Mass problem in the Standard Model

# Mass problem in the Standard Model

R. Martinez\fnsep Departamento de Física, Universidad Nacional de Colombia,
Ciudad Universitaria, Carrera 45 # 26-85, Bogotá D.C., Colombia.
S. F. Mantilla\fnsep Departamento de Física, Universidad Nacional de Colombia,
Ciudad Universitaria, Carrera 45 # 26-85, Bogotá D.C., Colombia.
###### Abstract

We propose a new 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 oscillation 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, reactor 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.

\wocname

EPJ Web of Conferences \woctitleICNFP 2017

## 1 Introduction

A large amount of high-energy phenomena have been understood in the context of the Standard Model (SM)SM (), 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 detectorshomestake (), 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 and , where the last one determines the ordering, normal ordering (NO) if and inverted ordering (IO) if nova (); neutrinodata ().

There are many models which propose different scenarios where FMH could be understood. One of the most important is the left-right model fritzsch1978 () from which the Fritzsch texture is obtainedtextures (). 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 fermionsfty1993 ().

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 cancellation. Their suitability has been shown in addressing quark massessomepheno (), dark matterDM-martinez-I (); DM-jhep (), scalar potential stabilityDM-martinez-II () and lepton massesmartinez1612 (), 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 chargesmantilla2017 (). 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 together with a discrete symmetry 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 boson , an extra singlet Higgs field is needed to break 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 field without VEV, whose charge is the same of but with the opposite parity available to do radiative corrections when they are deserved. Furthermore, the scalar sector shows the vacuum hierarchy (VH) , required to get the suited masses. , , and are at units of TeV, hundreds of GeV, units of GeV and hundreds of MeV, respectively.

On the other hand, the set of charges of the fermion sector is strongly constraint by the cancellation of chiral anomaliessomepheno (). Thus, in order to obtain nonuniversal charges and also cancel any chiral anomaly in the model, exotic quarks and leptons have been included, together with right-handed neutrinos and Majorana fermions so as inverse seesaw mechanism (ISS) is available to get small neutrino massesinverseseesaw (). All of them acquire mass through except which have their own Majorana mass . Additionally, the is in charge to make distinguishable the fermions with the same charges. It is outlined in the condensed notation 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=hee3ν¯¯¯¯¯ℓeL~Φ3νeR+heμ3ν¯¯¯¯¯ℓeL~Φ3νμR+heτ3ν¯¯¯¯¯ℓeL~Φ3ντR+hμe3ν¯¯¯¯¯ℓμL~Φ3νeR+hμμ3ν¯¯¯¯¯ℓμL~Φ3νμR+hμτ3ν¯¯¯¯¯ℓμL~Φ3ντR+gijχN¯¯¯¯¯¯¯¯¯νiCRχ∗NjR+12¯¯¯¯¯¯¯¯¯¯¯¯NiCRMijNNjR+h.c., (1)
 −LE=heμ3e¯¯¯¯¯ℓeLΦ3eμR+hμμ3e¯¯¯¯¯ℓμLΦ3eμR+hτe3e¯¯¯¯¯ℓτLΦ3eeR+hττ2e¯¯¯¯¯ℓτLΦ2eτR+he11E¯¯¯¯¯ℓeLΦ1E1R+hμ11E¯¯¯¯¯ℓμLΦ1E1R+g1eχe¯¯¯¯¯¯E1Lχ∗eeR+g2μχe¯¯¯¯¯¯E2LχeμR+g1χE¯¯¯¯¯¯E1LχE1R+g2χE¯¯¯¯¯¯E2Lχ∗E2R+h.c., (2)
 −LU=h113u¯¯¯¯¯q1L~Φ3u1R+h122u¯¯¯¯¯q1L~Φ2u2R+h133u¯¯¯¯¯q1L~Φ3u3R+h221u¯¯¯¯¯q2L~Φ1u2R+h311u¯¯¯¯¯q3L~Φ1u1R+h331u¯¯¯¯¯q3L~Φ1u3R+h12T¯¯¯¯¯q1L~Φ2TR+h21T¯¯¯¯¯q2L~Φ1TR+g1σu¯¯¯¯¯¯TLσu1R+g2χu¯¯¯¯¯¯TLχu2R+g3σu¯¯¯¯¯¯TLσu3R+gχT¯¯¯¯¯¯TLχTR+h.c., (3)
 −LD=h111J¯¯¯¯¯q1LΦ1J1R+h212J¯¯¯¯¯q2LΦ2J1R+h313J¯¯¯¯¯q3LΦ3J1R+h121J¯¯¯¯¯q1LΦ1J2R+h222J¯¯¯¯¯q2LΦ2J2R+h323J¯¯¯¯¯q3LΦ3J2R+h213d¯¯¯¯¯q2LΦ3d1R+h223d¯¯¯¯¯q2LΦ3d2R+h233d¯¯¯¯¯q2LΦ3d3R+h312d¯¯¯¯¯q3LΦ2d1R+h322d¯¯¯¯¯q3LΦ2d2R+h332d¯¯¯¯¯q3LΦ2d3R+g11σd¯¯¯¯¯¯¯J1Lσ∗d1R+g11σd¯¯¯¯¯¯¯J1Lσ∗d2R+g13σd¯¯¯¯¯¯¯J1Lσ∗d3R+g21σd¯¯¯¯¯¯¯J2Lσ∗d1R+g22σd¯¯¯¯¯¯¯J2Lσ∗d2R+g23σd¯¯¯¯¯¯¯J2Lσ∗d3R+g1χJ¯¯¯¯¯¯¯J1Lχ∗J1R+g2χJ¯¯¯¯¯¯¯J2Lχ∗J2R+h.c., (4)

