The origin of the first and third generation fermion masses in a technicolor scenario

0.27 MB·English

by  A. Doff

EPJ manuscript No.
(will be inserted by the editor)
The origin of the first and third generation fermion masses in a
technicolor scenario
A. Doff and A. A. Natale
4
0 Institutode F´ısica Te´orica, UNESP,RuaPamplona 145, 01405-900, S˜ao Paulo, SP,Brazil
0
2 Received: date/ Revised version: date

n
a Abstract. We argue that the masses of the first and third fermionic generations, which are respectively
J of the order of a few MeV up to a hundred GeV, are originated in a dynamical symmetry breaking

2 mechanism leading to masses of the order αµ, where α is a small coupling constant and µ, in the case
2 of the first fermionic generation, is the scale of the dynamical quark mass (≈ 250 MeV). For the third
fermion generation µ is the value of the dynamical techniquark mass (≈ 250 GeV). We discuss how this

4 possibility canbeimplementedinatechnicolorscenario,andhowthemassoftheintermediategeneration
v is generated.
6
6
PACS. 12.60.Nz Technicolor models – 12.10.Dm Unified theories and models of strong and electroweak
1
interactions – 14.80.Cp Non-standard-modelHiggs bosons
2
0
3
0 1 Introduction must be of order of a few MeV and a hundred GeV re-
/ spectively.InmodelswithafundamentalHiggsbosonthe
h Thestandardmodelisinexcellentagreementwiththeex- values of A and C are obtained due to adjusted vacuum
p
perimentaldata.Theonlystillobscurepartofthemodelis expectationvalues(vev)orYukawacouplings.Inthisway
-
p the oneresponsiblefor the massgeneration,i.e.the Higgs there is no natural explanation for the values of A and
e mechanism.Inordertomake the massgenerationmecha- C;they appear just as anadhoc choiceof couplings!The
h nismmorenaturalthereareseveralalternatives,wherethe questionthatwewouldlike todiscusshereishowwenat-
v: mostpopularonesaresupersymmetryandtechnicolor.In urally can generate the scales A and C? In order to do
i the firstone the mass generationoccursthroughthe exis- so let us recall which are the mass scales in the standard
X
tence of non-trivial vacuum expectation values of funda- model. In this model we have basically two natural mass
r mental scalar bosons while in the second case the bosons scales: µ ≈ 250 MeV, which is the quantum chromo-
a qcd
responsibleforthebreakingofgaugeandchiralsymmetry dynamics (qcd) dynamical quark mass scale and v ≈ 250
are composite. Up to now the fermionic mass spectrum is GeV, the vacuum expectation value of the fundamental
the strongest hint that we have in order to unravel the Higgs field responsible for the gauge symmetry breaking.
symmetry breaking mechanism. A simple and interesting As qcd is already an example of a theory with dynami-
way to describe the fermionic mass spectrum is to sup- cal symmetry breaking we will also assume that techni-
pose that the mechanism behind mass generation is able color (tc) models provide a more natural way to explain
to produce a non-diagonal mass matrix with the Fritzsch the gauge symmetry breaking [2,3], i.e. at this level all
texture [1] thesymmetrybreakingmechanismsshouldbedynamical.
Therefore we will not discuss about a fundamental scalar
0 A 0
field with vev v but of a composite scalar field character-
M = A 0 B . (1)
f  0∗ B C ized by µtc ≈250 GeV, which is the scale of the dynami-
∗ cal techniquark mass. Of course, at very high energies we
 
This matrix is similar for the charged leptons, 1/3 and possiblyhaveothernaturalmassscalesasthePlanckone,
2/3 charged quarks. The entry C is proportional to the a grand unified theory (gut) scale Mgut or a horizontal
mass of the third generation fermion, while the entry A (family) symmetry mass scale Mh, although it is far from
is proportional to the mass of the lighter first genera- clear how such scales interfere with the values of A and
tion. The diagonalization of such mass matrix will de- C. Finally, in tc models we may also have the extended
termine the CKM mixing angles and the resulting diag- technicolor (etc) mass scale Metc [4] upon which no con-
onal mass matrix should reproduce the observed current straint can be established above the 1 TeV scale [5]. In
fermion masses. There are other possible patterns for the this work we will build a model where the scales A and
massmatrixandwechoosethe oneofEq.(1)justforsim- C of Eq.(1) can be related respectively to the scales µqcd
plicity. We call attention to the values of A and C. They andµtc timessomesmallcouplingconstant.Thevaluesof
2 A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario
Eq.(1) will depend the least as possible on the very high which is the one obtained when the composite operator
energy mass scales like M , M , etc ... The model will ψ¯ψ ≡µ3 has canonical dimension and where i can in-
gut etc i i i
require a very peculiar dynamics for the tc theory as well dicatetcorqcd.Whenθ =0operatorsofhigherdimension
(cid:10) (cid:11)
asforqcd,andthispeculiarityinwhatconcernsqcddiffers may lead to the hard self-energy expression
the present approach from any other that may be found
in the literature. In the next section we discuss which is Σh(p)=µ[1+bgt2c(qcd)(µ2)ln(p2/µ2)]−γ, (4)
