Review Article Lepton Number Violation and Neutrino …bibliotecadigital.udea.edu.co/bitstream/10495/8795/1/PonceWilliam_2015_LeptonNumber...Review Article Lepton Number Violation

Review ArticleLepton Number Violation and Neutrino Masses in 3-3-1 Models

Richard H Benavides1 Luis N Epele2 Huner Fanchiotti2

Carlos Garciacutea Canal2 and William A Ponce3

1Facultad de Ciencias Exactas y Aplicadas Instituto Tecnologico Metropolitano Medellın Colombia2Laboratorio de Fısica Teorica Departamento de Fısica Universidad Nacional de La Plata IFLP CONICETCC 67-1900 La Plata Argentina3Instituto de Fısica Universidad de Antioquia AA 1226 Medellın Colombia

Correspondence should be addressed to Richard H Benavides ribebenavidesgmailcom

Received 21 December 2014 Accepted 24 February 2015

Academic Editor Niels Bjerrum-Bohr

Copyright copy 2015 Richard H Benavides et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

Lepton number violation and its relation to neutrino masses are investigated in several versions of the 119878119880(3)119888otimes 119878119880(3)

119871otimes 119880(1)

119909

model Spontaneous and explicit violation and conservation of the lepton number are considered In one of the models (the so-called economical one) the lepton number is spontaneously violated and it is found that the would be Majoron is not presentbecause it is gauged away providing in this way the longitudinal polarization component to a now massive gauge field

1 Introduction

The colorless and electric neutral charge distinguish the threeleft handed neutrinos ]0

119897119871 119897 = 119890 120583 120591 from the other fermions

of the 119878119880(3)119888otimes 119878119880(2)

119871otimes 119880(1)

119884Standard Model (SM) The

neutrino right handed components ]0119888119897119871

are not included inthe spectrum of the SM which has only one scalar Higgsdoublet implying massless neutrinos at the tree level Thisresult holds at all orders in perturbation theory and alsowhennonperturbative effects are taken into account due to theexistence of an exact baryon minus lepton number (119861 minus 119871)symmetry even if (119861 + 119871) is violated by weak sphaleronconfigurations Nevertheless neutrinos oscillate [1ndash5] whichimplies that at least two of them have small but nonzeromasses

Masses for neutrinos require physics beyond the SMconnected either to the existence of ]0119888

119897119871andor to the breaking

of the (119861 minus 119871) symmetry If right handed neutrinos exist theYukawa terms lead after electroweak symmetry breaking toDirac neutrino masses requiring Yukawa coupling constantsfor neutrinos ℎ120601] le 10

minus13 But ]0119888119897119871 singlets under the SMgauge

group can acquire large Majorana masses and turn on thesee-sawmechanism [6ndash10] an appealing and natural sceneryfor neutrino mass generation

Alternatively the left handed neutrinos members ofthe SM lepton doublets 120595

119897119871= (]

119897 119897minus

)119879

119871 can also acquire

a Majorana mass 119898] which carries weak isospin 119868 = 1

and violates lepton number 119871 by two units generated vianonrenormalizable operators of the form (120595

119888

119897119871120601lowast

)(120601dagger

1205951198971015840119871)

where 120601 = (120601+ 1206010) is the SM Higgs doublet and 120601 = 1198941205902120601lowast

This dimension five operator is able to generate type I typeII and type III see-saw mechanisms by using for heavy fieldsan 119878119880(2)

119871singlet fermion triplet scalar and triplet fermion

respectively (in this regard see [11 12])The mechanism of coupling two standard model lepton

doublets with a Higgs triplet Δ which in turn developsnonzero vacuum expectation values (VEV) breaking in thisway the lepton number spontaneously implies in turn theexistence of a Majoron [13] particle ruled out experimentallyby the 119885 line shape measurements [14 15] (a singlet Majoronmay still survive but with large constraints [16 17])

The variant Zee mechanism [18ndash20] can be implementedwhen the 119871 = 2 Lorentz scalar 120595

119897119871119862120595

1198971015840119871(with 119862 the charge

conjugation matrix) is coupled to an 119878119880(2)119871charged singlet

ℎ+ with 119871 = minus2 introducing next a new scalar doublet 1206011015840

and breaking the 119871 symmetry explicitly in the scalar potentialwith a term of the form1206011206011015840ℎ+ In this way neutrinoMajorana

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 813129 9 pageshttpdxdoiorg1011552015813129

2 Advances in High Energy Physics

masses are generated by one-loop quantum effects and theunwanted Majoron is not present

The second Higgs doublet 1206011015840 can be avoided by introduc-ing instead a double charged Higgs singlet 119896++ which couplesto the single charged one by the trilinear coupling 119896++ℎminusℎminusand to the right handed charged leptons singlets 119897minus

119877via a term

of the form 119897minus1198771198621198971015840minus

119877119896++ generating in this way Majorana small

masses via two-loop quantum effects by what is known as theZee-Babu mechanism [21 22]

More examples of generation of neutrino masses viaquantum effects can be found in the systematic study pre-sented in [23]

This situationmotivates us to perform an extensive analy-sis of the lepton number symmetry in the most relevant 3-3-1models In particular we are interested in the gauging awaymechanism of the Majoron that the so-called economicalmodel presents due to the subtle connection between thelepton number generator 119871 and one of the 119878119880(3)

119871generators

This paper is organized as follows In Section 2 we reviewthe charge assignment and the gauge boson content of the 3-3-1 models in general in Section 3 the four possibilities forlepton number violation in the context of theminimal versionof the 3-3-1 model are presented and in Section 4 we classifyall the 3-3-1 models without exotic electric charges and repeatthe analysis presented in Section 3 but now for the so-called3-3-1 model with right handed neutrinos Then in Section 5we do the general analysis for the 8 different 3-3-1 modelswithout exotic electric charges with 3 families In Section 6the so-called economical model is studied and finally ourconclusions are presented in Section 7

2 3-3-1 Models

Some interesting extensions of the SM are based on the localgauge group 119878119880(3)

119888otimes119878119880(3)

119871otimes119880(1)

119909(3-3-1 for short) inwhich

the weak sector of the SM is extended to 119878119880(3)119871otimes 119880(1)

119909

Severalmodels for this gauge structure have been constructedso far

For the 3-3-1 models the most general electric chargeoperator in the extended electroweak sector is

119876 = 1198861205823+1

radic31198871205828+ 119909119868

3 (1)

where 120582120572 120572 = 1 2 8 are the Gell-Mann matrices for

119878119880(3)119871normalized as Tr(120582

120572120582120573) = 2120575

120572120573and 119868

3= 119863119892(1 1 1) is

the diagonal 3times3 unit matrix 119886 = 12 if one assumes that theisospin 119878119880(2)

119871of the SM is entirely embedded in 119878119880(3)

119871 119887 is

a free parameter which defines the different possible modelsand the 119909 values are obtained by anomaly cancellation For119860120572

120583 the 8 gauge fields of 119878119880(3)

119871 119909 = 0 and thus we may write

sum

120572

120582120572119860120572

120583= radic2(

1198630

1120583119882+

120583119870(119887+12)

120583

119882minus

1205831198630

2120583119870(119887minus12)

120583

119870minus(119887+12)

120583119870minus(119887minus12)

1205831198630

3120583

) (2)

where 11986301120583= 119860

3

120583radic2 + 119860

8

120583radic6 1198630

2120583= minus119860

3

120583radic2 + 119860

8

120583radic6

and11986303120583= minus2119860

8

120583radic6 The upper indices on the gauge bosons

stand for the electric charge of the particles some of thembeing functions of the 119887 parameter

3 The Minimal Model

In [24 25] it has been shown that for 119887 = 32 the followingfermion structure is free of all the gauge anomalies 120595119879

119897119871=

(]0119897 119897minus

119897+

)119871sim (1 3 0) 119876119879

119894119871= (119889

119894 119906119894 119883

119894)119871sim (3 3

lowast

minus13)119876119879

3119871= (119906

3 1198893 119884) sim (3 3 23) where 119897 = 119890 120583 120591 is a family

lepton index 119894 = 1 2 for the first two quark families and thenumbers after the similarity sign mean 3-3-1 representationsThe right handed fields are 119906119888

119886119871sim (3

lowast

1 minus23) 119889119888119886119871

sim

(3lowast

1 13) 119883119888119894119871sim (3

lowast

1 43) and 119884119888119871sim (3

lowast

1 minus53) where119886 = 1 2 3 is the quark family index and there are two exoticquarks with electric charge minus43(119883

119894) and other with electric

charge 53(119884)This version is calledminimal in the literaturebecause its lepton content is just the one present in the SM

For this model the minimal scalar content required tobreak the symmetry giving a realisticmass spectrum consistsof three triplets and one sextet 120578119879 = (1205780 120578minus

1 120578+

2) sim (1 3 0)

120588119879

= (120588+

1205880

120588++

) sim (1 3 1) 120594119879 = (120594minus 120594minusminus 1205940) sim (1 3 minus1)and

119878 = (

1205900

1119904+

1119904minus

2

119904+

1119904++

11205900

2

119904minus

21205900

2119904minusminus

2

) sim (1 6lowast

0) (3)

The scalars have Yukawa couplings to the leptons and quarksas follows

L119897

1= ℎ

120578

1198971198971015840120578120595119897119871119862120595

1198971015840119871+ ℎ

119904

1198971198971015840120595119897119871119878119862120595

1198971015840119871+ ℎ119888

L119902

1= ℎ

119906

119894119886119876119879

119894119871120588119862119906

119888

119886119871+ ℎ

119889

119894119886119876119879

119894119871120578119862119889

119888

119886119871

+ ℎ119883

119894119895119876119879

119894119871120594119862119883

119888

119895119871+ ℎ

119889

3119886119876119879

3119871120588lowast

119862119889119888

119886119871

+ ℎ119906

3119886119876119879

3119871120578lowast

119862119906119888

119886119871+ ℎ

119884

119876119879

3119871120594lowast

119862119884119888

119871+ ℎ119888

(4)

with vacuum expectation values (VEV) given by ⟨1205780⟩ = V1

⟨1205880

⟩ = V2⟨1205940⟩ = V

3 ⟨1205900

1⟩ = V

4 and ⟨1205900

2⟩ = V1015840

4

One of the main characteristics of this model is the factthat the lepton number 119871 is not a good quantum numberbecause both the charged lepton and its antiparticle are inthe same multiplet as a consequence 119871 does not commutewith the electroweak extended gauge symmetry

