Table Of ContentA predictive 3-3-1 model with A flavor symmetry.
4
A.E.Ca´rcamoHerna´ndez
UniversidadTe´cnicaFedericoSantaMar´ıa
andCentroCient´ıfico-Tecnolo´gicodeValpara´ıso
Casilla110-V,Valpara´ıso,Chile.
R.Martinez
UniversidadNacionaldeColombia,DepartamentodeF´ısica,CiudadUniversitaria,Bogota´D.C.,Colombia.
6
1
0
2
r
p Abstract
A
WeproposeapredictivemodelbasedontheSU(3) SU(3) U(1) gaugegroupsupplemented
C⊗ L⊗ X
0 bytheA Z Z Z Z discretegroup,whichsuccessfullydescribestheSMfermionmassand
4 3 4 6 16
3 mixingpa⊗tter⊗n. T⊗hes⊗mallactiveneutrinomassesaregeneratedviainverseseesaw mechanism
with three very light Majorana neutrinos. The observed charged fermion mass hierarchy and
]
h quark mixing pattern are originated from the breaking of the Z Z Z discrete group at
4 6 16
⊗ ⊗
p very high scale. The obtained values for the physical observables for both quark and lepton
-
sectors are in excellentagreementwith the experimentaldata. The modelpredictsa vanishing
p
e leptonicDiracCPviolatingphaseaswellasaneffectiveMajorananeutrinomassparameterof
h neutrinolessdoublebetadecay,withvaluesm =2and48meVforthenormalandtheinverted
ββ
[
neutrinomasshierarchies,respectively.
3
Keywords: Fermionmassesandmixings,Discreteflavorsymmetries,3-3-1models,Models
v
7 beyondtheStandardModel.
3
9
5
0 1. Introduction
.
1
Despite the greatsuccess of the StandardModel(SM), recentlyconfirmedby the discoveryof
0
5 the 126 GeV Higgs boson by LHC experiments [1, 2, 3, 4], there are many aspects not yet
1 explainedsuchastheoriginofthefermionmassandmixinghierarchyaswellasthemechanism
v: responsibleforstabilizingtheelectroweakscale[5,6]. ThisdiscoveryoftheHiggsscalarfield
i allowstoconsiderextensionsoftheSMwithadditionalscalarfieldsthatcanbeusefultoexplain
X
theexistenceofDarkMatter[7].
r
a TheStandardModelisatheorywithmanyphenomenologicalachievements. Howeverinthe
YukawasectoroftheSMtherearemanyparametersrelatedwiththefermionmasseswithnoclear
dynamicalorigin.Becauseofthisreason,itisimportanttostudyrealisticmodelsthatallowtoset
uprelationsamongalltheseparametersoftheYukawasector. Discreteflavorsymmetriesallow
toestablishansatzthatexplaintheflavorproblem,forrecentreviewsseeRefs. [8,9,10]. These
discreteflavorsymmetriesmaybecrucialinbuildingmodelsoffermionmixingthataddressthe
flavorproblem.Nonabeliandiscreteflavorsymmetriesariseinstringtheoriesduetothediscrete
PreprintsubmittedtoNuclearPhysicsB May3,2016
featuresofthefixedpointsoftheorbifolds[11]. Forinstance,thediscreteD groupisoriginated
4
intheS1/Z orbifold[11].
2
Besides that, another of the greatest mysteries in particle physics is the existence of three
fermionfamiliesatlowenergies.Thequarkmixinganglesaresmallwhereastheleptonicmixing
anglesarelarge.ModelsbasedonthegaugesymmetrySU(3) SU(3) U(1) havethefeature
C L X
⊗ ⊗
of beingvectorlikewith three familiesof fermionsandare thereforeanomalyfree [12, 13, 14,
15,16]. WhentheelectricchargeisdefinedinthelinearcombinationoftheSU(3) generators
L
T andT ,itisafreeparameter,independentoftheanomalies(β). Thechoiceofthisparameter
3 8
definesthechargeoftheexoticparticles. Choosingβ = 1 ,thethirdcomponentoftheweak
−√3
leptontripletisaneutralfieldνC,whichallowstobuildtheDiracmatrixwiththeusualfieldν
R L
ofthe weakdoublet. Ifoneintroducesa sterile neutrino N inthemodel, thenitis possibleto
R
generatelightneutrinomassesviainverseseesawmechanism. The3-3-1modelswithβ = 1
−√3
havetheadvantangeofprovidinganalternativeframeworktogenerateneutrinomasses, where
theneutrinospectrumincludesthelightactivesub-eVscaleneutrinosaswellassterileneutrinos
which could be dark matter candidates if they are light enough or candidates for detection at
the LHC,if theirmassesareatthe TeVscale. Thisinterestingfeaturemakesthe3-3-1models
veryinteresting,sinceiftheTeVscalesterileneutrinosarefoundattheLHC,thesemodelscan
beverystrongcandidatesforunravelingthemechanismresponsibleforelectroweaksymmetry
breaking. Furthermore,the3-3-1modelscanprovideanexplanationforthe750GeVdiphoton
excessrecentlyreportedbyATLASandCMS[17]aswellasforthe2TeVdibosonexcessfound
byATLAS[18].
