Mass problem in the Standard Model

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©Owned by the authors, published by EDP Sciences, 2017

Mass problem in the Standard Model

R. Martinez1,a_{and S. F. Mantilla}1,b

1_{Departamento de Física, Universidad Nacional de Colombia,}

Ciudad Universitaria, Carrera 45 # 26-85, Bogotá D.C., Colombia.

Abstract.We propose a new SU(3)C⊗SU(2)L⊗U(1)Y⊗U(1)X gauge model which is

non universal respect to the three fermion families of the Standard Model. We introduce additional one top-like quark, two bottom-like quarks and three right handed neutrinos in order to have an anomaly free theory. We also consider additional three right handed neutrinos which are singlets respect to the gauge symmetry of the model to implement see saw mechanism and give masses to the light neutrinos according to the neutrino os-cillation phenomenology. In the context of this horizontal gauge symmetry we find mass ansatz for leptons and quarks. In particular, from the analysis of solar, atmospheric, re-actor and accelerator neutrino oscillation experiments, we get the allow region for the Yukawa couplings for the charge and neutral lepton sectors according with the mass squared differences and mixing angles for the two neutrino hierarchy schemes, normal

and inverted.

1 Introduction

A large amount of high-energy phenomena have been understood in the context of the Standard Model (SM)[1], nowadays one of the most successful frameworks in physics. Nevertheless, there are some observations which might be out of the scope of the SM such as the fermion mass hierarchy (FMH), the origin of the large differences among scales of fermion masses, from units of MeV to hundreds of GeV is not completely understood without using unpleasant fine-tunings. Moreover, according to neutrino detectors[2], the evidence of light neutrino masses from neutrino oscillations enlarges this issue by extending the mass scales until meV. However, in contrast with charged fermions, neutrino masses are known until their squared mass differences∆m2₁₂and∆m3, where the last one determines the ordering, normal ordering (NO) if∆m2₁₃>0 and inverted ordering (IO) if∆m2₂₃<0[3, 4].

There are many models which propose different scenarios where FMH could be understood. One of the most important is the left-right model SU(2)L⊗SU(2)R⊗U(1)B−L[5] from which the Fritzsch

texture is obtained[6]. Such a scheme was extended to leptons and neutrinos by Fukugita, Tanimoto and Yanagida, who also implemented Majorana masses so as active neutrinos can acquire tiny masses, with the respective lepton violation processes produced by the existence of Majorana fermions[7].

Among different schemes, abelian extensions of the SM show a fertile scenario where some issues can be understood by implementing simple tools such as symmetry breaking or chiral anomaly can-cellation. Their suitability has been shown in addressing quark masses[8], dark matter[9, 10], scalar

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potential stability[11] and lepton masses[12], where light fermions acquire masses through radiative corrections. However, by doing a slight modification in adding an additional doublet, a completely new scheme emerges with interesting texture matrices from which the FMH can be obtained naturally. The proceedings presents the particle content and the set of U(1)X charges[13]. Thereafter, the

mass matrices are shown from the Yukawa Lagrangians with their respective eigenvalues in which the FMH can be inferred. After that, the suitability of the model is checked by exploring the neutrino parameter space in order to fit neutrino oscillation data. Finally, some conclusions are outlined with a summary.

2 Particle content

The scheme proposed consists on extending the SM with a new nonuniversal abelian interaction U(1)X

together with a discrete symmetryZ2 which distinguish among families in such a way that fermion mass matrices can suggest the FMH without any kind of fine-tuning on Yukawa coupling constants. Moreover, because of the new gauge bosonZ_{, an extra singlet Higgs field}_χ_{is needed to break U(1)}

X

with its vacuum expectation value (VEV). Additionally, there are three Higgs doublets so as every fermion gets massive, reducing the need of radiative corrections to the minimum. There is also a scalar fieldσwithout VEV, whoseU(1)X charge is the same ofχ but with the opposite parityZ2 available to do radiative corrections when they are deserved. Furthermore, the scalar sector shows the vacuum hierarchy (VH)vχ > v1 > v2 > v3, required to get the suited masses. vχ,v1,v2andv3are at units of TeV, hundreds of GeV, units of GeV and hundreds of MeV, respectively.