The assignment of 119871 starts with the SM assignments [26]

119871 (119897minus

119871 ]119897119871) = minus119871 (119897

+

119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

(5)

Advances in High Energy Physics 3

then looking to the Yukawa interactions of the SM particlesand imposing 119871 = 0 in the covariant derivative imply

119871 (119870++

119870+

119884119871 119883

119888

119894119871) = minus2

119871 (119870minusminus

119870minus

119883119894119871 119884119888

119871) = 2

(6)

For the scalars 119871 is assigned by inspection of the Yukawacoupling constants and one finds

119871 (120594minus

120594minusminus

119904minusminus

2) = 2

119871 (120578+

2 120588++

1205900

1 119904+

1 119904++

1) = minus2

119871 (1205780

120578minus

1 1205940

120588+

1205880

1205900

2 119904minus

2) = 0

(7)

Notice that119883119894and119884 are bi-leptoquarks and119870+119870minus119870++ and

119870minusminus are bi-lepton gauge bosons Finally the physical gauge

bosons related to the neutral currents of the model have 119871 =0

It is interesting to notice that the above lepton numbers ofthe individual components of eachmultiplet can bewritten as[27]

119871 =21205828

radic3+L119868

3 (8)

whereL is a global symmetry of the Lagrangian which is notbroken by theVEV and is related to the following assignmentL(120595

119897119871) = 13L(119876

119894119871) = 23L(119876

3119871 120578 119878 120588) = minus23L(120594) =

43L(119883119888119894119871) = minus2L(119884119888

119871) = 2 andL(119906119888

119886 119889119888

119886 119860

120572120583) = 0

The former analysis shows that since 119871(1205780 1205940 1205880 12059002) =

0 the only place where the 119871 number can be spontaneouslyviolated is in 1205900

1 but it may be explicitly violated in the scalar

potential As a matter of fact a term like

119881119871119881= 119891

1120578119878120578 + 119891

2119878119878119878 + 120581

1(120594dagger

120578) (120588dagger

120578)

+ 1205812120578dagger

119878120594120588 + 1205813120594120588119878119878 + ℎ119888

(9)

explicitly violates ΔL = Δ119871 = plusmn2 when all the VEV are zeroleaving 120582

8unbroken Then the four possibilities of lepton

number violation in the context of this model are thus asfollows

(1) 119881119871119881= 0 and ⟨119878⟩ = 0This is theminimal 3-3-1 Pisano-

Pleitez-Frampton model where total lepton numberis conserved and neutrinos are massless particlesConsequently this version of the model is in conflictwith the existence of massive neutrinos

(2) 119881119871119881= 0 but ⟨1205900

1⟩ = 0 In this case the lepton number

is spontaneously broken leading to a triplet MajoronThis case has been analyzed in [28 29]

(3) 119881119871119881

= 0 and ⟨12059001⟩ = 0 119871 is violated explicitly and

nonzero masses for neutrinos can be generated fromquantum corrections

(4) The case for 119881119871119881

= 0 and ⟨12059001⟩ = 0 is also possible

with a rich phenomenology whichmay include a lightpseudo Goldstone Majoron [30]

Table 1 Anomalies for 3-3-1 fermion fields structures

Anomalies 1198781

1198782

1198783

1198784

1198785

1198786

[119878119880(3)119862]2

119880(1)119883

0 0 0 0 0 0[119878119880(3)

119871]2

119880(1)119883

minus23 minus13 1 0 0 minus1[119866119903119886V]2119880(1)

1198830 0 0 0 0 0

[119880(1)119883]3 109 89 minus129 minus69 69 129

[119878119880(3)119871]3 1 minus1 minus3 3 minus3 3

4 3-3-1 Models without ExoticElectric Charges

If one wishes to avoid exotic electric charges as the onespresent in the minimal model one must choose 119887 = 12 in(1) Following [31 32] we can find six sets of fermions whichcontain the antiparticles of the charged particles which are

(i) 1198781= [(]0

120572 120572minus

119864minus

120572) 120572

+

119864+

120572]119871with quantum numbers

(1 3 minus23) (1 1 1) and (1 1 1) respectively(ii) 119878

2=[(120572

minus

]120572 119873

0

120572) 120572

+

]119871with quantumnumbers (1 3lowast

minus13) and (1 1 1) respectively(iii) 119878

3= [(119889 119906 119880) 119906

119888

119889119888

119880119888

]119871with quantumnumbers (3

3lowast

13) (3lowast 1 minus23) (3lowast 1 13) and (3lowast 1 minus23)respectively

(iv) 1198784= [(119906 119889 119863) 119906

119888

119889119888

119863119888

]119871with quantum numbers

(3 3 0) (3lowast 1 minus23) (3lowast 1 13) and (3lowast

1 13)respectively

(v) 1198785= [(119890

minus

]119890 119873

0

1) (119864

minus

1198730

2 119873

0

3) (119873

0

4 119864+

119890+

)]119871with

quantum numbers (1 3lowast minus13) (1 3lowast minus13) and(1 3

lowast

23) respectively(vi) 119878

6= [(]

119890 119890minus

119864minus

1) (119864

+

2 119873

0

1 119873

0

2) (119873

0

3 119864minus

2 119864minus

3) 119890+ 119864+

1

119864+

3]119871with quantum numbers (1 3 minus23) (1 3 13)

(1 3 minus23) (111) (111) and (111) respectively

The different anomalies for these six sets are [31] found inTable 1

With this table anomaly-free models without exoticelectric charges can be constructed for one two or morefamilies

As noted in [31] there are eight three-family models thatare anomaly free which are

(i) Model A with right handed neutrinos 31198782+ 119878

3+ 2119878

4

(ii) Model B with exotic electrons 31198781+ 2119878

3+ 119878

4

(iii) Model C with unique lepton generation one (threedifferent lepton families) 119878

1+ 119878

2+ 119878

3+ 2119878

4+ 119878

5

(iv) Model D with unique lepton generation two 1198781+119878

2+

21198783+ 119878

4+ 119878

6

(v) Model E hybrid one (two different lepton structures)1198783+ 2119878

4+ 2119878

5+ 119878

6

(vi) Model F hybrid two 21198783+ 119878

4+ 119878

5+ 2119878

6

(vii) Model G carbon copy one (three identical families asin the SM) 3(119878

4+ 119878

5)

(viii) Model H carbon copy two 3(1198783+ 119878

6)


41 The 3-3-1 Model with Right Handed Neutrinos Intro-duced in [33 34] it has the following 3-3-1 anomaly-freefermion structure

120595119879

119897119871= (119897

minus

]0119897 119873

0

119897)119871

sim (1 3lowast

minus1

3) 119897

+

119871sim (1 1 1)

119876119879

119894119871= (119906

119894 119889119894 119863

119894)119871sim (3 3 0)

119876119879

3119871= (119889

3 1199063 119880)

119871sim (3 3

lowast

1

3)

(10)

where 119897 = 119890 120583 120591 is a family lepton index 1198730119897119871

stands forelectrically neutral Weyl state and 119894 = 1 2 for the first twoquark families The right handed quark fields are

119906119888

119886119871sim (3

lowast

1 minus2

3) 119889

119888

119886119871sim (3

lowast

11

3)

119863119888

119894119871sim (3

lowast

11

3) 119880

119888

119871sim (3

lowast

1 minus2

3)

(11)

where again 119886 = 1 2 3 is the quark family index and thereare two exotic quarks with electric charge minus13(119863

119894) and other

with electric charge 23(119880)The minimal scalar content required to break the sym-

metry giving a realistic mass spectrum consists now of onlythree triplets [33 34]

120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120578119879

= (120578minus

1 1205780

2 1205780

3) sim (1 3

lowast

minus1

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(12)

with VEV given by ⟨1205880⟩119879 = (V1 0 0) ⟨1205780⟩119879 = (0 V

2 0) and

⟨1205940

⟩119879

= (0 0 119881)A careful analysis of the Yukawa terms for the lepton

sector

L119884

lep = ℎ119890

1198971198971015840120588lowast

120595119897119871119862119897+1015840

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871 (13)

shows that 1198730119897119871 the third component of the fermion triplet

(1 3lowast

minus13) must be identified with ]0119888119897119871 the antiparticle of

]0119897119871 As a consequence 119871 is not a good quantum number in

the context of this model because 119871 does not commute withthe symmetry 119878119880(3)

119871otimes 119880(1)

119883

Doing a similar analysis to the one presented for theminimal model we obtain the following lepton numberassignments [35]

119871 (119897minus

119871 ]0119897119871) = minus119871 (119897

+

119871 ]0119888119897119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

119871 (119870+

1198700

119880119871 119863

119888

119894119871) = minus119871 (119870

minus

1198700

119863119894119871 119880119888

119871) = minus2

119871 (120594minus

1 1205940

2) = minus119871 (120588

+

3 1205780

3) = 2

119871 (1205880

1 120588+

2 120578minus

1 1205780

2 1205940

3) = 0

(14)

Notice that the new quarks 119863119894and 119880 are bileptoquarks

and119870+119870minus 1198700 and1198700 are bi-lepton gauge bosonsAgain (8) can be used to write the previous lepton

number assignment using now the following L valuesL(120595

119897119871) = 13 L(119876

119894119871) = 23 L(119876

3119871 120578 120588) = minus23

L(120594) = 43 L(119863119888119894119871) = minus2 L(119880119888

119871) = 2 L(119897+

119871) = minus1

and L(119906119888119886119871 119889119888

119886119871 119860

120572120583) = 0 values in agreement with the one

presented in [35]For this model the quark mass spectrum was analyzed

in [35] using only the following lepton number conservationYukawa potential

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ3119886120578lowast

1198763119871119862119906

119888

119886119871+ ℎ

119894119886120578119876

119894119871119862119889

119888

119886119871+ ℎ119888

(15)

which conserves both the global numbers 119871 and L But themost general Yukawa potential for quarks must also includethe following terms

