How confinement may affect technicolor?
aa r X i v : . [ h e p - ph ] O c t How confinement may affect technicolor?
A. Doff , F. A. Machado and A. A. Natale Universidade Tecnol´ogica Federal do Paran´a - UTFPR - COMATVia do Conhecimento Km 01, 85503-390, Pato Branco - PR, Brazil Instituto de F´ısica Te´orica, UNESP - Universidade Estadual Paulista,Rua Dr. Bento T. Ferraz, 271, Bloco II, 01140-070, S˜ao Paulo - SP, BrazilE-mail: [email protected]
Abstract.
Confinement has been introduced into the quark gap equation, asproposed by Cornwall, as a possible solution to the problem of chiral symmetrybreaking in QCD with dynamically massive gluons. We argue that the samemechanism can be applied for technicolor with dynamically massive technigluons.Within this approach both theories develop a hard self-energy dynamics, resulting froman effective four-fermion interaction, which does not lead to the known technicolorphenomenological problems. We outline a quite general type of technicolor modelwithin this proposal that may naturally explain the masses of different fermiongenerations.PACS numbers: 12.38, 12.60 ow confinement may affect technicolor?
1. Introduction
The nature of the electroweak symmetry breaking is one of the most important problemsin particle physics, and there are many questions that may be answered in the near futureby the LHC experiments, such as the following: Is the Higgs boson, if it exists at all,elementary or composite, and what are the symmetries behind the Higgs mechanism?The possibility that the Higgs boson is a composite state instead of an elementaryone is more akin to the phenomenon of spontaneous symmetry breaking that originatedfrom the Ginzburg-Landau Lagrangian, which can be derived from the microscopic BCStheory of superconductivity describing the electron-hole interaction (or the compositestate in our case). This dynamical origin of the spontaneous symmetry breaking has beendiscussed with the use of many models, the technicolor (TC) being the most popularone [1].Ordinary fermion masses ( m f ) result from the interaction of these fermions withtechnifermions through the exchange of a extended technicolor boson (ETC) and dependcrucially on the technifermion self-energy. In the early models this self-energy wasconsidered to be given by the standard operator product expansion (OPE) result [2]:Σ T C ( p ) ∝ D ¯ T f T f E /p , where D ¯ T f T f E is the TC condensate and of order of a few hundredGeV. With this self-energy the fermion masses are given by m f ≈ D ¯ T f T f E /M etc . Inorder to obtain the fermion masses of the second and third generations the ETC gaugeboson masses had to be very light. Since these bosons connect different fermionicgenerations and must be light, they may produce flavor changing neutral currents(FCNC) incompatible with the experimental data. A possible way out of this dilemmawas proposed by Holdom [3], remembering that the self-energy behaves asΣ T C ( p ) ≈ D ¯ T f T f E µ p − p µ ! γ m / , (1)where µ is the characteristic TC scale and γ m the anomalous dimension associated tothe condensate operator.If γ m ≥ γ m ) are quite desirable for technicolor phenomenology[1]. Lattice simulations were used to study many models that could have a large γ m ,regrettably some of these models do have an appreciable anomalous dimension but notlarge enough to solve the phenomenological problems of TC theories [5], indicating howdifficult is to build a realistic TC model.It is quite possible that the TC problems are related to the poorly known self-energyexpression, or the way chiral symmetry breaking (CSB) is realized in non-Abelian gaugetheories. Actually, the only known laboratory to test the CSB mechanism is QCD, andeven in this case, considering several recent results about dynamical mass generation inQCD that we shall discuss throughout the paper, imply that the dynamical quark mass ow confinement may affect technicolor? p ),may lead to models where both theories contribute to the ordinary fermion masses, anddo not lead to FCNC problems. Section V contains our conclusions.
