$S_3$ discrete group as a source of the quark mass and mixing pattern in $331$ models

0.21 MB·

by  A. E. Cárcamo Hernández

EPJ manuscript No.
(will be inserted by the editor)
S
discrete group as a source of the quark mass and mixing
3
331
pattern in models.
A. E. C´arcamo Hern´andez a, R. Martinez b and Jorge Nisperuza c
aUniversidad T´ecnica Federico Santa Mar´ıa and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso, Casilla 110-V, Valpara´ıso, Chile,
5 b,cDepartamento deF´ısica, Universidad Nacional deColombia,Ciudad Universitaria, Bogot´a D.C., Colombia.
1
0
Received: date/ Revised version: date
2

b
e Abstract. We propose a model based on the SU(3)C ⊗SU(3)L⊗U(1)X gauge symmetry with an extra
F S3⊗Z2⊗Z4⊗Z12 discretegroup,whichsuccessfullyaccountsfortheSMquarkmassandmixingpattern.

4 TheobservedhierarchyoftheSMquarkmassesandquarkmixingmatrixelementsarisesfromtheZ4 and
2
Z12 symmetries, which are broken at very high scale by the SU(3)L scalar singlets (σ,ζ) and τ, charged

] underthesesymmetries,respectively.TheCabbibomixingarisesfromthedowntypequarksectorwhereas
h
the up quark sector generates the remaining quark mixing angles. The obtained magnitudes of the CKM
p
- matrixelements,theCPviolatingphaseandtheJarlskoginvariantareinagreementwiththeexperimental
p
data.
e
h
[
PACS. PACS-key discribing text of that key – PACS-key discribing text of that key

5
v
7 1 Introduction SU(3) SU(2) U(1) gauge symmetry is unlikely
C L Y
3 ⊗ ⊗
to be a truly fundamental theory due to unexplained fea-
9
0 The discovery of a scalar field with a mass of 125 GeV tures [6,7]. Most of them are linked to the existence of
.
1 by LHC experiments [1–4] confirms that the Standard three families of fermionsas wellas the fermionmass and
0
Model (SM) is the right theory of electroweak interac- mixing hierarchy; problems presented in its quark and
4
1 tions and may provide an explanation for the origin of lepton sectors. Neutrino oscillation experiments provide
:
v mass of fundamental particles and for the spontaneous a clear indication that neutrinos are massive particles,
Xi symmetry breaking. Despite the success of the LHC ex- buttheseexperimentsdonotexplainneithertheneutrino
r periments, there are many aspects not yet explainedsuch masssquaredsplittingsnortheDiracorMajoranaidentity
a
asthefermionmasshierarchy.ThisdiscoveryoftheHiggs of neutrinos. While in the quark sector the mixing angles
scalar field opens the possibility to formulate theories be- aresmall,intheleptonsectortwoofthemixinganglesare
yondthe SMthatinclude additionalscalarfields thatcan large,and one mixing angle is small. This suggests differ-
be useful to explain the existence of Dark Matter [5]. ent mechanisms for the generation of mass in the quark
One of the outstanding unresolved issues in Particle and lepton sectors. Experiments with solar, atmospheric
Physicsistheoriginofthemassesoffundamentalfermions. and reactor neutrinos provide evidence of neutrino oscil-
Thecurrenttheoryofstrongandelectroweakinteractions, lations from the measured non vanishing neutrino mass
the Standard Model (SM), has proven to be remarkably squared splittings.
successful in passing all experimental tests. Despite its
One clear and outstanding feature in the pattern of
great success, the Standard Model (SM) based on the
quarkmassesis thatthey increasefromonegenerationto
a [email protected] the next spreading over a range of five orders of magni-
b [email protected] tude [7–9].Fromthe phenomenologicalpointofview,itis
2 A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3
possible to describe some features of the mass hierarchy presentsa discussionofthe stability conditions of the low
by assuming zero-texture Yukawa matrices [10–13]. Re- energy scalar potential.
cently,discretegroupshavebeenconsideredtoexplainthe
observed pattern of fermion masses and mixing [14–23].
Other models with horizontal symmetries have been pro- 2 The Model
posed in the literature [24].
Ontheotherhand,theoriginofthestructureoffermions WeconsideranextensionoftheminimalSU(3) SU(3)
C⊗ L⊗
can be addressed in family dependent models. Alterna- U(1) (331)modelwiththefullsymmetry experiencing
X G
tively, an explanation to this issue can also be provided a three-step spontaneous breaking:
by the models based on the gauge symmetry SU(3)
c
⊗
SU(3) U(1) ,alsocalled3-3-1models,whichintroduce =SU(3) SU(3) U(1) S Z Z Z
L⊗ X G C ⊗ L⊗ X ⊗ 3⊗ 2⊗ 4⊗ 12
a family non-universal U(1)X symmetry [25–28]. Models Λint
⇓
basedonthe gaugesymmetry SU(3) SU(3) U(1)
C L X
× ×
are very interesting since they predict the existence of SU(3) SU(3) U(1)
C ⊗ L⊗ X
threefamiliesfromthequiralanomalycancellation[29].In
v
these models, two families of quarks have the same quan- χ
⇓
tum numbers, which are associated to the two families of
SU(3) SU(2) U(1)
light quarks to correctly predict the Cabbibo mixing an- C ⊗ L⊗ Y
gle. The third family has different U(1) values and thus
X v ,v
η ρ
it is associated to the heavy quarks. Thus, the fact that ⇓
thethirdfamilyistreatedunderadifferentrepresentation, SU(3) U(1) (2.1)
C ⊗ Q
canexplainthelargemassdifferencebetweentheheaviest
quark family and the two lighter ones [30]. These models where the different symmetry breaking scales satisfy the
include a Peccei-Quinn symmetry that sheds light into following hierarchy Λ v v ,v .
int χ η ρ
≫ ≫
the strong CP problem [31]. The 331 models with sterile In our model 331 model, the electric charge is defined
neutrinos have weakly interacting massive fermionic dark in terms of the SU(3) generators and the identity by:
matter candidates [32].