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871

+ ℎ119880

120578lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120578119876

119894119871119862119863

119888

119895119871+ ℎ119888

(16)

which explicitly violates the global numbers 119871 and L Thisavoids the possible existence of a Majoron in the context ofthis model

The Yukawa Lagrangian for the neutral leptons extractedfrom (13) and in the basis (]

1 ]2 ]3 ]1198881 ]1198882 ]1198883) produces the

following tree level neutrino mass matrix

119872 =

(((((

(

0 0 0 0 119886 119887

0 0 0 minus119886 0 119888

0 0 0 minus119887 minus119888 0

0 minus119886 minus119887 0 0 0

119886 0 minus119888 0 0 0

119887 119888 0 0 0 0

)))))

)

(17)

where the entries are Dirac masses at the SM scale timesYukawa couplings with eigenvalues (0 0 plusmn119898] plusmn119898]) where119898] = radic119886

2 + 1198872 + 1198882 which stands for three Dirac neutrinosone massless and two degenerated The model is viable onlyfor very small Yukawa couplings constants and radiativecorrections able to remove the degeneracies (analysis done toa limited extent in [35])

In general 12059402and 1205780

3can have a VEV different from zero

which could imply spontaneous symmetry breaking of thelepton number 119871 But 119871 can also be broken explicitly in thescalar potential by terms of the form

1198811015840

119871119881= 120583120594

dagger

120578 + 120578dagger

120594 (1205811

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205812

10038161003816100381610038161205781003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2

)

+ 1205814

10038161003816100381610038161003816120594dagger

12057810038161003816100381610038161003816

2

+ 1205815(120578dagger

120588) (120588dagger

120594) + ℎ119888

(18)


which again satisfy ΔL = Δ119871 = plusmn2 when all the VEV arezero leaving 120582

8to be broken explicitly

As in the minimal model there are four different cases(1) 1198811015840

119881119871= 0 ⟨1205940

2⟩ = ⟨120578

0

3⟩ = 0 The total lepton number is

conserved and the neutrinos can pick up only Diractype masses

(2) 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 The lepton

number 119871 is now spontaneously violated This casehas been analyzed in [36] where a CP odd Majoronwas found

(3) 1198811015840119881119871

= 0 ⟨12059402⟩ = ⟨120578

0

3⟩ = 0 119871 is explicitly violated and

again nonzero masses for neutrinos can be generatedby quantum effects

(4) Again 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 is

also possible leading to a phenomenology with thepresence of a light pseudo Goldstone Majoron

5 The Neutral Sector

To present the kind of analysis we are aimed to let usconcentrate on Model D to start with

The lepton fields for this particular model are included inthe structure 119878

1+ 119878

2+ 119878

6which contains 21 two component

spinors including seven neutral Weyl states Let us writethem in the following way

1205951119871= (]

1 119897minus

1 119864minus

0)119871sim (1 3 minus

2

3)

119897+

1119871sim (1 1 1) 119864

+

0119871sim (1 1 1)

1205952119871= (119897

minus

2 ]2 119873

0

0)119871

sim (1 3lowast

minus1

3) 119897

+

2119871sim (1 1 1)

1205953119871= (]

3 119897minus

3 119864minus

1)119871sim (1 3 minus

2

3)

119897+

3119871sim (1 1 1) 119864

+

1119871sim (1 1 1)

1205954119871= (119864

+

2 119873

0

1 119873

0

2)119871

sim (1 31

3)

1205955119871= (119873

0

3 119864minus

2 119864minus

3)119871

sim (1 3 minus2

3) 119864

+

3119871sim (1 1 1)

(19)

with the 3-3-1 quantum numbers given in parenthesisUsing the scalars of (12) with the VEV as stated

the mass matrix for the neutral sector in the basis(]1 ]2 ]3 119873

0

0 119873

0

1 119873

0

2 119873

0

3) is now of the form

119872119899=

((((((((

(

0 0 0 0 119860 minus119886 0

0 0 0 0 119872 0 0

0 0 0 0 119861 minus119887 0

0 0 0 0 0 119872 0

119860 119872 119861 0 0 0 119866

minus119886 0 minus119887 119872 0 0 minus119889

0 0 0 0 119866 minus119889 0

))))))))

)

(20)

Table 2 Tree level neutrinos sectors

Model Number of Weylneutral states

MasslessWeyl states

Dirac states atthe EW scale

A 6 2 2B 3 3 0C 8 0 3D 7 3 0E 14 0 3F 13 0 1G 12 0 3H 15 0 4

where the 119872 value is related to a GUT mass scale comingfrom the bare mass term 120595

2119871119862120595

4119871+ ℎ119888 119860 119861 and 119862 are mass

terms at the TeV scale 119881 and 119886 119887 and 119888 are mass terms atthe electroweak scale V sim V

1sim V

2 The diagonalization of

the former mass matrix produces two Dirac massive spinorswith masses at the GUT scale and three Weyl massless statesthat we can associate with the detected solar and atmosphericoscillating neutrinos

So up to this point the model has the potential tobe consistent with the neutrino phenomenology But thequestion is if the three Weyl states remain massless or if theymay pick up small radiative masses in the context of themodel or a simple extension of it something out of the reachof the analysis presented here

51 General Analysis for 3 Families Analysis similar tothe previous one has been carried through for the neutralfermion sector of the eight anomaly-free lepton structuresenumerated in Section 4The results are presented in Table 2

According to this table only models B and D fulfill thenatural condition of having 3 tree-level zero mass neutrinoswhich may pick up nonzero masses via radiative correctionswith or without the addition of new ingredients Some otherstructures may become realistic if new fields are addedandor if some Yukawa coupling constants are fine tuned tovery small values andor if discreet symmetrieswhich forbidsYukawa terms are imposed and so forth

Let us see this in the following example

52 The 3-3-1 Model with Exotic Electrons To see what kindof new ingredients are needed in order to provide masses tothe neutral fields in these 3-3-1 models without exotic electriccharges let us briefly view the situation for model B whichwas introduced in the literature for the first time in [37] Theneutral fermion sector for this model has been studied insome detail in [38 39] but the approach here is simpler

The anomaly-free fermion structure for thismodel is [37]

120595119879

119897119871= (]0

119897 119897minus

119864minus

119897)119871

sim (1 3 minus2

3)

119897+

119871sim (1 1 1) 119864

+

119897119871sim (1 1 1)

119876119879

119894119871= (119889

119894 119906119894 119880119894)119871sim (3 3

lowast

1

3)


119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)

where 119897 = 119890 120583 120591 is a lepton family index 119864minus119897stands for three

exotic electron fields 119894 = 1 2 for the first two quark families119886 = 1 2 3 is again the quark family index and there are twoexotic quarks with electric charge 23(119880

119894) and other one with

electric charge minus13(119863) This model does not contain righthanded neutrino fields

The gauge boson and scalar sectors for this model areexactly the same ones to that for the model with right handedneutrinos [33] but the big differences are that now the leptonnumber 119871 is a good quantum number of the model and thegauge bosons do not carry lepton number at all neither theexotic quarksThe scalars (120578 120588 120594) introduced have also119871 = 0the lepton number cannot be broken spontaneously and asa consequence the neutrinos remain massless even with theinclusion of the radiative corrections

In what follows and in order to simplify matters andmake this model more predictable we consider only the setof two scalar triplets 120594 and 120588 instead of the set of threetriplets proposed in the original paper [37] or themuchmorecomplex structure introduced in [38] Also let us take theVEV to be ⟨120594⟩119879 = (0 V 119881) and ⟨120588⟩119879 = (V

1 0 0) (as in the

Economical 3-3-1 model [40] which is going to be studiednext) The Yukawa couplings of the leptons to this scalars arenow

L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)

which for 119897 1198971015840 = 119890 120583 120591 saturates all the entries of the 6 times 6charged leptonmass matrix and allows tree-level masses onlyfor charged leptons even though there are in (22) externallegs with neutrino fields of the form ]0

119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+

ℎ119888The possible inclusion of the scalar 120578 does not change thissituation at all

Masses for neutrinos can be obtained only by enlargingthe model For this purpose one can introduce a new scalartriplet 120601 = (120601++

1 120601+

2 120601+

3) sim (1 3 43) which couples to the

spin 12 leptons via a term in the Lagrangian of the form

L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes

Figure 1 Generation of the neutrino masses via the one loopradiative mechanism in the 3-3-1 model with exotic electrons

which implies lepton number values 119871(120601++1 120601+

2 120601+

3) = minus2 in

order to have it conserved in L119897

3 Notice that the expression

above also provides several external legs with neutrino fieldswhich can be used to generate masses to the neutral fermionsvia quantum effects

Since ⟨120601⟩ = (0 0 0) the new scalar fields are not ableto break spontaneously the lepton number But the point isthat the lepton symmetry is now explicitly broken in theLagrangian by a term in the scalar potential of the form120582(120601sdot120594)(120588

lowast

sdot120594)which violates lepton number by two units andturns on the Zee radiative mechanism in the context of this3-3-1 model with exotic electrons As a matter of fact all theprevious ingredients allow us to draw the diagram in Figure 1in the context of the field structure presented so far

Although the scalar sector has three independent fields(120594 120588 120601) its VEV structure is simpler than the one proposedin the original paper [37]

Neutrino masses in the context of the model analyzed inthis section were studied for the first time in [41] The maindifference between that paper and this one is that in [41] andin order to implement the Zee-Babu mechanism [18 19 21]for generating neutrino mass terms a double charged Higgsscalar 119878119880(3)

119871singlet 119896++ sim (1 1 2) was used instead of our

120601 scalar triplet which is the new and main ingredient of ouranalysis So both papers address the same problem from twodifferent points of view