Neutrino oscillation experiments [6, 19, 20, 21, 22, 23] indicate that there are at least two
massiveactiveneutrinosandatmostonemasslessactiveneutrino. Inthemasseigenstates,itis
necessaryforthesolarneutrinososcillationsthatδm2 = m2 = m2 m2 wherem2 m2 > 0.
sun 21 2− 1 2− 1
For the atmospheric neutrinos oscillations it is required that δm2 = m2 = m2 m2 where
atm 31 3 − 1
the difference can be possitive (normal hierarchy) or negative (inverted hierarchy). Neutrino
oscillationsdonotgiveinformationneitherontheabsolutevalueoftheneutrinomassnoronthe
MajoranaorDiracnatureoftheneutrino. Howeverthereareneutrinomassboundsarisingfrom
cosmology[24],tritiumbetadecay[25]anddoublebetadecay[26,27,28,29,30,31,32,34,33].
The neutrino masses and mixings are known from neutrino oscillations, which depend on
thesquaredneutrinomassdiferencesandnotontheabsolutevalueoftheneutrinomasses. The
globalfitsoftheavailabledatafromtheDayaBay[19],T2K[20],MINOS[21],DoubleCHOOZ
[22]andRENO[23]neutrinooscillationexperiments,constraintheneutrinomasssquaredsplit-
tingsandmixingparameters[35]. Thecurrentneutrinodataonneutrinomixingparameterscan
beverywellaccommodatedintheapproximatedtribimaximalmixingmatrix,
2 1 0
U = q13 √13 1 , (1)
TBM −−√√166 √√133 −√1√22
whichisconsistentwithtwolargemixinganglesandoneverysmalloneoforderzero. Specifi-
cally,themixinganglespredictedbythetribimaximalmixingmatrixsatisfy sin2θ = 1,
12 TBM 3
sin2θ = 1,and sin2θ = 0. However,the3-3-1modelisnotab(cid:16)letoge(cid:17)neratethe
23 TBM 2 13 TBM
(cid:16)tribimax(cid:17)imalmatrixstru(cid:16)cture.B(cid:17)ecauseofthisreason,discretesymmetrygroups[36,37,38,41,
39,40,42,43,44,45]thatactonthefermionfamiliesareimposedwiththeaimtogenerateansatz
2
thatreproducethesematrices. OneofthemostpromisingdiscreteflavorgroupsisA ,sinceitis
4
thesmallestsymmetrywithonethree-dimensionalandthreedistinctone-dimensionalirreducible
representations,wherethethreefamiliesoffermionscanbeaccommodatedrathernaturally.An-
otherapproachtodescribethefermionmassandmixingpatternconsistsinpostulatingparticular
massmatrixtextures(see Ref[46] forsomeworksconsideringtextures). Besidesthat, models
withMulti-Higgssectors,GrandUnification,ExtradimensionsandSuperstringsaswellaswith
horizontal symmetries have been proposed in the literature [8, 47, 48, 49, 50] to explain the
observedpatternoffermionmassesandmixings.
InthispaperweproposeaversionoftheSU(3) SU(3) U(1) modelwithanadditional
C L X
× ×
flavorsymmetrygroupA Z Z Z Z andanextendedscalarsectorneededinorderto
4 3 4 6 16
⊗ ⊗ ⊗ ⊗
reproducethespecificpatternsofmassmatricesinthefermionsectorthatsuccessfullyaccount
forfermionmassesandmixings.Theparticularroleofeachadditionalscalarfieldandthecorre-
spondingparticleassignmentsunderthesymmetrygroupofthemodelareexplainedindetailin
Sec. 2. Themodelwearebuildingwiththeaforementioneddiscretesymmetries,preservesthe
contentofparticlesoftheminimal3-3-1model,butwe addadditionalveryheavyscalarfields
with quantumnumbersthatallow to buildYukawatermsinvariantunderthelocalanddiscrete
groups.Thisgeneratesthepredictiveandviabletexturesthatexplainthe18physicalobservables
inthequarkandleptonsectors,i.e.,the9chargedfermionmasses,2neutrinomasssquaredsplit-
tings,3leptonmixingparameters,3quarkmixinganglesand1CPviolatingphaseoftheCKM
quarkmixingmatrix.OurmodelsuccessfullydescribestheprevailingpatternoftheSMfermion
massesandmixing.
Thecontentofthispaperisorganizedasfollows. InSec. 2weoutlinetheproposedmodel.In
Sec. 3wediscusssleptonmassesandmixingsandshowourcorrespondingresults. Ourresults
forthemassesandmixingsinthequarksectorfollowedbyanumericalanalysisarepresentedin
Sec. 4. FinallyinSec. 5,westateourconclusions.InAppendixAwepresentabriefdescription
oftheA group.
