LEPTON MASSES IN A NON UNIVERSAL U1 MODEL WITH THREE FAMILIES C. Cortes-Parra R. Martinezand J. S. Alvarado

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LEPTON MASSES IN A NON UNIVERSAL U(1) MODEL WITH

THREE FAMILIES

C. Cortes-Parra*, R. Martinez†and J. S. Alvarado‡

Departamento de Física, Universidad Nacional de Colombia, Carrera 30 No. 45 −03, Bogotá D.C, Colombia

October 27, 2022

ABSTRACT

We present an extension

U(1)X

to the Standard Model that reproduces the lepton mass

structures determined by the experiments. In the charged sector, we introduced effective

operators of dimension

n= 7

to generate the mass of the electron, which is null at tree-level

due to the

charge. In the neutral sector, we added three sterile right-handed neutrinos and

three Majorana neutrinos to generate the mass structure for the left-handed neutrinos, by

the inverse seesaw mechanism. The model free parameters were ﬁtted with the known mass

eigenvalues

me, mµ, mτ

, and with the most recent results of a global analysis of the data

from neutrino oscillation. From the adjustment of the free parameters, we obtained allowed

regions for the Yukawa coupling set.

1 INTRODUCTION

Recently various experiments such as Super-Kamiokande [

], SAGE [

], MINOS [

] and Double Chooz [

have conﬁrmed neutrino oscillation using different neutrino sources (solar, atmospheric, of reactors and

accelerators). Experimental data indicate that active neutrinos of the Standard Model (SM) have mass and

their ﬂavor ﬁelds are given by a combination of mass eigenstates (species)

|νa

Li=X

i=1,2,3

Uai νi

L, a =e, µ, τ (1)

where

U≡UPMNS = (Vl

L)†Vν

is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [

]. The matrices

and

Vν

diagonalize the mass matrix of charged leptons and active neutrinos, respectively. In general,

if neutrinos are Dirac particles the PMNS matrix can be parameterized in terms of three mixing angles

θ12, θ13 θ23 and one CP violating phase δ[7]:

UPMNS =



c12c13 s12c13 s13e−iδ

−s12c23 −c12s23s13eiδ c12c23 −s12s23s13eiδ s23c13

s12s23 −c12s23s13eiδ −c12c23 −s12s23s13eiδ c23c13 

,(2)

with

sij = sin θij

and

cij = cos θij

. If neutrinos are Majorana particles, the PMNS matrix includes two

additional phases (α, β) but they have no inﬂuence on neutrino oscillation.

*camacortespar@unal.edu.co

†remartinezm@unal.edu.co

‡jsalvaradog@unal.edu.co

arXiv:2210.14407v1 [hep-ph] 26 Oct 2022

PREPRINT - OCTOBER 27, 2022

All data analyzed by the experiments mentioned and others can be described consistently by means of two

non-equivalent arrangements for mass eigenvalues [8]:

Normal Ordering: m1< m2< m3,(∆m2

32 '∆m2

31 >0),(3)

Inverted Ordering: m3< m1< m2,(∆m2

32 '∆m2

31 <0),(4)

where the squared-mass differences are

∆m2

ij =m2

i−m2

j(i, j = 1,2,3)

. From the latest global analysis

reported by Esteban et al. [

], experimental values of squared-mass differences are listed in Table 1.

Esteban et al. also determined upper and lower limits for each component of the PMNS matrix

|U|SK,3σ

PMNS =



0.801 →0.845 0.513 →0.579 0.143 →0.156

0.244 →0.499 0.505 →0.693 0.631 →0.768

0.272 →0.518 0.471 →0.669 0.623 →0.761

.(5)

Normal Ordering (NO) Inverted Ordering (IO)

∆m2

10−5eV26.82 →8.04 6.82 →8.04

∆m2

10−3eV22.430 →2.593 −2.574 → −2.410

Table 1: Squared-mass differences at

3σ

reported by Esteban et al., using Super Kamiokande data.

l= 1

for

NO and l= 2 for IO.

Seeing this, what theoretical frameworks can we use for the explanation of the neutrinos small mass and

neutrino oscillation? The most studied method that answers our question is the seesaw mechanism, which

adds right-handed neutrinos to the SM to give mass to left-handed neutrinos. Since the new energy scale

associated with the new ﬁelds is high (

∼1014 −1016

GeV [

]), the seesaw mechanism can not be tested by

the experiments. Therefore in this model we use the inverse seesaw mechanism (ISS) which adds very light

right-handed Majorana neutrinos to the SM, such that in the basis

(νL, νC

R, NC

the mass matrix has the form:

Mν=



0mT

ν0

mν0mT

0mNMN

,(6)

where the matrix block

has component of the TeV scale order,

is in KeV scale and

mν

is in the

electroweak scale. In this way the active neutrinos are obtained in sub-eV scale.