1
In this paper we propose a version of the SU(3)C × Q=T3− √3T8+XI, (2.2)
SU(3) U(1) model with an additional discrete sym-
L X
×
metry group S Z Z Z and an extended scalar with I = Diag(1,1,1), T = 1Diag(1, 1,0) and T =
3⊗ 2⊗ 4⊗ 12 3 2 − 8
sector needed in order to reproduce the specific patterns ( 1 )Diag(1,1, 2).
2√3 −
of mass matrices in the quark sector that successfully ac- The anomalycancellationof SU(3) requires thatthe
L
count for the quark mass and mixing hierarchy. The par- two families of quarks be accommodatedin 3 irreducible
∗
ticular role of each additional scalar field and the corre- representations (irreps). From the quark colors, it follows
sponding particle assignments under the symmetry group that the number of 3 irreducible representations is six.
∗
ofthe modelareexplainedin details inSec. 2.Our model Theotherfamilyofquarksisaccommodatedwithitsthree
successfully describes the prevailing pattern of the SM colors,intoa3irreduciblerepresentation.Whenincluding
quark masses and mixing. the three families of leptons, we have six 3 irreps. Conse-
This paper is organized as follows. In Sec. 2 we out- quently, the SU(3) representations are vector like and
L
line the proposed model. In Sec. 3 we present our results anomalyfree.Inorderto haveanomalyfree U(1) repre-
X
in terms of quark masses and mixing, which is followed sentations, one needs to assign quantum numbers to the
by a numerical analysis. In Sec. 4, we discuss the scalar fermion families in such a way that the combination of
mass spectrum resulting from the low energy scalar po- the U(1) representations with other gauge sectors be
X
tential. Finally in Sec. 5, we state our conclusions. In the anomalyfree.Therefore,fromtherequirementofanomaly
appendixeswepresentseveraltechnicaldetails.Appendix cancellationwegetthefollowing(SU(3) ,SU(3) ,U(1) )
C L X
A gives a brief description of the S group. Appendix B left handed fermionic representations:
3
A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3 3
The[SU(3) ,U(1) ]groupstructureofthescalarfields
L X
of our model is:
χ0
1
χ= χ−2 :(3,−1/3), ξ1 :(1,0),
D1,2 √12(υχ+ξχ±iζχ)
Q1L,2 =−U1,2 :(3,3∗,0),  ρ+1 
J1,2 ρ= 1 (υ +ξ iζ ) :(3,2/3), ξ :(1,0),
 L √2 ρ ρ± ρ  2
U3 ρ+
 3 
Q3L =D3 :(3,3,1/3), √12(υη+ξη±iζη)
T
 L η = η−2 :(3,−1/3), σ :(1,0),
ν1,2,3 η0
 3 
L1,2,3 = e1,2,3 :(1,3, 1/3), (2.3)  
L   − τ :(1,0), ζ1 :(1,0), ζ2 :(1,0). (2.4)
(ν1,2,3)c
 L
 
Wegroupthescalarfieldsintodoubletandsingletrep-
resentions of S . The S Z Z Z assignments of
3 3 2 4 12
⊗ ⊗ ⊗
the scalar fields are:
Let’s note that the right-handedsector transforms as sin-
glets under SU(3) . The right handed up and down type
L
SM quarks transform under (SU(3) ,SU(3) ,U(1) ) as
C L X Φ =(η,χ) (2,1,1,1), ρ (1,1,1,1),
′
U1,2,3 :(3 ,1,2/3) and D1,2,3 :(3 ,1, 1/3), respectively. ∼ ∼
InRaddition∗,weseethatthRemodel∗has−thefollowingheavy σ ∼(1,−1,i,1), τ ∼ 1,1,1,ω−14 ,
fermions:a single flavorquark T with electric charge2/3, ξ =(ξ1,ξ2)∼(2,−1,1,1), (cid:16) (cid:17) (2.5)
twoflavorquarksJ1,2withcharge 1/3.Therighthanded ζ =(ζ ,ζ ) (2, 1,i,1). (2.6)
− 1 2 ∼ −
sector of the exotic quarks transforms as T : (3 ,1,2/3)
R ∗
and J1,2 : (3 ,1, 1/3). In the concerning to the lep-
R ∗ − where ω =e2πi/3.