On the other hand, the set of charges U(1)X of the fermion sector is strongly constraint by the

cancellation of chiral anomalies[8]. Thus, in order to obtain nonuniversal charges and also cancel any chiral anomaly in the model, exotic quarksT,_J1,2_{and leptons}_E1,2_{have been included, together with} right-handed neutrinosνe_R,µ,τand Majorana fermionsN1,2,3

R so as inverse seesaw mechanism (ISS) is

available to get small neutrino masses[14]. All of them acquire mass throughχexceptN1,2,3 _which have their own Majorana massMN. Additionally, the Z2 is in charge to make distinguishable the fermions with the same U(1)X charges. It is outlined in the condensed notationX±in tab. 1 of the

particle content.

The resulting Yukawa Lagrangian of the model for the neutral, charged lepton, up-like and down-like sectors are given by, respectively

−LN=hee₃νeLΦ˜3νeR+he3µνeLΦ˜3νµR+he3τνLeΦ˜3ντR+hµ3eν µ

LΦ˜3νeR+hµµ3ν µ

LΦ˜3νµR+hµτ3ν µ LΦ˜3ντR

+gi j_χ_Nνi C_R χ∗_N_Rj+1 2NRi CM

i j

NN

j

R+h.c., (1)

−LE=he₃µ_ee_LΦ3eµR+h µµ

3e µ LΦ3e

µ

R+hτ3eeτLΦ3eeR+hττ2eτLΦ2eτR+he11EeLΦ1E1R+hµ11E µ LΦ1E1R

+g1χeeE1Lχ∗eeR+g2χµeE2_Lχeµ_R+g1χEE1LχE1R+g2χEE2Lχ∗E2R+h.c., (2)

−LU=h11₃uq1LΦ˜3u1R+h122uq1LΦ˜2u2R+h133uq1LΦ˜3uR3 +h221uq2LΦ˜1u2R+h311uq3LΦ˜1u1R+h133uq3LΦ˜1u3R

+h1₂_Tq1_LΦ˜₂TR+h2₁_Tq2LΦ˜1TR+g1σuTLσuR1+g2χuTLχu2R +g3σuTLσu3R +gχTTLχTR+h.c., (3)

−LD=h11₁_Jq_L1Φ1JR1+h212Jq2LΦ2JR1+h313Jq3LΦ3JR1+h121Jq1LΦ1JR2+h222Jq2LΦ2JR2+h332Jq3LΦ3JR2

+h21₃_dq2_LΦ₃d_R1+h22₃_dq2_LΦ₃d2_R+h₃23_dq2_LΦ₃d3_R+h31₂_dq3_LΦ₂d1_R+h32₂_dq_L3Φ₂d2_R+h₂33_dq3_LΦ₂d3_R

+g11σdJL1σ∗dR1+g11σdJL1σ∗d2R+g13σdJL1σ∗d3R+g21σdJL2σ∗dR1+g22σdJL2σ∗d2R+gσ23dJL2σ∗d3R

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potential stability[11] and lepton masses[12], where light fermions acquire masses through radiative corrections. However, by doing a slight modification in adding an additional doublet, a completely new scheme emerges with interesting texture matrices from which the FMH can be obtained naturally. The proceedings presents the particle content and the set of U(1)X charges[13]. Thereafter, the

mass matrices are shown from the Yukawa Lagrangians with their respective eigenvalues in which the FMH can be inferred. After that, the suitability of the model is checked by exploring the neutrino parameter space in order to fit neutrino oscillation data. Finally, some conclusions are outlined with a summary.

2 Particle content

The scheme proposed consists on extending the SM with a new nonuniversal abelian interaction U(1)X

together with a discrete symmetryZ2 which distinguish among families in such a way that fermion mass matrices can suggest the FMH without any kind of fine-tuning on Yukawa coupling constants. Moreover, because of the new gauge bosonZ_{, an extra singlet Higgs field}_χ_{is needed to break U(1)}

X

with its vacuum expectation value (VEV). Additionally, there are three Higgs doublets so as every fermion gets massive, reducing the need of radiative corrections to the minimum. There is also a scalar fieldσwithout VEV, whoseU(1)X charge is the same ofχ but with the opposite parityZ2 available to do radiative corrections when they are deserved. Furthermore, the scalar sector shows the vacuum hierarchy (VH)vχ > v1 > v2 > v3, required to get the suited masses. vχ,v1,v2andv3are at units of TeV, hundreds of GeV, units of GeV and hundreds of MeV, respectively.