4
2. Themodel
WeextendtheSU(3) SU(3) U(1) groupoftheminimal3-3-1modelbyaddinganextra
C L X
⊗ ⊗
flavorsymmetrygroupA Z Z Z Z ,insuchawaythatthefullsymmetry experi-
4 3 4 6 16
⊗ ⊗ ⊗ ⊗ G
encesathree-stepspontaneousbreaking,asfollows:
=SU(3) SU(3) U(1) A Z Z Z Z (2)
G C⊗ L⊗ X ⊗ 4⊗ 3⊗ 4⊗ 6⊗ 16
Λint SU(3) SU(3) U(1) Z vχ SU(3) SU(2) U(1)
−−→ C⊗ L⊗ X ⊗ 3−→ C⊗ L⊗ Y
vη,vρ SU(3) U(1) ,
−−−→ C⊗ Q
wherethedifferentsymmetrybreakingscalessatisfythefollowinghierarchyv ,v v Λ .
η ρ χ int
≪ ≪
We define the electric charge in our 3-3-1 model in terms of the SU(3) generators and the
identity,asfollows:
1
Q=T T +XI, (3)
3 8
− √3
withI = Diag(1,1,1),T = 1Diag(1, 1,0)andT =( 1 )Diag(1,1, 2).
3 2 − 8 2√3 −
3
TheanomalycancellationofSU(3) requiresthatthetwofamiliesofquarksbeaccommodated
L
in 3 irreducible representations(irreps). Besides that, the number of 3 irreducible represen-
∗ ∗
tations is six, as follows from the quark colors. We accommodate the other family of quarks
into a 3 irreducible representation. Furthermore, we have six 3 irreps taking into account the
threefamiliesofleptons.Thus,theSU(3) representationsarevectorlikeandfreeofanomalies.
L
HavinganomalyfreeU(1) representationsrequiresthatthequantumnumbersforthefermion
X
familiesbeassignedinsuchawaythatthecombinationoftheU(1) representationswithother
X
gaugesectorscancelsanomalies. Consequently,toavoidanomalies,thefermionshavetobeac-
commodatedintothefollowing(SU(3) ,SU(3) ,U(1) )left-andright-handedrepresentations:
C L X
D1,2 U3 ν1,2,3
Q1L,2 =D1−,2JU:1,(123,2,1L,:(13/,33)∗,,0), UQ33L:=(3,D1T,32L/3:)(,3,3,1/3), L1L,2,3 =(νe11,,22,,33)cL :(1,3,−1/3),
R − R
U1,2 :(3,1,2/3), D3 :(3,1, 1/3),
R R −
JR1,2 :(3,1,−1/3), TR :(3,1,2/3),
e :(1,1, 1), µ :(1,1, 1), τ :(1,1, 1),
R − R − R − (4)
N1 :(1,1,0), N2 :(1,1,0), N3 :(1,1,0),
R R R
where Ui and Di for i = 1,2,3 are three up- and down-type quark components in the flavor
L L
basis,whileνi andei (e ,µ ,τ )aretheneutralandchargedleptonfamilies. Theright-handed
L L L L L
fermionsareassignedasSU(3) singletsrepresentationshavingU(1) quantumnumbersequal
L X
to their electric charges. Furthermore, the fermion spectrum of the model includes as heavy
fermions: a single flavor quark T with electric charge 2/3, two flavor quarks J1,2 with charge
1/3, three neutral Majorana leptons (ν1,2,3)c and three right-handed Majorana leptons N1,2,3
− L R
(see Ref. [51] for a recent discussion about neutrino masses via double and inverse see-saw
mechanismfora3-3-1model).
The 3-3-1modelsextend the scalar sector of the SM into three 3’s irrepsof SU(3) , where
L
oneheavytripletχacquiresavaccuumexpectationvalue(VEV)attheTeVscale,v ,breaking
χ
theSU(3) U(1) symmetrydowntotheSU(2) U(1) electroweakgroupoftheSMand
L X L Y
× ×
thengivingmassestothenonSMfermionsandgaugebosons;andtwolightertripletfieldsηand
ρthatgetVEVsv andv ,respectively,attheelectroweakscalethusgeneratingthemassforthe
η ρ
fermionandgaugesectoroftheSM.Weenlargethescalarsectoroftheminimal3-3-1modelby
introducing14SU(3) scalarsinglets,namely,ξ ,ζ ,S ,ϕ,∆,φ,τandσ(j=1,2,3).
L j j j
The scalars of our modelare accommodatedinto the following[SU(3) ,U(1) ] representa-
L X
tions:
χ0
1
χ=√12(υχ+ρχ+ξ−2χ±iζχ):(3,−1/3), ξj :(1,0), τ:(1,0), ϕ:(1,0), j=1,2,3,
1
ρ=√112((υυρ++ρ+3ξξρ±iiζζρ)):(3,2/3), ζj :(1,0), φ:(1,0), ∆:(1,0), j=1,2,3,
√2 η η± η
η= ηη−203 :(3,−1/3), Sj :4(1,0), σ∼(1,0), j=1,2,3. (5)
Thescalarfieldsaregroupedintotripletandsingletrepresentionsof A . Thescalarfieldsof
4
ourmodelhavethefollowingassignmentsunderA Z Z Z Z :
4 3 4 6 16
⊗ ⊗ ⊗ ⊗
η 1,e−23iπ,1,1,1 , ρ 1,e23iπ,1,1,1 , χ (1,1,1,1,1),
∼ ∼ ∼
(cid:16) (cid:17) (cid:16) (cid:17)
ξ (3,1,1,1, 1), ζ 3,1,1,1,ei8π , S 3,e−23iπ,1,1,ei8π , σ 1,1,1,1,e−i8π ,
∼ − ∼ ∼ ∼
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
ϕ 1,1,1,e−i3π,1 , ∆ 1,1, 1,e−i3π,1 , φ 1′,1,i,1,1 , τ 1′′,1,i,1,1 , (6)
∼ ∼ − ∼ ∼
(cid:16) (cid:17) (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1)
wherethenumbersinboldfacearedimensionsoftheA irreduciblerepresentations.
