Fermion Mass Hierarchy from Nonuniversal Abelian Extensions of the Standard Model

PHYSICAL REVIEW D 98, 015038 (2018)

Fermion mass hierarchy from nonuniversal Abelian extensions of the standard model

† ‡ Carlos E. Diaz,* S. F. Mantilla, and R. Martinez Departamento de Física, Universidad Nacional de Colombia, Ciudad Universitaria, K. 45 No. 26-85, 111321 Bogotá D.C., Colombia

(Received 24 January 2018; published 31 July 2018) 1 A nonuniversal Abelian extension Uð ÞX free from chiral anomalies is introduced into the standard model (SM), in order to evaluate its suitability in addressing the fermion mass hierarchy (FMH) by using seesaw mechanisms (SSM). In order to break the electroweak symmetry, three Higgs doublets are introduced, which give mass at tree level to the top and bottom quarks, and the muon lepton. With an 1 electroweak singlet scalar field, the Uð ÞX symmetry is broken and the exotic particles acquire masses. The light particles in the SM obtain their masses via SSM and Yukawa couplings differences. Active neutrino masses are generated through inverse seesaw mechanisms (ISM). Additionally, the algebraic expressions for the mixing angles for quarks and leptons are also shown in the article.

DOI: 10.1103/PhysRevD.98.015038

I. INTRODUCTION the left-right model proposed by H. Fritzsch whose mass matrices have suited textures to understand the existence of The current phenomenological data in high-energy phys- three mass scales in the fermionic spectrum [8].C.Froggat ics is consistent with the existence of twelve fundamental and H. Nielsen presented a model in which the heaviest fermions divided into quarks and leptons with their masses fermions acquire mass through the VEV of the Higgs field, ranging from units of MeV to hundreds of GeV [1], and a while the lightest ones get massive through radiative cor- large gap until thousandths of eV according to neutrino rections by employing degrees of freedom heavier than the oscillation data [2,3]. Such particles are grouped in three SM particles [9]. Another remarkable way to understand chiral anomaly-free families under the gauge symmetries 3 ⊗ 2 ⊗ the hierarchy mechanism is to assume that light neutrinos of the standard model (SM) GSM ¼ SUð ÞC SUð ÞL 1 acquire their masses through radiative corrections [10]. Uð ÞY [4]. However, although the electroweak spontaneous symmetry breaking (SSB) accounts for the mass Similar methodologies are shown in Ref. [11] using multiple acquisition of fermions, it is not understood how fermion scalar fields to achieve FMH. As a special case, Y. Koide addressed the model in a geometrical shape, yielding the masses cover so many orders of magnitude despite there τ being only one vacuum expectation value (VEV) in the well-known Koide formula to obtain the lepton mass [12], current SM. Moreover, the hierarchy among mixing angles while Z. Xing analyzed the quark spectrum involving masses of quark and lepton observed in the Cabbibo-Kobayashi- and mixing angles of the CKM matrix [13]. Thereafter, extra Maskawa (CKM) [5] and Pontecorvo-Maki-Nakagawa- dimensions were introduced in new scenarios which yielded Sakata (PMNS) matrices [6], respectively, has not been different behaviors for each family so that their Yukawa well understood. This issue, called the “fermion mass coupling constants can account for the mass hierarchy [14]. hierarchy” (FMH) [7], is a motivation to extend the SM Similarly, Randall-Sundrum models offer scenarios were the by adding new particles or symmetries. delocalization of fermion wavefunctions between the – The FMH problem has been addressed from different Planck-brane and the TeV-brane in an anti de Sitter space points of view in order to achieve the simplest model beyond could account for FMH [15]. the SM. One of the first and most important schemes is Therefore, the confirmation of neutrino oscillations open new possibilities BSM. The smallness of neutrino masses is traditionally explained by the seesaw mechanism (SSM) *[email protected] † [16], which adds Majorana fermions N as right-handed [email protected] R ‡ μ 1014 [email protected] neutrinos whose masses are at N ¼ GeV such that SM neutrinos get light masses in accordance with exper- 2 Published by the American Physical Society under the terms of imental upper limits and square mass differences, Δm12 and the Creative Commons Attribution 4.0 International license. Δ 2 Further distribution of this work must maintain attribution to m3l, from oscillation data. However, such a shocking mass the author(s) and the published article’s title, journal citation, can be lower by including a second set of right-handed 3 and DOI. Funded by SCOAP . neutrinos νR which acquire mass at units of TeV so that the

2470-0010=2018=98(1)=015038(12) 015038-1 Published by the American Physical Society DIAZ, MANTILLA, and MARTINEZ PHYS. REV. D 98, 015038 (2018) inverse SSM (ISS) [17] can is implemented, and the because of the existence of fundamental scalar fields in Majorana mass turns out to be at units of keV Nature. In this way, the FMH can be understood in two Higgs μN ∼ 1 keV. Similarly, the large lepton mixings have been doublet models (2HDM) [20] and next-to-minimal 2HDMs addressed with new methods. N. Haba and H. Murayama (N2HDM) [21] and N3HDM [22]. Thus, this article is presented a model on neutrino masses through anarchic oriented in nonuniversal Abelian extensions of the SM ⊗ 1 mass textures, i.e., without any particular structure [18], GSM Uð ÞX whose charges are different among families, and discrete symmetries such as A4 were also employed to and its symmetry breaking is ensured by a Higgs singlet χ achieve fermion masses and mixings [19]. at TeV scale. As a consequence, chiral anomalies from Lastly, the detection of the Higgs boson has encouraged triangle diagrams could emerge, so it is important search for new schemes with extended scalar sectors and gauge groups solutions of chiral anomaly equations that cancel them all,

X X 3 2 1 → − ½SUð ÞC Uð ÞX AC ¼ XQL XQR XQ QX 2 2 1 → 3 ½SUð ÞL Uð ÞX AL ¼ XlL þ XQL l X Q X 2 2 2 2 2 ½Uð1Þ Uð1Þ → A 2 ¼ ½Y Xl þ 3Y X − ½Y X þ 3Y X Y X Y lL L QL QL lR LR QR QR l l X;Q X;Q 2 2 2 2 2 Uð1Þ ½Uð1Þ → A ¼ ½Yl X þ 3Y X − ½Yl X þ 3Y X Y X Y L lL QL QL R lR QR QR l l X;Q X ;Q ½Uð1Þ 3 → A ¼ ½X3 þ 3X3 − ½X3 þ 3X3 X X lL QL lR QR l l X;Q X;Q 2 1 → 3 − 3 ½Grav Uð ÞX AG ¼ ½XlL þ XQL ½XlR þ XQR l;Q l;Q