6 The Economical 3-3-1 Model

The model was introduced for the first time [40 42] and thequark and lepton content corresponds to the 3-3-1model withright handed neutrinos presented above but the scalar sectoris modified becoming minimal in the sense that only twoscalar triplets (with a modified VEV structure) are used inorder to break the symmetry They are

120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)

with the VEV given now by ⟨120588⟩119879 = (V1 0 0) and ⟨120594⟩119879 =

(0 V 119881)The lepton number 119871 and the global symmetry L are as

given for the model with right handed neutrinos and (8) andthe lepton number assignment in (14) still hold


This model has been the subject of several recent studies[42ndash44] and it has the peculiarity that the lepton number 119871 isspontaneously broken due to the fact that 119871(1205940

2) = 2

Since the scalar sector is very simple now the model ishighly predictable As amatter of fact the full scalar potentialconsists only of the following six terms [40]

119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)

A simple calculation shows that both L and the leptonnumber 119871 are conserved by 119881(120594 120588) and also by the fullLagrangian except for some of the following Yukawa inter-actions which induce masses for the fermions

L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)

where the subscripts LNC and LNV indicate lepton numberconserving and lepton number violating term respectivelyAs a matter of fact L119884

LNV violates explicitly L and 119871 by twounits

After spontaneous breaking of the gauge symmetry thescalar potential develops the following lepton number violat-ing terms

119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)

where we have defined as usual 12059402= V + (119867

120594+ 119894119860

120594)radic2

119867120594and119860

120594are the so-called CP even and CP odd (scalar and

pseudo scalar) components and for simplicity we are takingreal VEV (CP violation through the scalar exchange has notbeen considered here)

Notice that the lepton number violating part in (28) istrilinear in the scalar fields and as expected 119881LNV = 0 forV = 0 From the former expression we can identify 119860

120594as the

only candidate for a Majoron in this modelThe minimization of the scalar potential has been done

in full detail in [40] (reproduced also in the second paper in[42]) For that purpose twomore definitionswere introduced1205880

1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An

outline of the main results in [40] important for our presentdiscussion are as follows

(i) The three CP odd pseudo scalars1198601205941198601015840

120594 and119860

120588 the

would beGoldstone bosons are eaten upby1198851198851015840 and(1198700

+ 1198700

)radic2 the real part of the neutral bi-leptongauge boson

(ii) Out of the three CP even scalars (V1198671015840120594minus 119881119867

120594)

radicV2 + 1198812 becomes a would be Goldstone boson eatenup by 119894(1198700minus1198700)radic2 the imaginary part of the neutralbi-lepton gauge boson which picks up 119871 = 2 via 119867

120594

The other two CP even scalars become the SM Higgsboson and one extra Higgs boson with a heavy massof order 119881 respectively

(iii) In the charged scalar sector (120588plusmn2 120594plusmn

1 120588plusmn

3) there are four

would be Goldstone bosons two of them are (119881120594plusmn2minus

V1120588plusmn

3)radic1198812 + V2

1with 119871 = plusmn2 eaten up by 119870plusmn and

other two with 119871 = 0 eaten up by119882plusmn(iv) Two charged scalars remain as physical states

Counting degrees of freedom tell us that there are in 120594 and120588 twelve ones namely three CP even three CP odd and sixcharged ones Eight of them are eaten up by the eight gaugebosons 119882plusmn 119870plusmn 1198700 1198700 119885 and 1198851015840 Four scalars remain asphysical states one of them being the SM Higgs scalar

Since 119871 is explicitly broken in the context of this modelthe most outstanding result in our analysis is that thewould be pseudo Goldstone Majoron 119860

120594 the only CP odd

electrically neutral scalar with 119871 = 2 has been eaten up by(1198700

+ 1198700

)radic2 the real part of the bi-lepton gauge boson Aclever way to avoid an unwanted Majoron

A variant of this model was considered in [45] where thefermion mass spectrum was studied with the inclusion of a1198852discrete symmetry which excludes the LNV interactions

in the Yukawa potential in (27) For this variant of the modelL is conserved through the entire Lagrangian the leptonnumber 119871 is only spontaneous violated by 119881LNV in (28) andthe would be Majoron 119860

120594is gauge away eaten up by (1198700 +

1198700

)radic2 Notice that being L a good quantum number thespontaneous violation of 119878119880(3)

119871implies the spontaneous

violation of119871 via (8) something that it is now allowed becausethe fermion sector for119871 is vector like and thus nonanomalous

The economical scalar structure presented here is not ableto reproduce a consistent quark mass spectrum at tree levelBy fortune a careful analysis combining the renormalizableYukawa interactions in (26) and (27) and the effectivedimension-five operators

LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)

are able to remove the zero quark masses But the imple-mentation of LNR in the former expression requires theintroduction of new and heavy scalar fields

But there remains the question of the quantum effectsA careful analysis shows that the conclusion in [46] relatedwith the quark mass matrices is true that is the inclusion ofall the one-loop diagrams with the proper Yukawa couplingsstill leaves the quark mass matrices with determinant equal


to zero So contrary to what is stated in [42 47] theone-loop diagrams are not able by themselves to providea consistent mass spectrum for the quarks in the contextof this economical model But it does not mean that thereis a remanent 119880(1) symmetry in the full Lagrangian as itis erroneously stated in [46] (in fact in [47] it is clearlyproved that such a 119880(1) symmetry does not exist at all) Thesolution to this puzzle and to the controversy raised between[46 47] lies in the two-loop quantum effects which providesa consistent quark mass spectrum via Babu type mechanisms[21] But this analysis lies outside the scope of this paper andit will be presented elsewhere

To conclude this section let us mention that the versionof this economical 3-3-1 model developed in the context ofthe model with right handed neutrinos can be extended toany one of the eight 3 family models presented in Section 4

7 Conclusions

The main motivation of our study was to investigate theneutrino mass spectrum in the framework of the local gaugestructure 119878119880(3)

119888otimes 119878119880(3)

119871otimes 119880(1)

119909

To summarize we have carried out an extensive analysisof the lepton number symmetry in the context of the bestknown versions of the 3-3-1 model It is interesting to remarkthat in one of these versions namely the so-called economicalmodel one explicitly finds the quite unusual situation of thegauging away of the would beMajoron providing in this waythe longitudinal polarization component to a now massivegauge field

This rare but quite unusual mechanism is related to thefact that the lepton number generator 119871 is connected with the1205828generator of 119878119880(3)

119871 as shown in (8)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Enrico Nardi for a written communicationand Vicente Vento for his comments William A Ponceand Richard H Benavides thank the ldquoLaboratorio de FısicaTeoricardquo from U de La Plata in Argentina for the warmhospitality during the initial stages of the work which hasbeen partially supported by ldquoSostenibilidad U de A 2014rdquoand ldquoCentro de Investigaciones del ITMrdquo

References

[1] S Fukuda Y Fukuda M Ishitsuka et al ldquoConstraints onneutrino oscillations using 1258 days of Super-Kamiokandesolar neutrino datardquo Physical Review Letters vol 86 no 25 pp5656ndash5660 2001

[2] Q R Ahmad R C Allen T C Andersen et al ldquoDirect evidencefor neutrino flavor transformation from neutral-current inter-actions in the sudbury neutrino observatoryrdquo Physical ReviewLetters vol 89 no 1 Article ID 011301 6 pages 2002

[3] K Eguchi S Enomoto K Furuno et al ldquoFirst results fromKamLAND evidence for reactor antineutrino disappearancerdquoPhysical Review Letters vol 90 no 2 Article ID 021802 6 pages2003

[4] YAshie JHosakaK Ishihara et al ldquoEvidence for an oscillatorysignature in atmospheric neutrino oscillationsrdquo Physical ReviewLetters vol 93 Article ID 101801 2004

[5] K Nakamura K Hagiwara K Hikasa et al ldquoReview of particlephysicsrdquo Journal of Physics G vol 37 no 7 Article ID 0750212010

[6] PMinkowski ldquo120583 rarr 119890120574 at a rate of one out of 109muondecaysrdquoPhysics Letters B vol 67 no 4 pp 421ndash428 1977

[7] M Gell-Mann P Ramond and R Slansky Supergravity editedby Pvan Nieuwenhuizen and D Z Freedman 1980

[8] T Yanahida in Proceedings of the Workshop on the UnifiedTheory and the Baryon Number in the Universe O Sawada andA Sugamoto Eds p 95 KEK Tsukuba Japan 1979

[9] R Mohapatra and G Senjanovic ldquoNeutrino mass and sponta-neous parity nonconservationrdquo Physical Review Letters vol 44no 14 pp 912ndash915 1980

[10] R N Mohapatra and G Senjanovic ldquoNeutrino masses andmixings in gauge models with spontaneous parity violationrdquoPhysical Review D vol 23 p 165 1981

[11] S Weinberg ldquoBaryon- and lepton-nonconserving processesrdquoPhysical Review Letters vol 43 no 21 pp 1566ndash1570 1979

[12] F Bonnet M Hirsch T Ota and W Winter ldquoSystematic studyof the 119889 = 5Wein-berg operator at one-loop orderrdquo Journal ofHigh Energy Physics vol 2012 no 7 article 153 2012

[13] G B Gelmini and M Roncadelli ldquoLeft-handed neutrino massscale and spontaneously broken lepton numberrdquo Physics LettersB vol 99 no 5 pp 411ndash415 1981

[14] J Steinberger ldquoFirst results at the LEP 119890+119890minus colliderrdquo PhysicsReports vol 203 p 345 1991

[15] E Accomando A Andreazza H Anlaufc et al ldquoPhysics with119890+

119890minus linear collidersrdquo Physics Reports vol 299 pp 1ndash78 1998

[16] M Gunther J Hellmig G Heusser et al ldquoBounds on newMajoron models from the Heidelberg-Moscow experimentrdquoPhysical Review D vol 54 p 3641 1996

[17] R Tomas H Pas and J W F Valle ldquoGeneralized bounds onMajoron-neutrino couplingsrdquo Physical Review D vol 64 no 9Article ID 0950005 7 pages 2001

