Fixing c for allowed Amij

To fix the parameter c, we consider the second relation in Eq. (3.27) where we have two unknown parameters (namely m0 and c). Thus, we plot in Fig. 3 Am31 as a function of m0 with the parameter c presented in the palette on the right. In the left panel of Fig. 3, Am31 vary within its 3a allowed range for the normal hierarchy case; see Table 1. For the rest of the parameters of Eq. (3.27), we have earlier fixed the parameter e in the range of O(-1 ), and from Eqs. (3.37), (3.38), (3.39), and (3.40) we have fixed the parameter a in the interval [-0.25 : 0.25]. We also have restricted the parameter c in the range [-1 : 1] in Eq. (3.28). Gathering all these restrictions, we observe from the color palette in the left panel of Fig. 3 that c can take any value in the range [-1 : 1]—except the zero value which is easy to notice from the second relation in Eq. (3.27)). One can also see that for the 3a allowed range of Am31, the values of c close to zero—presented by the green light color—corresponds to the values of m0 close to zero, and as m0 increases—say m0 > 0.03 eV—the parameter c vary from large negative (blue-purple colors) to large positive values (orange-red colors).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.

Fig. 3. Left: Am31 [eV] as a function of mQ [eV] with the parameter c presented in the palette on the right for normal hierarchy. Right: same variation in the left panel but for inverted hierarchy. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

Fig. 4. Amu [eV] as a function of m0 [eV] with the parameter c presented in the palette on the right.

For the right panel, Am3i vary within its 3a allowed range for the inverted hierarchy case, and the parameters m0 and c vary in the same ranges as in the left panel. One can see approximately the same distribution of colors as in the left panel, and the only difference is the 3a allowed range of Am31. Now we consider the first relation in Eq. (3.27) to get the allowed ranges of m0 and c in the case of Am21, thus, we plot in Fig. 4 Am21 as a function of m0 with the parameter c presented in the palette on the right. We observe that range of the parameter c is reduced to

—0.6 < c < 0.6 and the range of the parameter m0 is reduced to 0.00018 eV < m0 < 0.77 eV

(3.41)

(3.42)

4. Charged fermions in SU(5) x D4 x U(1) model

In this section we give the invariant operators under SU(5) x D4 x U(1) that determine the mass matrices of the up-, down-quarks and the charged leptons. Moreover we add operator which contain the 45-dimensional Higgs in order to avoid the bad relation between the down quarks and the leptons Yd = predicted in the GUT scale. Recall that the mass matrices of the quarks and charged leptons can be embedded in the Yukawa couplings given by

10m.10M.5Ru D QLUcHu (4.1)

for the up-quarks type, and

10m.5m.5h, D QLdcHd + LecHd (4.2)

for the down-quarks and charged leptons.

4.1. Up quark sector

We start with the mass matrix of the up quark which originate from the up-type Yukawa couplings 10.10.5 = T.T.Hu. The leading order (LO) D4 x U(1) invariant superpotential giving rise to the mass matrix of the up quarks reads

Wup = ^ T1T1UH5 + ^ T2T2TH5 + ^ T3T3FH5 (4.3)

y1 y2 y3

y1 T1T1VH5 + y2 T2T2TH5 + y3: AAA

where y1, y2 and y3 are the Yukawa coupling constants and A is the cutoff scale of the model. The superpotential Wup decompose into the SM Yukawa couplings as follows

Wup = A (QL1 uc) VHu + A2 (QL2cc) THu + A3 (QL3tc) FHu (4.4)

When the flavon develop their VEVs as

<Q> = uq, <T> = ur, <F> = vF (4.5)

and the Higgs as usual <Hu> = vu, this leads to a diagonal up quark mass matrix given by

(AvQ 0 0 \

Mup = Vu

0 a vr 0 V 0 0 A UF/

where the eigen-masses are

y1UQ y2UT y3UF -

mu = Uumc = Uumt = Uu(4.7)

By using the experimental values of the up quark, the charm quark and the top quark masses as given by the Particle Data Group [39] namely mu ~ 2.3 MeV, mc ~ 1.275 GeV and mt ~ 173.21 GeV, and by taking the VEV uu « 174 GeV we obtain the following constraints

y1 uq « 1.32 x 10-5A

y2UT « 7.32 x 10-3A (4.8)

y3UF « 0.995A

Notice that if we assume the coupling constant y3 « O(1), the VEV uf should be close to the cutoff scale A in order to accomodate the numerical value of the top quark mass.