Such a requirement, together with nonuniversality, is eigenvalues and mixing angles. The model achieves in a satisfied by adding new isospin singlet exotic fermions natural way the fermionic mass hierarchy through seesaw T , J and E to the model. Furthermore, these new fermions mechanisms (SSM), with the existence of elements at might contribute to mass acquisition so that FMH can be two different orders of magnitude in a very special location obtained by avoiding unpleasant fine-tunings. Additionally, inside the mass matrix. the nonuniversality in the set of X charges which cancels The simplest example of the SSM comprises two the chiral anomalies could imply flavor violation processes fermions f and F coupled by two Higgs scalars ϕ1;2. − such as Bþ → Kþlþl [23] or h → μτ [24]. The Yukawa Lagrangian is The present article shows a nonuniversal Abelian extension which address FMH. First of all, Sec. II presents the −LY ¼ h1ffLϕ1fR þ h1F fLϕ1F R seesaw mechanism which employs the vacuum hierarchy F ϕ F ϕ F (VH) among scalars to yield algebraic expressions of þ h2f L 2fR þ h2F L 2 R; ð1Þ fermion masses which suggest the existence of lighter and heavier fermions than the original VEVs. After that, and the corresponding mass matrix after evaluating at the Sec. III introduces a nonuniversal model corresponding to VEVs is one solution of the chiral anomaly equations (1) with the corresponding analysis on the different hidden flavor h1fv1 h1F v1 symmetries and its consequences in mass acquisition MSSM ¼ : ð2Þ h2fv2 h2F v2 mechanisms. Then, Sec. IV employs special textures to get a suited fermionic spectrum which might account for † † The diagonalization may be done on either MM or M M. FMH. Lastly, the important features of each model are Both matrices give the mass eigenvalues presented and compared in Sec. V with some conclusions outlined at the end of the article. 2 2 2 h1 h2F − h2 h1F v v 2 ≈ ð f f Þ 1 2 mf 2 2 2 2 2 2 ; ð3Þ II. SEESAW MECHANISM ½ðh1fÞ þðh1F Þ v1 þ½ðh2fÞ þðh2F Þ v2 The majority of textures propose finite and null compo- 2 ≈ 2 2 2 2 2 2 nents of the mass matrices in order to get the suited mass mF ½ðh1fÞ þðh1F Þ v1 þ½ðh2fÞ þðh2F Þ v2: ð4Þ

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Regarding mixing angles, the matrix MM† yields the left- In both scenarios the left-handed mixing angle gets sup- handed mixing angle, while M†M yields the right-handed pressed by the specific VH between the Higgs doublets. one. They are given by On the contrary, the right-handed mixing angle results as the ratio among the Yukawa coupling constants of the h1 h2 h1F h2F v1v2 dominant VEV. θ ≈ ½ f f þ tan L 2 2 2 2 2 2 ; ð5Þ ½ðh2 Þ þðh2F Þ v − ½ðh1 Þ þðh1F Þ v The following sections show the Abelian extensions f 2 f 1 1 Uð ÞX with nonuniversal sets of charges and the fermion 2 2 h1 h1F v h2 h2F v mass acquisition in the quark and lepton sectors. θ ≈ f 1 þ f 2 tan R 2 2 2 2 : ð6Þ ðh1F Þ v1 þðh2F Þ v2 III. NONUNIVERSAL ABELIAN EXTENSIONS, There are some remarkable features in the expressions PARTICLE CONTENT AND FLAVOR obtained above. The first and most important is the SYMMETRIES suppression in the first eigenvalue of the matrix through the difference between Yukawa coupling constants which The model presented in this article is a nonuniversal are all assumed at the order one, and the seesaw with the Abelian extension of the SM, in which a new gauge symmetry Uð1Þ is added to the SM gauge group G . nonsuppressed second eigenvalue mF dividing such a X SM difference. Regarding mixing angles, the left-handed angle Such an extension is broken to the SM by an additional Higgs scalar singlet χ whose VEV vχ lies at TeV. Then, the θL is suppressed because of the VEVs, while the right- handed one does not, as it is shown below by choosing one electroweak SSB is done by three Higgs scalar doublets Φ 2 2 2 2 2462 2 of the two possible VHs: when v1

2 † 2 † 2 † 2 2 † † 2 † † 2 † † V ¼ μ1Φ1Φ1 þ μ2Φ2Φ2 þ μ3Φ3Φ3 þ μχχ χ þ μ12ðΦ1Φ2 þ Φ2Φ1Þþμ13ðΦ1Φ3 þ Φ3Φ1Þþμ23ðΦ2Φ3 þ Φ3Φ2Þ

f † † 2 † 2 † 2 2 † † þ pffiffiffi ðΦ Φ3χ þ H:cÞþλ11ðΦ Φ1Þ þ λ22ðΦ Φ2Þ þ λ33ðΦ Φ3Þ þ λχðχ χÞ þ 2λ12ðΦ Φ1ÞðΦ Φ2Þ 2 1 1 2 3 1 2 0 † † † † 0 † † † † 0 † † þ 2λ12ðΦ1Φ2ÞðΦ2Φ1Þþ2λ13ðΦ1Φ1ÞðΦ3Φ3Þþ2λ13ðΦ1Φ3ÞðΦ3Φ1Þþ2λ23ðΦ2Φ2ÞðΦ3Φ3Þþ2λ23ðΦ2Φ3ÞðΦ3Φ2Þ † † † þ 2ðλ1χΦ1Φ1 þ λ2χΦ2Φ2 þ λ3χΦ3Φ3Þð χ χÞ; ð9Þ