thedynamicsofnon-Abeliantheoriesthatwillleadtothe
desired relation between A(C) and µqcd(µtc). In Section where γ must be larger than 1/2 and the self-energy be-
III we introducea modelassumingthatits stronglyinter- haves like a bare mass [7]. Therefore no matter is the di-
acting sector has the properties described in the previous mensionalityoftheoperatorsresponsibleforthemassgen-
section,andshow thatthe intermediate massscale(B) of eration in technicolor theories the self-energy can always
Eq.(1) appears naturally in such a scheme. In Section IV be described by Eq.(2). In the above equations g is
tc(qcd)
we compute the fermion mass matrix. Section V contain the technicolor(qcd) coupling constant and γ = 3ctc(qcd),
some brief comments about the pseudo goldstone bosons 16π2b
wherec = 1[C (R )+C (R )−C (R )],withthe
that appear in our model and we draw our conclusions in tc(qcd) 2 2 1 2 2 2 ψψ
the last section. quadratic Casimir operators C2(R1) and C2(R2) associ-
atedtotheR.H andL.H fermionicrepresentationsofthe
technicolor(qcd)group,andC (R )isrelatedtothecon-
2 ψψ
2 The self-energy of quarks and techniquarks densate representation. b is the g3 coefficient of the
tc(qcd)
technicolor(qcd)groupβ function.Thecompleteequation
Intcmodelstheordinaryfermionmassisgeneratedthrough for the dynamical fermion mass displayed in Fig.(1) is
the diagram shown in Fig.(1). In Fig.(1) the boson indi-
AcabteeldiabnyboSsUon(k,)wcitohrrceosuppolnindgs αtok tthoefeerxmchioannsg(efo)foar tneochn-- mf =ak dq4 µq22 θ gk2(q)([q12++bMtc(2q)c(dq)g2t2+c(qµcd2)ln(µq)22)]−δ,
nifermions (T). In the models found in the literature the Z (cid:18) (cid:19) k tc(qcd)
(5)
role of the SU(k) group is performed by the extended
technicolor group and the boson mass is given by M where we define a = 3C2kµtc(qcd). In the last equation
etc k 16π4
To perform the calculation of Fig.(1) we can use the fol- C is the Casimir operator related to the fermionic rep-
2k
resentations of the SU(k) (or etc) group connecting the
differentfermions(tcorqcd),g andM aretherespective
k k
coupling constantandgauge bosonmass,a factor µ
SU(k) tc(qcd)
remained in the fermion propagator as a natural infrared
regulator and δ = γcosθπ, g2(q) is assumed to be giving
k
by
f αk Tf,f Tf,f αk f gk2(q2)≃ (1+b gg2k2((MM2k2))ln( q2 )). (6)
k k k M2
k
Fig.1. Typicaldiagramcontributingtothefermionmassesof Note that in Eq.(5) we have two terms of the form [1+
thefirstandthirdgeneration.Theexchangeofthebosonindi- b g2lnq2] where the index i can be related to tc(qcd) or
i i
catedbySU(k)playsthesameroleofanextendedtechnicolor SU(k). To obtain an analytical formula for the fermion
boson. mass we will consider the substitution q2 → xMk2, and we
µ2
i
will assume that b g2(M ) ≈ b g2 (M ) , what
k k k tc(qcd) tc(qcd) k
lowing general expression for the techniquark (or quark) will simplify considerably the calculation. Knowing that
self-energy [6] the SU(k) group usually is larger than the tc(qcd) one,
we computed numerically the error in this approximation
µ2 θ for few examples found in the literature. The resulting
Σ(p)g =µ p2 [1+bgt2c(qcd)(µ2)ln(p2/µ2)]−,γcos(θπ) expressionfor mf will be overestimated by a factor 1.1−
(cid:18) (cid:19) 1.3 and is giving by
(2)
whereinthelastequationweidentifiedγ =γ asthe
canonicalanomalous dimension of the tc(qcd)tcm(qcads)s oper- m ≃ 3C2kgk2(Mk)µ µ2 θ1+b g2 lnMk2 −Iδ,
ator,andµisthedynamicalfermion(tcorqcd)mass.The f 16π2 M2 tc(qcd) tc(qcd) µ2
(cid:18) k(cid:19)(cid:20) (cid:21)
advantage of using such expression is that it interpolates (7)
betweentheextremepossibilitiesforthetechnifermion(or where
quark) self-energy, i.e. when θ = 1 we have the soft self- I = 1 ∞dσσǫ 1e σ 1 .
− −
energy giving by Γ(σ) θ+ρσ
Z0
Σ (p)= µ3[1+bg2 (µ2)ln(p2/µ2)]γ, (3) with ρ = btc(qcd)gt2c(qcd)(Mk) and ǫ = δ +1 = γcosθπ +
s p2 tc(qcd) 1. To obtain Eq.(7) we made use of the following Mellin
A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario 3
transform obtain. Finally, this is our only working hypothesis and
willleadustothefollowingproblem:Howcanweprevent
ǫ σκ
1+κln x −= 1 ∞dσe σ x −σǫ 1. (8) the coupling of the first and second fermionic generations
− −
µ2 Γ(ǫ) µ2 tothetechnifermions?Amodelalongthislineisproposed
(cid:20) (cid:21) Z0 (cid:18) (cid:19)
in the next section.
Finally, we obtain
3C g2(M )µ µ2 θ
m ≃ 2k k etc F(cosθπ,γ,ρ). (9) 3 The model
f 16π2 M2
(cid:18) etc(cid:19)
where 3.1 The fermionic content and couplings
γcos(θπ)
θ θ 1 θ
According to the dynamics that we proposed in the pre-
F(cosθπ,γ,ρ)= Γ(−γcos(θπ), )exp( )
ρ ρ ρ ρ vious section, which consists in a self-energy with θ = 0
(cid:18) (cid:19)
M2 −γcos(θπ) inEq.(2),andasthe differentfermionmasseswillbegen-
1+b g2 ln k . erated due to the interaction with different strong forces,
tc(qcd) tc(qcd) µ2
(cid:20) (cid:21) we must introduce a horizontal (or family) symmetry to
preventthe first andsecond generationordinaryfermions
Simple inspection of the above equations shows that
to couple to technifermions at leading order. The lighter
θ =0 lead us to the relation that we are looking for i.e.
