Quark masses without Yukawa hierarchies

0.17 MB·English

by  H. Fanchiotti

Europhysics Letters PREPRINT
Quark masses without Yukawa hierarchies
H. Fanchiotti1, C. Garc´ıa-Canal1 and W. A. Ponce2( )
∗
1 LaboratoriodeF´ısicaTeo´rica, DepartamentodeF´ısica, FacultaddeCiencias Exactas,
Universidad Nacional de La Plata, C.C. 67-1900, La Plata Argentina.
2 Instituto de F´ısica, Universidad de Antioquia, A.A. 1226, Medell´ın, Colombia.
6
0
PACS.12.60.Cn – Extensions of electroweak gauge sector.
0
PACS.12.15.Ff – Quark and lepton masses and mixing.
2

n
a
J
2 Abstract. – A model based on the local gauge group SU(3)c ⊗SU(3)L ⊗U(1)X without
1 particleswithexoticelectricchargesisshowntobeabletoprovidethequarkmassspectrumand
their mixing, by means of universal see-saw mechanisms, avoiding a hierarchy in the Yukawa

1 coupling constants.
v
1
0
1
1 The Standard Model (SM), with all its successes, is in the unaesthetic position of having
0 no explanation of fermion masses and mixing angles, both in the quark and lepton sectors.
6 Besides, recent experimental results on neutrino oscillations [1], which imply physics beyond
0
the SM, call for extensions of the model. In this regard, models based on the local gauge
/
h groupSU(3) SU(3) U(1) (namedinthe literature3-3-1models)havebeenadvocated
c L X
p ⊗ ⊗
recently,duetothefactthatseveralversionsofthemodelcanbeconstructedsothatanomaly
-
p cancellation is achieved [2,3] under the condition that, the number of families Nf equals the
e number of colors N =3. Among those models we have chosen to work with a particular one
c
h
that avoids the inclusion of fermion fields with exotic electric charges.
:
v As it has been recentlypointed out[4],the appearanceofsee-sawmechanismscouldbe in
i itself a guiding principle to distinguish between fundamental scales and those which are not;
X
if this is so, then the explanation of the five orders of magnitude spanned by the quark mass
r
a spectrum wouldrequirea new mass scaleunconnectedwith the electroweaksymmetry break-
ing mass scale, which may come from new physics, e.g.: supersymmetry, left-right symmetric
models,orsomethingelse. See-saworiginforallfermionmasseshasbeenanalyzedinthepast
in the context of several models [5].
Inthe frameworkprovidedbya3-3-1model,andbyusinga convenientsetofHiggsfields,
we show that one can avoidhierarchiesin the Yukawacouplings. The presence of a new scale
V >>v relatedtothebreakingofSU(3) U(1) ,triggerssee-sawmechanismsthatprovide
L X
⊗
a sensible mass spectrum for quarks. At the same time, these mechanisms provide relations
between the mass eingenstates and the weak interaction eingenstates, and thus a Cabbibo
Kobayashi Maskawa (CKM) mixing matrix emerges.
ThemodelbasedonthelocalgaugegroupSU(3) SU(3) U(1) has17gaugeBosons:
c L X
⊗ ⊗
one gauge field Bµ associated with U(1) , the 8 gluon fields Gµ associated with SU(3)
X c
(∗)E-mail: [email protected]
(cid:13)c EDPSciences
2
EUROPHYSICSLETTERS
which remain massless after breaking the symmetry, and another 8 gauge fields associated
with SU(3) and that we write for convenience as [6]
L
Dµ W+µ K+µ
1 1 1
λ Aµ = W µ Dµ K0µ ,
2 α α √2 K−µ K¯02µ Dµ 
− 3
 