ton sector, we have three right handed charged leptons
e1,2,3 : (1,1, 1) and three right-handed Majorana lep- Regardingthe quarksector,we assignthe quark fields
R −
tons N1,2,3 : (1,1,0) (recently, a discussion about neu- intrivialandnontrivialsingletrepresentionsofS3.Weas-
R
trino masses via double and inverse see-saw mechanism sumedthatalllefthandedquarksandrighthandedquarks
was perform in ref. [33]). are assigned to S3 trivial singlets excepting, UR1, UR2, UR3,
T , D3, J1 and J2, which are assumed to be non trivial
R R R R
singlets.Thequarkassignmentsunder S Z Z Z
3 2 4 12
⊗ ⊗ ⊗
The scalar sector of the 331 model includes three 3’s are:
irreps of SU(3) , where one triplet χ acquires a vacuum
L
expectationvalue(VEV)athighenergyscale,v ,respon-
χ
sible for the breaking of the SU(3) U(1) symmetry
downtotheSU(2)L U(1)Y electrowLe×akgrouXpoftheSM; Q1L ∼ 1,1,1,ω−12 , Q2L ∼ 1,1,1,ω−41 ,
×
and two light triplet fields η and ρ get VEVs vη and vρ, Q3L ∼ (cid:16)(1,1,1,1), (cid:17) UR1 ∼(cid:16) (1′,1,−1,ω(cid:17)),
respectively,atthe electroweakscaleandgivemasstothe UR2 ∼ 1′,1,−1,ω14 , UR3 ∼(1′,−1,−i,1), (2.7)
fermion and gauge sector. In addition to the aforemen-
D1 (cid:16)(1, 1,1,ω),(cid:17) D2 (1, 1,1,i),
tioned scalar spectrum, we introduce six SU(3) scalar R ∼ − R ∼ −
singlets, namely, ξ1, ξ2, ζ1, ζ2, σ and τ. Their rLole and DR3 ∼ (1′,1,1,i), TR ∼(1′,−1,1,1),
importance will be explained later in this section. JR1 ∼ 1′,−1,1,ω−21 , JR2 ∼ 1′,−1,1,ω−14 .
(cid:16) (cid:17) (cid:16) (cid:17)
4 A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3
Withtheaboveparticlecontent,thefollowingrelevant WeassumethefollowingVEVpatternfortheSU(3)
L
Yukawa terms for the quark sector arise: singlet scalar fields:
−LY =y1(U1)Q1Lρ∗UR1σΛ2τ86 +y2(U2)Q2Lρ∗UR2σΛ2τ42 hξi=vξ(1,0), hτi=vτ
ζ =v (0,1), σ =v . (2.9)
+y(U)Q1ρ U3στ2 +y(U)Q2ρ U3στ h i ζ h i σ
13 L ∗ R Λ3 23 L ∗ RΛ2
+y(U)Q3ΦU3 ζ +y(T)Q3ΦT ξ i.e. the VEVs of ξ and ζ are aligned as (1,0) and (0,1) in
33 L RΛ L RΛ the S3 directions, respectively. Besides that, the SU(3)L
+y(D)Q1Φ D1 ξτ6 +y(D)Q1Φ D2 ξτ5 scalar singlets, ξ1, ζ2, σ and τ, are assumed to acquire
11 L ∗ R Λ7 12 L ∗ R Λ6 VEVsatascaleΛ muchlargerthanυ inordertobreak
int χ
+y2(D2)Q2LΦ∗DR2 ξΛτ54 +y2(D1)Q2LΦ∗DR1 ξΛτ65 the G =SU(3)C⊗SU(3)L⊗U(1)X⊗S3⊗Z2⊗Z4⊗Z12
symmetrygroupdowntoSU(3) SU(3) U(1) .Let
+y3(D3)Q3LρDR3 Λτ33 +y1(J)Q1LΦ∗JR1Λξ us note that the S3 doublet SUC(3⊗)L singleLt⊗scalarsXξ and
+y(J)Q2Φ J2 ξ (2.8) ζ are the only scalar fields odd under the Z2 symmetry.
2 L ∗ RΛ Furthermore the only scalar fields charged under the Z4
and Z symmetries are the SU(3) singlet scalars σ, ζ
12 L
where the dimensionless couplings y(U,D) (i,j = 1,2,3), and τ, respectively. Thus, the breaking of the Z , Z and
ij 2 4
y(T), y(J) are (1) parameters. Z symmetriesis causedbythe scalarfields ξ,(σ,ζ)and
1,2 O 12
To explain the fermion mass hierarchy it is necessary τ, respectively, acquiring VEVs at a very high scale. It is
toassumeanansatzfortheYukawamatrices.Acandidate worthmentioningthatwehavechosenaVEVpatternsfor
for generating specific Yukawa textures is the S3 Z2 theS3doubletsSU(3)Lsingletscalarξandζ,inthe(1,0)
⊗ ⊗
Z4 Z12 discrete group that can explain the prevailing and (0,1) S3 directions, respectively, as indicated by Eq.
⊗
pattern of fermion masses and mixing. The S discrete (2.9), in order to decouple the heavy exotic quarks from
3
symmetry is the smallest non-Abelian discrete symmetry the SM quarks. Due to the aforementioned choice of the
grouphavingthreeirreduciblerepresentations(irreps),ex- VEV pattern of ξ, only the SU(3)L scalar triplet χ par-
plicitly two singlets and one doublet irreps. The Z sym- ticipates in the Yukawa interactions giving masses to the
2
metrydeterminestheallowedYukawatermsforthequark exotic T, J1 and J2 quarks. Furthermore, the masses of
sector, thus resulting in a reduction of model parameters theSMquarkswillarisefromtheYukawatermsinvolving
and allowing one to decouple the bottom quark from the the SU(3)L scalar triplets η and ρ.
light down and strange quarks. The Z and Z symme- Considering that the quark mass and mixing pattern
4 12
tries shape the hierarchical structure of the quark mass arises from the Z and Z symmetries, and in order to
4 12
matrices that yields a realistic pattern of quark masses relate the quark masses with the quark mixing parame-
andmixing.ItisnoteworthythatthepropertiesoftheZ ters, we set the VEVs of the SU(3) singlet scalar fields
N L
groupsimplythattheZ andZ symmetriesarethelow- excepting v as follows:
4 12 ζ
estcyclicsymmetriesthatallowonetobuidYukawaterms
of dimensions six and ten, respectively. Consequently,the
v =v =v =Λ =λΛ, (2.10)
Z Z symmetry is the lowest cyclic symmetry from ξ τ σ int
4 12
⊗
which a 12 dimensional Yukawa term can be built, cru-
where λ = 0.225 is one of the parameters of the Wolfen-
cial to get the required λ8 supression in the 11 entry of
stein parametrization and Λ is the cutoff of our model.