generationswillcoupleonlytotheqcdcondensateoronly
C ∝g2µ , (10) at higher loop order in the case of the tc condensate. Us-
k tc ing the hard expression for the self-energy (Eq.(4)) the
whichgivemassesofO(GeV).IftheSU(k)(oretc)bosons fermion masses will depend only logarithmically on the
connect quarks to other ordinary fermions we also have masses of the gauge bosons connecting ordinary fermions
to technifermions. Therefore we may choose a scale for
A∝g2µ , (11) these interactions of the order of a gut scale, without the
k qcd
introduction of large changes in the value of the fermion
which are masses of a few MeV. To obtain Eqs.(10) and masses. We stress again that the only hypothesis intro-
(11) we neglected the logarithmic term that appears in duced up to now is the dynamics described in the pre-
Eq.(9). In principle there is no problemto assume the ex- vious section. On the other hand, as we shall see in the
istenceofatcdynamicalself-energywithθ =0.Thereare sequence,wewillsubstitutethe needofanextendedtech-
tc models where it has been assumed that the self-energy nicolor group by the existence of a quite expected unified
isdominatedbyhigherorderinteractionsthatarerelavant theory containing tc and the standard model (SM) at a
atorabovethetcscaleleadingnaturallytoaveryharddy- gut scale. There is also another advantage in our scheme:
namics [8,9]. The existence of a hard self-energy in qcd is It will be quite independent of the physics at this “uni-
the unusualingredientthat we areintroducing here.Usu- fication” scale and will require only a symmetry (hori-
ally it is assumed that such solutionis not alloweddue to zontal) preventing the leading order coupling of the light
a standard operator product expansion (OPE) argument fermion generations to technifermions. Finally, the hori-
[10]. This argument does not hold if there are higher or- zontal symmetry will be a local one, although we expect
der interactions in the theory or a nontrivial fixed point that a globalsymmetry will also lead to the same results.
of the qcd (or tc) β function [11]. There are many pros We consider a unified theory based on the SU(9) gauge
and cons in this problem which we will not repeat here group,containing a SU(4) tc group (stronger than qcd)
tc
[12], but we just argue that several recent calculations of and the standard model, with the following anomaly free
the infraredqcd (or any non-Abelian theory)are showing fermionic representations [17]
theexistenceofanIRfixedpoint[13]andtheexistenceof
a gluon (or technigluon) mass scale which naturally leads 5⊗[9,8]⊕1⊗[9,2] (12)
to an IR fixed point [14]. The existence of such a mass
scale seems to modify the structure of chiral symmetry where the [8] and [2] are antisymmetric under SU(9).
breaking [15]. This fact is not the only one that may lead Therefore the fermionic content of these representations
to a failure of the standard OPE argument. For instance, can be decomposed according to the group product
the effect of dimension two gluon condensates, if they ex- SU(4) ⊗ SU(5) (SU(5) is the standard Georgi-
tc gg gg
ist, [16] can also modify the dynamics of chiral symmetry Glashow gut [18]) as:
breakingandthis possibilityhasnotbeeninvestigatedup
to now.Therefore it seems that we still do not have a full [9,2]
understanding of the IR behaviorof the non-Abelian the-
0 u¯ −u¯ −u −d
iB iY iR iR
ories, which can modify the behavior of the self-energies
−u¯ 0 u¯ −u −d
iB iR iY iY
that we are dealing with. According to this we will just (1,10)= u¯ −u¯ 0 −u −d 
iY iR iB iB
assume that such behavior can happen in tc as well as in
u u u 0 e¯
qcd.Howmuchthisisabadorgoodassumptionitwillbe  iR iY iB i 
 d d d −e¯ 0 
certainlyreflectedinthefermionicspectrumthatweshall  iR iY iB i 
 
4 A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario
QiR zontal symmetry SU(3)H are shown in Fig.(2). With the
Q
iY
(4,5)= QiB , (¯6,1)=Ni
 L¯i  Qi ui
 N¯i TC ui ui
  SU(9) SU(5)
[9,8]
d¯ X¯
(1,¯5)= dd¯¯iiYR , (1,¯5)=XX¯¯RYkk di Li ui di
iB Bk SU(9) SU(5)
e E
 i   k 
 ν  N 
 ei   Eki Ni ei
di di
SU(9) SU(5)
(¯4,1)= Q¯ ,L ,N ,
iε i iL
where ε= 1..3 is a color index and k = 1..4 indicates the Li di
generation number of exotic fermions that must be intro- ui di
duced in order to render the model anomaly free. These SU(9) SU(3)
fermionswillacquiremassesoftheorderofthegranduni-
fiedscale.Wearealsoindicatingageneration(orhorizon-
tal) index i = 1..3, that will appear due to the necessary Qi ui
e u
replication of families associated to a SU(3) horizon- i i
H SU(9) SU(3)
tal group. This model is a variation of a model proposed
by one of us many years ago [19]. The mass matrix of
Eq.(1) will be formed according to the representations of Qi Li
the strongly interacting fermions of the theory under the di ei
SU(3)
SU(3) group. The technifermions form a quartet under SU(3)
H
SU(4) and the quarks are triplets of qcd. The techni-
tc
color and color condensates will be formed at the scales
Fig. 2. Couplings of ordinary fermions and technifermions to
µ and µ in the most attractive channel (mac) [20] of
thtce produqcctds ¯4⊗4 and ¯3⊗3 of each strongly interact- the gauge bosons of SU(9), SU(5)gg and SU(3)H which are
relevant for the generation of fermion masses.
ing theory. We assign the horizontal quantum numbers
to technifermions and quarks such that these same prod-
uctscanbedecomposedinthefollowingrepresentationsof
SU(3) :6inthecaseofthetechnicolorcondensate,and3 couplingsshowninFig.(2)wecandeterminethediagrams
H
that are going to contribute to the 2/3 and 1/3 charged
inthecaseoftheqcdcondensate.Forthisitisenoughthat
quark masses as well as to the charged lepton masses.