On the other hand, the set of charges U(1)X of the fermion sector is strongly constraint by the

cancellation of chiral anomalies[8]. Thus, in order to obtain nonuniversal charges and also cancel any chiral anomaly in the model, exotic quarksT,_J1,2_{and leptons}_E1,2_{have been included, together with} right-handed neutrinosνe_R,µ,τand Majorana fermionsN1,2,3

R so as inverse seesaw mechanism (ISS) is

available to get small neutrino masses[14]. All of them acquire mass throughχexceptN1,2,3_which have their own Majorana massMN. Additionally, the Z2 is in charge to make distinguishable the fermions with the same U(1)X charges. It is outlined in the condensed notationX±in tab. 1 of the

particle content.

The resulting Yukawa Lagrangian of the model for the neutral, charged lepton, up-like and down-like sectors are given by, respectively

−LN=hee₃νeLΦ˜3νeR+he3µνeLΦ˜3νµR+he3τνLeΦ˜3ντR+hµ3eν µ

LΦ˜3νeR+hµµ3ν µ

LΦ˜3νµR+hµτ3ν µ LΦ˜3ντR

+gi j_χ_Nνi C_R χ∗_N_Rj+1 2NRi CM

i j

NN

j

R+h.c., (1)

−LE=he₃µ_ee_LΦ3eµR+h µµ

3e µ LΦ3e

µ

R+hτ3eeτLΦ3eeR+hττ2eτLΦ2eτR+he11EeLΦ1E1R+hµ11E µ LΦ1E1R

+g1χeeE1Lχ∗eeR+g2χµeE2_Lχeµ_R+g1χEE1LχE1R+g2χEE2Lχ∗E2R+h.c., (2)

−LU=h11₃uq1LΦ˜3u1R+h122uq1LΦ˜2u2R+h133uq1LΦ˜3uR3 +h221uq2LΦ˜1u2R+h311uq3LΦ˜1u1R+h133uq3LΦ˜1u3R

+h1₂_Tq1_LΦ˜₂TR+h2₁_Tq2LΦ˜1TR+g1σuTLσuR1+g2χuTLχu2R +g3σuTLσu3R +gχTTLχTR+h.c., (3)

−LD=h11₁_Jq1_LΦ1JR1+h212Jq2LΦ2JR1+h313Jq3LΦ3JR1+h121Jq1LΦ1JR2+h222Jq2LΦ2JR2+h332Jq3LΦ3JR2

+h21₃_dq2_LΦ₃d_R1+h22₃_dq2_LΦ₃d_R2+h₃23_dq2_LΦ₃d_R3+h31₂_dq3_LΦ₂d_R1+h32₂_dq_L3Φ₂d2_R+h₂33_dq3_LΦ₂d3_R

+g11σdJL1σ∗d1R+g11σdJL1σ∗d2R+g13σdJL1σ∗dR3+g21σdJL2σ∗dR1+g22σdJL2σ∗d2R+gσ23dJL2σ∗d3R

+g1χJJL1χ∗JR1+gχ2JJL2χ∗JR2+h.c., (4)

Bosons X± _Quarks _X± _Leptons _X±

Scalar Doublets SM Fermionic Doublets

Φ1=

  φ+

1 h1+v_√1+iη1

2



 +2/3+ _q1 L= u1 d1 L +1

/3+ e L= νe ee L 0 +

Φ2=

  φ+

2 h2+v_√2+iη2

2



 +1/3− _q2 L= u2 d2 L 0 − µ L= νµ eµ L 0 +

Φ3=

  φ+

3 h3+v_√3+iη3

2



 +1/3+ _q3 L= u3 d3 L

0+ τ

L= ντ eτ L − 1+

Scalar Singlets SM Fermionic Singlets

χ=ξχ+v_√χ+iζχ

2

σ

−1/3+ −1/3−

u1,3 R

u2 R

d1,2,3 R

+2/3+

+2/3−

−1/3−

ee R eµ R eτ R

−4/3+ −1/3+ −4/3−

Gauge bosons Non-SM Quarks Non-SM Leptons

W±

µ

W3

µ

0+

0+ T_TL

R

+1/3−

+2/3− ν e,µ,τ R Ne,µ,τ

R

+1/3+

0+

Bµ 0+ J1L,2 0+ E1L,E2R −1+

Ξµ 0+ JR1,2 −1/3+ E1R,E2L −2/3+

Table 1.Non-universalXquantum number andZ₂parity for SM and non-SM fermions.

whereΦ =iσ2Φ∗are the scalar doublet conjugates.