4.2. Down quark and charged lepton sector

The D4 x U(1) invariant superpotential generating the masses of the down quarks and charged leptons is given by

Wed = A4> T2F1QQH5 + A T1 (F2,3P) TH5 + AT3 (Fz,30) H5 (4.9)

where y4, y5 and y6 are the Yukawa coupling constants associated to the down quarks and charged leptons sector. The masses of the down quarks and charged leptons are generated from the same down-type Yukawa couplings (namely 10M-5M-5Hd) leading to the GUT mass relations

mb = mx , ms = mM , md = me (4.10)

which are acceptable for the third generation at the GUT scale but fails for the first and second generations due to their inconstancy with the experimental values; so the alternative relations which are much closer to the present data are the well known Georgie-Jarskog (GJ) [37] formulas given by

mb = mr , md = 3me , 3ms = mM (4.11)

These relations may be predicted by allowing additional couplings to the Higgs field that belongs to the 45-dimensional representation of SU (5). The Higgs H45 couple to operators T:F: and lead to different mass matrices of the down quarks and the charged leptons. Moreover, in additional to the GJ formulas, several relations between the down quarks and charged leptons are possible by considering Higgses that belong to different SU(5) representations [38]. In order to reproduce the difference between the charged lepton mass and the down type quark mass in our model, we introduce the 45-dimensional Higgs denoted as H45 which transform as non-trivial singlet under D4 flavor symmetry (namely H45~ 1+,-) as well as carrying the U(1) charge qU(\) = 10, this Higgs is antisymmetric and satisfy the following relations

(H^f = -(H-45)bf , (H^)f = 0

(H45);:5) = U45 , i = 1,2,3 (4.12)

((H4S)45) = -3U45

With respect to the invariance under SU (5) x D4 x U(1) symmetry model, the H45 Higgs can only combine with the operator given by

W45d = y7 T2 (F2,3<P) H455 (4.13)

Thus, the total superpotential of the down quarks and charged leptons reads as

Wed = || T2F1 ^H5 + ~y~2 T1 (F2,3cp) TH5 + y6 T3 (F2,30) H5 + y7 T2 (Fz,3^) H45

(4.14)

which becomes after performing tensor product under D4 as

Wed = T2F1 + T1 (F2P2 + F3P1) TH5 + y6T3 (F202 + F30x) H5

+ y7 T2 (F2P2 + F3P1) H45 (4.15)

• Down mass matrix

Using Eq. (4.14), the D4 x U(1) invariant superpotential of the down quarks in terms of the SM Yukawa couplings reads

Wd = (QL2dc) WHd + Qh (scP2 + bcn) THd + y6 QL3 (sc02 + Hd

+ y7 QL2 (scP2 + bccpx) h455 (4.16)

where Hd and ^55, are the doublet components of the SU (5) Higgses H5 and H45 respectively. Taking the VEVs of the Hd as usual—(Hd} = vd —and the flavons V and ^ as in Eq. (4.5), and assuming the VEVs of $ and p as

($} = (u$, 0)T , (p} = (0,up)T (4. 1 7)

the mass matrix of the down quarks is given by

0 Uda 0

Udß h 0

0 0 UdS

ß = ^4

a = y5

S = y6

h = y7

U45Uy A

(4. 1 8)

(4. 1 9)

• Leptons mass matrix

Using Eq. (4.14), the D4 x U(1) invariant superpotential of the charged leptons in terms of the SM Yukawa couplings reads

We = AI (LiMc) 22Hd + y52 (¿2V2 + L3V1) ecTHd

+ — (¿202 + ¿301) TcHd — 3 — (L2V2 + L3Ç1) ßch455 A A 45

(4.20)

As the flavons VEVs are the same as in the down sector, we find the following charged leptons mass matrix

0 udß 0 Me = I vda —3h 0 0 0 VdSt

(4.21)

where a, 3 and 8 are the same as in Eq. (4.19). Recall that the Higgs H45 contribute to the element 2-2 for both down quark and charged lepton mass matrices with the factor -3 in Me to differentiate between the two sectors, this factor is an SU(5) Clebsch-Gordan coefficient which come from the properties of the Higgs H45 given in Eq. (4.12). Diagonalizing the mass matrices Md and Me, the down-type quark masses are given by

2 h — h2 + 4Ud2aß

2h + Uh2 + Avjaß

22 y4y5 VdVrVQ

y7 U45A3 V45Vp

22 y4y5 VdVrVQ

(4.22)

mb = |udS| =

while for the charged leptons masses, we find

— 3 h + 9h2 + 4v2aß

— 3 h — U9h2 + 4Ud2aß

y4y5 22 VdVrUQ

3y7 V45A3

, U45U^ y4y5 VdvrvQ 3y7—:--+

3y7 U45A3

(4.23)

mt = \vdS\ =

Thus, the masses of the quarks and charged leptons of the first and the second family are successfully differentiated by 45-dimensional Higgs H45, and the GJ relations are guaranteed if we assume

h > uda & udß

(4.24)