[18] A Zee ldquoA theory of lepton number violation and neutrinoMajorana massesrdquo Physics Letters B vol 93 no 4 pp 389ndash3931980

[19] A Zee ldquoCharged scalar field and quantum number violationsrdquoPhysics Letters B vol 161 no 1ndash3 pp 141ndash145 1985

[20] D Chang and A Zee ldquoRadiatively induced neutrino Majoranamasses and oscillationrdquo Physical Review D vol 61 Article ID071303R 2000

[21] A Zee ldquoQuantum numbers of Majorana neutrino massesrdquoNuclear Physics B vol 246 p 99 1986

[22] K S Babu ldquoModel of rdquocalculablerdquo Majorana neutrino massesrdquoPhysics Letters B vol 203 no 1-2 pp 132ndash136 1988

[23] D Restrepo O Zapata and C E Yaguna ldquoModels withradiative neutrino masses and viable dark matter candidatesrdquoJournal of High Energy Physics vol 2013 article 11 2013

[24] F Pisano andV Pleitez ldquoAn 119878119880(3)otimes119880(1)model for electroweakinteractionsrdquo Physical Review D vol 46 p 410 1992

[25] P Frampton ldquoChiral dilepton model and the flavor questionrdquoPhysical Review Letters vol 69 no 20 pp 2889ndash2891 1992


[26] J T Liu and D Ng ldquoLepton-flavor-changing processes and CPviolation in the SU(3)

119888timesSU(3)

119871timesU(1)

119883modelrdquo Physical Review

D vol 50 p 548 1994[27] M B Tully and G C Joshi ldquoGenerating neutrinomass in the 3-

3-1modelrdquo Physical ReviewD vol 64 Article ID 011301 4 pages2001

[28] J C Montero C A D S Pires and V Pleitez ldquoCommenton lsquoMajoron emitting neutrinoless double beta decay in theelectroweak chiral gauge extensionsrsquordquo Physical ReviewD vol 60Article ID 098701 1999

[29] J C Montero C A D S Pires and V Pleitez ldquoSpontaneousbreaking of a global symmetry in a 3-3-1modelrdquoPhysical ReviewD vol 60 Article ID 115003 1999

[30] L F Li Y Liu and L Wolfenstein ldquoHidden higgs particlesrdquoPhysics Letters B vol 159 no 1 pp 45ndash48 1985

[31] WA Ponce J B Florez and L A Sanchez ldquoAnalysis of SU(3)119888times

SU(3)119871times U(1)

119883local Gauge theoryrdquo International Journal of

Modern Physics A vol 17 p 643 2002[32] D L Anderson and M Sher ldquo3-3-1 models with unique lepton

generationsrdquo Physical ReviewD vol 72 no 9 Article ID 0950149 pages 2005

[33] J C Montero F Pisano and V Pleitez ldquoNeutral currentsand Glashow-Iliopoulos-Maiani mechanism in SU(3)

119871otimesU(1)

119873

models for electroweak interactionsrdquo Physical Review D vol 47no 7 pp 2918ndash2929 1993

[34] R Foot H N Long and T A Tran ldquo119878119880(3)119871otimes 119880(1)

119873and

119878119880(4)119871otimes 119880(1)

119873gauge models with right-handed neutrinosrdquo

Physical Review D vol 50 no 1 pp R34ndashR38 1994[35] D Chang and H N Long ldquoInteresting radiative patterns of

neutrino mass in an SU(3)119862otimes SU(3)

119871otimes U(1)

119883model with right-

handed neutrinosrdquo Physical Review D vol 73 no 5 Article ID053006 17 pages 2006

[36] C A S de Pires and P S R da Silva ldquoSpontaneous breaking ofthe lepton number and invisible majoron in a 3-3-1 modelrdquoTheEuropean Physical Journal CmdashParticles and Fields vol 36 no 3pp 397ndash403 2004

[37] M Ozer ldquoSU(3)119871times U(1)

119883model of electroweak interactions

without exotic quarksrdquo Physical Review D vol 54 no 1 pp1143ndash1149 1996

[38] W A Ponce andO Zapata ldquoLeptonmasses andmixing withoutYukawa hierarchiesrdquo Physical ReviewD vol 74 no 9 Article ID093007 7 pages 2006

[39] J C SalazarWA Ponce andDA Gutierrez ldquoPhenomenologyof the 119878119880(3)

119888⨂119878119880(3)

119871⨂119880(1)

119883model with exotic charged

leptonsrdquo Physical Review D vol 75 no 7 Article ID 075016 17pages 2007

[40] W A Ponce Y Giraldo and L A Sanchez ldquoMinimal scalarsector of 3-3-1 models without exotic electric chargesrdquo PhysicalReview D vol 67 no 7 Article ID 075001 10 pages 2003

[41] T Kitabayashi ldquoRemark on neutrino masses and oscillationsin an SU(3)

119871times U(1)

119873model with the radiative mechanismrdquo

Physical Review D vol 64 no 5 Article ID 057301 4 pages2001

[42] P V Dong H N Long D T Nhung and D V Soa ldquoSU(3)119888times

SU(3)119871timesU(1)

119883model with two Higgs tripletsrdquo Physical Review

D vol 73 Article ID 035004 2006[43] P V Dong H N Long and D V Soa ldquoInteresting radiative

patterns of neutrinomass in an SU(3)119862timesSU(3)

119871timesU(1)

119883model

with right-handed neutrinosrdquo Physical Review D vol 73 no 5Article ID 053006 17 pages 2006

[44] P V Dong H N Long and D T Nhung ldquoAtomic parityviolation in the economical 3-3-1 modelrdquo Physics Letters B vol639 no 5 pp 527ndash533 2006

[45] D A Gutierrez W A Ponce and L A Sanchez ldquoStudy of theSU(3)

119862otimesSU(3)

119871otimesU(1)

119883model with theminimal scalar sectorrdquo

International Journal of Modern Physics A vol 21 no 10 p 22172006

[46] J C Montero and B L Sanchez-Vega ldquoNatural Peccei-Quinnsymmetry in the 3-3-1 model with a minimal scalar sectorrdquoPhysical Review D vol 84 Article ID 055019 2011

[47] P V Dong H N Long and H T Hung ldquoQuestion of Peccei-Quinn symmetry and quark masses in the economical 3-3-1modelrdquo Physical Review D vol 86 Article ID 033002 2012

Submit your manuscripts athttpwwwhindawicom

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masses are generated by one-loop quantum effects and theunwanted Majoron is not present

The second Higgs doublet 1206011015840 can be avoided by introduc-ing instead a double charged Higgs singlet 119896++ which couplesto the single charged one by the trilinear coupling 119896++ℎminusℎminusand to the right handed charged leptons singlets 119897minus

119877via a term

of the form 119897minus1198771198621198971015840minus

119877119896++ generating in this way Majorana small

masses via two-loop quantum effects by what is known as theZee-Babu mechanism [21 22]

More examples of generation of neutrino masses viaquantum effects can be found in the systematic study pre-sented in [23]

This situationmotivates us to perform an extensive analy-sis of the lepton number symmetry in the most relevant 3-3-1models In particular we are interested in the gauging awaymechanism of the Majoron that the so-called economicalmodel presents due to the subtle connection between thelepton number generator 119871 and one of the 119878119880(3)

119871generators

This paper is organized as follows In Section 2 we reviewthe charge assignment and the gauge boson content of the 3-3-1 models in general in Section 3 the four possibilities forlepton number violation in the context of theminimal versionof the 3-3-1 model are presented and in Section 4 we classifyall the 3-3-1 models without exotic electric charges and repeatthe analysis presented in Section 3 but now for the so-called3-3-1 model with right handed neutrinos Then in Section 5we do the general analysis for the 8 different 3-3-1 modelswithout exotic electric charges with 3 families In Section 6the so-called economical model is studied and finally ourconclusions are presented in Section 7

2 3-3-1 Models

Some interesting extensions of the SM are based on the localgauge group 119878119880(3)

119888otimes119878119880(3)

119871otimes119880(1)

119909(3-3-1 for short) inwhich

the weak sector of the SM is extended to 119878119880(3)119871otimes 119880(1)

119909

Severalmodels for this gauge structure have been constructedso far

For the 3-3-1 models the most general electric chargeoperator in the extended electroweak sector is

119876 = 1198861205823+1

radic31198871205828+ 119909119868

3 (1)

where 120582120572 120572 = 1 2 8 are the Gell-Mann matrices for

119878119880(3)119871normalized as Tr(120582

120572120582120573) = 2120575

120572120573and 119868

3= 119863119892(1 1 1) is

the diagonal 3times3 unit matrix 119886 = 12 if one assumes that theisospin 119878119880(2)

119871of the SM is entirely embedded in 119878119880(3)

119871 119887 is

a free parameter which defines the different possible modelsand the 119909 values are obtained by anomaly cancellation For119860120572

120583 the 8 gauge fields of 119878119880(3)

119871 119909 = 0 and thus we may write

sum

120572

120582120572119860120572

120583= radic2(

1198630

1120583119882+

120583119870(119887+12)

120583

119882minus

1205831198630

2120583119870(119887minus12)

120583

119870minus(119887+12)

120583119870minus(119887minus12)

1205831198630

3120583

) (2)

where 11986301120583= 119860

3

120583radic2 + 119860

8

120583radic6 1198630

2120583= minus119860

3

120583radic2 + 119860

8

120583radic6

and11986303120583= minus2119860

8

120583radic6 The upper indices on the gauge bosons

stand for the electric charge of the particles some of thembeing functions of the 119887 parameter

3 The Minimal Model

In [24 25] it has been shown that for 119887 = 32 the followingfermion structure is free of all the gauge anomalies 120595119879

119897119871=

(]0119897 119897minus

119897+

)119871sim (1 3 0) 119876119879

119894119871= (119889

119894 119906119894 119883

119894)119871sim (3 3

lowast

minus13)119876119879

3119871= (119906

3 1198893 119884) sim (3 3 23) where 119897 = 119890 120583 120591 is a family

lepton index 119894 = 1 2 for the first two quark families and thenumbers after the similarity sign mean 3-3-1 representationsThe right handed fields are 119906119888