2. Technicolor with dynamically massive technigluons
Many years ago Cornwall proposed that a dynamical gluon mass could be generatedin QCD [9]. Only recently this possibility was confirmed by lattice simulations [10]and checked rigorously through Schwinger-Dyson equations (SDE) [11]. It seems thatthis is a general property of non-Abelian gauge theories [12]. There is no reason tobelieve that the same mechanism does not happen in TC theories. The only possibilityfor technigluons not acquiring a dynamical mass that we can think of is the case ofa conformal or non-asymptotically free TC model, where the effect of technifermionloops in the Schwinger-Dyson equations cancel the gauge loop effects responsible for thedynamical technigluon mass.We assume a TC theory based, for instance, on a SU ( N ) gauge group, with afermion content such that the theory is asymptotically free and is not almost conformal(or not near a perturbative fixed point). We also assume that in this theory the ow confinement may affect technicolor? T C ≡ M ( p ) = C (2 π ) Z d k ¯ g tc ( p − k )3 M ( k )[( p − k ) + m tg ( p − k )][ k + M ( k )] , (2)where we consider the Landau gauge, C is the Casimir operator for the fermionicrepresentation, m tg ( k ) is the dynamical technigluon mass and ¯ g tc the effective TCcoupling constant. First, we must say that, as far as we know, the CSB mechanism inTC models has not been studied up to now in the presence of dynamical technigluonmass generation. Secondly, to understand what may happen in a TC theory we willrecall some QCD results.When dynamical gauge boson masses are generated in any asymptotically freenon-Abelian gauge theory we also expect that the coupling constant develops a non-perturbative infrared fixed point [13]. In QCD it was predicted many years ago that thecoupling constant would behave as [9]¯ g QCD ( k ) = 1 b ln[( k + 4 m g ) / Λ QCD ] , (3)where b is the first β function coefficient, and m g ≡ m g ( k = 0) ≈ QCD ≈ − α s (0) ≡ ¯ g (0) / π ) is of order 0 .
5. Thisnumber may be considered surprisingly small but there are several phenomenologicalcalculations indicating that this value should not be larger than 1; see, for instance, acompilation of infrared values of the QCD coupling constant shown in Ref.[15]. Now,the gluon propagator in the fermionic SDE kernel, no longer behaves as 1 /k but as1 / ( k + m g ) in the infrared, what diminishes the strength of the interaction, and we alsoadd to this fact the damping caused by the small value of the infrared coupling constant(¯ g (0)). The consequence is that we do not generate dynamical quark masses ( M ( k ))(or quark condensates) compatible with the experimental data in QCD for quarks inthe fundamental representation [16]! QCD could only generate CSB if quarks were inhigher dimensional representations, i.e. with higher values for the Casimir operator inorder to compensate the infrared damping discussed above [17].TC theories will also present dynamical technigluon mass generation, and for thesame reasons that we discussed in the QCD case, i. e. a small infrared TC couplingconstant and the damping caused by the 1 /m tg infrared value of the technigluonpropagator, we do not expect that they will develop enough chiral symmetry breakingto form the TC condensates. In this work we will follow the idea of Ref.[6, 7] thatconfinement is necessary and sufficient to promote CSB and develop the expected(techni)quark condensates. Actually, our next section will start discussing evidencesfor a relation between CSB and confinement. ow confinement may affect technicolor?
3. Chiral symmetry breaking as a consequence of confinement
The majority of studies about CSB in gauge theories, no matter if in QCD or TC, reliedon the one-gauge boson exchange. If we deal with dynamically massive gauge bosons, asdiscussed in the previous section, CSB will not be achieved at least if we have fermionsin the fundamental representation. We will than consider the case where confinement isnecessary for CSB, and in order to emphasize this possibility we will review some QCDaspects that point out in this direction. These arguments are going to be used in orderto justify that confinement is also necessary for the TC chiral symmetry breaking.In Ref.[9] it was proposed the following scenario for QCD: a) Gluons acquire adynamical mass, b) The theory with dynamically massive gluons generate vortices, andc) These center vortices generate confinement. Lattice simulations are showing evidencesfor a relation between CSB and confinement, where center vortices play a fundamentalrole. In the SU (2) case the artificial center vortices removal also implies recovery ofthe chiral symmetry [18, 19, 20]! We also have another lattice result indicating theimportance of the deep infrared region for CSB in QCD [21]. In this simulation the quarkcondensate h ¯ qq i is drastically reduced ( ≈ V F ( r ) = K F r − α s r , (4)where the first (confining) term is linear with the distance and proportional to thestring tension K F . The second term, that is of order α s , the strong coupling constant,describes the one gluon exchange contribution. The classical potential between staticquark charges is related to the Fourier transform of the time-time component of the fullgluon propagator in the following way V ( r ) = − C π Z d q α s ( q )∆ ( q ) exp ı q . r , (5)where the bold terms, q and r , are 3-vectors. As noticed in Ref.