the up-type quarkmassmatrix,whereλ=0.225is one of
To reproduce the right value of the top quark mass
the Wolfenstein parameters. Furthermore, thanks to the
while keeping y(U) (1) . √4π as required by pertur-
Z Z symmetry, the lowest down-type quark Yukawa 33 ∼ O
4 12
⊗ bativity, we set v in the following range:
ζ
term contributing to the 11 entry of the down-type quark
massmatrixhasdimension11.Thus,theZ andZ sym-
4 12
metries are crucial to explain the smallness of the up and √2mt
Λ<v <Λ. (2.11)
ζ
down quark masses. √4πvη
A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3 5
3 Quark masses and mixing.
v v
m a(U)λ8 , m a(U)λ4 ,
u ≃ 11 √2 c ≃ 22 √2
Using Eq. (2.8) and considering that the VEV pattern of
v
(U)
the SU(3)L singlet scalar fields satisfies Eq. (2.9) with mt ≃ a33 √2, (3.4)
the nonvanishing VEVs set to be equal to λΛ (being Λ v
m a(D)a(D) a(D)a(D) λ7 ,
the cutoff of our model) as indicated by Eq. (2.10), we d ≃ 11 22 − 12 21 √2
(cid:12) v (cid:12) v
findthattheSMquarksdonotmixwiththe heavyexotic m a(cid:12)(D)λ5 , m a(cid:12)(D)λ3 .
s ≃ (cid:12)22 √2 b ≃ (cid:12)33 √2
quarksandthatthe massmatricesforup- anddown-type
SM quarks are
We also find that the CKM quark mixing matrix is
approximatelly given by
a(U)λ8 0 a(U)λ3
11 13 v
M = 0 a(U)λ4 a(U)λ2 , (3.1)
U  22 23 √2
0 0 a(U)
 33 
a(D)λ7 a(D)λ6 0  VCKM (3.5)
11 12 v c c c s eiδs
M = a(D)λ6 a(D)λ5 0 , 12 13 13 12 13
D  21 22 √2 e iδc s s c s c c +e iδs s s c s ,
0 0 a(D)λ3 ≃  − 12 13 23− 23 12 12 23 − 12 13 23 − 13 23
 33  s s e iδc c s c s e iδc s s c c
  − 12 23− − 12 23 13 12 23− − 23 12 13 13 23 
 
where λ = 0.225 is one of the Wolfenstein parameters,
where c =cosθ , s =sinθ (with i=j and i,j =
v = 246 GeV the symmetry breaking scale, and a(U,D) ij ij ij ij 6
ij
1,2,3), θ and δ being the quark mixing angles and the
(i,j = 1,2,3) are (1) parameters. From the SM quark ij
O CPviolatingphase,respectively.Thequarkmixingangles
mass matrix textures given by Eq. (3.1), it follows that
and the CP violating phase are given by
the Cabbibo mixing arisesfromthe down-type quark sec-
tor whereas the up quark sector generates the remaining
quark mixing angles. The (1) dimensionless couplings
O a(D) a(U)
a(U,D) (i,j = 1,2,3) in Eq. (3.1) are given by the follow- tanθ 12 λ, tanθ 23 λ2, (3.6)
ij 12 ≃ a(D) 23 ≃ a(U)
ing relations: 22 33
a(U)
tanθ 13 λ3, δ = arg a(U) .
v v v 13 ≃ (cid:12)(cid:12)(cid:12)a3(U3)(cid:12)(cid:12)(cid:12) − (cid:16) 13 (cid:17)
a(U) =y(U) ρ, a(U) =y(U) ρ, a(U) =y(U) ρ,
11 11 v 22 11 v 13 13 v
Hereweassumethatthe (1)dimensionlesscouplings
v v v
a2(U3) =y2(U3) vρ, a(3U3) =y3(U3) vζΛη, a(ijU,D) (i,j =1,2,3)inEq.(3O.1)arerealexceptfora(1U3).It
v v v
a(D) =y(D) η, a(D) =y(D) η, a(D) =y(D) η, isnoteworthythatEqs.(3.4)-(3.6)giveanelegantdescrip-
11 11 v 12 12 v 21 21 v
tionoftheSMquarkmassesandmixinganglesintermsof
v v
(D) (D) η (D) (D) ρ
a =y , a =y . (3.2)
22 22 v 33 33 v theWolfensteinparameterλ=0.225andofparametersof
orderunity.Itisworthcommentingthattheobservablesin
Furthermore,wefindthatthe exoticquarkmassesare the quark sectorareconnectedwith the electroweaksym-
metry breaking scale v = 246 GeV through their power
dependenceontheWolfensteinparameterλ=0.225,with
mT = λy(T) vχ , (3.3) O(1) coefficients.