2 where the terms with μ23 break softly the discrete symmetry The other conditions for the electroweak VEV are Z μ2 μ2 1 2 and the ones with 12, 13 the Uð ÞX gauge symmetry softly [25]. After symmetry breaking the minimal condition χ 2 2 ˜ 2 ˜ 2 2 vχv3 for the VEV of the scalar field is given by v1 μ1 þ λ11v1 þ λ12v2 þ λ13v3 þ λ1χvχ þ f v1 2 2 2 2 2 2 2 v1v3 þ μ12v2 þ μ13v3 ¼ 0 μχ þ λχvχ þ λ1χv1 þ λ2χv2 þ λ3χv3 þ f ¼ 0: ð10Þ vχ 2 2 ˜ 2 ˜ 2 2 2 v2fμ2 þ λ22v2 þ λ12v1 þ λ23v3 þ λ2χvχgþμ12v1 2 þ μ23v3 ¼ 0 The VEV vχ breaks the symmetry beyond the SM, giving vχv1 masses to the exotic fermions. Therefore, since vχ ≫ 2 2 ˜ 2 ˜ 2 2 v3 μ3 þ λ33v3 þ λ13v1 þ λ23v2 þ λ3χvχ þ f v1;v2;v3, v3 2 2 þ μ23v2 þ μ13v1 ¼ 0: ð12Þ μ2 2 ≈− x vχ λ ð11Þ χ For v1, v2, v3, the constraints are given by

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2 2 2 2 vχ v3 μ λ˜ λ˜ λ Q Z2 1 þ 12v2 þ 13v3 þ 1χvχ þ f 2 1 3 ⊗ 2 2 3 2 v1 GFlavor ! SUð Þu ; SUð Þd ; ; ð17Þ v1 ≈− R R λ11 μ4 v2 v2 ≈ 12 1 breaking the universality in the up quark right-handed 2 μ2 λ˜ 2 λ˜ 2 λ 2 2 ð 2 þ 12v1 þ 23v3 þ 2χvχÞ sector and ensuring the complete acquisition of masses in μ4 2 the fermionic sector. 2 ≈ 13v1 v3 2 ˜ 2 ˜ 2 2 vχ v1 2 ; ð13Þ Section IV is focused on the fermion mass acquisition, ðμ3 þ λ13v1 þ λ23v2 þ λ3χvχ þ f Þ v3 mixing angles and the different mechanisms involved to λ˜ ≡ λ λ0 obtain the FMH. where ij ij þ ij. The condition for the soft symmetry 2 2 2 2 2 2 breaking reads jμ1j; jμ2j; jμ3j ≫ jμ12j; jμ13j; jμ23j, therefore v1 ≫ v2;v2. Comparing the second and third equations IV. MASS HIERARCHY ACQUISITION from (13), it can be seen that there is an additional factor vχ v1 2 As was mentioned in Sec. III, the model lacks of flavor f in the denominator for v3, which does not appear in v3 2 global symmetries. This feature implies that all fermions the expression for v . Consequently, the space of param- 2 acquire mass at tree level. eters permits naturally the assumption of the following VH The hadronic sector of the model contains the SM fields 2 2 2 T 1 T 2 v1 ≫ v2 ≫ v3: with four exotic chiral quarks: two uplike quarks , and two downlike quarks J 1, J 2. The leptonic sector of Lastly, the scalar sector, together with the Majorana mass the model contains the SM fields with three exotic chiral μ scale N , exhibits the VH: charged leptons E1, E2, E3 and three Majorana fermions pffiffiffi N N 1 N 2 N 3 R ¼ð ; ; RÞ. The nonuniversal quantum num- v1 ¼ 245.7 GeV ∼ mt 2;vχ ¼ 2.5 TeV; pffiffiffi bers and parities are shown in Table III. v2 ¼ 12.14 GeV ∼ m 2; μ ∼ 1 keV; The Yukawa Lagrangians under the symmetry Uð1Þ ⊗ b ffiffiffi N X p Z2 in the quark sector are v3 ¼ 250 MeV ∼ mμ 2: ð14Þ

This choice of VEVs of the Higgs doublets Φ1;2;3 plays a −L 11 1 Φ˜ 1 12 1 Φ˜ 2 13 1 Φ˜ 3 fundamental role in the model, because it sets the energy U ¼ h3uqL 3uR þ h2uqL 2uR þ h3uqL 3uR scale for the charm and top quarks and the muon. All other 11 1 ˜ 1 12 2 ˜ 2 21 2 ˜ 1 þ h q Φ2T þ h q Φ1u þ h q Φ1T SM particles acquire masses through suppression mecha- 2T L R 1u L R 1T L R nisms along with such introduced energy scales. 31 3 Φ˜ 1 33 3 Φ˜ 3 12T 1 χ 2 þ h1uqL 1uR þ h1uqL 1uR þ gχu L uR On the other hand, the fermionic sector comprises the SM 11 1 1 22 2 2 21 2 1 þ gχ T χ T þ gχ T χu þ gχ T χT fermions (including right-handed neutrinos νR) and an T L R u L R T L R T exotic sector composed by uplike quarks , downlike 22 T 2 χT 2 þ gχT L R þ H:c:; ð18Þ quarks J , charged leptons E and Majorana fermions N R. The exotic sector of model has two uplike quarks T 1;2,two downlike quarks J 1;2, three charged leptons E1;2;3 and three 1 2 3 Majorana masses N ; ; . All the exotic fermions are isospin −L 11 1 Φ 1 13 1 Φ J 1 22 2 Φ 2 R D ¼ h3dqL 3dR þ h3JqL 3 R þ h3dqL 3dR singlets, so they acquire mass through the Higgs singlet vχ. 23 2 Φ 3 32 3 Φ 2 33 3 Φ 3 The addition of the exotic sector not only ensures the þ h3dqL 3dR þ h2dqL 2dR þ h2dqL 2dR cancellation of chiral anomalies, which would not be 11 1 1 11 1 1 22 2 2 þ gχ J χd þ gχ J χJ þ gχ J χ J canceled by only SM fermions, but also contributes to the d L R J L R J L R 1 : :; mass acquisition mechanisms of fermions. The set of Uð ÞX þ H c ð19Þ charges is shown in Table II. Z Before acting 2, the quark sector exhibits the global 1 ⊗ Z symmetry and, under the symmetry Uð ÞX 2, the Yukawa Lagrangians in the lepton sector are Q G ¼ SUð2Þ 2;3 ⊗ SUð3Þ 1;2;3 ⊗ SUð2Þ 2;3 ; ð15Þ Flavor qL uR dR −L eμle Φ˜ νμ eτ le Φ˜ ντ μelμ Φ˜ νe while the lepton sector has the flavor symmetry N ¼ h2ν L 2 R þ h1ν L 1 R þ h2ν L 2 R μμ μ ˜ μ ττ ¯τ ˜ τ ei e C i L þ h l Φ1ν þ h l Φ3ν þ g ν χN G ¼ SUð2Þ 1;3 ; ð16Þ 1ν L R 3ν L R χN E R Flavor eR μ μ ν CM i N i gτi ντ CχN i which does not show any universal symmetry, so it does not þ R N R þ χN R R need breaking. After the action of the discrete symmetry, 1 N iC ijN j the quark global symmetry turns out to be þ 2 R MN R þ H:c:; ð20Þ