To get the experimental values of down quark masses taking into account the GJ relation between the down quarks and charged leptons, we take several estimations of the mass parameters in (4.22). Taking into consideration the estimations assumed in the up quark sector (see Eq. (4.8)), to reach the numerical values of the down, strange and bottom quark masses as given by the Particle Data Group [39], namely md — 4.8 MeV, ms — 95 MeV and mb — 4.66 GeV, we assume that vd « 174 GeV and

= 12.45 x 104 GeV-1

y7y2y2 u45 ynu45uv = 90.2A MeV y6u^ = 2.67 x 10-2A

(4.25)

4.3. Quark mixing matrix

Regarding the mixing matrix of the quark sector, the unitary matrix that diagonalizing the up quark mass matrix is the identity matrix UUp = Iid since the up quark matrix obtained is diagonal (4.6), in the other hand, the down quark mass matrix (4.18) is diagonalized by the unitary matrix

UDown —

-h-J h2+4aßu2

-h+J h2+4aßu2

ßud 4+ \

h2+4aßu

h2+4qßi

,Jh2+4aßu ß^d

h2+4aßu ß^d

and consequently the total mixing matrix for the quark sector is given by

\uq\ =

-h-^J h2 +4aßuj

ßud 4^

-h+^J h2+4aßuj

ßud / h-V h2+4*ßv2Y H ^ )

( h+. ¡h2+4aßv 2

^ ßud d 0

Using the estimations in Eqs. (4.8)-(4.25) and assuming

-y' h2+4aßv2 ßÜl

a & ß ~ 12.6 x 10

(4.26)

(4.27)

(4.28)

we obtain the total quark mixing matrix as follows

\UQ I =

Uu pUDown

'0.9743 0.225 0' 0.225 0.9743 0 | (4.29)

which are reasonably close to the experimental values— | Uq | ~ | UCKMI, especially the elements \UudI, IUus|, |Ucd| and |Ucs|, while the zero mixing elements predicted in (4.29), have non-zero but small values comparing to the observed values given by [39]

/0.97427 0.22536 0.00355\ Uckm | = 0.22522 0.97433 0.0414 (4.30)

\0.00886 0.0405 0.99914 J

We end this section by noticing that spontaneous breaking of discrete symmetry leads in general to cosmological domain walls [40]. To avoid this problem, various scenarios have been proposed, the most common ones are either based on inflation ideas [43] or by using explicit symmetry breaking which is used in several models such as the minimally extended supersymmetric standard model (NMSSM) and string theory inspired prototypes [41,42]. The inflation based scenario might be a nice solution of domain walls problem for GUT models provided the inflationary scale is big; say around 0(1016) GeV [43]; at this scale, the topological defects are formed before the end of inflation. This is the case in our SUSY GUT model where the discrete symmetry D4 is broken by the flavon superfields getting their VEVs at the GUT scale, and consequently the domain walls are inflated away. Notice by the way that the greatest danger of domain walls arises for broken symmetry at lower scale as topological defects may occur after the inflationary stage. For example, in the model proposed in Ref. [44] with superpotential W (X) having Z„+3 as discrete symmetry, the domain walls problem occurs in the degenerate minima of W (X); and it has been suggested that the annihilation of such walls as due to a small deformation of the superpotential that breaks explicitly Z„+3 symmetry. This idea is realized by adding to W (X) a small deformation term 8W = aX linear in the chiral superfield X which breaks Z„+3 symmetry explicitly, for further details see [44].

5. Conclusion and numerical results

In this paper we have constructed a supersymmetric SU(5) x D4 x U(l)f GUT model providing a good description of quarks and leptons mass hierarchies and neutrino mixing properties. Besides the bosonic gauge field degrees of freedom and their superpartners described by vector superfields V valued in the Lie algebra of SU (5), the supersymmetric GUT model has also chiral superfields {$} that play a basic role in this construction; they can be classified into three kinds as follows:

(a) matter sector described by the generation superfields (Ti,Fi,Ni) carrying quantum numbers under the gauge symmetry as Ti ~ 10i, Fi ~ 5i and Ni ~ 1i; but also under the flavor symmetry Gf = D4 x U(1)f as in (2.7)-(2.8).