√2
The Wolfenstein parameterization [35] of the CKM
mJ1,2 = λy1(J,2)√vχ2 = yy(1(TJ,2))mT. matrix is given by:
1 λ2 λ Aλ3(ρ iη)
− 2 −
FromEq.(3.1)wefindthattheup-anddown-typeSM V λ 1 λ2 Aλ2 , (3.7)
W ≃ − − 2 
quark masses are approximatelly given by Aλ3(1 ρ iη) Aλ2 1
 − − − 
 
6 A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3
with Observable Model value Experimental value
λ =0.22535 0.00065, A=0.811+0.022, (3.8) mu(MeV) 1.47 1.45+−00..5465
± −0.012 mc(MeV) 641 635±86
ρ =0.131+−00..002163, η =0.345+−00..001134, (3.9) mt(GeV) 172.2 172.1±0.6±0.9
ρ ρ 1 λ2 , η η 1 λ2 . (3.10) md(MeV) 2.2 2.9+−00..54
≃ (cid:18) − 2 (cid:19) ≃ (cid:18) − 2 (cid:19) ms(MeV) 60.0 57.7−+1165..87
The comparison with Eq. (3.7) leads to the following re- mb(GeV) 2.82 2.82+−00..0094
lations: (cid:12)(cid:12)Vud(cid:12)(cid:12) 0.97419 0.97427±0.00015
(cid:12)(cid:12)Vus(cid:12)(cid:12) 0.22572 0.22534±0.00065
a(3U3) ≃ 1, a2(U3) ≃0.81, a(1U3) ≃−0.3eiδ, δ =67◦, (cid:12)(cid:12)Vub(cid:12)(cid:12) 0.00351 0.00351+−00..0000001154
a(U) mc 1.43, a(U) mu 1.27, (3.11) (cid:12)(cid:12)Vcd(cid:12)(cid:12) 0.22548 0.22520±0.00065
22 ≃ λ4mt ≃ 11 ≃ λ8mt ≃ (cid:12)(cid:12)Vcs(cid:12)(cid:12) 0.97338 0.97344±0.00016
then it follows that a(U) is required to be complex, as (cid:12)(cid:12)Vcb(cid:12)(cid:12) 0.0411 0.0412+−00..00001015
previouslyassumedand13itsmagnitudeisabitsmallerthan (cid:12)(cid:12)Vtd(cid:12)(cid:12) 0.0110 0.00867+−00..0000002391
the remaining (1) coefficients. (cid:12)(cid:12)Vts(cid:12)(cid:12) 0.0398 0.0404+−00..00001015
Assuming tOhat the hierarchy of the SM quark masses (cid:12)(cid:12)Vtb(cid:12)(cid:12) 0.999147 0.999146+−00..000000002416
J 2.96×10−5 (2.96+0.20)×10−5
andquarkmixing matrixelements arisesfromthe Z and −0.16
Z symmetries, we set a(D) = a(D). We fit the par4ame- δ 68◦ 68◦
12 21 22
ters a(D) (i=j) in Eq. (3.1) to reproduce the down-type
ij 6 Table 1. Model and experimental values of thequark masses
quark masses and quark mixing parameters. The results
and CKM parameters.
are shown in Table 1 for the following best-fit values:
a(D) 0.84, a(D) 0.4, a(D) 0.57, a(D) 1.42. 1. With the multiplication rules of the S group given
11 ≃ 12 ≃ 22 ≃ 33 ≃ ′ 3
(3.12) in the appendix, we have to assign to the scalar fields in
The obtained quark masses and CKM parameters are the S irrepsandbuildthe correspondingscalarpotential
3
consistentwiththeexperimentaldata.Thevaluesofthese invariant under the symmetry group.
observablesas wellas the quarkmasses together with the Since all singlet scalars acquire VEVs at a scale much
experimental data are shown in Table 1. The experimen- larger than v , they are very heavy and thus the mix-
χ
talvaluesofthe quarkmasses,whicharegivenatthe M ing between these scalar singlets and the SU(3) scalar
Z L
scale, have been taken from Ref. [36] (which are similar triplets can be neglected. For the sake of simplicity we
to those in [37]), whereas the experimental values of the assume a CP scalar potential with only real couplings as
CKM matrix elements and the Jarlskog invariant J are doneinRefs.[28,34].Then,therenormalizablelowenergy
taken from Ref. [7]. As seen from Table 1, all observables scalar potential of the model is constructed with the S
3
in the quark sector are in excellent agreement with the doublet Φ = (η,χ) and the non-trivial S singlet ρ fields,
3
experimental data, excepting V , which turns out to be inthewayinvariantunderthegroupSU(3) SU(3)
td C⊗ L⊗
larger by a factor 1.3 than its corresponding experi- U(1) S .The renormalizablelowenergyscalarpoten-
∼ (cid:12)(cid:12) (cid:12)(cid:12) X⊗ 3
mental value, and naively deviated 8 sigma away from it. tial is given by:
V =µ2(ρ ρ)+µ2 Φ Φ +λ (ρ ρ)(ρ ρ)
H ρ † Φ † 1 1 † †
+λ2 Φ†Φ 1 Φ†(cid:0)Φ 1+(cid:1)λ3 Φ†Φ 1′ Φ†Φ 1′
4 The scalar potential
+λ4(cid:0)Φ†Φ(cid:1)2(cid:0)Φ†Φ(cid:1)2+λ5((cid:0)ρ†ρ)(cid:1)Φ†(cid:0)Φ 1 (cid:1)
TobuildaSU(3) SU(3) U(1) S invariantscalar +λ6(cid:0)(ρ†Φ(cid:1)) (cid:0)Φ†ρ (cid:1)1+f εijk(Φ(cid:0)iΦj)(cid:1)1′ρk+h.c ,
C⊗ L⊗ X⊗ 3 (4.1)
potential it is necessary to decompose the direct product (cid:0) (cid:0) (cid:1)(cid:1) (cid:2) (cid:3)
of S representations into irreducible S representations. where Φ =(η ,χ ) is a S doublet with i=1,2,3.