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−L ee e Φ e eτ ¯e Φ τ e2 ¯e Φ E2 U 1 2 3 T 1 E ¼ h3elL 3eR þ h3elL 3eR þ h3ElL 3 R ¼ðu ;u ;u ; Þ; μμ μ Φ μ τe τ Φ e ττ τ Φ τ u u; c; t; T1 : 22 þ h3elL 3eR þ h1elL 1eR þ h1elL 1eR ¼ð Þ ð Þ τ2 τ 2 11 1 1 13 1 3 þ h l Φ1E þ g E χ E þ g E χ E 1E L R χE L R χE L R The mass term in the flavor basis turns out to be 2eE2 χ e 2τ E2 χ τ 22 E2 χE2 þ gχe L eR þ gχe L eR þ gχE L R 2 2 31 3 1 33 3 3 −L U M U T T E χE E χE U ¼ L U R þ mT2 L R þ H:c:; ð23Þ þ gχE L R þ gχE L R þ H:c: ð21Þ 1 M The assignation of the Uð ÞX charges for the fermions where U is the uplike quarks mass matrix determines the X charges’ configuration for the scalar fields 0 1 in order to build the Yukawa Lagrangian in the fermion 11 12 13 11 h v3 h v2 h v3 h v2 sector. B 3u 2u 3u 2T C B 0 21 0 21 C For the ordinary fermionic sector, if only a Higgs doublet 1 B h1uv1 h1Tv1 C M ¼ pffiffiffi B C ð24Þ Φ1 with charge X ¼ 1=3 is regarded, then only two uplike U 2 B 31 0 33 0 C @ h1uv1 h1uv1 A quarks acquire masses, as can be seen from Eq. (18),or 0 12 0 11 in the following lines, from the mass matrix shown in gχuvχ gχTvχ Eq. (24). Every downlike quark remains massless and just one of the charged leptons obtains mass, as can be seen Since the determinant of MU is nonvanishing, all uplike from Eqs. (33) and (54), respectively. quarks acquire mass. Then, the mass eigenvalues corre- Φ Additionally, when a second Higgs doublet 2 with sponding with the SM quark masses are X ¼ 2=3 is introduced, a downlike quark acquires mass at tree level. In the quark sector, the doublets Φ1 and Φ2 give h11h33 − h13h31 2 v2 partial masses to the uplike and downlike quarks. It could 2 ≈ ð 3u 1u 3u 1uÞ 3 mu 31 2 33 2 2 ; be thought that there is a similarity with a two Higgs ðh1uÞ þðh1uÞ Z 21 12 21 11 2 2 doublet model type II, where the discrete symmetry 2 is ðh1 gχu − h1 gχ Þ v 1 m2 ≈ u T T 1 ; replaced by the charges X from Uð ÞX. In this model a c 12 2 11 2 2 ðgχuÞ þðgχTÞ parameter tanðβÞ¼v1=v2 ≈ mt=mb can be defined. 2 In order to obtain masses for the rest charged leptons, it is 2 31 2 33 2 v1 mt ≈ ½ðh1 Þ þðh1 Þ Þ ; ð25Þ necessary to add a third Higgs doublet Φ3 with X ¼ 2=3, u u 2 and in this situation two of this particles obtain masses. Moreover, with this addition, the rest downlike quarks and while the masses of the exotic uplike quarks are an uplike acquire masses. The seesaw mechanisms with exotic particles also 2 v generates fine-tunning via Yukawa coupling differences 2 ≈ 12 2 11 2 χ mT1 ½ðgχuÞ þðgχTÞ 2 from first order, which suppress some mass eigenvalues and 2 explain how the FMH is addressed with the Higgs doublet’s 2 22 2 vχ m ≈ ðg Þ : ð26Þ VEV spectrum and the singlet VEV. Furthermore, it is also T2 χT 2 possible to explain the mixing angles of quarks and leptons, as it was already shown in a similar model [22]. The corresponding left-handed rotation matrix can be Therefore, the X charges assignation for the scalar sector expressed by is directly derived from the structure of X charges in the fermionic sector, which gives a free anomaly Uð1Þ model. X V U ¼ V U V U ; ð27Þ The simplest scenario to obtain mass spectrum is to assign L L;SS L;B values to v1, v2 and v3 using the masses of top, bottom and muon, respectively. where the seesaw angle is Next, the Yukawa Lagrangians are evaluated at the 0 12 12 12 11 1 h gχ h g VEVs, yielding the mass matrices whose eigenvalues, ð 2u uþ 2T χT Þ v2 iðb −e Þ 12 2 11 2 e U U vχ as well as their mixing angles, are shown below. The B ðgχuÞ þðgχT Þ C U† B 21 12 21 11 C diagonalization procedure is shown in the appendix. Θ B ðh1 gχuþh1 gχ Þ C L ¼ u T T v1 iðcU−eUÞ ; ð28Þ @ 12 2 11 2 e A ðgχuÞ þðgχT Þ vχ A. Uplike quarks 0 The uplike quark sector is described in the bases U and V U u, where the former is the flavor basis while the latter is the while L;B diagonalizes only the SM-up quarks. Its angles mass basis are given by