where Dµ = Aµ/√2+Aµ/√6, Dµ = Aµ/√2+Aµ/√6, and Dµ = 2Aµ/√6. λ , i =
1 3 8 2 − 3 8 3 − 8 i
1,2,...,8, are the eight Gell-Mann matrices normalized as Tr(λ λ )=2δ .
i j ij
The charge operator associated with the unbroken gauge symmetry U(1) is given by
Q
Q=λ /2+λ /(2√3)+XI whereI =Diag.(1,1,1)(the diagonal3 3unitmatrix),and
3L 8L 3 3
×
the X values, related to the U(1) hypercharge,are fixed by anomaly cancellation. The sine
X
squareofthe electroweakmixing angleis givenby S2 =3g2/(3g2+4g2)where g andg are
W 1 3 1 1 3
the coupling constants of U(1) and SU(3) respectively, and the photon field is given by
X L
T
Aµ =S Aµ+C WAµ+ (1 T2 /3)Bµ , (1)
0 W 3 W √3 8 − W
(cid:20) q (cid:21)
where C and T are the cosine and tangent of the electroweak mixing angle, respectively.
W W
The two weak flavor diagonal neutral currents in the model are coupled to the gauge
Bosons
T T
Z0µ =CWAµ3 −SW √W3Aµ8 + (1−TW2 /3)Bµ ; Z0′µ =− (1−TW2 /3)Aµ8 + √W3Bµ, (2)
(cid:20) q (cid:21) q
where Zµ coincides with the neutral gauge boson of the SM [3]. There is also an electrically
0
neutral current associated with the flavor non diagonal gauge boson K0µ which is chargedin
the sense that it has a kind of weak V isospin charge.
The quark content of the model is [3,6]: Qi = (ui,di,Di) (3,3,0), i = 1,2 for
L L ∼
two families, where Di are two extra quarks of electric charge 1/3 (numbers inside the
L −
parenthesis stand for the [SU(3) ,SU(3) ,U(1) ] quantum numbers); Q3 = (d3,u3,U)
c L X L L ∼
(3,3 ,1/3), where U is an extra quark of electric charge 2/3. The right handed quarks are
∗ L
uac (3 ,1, 2/3), dac (3 ,1,1/3) with a = 1,2,3, a family index, Dic (3 ,1,1/3), i =
L ∼ ∗ − L ∼ ∗ L ∼ ∗
1,2, and Uc (3 ,1, 2/3). The lepton content of the model is: L = (e ,ν0,N0)
L ∼ ∗ − aL −a a a L ∼
(1,3 , 1/3), for a = 1,2,3 = e,µ,τ respectively [three SU(3) anti-triplets], and the three
∗ L
singlet−se+ (1,1,1),withν0 theneutrinofieldassociatedwiththeleptone andN0 playing
aL ∼ a a a
the role of the corresponding right-handed neutrinos. There are not exotic charged leptons,
anduniversalityforthe knownleptonsinthethreefamiliesispresentattreelevelinthe weak
basis. With the former quantum numbers the model is free of all the gauge anomalies [6].
Instead of using the set of Higgs fields introduced in the original papers [3], we use the
following set of four scalar triplets, with their Vacuum Expectation Values (VEV) as stated:
φT = (φ+,φ0,φ′0) = (0,0,v ) (1,3,1/3);
h 1i h 1 1 1 i h 1 i∼
φT = (φ+,φ0,φ′0) = (0,v ,0) (1,3,1/3);
h 2i h 2 2 2 i h 2 i∼
hφT3i = h(φ03,φ−3,φ′3−)i=h(v3,0,0)i∼(1,3,−2/3);
φT = (φ+,φ0,φ′0) = (0,0,V) (1,3,1/3),
h 4i h 4 4 4 i h i∼
with the hierarchy v v v v 102 GeV << V TeV. The analysis shows that this
1 2 3
∼ ∼ ∼ ∼ ∼
setofVEVbreakstheSU(3) SU(3) U(1) symmetryintwostepsfollowingthescheme
c L X
⊗ ⊗
V v
SU(3) SU(3) U(1) SU(3) SU(2) U(1) SU(3) U(1) ,
c L X c L Y c Q
⊗ ⊗ −→ ⊗ ⊗ −→ ⊗
H. Fanchiotti et al.: Quark masses without Yukawa hierarchies 3
where the first scale comes from V +v V and the second one from v +v v. The
1 2 3
≈ ≈
breaking allows for the matching conditions: g = g and 1/g2 = 1/g2+1/(3g2), where g
2 3 Y 1 2 2
and g are the gauge coupling constants of the SU(2) and U(1) gauge groups in the SM.
Y L Y
Related models to this, with the same fermion content but different scalar sector (φ is
1
absent) are analyzed in the papers in Refs. [3]. Other 3-3-1 models without exotic electric
charges, but with different fermion contents, can be found in Refs. [7]
The Higgsscalarsintroducedaboveareusedto writethe Yukawatermsfor the quarks. In
the case of the Up quark sector, the most general invariant Yukawa Lagrangianis given by
3 2 3
u = Q3φ C(hUUc + hu uac)+ Qi φ C( huuac+hUUc)+h.c., (3)
LY L α α L aα L L ∗3 ia L ′i L
α=1,2,4 a=1 i=1 a=1
X X X X
where the hu,U s are couplings that we assume of order one. C is the charge conjugation
′
operator. InordertorestrictthenumberofYukawacouplings,andproducearealisticfermion
mass spectrum, we introduce the following anomaly-free [8] discrete Z symmetry
2
Z (Qa,φ ,φ ,φ ,uic,dac)=1 ; Z (φ ,u3c,Uc,Dic,L ,e+ )=0, (4)
2 L 2 3 4 L L 2 1 L L L aL aL
where a=1,2,3(=e,µ,τ for the leptons) and i=1,2 are family indices.
Then in the basis (u1,u2,u3,U) we get, from Eq.(3-4), the following tree-level Up quark
mass matrix:
0 0 0 hu v
11 1
0 0 0 hu v
Mu = hu v hu v hu v h2u1V1 , (5)
13 3 23 3 32 2 34
 h′1Uv3 h′2Uv3 hU2v2 hU4V 
 