the standard left-handed (right-handed) fermions trans-
ThesediagramsarerespectivelyshowninFig.(3)to(5).It
form as triplets (antitriplets) under SU(3) , assuming
H
thatthetcandqcdcondensatesareformedinthe6andin is importantto observethe followinginthe abovefigures:
the3oftheSU(3) group.Thisisconsistentwiththemac The second generation fermions obtain masses only at a
H
twolooporder.Thismasswillbeproportionaltoµ times
hypothesis [20] althougha complete analysis ofthis prob- tc
two small couplings (g and g , respectively the SU(3)
lem is out of the scope of this work. The above choice for h 9 H
and SU(9) coupling constants). It will also be nondiago-
thecondensationchannelsiscrucialforourmodel,because
thetccondensateintherepresentation6(ofSU(3) )will nal in the SU(3)H indices. The first generation fermions
H
obtain masses only due to the qcd condensate whereas
interactonlywiththethirdfermionicgenerationwhilethe
3 (the qcd condensate) will interact only with the first the third generation ones couple directly to the tc con-
densates. Due to the particular choice of representations
generation.Inthis waywe cangeneratethe coefficients C
under the unified theory containing tc and the standard
and A respectively of Eq.(1), because when we add these
model we end up with more than one mass diagram for
condensates (vevs) and write them as a 3×3 matrix we
will end up (at leading order) with severalfermions. It is particularly interesting the way the
fermions of the first generation obtain masses. In some of
0 A 0 the diagrams of the above figures we show a boson that
M = A 0 0 . (13) is indicated by SU(5). This boson belongs to the SU(9)
f ∗
 0 0 C group, but would also appear in the standard SU(5)gg
gut.Forexample,theelectrononlycouplestothedquark
 
ThisproblemisverysimilartotheoneproposedbyBerezhi- (andtotheqcdcondensate)throughaSU(5) gaugebo-
gg
ani andGelmini et al. [21] where the vevs of fundamental son existent in the Georgi-Glashowminimal gut, whereas
scalarsaresubstitutedbycondensates.Thenewcouplings theuanddcanconnectto thesecondgenerationthrough
generated by the unified SU(9) group and by the hori- the horizontal symmetry gauge bosons. We also expect
A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario 5
other diagrams at higher order in gh and/or g9 that are SU(5)
not drawn in these figures.
e d d e
(g)
SU(5) SU(3)H SU(3)H SU(3)H
SU(9) SU(3)
H
u u,d d,u u (a) u c c u
SU(3) SU(3) µ τ Q Q τ µ (h) µ τ L L τ µ
H H
SU(9) SU(9)
SU(9) SU(3)H
c t Q Q t c c t L L t c
(b)
τ Q Q τ (i) τ L L τ
SU(9) SU(9)
Fig. 5. Diagrams contributing tolepton masses.
t Q Q t (c) t L L t
Fig.3.Diagramscontributingtothecharge2/3quarkmasses. to the qcd andtc vacuum condensates.The bosonsrepre-
In (a) we indicate by SU(5) the exchange of a boson that sented by η and ϕ, respectively, are related to the system
belongstotheSU(9)group,butthatwouldalsoappearinthe of composite Higgs bosons formed in the representations
minimal SU(5) gut. 3and6ofthe horizontalgroup.Suchsuppositionisquite
plausible if we consider the results of Ref.[8,9], where it
was shown that the interactions of a composite Higgs bo-
son is very similar to the ones of a fundamental boson.
Our intention is to show that such system leads to an in-
SU(5) SU(3)H termediate mass scale and to a mass matrix identical to
Eq.(1).
d u u d d s s d The vevs of qcd and technicolor, due to the horizon-
(d)
talsymmetry,canbe writtenrespectivelyin the following
SU(3) SU(3)
H H form [21]
SU(9) SU(3)H
0 00 0
s b N,L L,N b s (e) s b Q Q b s hηi∼ 0 , hϕi∼ 00 0 , (16)
   
v 00v
η ϕ
SU(9) SU(3)H    
and will be of the order of 250 MeV and 250 GeV. It is
b L,N N,L b (f) b Q Q b instructive at this point to observe what fermionic mass
matrix we can obtain with the vevs of Eq.(16). We can
Fig. 4. Diagrams contributing to the mass generation of 1/3 assume that the composite scalars η and ϕ have ordinary
charged quarks. Yukawa couplings [1,21] to fermions described by the fol-
lowing effective Yukawa lagrangian
L =aΨ¯i ηkUjǫ +bΨ¯i ϕijUj, (17)
Y Lλ λ R ijk Lλ R
3.2 The composite Higgs system where Ψ and U are the ordinary fermion fields. λ is a
weak hypercharge (SU(2) ) index, for instance, λ = 1
w
We can also observe that the second generation fermions
represents charge 2/3 quarks and λ = 2 correspond to
will be massive not looking at the diagrams of Fig.(3) to
the charge 1/3 quarks, i,j e k indicate the components
(5), but studying the composite Higgs system. With this of the composite scalar bosons of the representations 3
we meanthat the qcdandtc condensatesactasif wehad and 6 of SU(3) and a and b are the coupling constants.
H
twocompositebosonsrepresentedbythefieldsηandϕ.In
SubstitutingthevevsofEq.(16)intheYukawalagrangian
principle this system could be described by the following
for the charge 2/3 quarks, we obtain
effective potential
L =ac¯ v u −au¯ v c +bt¯ v t , (18)
V(η,ϕ)=µ2ηη†η+λη(η†η)2+µ2ϕϕ†ϕ+λϕ(ϕ†ϕ)2, (14) Y L η R L η R L ϕ R
in such a way that we can identify the vevs (given by the leading to a mass matrix in the (u,c,t) basis which is
ratio of masses and couplings) given by
0 −av 0
η
v2 =−µ2η , v2 =−µ2ϕ, (15) m32 =avη 0 0 . (19)
η λη ϕ λϕ 0 0 bvϕ
 
6 A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario
The main point of the model is that the fermions of the This coefficient can be calculated if we assume that hηi is
third generation obtain large masses because they couple givenbyEq.(21),hϕiisthesamevevdescribedbyEq.(16)
directly to technifermions, while the ones of the first gen- and both are related to the tc and qcd condensates. We
eration obtain masses originated by the ordinary conden- will neglect δ compared to Π in Eq.(20), what is reason-
sation of qcd quarks. Having this picture on mind we can able if we look at Fig.(6) (Π is given by the first dia-
nowseethatthemostgeneralvevforthissystemincludes gram). The coupling Π is computed from the first dia-
the mass generation for the intermediate family. gram of Fig.(6) using the effective vertex χχWW shown
Itisimportanttoverifythatthereisnowaytoprevent in Fig.(7), where an ordinary fermion runs in the loop,
the coupling at higher order of the different composite where the χ field may indicate technicolor (χ=η) or qcd
scalar bosons with SU(3) quantum numbers. Examples
H
of such couplings are shown in Fig.(6)
χ
W
η ϕ
η η
gw2 W− gw2
W+
gw2 gw2 + W+ W+ . . .
