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Electroweak Symmetry Breaking via a

arXiv:hep-ph/9203218v2 21 Mar 199216 July 2021LBL-32089Electroweak Symmetry Breaking via aTechnicolor Nambu–Jona-Lasinio ModelMarkus A. LutyTheoretical Physics GroupLawrence Berkeley Laboratory1 Cyclotron RoadBerkeley, California 94720AbstractWe consider a theory of gauge fields and fermions which we argue gives rise to dynamicssimilar to that of the Nambu–Jona-Lasinio (NJL) model when a gauge coupling constantis appropriately fine-tuned. We discuss the application of this model to dynamical elec-troweak symmetry breaking by a top-quark condensate.

In this model, custodial symmetryis violated solely by perturbatively weak interactions, and the top–bottom mass splitting isdue to the enhanced sensitivity to custodial symmetry violation near the critical point. Wealso consider models in which electroweak symmetry is broken by new strongly-interactingfermions with NJL-like dynamics.

We argue that these models require additional fine-tuning in order to keep corrections to the electroweak ρ-parameter acceptably small.

1. IntroductionIn the last few years, there has been a revived interest in the Nambu–Jona-Lasinio(NJL) model [1] as an effective description of electroweak symmetry breaking via a top-quark condensate [2][3].

Numerous variations on this basic theme have also been explored[4]. More recently, several groups have constructed renormalizable models which are sup-posed to give rise to top-quark condensates and NJL-like dynamics.

[5]. In this paper, wewill consider a model of asymptotically free gauge fields and fermions which we argue hasdynamics similar to that of the NJL model.

In this model, all symmetry breaking is drivenby strongly-coupled SU(N) gauge groups with fermions in the fundamental representa-tion. We can therefore use our understanding of chiral symmetry breaking in QCD-liketheories to analyze the model.The only new dynamical assumption is that the chiralsymmetry breaking phase transition is second order.

We will consider models in whichthe electroweak symmetry is broken by top-quark condensates, as well as models in whichelectroweak symmetry is broken by condensates of new fermions.The plan of this paper is as follows: In section 2, we will review some basic resultsconcerning the NJL model, and discuss the sense in which the NJL model can be viewed asa low-energy effective lagrangian. In section 3, we will present a model of asymptotically-free gauge fields and fermions and give qualitative arguments that this model gives rise todynamics similar to the NJL model.

In section 4, we flesh out the dynamical picture of theprevious section by analyzing the model using the Schwinger–Dyson equations in ladderapproximation. In section 5, we apply the model to electroweak symmetry breaking, andsection 6 contains our conclusions.2.

The NJL modelWe begin by treating the NJL model as a toy model with no reference to electroweakphysics.The NJL model we will consider has F “flavors” and N “technicolors.” Thelagrangian isLNJL = ψi/∂ψ + GN (ψLjaψRka)(ψRkbψLjb),(1)where j, k = 1, . .

., F are flavor indices and a, b = 1, . .

., N are technicolor indices. Thistheory has a global chiral symmetrySU(N)T C × U(F)ψL × U(F)ψR,(2)in an obvious notation.

The theory is nonrenormalizable, and must therefore be interpretedas an effective theory equipped with an energy cutoffΛ. In order to describe physics at

scales above Λ, we must construct a more fundamental theory which reduces to the NJLmodel at low energies.In the limit N →∞(with F held finite), this model can be solved exactly.Thequalitative features are well known: if G is less than a critical value Gcrit, then the chiralsymmetry of eq. (2) is unbroken, and the effective theory below the scale Λ simply describesmassless fermions interacting through the contact interaction of the form eq.

(1). However,if G > Gcrit, then the chiral symmetry is partially broken: the condensate⟨ψLjaψRja⟩̸= 0(3)results in the symmetry-breaking patternU(F)ψL × U(F)ψR −→U(F)ψL+ψR,(4)where U(F)ψL+ψR is the vector-like subgroup of U(F)ψL × U(F)ψR.If G is tuned to be very close to Gcrit,G = Gcrit1 + O(Gm2)withm2 ≪G−1,(5)then the effective theory near the scale m is precisely a U(F) × U(F) linear sigma modelcoupled to the fermions ψ.