which is a see-saw type mass matrix, with one eigenvalue equal to zero.
On the other hand, the Yukawa terms for the Down quark sector, using the four Higgs
scalars introduced in Eq. (3), are:
LdY = QiLφ∗αC( hdiaαdaLc+ hDijαDLjc)+Q3Lφ3C( hDi DLic+ hdadaLc)+h.c..
α=1,2,4 i a j i a
X X X X X X
(6)
Inthebasis(d1,d2,d3,D2,D3)andusingthe discretesymmetryZ ,the formerexpression
2
produces the following tree-level Down quark mass matrix:
0 0 0 hd v hd v
11 1 21 1
0 0 0 hd v hd v
 12 1 22 1 
M = 0 0 0 hd v hd v , (7)
d 13 1 23 1
 hDv hDv hDv hD V hD V 
 11 2 21 2 1 3 114 214 
 hDv hDv hDv hD V hD V 
 12 2 22 2 2 3 124 224 
 
where we have used hD(d)v = hD(d)v . The mass matrix M is again a see-saw type, with
iaα α ia α d
at least one eigenvalue equal to zero.
BeforeenteringintoamoredetailedanalysisofM andM ,letusinsistintheresultingsee-
u d
saw characterof these matrices. In both cases there is a zero eigenvalue that we immediately
identify with the u and d quarks of the first family, respectively. Then, they are massless at
tree-level in the model considered here. In the U sector, the c quark acquires a see-saw mass,
4
EUROPHYSICSLETTERS
while in the D sector, both s quark and b quark have see-saw masses (nevertheless, with a
particular election of parameters,one can end up with a massless s quark too). The U sector
structure is in some sense singular because the top mass is of the order of the electroweak
scale; in fact it gets already a tree level mass of this order.
Afurthernumericalcheckofthematricesisdefinitiveinthesensethatthemodelprovides
a see-saw mass hierarchy defined by the relationship between the symmetry breaking scales
v/V. In what follows, and without loss of generality, we are going to impose the condition
v = v = v v << V, with the value for v fixed by the mass of the charged weak gauge
1 2 3
≡
boson M2 = g2(v2 +v2)/2 = g2v2 which implies v = 246/2 = 123 GeV (v is associated
W± 3 2 3 2 1
with an SU(2) singlet and does not contribute to the W mass).
L ±
StartingwiththeU matrix,theanalysisshowsthatM M hasonezeroeigenvalue,related
u† u
to the eigenvector [(hu hU hu hU),(hu hU hu hU),(hu hU hu hU),0], that we may
32 ′2 − 23 2 13 2 − 32 ′1 23 ′1 − 13 ′2
identify with the up quark u in the first family, which remains massless at tree-level.
In order to simplify the otherwise cumbersome calculations and to avoid the proliferation
ofunnecessaryparametersatthisstageofthe analysis,weproposetostartwiththe following
simple matrix
0 0 0 1
0 0 0 1
Mu′ =hv 1 1 hu /h δ 1 , (8)
32 −
 1 1 1 δ−1 
 