3 Mass matrices

The up-like quark sector is described in the basesU =(u1,u2_,_u3_,_T_{) and}_u ₌₍_u_,_c_,_t_,_T_{), where the} former is the flavor basis while the latter is the mass basis. The mass term in the flavor basis turns out to be

−LU=ULMUUR+h.c., (5)

whereMUis

MU= √1

2     h11

3uv3 h122uv2 h133uv3 h12Tv2 0 h22

1uv1 0 h21Tv1 h31

1uv1 0 h331uv1 0

0 g2_χ_uvχ 0 gχTvχ

  

. (6)

Since the determinant ofMUis non-vanishing, the four up-like quarks acquire masses. The four mass

eigenvalues are

m2

u=

h11

3uh331u−h133uh311u

2

(h33

1u)2+(h311u)2 v2₃

2, m2c=

h22

1ugχT −h21Tg2χu

2

(gχT)2+(g2χu)2 v2₁

2,

m2

t =

(h33

1u)2+(h311u)2

v2₁

2, m2T =

(gχT)2+(g2χu)2

v2χ

2, (7)

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The down-like quarks are described in the basesD=(d1,d2,d3,J1_,_J2_{) and}_d₌₍_d_,_s_,_b_,_J1_,_J2_), where the former is the flavor basis while the latter is the mass basis. The matrix in the flavor basis is

−LD=DLMDDR+h.c., (8)

whereMDturns out to be

MD= √1

2

     

0 0 0 h11

1Jv1 h121Jv1 h21

3dv3 h223dv3 h233dv3 h212Jv2 h222Jv2 h31

2dv2 h322dv2 h332dv2 h313Jv3 h323Jv3 0 0 0 g1_χ_Jvχ 0 0 0 0 0 g2_χ_Jvχ

     

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AlthoughMDhas a vanishing determinant and the quarkdremains massless, it can acquire a small

mass with radiative corrections, given by the expression

Σ1_dk=

i=1,2

fσgi_σ1_d∗hki_k_Jvk (4π)2mJi C0

mσ

mJi,

mhk

mJi

(10)

wherek=1,2,3, fσis the trilinear coupling constant involvingσandΦ1,2,3, andC0(x, y) is[15]

C0(x, y)=₍₁ 1

−x2₎₍₁₋_y2₎₍_x2₋_y2₎

x2_y2_lnx2

y2

−x2_ln_x2₊_y2_ln_y2 ₍₁₁₎

Thus, up to one-loop correction the mass matrix is

MD= √1

2

     

Σ11_d Σ12_d Σ13_d h11₁_Jv1 h12₁_Jv1 h21

     

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whose determinant does not vanish. The eigenvalues are, according to VH,

m2

d=

Σ11_d h22₃_d₋Σ12_d h21₃_dh33₂_d+Σ13_d h21₃_d₋Σ11_d h23₃_dh32₂_d+Σ12_d h23₃_d₋Σ13_d h22₃_dh31₂_d2

(h21

3d)2+(h223d)2

(h33 2d)2+

(h23

3d)2+(h213d)2

(h32 2d)2+

(h22

3d)2+(h233d)2

(h31 2d)2

,

m2

s=

(h21

3d)2+(h223d)2

(h33 2d)2+

(h23

3d)2+(h213d)2

(h32 2d)2+

(h22

3d)2+(h233d)2

(h31 2d)2

(h33

2d)2+(h322d)2+(h312d)2

v2₃

2, (13)

while the masses ofb,J1_and_J2_are

m2

b=

(h33

2d)2+(h322d)2+(h312d)2

v2₂

2, m2J1=(g1χJ)2

v2χ

2, m2J2=(g2χJ)2

v2χ

2. (14) The heaviest quarksJ1_and_J2_{acquire masses at TeV scale with}_v

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The down-like quarks are described in the basesD=(d1,d2,d3,J1_,_J2_{) and}_d₌₍_d_,_s_,_b_,_J1_,_J2_), where the former is the flavor basis while the latter is the mass basis. The matrix in the flavor basis is