(b) Higgs sector described by the superfields (H5, H5, H45) transform under the gauge symmetry as H5 ~ 5H, H5 ~ 5H and H45 ~ 45H; and they carry as well non-trivial quantum number under Gf = D4 x U(1)f as in (2.8)-(2.12).

(c) Flavons sector described by eleven chiral superfields; they are scalars under SU(5) gauge invariance; but distinguished by quantum numbers under flavor symmetry Gf = D4 x U(1)f as shown on Tables (2.13)-(2.14).

The invariant chiral superpotential W($) of the model has twenty eight free parameters in which we need to fix eighteen in order to produce the approximative experimental values of the physical parameters in the quark and lepton sectors as given by tables reported below; see Tables (5.2)-(5.3) and Tables (5.5)-(5.9). The total superpotential W($) = Wch + Wchs of the model has a contribution Wch coming from the charged sector and another Wchs from the charge-less sector; they are as follows

Wch = Wup + Wed

Wchs = Wd + Wm + SWm .

where the superpotentials Wup and We,d of the charged sector are given in Eqs. (4.3)-(4.15) and the superpotentials of the chargeless sector WD, WM and SWM are given in Eqs. (3.2), (3.5) and (3.14).

Notice that the role of the discrete D4 dihedral group factor in the flavor symmetry Gf may be compared with the role of the alternating group A4 used in other SU (5) based GUT models building; see for instance [21]. Here D4 has been motivated by its natural description of /-t symmetry as well as by the wish to complete partial results in supersymmetric GUTs. The extra continuous global U(1) f invariance is necessary to control the superpotential W ($) of the GUT model and also to forbid higher dimensional operators that yields to rapid proton decay.

Among the key steps of this work, we mention the following ones: First, we have required a scale difference among the VEVs of the flavons T, ^ and F to fulfill the hierarchy among the three generations of up quarks. We then allowed for the presence of the flavon superfields y and 0 along with the flavons T and ^ used in the up sector, and the 45-dimensional Higgs in the down quarks-charged leptons sector in order to reconcile with the GJ relations which allow to distinguish between the two sectors. Next, we have studied the neutrino sector where the effective light neutrino mass matrix arise at LO through the type I seesaw mechanism; and by using the D4 representation properties, the Dirac mass matrix was found diagonal thus allowing the Majorana mass matrix to control the TBM matrix. Finally, in order to generate a non-zero reactor angle, we have added four extra flavon superfields to induce the deviation from TBM pattern.

We end this study by giving comments and a summary of the numerical results obtained in the charged and chargeless fermion sectors. As noticed before, our model involves in total twenty eight free parameters in which we need to fix eighteen to produce the approximative experimental values of the physical parameters in the quark and lepton sectors.

5.1. Numerical results

First we give numerical results for the chargeless sector; see Tables (5.2) and (5.3); then we turn to give numerical estimations of flavon VEVs that lead to masses of the quarks and charged leptons; see Tables (5.5)-(5.9).

5.1.1. Neutrino sector

The neutrino sector in our model involves fourteen free parameters in which we have fixed ten parameters to reproduce the experimental values of the physical parameters in the allowed

0.0515 .v 4 ' -s

0.0505

0.05 ■-.s*:

0.0495

0.0485

0.0475

i 0.009

0.8 0.6 0.0089

0.4 0.0088

0 o -0.2 -0.4 CM E < 0.0087 0.0086

-0.6 0.0085

-1 0.0084

O 0.05 0.1 0.15 0.2 0.25 0.3 0.35 m0

Fig. 5. Left: Amji [eV] as a function of rag [eV] and the parameter c presented in the palette on the right for NH with the parameters a, s, sin$13 and sin$23 as inputs. Right: same variation in the left panel but for Am2i [eV].

ranges. To produce the TBM pattern in the neutrino mass matrix as well as generating the nonzero reactor angle $l3, we have fixed six parameters by imposing the constraints in Eqs. (3.9), (3.17), (3.18). The four remaining parameters to fix (namely s, a, c and mo), come from the pa-rameterizations used in Eqs. (3.19)-(3.21). These four parameters are successfully confined to produce the physical parameters Amij and sin 0ij in the neutrino sector.

As we have mentioned in section 3, the parameter of deviation s is fixed in the range [0 : 0.I], while the parameter a is fixed as in Eqs. (3.38)-(3.39). In the other hand, the remaining two parameters c and m0 are fixed using the 3a allowed ranges of Am3l and Am2l (see Figs. 3-4).