4
TheleptonstransformunderA Z Z Z Z as:
4 3 4 6 16
⊗ ⊗ ⊗ ⊗
LL ∼ 3,e23iπ,1,1,−1 , eR ∼ 1,1,1,1,e78iπ , µR ∼ 1′,1,1,1,i ,
τR (cid:16)1′′,1,1,1,ei4π (cid:17), NR (cid:16)3,e23iπ,1,1, 1(cid:17) . (cid:0) (cid:1) (7)
∼ ∼ −
(cid:16) (cid:17) (cid:16) (cid:17)
Note that left handed leptons are unified into a A triplet representation 3, whereas the right
4
handedchargedleptonsareassignedintodifferentA singlets,i.e,1,1 and1 . Furthermore,the
4 ′ ′′
righthandedMajorananeutrinosareunifiedintoaA tripletrepresentation.
4
TheA Z Z Z Z assignmentsforthequarksectorare:
4 3 4 6 16
⊗ ⊗ ⊗ ⊗
Q1L ∼ 1,1,1,1,e−i8π , Q2L ∼ 1′,1,1,1,1 , Q3L ∼ 1′′,1,1,1,1 ,
UR1 ∼ (cid:16)1,e−23πi,1,1,e(cid:17)78iπ , UR2(cid:0)∼ 1′,e−23πi,(cid:1)1,1,i , U(cid:0)R3 ∼ 1′′,e−(cid:1)23πi,1,1,1 ,
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
D1R ∼ 1,e23πi,1,−1,ei8π , D2R ∼ 1,e23πi,1,−1,1 , D3R ∼ 1′′,e23πi,1,−1,1
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
T 1 ,1,1,1,1 , J1 1,1,1,1,1 J2 1 ,1,1,1,1 . (8)
R ∼ ′′ R ∼ ′ R ∼ ′′
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
With the aboveparticle content, the followingrelevantYukawa termsfor the quarkand lepton
sectorarise:
(Q) = y(U)Q1ρ U1σ8 +y(U)Q2ρ U2σ4 +y(U)Q3ηU3
−LY 11 L ∗ RΛ8 22 L ∗ RΛ2 33 L R
+y(D)Q1η D1τφ∆3σ2 +y(D)Q1η D2τφ∆3σ +y(D)Q1η D3φ2∆3σ
11 L ∗ R Λ7 12 L ∗ R Λ6 13 L ∗ R Λ6
+y(D)Q2η D1τ2∆3σ +y(D)Q2η D2τ2∆3 +y(D)Q2η D3φ2∆3
21 L ∗ R Λ6 22 L ∗ R Λ5 23 L ∗ R Λ5
+y(D)Q3ρD1φ2∆3σ +y(D)Q3ρD2φ2∆3 +y(D)Q3ρD3 ϕ3
31 L R Λ6 32 L R Λ5 33 L RΛ3
+y(T)Q3χT +y(J)Q1χ J1+y(J)Q2χ J2 (9)
L R 1 L ∗ R 2 L ∗ R
σ7 σ4 σ2
(L) = h(L) L ρξ e +h(L) L ρξ µ +h(L) L ρξ τ
−LY ρe L 1 RΛ8 ρµ L 1 RΛ5 ρτ L 1 RΛ3
(cid:16) (cid:17) (cid:16) (cid:17) ′′ (cid:16) (cid:17) ′
1 η† η∗ +x ρT ρ Sσ
+ h(L) L χN + h N NC · · +h N NC
χ L R 1 2 1N R R 1 (cid:16) (cid:17) Λ (cid:16) (cid:17) 2N R R 3s Λ
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
+ h ε La LC b (ρ )c ζσ +H.c, (10)
ρ abc L L ∗ Λ2
(cid:18) (cid:16) (cid:17) (cid:19)3s 5
where the dimensionless couplings y(U), y(D) (i, j = 1,2,3), y(T), y(J), y(J), h(L), h(L), h(L), h(L),
ii ij 1 2 ρe ρµ ρτ χ
h , x, h and h are (1) parameters. Here we assumed that all Yukawa couplingsare real,
1N 2N ρ
O
exceptingy(D),y(D) andh(L) whichareassumedtobecomplex.