−LD=DLMDDR+h.c., (8)

whereMDturns out to be

MD= √1

2      

0 0 0 h11

1Jv1 h121Jv1 h21

      (9)

AlthoughMDhas a vanishing determinant and the quarkdremains massless, it can acquire a small

mass with radiative corrections, given by the expression

Σ1_dk=

i=1,2

fσgi_σ1_d∗hki_k_Jvk (4π)2mJi C0

mσ

mJi,

mhk

mJi

(10)

wherek=1,2,3, fσis the trilinear coupling constant involvingσandΦ1,2,3, andC0(x, y) is[15]

C0(x, y)=₍₁ 1

−x2₎₍₁₋_y2₎₍_x2₋_y2₎

x2_y2_lnx2

y2

−x2_ln_x2₊_y2_ln_y2 ₍₁₁₎

Thus, up to one-loop correction the mass matrix is

MD= √1

2      

Σ11_d Σ12_d Σ13_d h11₁_Jv1 h12₁_Jv1 h21

      (12)

whose determinant does not vanish. The eigenvalues are, according to VH,

m2

d=

Σ11_d h22₃_d₋Σ12_d h21₃_dh33₂_d+Σ13_d h21₃_d₋Σ11_d h23₃_dh32₂_d+Σ12_d h23₃_d₋Σ13_d h22₃_dh31₂_d2

(h21

3d)2+(h223d)2

(h33 2d)2+

(h23

3d)2+(h213d)2

(h32 2d)2+

(h22

3d)2+(h233d)2

(h31 2d)2

,

m2

s=

(h21

3d)2+(h223d)2

(h33 2d)2+

(h23

3d)2+(h213d)2

(h32 2d)2+

(h22

3d)2+(h233d)2

(h31 2d)2

(h33

2d)2+(h322d)2+(h312d)2

v2₃

2, (13)

while the masses ofb,J1_and_J2_are

m2

b=

(h33

2d)2+(h322d)2+(h312d)2

v2₂

2, m2J1=(g1χJ)2

v2χ

2, m2J2=(g2χJ)2

v2χ

2. (14) The heaviest quarksJ1_and_J2_{acquire masses at TeV scale with}_v

χ, whilebquark get mass throughv2 at GeV. The strange quark acquire mass withv3at hundreds of MeV, and the lightestddid not acquire mass at tree-level but at one-loop, where radiative corrections supress its mass.

The neutrinos involve both Dirac and Majorana masses in their Yukawa Lagrangian. The flavor and mass basis areNL=(νe_L,µ,τ, νe_R,µ,τC,_N_Re,µ,τC) andnL=(ν1_L,2,3,N1,2,3

L ,N˜L1,2,3), respectively. The mass

term expressed in the flavor basis is

−LN =1

2NCLMNNL, (15)

where the mass matrix has the following block structure

MN =

  

0 MT

ν 0

Mν 0 MT_N

0 MN MN

 

, (16)

withMN=diag

h1

N,h2N,h3N

_√vχ

2 the Dirac mass in the (ν

C

R,NR) basis, and

Mν= v3 √ 2    hee

3ν h eµ

3ν he3τν

hµe

3ν h µµ

3ν h µτ

3ν

0 0 0

 

, (17)

is a Dirac mass matrix for (νL,νR).MN=µNI3×3is the Majorana mass ofNR. The ISS, together with

the VHvχv3 |MN|yields

VN_L_,_SS†MNVNL,SS=

  

mν 0 0

0 mN 0

0 0 mN˜

 

 (18)

where the resultant 3×3 blocks are[16]

mν=MTν(MN)−1MN

MTN

−1

Mν, MN ≈ MN−MN, MN˜ ≈ MN+MN. (19)

The charged leptons are described in the bases of flavor E = (ee,eµ_,_eτ_,_E1_,_E2_{) and mass} _e ₌ (e, µ, τ,E1_,_E2_{). The mass term obtained from the Yukawa Lagrangian is}

−LE =ELMEER+h.c. (20)

whereME turns out to be

ME = √1

2      

0 heµ

3ev3 0 he11Ev1 0 0 hµµ

3ev3 0 h µ1 1Ev1 0 hτe

3ev3 0 hττ2ev2 0 0

g1χeevχ 0 0 g1_χ_Evχ 0

0 g2χµevχ 0 0 g2_χ_Evχ

      (21)

The determinant ofME is non-vanishing ensuring that the five charged leptons acquire masses.