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11 31 13 33 2 U;L h3uh1 þ h3 h1 v3 − 2 11 2 11 2 vχ tan θ ≈ u u u eiðaU dUÞ m ≈ g g 13 31 2 33 2 J1 ½ð χdÞ þð χJ Þ 2 ðh1uÞ þðh1uÞ v1 2 U;L vχ tan θ23 ≈ 0 2 ≈ 22 2 mJ2 ðgχJ Þ 2 : ð34Þ 11 12 − 12 11 U;L ðh2Tgχu h2ugχTÞ v2 − tan θ ≈ eiðbU cUÞ: ð29Þ 12 21 12 − 22 11 The corresponding left-handed rotation matrix is ðh2Tgχu h2ugχTÞ v1 V D V D V D 1 2 L ¼ L; L;B; ð35Þ The exotic species T and T got masses through vχ at SS units of TeV. The SM t quark has acquired mass with v1 1 2 where the seesaw angle which rotates out species J ; is without any suppression, so its mass remains at the scale of v1, hundreds of GeV. On the contrary, the c quark have 0 11 11 11 11 1 h3dgχdþh3J gχJ v3 11 2 11 2 acquired mass with v1 and v2 but yielding the suppressed ðg Þ þðg Þ vχ B χd χJ C ΘD† @ A mass of the c quark because of the existance of a SSM L ¼ 0 ; ð36Þ together with the T1 exotic quark. Finally, the u quark has 0 acquired mass through v3 with a similar SSM as for c quark, but with the t quark instead of T1. and the SM angles of V D are given by Consequently, the mass of the u quark gets suppressed L;B by the top quark t. D;L tan θ13 ≈ 0; h22h32 h23h33 B. Downlike quarks θD;L ≈ ð 3d 2d þ 3d 2dÞ v3 tan 23 23 2 33 2 ; h h v2 The downlike quarks are described in the bases D and d, ð 2dÞ þð 2dÞ D;L where the former is the flavor basis while the latter is the tan θ12 ≈ 0 ð37Þ mass basis The heaviest quarks J1 and J2 acquired mass at TeV scale D ¼ðd1;d2;d3; J 1Þ; due to vχ, while the b quark obtained its mass through v2 at d 1 units of GeV. The s quark has acquired its mass through v3 ¼ðd; s; b; J Þ: ð30Þ at hundreds of MeV with the corresponding SSM with the bottom quark b. The mass term in the flavor basis is Similarly, the quark d got its mass through the SSM with the exotic species J1. −L D M D J 2 J 2 D ¼ L D R þ mJ2 L R þ H:c:; ð31Þ C. Neutral leptons M where D turns out to be Neutrinos involve Dirac and Majorana masses in their N i 0 1 Yukawa Lagrangian. Since R are Majorana fermions, the 11 11 h v3 00h v3 bases are chiral and the mass basis describes Majorana B 3d 3J C B 22 23 C neutrinos. The flavor and mass bases are, respectively, 1 B 0 h3 v3 h3 v3 0 C M ¼ pffiffiffi B d d C ð32Þ D 2 B 0 32 33 0 C N νe;μ;τ νe;μ;τ C N e;μ;τ C @ h2dv2 h2dv2 A L ¼ð L ; ð R Þ ; ð R Þ Þ; 11 0011 n ν1;2;3 1;2;3 C ˜ 1;2;3 C gχdvχ gχJ vχ L ¼ð L ; ðNR Þ ; ðNR Þ Þ: ð38Þ

Thus, the mass eigenvalues of the SM quarks are The mass term expressed in the flavor basis is 1 11 11 11 11 2 2 −L NCM N ðh g − h g Þ N ¼ L N L; ð39Þ 2 ≈ 3J χd 3d χJ v3 2 md 11 2 11 2 ; ðgχ Þ þðgχJ Þ 2 d where the mass matrix has the following block structure ðh23h32 − h22h33Þ2 v2 m2 ≈ 3d 2d 3d 2d 3 ; 0 1 s 32 2 33 2 2 0 MT 0 ðh2dÞ þðh2dÞ ν 2 B T C M ¼ @ Mν 0 M A; ð40Þ 2 ≈ 32 2 33 2 v2 N N mb ½ðh2dÞ þðh2dÞ ð33Þ 2 0 MN MN and the masses of the exotic species are given by with Mν as the Dirac mass matrix between νL and νR

015038-6 FERMION MASS HIERARCHY FROM NONUNIVERSAL … PHYS. REV. D 98, 015038 (2018) 0 1 eμ eτ 0 h v2 h v1 The left-handed rotation matrix can be expressed by 1 2ν 1ν B μe μμ C Mν ¼ pffiffiffi @ h v2 h v1 0 A; ð41Þ 2 2ν 1ν V N V N V N ττ L ¼ L;SS L;B; ð48Þ 00h3νv3 where the seesaw angle is M the Dirac mass matrix between νC and N N R R 0 1 μe μ μμ μ eτ μ 0 1 h2ν GN4v2 N h1ν GN4v1 N h1νGN1v1 N 1 μ2 2 1 μ2 2 e1 2 2 1 2 3 e e g vχ e e e B gχN gN vχ gχN gN vχ ð χN Þ C gχN gχN gχN B C B C B μe μ μμ μ eτ μ C vχ B μ1 μ2 μ3 C ΘN† h2ν GN2v2 N h1ν GN2v1 N h1νGN4v1 N M ffiffiffi B C ¼ B μ2 2 2 μ2 2 2 e1 μ2 2 C ð49Þ N ¼ p gN gN gN ; ð42Þ L B ðg Þ vχ ðg Þ vχ g g vχ C 2 @ A @ N N χN N A τ1 τ2 τ3 ττ μ g g g h3νGN3v3 N χN χN χN 00τ3 2 2 ðgχN Þ vχ ffiffiffi μi p μi where g ¼ 2M =vχ, and MN ¼ G μN is the Majorana V E N N N and L;SM, contained in the block-diagonal mixing matrix mass of N . V E R L;B after rotating out the heavy species has the angles By employing the inverse SSM because of the VH in Eq. (14), it is found that eμ μe 2 θN;L ≈ h2νh2νv2 0 1 tan 13 eτ μμ 2 ; h1νh1ν v1 mν 00 μ μ2 † B C eτ e e1 V N M V N @ 0 0 A h1νh2νgχN gN GN4 v2 ð L;SSÞ N L;SS ¼ mN ð43Þ θN;L ≈ tan 23 τ 2 μ2 2 μμ 2 1 2 00 e − e v1 mN˜ ðh1νÞ ðgN Þ GN1 ðh1ν Þ ðgχN Þ GN2 μe 3 3 N;L h2νv2 where the new × blocks are [26] tan θ12 ≈ μμ : ð50Þ h1ν v1 MT MT −1 M −1M mν ¼ ν ð N Þ MN ð N Þ ν; ≈ M − ≈ M MN N MN ;MN˜ N þ MN : ð44Þ D. Charged leptons The charged leptons are described in the bases E and e, It was assumed M diagonal and N where the former is the flavor basis while the latter is the 0 1 mass basis GN1 GN4 0 B C G @ 0 A e μ τ 1 2 3 N ¼ GN4 GN2 ð45Þ E ¼ðe ;e ;e ; E ; E ; E Þ; 00 GN3 e ¼ðe; μ; τ;E1;E2;E3Þ: ð51Þ so that it can yield the adequate mixing angles to fit PMNS The mass term obtained from the Yukawa Lagrangian is matrix. By rejecting terms proportional to v3 in mν, the ν1 neutrino L turns out to be massless, the masses of the other −L ¼ E M E þ H:c: ð52Þ two neutrinos are E L E R M μe 2 2 where E turns out to be ðh Þ G 2 μ v ≈ 2ν N N 1 0 1 mν2 μ2 2 2 ; vχ ðgN Þ ee 0 eτ 0 e2 3 0 Bh3ev3 h3ev3 h3E v3 C eμ 2 2 B C ðh2νÞ GN1 μNv1 μμ ≈ B 0 h3 v3 00 0 0C mν3 e1 2 2 ð46Þ B e C ðgχN Þ vχ B τe ττ τ2 C 1 Bh1 v1 0 h1 v1 0 h1Ev1 0 C ffiffiffiB e e C ME ¼ p B 11 13 C and the masses of the exotic species are 2B 000gχEvχ 0 gχEvχ C B C B 2e 0 2τ 0 22 0 C e1 B gχevχ gχevχ gχEvχ C gχN vχ G 1μ @ A ffiffiffi N N 31 33 mN1 ¼ p ; 000 0 R 2 2 gχEvχ gχEvχ μ2 g vχ μ N ffiffiffi GN2 N ð53Þ mN2 ¼ p ; R 2 2 τ3 The determinant of ME is nonvanishing ensuring that all gχN vχ GN3μN 3 ffiffiffi charged leptons acquire mass. Thus, the eigenvalues of the mN ¼ p : ð47Þ R 2 2 mass matrix yields the masses of the SM leptons