W−
η ϕ ϕ gw2 W− gw2 ϕ χ W
Fig.6.Higherordercorrectionscouplingtheηandϕcompos-
ite bosons. Fig.7. Diagramleadingtothecouplingbetweentwocompos-
itescalar bosons and two gauge bosons
ThediagramsofFig.(6)willproducenewtermsforthe
effective potential of our composite system, therefore we (χ = ϕ) composites scalar bosons. To compute Fig.(7)
must add to Eq.(14) the following terms we also need the effective coupling between the compos-
ite scalarsboson and the ordinary fermions. This one has
V2(η,ϕ)=Πη†ηϕ†ϕ+δη†ϕηϕ†+... (20) beencalculatedin the workof Carpenteret al.[8,9]some
yearsagoanditisshowninFig.(8).Afteraseriesofsteps
The introduction of this expression in the potential of
Eq.(14)willshiftthevevsgeneratedbytheeffectivefields
η and ϕ, and the vev associated to the field η will be
shifted to
ε g
hηi∼ 0 . (21) WΣ
2Μ
 
v W
η
 
We do not include the shift in the vev of ϕ, because
v ≪v and the modification is negligible. Note that the
η ϕ
Yukawa lagrangian that we discussed in Eq.(18) in terms Fig. 8. Vertex coupling a scalar composite boson to ordinary
of the new vevs can be written as fermions
L = ac¯ v u −au¯ v c +bt¯ v t −ac¯ εt +at¯ εc .
Y L η R L η R L ϕ R L R L R
(22) the calculation of the diagram of Fig.(7) will be given by
Therefore,in the (u,c,t) basis, the structure of the mass
matrix now is g4 δab gµν Σ2
Π ∼− W d2q χ. (25)
χχWW M2 32π2 q2
0 −avη 0 W Z
m32 =avη 0 aε . (23) Following closely the procedure adopted by Carpenter et
0 −aε bv
ϕ ς
al.[9]wemayapproximatetheselfenergybyΣ ∼µ q2 −,
  χ χ µ2
This example was motivated by a system of fundamental χ
Higgsbosons[21].Butthemostremarkablefactisthatwe where ςχ = 3C162πχ2gχ2 , to obtain the following couplin(cid:16)g be(cid:17)-
canreproducethisresultwithacompositesystemformed tween two composite scalars and the intermediate gauge
by the effective low energy theories coming from qcd and bosons of the weak interaction
tc as we shall see in the following. The coefficient ε in
M2 δabG2µ2
Eq.(21) will result from the minimization of the full po- Π ∼− W F χgµν. (26)
tential χχWW 2π2 ςχ
V(η,ϕ)= µ2ηη†η+λη(η†η)2+µ2ϕϕ†ϕ+λϕ(ϕ†ϕ)2+ In Eq.(26) we made use of the relation GF = gW2 . Note
√2 8MW2
Πη ηϕ ϕ+δη ϕηϕ . (24) that the coupling between scalars and gauge bosons is
† † † †
A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario 7
dominated by the ultravioletlimit, where the approxima- 4 Computing the mass matrix
tion for the self energy discussed above is also valid. The
effective coupling Π in Eq.(24) is equivalent to the cal- We can now compute the mass matrix. Let us first con-
culation of the first diagram of Fig.(6). Using Eq.(26) we sider only the 2 charged quarks and verify their different
3
will come to the following expression contributions to the matrix in Eq.(1). These will come
from the diagrams labeled (a), (b) and (c) in Fig.(3) and
MW4 G4Fµ2tcµ2qcd are equal to
Π = . (27)
ηηϕϕ 32π8ς ς
tc qcd
µ M2 −γqcd+1
We can now approximately determine the value of ε A = 10c qcαd 1+bgq2cdlnµ25 +
assuming that the potential of Eq.(14) has a minimum qcd qcd " qcd#
describedbythevevshϕi,Eq.(16),andhηi,Eq.(21),what 4µ M2 −γqcd+1
leadustothefollowingvalueofthepotentialatminimum qcd 1+bg2 ln h ,
135c α qcd µ2
qcd qcd " qcd#
V(η,ϕ)| =µ2v2+λ v4+µ2v2 +λ v4 +λ ε4. (28)
min η η η η ϕ ϕ ϕ ϕ η B = 28µtc 1+bg2lnM92 −,γtc+1
We then compare the minimum of this potential with the 675πc α tc µ2
tc tc (cid:20) tc(cid:21)
one obtained from Eq.(24), where the term proportional
to δ is neglected in comparison to the one proportional C = 2µtc 1+bg2lnM92 −.γtc+1 (31)
to Π. This is equivalent to say that the second diagram 15c α tc µ2
tc tc (cid:20) tc(cid:21)
of Fig.(6) is much smaller then the first diagram,and the
vevsenteringinEq.(24)aretheunperturbedonesbecause WherethecontributionsforA,BandC comerespectively
the perturbation will enter through the Π term. Finally, from the diagrams (a), (b) and (c) displayed in Fig.(3).