By using the VH, the eigenvalues are

m2

e=

heµ

3eh µ1 1E−h

µµ

3ehe11E

2

(he1 1E)2+(h

µ1 1E)2

v2₃

2, m2µ=

heµ

3ehe11E+h µµ

3eh µ1 1E

2

(he1 1E)2+(h

µ1 1E)2

v2₃

2 +

hτe

3e

2

v2₃

2 , m2 τ= hττ 2e 2

v2₂

2 , m2E1=

g1_χ_E2+g1χee

2v2χ

2, m2E2=

g2_χ_E2+g2χµe

2v2χ

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θE₁₂ he

ν h

µ

ν hτν θ

µ

ν θτν−θ µ ν

Normal Ordering

0o ₀_.₂₇₀_±₀_.₀₀₇ ₀_.₇₃₈_±₀_.₀₄₀ ₀_.₇₄₇_±₀_.₀₄₀ _±₍₃₉_.₄₉_±₂_.₉₉₎ _±₍₃₈_.₇₉_±₀_.₇₈₎

15o ₀_.₂₉₄_±₀_.₀₀₈ ₀_.₇₃₇_±₀_.₀₄₅ ₀_.₇₃₈_±₀_.₀₄₃ _±₍₆₆_.₇₃_±₁_.₀₂₎ _±₍₃₃_.₃₂_±₀_.₈₁₎

30o ₀_.₄₀₀_±₀_.₀₀₈ ₀_.₆₈₉_±₀_.₀₃₅ ₀_.₇₃₄_±₀_.₀₃₅ _±₍₄₆_.₃₈_±₁_.₉₁₎ _±₍₂₇_.₅₃_±₁_.₁₂₎

45o ₀_.₄₉₅_±₀_.₀₀₃ ₀_.₅₄₈_±₀_.₀₀₄ ₀_.₇₉₆_±₀_.₀₀₅ _±₍₄₂_.₆₁_±₀_.₈₂₎ _±₍₁₉_.₁₀_±₀_.₇₅₎

Inverted Ordering

0o ₀_.₉₈₄_±₀_.₀₀₆ ₀_.₇₂₅_±₀_.₀₃₁ ₀_.₇₀₀_±₀_.₀₃₂ _±₍₈₁_.₈₈_±₀_.₈₄₎ _∓₍₁₆₃_.₁₇_±₀_.₅₆₎

1o ₀_.₉₈₂_±₀_.₀₀₆ ₀_.₇₃₂_±₀_.₀₃₀ ₀_.₆₉₅_±₀_.₀₃₁ _±₍₈₁_.₅₇_±₀_.₇₄₎ _±₍₁₆₁_.₉₁_±₀_.₅₅₎

2o ₀_.₉₈₀_±₀_.₀₀₆ ₀_.₇₄₇_±₀_.₀₂₂ ₀_.₆₈₁_±₀_.₀₂₂ _±₍₈₁_.₅₄_±₀_.₅₁₎ _∓₍₁₆₀_.₇₇_±₀_.₅₀₎

3o ₀_.₉₇₈_±₀_.₀₀₆ ₀_.₇₅₉_±₀_.₀₁₄ ₀_.₆₇₁_±₀_.₀₁₃ _±₍₈₁_.₅₁_±₀_.₃₂₎ _∓₍₁₅₉_.₅₆_±₀_.₄₆₎

Table 2.Some parameter domains which reproduce data for NO and IO from [4].θe_ν=0.

The exoticE1_and_E2_{leptons acquired mass at the TeV scale.}_τ_{got mass at the GeV scale with}_v 2.µ andehave acquired mass throughv3, the smallest VEV at hundreds of MeV. Moreover, theemass is suppressed by the difference between the Yukawa coupling constants, which can be assumed to be at the same order of magnitude. Finally, because the mixing angle betweeneandµis not small and can contribute importantly to the PMNS matrix, is set as a free parameter namedθ₁₂E.