As a final comment, notice that more precise ranges of the parameters c and m0 may be obtained if we consider their compatibility with the mixing angles sin $l3 and sin $23. We distinguish two cases as follows:

i) m0 and c for allowed Am3l, sin $l3 and sin $23

We plot in the left panel of Fig. 5 Am3l as a function of m0, with c presented in the palette on the right, while the 3a allowed ranges of sin $l3 and sin $23 are included as input parameters. This inclusion of the mixing angles has reduced the allowed values of m0 and c as can be seen in the left panel of Fig. 5. Since Am3l, sin$l3 and sin$23 depend also on the parameters a and s, their values get also restricted. To summarize, we take few examples of the allowed values of a and s that are compatible with the mixing angles sin $l3 and sin $23 and the parameters c, m0 and Am3l as shown in the left panel of Fig. 5 (see Table (5.2)).

Free parameters Observables

£ a c M0 [eV] Sin 013 sin 023 Affl3i [eV]

0.0647 0.149 -0.732 0.0434 0.153 0.630 0.0484

0.0906 0.214 -0.951 0.0542 0.149 0.632 0.0495

0.0801 -0.199 0.819 0.0350 0.142 0.778 0.0505

0.0566 -0.142 0.903 0.0493 0.140 0.777 0.0492

ii) m0 and c for allowed Am2l, sin $l3 and sin $23

We plot in the right panel of Fig. 5 the same as in the left panel but for Am2l instead of Am3l; hence, we repeat the same study as in the previous case, and we take a few examples of the allowed values of a and s that are compatible with the mixing angles sin $l3 and sin $23 and the parameters c, m0 and Am3l as shown in the right panel of Fig. 5 (see Table (5.3)).

Free parameters Observables

£ a c ®0 [eV] Sin 013 sin 023 Am21 [eV]

0.0958 -0.215 0.284 0.0062 0.157 0.785 0.00860

0.0969 -0.240 0.387 0.0143 0.142 0.778 0.00892

0.0779 0.193 -0.244 0.00179 0.142 0.635 0.00877

0.0824 0.207 -0.443 0.0222 0.140 0.636 0.00899

5.1.2. Quarks and charged leptons sectors

The quarks and charged leptons mass matrices in (4.6), (4.18), (4.21) involve in total fourteen free parameters that we collect hereafter

yi, y2, y3, y4, ys, y6, yi (54)

uq, vr, vF, U4s, a, p .

From these free parameters we need to fix eight of them in order to reproduce the phenomenolog-ical charged fermion masses by taking into account the GJ relations as well as the quark mixing matrix. The choice of the parameters is done in three steps as follows:

• In the up quark sector, we have fixed three parameters as in Eq. (4.8) to generate the phenomenological masses of the three up-type quarks. To have masses agreeing with experimental values taken from Ref. [39]

Observables Model values Experimental values

mu 2.3 MeV 2.3-0.5 MeV

mc 1.275 GeV 1.275 ± 0.025 GeV

mt 173.21 GeV 173.21 ± 0.51 ± 0.71 GeV

we need to fix the VEVs of the flavons Q, V and F as follows yi f « 1.32 x 10-s

y2 f « 1.32 x 10-3 (5.6)

y3 f « 0.995

• In the down quarks-charged leptons sector, besides Eq. (4.8) used in the up-quark sector, we have fixed four parameters as in Eqs. (4.24)-(4.25) to establish the numerical masses of the down quarks. To ensure the values

Down quarks Model values Experimental values

md 4.8 MeV 4.8-0.3 MeV

ms 95 MeV 95 ± 5 MeV

mb 4.66 GeV 4.66 ± 0.03 GeV

we have used the following

-L « 12.45 x 104 GeV-1

ny2y\ U45

y7 ^ « 90.2 MeV y6 % « 2.67 x 10-2 (5'8)

a « P h > vda

• In addition to Eqs. (4.8), (4.24), (4.25) used to generate the phenomenological masses of the charged fermions, we have also imposed a « P ~ 12.6 x 10-5 fixing one more parameter of the GUT model. This choice allowed us to obtain approximately the experimental values of the CKM elements I Uj I collected in following table

Observables Model values Experimental values

|Uud | 0.9743 0.97427 ± 0.00014

|Uu, | 0.225 0.22536 ± 0.00061

|Ucd | 0.225 0.22522 ± 0.00061

|U„ | 0.9743 0.97343 ± 0.00015

Uub | 0 0.00413 ± 0.000049

|Ucb| 0 0.0414 ± 0.0012

|Uib| 1 0.99914 ± 0.00005

|U„| 0 0.0405-0.0012

|U,d| 0 0.00886-0.00032

Appendix A. Proton decay in SU(5) x D4 x U(1) model

In this appendix we provide a discussion concerning the proton decay in our model SU(5) x D4 x U(1); it is organized into two sub-subsections: the first part concerns the usual 4 and 5 dimensional operators yielding to fast proton decay. The second part deals with those 7 and 8 operators induced by integrating out the colored Higgs triplets Au and Ad from the superpotential (4.3), (4.16).