[23] the linear confiningterm of the potential ( K F r ) cannot be obtained from the gluon propagator determinedin the lattice or from the gluonic SDE , i.e. we could roughly say that the dynamicallymassive gluon propagator also does not lead to quark confinement as it may not leadto CSB for fermions in the fundamental representation. The existence of a linearconfining potential felt by quarks is supported by lattice simulations [24], and is astrong justificative for a confining effective propagator. This linear confining part ofthe potential must also show a cutoff at some distance. For n f = 2 quarks in the ow confinement may affect technicolor? r c ≈ . f m , (6)which corresponds to a critical mass (or momentum), compatible with the m valuenecessary to generate the expected amount of CSB in the gap equation. This distancemay change with the fermionic representation (because the string tension changes withthe fermionic representation [24]), but there shall always be a critical value associatedto the string breaking or to the force screening.All the above facts were collected in order to show that a theory with dynamicallymassive gauge bosons, as expected for any asymptotically free non-Abelian gauge theory,may not have enough strength to generate CSB with fermions in the fundamentalrepresentation. Of course, for large fermionic representations, with a large value for theCasimir operator [ C in Eq.(2)] this may not be true [17]. Confinement and CSB seemto be intimately connected. The Fourier transform (Eq.(5)) of a dynamically massivegauge boson propagator does not lead to a confining potential ( ∝ K F r ), although itcan explain its short distance behavior ( ∝ α s /r ) [23]. In some way confinement mustalso be limited to some scale as described by Eq.(6). Therefore, to model CSB inQCD or TC, as we intend to do in a Schwinger-Dyson equation approach, we mustintroduce confinement explicitly and also consider the one-gauge dynamically massiveboson exchange. The propagators that we shall use in the fermionic Schwinger-Dysonequation, when plugged into Eq.(5), have to reproduce at some extent the behavior ofEq.(4) and the confining contribution has to reflect the limit shown in Eq.(6). Theseideas that were introduced in Ref.[6] and applied phenomenologically in Ref.[8] in theQCD case, are going to be extended to TC theories in this work.Cornwall introduced a confinement effect explicitly into the gap equation throughthe following effective propagator , which is not at all related to the propagation of astandard quantum field [6]: D µνeff ( k ) ≡ δ µν D eff ( k ); D eff ( k ) = 8 πK F ( k + m ) , (7)where K F is the string tension. In the m → πK F δ µν /k that yields approximately an area law for the Wilsonloop. This propagator has an Abelian gauge invariance that appears in the quark actionobtained by integrating over quark world lines implying an area-law action [6]. We mustnecessarily have a finite m = 0 value due to entropic reasons as demonstrated in Ref.[6],and its value is related to the dynamical quark mass ( m ≈ M (0)), as required bygauge invariance, originating a negative term − K F /m in the static potential in orderto generate the Goldstone bosons associated to the chiral symmetry breaking.We can now turn to TC and write down what we may expect for the gap equation.As happens in the QCD case, the technifermion SDE can be modeled by the sum of a partcontaining the confining effective propagator plus another contribution with a massiveone-techni-gluon exchange [6, 8], which, in the Abelian techni-gluon approximation, is ow confinement may affect technicolor? M ( p ) = 1(2 π ) Z d k D tceff ( p − k ) 4 M ( k ) k + M ( k ) + C (2 π ) Z d k ¯ g tc ( p − k )3 M ( k )[( p − k ) + m tg ( p − k )][ k + M ( k )] , (8)where M ( p ) = M c ( p ) + M tg ( p ) is the dynamical techni-quark mass generated bythe effective confining and the dressed techni-gluon contributions. This last equationis the basic one that we shall explore in this work. Note that the effective propagatorin the first integral of Eq.(8) leads to a confining potential ( ∝ K tc r ) and the massivetechni-gluon exchange to the short distance contribution ( ∝ α tc /r ) of the static TCpotential. We have just replaced the QCD quantities ( K F , ¯ g QCD and m g ) by theequivalent TC quantities ( K tc , ¯ g tc and m tg ). In the following we also assume thatthe string tension in the confining propagator has also to be changed according to thefermionic representation [8], but much of our discussion will be related to fermions inthe fundamental representation.If the TC theory contains fermions in the fundamental representation it can beshown that just the first integral on the right hand side of Eq.(8), i.e. the gap equationwithout the massive technigluon exchange, is enough to generate the desirable amount ofchiral symmetry breaking (with appropriate values K tc and m ≈ M (0)). The asymptoticbehavior of the self-energy in this case is M ( p ) | p →∞ ∝ /p , which is a very soft behavior. The one-technigluon exchange enters only to modify theasymptotic behavior of the gap equation as happens in the QCD case [8].The full gap equation can be transformed into a differential equation and it ispossible to verify that the solution is a linear combination of two independent solutionsof the form f ( x ) = b f reg ( x ) + b f irreg ( x ), where b and b are determined by theboundary conditions. The asymptotic behavior is dominated by the one-technigluonexchange contribution, whereas the effects of the confining propagator enter only throughthe boundary conditions [8]. In Ref.