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Fermion Mass Hierarchy from Nonuniversal Abelian Extensions of the Standard Model

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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 sponta- neous 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 exten- sion 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 015038-2 FERMION MASS HIERARCHY FROM NONUNIVERSAL … PHYS. REV. D 98, 015038 (2018) 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.
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