A.1. Four and five dim operators leading to proton decay

We start by recalling that in SU(5) based GUT models, there are several baryon number violating terms leading to nucleon decay. The present experimental bounds come from Super-Kamiokand where the lower limit of lifetime for p ^ e+n0 is x(p ^ e+n0) > 1.4 x 1034 years and the lifetime limit for p ^ vK+ is obtained as 5.9 x 1033 years [45]. In supersymmetric SU(5) model, the dangerous proton decay terms arise from the dimension 4 and dimension 5 operators which have the form

10m.5m.5m ^ ^QLd(QhFdc) + Xudd (ucdcdc) + Xeii(ecLL)

10m. 10m. 10m.5m ^ ^qqql(QlQlQlL) + Xuude(ucucdcec)

Regarding the dimension 4 operators 10m.5M-5m, interaction processes involving violating lep-ton number term (QuLjdck) and the violating baryon number (u^d^) lead to rapid proton decay with family indices as i = 1, 2; j = 1, 2 and k = 2, 3. As we mentioned in subsection 2.2, the matter superfields Ti = 10m are assigned into the D4 representations l+,-, l+,-

and l+,+ respectively; while the Fi = I5m matter superfields are hosted by the D4 singlet l+,-and the D4 doublet 20,0. Therefore, the dimension 4 operators yielding to proton decay in SU(5) x D4 x U(l) model are given by

Tl-Fl-F2,3 , T2-Fl-F2,3 (A2)

Tl.F2,3.F2,3 , T2.F23-F2,3

The operator couplings in the first row of (A 2) are forbidden by D4 discrete symmetry while those of the second row are permitted . This feature may be exhibited by taking the tensor products of D4 representations . For Tl.Fl.F2,3 and T2.Fl.F2,3 we have l+,- ® l+,- ® 20,0 behaving as a doublet. The undesired couplings Ti.F2,3.F2,3 and T2.F2,3.F2,3 are eliminated by the global U(l) symmetry (see Table (A.4)). As for the dimension 5 operators 10m.10m.10m.5m which are given in the second line in Eq. (A.l) are generically generated via color triplet Higgsino exchange [48]. For instance, the following dimension 5 operators lead to rapid proton decay

TI.TI.T3.F2,3 , TI.TI.T2.F2,3 , T1.T1.T2.F1 (A.3)

The first two couplings in Eq. (A.3) are excluded by the D4 symmetry while the third one is invariant under D4, but is ruled out by the global U(l) symmetry since its charge is qu(\) = 45 and therefore is absent. The dimension 4 and 5 operators leading to rapid proton decay and suppressed by D4 symmetry and global U(l) are listed in the following table:

4- and 5-dim operators D4 invariance U(l)

Tl-Fl-F2,3 No 40

T2-F1-F23 No 35

Tl-F2,3-F2,3 Yes 40

T2 -F2,3-F2,3 Yes 35

Ti-Ti -T3-F23 No ll

Tl-Tl-T2-F2,3 No 45

Ti-Ti .T2.F1 Yes 45

Notice that in our SU(5) x D4 x U(l) model, there are also operators with dimension2 equal to 6 involving flavon superfields as

Ti-Fi-F2,3-v-& , Ti.F2,3.F2,3.p.^ , Ti-F^-F^-p'-^

and may lead to rapid nucleon decay; but can be eliminated by the usual R-parity [53]; this discrete symmetry is known to avoid all renormalizable baryon and lepton number violating operators such as Ti.Fj.Fk in SUSY models. Concerning operators of dimension 5 (A.3), their couplings with the flavon superfields to form operators of dimension 6 are forbidden by the

2 The 6-dimension operators are the highest dimensional couplings used in our model (except for the operators in Eq. (A.8) derived from the Yukawa superpotential), thus, we restrict our discussion concerning the higher couplings leading to fast proton decay to the 6 dimensional operators.

global U(1) symmetry. Finally, notice that the MSSM ^-term ^HuHd coming from the coupling between the SU(5) Higgses 5Hu and 5Hd is forbidden by the D4 discrete symmetry

A.2. More on proton decay suppression

Here we first examine the seven and eight dimensional couplings inherited from Wup and Wd superpotentials given by Eqs. (4.3), (4.16); these couplings are mediated by colored Higgsino triplet A and are relevant to nucleon decay after including the dressing procedure [54]. Then, we discuss the effect of the dressing through the exchange of charged winos W± and higgsinos h±.