3 3 i i i 3
The S group has three irreducible representations that The S symmetry in the quadratic term of the scalar
3 3
can be characterized by their dimension, i.e., 2, 1 and potentialissoftlybrokenbecausethevacuumexpectation
A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3 7
values of the scalar fields η and χ contained in the S have the following masses:
3
doubletΦsatisfythehierarchyv v .Then,weinclude
tµh2e quχaηdra+tich.Sc3assodfto-bnreeiankiRngef.teχ[r3m3≫]s, aηµn2ηd−usµe2χtheηS†η maunld- m2h0 ≃ (cid:2)8λ4λ5vη2vρ2+146(λλ22λ4+vη4λ4+)(cid:0)v4η2λ1+(λvρ22+λ4)−λ25(cid:1)vρ4(cid:3),
ηχ † (cid:0) (cid:1)(cid:0) 3(cid:1) fv v v
taisplfio(cid:0)clalotwios(cid:1)n:rules to rewrite the lowenergy scalarpotential m2H10 ≃ √2χ (cid:18)vηρ + vηρ(cid:19), (cid:0) (cid:1)
fv v v
m2 χ η + ρ ,
VH =µ2ρ(ρ†ρ)+µ2η η†η +µ2χ χ†χ +µ2ηχ χ†η + η†χ A0 ≃ √2 (cid:18)vρ vη(cid:19) v
m2 =m2 2λ v2 +√2fv ρ,
+λ1(ρ†ρ)2+(λ(cid:0)2+λ(cid:1)4) χ†(cid:0)χ 2+(cid:1) (η†η)2(cid:2)(cid:0) (cid:1) (cid:0) (cid:1)(cid:3) H20 H02 ≃ 4 χ χvη
+λ5 (ρ†ρ)(χ†χ)+(ρ†ρh)(cid:0)(η†η)(cid:1) i (4.2) m2H30 ≃(λ2+λ4)vχ2,
v v
++2λ6(λ(cid:2)2χ−†ρλ4()ρ(cid:0)†χχ†)χ+(cid:1)(cid:0)η熆ηρ(cid:1)+(ρ2†η((cid:3))λ4−λ3)(cid:0)χ†η(cid:1)(cid:0)η†χ(cid:1) m2H1± ≃√λ2(cid:18)vηρ + vηρ(cid:19)vfvχ,
m2 6υ2 +√2fv η,
+(λ3(cid:2)(cid:0)+λ4(cid:1)) χ†η 2(cid:0)+ η(cid:1)†χ 2 +(cid:3) 2f εijkηiχjρk+h.c . H2± ≃ 2 χ χvρ
Itisnoteworthyht(cid:0)hatt(cid:1)heS(cid:0)soft(cid:1)-bireaking(cid:0)termµ2 χ η +(cid:1) m2G01 =m2G02 =m2G02 =m2G03 =m2G±1 =m2G±2 =0. (4.5)
3 ηχ †
h.c does not play an important role neither for the mini- It is noteworthy that the physical scalar spectrum at
(cid:0) (cid:1)
mization of the scalar potential nor for the generation of low energies of our model includes: four massive charged
the physical scalar masses Ref. [33]. Higgs (H1±, H2±), one CP-odd Higgs (A0), three neutral
Fromthe previousexpressionsandfromthe scalarpo- CP-evenHiggs(h0,H10,H30)andtwoneutralHiggs(H20,H02)
tentialminimizationconditions,thefollowingrelationsare bosons. Here we identify the scalar h0 with the SM-like
obtained: 126 GeV Higgs boson observed at the LHC. Let us note
that the neutral Goldstone bosons G0, G0, G0 , G0 are
1 3 2 2
associated to the longitudinal components of the Z, Z ,
′
λ fv v
−µ2χ =(λ2+λ4)vχ2 + 25vρ2+(λ2−λ4)vη2−√2 υρ η, K0 and K0gauge bosons, respectively. Besides that, the
χ
λ fv v charged Goldstone bosons G±1 and G±2 are associated to
−µ2η =(λ2+λ4)vη2+ 25vρ2+(λ2−λ4)vχ2 −√2 vχ ρ, the longitudinal components of the W± and K± gauge
η
bosons, respectively [25,28].
λ fv v
µ2 =λ v2+ 5 v2 +v2 √2 χ η. (4.3)
− ρ 1 ρ 2 χ η − v In Appendix B we employ the method of Ref. [38] to
ρ
(cid:0) (cid:1) show that the low energy scalar potential is stable when
Considering the quartic scalar couplings of the same
the following conditions are fulfilled:
order of magnitude, we find from the previous relations
that the trilinear scalar coupling f has to be of the order
of vχ. Furthermore, from Eq. 4.3, we get the following λ1 >0, λ2 >0, λ4 >0, λ6 >0, f >0,
relation: λ >λ , λ +λ >0, λ +λ >0,
2 3 2 4 5 6
µ2 µ2 + 2λ +√2vρ υ2 v2 =0. (4.4) λ1(λ2+λ4)>λ25, λ5+λ6 >2 λ1(λ2+λ4). (4.6)
χ− η 4 v χ− η
(cid:18) η(cid:19) p
(cid:0) (cid:1)
The previous relations imply that the negative quadratic 5 Conclusions
couplings should satisfy µ2 v2 and µ2 µ2 v2
χ ∼− χ ρ ∼ η ∼− ρ ∼
v2 v2, being v = 246 GeV. Therefore, the negative Inthispaperweproposedamodelbasedonthesymmetry
− η ∼ −
quadratic coupling for the SU(3) scalar triplet χ is of groupSU(3) SU(3) U(1) S Z Z Z ,
L C ⊗ L⊗ X ⊗ 3⊗ 2⊗ 4⊗ 12
the order of its squared VEV. The remaining negative which is an extension of the 331 model with β = 1
−√3
quadratic couplings are of the order of the squaredVEVs of Ref. [33]. Our model successfully accounts for the ob-
of the SU(3) scalar triplets η and ρ. served SM quark mass and mixing pattern. The S and
L 3
From the low energy scalar potential given by Eq. Z symmetries are crucial for reducing the number of pa-
2
(4.2),wefindthatthephysicalscalarfieldsatlowenergies rameters in the Yukawa terms for the quark sector and
8 A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3
decoupling the bottom quark from the light down and With these multiplication rules we have to assign to the
strange quarks. The observed hierarchy of the SM quark scalar fields in the S irreps and build the corresponding
3
masses andquark mixing matrix elements arisesfrom the scalar potential invariant under the symmetry group.