4 PMNS fitting

By replacing the Dirac mass matrix from (17) into the light mass eigenvalues in (19) and redefining the Yukawa coupling constants by a polar parametrization

Mν= v3

√

2ρ

  

ρheνceν ρh µ

νcµν ρhτνcτν

he νse h

µ

νsµ hτνsτ

0 0 0

 

, (23)

the effective mass matrix for the light neutrinos is

mν= µNv2₃

h1 N

2

v2

χ

   

_he

ν2 heνh µ

νceνµ heνhτνceντ

he νh

µ νceνµ

hµ ν

2 hµ

νhτνc µτ ν

he

νhτνceντ h µ νhτνc

µτ

ν hτν2

  

, (24)

wherecαβ

ν =cos(θαν −θ β

ν). While the mass scale is fixed by the constrainµNv23/

h1 Nvχ

2

=50 meV, the other parameters can be explored with Montecarlo procedures in order to reproduce neutrino oscillation data[4]. Sincemνonly depends on angle differences,θeνis set null. Tab. 2 presents some domains were neutrino oscillation data are reproduced at 3σ[4] in function of the charged lepton mixing angleθE₁₂. Thus, the model is able to reproduce neutrino oscillation data in NO and IO schemes.

5 Discussion and Conclusions

The model proposes an extension to the SM by including a new nonuniversal abelian interaction U(1)X

(7)

θE₁₂ he

ν h

µ

ν hτν θ

µ

ν θντ−θ µ ν

Normal Ordering

0o ₀_.₂₇₀_±₀_.₀₀₇ ₀_.₇₃₈_±₀_.₀₄₀ ₀_.₇₄₇_±₀_.₀₄₀ _±₍₃₉_.₄₉_±₂_.₉₉₎ _±₍₃₈_.₇₉_±₀_.₇₈₎

15o ₀_.₂₉₄_±₀_.₀₀₈ ₀_.₇₃₇_±₀_.₀₄₅ ₀_.₇₃₈_±₀_.₀₄₃ _±₍₆₆_.₇₃_±₁_.₀₂₎ _±₍₃₃_.₃₂_±₀_.₈₁₎

30o ₀_.₄₀₀_±₀_.₀₀₈ ₀_.₆₈₉_±₀_.₀₃₅ ₀_.₇₃₄_±₀_.₀₃₅ _±₍₄₆_.₃₈_±₁_.₉₁₎ _±₍₂₇_.₅₃_±₁_.₁₂₎

45o ₀_.₄₉₅_±₀_.₀₀₃ ₀_.₅₄₈_±₀_.₀₀₄ ₀_.₇₉₆_±₀_.₀₀₅ _±₍₄₂_.₆₁_±₀_.₈₂₎ _±₍₁₉_.₁₀_±₀_.₇₅₎

Inverted Ordering

0o ₀_.₉₈₄_±₀_.₀₀₆ ₀_.₇₂₅_±₀_.₀₃₁ ₀_.₇₀₀_±₀_.₀₃₂ _±₍₈₁_.₈₈_±₀_.₈₄₎ _∓₍₁₆₃_.₁₇_±₀_.₅₆₎

1o ₀_.₉₈₂_±₀_.₀₀₆ ₀_.₇₃₂_±₀_.₀₃₀ ₀_.₆₉₅_±₀_.₀₃₁ _±₍₈₁_.₅₇_±₀_.₇₄₎ _±₍₁₆₁_.₉₁_±₀_.₅₅₎

2o ₀_.₉₈₀_±₀_.₀₀₆ ₀_.₇₄₇_±₀_.₀₂₂ ₀_.₆₈₁_±₀_.₀₂₂ _±₍₈₁_.₅₄_±₀_.₅₁₎ _∓₍₁₆₀_.₇₇_±₀_.₅₀₎

3o ₀_.₉₇₈_±₀_.₀₀₆ ₀_.₇₅₉_±₀_.₀₁₄ ₀_.₆₇₁_±₀_.₀₁₃ _±₍₈₁_.₅₁_±₀_.₃₂₎ _∓₍₁₅₉_.₅₆_±₀_.₄₆₎

Table 2.Some parameter domains which reproduce data for NO and IO from [4].θe_ν=0.

The exoticE1_and_E2_{leptons acquired mass at the TeV scale.}_τ_{got mass at the GeV scale with}_v 2.µ andehave acquired mass throughv3, the smallest VEV at hundreds of MeV. Moreover, theemass is suppressed by the difference between the Yukawa coupling constants, which can be assumed to be at the same order of magnitude. Finally, because the mixing angle betweeneandµis not small and can contribute importantly to the PMNS matrix, is set as a free parameter namedθE₁₂.