assuming that the coefficient describing the coupling be- The values A, B and C correspond to the nondiagonal
tweenfourscalarbosonsthatareformedinthechiralsym- massesinthehorizontalsymmetrybasis.Tocometothese
metry breaking of QCD is given by[9] valuesweassumedα (=α =α =α )∼ 1 attheunifi-
k 9 h 5 45
cationscale.We alsoassumed,when computing diagrams
G2Fµ4qcdcqcdαqcd involvingthetechnileptonsandtechniquarkscondensates,
λ = , (29)
η the following relation
π
andwecanobtainasimilarexpressionforλϕ afterchang- hL¯Li= 1hQ¯Qi, (32)
ing the indices qcd by tc. Equalizing vη and vϕ to the 3
known qcd and tc condensates (assuming ψ¯ψ = v3 ≈
i i
becausethetechniquarkscarryalsothethreecolordegrees
µ3 [22]), we conclude that
i (cid:10) (cid:11) of freedom. As the mass matrix is the same obtained in
Ref.[1] we can use the same diagonalization procedure to
1
ε∼B ∼ MW4 G2Fµ4tc 4 GeV ∼16.8GeV. (30) obtain the t, c and u quark masses, which is given by
18π3c α
Thesurprising(cid:18)factinttchistcc(cid:19)alculationisthatthe coupling Mf23Diag =R−1Mf32R, (33)
ofthedifferentscalarbosonshasbeendetermineddynam-
where R is a rotation matrix described in Ref.[1]. After
ically and gives exactly the expected value for the nondi-
diagonalizationwe obtain
agonal coefficient B. In models with fundamental scalar
bosons this value results from one ad hoc choice. In this |A|2 |B |2
section we presented our model, determined the main di- m ∼ |C | , m ∼ and m ∼ |C |,
agrams contributing to the fermion masses and showed u |B |2 c |C | t
that this scenario naturally leads to a fermion mass ma- (34)
trix with the Fritzsch texture. We have not tested many where the values of A, B and C are the ones shown in
othermodels,butitseemsthatwemayhaveafullclassof Eq.(31).We willalsoassumethe unificationmassscaleas
modelsalongthelinethatweareproposinghere.Because M9 =M5 ∼1016GeVandthehorizontalmassscaleequal
of the peculiar dynamics that we are assuming we need to Mh ∼ 1013 GeV. The several constants contained in
only a horizontal symmetry and a partial unification of Eq.(31)arebtc = 161π2236,bqcd = 167π2,γtc = 1253 andγqcd =
thestandardmodelandthevalueoftheirmassscaleswill 4. We remember again that we assumed α ∼ 1 , µ =
7 k 45 tc
not strongly modify our predictions (although the chosen 250GeV and µ = 250MeV. The fermion masses come
qcd
horizontal symmetry will). Of course, the breaking of the out as a function of the parameter c α . For
(tc,qcd) (tc,qcd)
unifiedand/orhorizontalsymmetrywillhappenatavery simplicity (as well as a reasonable choice) we will define
high energy scale and will not be discussed here. In par- cα=c α =c α =0.5.
tc tc qcd qcd
ticular,this symmetry breaking canbe evenpromotedby We display in Table 1 the fermionic mass spectrum
fundamental scalars which naturally can appear near the obtained in this model. Some of the values show a larger
Planck scale. disagreement in comparison to the experimental values,
8 A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario
mt 160.3 GeV mb 113 GeV mτ 131.2 GeV LHC). We can list the possible pseudo Goldstone bosons
mc 1.57 GeV ms 1.10 GeV mµ 1.30 GeV according to their different quantum numbers:
mu 29.6 Mev md 15.6 Mev me 5.5 Mev Colored pseudos: They carrycolordegreesoffreedom
and can be divided into the 3 or 8 color representations.
Table 1. Approximate values for quarks and leptons masses We can indicate them by
accordingtothechosenvaluesofcouplingsandstronglyinter-
acting mass scales. Πa ∼Q¯γ λaQ.
5
Chargedpseudos:Theseonescarryelectricchargeand
and others show a quite reasonable agreement if we con- we can take as one example the following current
sider all the approximations that we have performed and
Π+ ∼L¯γ Q,
the fact that we have a totally dynamical scheme. 5
ItisalsoimpressivethatB inEq.(31),neglectingloga-
whereQ(L)indicatethetechniquark(technilepton)fields.
rithimicterms,isroughlygivingbyB ∼14α m /π which
h t Neutral pseudos: They do not carry color or charge
isoforderof17GeV.Thisistheexpectedvalueaccording
and one example is
to the estimative of the previous section (see Eq.(30)). In
some way this is also expected in a mechanism where one Π0 ∼N¯γ N.