13 31 ρτ
Although the flavor discrete groups in Eq. (2) look rather sofisticated, each discrete group
factorplaysitsownroleingeneratingpredictivefermiontexturesthatsuccessfullyaccountfor
thelowenergyfermionflavordata.Todescribethepatternoffermionmassesandmixingangles,
one needs to postulate particular Yukawa textures. As we will see in the next sections, the
predictivetexturesfortheleptonandquarksectorswillgiverisetotheexperimentallyobserved
deviationofthetribimaximalmixingpatternandtoquarkmixingemergingonlyfromthedown
type quark sector, respectively. A candidate for generating specific Yukawa textures is the A
4
flavorsymmetry,whichneedstobesupplementedbytheZ Z Z Z discretegroup.Aswe
3 4 6 16
⊗ ⊗ ⊗
willseeinthenextsections,thispredictivesetupcansuccessfullyaccountforfermionmassesand
mixings.TheinclusionoftheA discretegroupreducesthenumberofparametersintheYukawa
4
andscalarsectoroftheSU(3) SU(3) U(1) modelmakingitmorepredictive.Wechoose
C L X
⊗ ⊗
A sinceitisthesmallestdiscretegroupwithathree-dimensionalirreduciblerepresentationand
4
3 distinctone-dimensionalirreduciblerepresentations,whichallowsto naturallyaccommodate
the three fermion families. We unify the left-handed leptons in the A triplet representation
4
andtheright-handedleptonsareassignedto A singlets. Regardingthequarksector,weassign
4
quarks into A singlet representations. In what follows we describe the role of each discrete
4
cyclic groupfactor introducedin our model. The Z symmetry separates the A scalar triplets
3 4
participatingintheYukawainteractionsforchargedleptonsfromthoseonesparticipatinginthe
neutrinoYukawainteractions.Besidesthat,theZ symmetryavoidsmixingsbetweenSMquarks
3
andexoticquarkssincetherighthandedexoticquarksareneutralunderthissymmetrywhereas
the righthandedSM quarkshave non trivialZ charges. Thusthe Z symmetry decouplesthe
3 3
SM quarks from the exotic quarks resulting in a reduction of quark sector model parameters.
Furthermore,theZ symmetryisalsoimportantforreducingthenumberofquarksectormodel
4
parameters, since due to this symmetry, the SU(3) scalar singlets A nontrivial singlets only
L 4
appear in the down type quark Yukawa terms. Consequently this Z symmetry together with
4
the A assignments for quarks described in Eq. (8), results in a diagonal up type quark mass
4
matrix,thusgivingrisetoaquarkmixingonlyemergingfromthedowntypequarksector. The
Z symmetry is crucial for explaining the hierarchy between the SM down and SM up type
6
quarkswithouttuningtheSMdowntypequarkYukawacouplings,sinceitisthesmallestcyclic
symmetrythatallows ϕ3 intheYukawatermthatgeneratesthebottomquarkmass,whichisλ3 v
Λ3 √2
(λ = 0.225 is one of the Wolfenstein parameters) times a (1) parameter. The Z symmetry
16
O
gives rise to the observed hierarchyamong chargedfermion masses and quark mixing angles.
It is worth mentioningthat the propertiesof the Z groupsimply that the Z symmetryis the
N 16
smallestcyclicsymmetrythatallowstobuildtheYukawatermQ1ρ U1σ8 ofdimensiontwelve
L ∗ RΛ8
from a σ8 insertionon the Q1ρ U1 operator,crucialto getthe requiredλ8 suppression(where
Λ8 L ∗ R
λ=0.225isoneoftheWolfensteinparameters)neededtonaturallyexplainthesmallnessofthe
upquarkmass. Regardingthechargedleptonsector,letusnotethatthefivedimensionalYukawa
operators 1 L ρξ e , 1 L ρξ µ and 1 L ρξ τ areA invariantbutdonotpreservethe
Λ L 1 R Λ L 1 R Λ L 1 R 4
Z symmetr(cid:16)y,asf(cid:17)ollowsf(cid:16)romth(cid:17)e′′chargesas(cid:16)signm(cid:17)en′tsgivenbyEqs. (6)and(7).
16
InwhatfollowswecommentaboutthepossibleVEVspatternsforthe A scalartripletsξ, ζ
4
andS. HereweassumeahierarchybetweentheVEVsofthe A scalartripletsξ,ζ andS,i.e.,
4
v << v << v , which implies that the mixing angles of these scalar triplets are very small
S ζ ξ
6
sincetheyaresuppressedbytheratiosoftheirVEVs,whichisaconsequenceofthemethodof
recursiveexpansionproposedin Ref. [52]. Consequently, we can neglectthe mixing between
the A scalar triplets ξ, ζ and S, and treat their correspondingscalar potentials independently.
4
TherelevanttermsdeterminingtheVEVdirectionsofanyA acalartripletare:
4
V(Σ) = µ2(ΣΣ ) +κ (ΣΣ ) (ΣΣ ) +κ (ΣΣ) (Σ Σ ) +κ (ΣΣ ) (ΣΣ )
− Σ ∗ 1 Σ,1 ∗ 1 ∗ 1 Σ,2 1 ∗ ∗ 1 Σ,3 ∗ 1′ ∗ 1′′
+κ (ΣΣ) (Σ Σ ) +h.c +κ (ΣΣ) (Σ Σ ) +h.c
Σ,4 1 ∗ ∗ 1 Σ,5 1 ∗ ∗ 1
′ ′′ ′′ ′
+κ (cid:2)(ΣΣ ) (ΣΣ ) +κ ((cid:3)ΣΣ) (Σ(cid:2) Σ ) . (cid:3) (11)
Σ,6 ∗ 3s ∗ 3s Σ,7 3s ∗ ∗ 3s
whereΣ=ξ,ζ,S.