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e2 ττ eτ τ2 2e e2 τe ee τ2 2τ eτ τe ee ττ 22 2 2 ½ðh3Eh1 − h3 h1EÞgχe − ðh3Eh1 − h3 h1EÞgχe þðh3 h1 − h3 h1 ÞgχE v m2 ≈ e e e e e e e e 3 ; e ττ 2e − τe 2τ 2 τ2 2e − τe 22 2 τ2 2τ − ττ 22 2 2 ðh1egχe h1egχeÞ þðh1Egχe h1egχEÞ þðh1Egχe h1egχEÞ 2 2 ≈ μμ 2 v3 mμ ðh3eÞ 2 ; ττ 2e − τe 2τ 2 τ2 2e − τe 22 2 τ2 2τ − ττ 22 2 2 2 ðh1egχe h1egχeÞ þðh1Egχe h1egχEÞ þðh1Egχe h1egχEÞ v1 mτ ≈ ; 54 2e 2 2τ 2 22 2 2 ð Þ ðgχeÞ þðgχeÞ þðgχEÞ and the masses of the new exotic charged leptons V. DISCUSSION AND CONCLUSIONS

2 The article describes a nonuniversal Abelian extension to 2 11 2 13 2 vχ the SM G ⊗ Uð1Þ to address FMH. The set of X m 1 ≈ ½ðgχEÞ þðgχEÞ ; SM X E 2 charges, shown in Table I, is a solution of the chiral 2 vχ anomaly equations (1) with different exotic sectors, com- 2 ≈ 2e 2 2τ 2 22 2 mE2 ½ðgχeÞ þðgχeÞ þðgχEÞ 2 ; posed by uplike T and downlike quarks J , charged leptons 1 2 2 E and Majorana fermions N . The model contains two T ; , 31 33 vχ m2 ≈ ½ðg Þ2 þðg Þ2 : ð55Þ J 1;2 E1;2;3 N 1;2;3 E3 χE χE 2 two ,two and three R . All of them acquire masses at TeV scale through the VEV vχ. The left-handed rotation matrix can be expressed by The t quark acquires mass through v1 without any kind of suppression, so its mass remains at hundreds of GeV. On c V E ¼ V E V E ; ð56Þ the contrary, the quark mass turns out suppressed because L L;SS L;B of the SSM involving the c quark and the exotic species T 1 present in the mass matrix M , given by the sub-block where the seesaw angle is U 0 ee 2e eτ 2τ e2 22 1 21 21 h3egχeþh3egχeþh3E gχE v3 − 1 h1 v1 h1 v1 0 iðaE eEÞ 0 u T 2e 2 2τ 2 22 2 e ffiffiffi ; B ðgχeÞ þðgχeÞ þðgχE Þ vχ C p 12 11 2 gχ vχ gχ vχ B μμ 2μ C u T † B h gψ C E 3e e iðbE−fEÞ Θ ¼ B 0 2e 2 2τ 2 22 2 e 0 C; ð57Þ L B ðgχeÞ þðgχeÞ þðgχE Þ C @ A τe 2e ττ 2τ τ2 22 which yields the suppression of the c quark mass, h1 gχeþh1 gχeþh3E gχE e e iðcE−eEÞ 0 2e 2 2τ 2 22 2 e 0 ðgχeÞ þðgχeÞ þðgχE Þ 21 12 21 11 2 2 ðh1 gχu − h1 gχ Þ v V E V E m2 ≈ u T T 1 ; and L;SM, contained in L;B has the mixing angles c 12 2 11 2 2 ðgχuÞ þðgχTÞ ee τe eτ ττ e2 τ2 E;L ðh3eh1e þ h3eh1e þ h3Eh1EÞ v3 − tan θ ≈− eiðaE cEÞ 13 τe 2e ττ 2τ τ2 22 from hundreds to units of GeV since it acquires mass via v1, ðh1 gχe þ h1 gχe þ h1EgχEÞ v1 e e in accordance with experimental observations. Similarly, E;L tan θ23 ¼ 0 the u quark mass is suppressed by the SSM involving u and E;L t in the following sub-block of the uplike quarks mass tan θ12 ¼ 0 ð58Þ matrix, The exotic charged leptons E1;2;3 have acquired mass at TeV scale. The heaviest SM lepton τ acquired a supressed TABLE I. Scalar content of the model, nonuniversal X quantum Z mass at GeV scale through v1 so that it does not acquire number and 2 parity. mass at hundreds of GeV, but at units of GeV. The lepton μ Scalar doublets X Scalar singlets X has acquired mass through v3 without any suppression, so þ þ ξχ þvχ þiζχ þ ϕ1 þ1=3 χ pffiffi −1=3 its mass remains at hundreds of MeV. Finally, the lightest ¼ 2 Φ1 h1 v1 iη1 ¼ þpffiffiþ lepton, e, got its mass through a Yukawa suppression. 2 þ − Summarizing, the FMH is achieved by the implementa- ϕ2 þ2=3 Φ2 ¼ h2 v2 iη2 tion of the VH together with the mass matrices obtained þpffiffiþ 2 þ þ from the Yukawa Lagrangian, whose terms are constrained ϕ3 þ2=3 1 Z Φ3 ¼ h3 v3 iη3 by the nonuniversal Uð ÞX gauge and 2 discrete sym- þpffiffiþ metries. The fermion masses are outlined in Table III. 2