 
where δ = v/V is the expansion parameter and h is a parameter that can take any value of
order 1. The results below show that this matrix has the necessary ingredients to produce a
consistent mass spectrum.
Neglecting terms of order δ5 and higher, the four eigenvalues of M M are: one zero
u′† u′
eigenvalue related to the eigenstate (u1 u2)/√2 (notice the maximal mixing present); a
−
see-saw eigenvalue 4h2V2δ4 = 4h2v2δ2 m2 associated to the charm quark, and the other
≈ c
two
h2V2δ2 v2
[e2 +δ2e2(4 e2)/4] (h hu )2 m2
2 − + − − ≈ 2 − 32 ≈ t
h2V2[2+δ2(6+e /2)+δ4(4e2 e2e2 32)/8] m2
+ +− + −− ≈ U
wheree =(1 hu /h). The eigenvectorsaregivenbythe rowsofthe following4 4unitary
± ± 32 ×
matrix:
1 1 0 0
√2 −√2
UU = C√η21 C√η21 0 −S∆η1 , (9)
L 0 0 ∆ 1 δe+
 − − 2∆ 
 S√η21 S√η21 Cη21∆δe+ C∆η1 
 
where C and S √2δ(1 3δ2) are the cosine and sine of a mixing angle η , and ∆ =
η1 η1 ≈ − 1
(1+δ2e /4).
+
So, in the Up quark sector the heavy quark gets a large mass of order V, the top quark
p
gets a mass at the electroweak scale (times a difference of Yukawas that in the general case
of matrix (5) is (hU hu )), the charm quark gets a see-saw mass, and the first family up
2 − 32
quark u remains massless at tree-level. From the former expressions we get hU hu 2
| 2 − 32| ∼
and m 2hv2/V, which in turn implies V hm2/m 19.4h TeV., fixing in this way an
c ≈ ≈ t c ≈
upper limit for the 3-3-1 mass scale V (experimental values are taken from Ref. [9]).
We go now to the D quark mass matrix. This matrix is full of physical possibilities,
dependinguponthe particularvaluesassignedto theYukawacouplings. Forexample,ifallof
H. Fanchiotti et al.: Quark masses without Yukawa hierarchies 5
themaredifferentfromeachother,thenthematrixMd†Md hasrankonewithazeroeigenvalue
related to the eingenvector [(hDhD hDhD),(hDhD hDhD),(hDhD hDhD),0,0], that
22 1 − 21 2 11 2 − 12 1 21 12− 11 22
wemayidentifywiththedownquarkdinthefirstfamily(whichinanycaseremainsmassless
at tree-level); for this case the general analysis shows that we have two see-saw eigenvalues
associated with the bottom b and strange s quarks.
In the particular case when all the Yukawas are equal to one but hD = hD = HD =
114 224 6
1, the null space of Md†Md has rank two, with the eigenvectors associated with the zero
eigenvalues given by [ 2,1,1,0,0]/√6 and [0, 1,1,0,0]/√2, which in turn implies only one
− −
see-saweigenvalueassociatedwiththebottomquarkb,withavalueforitsmassapproximately
equal to 6vδ/(1+HD), with masses for the two heavy states of the order of V(1 HD).
±
Forthefirstcasementioned,thechiralsymmetryremainingattree-levelisSU(2) (quarks
f
u and d massless), and for the second case the chiral symmetry is SU(3) (quarks u, d and s
f
are massless). In both cases the chiral symmetry is broken by the radiative corrections.
In any way, a realistic analysis of the down sector requires to have in mind the mixing
matrix (9)ofthe upquarksectorandthe factthatthe CKMmixing matrixis almostunitary
and diagonal. Aiming at this and in order to avoid again a proliferation of parameters, let
us analyze the particular case given by the following left-right symmetric (hermitian) down
quark mass matrix:
0 0 0 1 1
0 0 0 1 1
 