119886119871sim (3

lowast

1 minus23) 119889119888119886119871

sim

(3lowast

1 13) 119883119888119894119871sim (3

lowast

1 43) and 119884119888119871sim (3

lowast

1 minus53) where119886 = 1 2 3 is the quark family index and there are two exoticquarks with electric charge minus43(119883

119894) and other with electric

charge 53(119884)This version is calledminimal in the literaturebecause its lepton content is just the one present in the SM

For this model the minimal scalar content required tobreak the symmetry giving a realisticmass spectrum consistsof three triplets and one sextet 120578119879 = (1205780 120578minus

1 120578+

2) sim (1 3 0)

120588119879

= (120588+

1205880

120588++

) sim (1 3 1) 120594119879 = (120594minus 120594minusminus 1205940) sim (1 3 minus1)and

119878 = (

1205900

1119904+

1119904minus

2

119904+

1119904++

11205900

2

119904minus

21205900

2119904minusminus

2

) sim (1 6lowast

0) (3)

The scalars have Yukawa couplings to the leptons and quarksas follows

L119897

1= ℎ

120578

1198971198971015840120578120595119897119871119862120595

1198971015840119871+ ℎ

119904

1198971198971015840120595119897119871119878119862120595

1198971015840119871+ ℎ119888

L119902

1= ℎ

119906

119894119886119876119879

119894119871120588119862119906

119888

119886119871+ ℎ

119889

119894119886119876119879

119894119871120578119862119889

119888

119886119871

+ ℎ119883

119894119895119876119879

119894119871120594119862119883

119888

119895119871+ ℎ

119889

3119886119876119879

3119871120588lowast

119862119889119888

119886119871

+ ℎ119906

3119886119876119879

3119871120578lowast

119862119906119888

119886119871+ ℎ

119884

119876119879

3119871120594lowast

119862119884119888

119871+ ℎ119888

(4)

with vacuum expectation values (VEV) given by ⟨1205780⟩ = V1

⟨1205880

⟩ = V2⟨1205940⟩ = V

3 ⟨1205900

1⟩ = V

4 and ⟨1205900

2⟩ = V1015840

4

One of the main characteristics of this model is the factthat the lepton number 119871 is not a good quantum numberbecause both the charged lepton and its antiparticle are inthe same multiplet as a consequence 119871 does not commutewith the electroweak extended gauge symmetry

The assignment of 119871 starts with the SM assignments [26]

119871 (119897minus

119871 ]119897119871) = minus119871 (119897

+

119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

(5)



119871 (119870++

119870+

119884119871 119883

119888

119894119871) = minus2


119870minus

119883119894119871 119884119888

119871) = 2

(6)


119871 (120594minus

120594minusminus

119904minusminus

2) = 2

119871 (120578+

2 120588++

1205900

1 119904+

1 119904++

1) = minus2

119871 (1205780

120578minus

1 1205940

120588+

1205880

1205900

2 119904minus

2) = 0

(7)





119871 =21205828

radic3+L119868

3 (8)


119897119871) = 13L(119876

119894119871) = 23L(119876

3119871 120578 119878 120588) = minus23L(120594) =

43L(119883119888119894119871) = minus2L(119884119888

119871) = 2 andL(119906119888

119886 119889119888

119886 119860

120572120583) = 0





119881119871119881= 119891

1120578119878120578 + 119891

2119878119878119878 + 120581

1(120594dagger

120578) (120588dagger

120578)

+ 1205812120578dagger

119878120594120588 + 1205813120594120588119878119878 + ℎ119888

(9)






(2) 119881119871119881= 0 but ⟨1205900



(3) 119881119871119881



(4) The case for 119881119871119881




Anomalies 1198781

1198782

1198783

1198784

1198785

1198786

[119878119880(3)119862]2

119880(1)119883

0 0 0 0 0 0[119878119880(3)

119871]2

119880(1)119883


1198830 0 0 0 0 0

[119880(1)119883]3 109 89 minus129 minus69 69 129




(i) 1198781= [(]0

120572 120572minus

119864minus

120572) 120572

+

119864+



2=[(120572

minus

]120572 119873

0

120572) 120572

+



3= [(119889 119906 119880) 119906

119888

119889119888

119880119888


3lowast


(iv) 1198784= [(119906 119889 119863) 119906

119888

119889119888

119863119888



1 13)respectively

(v) 1198785= [(119890

minus

]119890 119873

0

1) (119864

minus

1198730

2 119873

0

3) (119873

0

4 119864+

119890+

)]119871with


lowast


6= [(]

119890 119890minus

119864minus

1) (119864

+

2 119873

0

1 119873

0

2) (119873

0

3 119864minus

2 119864minus

3) 119890+ 119864+

1

119864+







3+ 2119878

4


3+ 119878

4


1+ 119878

2+ 119878

3+ 2119878

4+ 119878

5


2+

21198783+ 119878

4+ 119878

6


4+ 2119878

5+ 119878

6


4+ 119878

5+ 2119878

6


4+ 119878

5)


6)



120595119879

119897119871= (119897

minus

]0119897 119873

0

119897)119871

sim (1 3lowast

minus1

3) 119897

+

119871sim (1 1 1)

119876119879

119894119871= (119906

119894 119889119894 119863

119894)119871sim (3 3 0)

119876119879

3119871= (119889

3 1199063 119880)

119871sim (3 3

lowast

1

3)

(10)



119906119888

119886119871sim (3

lowast

1 minus2

3) 119889

119888

119886119871sim (3

lowast

11

3)

119863119888

119894119871sim (3

lowast

11

3) 119880

119888

119871sim (3

lowast

1 minus2

3)

(11)


119894) and other



120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120578119879

= (120578minus

1 1205780

2 1205780

3) sim (1 3

lowast

minus1

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(12)


2 0) and

⟨1205940

⟩119879


sector

L119884

lep = ℎ119890

1198971198971015840120588lowast

120595119897119871119862119897+1015840

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871 (13)


(1 3lowast




119871otimes 119880(1)

119883


119871 (119897minus

119871 ]0119897119871) = minus119871 (119897

+

119871 ]0119888119897119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

119871 (119870+

1198700

119880119871 119863

119888

119894119871) = minus119871 (119870

minus

1198700

119863119894119871 119880119888

119871) = minus2

119871 (120594minus

1 1205940

2) = minus119871 (120588

+

3 1205780

3) = 2

119871 (1205880

1 120588+

2 120578minus

1 1205780

2 1205940

3) = 0

(14)




119897119871) = 13 L(119876

119894119871) = 23 L(119876

3119871 120578 120588) = minus23

L(120594) = 43 L(119863119888119894119871) = minus2 L(119880119888

119871) = 2 L(119897+

119871) = minus1

and L(119906119888119886119871 119889119888

119886119871 119860




L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ3119886120578lowast

1198763119871119862119906

119888

119886119871+ ℎ

119894119886120578119876

119894119871119862119889

119888

119886119871+ ℎ119888

(15)


L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871

+ ℎ119880

120578lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120578119876

119894119871119862119863

119888

119895119871+ ℎ119888

(16)



1 ]2 ]3 ]1198881 ]1198882 ]1198883) produces the


119872 =

(((((

(

0 0 0 0 119886 119887

0 0 0 minus119886 0 119888

0 0 0 minus119887 minus119888 0

0 minus119886 minus119887 0 0 0

119886 0 minus119888 0 0 0

119887 119888 0 0 0 0

)))))

)

(17)






1198811015840

119871119881= 120583120594

dagger

120578 + 120578dagger

120594 (1205811

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205812

10038161003816100381610038161205781003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2

)

+ 1205814

10038161003816100381610038161003816120594dagger

12057810038161003816100381610038161003816

2

+ 1205815(120578dagger

120588) (120588dagger

120594) + ℎ119888

(18)





119881119871= 0 ⟨1205940

2⟩ = ⟨120578

0



(2) 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 The lepton


(3) 1198811015840119881119871

= 0 ⟨12059402⟩ = ⟨120578

0



(4) Again 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 is





1+ 119878

2+ 119878



1205951119871= (]

1 119897minus

1 119864minus

0)119871sim (1 3 minus

2

3)

119897+

1119871sim (1 1 1) 119864

+

0119871sim (1 1 1)

1205952119871= (119897

minus

2 ]2 119873

0

0)119871

sim (1 3lowast

minus1

3) 119897

+

2119871sim (1 1 1)

1205953119871= (]

3 119897minus

3 119864minus

1)119871sim (1 3 minus

2

3)

119897+

3119871sim (1 1 1) 119864

+

1119871sim (1 1 1)

1205954119871= (119864

+

2 119873

0

1 119873

0

2)119871

sim (1 31

3)

1205955119871= (119873

0

3 119864minus

2 119864minus

3)119871

sim (1 3 minus2

3) 119864

+

3119871sim (1 1 1)

(19)



0

0 119873

0

1 119873

0

2 119873

0


119872119899=

((((((((

(

0 0 0 0 119860 minus119886 0

0 0 0 0 119872 0 0

0 0 0 0 119861 minus119887 0

0 0 0 0 0 119872 0

119860 119872 119861 0 0 0 119866


0 0 0 0 119866 minus119889 0

))))))))

)

(20)



MasslessWeyl states


A 6 2 2B 3 3 0C 8 0 3D 7 3 0E 14 0 3F 13 0 1G 12 0 3H 15 0 4


2119871119862120595

4119871+ ℎ119888 119860 119861 and 119862 are mass


1sim V









120595119879

119897119871= (]0

119897 119897minus

119864minus

119897)119871

sim (1 3 minus2

3)

119897+

119871sim (1 1 1) 119864

+

119897119871sim (1 1 1)

119876119879

119894119871= (119889

119894 119906119894 119880119894)119871sim (3 3

lowast

1

3)


119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)







1 0 0) (as in the


L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)


119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+



1 120601+

2 120601+



L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes



2 120601+

3) = minus2 in





lowast








120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)