Z and Z symmetries, which are broken at a very high
4 12
scale by the SU(3) scalar singlets (σ,ζ) and τ, charged
L
B : Stability conditions of the low energy
under these symmetries, respectively. The Cabbibo mix-
ingarisesfromthedown-typequarksectorwhereastheup scalar potential
quark sector generates the remaining mixing angles. The
Inthissubsectionwearegoingtodeterminetheconditions
SM quark masses are generated from Yukawa terms in-
requiredto havea stable scalarpotentialby followingthe
volving the SU(3) scalar triplets η and ρ, which acquire
L
method described in Ref. [38]. The gauge invariant and
VEVsattheelectroweakscalev =246GeV.Ontheother
renormalizablelowenergyscalarpotentialasafunctionof
hand, the exotic quark masses arise from Yukawa terms
the fields φ =χ, φ =η and φ =ρ is a linear hermitian
involving the SU(3) scalar triplet χ, which acquires a 1 2 3
L
combination of the following terms:
VEV at the TeV scale. The obtained values of the quark
masses,the magnitudes ofthe CKMmatrix elements,the
φ φ , φ φ φ φ (B.1)
i j i j k l
CP violating phase, and the Jarlskog invariant are con-
sistent with the experimental data. The complex phase where i,j,k,l=φ , φ and φ . To discuss the stability of
1 2 3
responsible for CP violation in the quark sector has been the potential, its minimum, and its gauge invariance one
assumedtocomefromasevendimensionalup-typequark canmakethefollowingarrangementofthescalarfieldsby
Yukawa term. using 2 2 hermitian matrices as follows:
×
K = φ†iφi φ†iφj ,
Acknowledgements (φiφj) φ†jφi φ†jφj!
e 1
A.E.C.H was supported by Fondecyt (Chile), Grant No. = 2 K0(φiφj)12×2+Ka(φiφj)σa (B.2)
11130115 and by DGIP internal Grant No. 111458. R.M. (cid:16) (cid:17)
where (φ φ ) = ρη,ρχ,ηχ, σa (a = 1,2,3) are the Pauli
wassupportedbyCOLCIENCIASandbyFondecyt(Chile), i j
matrices and 1 is the identity matrix. From the previ-
Grant No. 11130115. 2×2
ous expressionsone can build the followingbilinear terms
as functions of the scalar fields:
Appendices
K0(φ φ ) = φ†iφi+φ†jφj,
i j
A : The product rules for S3 Ka(φ φ ) = φ†iφj σaij. (B.3)
i j
Xi,j (cid:16) (cid:17)
The S group has three irreducible representations that
3
Thepropertiesofthepotentialcanbeanalyzedinterms
can be characterizedby their dimension, i.e., 2, 1 and 1.
′
of K and K with φ φ = ρη,ρχ,ηχ in the
Considering two S doublet representations (x ,x ) and 0(φ φ ) (φ φ ) i j
3 1 2 i j i j
domain K 0 y K2 K2. Defining κ = K/K the
(y1,y2), the direct product can be decomposed as follows 0 ≥ 0 ≥ 0
potential can be written as
[15]:
x1 y1 V = V2+V4,
=(x y +x y ) +(x y x y )
1 1 2 2 1 1 2 2 1 1′
x2!2⊗ y2!2 − V2 = K0(φiφj)J2(φiφj)(κ),
x y x y (φXiφj)
1 1 2 2
+ − , (A.1) J (κ)= ξ +ξT κ ,
x1y2+x2y1!2 2(φiφj) 0(φiφj) (φiφj) (φiφj)
V = K2 J (κ), (B.4)
4 0(φiφj) 4(φiφj)
x1 x2y (φXiφj)
(y) = − ,
x2!2⊗ 1′ x1y !2 J4(φiφj)(κ)= η00(φiφj)+2ηT(φiφj)κ(φiφj)
(x) (y) = (xy) . (A.2) +κT E κ ,
1′ ⊗ 1′ 1 (φiφj) (φiφj) (φiφj)
A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3 9
whereE isa3 3matrixandthefunctionsJ (κ) in the following matrices:
(φ φ ) 2(φ φ )
i j × i j
and J (κ) are defined in the domain κ 1. The
stabili4t(yφioφfj)the scalar potential requires that| i|t≤has to be Kρη = ρρ††ηρ η熆ηρ!= 12 K0(ρη)12×2+Ka(ρη)σa ,
boundedfrombelow.Thestabilityisdeterminedfromthe (cid:0) (cid:1)
e ρ ρ χ ρ 1
behavior of V in the limit K0 →∞, i.e., Kρχ = ρ††χ χ††χ!= 2 K0(ρχ)12×2+Ka(ρχ)σa ,