[8] we verified that the irregular solution dominateswhen a cutoff Λ ≈ m is introduced. In a SU ( N ) technicolor theory this ultravioletbehavior would be equal to [8] M ( p ) | p →∞ ∝ M (ln p /M ) − d , (9)where d = 9 C / (11 N − n f ) for n f flavors. This solution minimizes the vacuum energyand has a vacuum expectation value proportional to 1 /g [26]. All the above commentis just to recall how the boundary conditions may affect the behavior of the self-energy.We shall not consider Λ ≈ m in the sequence, but we will argue that the integrals inEq.(8) should be performed in different momentum regions.We now suppose that the confining propagator is limited to a specific momentuminterval. The confining propagator that we are discussing here is not the one ofa fundamental field, therefore we argue that it must specify a certain region where ow confinement may affect technicolor? r c ), and if the phenomenological classical potential between static quark charges is givenby the Fourier transform of the time-time component of this confining propagator, theconfining propagator will not reflect this breaking unless we cut the momentum upto a maximum value where the confinement region exists, or we can understand themomentum flowing in the confining propagator as the energy that may flow betweenconfined quarks. If this hypothesis is correct it is natural to have the following four-fermion approximation [8]: M ( p ) ≈ M f ( p ) =2 π K R m tc Z d k M f ( k ) θ ( m tc − k ) k + M f ( k ) + C (2 π ) × Z Λ d k ¯ g tc ( p − k )3 M f ( k )[( p − k ) + m tg ( p − k )][ k + M f ( k )] . (10)In Ref.[8] we verified that the critical behavior of Eq.(10) and the one of Eq.(8) arebasically the same in what concerns the critical values of the “constants” K F,tc and m ,with the massive one-gauge boson exchange barely affecting the symmetry breaking.The value of the chiral parameters, like the dynamical fermion mass and condensates,are not so much different, implying that the approximation is quite reasonable. This isa consequence of the very strong confining force and the fact that most of the symmetrybreaking is dominated by the physics at very low momenta.The solution of Eq.(10) has a slow decrease with the momentum [8] and is typicalof the gauged Nambu-Jona-Lasinio (NJL) type of models [27]. The dressed one-gluonexchange has not enough strength to generate such type of four-fermion interaction [8],which occurs only due to the large ratio between the string tension and the factor m in the confining potential. Actually, we have a simple reasoning to explain why theself-energy solution is the one corresponding to what is called irregular behavior (orNJL type of solution). Eq.(8) is a particular case of the following equation: M ( p ) ≈ β Z m d kk G ( p, k ) M ( k ) k + M ( k )+ α Z Λ d k ¯ g ( p − k ) M ( k )[( p − k ) ][ k + M ( k )] , (11)where Λ is an ultraviolet cutoff, G ( p, k ) is an integrable function in the interval [0 , m ],where the interval is understood for p and k , and we have chosen arbitrarily m as themomentum limit to where confinement is propagated. M ( k ) is a well behaved functionin the infrared with M (0) ≈ m . We can verify that the ultraviolet boundary conditionbehavior ( p → Λ ) of Eq.(11) is given by M (Λ ) ∝ β Z m d kk M ( k ) k + M ( k ) , (12)which is a constant and not different from a bare mass in the gap equation, leadingto a hard behavior for the dynamical mass. Another argument in favor of limiting ow confinement may affect technicolor? ⊗ propagator”, and we know from Eq.(3) that the 1-(techni)gluon exchangehas not enough strength to generate such effective coupling. On the other hand theconfining effective propagator, with the usual values for the string tension, is strongenough to generate the effective gap equation (10)! Apart from the (techni)gluonmass effect appearing in the 1-(techni)gluon contribution, Eq.(10) has been extensivelystudied in Ref. [27], and it does lead to a self-energy solution that decreases slowlywith the momentum, although the origin and the cutoffs are totally different. Thiscan also be verified comparing the -fermion coupling constant ( λ ) of Ref.[27] with our“effective coupling” K R /m tc , related to the representation R of the TC group. Thefermion condensate in a given representation R obtained from Eq.(10), as shown inRef.[8], has the same form found by Takeuchi (Eq.(6) of Ref.[27]) h ¯ qq i R ( m tc ) ≈ − N R π m tc K R M R ( m tc ) , (13)corresponding to a broken-symmetry phase characterized by K R /m tc > λ > m ≈ M (0) into the confining propagator is necessary for entropic reasons,otherwise it would be extremely difficult to generate the Goldstone bosons associatedto the chiral symmetry breaking [6]. Within the approximations discussed here theseresults are valid for any non-Abelian gauge theory in the confining phase. ow confinement may affect technicolor?