2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119871 (119870++

119870+

119884119871 119883

119888

119894119871) = minus2


119870minus

119883119894119871 119884119888

119871) = 2

(6)


119871 (120594minus

120594minusminus

119904minusminus

2) = 2

119871 (120578+

2 120588++

1205900

1 119904+

1 119904++

1) = minus2

119871 (1205780

120578minus

1 1205940

120588+

1205880

1205900

2 119904minus

2) = 0

(7)





119871 =21205828

radic3+L119868

3 (8)


119897119871) = 13L(119876

119894119871) = 23L(119876

3119871 120578 119878 120588) = minus23L(120594) =

43L(119883119888119894119871) = minus2L(119884119888

119871) = 2 andL(119906119888

119886 119889119888

119886 119860

120572120583) = 0





119881119871119881= 119891

1120578119878120578 + 119891

2119878119878119878 + 120581

1(120594dagger

120578) (120588dagger

120578)

+ 1205812120578dagger

119878120594120588 + 1205813120594120588119878119878 + ℎ119888

(9)






(2) 119881119871119881= 0 but ⟨1205900



(3) 119881119871119881



(4) The case for 119881119871119881




Anomalies 1198781

1198782

1198783

1198784

1198785

1198786

[119878119880(3)119862]2

119880(1)119883

0 0 0 0 0 0[119878119880(3)

119871]2

119880(1)119883


1198830 0 0 0 0 0

[119880(1)119883]3 109 89 minus129 minus69 69 129




(i) 1198781= [(]0

120572 120572minus

119864minus

120572) 120572

+

119864+



2=[(120572

minus

]120572 119873

0

120572) 120572

+



3= [(119889 119906 119880) 119906

119888

119889119888

119880119888


3lowast


(iv) 1198784= [(119906 119889 119863) 119906

119888

119889119888

119863119888



1 13)respectively

(v) 1198785= [(119890

minus

]119890 119873

0

1) (119864

minus

1198730

2 119873

0

3) (119873

0

4 119864+

119890+

)]119871with


lowast


6= [(]

119890 119890minus

119864minus

1) (119864

+

2 119873

0

1 119873

0

2) (119873

0

3 119864minus

2 119864minus

3) 119890+ 119864+

1

119864+







3+ 2119878

4


3+ 119878

4


1+ 119878

2+ 119878

3+ 2119878

4+ 119878

5


2+

21198783+ 119878

4+ 119878

6


4+ 2119878

5+ 119878

6


4+ 119878

5+ 2119878

6


4+ 119878

5)


6)



120595119879

119897119871= (119897

minus

]0119897 119873

0

119897)119871

sim (1 3lowast

minus1

3) 119897

+

119871sim (1 1 1)

119876119879

119894119871= (119906

119894 119889119894 119863

119894)119871sim (3 3 0)

119876119879

3119871= (119889

3 1199063 119880)

119871sim (3 3

lowast

1

3)

(10)



119906119888

119886119871sim (3

lowast

1 minus2

3) 119889

119888

119886119871sim (3

lowast

11

3)

119863119888

119894119871sim (3

lowast

11

3) 119880

119888

119871sim (3

lowast

1 minus2

3)

(11)


119894) and other



120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120578119879

= (120578minus

1 1205780

2 1205780

3) sim (1 3

lowast

minus1

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(12)


2 0) and

⟨1205940

⟩119879


sector

L119884

lep = ℎ119890

1198971198971015840120588lowast

120595119897119871119862119897+1015840

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871 (13)


(1 3lowast




119871otimes 119880(1)

119883


119871 (119897minus

119871 ]0119897119871) = minus119871 (119897

+

119871 ]0119888119897119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

119871 (119870+

1198700

119880119871 119863

119888

119894119871) = minus119871 (119870

minus

1198700

119863119894119871 119880119888

119871) = minus2

119871 (120594minus

1 1205940

2) = minus119871 (120588

+

3 1205780

3) = 2

119871 (1205880

1 120588+

2 120578minus

1 1205780

2 1205940

3) = 0

(14)




119897119871) = 13 L(119876

119894119871) = 23 L(119876

3119871 120578 120588) = minus23

L(120594) = 43 L(119863119888119894119871) = minus2 L(119880119888

119871) = 2 L(119897+

119871) = minus1

and L(119906119888119886119871 119889119888

119886119871 119860




L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ3119886120578lowast

1198763119871119862119906

119888

119886119871+ ℎ

119894119886120578119876

119894119871119862119889

119888

119886119871+ ℎ119888

(15)


L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871

+ ℎ119880

120578lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120578119876

119894119871119862119863

119888

119895119871+ ℎ119888

(16)



1 ]2 ]3 ]1198881 ]1198882 ]1198883) produces the


119872 =

(((((

(

0 0 0 0 119886 119887

0 0 0 minus119886 0 119888

0 0 0 minus119887 minus119888 0

0 minus119886 minus119887 0 0 0

119886 0 minus119888 0 0 0

119887 119888 0 0 0 0

)))))

)

(17)






1198811015840

119871119881= 120583120594

dagger

120578 + 120578dagger

120594 (1205811

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205812

10038161003816100381610038161205781003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2

)

+ 1205814

10038161003816100381610038161003816120594dagger

12057810038161003816100381610038161003816

2

+ 1205815(120578dagger

120588) (120588dagger

120594) + ℎ119888

(18)





119881119871= 0 ⟨1205940

2⟩ = ⟨120578

0



(2) 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 The lepton


(3) 1198811015840119881119871

= 0 ⟨12059402⟩ = ⟨120578

0



(4) Again 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 is





1+ 119878

2+ 119878



1205951119871= (]

1 119897minus

1 119864minus

0)119871sim (1 3 minus

2

3)

119897+

1119871sim (1 1 1) 119864

+

0119871sim (1 1 1)

1205952119871= (119897

minus

2 ]2 119873

0

0)119871

sim (1 3lowast

minus1

3) 119897

+

2119871sim (1 1 1)

1205953119871= (]

3 119897minus

3 119864minus

1)119871sim (1 3 minus

2

3)

119897+

3119871sim (1 1 1) 119864

+

1119871sim (1 1 1)

1205954119871= (119864

+

2 119873

0

1 119873

0

2)119871

sim (1 31

3)

1205955119871= (119873

0

3 119864minus

2 119864minus

3)119871

sim (1 3 minus2

3) 119864

+

3119871sim (1 1 1)

(19)



0

0 119873

0

1 119873

0

2 119873

0


119872119899=

((((((((

(

0 0 0 0 119860 minus119886 0

0 0 0 0 119872 0 0

0 0 0 0 119861 minus119887 0

0 0 0 0 0 119872 0

119860 119872 119861 0 0 0 119866


0 0 0 0 119866 minus119889 0

))))))))

)

(20)



MasslessWeyl states


A 6 2 2B 3 3 0C 8 0 3D 7 3 0E 14 0 3F 13 0 1G 12 0 3H 15 0 4


2119871119862120595

4119871+ ℎ119888 119860 119861 and 119862 are mass


1sim V









120595119879

119897119871= (]0

119897 119897minus

119864minus

119897)119871

sim (1 3 minus2

3)

119897+

119871sim (1 1 1) 119864

+

119897119871sim (1 1 1)

119876119879

119894119871= (119889

119894 119906119894 119880119894)119871sim (3 3

lowast

1

3)


119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)







1 0 0) (as in the


L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)


119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+



1 120601+

2 120601+



L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes



2 120601+

3) = minus2 in





lowast








120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)






2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120595119879

119897119871= (119897

minus

]0119897 119873

0

119897)119871

sim (1 3lowast

minus1

3) 119897

+

119871sim (1 1 1)

119876119879

119894119871= (119906

119894 119889119894 119863

119894)119871sim (3 3 0)

119876119879

3119871= (119889

3 1199063 119880)

119871sim (3 3

lowast

1

3)

(10)



119906119888

119886119871sim (3

lowast

1 minus2

3) 119889

119888

119886119871sim (3

lowast

11

3)

119863119888

119894119871sim (3

lowast

11

3) 119880

119888

119871sim (3

lowast

1 minus2

3)

(11)


119894) and other



120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120578119879

= (120578minus

1 1205780

2 1205780

3) sim (1 3

lowast

minus1

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(12)


2 0) and

⟨1205940

⟩119879


sector

L119884

lep = ℎ119890

1198971198971015840120588lowast

120595119897119871119862119897+1015840

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871 (13)


(1 3lowast




119871otimes 119880(1)

119883


119871 (119897minus

119871 ]0119897119871) = minus119871 (119897

+

119871 ]0119888119897119871) = 1

119871 (119906119886119871 119906119888

119886119871 119889119886119871 119889119888

119886119871119882

plusmn

120583 119863

0

1120583 119863

0

2120583 119863

0

3120583) = 0

119871 (119870+

1198700

119880119871 119863

119888

119894119871) = minus119871 (119870

minus

1198700

119863119894119871 119880119888

119871) = minus2

119871 (120594minus

1 1205940

2) = minus119871 (120588

+

3 1205780

3) = 2

119871 (1205880

1 120588+

2 120578minus

1 1205780

2 1205940

3) = 0

(14)




119897119871) = 13 L(119876

119894119871) = 23 L(119876

3119871 120578 120588) = minus23

L(120594) = 43 L(119863119888119894119871) = minus2 L(119880119888

119871) = 2 L(119897+

119871) = minus1

and L(119906119888119886119871 119889119888

119886119871 119860




L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ3119886120578lowast

1198763119871119862119906

119888

119886119871+ ℎ

119894119886120578119876

119894119871119862119889

119888

119886119871+ ℎ119888

(15)


L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871

+ ℎ119880

120578lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120578119876

119894119871119862119863

119888

119895119871+ ℎ119888

(16)



1 ]2 ]3 ]1198881 ]1198882 ]1198883) produces the


119872 =

(((((

(

0 0 0 0 119886 119887

0 0 0 minus119886 0 119888

0 0 0 minus119887 minus119888 0

0 minus119886 minus119887 0 0 0

119886 0 minus119888 0 0 0

119887 119888 0 0 0 0

)))))