5
fermionicgenerationobtainamassat1-looplevelcoupling
to the next higher generation fermion (see, for instance,
Following closely Ref.[25] the standard procedure to
Ref.[23]). The values of the u and e masses can be easily
determine the SU(3) contribution to the mass (M ) of
qcd c
loweredwithasmallervalueofµ .Ofcourse,wearealso
qcd a colored pseudo Goldstone boson gives
assumingaveryparticularformforthemassmatrixbased
in one particular family symmetry. Better knowledge of 1
the symmetry behind the mass matrix, and a better un- M ∼ C2(R)αc(µ) 2 FΠ35.5MeV
c
α f
derstandingofthestronginteractiongroupalignmentwill (cid:18) el (cid:19) π
certainly improve the comparisombetween data and the- ∼170 C (R)GeV ∼O(300)GeV. (35)
2
ory. The high value for the masses obtained for some of
p
the second generation fermions also come out from the While the electromagneticcontributionto the mass of
overestimation of the b and τ masses. The mass splitting the charged pseudos Goldstone bosons is estimated to be
betweenthetandbquarks,whichisfarfromthedesirable [25]
result,isaproblemthathasnotbeensatisfactorilysolved
in most of the dynamical models of mass generation up FΠ
M ∼Q 35.5MeV ∼Q 47GeV ∼O(50GeV),
to now. It is possible that an extra symmetry, preventing em ps f ps
π
these fermions to obtain masses at the leading order as (36)
suggested by Raby [24] can be easily implemented in this in the equations above we assumed that the technipion
model. We will discuss these points again in the conclu- andpiondecayconstantsaregivenbyF ≈125GeV and
Π
sions. Finally considering that we do not have any flavor f ≈ 95MeV, Q is the electric charge of the pseudo-
π ps
changing neutral current problems [26], because the in- Goldstone boson,and C (R) is the quadratic Casimir op-
2
teraction between fermion and technifermions has been erator in the representation R of the pseudo-Goldstone
pushed to very high energies, and that we assume only bosonunderthe tc group.Thereis notmuchto changein
theexistenceofquiteexpectedsymmetries(agaugegroup these standard calculations, except that due to the par-
containing tc and the standard model and a horizontal ticular form of the technifermion self energy the tech-
symmetry) the model does quite well in comparison with nifermion will acquire large current masses, and subse-
many other models. quently the pseudos-Goldstone bosons formed with these
ones. We know that any chiral current Πf can be writ-
ten as a vacuum term m hψ¯ ψ i plus electroweak (color,
f f f
...) corrections [27], where m is the current mass of the
5 Pseudo-Goldstone boson masses f
fermion ψ participating in the composition of the cur-
f
rentΠf, neglectingthe electroweakcorrectionsandusing
Anotherproblemintechnicolormodelsistheproliferation
PCAC in the case of qcd we obtain the Dashen relation
ofpseudo-Goldstonebosons[2,3,25].Afterthechiralsym-
metry breaking of the strongly interacting sector a large m hq¯qi
m2 ≈ q , (37)
number of Goldstone bosons are formed, and only few of π f2
π
these degrees of freedom are absorbed by the weak inter-
actiongaugebosons.Theothersmayacquiresmallmasses wherehq¯qiisthequarkcondensate.Ofcoursethisrelation
resulting in light pseudo-Goldstone bosons that have not is valid for any chiral current and in particular for the
been observed experimentally. In our model these bosons technifermions we can write
obtain masses that are large enough to have escaped de-
tection at the present accelerator energies, but will show M2 ≈ MTfhT¯fTfi, (38)
up at the next generation of accelerators (for instance, Π F2
Π
A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario 9
whereM isthetechnifermioncurrentmass.Intheusual the model, all the others (unification of tc and the stan-
Tf
models (with the self-energy given by Eq.(3)) the tech- dard model and the existence of a horizontal symmetry)
nifermions are massless or acquire very tiny masses lead- are naturally expected in the current scenario of particle
ing to negligible values for M . In our model this is not physics. One of the characteristics of the model is that
Π
true. All technifermions acquire masses due to the self- the first fermionic generationbasically obtain masses due
interactionwith their owncondensates throughthe inter- to the interaction with the qcd condensate, whereas the
change of SU(9) bosons. third generation obtain masses due to its coupling with
ThereareseveralbosonsintheSU(9)(andalsointhe the tc condensate.The reasonfor this particularcoupling
SU(3) ) theory connecting to technifermions and gener- and for the alignment of the strong theory sectors gen-
H
ating a current mass as is shown in Fig.(9). erating intermediate masses is provided by the SU(3)
H
horizontalsymmetry.Ofcourse,ourmodel is notsuccess-
ful in predicting all the fermion masses although it has a
SU(9) seriesofadvantages.Itdoesnotneedthepresenceofmany
etc boson masses to generate the different fermionic mass
scales. The etc theory is replaced by an unified and hori-
zontal symmetries. It has no flavor changing neutral cur-
Tf α9 Tf Tf α9 Tf raernetms aonryunpwoiannttsedthlaigthsttipllsneueeddo-sGoomldeswtoonrekbinostohniss.lTinheeroef
Fig. 9. Diagram responsible for the technifermion mass gen- model.The breakingoftheSU(9)andhorizontalsymme-
eration. tries is not discussed, and just assumed to happen near
the Planck scale and possibly could be promoted by fun-
damental scalar bosons. The mass splitting in the third
A simple estimative, based on Eq.(4), of the contribu- generation could be produced with the introduction of a
tion of Fig.(9) to the technifermion masses gives new symmetry. For instance, if in the SU(9)breaking be-
sides the standard model interactions and the tc one we
MTfSU(9) ∼> O(80−130)GeV. (39) leaveanextra U(1), maybe we could havequantum num-
bers such that only the top quark would be allowed to
If we also include the contribution of the same diagram couple to the tc condensate at leading order. This possi-
where the exchangedbosonis a horizontalSU(3) boson bility shouldbe further studiedbecause it alsomay intro-
H
coupling technifermions of different generations, we must ducelargequantumcorrectionsinthemodel.Iftheunified
add to the above value the following one group (SU(9) in our case) is not broken by a dynamical
mechanism, i.e. we do not need that this group tumbles
MTfSU(3)H ∼> O(10−40)GeV. (40) dbyowonnetosmSaUll(e4r)tgcr⊗ouSpM(p,erthhaenpswSeUc(o2u)ld)rwephliacchebSeUco(m4)etcs
tc
Therefore, we expect that the technifermion current stronger at the scale µ ≈ 250 GeV. In this class of mod-
masses are at least of the order of M ≈ O(100)GeV. els we can choose different groups containing tc and the
Tf
Now, according to Eq.(38) and assuming hT¯T i ∼ F3 standardmodel,aswellasdifferenthorizontalsymmetries
f f Π
wehavethefollowingestimativeforthepseudo-Goldstone withdifferenttexturesforthemassmatrix.Thesewillcer-
boson masses tainly modify the values of the fermion masses that we
M > O(100)GeV. (41) have obtained. The alignment of the strongly interacting
Π
∼ sectors can be studied only with many approximations,
Note that in this calculation we have not considered
but it is quite possible that it generates more entries to
the qcd or electroweak corrections discussed previously.
the massmatrixthanonly the termB.Another greatad-
Therefore, even if the pseudo-Goldstones bosons do not
vantageofthe modelis thatitis quite independent ofthe
acquiremassesdue toqcdorelectroweakcorrectionsthey
veryhigh energyinteractions (like SU(9) or SU(3) ), al-
H
will at least have masses of order of 100 GeV because of
thoughthe horizontalsymmetry isfundamentalto obtain
the“current”technifermionmassesobtainedattheSU(9)
the desired mass matrices, and we believe that variations
(or SU(3) ) level.