Md′ =h′v 0 0 0 f g , (10)
 1 1 f HDδ−1 δ−1 
 
 1 1 g δ−1 HDδ−1 
 
where f and g are parameters of order one. This is the most general hermitian mass matrix
with only one eigenvalue equal to zero, related with the state (d1 d2)/√2, as required in
−
order to end up with an almost diagonal CKM mixing matrix.
The two see-saw exact eigenvalues of M are:
d′
δ (f g)2 8+(f +g)2 (f g)2 8+(f +g)2 2 8(f g)2
−h′v 4 HD− 1 + 1+HD ±s HD− 1 + 1+HD − 1 (−HD)2. (11)
(cid:20) − (cid:21) (cid:20) − (cid:21) − 
Moreover,notice thatfor the particular caseg = f, the five eigenvaluesofthe hermitian
 
−
matrix above get the following simple exact analytical expressions
h δ 1v
′ − 0,HD(1 1+16δ2/(HD)2), HD(1 1+8f2δ2/(HD)2) , (12)
2 + ± + − ± −
(cid:20) q q (cid:21)
where HD = HD 1. The see-saw values are thus 4δhv/HD and 2δf2h v/HD; they
imply f2±h/h m±HD/m 3HD and2h/h HDm−/m′,tha+tcanb−e seenas′either−amild
hierarchy b′ etw≈eenbh a−nd hc,≈or im−plying a′deta≈iled+tunsing cof some of the parameters of order
′
one (inconvenience that could be avoided by working in a frame where SU(3) becomes the
f
original chiral symmetry).
The eigenvectors are now given by the rows of the following 5 5 unitary matrix
×
1 1 0 0 0
−
C C 0 S S
UD = 1  0η2 0η2 √2C −Sη2 Sη2 , (13)
L √2 η3 − η3 − η3
 Sη2 Sη2 0 Cη2 −Cη2 
 0 0 √2S C C 
 η3 η3 η3 
 