where are the scalar doublet conjugates.

## 3 Mass matrices

The up-like quark sector is described in the bases and , 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

 (5)

where is

 MU=1√2⎛⎜ ⎜ ⎜ ⎜ ⎜⎝h113uv3h122uv2h133uv3h12Tv20h221uv10h21Tv1h311uv10h331uv100g2χuvχ0gχTvχ⎞⎟ ⎟ ⎟ ⎟ ⎟⎠. (6)

Since the determinant of is non-vanishing, the four up-like quarks acquire masses. The four mass eigenvalues are

 m2u=(h113uh331u−h133uh311u)2(h331u)2+(h311u)2v232,m2c=(h221ugχT−h21Tg2χu)2(gχT)2+(g2χu)2v212,m2t=[(h331u)2+(h311u)2]v212,m2T=[(gχT)2+(g2χu)2]v2χ2, (7)

where the VH has been employed. The heavy quarks and acquire masses through and , respectively. The quark acquire mass also through , however, this exhibits the difference of the Yukawa coupling constants because of the see-saw with the exotic quark . Finally, the quark acquire mass through with the same suppression mechanisms of quark but with instead of .

The down-like quarks are described in the bases and , where the former is the flavor basis while the latter is the mass basis. The matrix in the flavor basis is

 (8)

where turns out to be

 MD=1√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝000h111Jv1h121Jv1h213dv3h223dv3h233dv3h212Jv2h222Jv2h312dv2h322dv2h332dv2h313Jv3h323Jv3000g1χJvχ00000g2χJvχ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (9)

Although has a vanishing determinant and the quark remains massless, it can acquire a small mass with radiative corrections, given by the expression

 Σ1kd=∑i=1,2fσgi1σd∗hkikJvk(4π)2mJiC0(mσmJi,mhkmJi) (10)

where , is the trilinear coupling constant involving and , and ismartinez-rad-corr-331 ()

 C0(x,y)=1(1−x2)(1−y2)(x2−y2){x2y2ln(x2y2)−x2lnx2+y2lny2} (11)