(cid:0) (cid:1)
J4(φiφj)(κ)≥0, (B.5) Ke = η†η χ†η = 1 K 1 +K σa ,
for all κ 1. To impose J (κ) to be positively ηχ η†χχ†χ! 2 0(ηχ) 2×2 a(ηχ)
| | ≤ 4(φiφj) e (cid:0) (B(cid:1).11)
defined it is enough to consider the values of all station-
ary points in the domain κ < 1 and κ = 1. This re- where σa (a = 1,2,3) are the Pauli matrices and 1 is
| | | | 2 2
×
sultsinaboundforη00(φiφj),η0(φiφj) andE(φiφj),which the 2×2 identity matrix. From the previous expressions,
parametrizethequartictermsofthe potentialincludedin we find that the bilinear combinations of the scalar fields
V4. appearing in Eq. (B.11) are given by:
For κ <1 the stationary points should satisfy
| | K =ρ ρ+η η, K =ρ ρ+χ χ,
0(ρη) † † 0(ρχ) † †
Eκ = η , κ <1. (B.6) K =η η+χ χ, (B.12)
(φiφj) − (φiφj) | | 0(ηχ) † †
K = ρ ρ σa + η η σa + ρ η σa + η ρ σa ,
For the case where detE =0, the following relation is a(ρη) † 11 † 22 † 12 † 21
obtained: 6 Ka(ρχ) =(cid:0)ρ†ρ(cid:1)σa11+(cid:0)χ†χ(cid:1) σa22+(cid:0)ρ†χ(cid:1) σa12+(cid:0)χ†ρ(cid:1) σa21,
Ka(ηχ) =(cid:0)η†η(cid:1)σa11+(cid:0)χ†χ(cid:1)σa22+(cid:0)η†χ(cid:1)σa12+(cid:0)χ†η(cid:1)σa21.
J (κ) =η ηT E 1η . (B.7)
4(φiφj) |est 00(φiφj)− (φiφj) − (φiφj) Sincet(cid:0)hest(cid:1)abilityo(cid:0)fthe(cid:1)scalarp(cid:0)oten(cid:1)tialisd(cid:0)eterm(cid:1)ined
For κ = 1 the stationary points are obtained from fromits quartic terms,the stationarysolutionsconsistent
| |
the function: with a stable scalar potential are described by the follow-
ing functions:
F (κ)=J (κ)+u(1 κ2), (B.8)
4(φiφj) 4(φiφj) − fρη(u)= u+E00(ρη)−Ea(ρη) E(ρη)−u13×3 −ab1Eb(ρη),
1
where u is a Lagrange multiplier that satisfies the follow- fρχ(u)= u+E00(ρχ)−Ea(ρχ)(cid:0)E(ρχ)−u13×3(cid:1)−ab Eb(ρχ),
ing condition 1
fηχ(u)= u+E00(ηχ)−Ea(ηχ)(cid:0)E(ηχ)−u13×3(cid:1)−ab Eb(ηχ),
(E u)κ= η , (cid:0) (cid:1) (B.13)
(φ φ ) (φ φ )
i j − − i j
J (κ) = u+η (B.9) where, for the ρ and η fields, we have
4(φiφj) |est 00(φiφj)
−ηT(φiφj)(E(φiφj)−u)−1η(φiφj). E00(ρη) = λ1+λ2+4 λ4+λ5,
The stationary points of J4(φiφj)(κ) for |κ| ≤ 1 can be Ea(ρη) = λ1−λ42−λ4δa3,
obtained from:
λ 0 0
6
1
f (u)= J (κ) >0, E = 0 λ 0 , (B.14)
(φiφj) 4(φiφj) |est (ρη) 4 6 
f(′φ φ )(u)> 0. (B.10)  0 0 λ1+λ2+λ4−λ5
i j  
In the same manner, for the multiplets ρ and χ, the
Considering that the quartic terms of the scalarpotential
expressions are
are dominant when the vacuum expectation values of the
λ +λ +λ +λ
1 2 4 5
scalarfieldstakelargevalues,thesetermswillbethemost E00(ρχ) = 4 ,
relevant to analyze the stability of the scalar potential. λ λ λ
1 2 4
E = − − δ ,
Following the method described in Ref. [38], we proceed a(ρχ) 4 a3
torewritethequartictermsofthescalarpotentialinterms λ 0 0
6
1
of bilinear combinations of the scalar fields. To this end, E = 0 λ 0 . (B.15)
(ρχ) 4 6 
the bilinear combinations of the scalar fields are included 0 0 λ +λ +λ λ
 1 2 4− 5
 
10 A. C´arcamo, R.Martinez, J. Nisperuza: Quark masses and mixings in 331S3
Similarly, for the η and χ fields, we find:
λ4 0 0 λ1 >0, λ2 >0, λ4 >0, λ6 >0, f >0,
E =λ , E =0, E = 0 λ 0 .
00(ηχ) 2 a(ηχ) (ηχ)  3  λ >λ , λ +λ >0, λ +λ >0,
− 2 3 2 4 5 6
0 0 λ
 4 λ (λ +λ )>λ2, λ +λ >2 λ (λ +λ ). (B.22)
 (B.16) 1 2 4 5 5 6 1 2 4
p
Following Ref. [38], we determine the stability of the
scalar potential from the conditions:
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