)

(17)






1198811015840

119871119881= 120583120594

dagger

120578 + 120578dagger

120594 (1205811

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205812

10038161003816100381610038161205781003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2

)

+ 1205814

10038161003816100381610038161003816120594dagger

12057810038161003816100381610038161003816

2

+ 1205815(120578dagger

120588) (120588dagger

120594) + ℎ119888

(18)





119881119871= 0 ⟨1205940

2⟩ = ⟨120578

0



(2) 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 The lepton


(3) 1198811015840119881119871

= 0 ⟨12059402⟩ = ⟨120578

0



(4) Again 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 is





1+ 119878

2+ 119878



1205951119871= (]

1 119897minus

1 119864minus

0)119871sim (1 3 minus

2

3)

119897+

1119871sim (1 1 1) 119864

+

0119871sim (1 1 1)

1205952119871= (119897

minus

2 ]2 119873

0

0)119871

sim (1 3lowast

minus1

3) 119897

+

2119871sim (1 1 1)

1205953119871= (]

3 119897minus

3 119864minus

1)119871sim (1 3 minus

2

3)

119897+

3119871sim (1 1 1) 119864

+

1119871sim (1 1 1)

1205954119871= (119864

+

2 119873

0

1 119873

0

2)119871

sim (1 31

3)

1205955119871= (119873

0

3 119864minus

2 119864minus

3)119871

sim (1 3 minus2

3) 119864

+

3119871sim (1 1 1)

(19)



0

0 119873

0

1 119873

0

2 119873

0


119872119899=

((((((((

(

0 0 0 0 119860 minus119886 0

0 0 0 0 119872 0 0

0 0 0 0 119861 minus119887 0

0 0 0 0 0 119872 0

119860 119872 119861 0 0 0 119866


0 0 0 0 119866 minus119889 0

))))))))

)

(20)



MasslessWeyl states


A 6 2 2B 3 3 0C 8 0 3D 7 3 0E 14 0 3F 13 0 1G 12 0 3H 15 0 4


2119871119862120595

4119871+ ℎ119888 119860 119861 and 119862 are mass


1sim V









120595119879

119897119871= (]0

119897 119897minus

119864minus

119897)119871

sim (1 3 minus2

3)

119897+

119871sim (1 1 1) 119864

+

119897119871sim (1 1 1)

119876119879

119894119871= (119889

119894 119906119894 119880119894)119871sim (3 3

lowast

1

3)


119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)







1 0 0) (as in the


L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)


119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+



1 120601+

2 120601+



L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes



2 120601+

3) = minus2 in





lowast








120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)






2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119881119871= 0 ⟨1205940

2⟩ = ⟨120578

0



(2) 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 The lepton


(3) 1198811015840119881119871

= 0 ⟨12059402⟩ = ⟨120578

0



(4) Again 1198811015840119881119871

= 0 ⟨12059402⟩ = 0 andor ⟨1205780

3⟩ = 0 is





1+ 119878

2+ 119878



1205951119871= (]

1 119897minus

1 119864minus

0)119871sim (1 3 minus

2

3)

119897+

1119871sim (1 1 1) 119864

+

0119871sim (1 1 1)

1205952119871= (119897

minus

2 ]2 119873

0

0)119871

sim (1 3lowast

minus1

3) 119897

+

2119871sim (1 1 1)

1205953119871= (]

3 119897minus

3 119864minus

1)119871sim (1 3 minus

2

3)

119897+

3119871sim (1 1 1) 119864

+

1119871sim (1 1 1)

1205954119871= (119864

+

2 119873

0

1 119873

0

2)119871

sim (1 31

3)

1205955119871= (119873

0

3 119864minus

2 119864minus

3)119871

sim (1 3 minus2

3) 119864

+

3119871sim (1 1 1)

(19)



0

0 119873

0

1 119873

0

2 119873

0


119872119899=

((((((((

(

0 0 0 0 119860 minus119886 0

0 0 0 0 119872 0 0

0 0 0 0 119861 minus119887 0

0 0 0 0 0 119872 0

119860 119872 119861 0 0 0 119866


0 0 0 0 119866 minus119889 0

))))))))

)

(20)



MasslessWeyl states


A 6 2 2B 3 3 0C 8 0 3D 7 3 0E 14 0 3F 13 0 1G 12 0 3H 15 0 4


2119871119862120595

4119871+ ℎ119888 119860 119861 and 119862 are mass


1sim V









120595119879

119897119871= (]0

119897 119897minus

119864minus

119897)119871

sim (1 3 minus2

3)

119897+

119871sim (1 1 1) 119864

+

119897119871sim (1 1 1)

119876119879

119894119871= (119889

119894 119906119894 119880119894)119871sim (3 3

lowast

1

3)


119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)







1 0 0) (as in the


L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)


119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+



1 120601+

2 120601+



L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes



2 120601+

3) = minus2 in





lowast








120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)






2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




119876119879

3119871= (119906

3 1198893 119863) sim (3 3 0)

119906119888

119886119871sim (3

lowast

1 minus2

3) 119880

119888

119894119871sim (3

lowast

1 minus2

3)

119889119888

119886119871sim (3

lowast

11

3) 119863

119888

119871sim (3

lowast

11

3)

(21)







1 0 0) (as in the


L119897

2= sum

1198971198971015840

[(120595119879

119897119871sdot 120594) 119862 (ℎ

119890

1198971198971015840 1198971015840+

119871+ ℎ

119864

1198971198971015840119864+

1198971015840119871)] + ℎ119888 (22)


119897119871120594minus

1119862(ℎ

119890

1198971198971015840 1198971015840+

119871+ℎ

119864

1198971198971015840119864+

1198971015840119871)+



1 120601+

2 120601+



L119897

3= 120598

119899119898119901sum

1198971198971015840

ℎ]1198971198971015840120601119899

120595119898

119897119871119862120595

119901

1198971015840119871

+ ℎ119888

= sum

1198971198971015840

ℎ]1198971198971015840 [120601

++

1(119897minus

119871119864minus

1198971015840119871minus 1198971015840minus

119871119864minus

119897119871)

+ 120601+

2(119864minus

119897119871]1198971015840119871minus 119864

minus

1198971015840119871]119897119871) + 120601

+

3(]1198971198711198971015840minus

119871minus ]

1198971015840119871119897minus

119871)]

+ ℎ119888

(23)

0

lL0

lL

120594+

1

120582V1

120601minus

3

he

ll998400 h

e

l998400l

l998400minus

Lh

e

l998400l998400 l

998400+

L

otimes



2 120601+

3) = minus2 in





lowast








120588119879

= (1205880

1 120588+

2 120588+

3) sim (1 3

lowast

2

3)

120594119879

= (120594minus

1 1205940

2 1205940

3) sim (1 3

lowast

minus1

3)

(24)






2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





2) = 2


119881 (120594 120588) = 1205832

1

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205832

2

10038161003816100381610038161205881003816100381610038161003816

2

+ 1205811

10038161003816100381610038161003816120594dagger

12059410038161003816100381610038161003816

2

+ 1205812

10038161003816100381610038161003816120588dagger

12058810038161003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205941003816100381610038161003816

2 10038161003816100381610038161205881003816100381610038161003816

2

+ 1205814

10038161003816100381610038161003816120594dagger

12058810038161003816100381610038161003816

2

+ ℎ119888

(25)


L119884

=L119884

LNC +L119884

LNV

L119884

LNC = ℎ119880

120594lowast

1198763119871119862119880

119888

119871+ ℎ

119863

119894119895120594119876

119894119871119862119863

119888

119895119871

+ ℎ119889

119886120588lowast

1198763119871119862119889

119888

119886119871+ ℎ

119906

119894119886120588119876

119894119871119862119906

119888

119886119871

+ ℎ119890

1198971198971015840120588lowast

1205951198971198711198621198971015840+

119871+ ℎ

1198971198971015840120588120595

119897119871119862120595

1198971015840119871+ ℎ119888

(26)

L119884

LNV = ℎ119906

119886120594lowast

1198763119871119862119906

119888

119886119871+ ℎ

119889

119894119886120594119876

119894119871119862119889

119888

119886119871

+ ℎ119863

119894120588lowast

1198763119871119862119863

119888

119894119871+ ℎ

119880

119894120588119876

119894119871119862119880

119888

119871+ ℎ119888

(27)




119881LNV = V [radic2119867120594(1205811

10038161003816100381610038161205941003816100381610038161003816

2

+ 1205813

10038161003816100381610038161205881003816100381610038161003816

2

)]

+ V1205814[120588minus

1(120594dagger

120588) + 120588+

1(120588dagger

120594)]

(28)


120594+ 119894119860

120594)radic2

119867120594and119860




120594as the



1= V

1+ (119867

120588+ 119894119860

120588)radic2 and 1205940

3= 119881 + (119867

1015840

120594+ 119894119860

1015840

120594)radic2 An



120594 and119860

120588 the


+ 1198700



120594)


120594



1 120588plusmn

3) there are four


V1120588plusmn

3)radic1198812 + V2







+ 1198700





1198700





LNR =120598119899119898119901

Λ[120594

119899

120588119898

119876119901

3119871119862(120582

119880

3119880119888

119871+

3

sum

119886=1

120582119906

119886119906119888

119886119871)

+ 120594lowast119899

120588lowast119898

2

sum

119894=1

119876119901

119894119871119862(120582

119889

119894119863119888

119871+

3

sum

119886=1

120582119889

119894119886119889119888

119886119871)]

(29)






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






7 Conclusions


119888otimes 119878119880(3)

119871otimes 119880(1)

119909






Acknowledgments


References





























119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119888timesSU(3)

119871timesU(1)








SU(3)119871times U(1)





119871otimesU(1)

119873



119873and

119878119880(4)119871otimes 119880(1)




119871otimes U(1)









119888⨂119878119880(3)

119871⨂119880(1)





119871times U(1)








119871timesU(1)

119883model




119862otimesSU(3)

119871otimesU(1)










FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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