4. Technicolor models with dynamically massive technigluons andconfinement effects
It is possible to outline a class of TC models that can be built based on the irregularsolution for the self-energy, with the main advantage that QCD and TC have the sametype of self-energy and participate equally in the mass generation mechanism for theknown fermions [29]. Considering that QCD and TC have the so called “irregular”self-energy [8, 27], which will be parameterized as [29, 30]Σ( p ) ∼ µ h bg ln (cid:16) p /µ (cid:17)i − γ , (14)where µ is the characteristic scale of mass generation (QCD or TC), γ = 3 c/ π b and c = [ C ( R ) + C ( R ) − C ( R )] . C ( R i ) are the Casimir operators for fermions in therepresentations R and R that condense in the representation R , when we computethe ordinary fermion mass ( m f ) we obtain [29]: m f ≈ g etc µ T C ( QCD ) × h b T C ( QCD ) g T C ( QCD ) ln (cid:16) M etc /µ T C ( QCD ) (cid:17)i − γ . (15)In the above equation µ T C ( QCD ) is the characteristic TC(QCD) chiral symmetry breakingscale, g etc is the ETC coupling constant, b T C ( QCD ) the first β function coefficient, g T C ( QCD ) is the TC(QCD) coupling constant, M etc the ETC boson mass, and we alsoneglected some constants. Three points are very important to be noticed: a) The fermionmasses depend quite weakly on the ETC boson mass, which may have very large valuesnot leading to FCNC problems, b) Small fermion masses are generated when the chiralsymmetry breaking is due to QCD. This is quite different from the usual models whereit is assumed that QCD has a very soft solution for the self-energy, c) The largest massthat we can generate, if µ T C is of the order of a few hundred GeV, is roughly of orderof g etc µ T C and not too much different from the top quark mass [30].According to the previous paragraph we can say that we may generate two differentmass values for the ordinary fermions: m lightf ≈ g etc µ QCD , m heavyf ≈ g etc µ T C , (16)where we neglected the (small) contribution of the term between brackets in Eq.(15). Ifwe compute the condensates we also can verify that we have a QCD and TC condensateswith scales separated by an O (10 ). The light masses are of order of the first generationfermion masses, while the heavier are of the order of the third generation masses [29].But how is it possible to prevent light fermions to acquire heavy masses? This can besolved with the help of a family, or horizontal, symmetry.In the sequence we sketch a scheme quite similar to the one proposed by Berezhianiand Gelmini et al. [31] where their vevs of fundamental scalars are substituted byQCD and TC condensates [29]. Let us suppose that we have a horizontal symmetrybased on the SU (3) H group and the TC theory has technifermions in the fundamentalrepresentation of SU (4) T C . The technifermions form a quartet under SU (4) T C and thequarks are triplets of QCD. The technicolor and color condensates will be formed at the ow confinement may affect technicolor? µ T C and µ QCD in the most attractive channel (mac) of the products ¯4 ⊗ and ¯3 ⊗ of each strongly interacting theory. We assign the horizontal quantum numbersto technifermions and quarks such that these same products can be decomposed in thefollowing representations of SU (3) H : in the case of the TC condensate, and in thecase of the QCD condensate. For this it is enough that the standard left-handed (right-handed) fermions transform as triplets (antitriplets) under SU (3) H , assuming that theTC and QCD condensates are formed in the and in the of the SU (3) H group. This isconsistent with the mac hypothesis although a complete analysis of this problem is outof the scope of this work. The above choice for the condensation channels is crucial forour model, because the TC condensate in the representation (of SU (3) H ) will interactonly with the third fermionic generation while the (the QCD condensate) will interactonly with the first generation. In this way we can generate the coefficients C and A respectively of a Fritzsch type matrix [32], because when we add these condensates(vevs) and write them as a 3 × at leading order ) with M f = A A ∗ C . (17)The points that must still be discussed are how we generate the intermediate massesand why the contribution of the term between brackets in Eq.