On the other hand, we can use the effective ﬁeld theory as a theoretical framework to explain some

experimental results. In this scenario we propose a dimensional expansion in the Lagrangian of the theory

L=L0+L1

Λ+L2

Λ2+··· ,(7)

where conventional renormalizable interactions are considered,

, and non-renormalizable interactions are

added,

(

n≥1

) [

], which are described by operators of

n+ 4

dimension suppressed by the new physics

energy scale

Λn

. These effective operators add high-energy effects that can be measured on a low-energy

scale.

In this work we use a next-to-minimal two Higgs double model (N2HDM) [

], which adds elementary

particles to the SM under a new symmetry

U(1)X

. This symmetry is one of the most studied of the SM, as can

be seen in the reference [14].

The paper is organized as follows. In the next section, we introduce the extension

U(1)X

to the SM, its

particle content with their respective charge

and hypercharge

values, which lead to zero the anomaly

equations. In section

we show how mass structures in the leptonic sector are predicted by the model. The

mass of electron, which is massless at tree-level, is generated by effective operators of dimension

n= 7

introducing a Lambda scale, and the mass matrix of the active neutrinos is determined by the inverse seesaw

mechanism. In section

, we present the parameter space of the model and the numerical formalism. The

free parameters are ﬁtted with the neutrino oscillation data available in NuFIT [

]. Results are showed and

analyzed in the section 5.

PREPRINT - OCTOBER 27, 2022

2 THE U(1)XEXTENSION

Under inclusion of a new non-universal gauge group

U(1)X

, Alvarado et al. [

] and Mantilla et al. [

]

proposed the next extension to the scalar sector of the SM:

Scalar bosons X Z2Y

Doublets

φ1= φ+

h1+v1+iη1

√2!+2/3 + +1

φ2= φ+

h2+v2+iη2

√2!+1/3 - +1

Singlet

χ=ξχ+vχ+iζχ

√2-1/3 + 0

Table 2: Bosonic content of the model with their respective charge X, hypercharge Yand parity Z2values.

The scalar doublets

φ1, φ2

have vacuum expectation values (VEV) that relate to the electroweak VEV by

v=pv2

1+v2

. The internal symmetry

is introduced to obtain matrices with suitable textures. The scalar

singlet

with VEV

vχ

is used for spontaneous symmetry breaking (SSB) of

U(1)X

and also for the mass

generation of the exotic fermions in the model. We assume

vχv

because

vχ

gives mass to the gauge ﬁeld

associated to the symmetry

U(1)X

, and from the non-observations of the LHC, there is a lower bound for

the Z0

µmass (4.5TeV < MZ0).

The fermionic sector of the proposed model [

] is presented in Table 3, where we use the following

notation:

U1,2,3= (u, c, t), D1,2,3= (d, s, b), ee,µ,τ = (e, µ, τ), νe,µ,τ = (νe, νµ, ντ).(8)

Quarks X Z2Leptons X Z2

L=U1

D1L

+1/3 + le

L=νe

eeL

0 +

L=U2

D2L

0 - lµ

L=νµ

eµL

0 +

L=U3

D3L

0 + lτ

L=ντ

eτL

-1 +

U1,3

R+2/3 + ee,τ

R-4/3 -

R+2/3 - eµ

R-1/3 -

D1,2,3

R-1/3 -

Extension

TL+1/3 - νe,µ,τ

R+1/3 -

TR+2/3 - Ne,µ,τ

R0 -

J1,2

L0 + EL,ER-1 +

J1,2

R-1/3 + ER,EL-2/3 +

Table 3: Fermionic content of the model with their respective charge Xand parity Z2values.

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LEPTONMASSESINANONUNIVERSALU(1)MODELWITHTHREEFAMILIESC.Cortes-Parra*,R.MartinezandJ.S.AlvaradoDepartamentodeFísica,UniversidadNacionaldeColombia,Carrera30No.4503,BogotáD.C,ColombiaOctober27,2022ABSTRACTWepresentanextensionU(1)XtotheStandardModelthatreproducestheleptonmassstructuresdeterminedbythee...

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LEPTON MASSES IN A NON UNIVERSAL U1 MODEL WITH THREE FAMILIES C. Cortes-Parra R. Martinezand J. S. Alvarado

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