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

 MD=1√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝Σ11dΣ12dΣ13dh111Jv1h121Jv1h213dv3h223dv3h233dv3h212Jv2h222Jv2h312dv2h322dv2h332dv2h313Jv3h323Jv3000g1χJvχ00000g2χJvχ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (12)

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

 m2d=[(Σ11dh223d−Σ12dh213d)h332d+(Σ13dh213d−Σ11dh233d)h322d+(Σ12dh233d−Σ13dh223d)h312d]2[(h213d)2+(h223d)2](h332d)2+[(h233d)2+(h213d)2](h322d)2+[(h223d)2+(h233d)2](h312d)2,m2s=[(h213d)2+(h223d)2](h332d)2+[(h233d)2+(h213d)2](h322d)2+[(h223d)2+(h233d)2](h312d)2(h332d)2+(h322d)2+(h312d)2v232, (13)

while the masses of , and are

 (14)

The heaviest quarks and acquire masses at TeV scale with , while quark get mass through at GeV. The strange quark acquire mass with at hundreds of MeV, and the lightest did 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 are and , respectively. The mass term expressed in the flavor basis is

 (15)

where the mass matrix has the following block structure

 MN=⎛⎜ ⎜⎝0MTν0Mν0MTN0MNMN⎞⎟ ⎟⎠, (16)

with the Dirac mass in the (, ) basis, and

 Mν=v3√2⎛⎜ ⎜⎝hee3νheμ3νheτ3νhμe3νhμμ3νhμτ3ν000⎞⎟ ⎟⎠, (17)

is a Dirac mass matrix for (, ). is the Majorana mass of . The ISS, together with the VH yields

 (VNL,SS)†MNVNL,SS=⎛⎜⎝mν000mN000m~N⎞⎟⎠ (18)

where the resultant blocks arecatano2012 ()

 (19)

The charged leptons are described in the bases of flavor and mass . The mass term obtained from the Yukawa Lagrangian is

 (20)

where turns out to be

 ME=1√2⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝0heμ3ev30he11Ev100hμμ3ev30hμ11Ev10hτe3ev30hττ2ev200g1eχevχ00g1χEvχ00g2μχevχ00g2χEvχ⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠ (21)

The determinant of is non-vanishing ensuring that the five charged leptons acquire masses. By using the VH, the eigenvalues are

 (22)

The exotic and leptons acquired mass at the TeV scale. got mass at the GeV scale with . and have acquired mass through , the smallest VEV at hundreds of MeV. Moreover, the mass 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 between and is not small and can contribute importantly to the PMNS matrix, is set as a free parameter named .

## 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⎛⎜⎝ρheνceνρhμνcμνρhτνcτνheνsehμνsμhτνsτ000⎞⎟⎠, (23)

the effective mass matrix for the light neutrinos is

 mν=μNv23(h1N)2v2χ⎛⎜ ⎜ ⎜⎝(heν)2heνhμνceμνheνhτνceτνheνhμνceμν(hμν)2hμνhτνcμτνheνhτνceτνhμνhτνcμτν(hτν)2⎞⎟ ⎟ ⎟⎠, (24)

where . While the mass scale is fixed by the constrain , the other parameters can be explored with Montecarlo procedures in order to reproduce neutrino oscillation dataneutrinodata (). Since only depends on angle differences, is set null. Tab. 2 presents some domains were neutrino oscillation data are reproduced at 3neutrinodata () in function of the charged lepton mixing angle . 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 and a discrete symmetry , 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 suited Yukawa coupling constants yield mass matrices whose eigenvalues suggest the presence of suppression 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 the and quarks is

 Mut∝⎛⎜ ⎜⎝h113uv3|h133uv3\textemdash\textemdash\textemdashh311uv1|h331uv1⎞⎟ ⎟⎠,

whose eigenvalues turn out to be (with the assumption from VH)

 m2u=(h113uh331u−h133uh311u)2(h331u)2+(h311u)2v232,m2t=[(h331u)2+(h311u)2]v212+(h113uh331u+h133uh311u)2(h331u)2+(h311u