ThepartofthescalarpotentialforeachA scalartriplethas8freeparameters: 1bilinearand7
4
quarticcouplings. Theminimizationconditionsofthescalarpotentialfora A tripletyieldthe
4
followingrelations:
∂ V(Σ)
h∂vΣ1 i = −2vΣ1µ2Σ+4κΣ,1vΣ1(cid:16)v2Σ1 +v2Σ2 +v2Σ3(cid:17)+2κΣ,3vΣ1(cid:16)2v2Σ1 −v2Σ2 −v2Σ3(cid:17)
+4κ v v2 +v2 cos 2θ 2θ +v2 cos 2θ 2θ +8κ v v2 +v2
Σ,2 Σ1 Σ1 Σ2 Σ1 − Σ2 Σ3 Σ1 − Σ3 Σ,7 Σ1 Σ2 Σ3
+4 κ +hκ v 2v2 (cid:0) v2 cos 2(cid:1)θ 2θ (cid:0) v2 cos 2(cid:1)θi 2θ (cid:16) (cid:17)
Σ,4 Σ,5 Σ1 Σ1 − Σ2 Σ1 − Σ2 − Σ3 Σ1 − Σ3
+4κ(cid:0) v v2 (cid:1)1+hcos 2θ 2θ (cid:0) +v2 1+(cid:1)cos 2θ (cid:0) 2θ (cid:1)i
Σ,6 Σ1 Σ2 Σ1 − Σ2 Σ3 Σ1 − Σ3
= 0, h (cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9)i
∂ V(Σ)
h∂vΣ2 i = −2vΣ2µ2Σ+4κΣ,1vΣ2(cid:16)v2Σ1 +v2Σ2 +v2Σ3(cid:17)+2κΣ,3vΣ2(cid:16)2v2Σ2 −v2Σ1 −v2Σ3(cid:17)
+4κ v v2 +v2 cos 2θ 2θ +v2 cos 2θ 2θ +8κ v v2 +v2
Σ,2 Σ2 Σ2 Σ1 Σ2 − Σ1 Σ3 Σ2 − Σ3 Σ,7 Σ2 Σ1 Σ3
+4 κ +hκ v 2v2 (cid:0) v2 cos 2(cid:1)θ 2θ (cid:0) v2 cos 2(cid:1)θi 2θ (cid:16) (cid:17)
Σ,4 Σ,5 Σ2 Σ2 − Σ1 Σ2 − Σ1 − Σ3 Σ2 − Σ3
+4κ(cid:0) v v2 (cid:1)1+hcos 2θ 2θ (cid:0) +v2 1+(cid:1)cos 2θ (cid:0) 2θ (cid:1)i
Σ,6 Σ2 Σ1 Σ2 − Σ1 Σ3 Σ2 − Σ3
= 0, h (cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9)i
∂ V(Σ)
h∂vΣ3 i = −2vΣ3µ2Σ+4κΣ,1vΣ3(cid:16)v2Σ1 +v2Σ2 +v2Σ3(cid:17)+2κΣ,3vΣ3(cid:16)2v2Σ3 −v2Σ1 −v2Σ2(cid:17)
+4κ v v2 +v2 cos 2θ 2θ +v2 cos 2θ 2θ +8κ v v2 +v2
Σ,2 Σ3 Σ2 Σ1 Σ1 − Σ2 Σ2 Σ3 − Σ2 Σ,7 Σ3 Σ1 Σ2
+4 κ +hκ v 2v2 (cid:0) v2 cos 2(cid:1)θ 2θ (cid:0) v2 cos 2(cid:1)θi 2θ (cid:16) (cid:17)
Σ,4 Σ,5 Σ3 Σ3 − Σ1 Σ1 − Σ2 − Σ2 Σ3 − Σ2
+4κ(cid:0) v v2 (cid:1)1+hcos 2θ 2θ (cid:0) +v2 1+(cid:1)cos 2θ (cid:0) 2θ (cid:1)i
Σ,6 Σ3 Σ1 Σ1 − Σ2 Σ2 Σ3 − Σ2
= 0. h (cid:8) (cid:0) (cid:1)(cid:9) (cid:8) (cid:0) (cid:1)(cid:9)i (12)
where Σ = vΣ eiθΣ1,vΣ eiθΣ2,vΣ eiθΣ3 . Hereinordertosimplifytheanalysis,werestricttothe
h i 1 2 3
simplestcase(cid:16)ofzerophasesintheVEV(cid:17) patternsoftheA tripletscalars,i.e.,θ =θ =θ =0.