(15) is indeed small andcan be neglected.In the scenario that we shall consider the ETC group can connect all fermionsand contain the TC and QCD interactions. Actually the ETC role can be played by agrand unified theory (GUT), which has exactly these characteristics. This is possiblebecause the fermion mass barely depend on the ETC or GUT gauge boson masses, ascan be verified from Eq.(15). These ETC or GUT bosons can intermediate neutral flavorchanging interactions, however they can be very heavy in order to be consistent with allexperimental constraints on FCNC interactions [29]. We can build a TC model basedon a GUT such that G gut ⊃ G SM ⊗ SU ( N ) T C ⊗ G H , (18)where G SM is the Standard Model group, SU ( N ) T C is the TC group and G H correspondsto a horizontal symmetry, which is not necessarily a local one, but with a characteristicscale of the order of the GUT scale, and, for simplicity, couplings are assumed to beof the same order (i.e. g H ≈ g gut ). TC should condensate at TeV scale. All groupsare embedded into the GUT, therefore we may have all kind of neutral flavor changinginteractions but at the GUT scale, since this theory will play the role of the ETC theory[1]. In the example that we discussed before, where G H ≡ SU (3) H with technifermionscondensing in the ¯6 and quarks condensing in the representations of the horizontal ow confinement may affect technicolor? M f = A A ∗ B B ∗ C , (19)where A ∝ g gut µ QCD ≈ O (MeV) and C ∝ g gut µ T C ≈ O (GeV). The B term has anintermediate value naturally generated by the effective potential of the composite ¯6 and Higgs system as shown in [29]. Notice that we can only obtain a mass as heavy as thetop quark one in TC models with the use of Eq.(14) [30].To show that a mass matrix like the one of Eq.(19) is a feasible one, we can usethe technique of effective potential for composite operators as discussed in Ref.[33],and verify that QCD and TC lead to a two composite Higgs boson system indicatedrespectively by η and φ , with, due to the horizontal symmetry, the following vacuumexpectation values (vevs): h η i ≈ v η , h φ i ≈ v φ , (20)where the first vev will be of the order of 250 MeV and the second one of order 250GeV. The intermediate term in Eq.(19) will be originated by mixed terms in the effectivepotential of our composite system [29]. These terms will come out naturally from one-loop standard model interactions connecting η and φ (or quarks and techniquarks scalarcomposites), being of the following type V ( η, φ ) = ǫη † ηφ † φ + δη † φηφ † + ... (21)The details of how this effective potential contribution originates in such type of modelswere worked out in Ref.[29].Let us summarize why we consider this a quite general type of model. First, due tothe fact that confinement is responsible for a dynamical mass typical of a NJL gaugedmodel, or of the irregular type, we end up with two scales of ordinary fermion masses:QCD and TC. Secondly, due to the form of the self-energy the fermion masses barelydepend on the ETC gauge boson masses, and these can be quite heavy and do notgenerate FCNC problems. Finally, to generate reasonable fermion mass matrix weonly need a horizontal or family symmetry. There are possibly many theories basedon different groups that may fit into this scheme, and we do not need to appeal totechnifermions that belong to higher TC representations.
5. Conclusions
We initiated our work calling attention to the fact that in asymptotically free non-Abelian gauge theories the gauge bosons may acquire naturally dynamical masses. Thisfact has been already checked for QCD through lattice simulations and Schwinger-Dysonequations. We expect that the same phenomenon occurs in TC theories if the theory has ow confinement may affect technicolor? ow confinement may affect technicolor?
Acknowledgments
This research was partially supported by the Conselho Nacional de DesenvolvimentoCient´ıfico e Tecnol´ogico (CNPq) (AD and AAN) and by Funda¸c˜ao de Amparo a Pesquisado Estado de S˜ao Paulo (FAPESP) (FAM).
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