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TABLE II. Particle content of the Abelian extensions, nonuni- TABLE III. Summary of fermion masses. versal X quantum number and Z2 parity for the model. Family Fermion Fermion Left-handed Right-handed X X SM uplike quarks SM downlike quarks

2 0 2 2 0 2 SM quarks h −h v3 h −h 1 u u u ffiffi d d d v3ffiffi p 1 p 1 ht 2 h 2 1 0þ 2=3þ J 1 u uR þ 2− 0 2 2− 0 2 q − 2 c hc hc v1ffiffi s hs hs v3ffiffi L ¼ 1 2 2=3 p p d L uR þ hT1 2 hb 2 − 3 h v1 h v2 2 1 3 2 3þ 3 t pt ffiffi b pb ffiffi 2 u þ = uR þ = 2 2 qL ¼ 2 1 −2 3þ d L dR = 3 þ 2 − SM neutral leptons SM charged leptons 3 u þ1=3 d −1=3 R 1 2 2 0 2 002 0002 0 q ¼ 3 3 − ν μN v 2 ðh −h Þh 2−ðh −h Þh v3 L −1 3 1 3 e e e E e e E2 pffiffi d dR = L 2 2 hν1 2 0 2 L ðhN3Þ vχ hτ −hτ 2 2 μ 2 hμvffiffi3 SM leptons 2 ν N v1 2 μ p L 2 2 hν2 2 ðh 2Þ vχ e N νe −2 3þ ν 1 3þ 3 μ 2 2− 0 2 = R þ = ν N v1 2 τ hτ hτ v1 le μ 3 L pffiffi ¼ − 2 2 hν3 h 2 2 L e ν 0 ðhN1Þ vχ E e L R νμ −1 3− ντ −1 3þ lμ = R = Exotic uplike quarks Exotic downlike quarks L ¼ μ e −4 3þ 1 1 e eR = 1 T hT1ffiffivχ J hJ1ffiffivχ L μ p p ντ −1þ e −1− 2 2 τ R 2 h 2vχ 2 h 2vχ l ¼ τ τ 2 T pT ffiffi J pJ ffiffi L e −4=3þ 2 2 e L R Non-SM quarks Exotic neutral leptons Exotic charged leptons 1 − 1 − 1 h 1vχ 1 h 1vχ T 1 3 T 2 3 1 N Npffiffi E pE ffiffi L þ = R þ = R 2 2 2 − 2 − 2 2 T 1 T 4=3 hN2vχ hE2vχ L þ R þ 2 NR pffiffi E pffiffi 1 1 2 2 J −1 3þ J −2 3þ 3 3 L = R = hN3vχ hE3vχ 3 NR pffiffi E pffiffi J 2 0þ J 2 1 3þ 2 2 L R þ = Non-SM leptons E1 1− E1 4 3− L þ R þ = E2 −1þ E2 −4 3þ L R = which gives the suppressed mass of the d quark E3 5 3− E3 4 3− L þ = R þ = 1 3 11 11 − 11 11 2 2 N ; 0þ ðh3J gχ h3 gχJ Þ v3 R m2 ≈ d d ; Majorana fermions N 2 0− d 11 2 11 2 2 R ðgχdÞ þðgχJ Þ

so that the d quark does not acquire mass at hundreds, but at 11 13 1 h v3 h v3 units of MeV in accordance with phenomenological data. ffiffiffi 3u 3u p 31 33 ; The model suppresses the mass of the s quark with the b 2 h v1 h v1 1u 1u quark because of the SSM obtaining the mass of the u quark through v3 with the 22 23 subtraction of Yukawa coupling constants 1 h3 v3 h3 v3 pffiffiffi d d 2 32 33 11 33 − 13 31 2 2 h2dv2 h2dv2 2 ≈ ðh3uh1u h3uh1uÞ v3 mu 31 2 33 2 2 ; ðh1uÞ þðh1uÞ yielding the mass eigenvalue of a light s quark lowering the mass from hundreds of to units of MeV. 23 32 − 22 33 2 2 Moreover, the top quark obtains its mass at the scale of 2 ðh3dh2d h3dh2dÞ v3 ms ¼ 32 2 33 2 2 : hundreds of GeV directly via v1 with no suppression ðh2dÞ þðh2dÞ mechanism: Within this SSM, the bottom quark acquires its mass v2 m2 ≈ 31 2 33 2 1 directly throught v2 t ½ðh1uÞ þðh1uÞ Þ 2 : v2 The d quark obtains its mass through v3 and an SSM 2 ≈ 32 2 33 2 2 mb ½ðh2dÞ þðh2dÞ with the exotic species J1, 2 11 11 For the neutral sector, light active neutrinos and two- 1 h3dv3 h3J v3 pffiffiffi ; folded sterile heavy neutrinos at TeV scale are obtained 2 11 11 gχdvχ gχJ vχ with the employment of ISS. Moreover, the model selects