• Operators mediated by colored Higgsino triplet First, recall that the minimal supersymmetric SU(5) GUT in the low scale SUSY suffers from several problems; and has been ruled out as it predicts a fast proton decay arising from the operators of dimension five which are mediated by colored Higgsino triplet A; these operators come from the Yukawa superpotential; see for instance [46,47]. In Ref. [47], after examining the RGEs for the gauge couplings at one loop, the mass of colored Higgs triplet is found to be MA < 3.6 x 1015 GeV which is less than the limit MA > 7.6 x 1016 GeV required to ensure the proton stability.

In our SU(5) x D4 x U(1) model, the operators mediated by the colored Higgsino triplet are inherited from the superpotentials Wup and Wd in Eqs. (4.3)-(4.16). These superpotentials, which have the same structure as homologue considered in [21], read in terms of colored Higgs triplets Au e H5 and Ad e H55 as follows

WUpp = J [QL1 Qh + ucec]^Au + J[QL2 QL2 + cVirAu

+ j[QL3 QL3 + tcrc]FAu (A.5)

Wd = J [QL2L1 + ccdc] QQAd + J [Qlj L2P2 + ucscV2] rAd

+ j Ql3L301 + tcbcfa] Ad (A.6)

Integrating out Au and Ad in Eqs. (A.5)-(A.6), the remaining operators relevant for nucleon decay are of dimension seven and eight as follows

W7,8 = [yjT (QL1 QL1 QL2L1 + ucecccdc)

+ jT (QL1 QL1 QL3L3 + ucectcbc)

+ jT (QL2 QL2 QL1L2 + oc^cucsc) (P2T2

+ jf (QL3 QL3 QL1L2 + tcTcucsc) P2rr] (A.7)

where MA is the colored Higgs triplet mass which is expected to be at the GUT scale; say O (1016). Notice that it is known in GUT literature that the Higgsino mediated proton decay is strongly associated with the so called "doublet-triplet splitting" (DTS) problem [50] on how the Higgs triplets Au and Ad acquire GUT-scale masses MA while leaving their doublet partners Hu and Hd with only weak-scale masses. Several ways have been proposed to resolve this problem

such as: (i) tuning the parameters in the Higgs superpotential, see for instance [49]; (ii) using the Missing Partner Mechanism [5l]; or (ii) using Double Missing Partner Mechanism [52]. In the present paper, the doublet-triplet splitting problem might be circumvented by using the Missing Partner Mechanism which is considered as the most used solution of DTS. The general idea of this mechanism relies on giving the colored Higgs triplet MA a mass involving additional Higgses sitting the 50, 50 and 75 representations of SU(5). We will not develop this issue here; we refer to literature where several papers using this approach have addressed this question; see for instance Ref. [21].

Returning to eqs. (A.7), the higher dimensional couplings in W7,8 may be exhibited by using the superfield assignments of SU(5) x D4 x U(l) model; we have

l yi y4 r 3

7174 Tl.Tl.T2.Fl.U3

[Ti-Ti-T3-F2,3-a4]

T2-T2-Ti-F2,3-<p.r2

MA A3 T3-T3-Tl-F2,3-(P-T-F]

By using Eqs. (4.8)-(4.25), it is clear that all the operators in the list (A.8) are highly suppressed by a factor proportional to

1 yiy4 <^>3 (A.9)

1 yiy6

1 y2y5

1 y3y5

MA A3 for the first coupling; and l yi y6

MA A6 <^> (A.l0)

for the second coupling; and

l y2 y5 o

<^><r>2 (A.ll)

for the third coupling; and finally

1 J3J5

toxrxn (A.12)

for the last coupling. Assuming the Yukawa couplings yi « O(1), the first coupling (A.9) is suppressed by 2^31q15 which is of order of 10-31 GeV-1; while the suppression of the remaining

couplings (A.10)-(A.12) are of order 10-23 GeV-1, 10-24 GeV-1 and 10-22 GeV-1 respectively. In what follows, we turn to study the contribution to proton decay coming from dressing diagrams with winos and higgsinos mediators.