The effective lagrangian at the scale m isLeff= ψi/∂ψ + yψLjΦjkψRk + h.c.+ tr ∂µΦ†∂µΦ −µ2Φ tr Φ†Φ −λ1tr Φ†Φ2 −λ2 tr Φ†ΦΦ†Φ + · · · ,(6)where the ellipses indicate terms containing higher-dimension operators. If G < Gcrit, thenµ2Φ > 0 and the theory is in the unbroken phase.

The fermions are then massless, andthere are 2F 2 physical scalars with equal masses of order m. If G > Gcrit, then µ2Φ < 0and the theory is in the broken phase. The fermions then have masses of order m, andthere are F 2 physical scalars with masses of order m and F 2 massless Nambu–Goldstonebosons.

Because the underlying theory eq. (1) has only one coupling constant G, therewill be non-trivial relations among the parameters of the effective low-energy lagrangianeq.

(6).An important feature of the NJL model in the large-N limit is that the low-energydynamics is continuous as a function of G near the critical point. An analogous assumptionin the context of our renormalizable model will play an important role in what follows.Before presenting the renormalizable model, we discuss the sense in which the NJLmodel can arise as an effective theory from a more fundamental theory.

When we write

an effective field theory which matches onto a more fundamental theory at a scale Λ, wemust include all possible interaction terms among the low-energy fields consistent with thesymmetries of the underlying theory. The coefficients of these operators are determinedby matching the predictions of the low-energy theory with that of the underlying theory.There will be an infinite number of operators in the low-energy effective lagrangian, butonly a few of these, the so-called “relevant” operators, will be important for describingthe physics at energies µ ≪Λ.

The effects of the remaining “irrelevant” operators at lowenergies will be suppressed by powers of µ/Λ. This is the reason for the power of theeffective-lagrangian approach.For weakly-coupled theories, dimensional analysis is sufficient to classify operators asrelevant or irrelevant: operators with engineering dimension 4 or less are relevant, and allothers are irrelevant.

As the NJL model illustrates, the classification becomes non-trivialwhen couplings become strong. We therefore ask if there are any other operators whichare relevant if they are added to the NJL lagrangian eq.

(1) when G is near its criticalvalue.A clue to the answer to this question can be obtained from the fact that the low-energylimit of the fine-tuned NJL model is a linear sigma model. This equivalence can be madeexplicit by rewriting the NJL interaction in terms of an auxilliary scalar field Φ:L′NJL = ψi/∂ψ −G−1Φ∗jkΦjk +ψLjaΦjkψRka + h.c..(7)This is the effective lagrangian at the scale Λ, in which Φ has no kinetic or self-interactionterms.

Φ can be explicitly integrated out to recover the NJL lagrangian eq. (1).

At scalesbelow Λ, Φ acquires both kinetic and self-interaction terms as in eq. (6), and can serve asan interpolating field for the physical scalar states.In this formulation of the NJL model, it is clear that all of the the dimension-fouroperators appearing in eq.

(6) are relevant. If these are added to the effective lagrangianat the scale Λ and the field Φ is integrated out, the result will be a non-local effectiveaction containing only the ψ fields.

If we expand this action in powers of derivatives of ψfields, it seems clear that there should be a small number of higher-dimension operatorsthat become relevant in the fine-tuned limit, and whose effects at low energies preciselyreproduce the effects of the operators of eq. (6).One such operator was considered in ref.

[6]. A more complete analysis was performedin ref.

[7], which explicitly identified operators that, when added to the effective lagrangianat the scale Λ, reproduced the entire parameter space of the linear sigma model. It maytherefore seem that the predictions of refs.