015038-9 DIAZ, MANTILLA, and MARTINEZ PHYS. REV. D 98, 015038 (2018) the normal ordering, with the lightest active neutrino ACKNOWLEDGMENTS 1 ν turning out massless when the smallest VEV v3 is L This work was supported by the El Patrimonio neglected. Autonomo Fondo Nacional de Financiamiento para la The model predicts the nonsuppressed mass of the μ pffiffiffi Ciencia, la Tecnología y la Innovación Francisco Joséde μμ 2 given by mμ ¼ h3ev3= , which turns out at hundreds of Caldas program of COLCIENCIAS in Colombia. MeV because of v3. Furthermore, the charged lepton sector τ shows the largest SSM in the article involving e, and the APPENDIX: GENERAL SCHEME FOR 2 exotic species E , DIAGONALIZATION OF MASS MATRICES 0 1 ee eτ e2 The fermions of each sector are described in two bases: h3 v3 h3 v3 h3E3v3 1 e e The flavor basis F and the mass basis f. The mass matrix B τe ττ τ2 C pffiffiffi @ h1 v1 h1 v1 h1Ev1 A M f 2 e e F written in the basis is found in the diagonal form, 2e 2τ 22 gχevχ gχevχ gχEvχ having as diagonal elements the masses of the fermions. By the implementation of spontaneous rupture of symmetry, L which gives the simultaneous suppression of the e and τ the Yukawa Lagrangian F can be expressed as masses, from hundreds to units of MeV and GeV, respec- 2e −L F M F H:c: A1 tively. By setting gχe null to simplify algebraic expressions, F ¼ L F R þ ð Þ the masses of the e and τ turn out to be The diagonalization of the mass matrix is made by a eτ τe ee ττ 22 e2 τe ee τ2 2τ 2 2 biunitary transformation, which reads ½ðh3 h1 − h3 h1 ÞgχE − ðh3Eh1 − h3 h1EÞgχe v m2 ≈ e e e e e e 3 ; e τ2 2τ − ττ 22 2 2 ðh1Egχe h1egχEÞ V F †M V F ð LÞ F R: ðA2Þ τ2 2τ − ττ 22 2 2 ðh1Egχe h1egχEÞ v1 2 ≈ F F mτ 2τ 2 22 2 ; The unitary matrices V and V relate the mass and the ðgχ Þ þðg Þ 2 R L e χE flavor bases for both chiralities It is remarkable how the exotic E2 suppresses the τ mass F ¼ V Ff F ¼ V Ff : ðA3Þ from hundreds to units of GeV, and in turn it suppresses the L L L R R R mass of the e from hundreds to units of MeV, as it is shown However, MF is not a symmetric matrix and two unitary in the above expressions. matrices have to be found. An alternative way is to diago- A summary of the fermion masses obtained from the nalize the symmetric constructions from M , which are model is shown in Table I. The nonuniversal Abelian F extensions can be considered one of the simplest schemes † † † M M → ðV FÞ M M V F ðA4Þ beyond SM because it only comprises one Abelian gauge F F L F F L group Uð1Þ . However, they give rich frameworks where † † X M M → V F †M M V F; A5 fundamental issues such as fermion mass hierarchy can be F F ð RÞ F F R ð Þ addressed with the suited particle content and couplings. where the diagonal form is found only by the mixing of left- Moreover, this article shows how previous schemes [22] M M† can be improved to avoid radiative corrections or fine- handed fermions, in the case of F F, and by the mixing M† M tunings and obtain in a natural way light and heavy of the right-handed fermions, for the matrix F F (for M fermions in accordance with experimental data. neutrinos the mass matrix N is already symmertic.). Both Regarding the global symmetries which appear from the quadratic mass matrices lead to the same eigenvalues, that 1 correspond with the square of the masses. Since the main assignation of Uð ÞX quantum numbers in Table III, the discrete Z2 breaks them in the following scheme, goal is to obtain the values of the fermion masses, only the M M† matrix F F was diagonalized, giving also the Yukawa 2 2 3 ⊗ 3 1 2 3 ⊗ 2 2 3 ⊗ 2 1 3 SUð Þq ; SUð Þu ; ; SUð Þd ; SUð Þe ; mixings of the left-handed fermions. The first step is to notice L R R R Msym ≡ M† M Z2 that the matrix F F F can be written in the ! SUð2Þ 1;3 ⊗ SUð2Þ 2;3 ⊗ SUð2Þ 1;3 ; ð59Þ uR dR eR following block form: which prevents the existence of massless fermions after Mf MfF Msym 3×3 3×n the SSB. F ¼ ; ðA6Þ MFf MF On the other hand, it is important to reinforce that the set n×3 n×n of chiral anomaly-free Uð1Þ quantum numbers constrains X fF T 1 where n is the number of exotic fermions and ðM3 Þ ¼ the set of Uð ÞX quantum numbers in the scalar sector in ×n MFf order to understand the masses and mixing angles observed n×3. Thus, a seesaw rotation was applied to separate the in the fermionic spectrum of the SM. SM from the exotic sector [27]

015038-10 FERMION MASS HIERARCHY FROM NONUNIVERSAL … PHYS. REV. D 98, 015038 (2018) F† 1 Θ The matrices Rij are complex rotations that read V F ¼ L ðA7Þ L;SS −ΘF 1 0 1 L ; F F c12 s12 0 B C ΘF Mf −1MFf θF δF F F with L ¼ð Þ . This leaves the matrix in a block- R12ð 12; 12Þ¼@ −s12 c12 0 A; diagonal form, as was required, 001 0 1 sym F 0 F m 03 c13 s13 F T sym F F;SM ×n ðV Þ M V ¼ ; ðA8Þ F F B C L;SS F L;SS 0 sym R13ðθ13; δ13Þ¼@ 010A; n×3 MF;exot F F −s13 0 c13 sym ≈ Mf MfF MF −1MfF 0 1 where mF;SM þ ð Þ is the mass 10 0 sym ≈ MF B C matrix for the SM and MF;exot the one for the exotic F F F F R23ðθ ; δ Þ¼@ 0 c23 s23 A; ðA11Þ sector. 23 23 − F F Then, a diagonalization for each sector was performed, s23 c23 summarized in the following matrix: F θF F θF δF where cij ¼ cos ij and sij ¼ sin ij exp ði ijÞ. The angles F 0 F V 3×n θ are determined, in the calculus, from their tangents in an V F ¼ SM ; ðA9Þ ij B 0 F approximate way. Thus, the unitary transformation that n×3 VExot leaves the symmetric mass matrix for the left-handed F where VSM is expressed by fermions in a diagonal form is

F θF δF θF δF θF δF V F V F V F VSM ¼ R13ð 13; 13ÞR23ð 23; 23ÞR12ð 12; 12Þ: ðA10Þ L ¼ L;SS L;B: ðA12Þ

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