• Dressing by higgsinos and winos exchange The dressing of dimension five proton decay operators via the exchange of gluino, charginos and neutralinos concerns the processus qq ^ Zq; and consists of converting the sleptons I and q squarks into leptons I' and quarks q'. In order for these operators to be relevant to proton decay, the bosons need to be transformed to fermions by a loop diagram through the gluino, neutralino,

Fig. 6. Dimension 5 operator diagram mediated by the colored Higgs triplet A. The superparticles (dashed lines) are transformed in particles via wino exchange. A similar diagram with higgsino exchange and others can also drawn; see appendix C of Ref. [56].

charginos dressing procedure; this leads to four-fermion interactions qqql and ucucdcec with baryon and lepton violating dimension six operators [55]. In SUSY SU(5) models, the dressing of the dimension five operators is studied in the limit where the dominant contribution to the qqql operator comes from a diagram with charged wino dressing while the dominant contribution to the ucucdcec operator arises from a charged higgsino dressing as illustrated in Fig. 6; see for instance Ref. [56] and the references therein.

In our SU(5) x D4 x U(1) model, the dressing of the operators QQQL and ucucdcec of (A.7) involves charged winos and higgsinos and an effective coupling depending on the flavon field VEVs. For clarity, we split the superpotential W7?s as the sum of two parts

Wy,s = WLs + WRs (A.13)

where the part WLS contains the operators of type QQQL coupled to flavons as follows

= Mk[Qli QLI QL2Li) + A6 (QLi QLi Ql3LB) Wi

+ ^ (Ql2 QL2 QLI L2) W2r2 + ^ (QL3 QL3 QLI L2) rF] (A.14)

and the WRs part contains the operators of type ucucdcec like

WRS = M- [^ (ucecccdc) to3 + A6 (ucectcbc) Qfa + Ap (cc^cucsc) ^2r2

+ {tcTcucsc) nr F] (A.15)

The two first operators in Eq. (A.14) are dressed by the charged winos and are significant for the decay mode p ^ K+v; this wino dressing contributes to the amplitude of nucleon decay with a factor proportional to

1 {a)J (A.16)

Ma A3 \m~hmq2 J

for the first operator and

1 y1y6,n\i,h\ ( mw \

Ma A2 \m?m

l3" '43,

for the second. The mw is the wino mass and m? and mq are the slepton and the squark masses respectively. If we take the masses of the sfermions and the charged winos as in Murayama and Pierce paper [47] (msf ~ O (1 TeV) and mw e [100,400] GeV), these extra contributions from

the ratio of the winos and superparticle masses are small and enhance the suppression of the factors in Eqs. (A.9)-(A.l0).

Regarding the first operator in Eq. (A.l5) which is dressed by charged higgsino is relevant to the same mode p ^ K + v, its contribution to the amplitude of nucleon decay is proportional to

yiy4 <^}3

(A.17)

where m-h is the charged higgsino mass and m-e and m-c are the masses of the selectron and the scharm respectively. Following [47], the mass of higgsino varies in the range m^ e [100, 1000] GeV; thus the ratio of the higgsino and the superparticle masses is also small and the contribution from charged higgsino dressing that arise in Eq. (A.l7) is also highly suppressed.

Appendix B. Dihedral group D4

0 0 0 l\ A 0 0 0

l 0 0 0 u_ 0 0 0 i

0 l 0 0 , b — 0 0 i 0

0 0 l 0 0 i 0 0

The Dihedral group D4 is a non-abelian discrete symmetry group generated by two non-commuting generators a and b obeying to the conditions a4 = b2 = e; they have the 4 x 4 matrix realization

The D4 discrete group consists of eight elements which could be classified in the five conjugacy classes as

Cl :{e}, C2 : {a, a3} , C3 : {a2} , C4 : {b,a2b}, C5 : jab,a3b}

It has five irreducible representations; four singlets l+,+, l+,-, l-,+ and l-,-, and one doublet 20,0 where the sub-indices on the representations refer to their characters under the two generators a and b as in the table

Xij e a b

Xl+,+ + l + l + l

xi+,- + l -l + l

Xl-,+ + l + l -l

xi-,- + l -l -l

X20,0 2 0 0

The Kronecker product of two doublets 2x = (xl, x2)T and 2y = (yl, y2)T in the D4 group is given by

2x x 2y = l+i,+i + l+i,-i + l-i,+i + l-i,-i , (B.4)

1 + 1, + 1 = xiy2 + X2yi,

1 + 1,-1 = xiyi + X2y2,

l —1, + 1 = xiy2 - X2yi,

l-l,-l = xiyi - X2y2,

and the singlets product are

1a,p x 1y,s = 1ay,ps with a,y,fi,8 = ±. (B.6)

For more details on the D4 Dihedral group see for instance Ref. [57].

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