4 Σ1 Σ2 Σ3
Then,fromthescalarpotentialminimizationequationsgivenbyEq.(12),thefollowingrelations
areobtained:
3κ 4 κ +κ +6 κ +κ v2 v2 = 0,
Σ,3− Σ,6 Σ,7 Σ,4 Σ,5 Σ1 − Σ2
(cid:2)3κ 4(cid:0)κ +κ (cid:1)+6(cid:0)κ +κ (cid:1)(cid:3)(cid:16)v2 v2 (cid:17) = 0,
Σ,3− Σ,6 Σ,7 Σ,4 Σ,5 Σ1 − Σ3
(cid:2)3κ 4(cid:0)κ +κ (cid:1)+6(cid:0)κ +κ (cid:1)(cid:3)(cid:16)v2 v2 (cid:17) = 0. (13)
Σ,3− Σ,6 Σ,7 Σ,4 Σ,5 Σ2 − Σ3
(cid:2) (cid:0) (cid:1) (cid:0) 7 (cid:1)(cid:3)(cid:16) (cid:17)
FromtherelationsgivenbyEq. (13)andsettingκ = 4 κ +κ 2 κ +κ ,weobtain
ζ,3 3 ζ,6 ζ,7 − ζ,4 ζ,5
thefollowingVEVpattern: (cid:16) (cid:17) (cid:16) (cid:17)
ξ = vξ (1,1,1), ζ = vζ (1,0,1), S = vS (1,1, 1). (14)
h i √3 h i √2 h i √3 −
Inthecaseofξ,thisisavacuumconfigurationpreservingaZ subgroupofA ,whichhasbeen
3 4
extensivelystudiedinmanyA flavormodels(forrecentreviewsseeRefs. [8,9,10]). TheVEV
4
patternforthe A tripletscalarζ issimilartotheonepreviouslystudiedinan A andT flavor
4 4 7
SU(5)GUTmodels[38,44]andina 6HDMwith A flavorsymmetry[37]. Aswewillsee in
4
the next section, the VEV patterns for the A triplets ξ, ζ and S given in Eq. 14) are crucial
4
to get a predictive model that successfully reproducesthe experimentalvalues of the physical
observablesintheleptonsector.
FurthermoreweassumethattheseSU(3) scalarsingletsgetVEVsatascaleΛ muchlarger
L int
thanv (whichisoftheorderoftheTeVscale),withtheexceptionofS (j=1,2,3),whichget
χ j
VEVsmuchsmallerthantheelectroweaksymmetrybreakingscalev = 246GeV.TheVEVsof
theξ (j = 1,2,3),ϕ,∆,φ,τandσscalarsingletsbreaktheSU(3) SU(3) U(1) A
j C⊗ L⊗ X⊗ 4⊗
Z Z Z Z symmetrydowntoSU(3) SU(3) U(1) Z atthescaleΛ .
3 4 6 16 C L X 3 int
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
Fromtheexpressionsgivenabove,andusingthevacuumconfigurationfortheA scalartriplets
4
giveninEq.(14),wefindthefollowingrelations:
2
µ2 = 3 κ +κ +4 κ +κ v2,
ξ 3 ξ,1 ξ,2 ξ,6 ξ,7 ξ
h (cid:16) (cid:17) (cid:16) (cid:17)i
2
µ2 = 3 κ +κ +4 κ +κ v2,
ζ 3 ζ,1 ζ,2 ζ,6 ζ,7 ζ
h (cid:16) (cid:17) (cid:16) (cid:17)i
2
µ2 = 3 κ +κ +4 κ +κ v2. (15)
S 3 S,1 S,2 S,6 S,7 S
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1)(cid:3)
TheseresultsshowthattheVEVdirectionsforthethreeA triplets,i.e.,ξ,ζ andS scalarsinEq.
4
(14),areconsistentwithaglobalminimumofthescalarpotential(11)ofourmodelforalarge
regionofparameterspace.
Besidesthat,asthehierarchyamongchargedfermionmassesandquarkmixinganglesemerges
from the breakingof the Z Z Z discrete group, we set the VEVs of the SU(3) singlet
4 6 16 L
⊗ ⊗
scalar fields ξ, ϕ, ∆, φ, τ and σ, with respect to the Wolfenstein parameterλ = 0.225and the
modelcutoffΛ,asfollows:
v v v v v v Λ =λΛ. (16)
ϕ τ φ ∆ ξ σ int
∼ ∼ ∼ ∼ ∼ ∼
Furthermore, we assume that the A scalar triplets participating in the neutrino Yukawa inter-
4
actionshaveVEVsmuchsmallerthantheelectroweaksymmetrybreakingscale. Besidesthat,
aspreviouslymentioned,weassumea hierarchyamongtheVEVsofthetwo A scalar triplets
4
participating in the neutrino Yukawa terms. Consequently, as we will see in the next section,
the Majorana neutrinos acquire very small masses and thus an inverse seesaw mechanism for
the generation of light active neutrino masses, takes place. Therefore, we have the following
hierarchyamongtheVEVsofthescalarfieldsinourmodel:
v <<v <<v v v<<v <<Λ . (17)
S ζ ρ η χ int
∼ ∼
8
In what follows, we briefly comment about the low energy scalar sector of our model. The
renormalizablelowenergyscalarpotentialofthemodelisgivenby:
V = µ2(χ χ)+µ2(η η)+µ2(ρ ρ)+ f χη ρ εijk+H.c. +λ (χ χ)(χ χ)
H χ † η † ρ † i j k 1 † †
(cid:16) (cid:17)
+λ (ρ ρ)(ρ ρ)+λ (η η)(η η)+λ (χ χ)(ρ ρ)+λ (χ χ)(η η)
2 † † 3 † † 4 † † 5 † †
+λ (ρ ρ)(η η)+λ (χ η)(η χ)+λ (χ ρ)(ρ χ)+λ (ρ η)(η ρ). (18)
6 † † 7 † † 8 † † 9 † †
Afterthesymmetrybreakingtakesplace,itisfoundthatthescalarmasseigenstatesarerelated
withtheweakscalarstatesby:[14,15]:
G ρ G0 ζ H0 ξ
±1 =R ±1 , 1 =R ρ , 1 =R ρ , (19)
H1±! βT η±2! A01! βT ζη! h0! αT ξη!