H of this model can be formulated.
6 Conclusions
Wehavepresentedatechnicolortheorybasedonthegroup 7 Acknowledgments
structure SU(9)×SU(3) . The model is based on a par-
H
ticular ansatz for the tc and qcd self energy. We argue
thatouransatzforqcd,inviewofthemanyrecentresults
about its infrared behavior, is a plausible one, but even Thisresearchwassupportedbythe ConselhoNacionalde
if it is considered as an “ad-hoc” choice for the self en- DesenvolvimentoCient´ıficoe Tecnol´ogico(CNPq)(AAN)
ergy the main point is that it leads to a consistent model andbyFundaca˜odeAmparo`aPesquisadoEstadodeSa˜o
for fermion masses. This is the only new ingredient in Paulo (FAPESP) (AD).
10 A. Doff and A. A.Natale: The origin of the first and third generation fermion masses in a technicolor scenario
References
1. H. Fritzsch, Nucl. Phys. B 155, 189 (1979); H. Fritzsch
and Z. Xing, Prog. Part. Nucl. Phys.45, 1 (2000).
2. S. Weinberg, Phys. Rev. D 13, 974 (1976); S. Weinberg,
Phys. Rev. D 19 1277 (1979); L. Susskind, Phys. Rev. D
20, 2619 (1979).
3. C. T. Hill and E. H. Simmons, hep-ph/0203079.
4. S. Dimopoulos and L. Susskind, Nucl. Phys. B 155, 237
(1979);
E. Eichten and K. Lane, Phys. Lett. B 90, 125 (1980).
5. F. Maltoni, J. M. Niczyporuk and S. Willenbrock, Phys.
Rev.D65, 033004 (2002).
6. A.Doff andA. A.Natale, Phys.Lett.B 537, 275 (2002).
7. K.Lane, Phys. Rev. D10, 2605 (1974).
8. J.D.Carpenter,R.E. NortonandA.Soni,Phys.Lett. B
212, 63 (1988).
9. J.Carpenter,R.Norton,S.Siegemund-BrokaandA.Soni,
Phys.Rev.Lett. 65, 153 (1990).
10. H.D. Politzer, Nucl. Phys.B 117, 397 (1976).
11. B. Holdom, Phys. Rev. D24, 1441 (1981).
12. P.Langacker,Phys.Rev.Lett.34,1592(1975);A.A.Na-
tale,Nucl.Phys.B226,365(1983);L.-N.ChangandN.-P.
Chang,Phys.Rev.D 29,312(1984);Phys.Rev.Lett.54,
2407 (1985); N.-P.ChangandD.X.Li,Phys.Rev.D 30,
790(1984);K.Stam,Phys.Lett.B 152,238(1985);J.C.
Montero,A.A.Natale,V.PleitezandS.F.Novaes,Phys.
Lett.B161, 151 (1985).
13. R.AlkoferandL.vonSmekal,Phys.Rep.353,281(2001);
A.C.Aguilar,A.MiharaandA.A.Natale,Phys.Rev.D
65, 054011 (2002).
14. A.C. Aguilar, A.A. Natale and P. S.Rodrigues da Silva,
Phys.Rev.Lett. 90, 152001 (2003).
15. J.C.Montero,A.A.NataleandP.S.RodriguesdaSilva,
Prog.Theor.Phys.96,1209(1996);Phys.Lett.B406,130
(1997); A. A. Natale and P. S. Rodrigues da Silva, Phys.
Lett.B390, 378 (1997).
16. Ph. Boucaud et al., Phys. Lett. B493, 315 (2000); Phys.
Rev.D63,114003(2001);seealsoR.E.BrowneandJ.A.
Gracey, hep-th/0306200 and K. Kondo, hep-th/0306195,
and references therein.
17. Paul H. Frampton, Phys. Rev. Lett. 43, 1912 (1979);
Michael T. Vaughn,J. Phys. G 5, 1371 (1979).
18. H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32, 438
(1974).
19. A.A.Natale,Z.Phys.C 21,273 (1984); A.A.Natale, Z.
Phys.C 30, 427 (1986).
20. J. M. Cornwall, Phys. Rev. D 10, 500 (1974); S. Raby,
S. Dimopoulos and L. Susskind, Nucl. Phys. B 169, 373
(1980).
21. G.BGelmini,J.M.G´erard,T.YanagidaandG.Zoupanos,
Phys. Lett. B 135, 103 (1984); F. Wilczek and A. Zee,
Phys.Rev.Lett42,421(1979);Z.Berezhiani,Phys.Lett.
B 129, 99 (1983); Z. Berezhiani, Phys. Lett. B 150, 177
(1985);Z.Berezhiani,J.Chkareuli,Sov.J.Nucl.Phys.37,
618(1983) inenglishtranslation -inrussian Yad.Fiz.37,
1043 (1983).
22. H.Georgi, Nucl.Phys. B156, 126 (1979).
23. S.M. Barr and A. Zee, Phys.Rev. D 17, 1854 (1978).
24. S.Raby,Nucl.Phys. B 187, 446 (1981).
25. S.Dimopoulos, Nucl. Phys. B 168, 69 (1980).
26. S.Dimopoulos,S.RabyandP.Sikivie,Nucl.Phys.B176,
449(1980);S.DimopoulosandJ.Ellis,Nucl.Phys.B182,
505 (1981).
27. B.Machet andA.A.Natale,Ann.Phys.160,114(1985).