6
EUROPHYSICSLETTERS
λ13v1v3 −f1v3(−f2v3)
⊗ ⊗
φ0∗ φ′0 φ0∗(φ0∗) φ′0∗
3 1 4 2 1
ujL h′jU ULc V(×ah)U4 UL hui1 uiLc diL hDij4(2) DLjc V(×hbDj)j4 DLj hdja daLc
Fig. 1 – Thefiveone-loop diagrams that producethe radiative masses for the quarksu and d.
where: C and S 2δ[1 δ2/(HD)2]/HD; C and S √2δf[1 3f2δ2/(HD)2]/HD
are the coηs2ines andη2s≈ines of −other tw+o mixin+g angηl3es η anηd3 ≈η . Notice−that up to th−is poin−t,
2 3
the CKM matrix UC(0K)M = ULu†ULd deviates from the identity just by terms of the order δ2
and higher; where Uu is the 3 3 upper sector of UU of eq.(9) and the same for Ud.
L × L L
The consistency of the model requires that one can identify mechanisms able to produce
masses for the first family, and to generate the CKM mixing angles. A detailed study of the
LagrangianfortheUpquarksector(3),thediscreteZ symmetry(4)andthescalarpotential,
2
allows us to draw the radiative diagram in Fig.(1a), which is the only diagram available to
produce a finite one-loop radiative correction in the quark subspace (u1,u2) of the Up quark
sector. The mixing of the HiggsBosonscomes froma termin the scalarpotential ofthe form
λ (φ φ )(φ φ ), which turns on the radiative correction.
13 ∗1 1 ∗3 3
In order to have a contribution different from zero we must avoid maximal mixing in the
first two weak interaction states, otherwise a submatrix of the democratic type arises. This
is done by taking hu = 1 k and hU = 1+k in matrix (8) insted of 1, where k must be a
11 − ′1
very small parameterinorder to guarantee the see-saw characterof the Up sector quark mass
matrix. Evaluate the contribution coming from the diagram in Fig. (1a) we get
∆ =N [M2m2ln(M2/m2) M2m2ln(M2/m2)+m2m2ln(m2/m2)], (14)
ji ji 1 1 − 3 3 3 1 1 3
where N =hUhuλ v v M/[16π2(m2 m2)(M2 m2)(M2 m2)], M =hUV is the mass
ji ′j i1 13 1 3 3− 1 − 1 − 3 4
oftheheavyUpquark,andm andm arethe massesofφ0 andφ0 respectively. Toestimate
1 3 ′1 3
the contribution given by this diagram we assume the validity of the “extended survival
hypothesis” [10] which in our case means m m v, implying
1 3
≈ ≈
m λ vδln(V/v)/8π2 0.85λ MeV, (15)
u 13 13
≈ ≈
whichforλ 2producesm 1.7MeV,whichisofthecorrectorderofmagnitude[9](result
13 u
∼ ≈
independentofthevalueofkinfirstapproximation). Duetothefactthattheparameterk =0,
6
thestaterelatedtotheuquarkloosesitsmaximalmixing,becomingnow (h hu )u1+[h
{− − 32 −
hu (1 k)]u2+ku3 /N, with N being the normalization factor. The value of k is estimated
32 − }
with the value of the Cabbibo angle to be k 0.1.
≈
Forthe Downquarksectortherearefourone-loopdiagrams,twoforD1 andothertwofor
D2 as depicted in Fig.(1b). The mixing in the Higgs sector comes from terms in the scalar
potential of the form f φ φ φ +f φ φ φ +h.c.. When the algebra is done we get
1 1 3 4 2 1 2 3
m 2(f +f )δln(V/v)/8π2, (16)
d 1 2
≈
whichforf =f v implies m 2m ,without introducing anew massscalein the model.
1 2 d u
≈ ≈
H. Fanchiotti et al.: Quark masses without Yukawa hierarchies 7
The discrete Z symmetry introduced eliminates possible tree-level lepton mass terms of
2
the form L φ Ce and L L φ . Then in order to generate masses for the leptons we
aL 3 bL aL bL 3
mustuse eitherleptoquarkHiggsFields ifwe intentto use the radiativemechanism,orexotic
leptonsifwewanttousesee-sawmechanisms. Fortheneutrinosforexamplethisanalysishas
been done in Ref. [11], where new SU(3) Higgs scalar multiplets are introduced.
L
Inamodellikethiswithfourscalartriplets,weshouldworryaboutpossibleflavorchanging
neutral current (FCNC) effects. First we notice that due to our Z symmetry, they do not
2
occur at tree-levelbecause each flavorcouples only to a single multiplet. They can enter as a
consequenceoftheviolationofunitarityoftheCKMmatrixU0 whichisa3 3submatrix
CKM ×
of a rectangular 4 5 matrix. The violation of unitarity in our analysis is proportional
×
to δ2, implying FCNC proportional to δ4. Then, a value of δ 10 2 is perfectly safe as
−
≈
far as violation of unitarity of the CKM matrix and possible FCNC effects are concerned.
ExperimentalconstraintsontheposibleviolationofunitarityoftheCKMmatrixarediscussed
in Section 11 of Ref. [9].
In several 3-3-1 models with three scalar triplets [2,3] a discrete symmetry can suppress
mass terms for the neutral Higgs bosons and to produce axion states [12]. The preliminary
analysis shows that the Z symmetry introduced in our model with four scalar triplets, pro-
2
vides only with the eight Goldstone Bosons needed, and nothing else.
Inconclusion,wehavepresentedamodelwithonlytwoenergyscales,thathasthepowerof
avoiding hierarchies among Yukawa couplings. Throughout the analysis, all the Yukawas are
oforderone,asalsoisthe caseforthedimensionlessHiggscouplingλ . Thenewingredients
13
ofthemodelare: themassscaleV usedtodefinetheexpansionparameterδ,anewsetofHiggs
scalars and VEV and the discrete anomaly-free symmetry Z . All this triggers generalized
2
see-saw mechanism in the Up and Down quark sectors.
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