G0 χ0 G0 ζ G χ
2 =R 1 , 3 =R χ , ±2 =R ±2 , (20)
H20! η03! H30! ξχ! H2±! ρ±3!
with
cosα (β ) sinα (β ) 1 0
RαT(βT) = −sinαTT(βTT) cosαTT(βTT) !, R= −0 1 !, (21)
wheretanβ =v /v ,andtan2α = M /(M M )with:
T η ρ T 1 2− 3
M = 4λ v v +2√2fv ,
1 6 η ρ χ
M = 4λ v2 √2fv tanβ ,
2 2 ρ− χ T
M = 4λ v2 √2fv /tanβ . (22)
3 3 η− χ T
Itisnoteworthytomentionthattheourmodelhasthefollowingscalarstatesatlowenergies: 4
massivechargedHiggs(H ,H ),oneCP-oddHiggs(A0),3neutralCP-evenHiggs(h0,H0,H0)
1± 2± 1 1 3
and2neutralHiggs(H0,H0)bosons.Weidentifythescalarh0withtheSM-like126GeVHiggs
2 2
bosondiscoveredattheLHC.LetusnotethattheneutralGoldstonebosonsG0,G0,G0,G0corre-
1 3 2 2
0
spondtothelongitudinalcomponentsoftheZ,Z ,K0andK gaugebosons,respectively.Besides
′
that,thechargedGoldstonebosonsG andG areassociatedtothelongitudinalcomponentsof
±1 ±2
theW andK gaugebosons,respectively[12,15].
± ±
9
3. Leptonmassesandmixings
From Eq. (10) and taking into account that the VEV pattern of the A triplet, SU(3) singlet
4 L
scalarfieldξ satisfiesEq. (14)withthenonvanishingVEVsoftheSU(3) singletscalarsξand
L
σ,settobeequaltoλΛ(beingΛthecutoffofourmodel)asindicatedbyEq. (16),wefindthat
thechargedleptonmassmatrixisgivenby:
1 1 1 1 0 0
1
Ml =R†lLPldiag(cid:16)me,mµ,mτ(cid:17), RlL = √3 11 ωω2 ωω2 , ω=e23πi, Pl = 00 10 e0i(α23),
beingαthecomplexphaseofh(L),andthechargedleptonmassesare:
ρτ
v v v
m =a(l)λ8 , m =a(l)λ5 , m =a(l)λ3 . (24)
e 1 √2 µ 2 √2 τ 3 √2
where λ = 0.225is one ofthe Wolfenstein parameters, v = 246GeV the scale of electroweak
symmetrybreakinganda(l) (i=1,2,3)are (1)parameters. Letusnotethatthechargedlepton
i O
massesarelinkedwiththescaleofelectroweaksymmetrybreakingthroughtheirpowerdepen-
dence on the Wolfenstein parameterλ = 0.225, with (1) coefficients. Furthermore,from the
O
lepton Yukawa terms given in Eq. (10) it is easy to see that our modeldoes notfeature flavor
changing leptonic neutral Higgs decays. Consequently, our model cannotexplain the recently
reportedanomalyin the h µτ decay, implyingthata measurementof its branchingfraction
→
willbedecisiveforitsexclusion.
Regardingtheneutrinosector,wecanwritetheneutrinomasstermsas:
ν
1 L
−L(mνa)ss = 2(cid:16) νCL νR NR (cid:17)Mν NνCRRC +H.c, (25)
wheretheneutrinomassmatrixisconstrainedfromtheA flavorsymmetryandhasthefollowing
4
form:
0 M 0
3 3 D 3 3
Mν = M×DT 03 3 M×χ , (26)
andthesubmatricesaregivenby: 03×3 M×χT MR
h v2η+xv2ρv h vSvσ h vSvσ
MD = hρv2ρΛv2ζvσ 1−0010 1100 −001 , MR = −1hNh22NNΛv√2Sv√3Sv3ΛσvΛσσ h−1hN2v2N2ηN+Λv√x2S√3vv32ρΛσΛvσ h1hN22vNN2ηΛ+v√√xS233vvΛΛ2ρσvσ ,
v
Mχ = h(χL) √χ2 00 10 01 . (27)
As previouslymentioned, we assume in our modelthat the SU(3) scalar singlet, A triplet S
L 4
interactingwiththerighthandedMajorananeutrinosgetsaverysmallvacuumexpectationvalue,
10