4 PMNS fitting

By replacing the Dirac mass matrix from (17) into the light mass eigenvalues in (19) and redefining the Yukawa coupling constants by a polar parametrization

Mν= v3 √ 2ρ   

ρheνceν ρh µ

νcµν ρhτνcτν

he νse h

µ

νsµ hτνsτ

0 0 0

 

, (23)

the effective mass matrix for the light neutrinos is

mν= µNv2₃

h1 N 2 v2 χ     _he

ν2 heνh µ

νceνµ heνhτνceντ

he νh

µ νceνµ

hµ ν

2 hµ

νhτνc µτ ν

he

νhτνceντ h µ νhτνc

µτ

ν hτν2

  

, (24)

wherecαβ

ν = cos(θαν −θ β

ν). While the mass scale is fixed by the constrainµNv23/

h1 Nvχ

2

=50 meV, the other parameters can be explored with Montecarlo procedures in order to reproduce neutrino oscillation data[4]. Sincemνonly depends on angle differences,θeνis set null. Tab. 2 presents some domains were neutrino oscillation data are reproduced at 3σ[4] in function of the charged lepton mixing angleθE₁₂. Thus, the model is able to reproduce neutrino oscillation data in NO and IO schemes.

5 Discussion and Conclusions

The model proposes an extension to the SM by including a new nonuniversal abelian interaction U(1)X

and a discrete symmetryZ₂, with extended fermion and scalar sectors such that chiral anomalies get cancelled and the majority of fermions can acquire mass. The existence of the VH, together with the

Leptons Quarks

Family Mass Mass

1 ν1_L µNv

2 3 h1 N 2 v2 χ h2

ν1 u h2

u−hu2 ht

v3 √

2

2 ν2_L µNv

2 3 h1 N 2 v2 χ h2

ν2 c h2

c−hc2 hT

v1 √

2

3 ν3_L µNv

2 3 h1 N 2 v2 χ h2

ν3 t

htv1 √

2

Exot Ni

L hi

Nvχ √

2 ∓µN T

hTvχ

√

2

1 e h

2

−h2

hv1

v3 √

2 d

Σdh2d h2

s+hs2

2 µ h

2

+h

2

hv1

v3 √

2 s

h2

s+hs2 hb

v3 √

2

3 τ h√τv2

2 b

hbv2 √

2

Exot E1 hE_√1vχ

2 J

1 hJ_√1vχ

2

Exot E2 hE_√2vχ

2 J

2 hJ_√2vχ

2

Table 3.Summary of fermion masses showing their VEVs as well as the suppression mechanism if it is

involved. The orders of magnitude ofvχ,v1,v2,v3andµNare units of TeV, hundreds of GeV, units of GeV,

hundreds of MeV and units of MeV, respectively.

suited Yukawa coupling constants yield mass matrices whose eigenvalues suggest the presence of sup-pression mechanisms which could offer a fermionic spectrum spanning different order of magnitude of mass, from units of MeV to units of TeV. Such mass eigenvalues are outlined in tab. 3.

These eigenvalues are produced by the very structure of the mass matrices where a heavier mass suppress a lighter one. For instance, the subblock involving theuandtquarks is

Mut∝

  

h11

3uv3 | h133uv3

— — —

h31

1uv1 | h331uv1

  ,

whose eigenvalues turn out to be (with the assumptionv3/v11 from VH)

m2u=

h11

3uh331u−h133uh311u

2

(h33

1u)2+(h311u)2 v2₃

2, m2t =

(h33₁u)2+(h311u)2

v2₁

2 +

h11

3uh331u+h133uh311u

2

(h33

1u)2+(h311u)2 v2₃

2.

This result is repeated with thecquark and thee. Thus, the FMH is addressed from the structure of the mass matrices, which is also produced by the nonuniversal interaction U(1)X and the discrete

symmetryZ₂. In contrast, thedquark acquires mass through radiative corrections involvingσ,Φ₁,2,3 and the exotic quarksJ1,2_.

(8)

More-over, the model is able to reproduce NO and IO schemes according to current neutrino oscillation data. The squared mass differences and PMNS mixing angles were obtained without fine-tunings.

Acknowledgment

This work was supported by COLCIENCIAS in Colombia.

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