[2][3][4] based on the NJL model are vacuous,but this is not so. The reason is that as long as Λ is many orders of magnitude above

the weak scale, and the linear sigma model has a large Yukawa coupling at scales justbelow Λ, the logarithmic evolution of the parameters of the effective lagrangian drives theparameters of the theory to an approximate infrared fixed point at the weak scale [8]. (This is true for all models which reduce to the standard model with one or two Higgsdoublets in the low-energy limit.) The role of the NJL-like dynamics in these models istherefore to explain why the top-quark coupling is singled out to be strong enough to liein the basin of attraction of the approximate infrared fixed point.The situation is clearly quite different if the compositeness scale is taken to be nearthe weak scale.

In this case, the couplings do not run over a sufficient range to be af-fected by the approximate infrared fixed point, and all the relevant operators are ´a prioriequally important. The construction of explicit renormalizable models is clearly especiallyinteresting in this case.If we consider some fundamental theory which gives rise to the NJL interaction eq.

(1),it is clear that all of the relevant operators discussed above will be generated. Therefore,in attempting to construct a renormalizable model which captures the physics of the NJLmodel, strictly speaking we can only expect to find a model with a fermion bilinear orderparameter whose low-energy limit is a linear sigma model.

Nonetheless, in the model weconstruct, we will find that the dynamics is very similar to the NJL model eq. (1).3.

A Renormalizable ModelThe NJL model described in the previous section bears a superficial resemblace toa model in which the four-fermion coupling in eq. (1) is replaced by SU(N)T C gaugeinteractions.

In the SU(N)T C gauge model, the gauge coupling becomes strong at somescale ΛT C, and a condensate of the form eq. (3) forms, giving rise to F 2−1 NGB’s.

(Recallthat the NJL model of eq. NJLL has F 2 NGB’s.) We can try to press the analogy furtherby using Fierz indentities to rewrite the NJL interaction as(ψLjaψRka)(ψRkbψLjb) = (ψLjγµTAψLj)(ψRkγµTAψRk) + O(1/N),(8)where the TA are SU(N)T C generators normalized so that tr TATB = δAB/2.

Eq. (8) is theoperator corresponding to the most attractive channel for one gauge boson exchange in theSU(N)T C theory.

However, the two models are clearly qualitatively different: The NJLmodel contains propagating fermions at low energies, while the fermions are confined inthe SU(N)T C gauge model. Also, the NJL model can be fine-tuned to make the scalar andfermion masses small compared to the cutoffscale Λ, while the SU(N)T C gauge model hasno adjustable parameter; all dimensionful quantities in the SU(N)T C model are of orderΛT C raised to the appropriate power.

We can, however, contruct a class of models whose low-energy dynamics interpolatescontinuously between that of the NJL model and the SU(N)T C gauge model. The mod-els we will construct consist entirely of gauge fields coupled to fermions, so the theoriesare well-behaved at high energies.

Chiral symmetry breaking in these models is triggeredby asymptotically-free gauge couplings, and the fine-tuning in the NJL limit can be un-derstood in terms of the fine-tuning of gauge couplings (defined at a suitable subtractionpoint) in the underlying gauge theory.The basic idea underlying the model is very simple. We assume that there is another“interloping” sector which breaks the SU(N)T C group completely, so that all the SU(N)T Cgauge bosons are massive.

If this symmetry breaking occurs at a scale where the SU(N)T Ccoupling is weak, the effective theory below this scale will consist of massless fermionsinteracting through contact interactions such as those of eq. (8), and ⟨ψψ⟩= 0.

On theother hand, if the SU(N)T C coupling gets strong at a scale higher than the scale at whichthe interloping sector becomes strong, then the SU(N)T C interactions will break the ψchiral symmetry and ⟨ψψ⟩∼Λ3T C. In this case, the low-energy effective theory containsonly the Nambu–Goldstone bosons resulting from the chiral symmetry breaking. If weassume that the transition between the two limits just described is smooth, then we canobtain a model in which ⟨ψψ⟩≪Λ3T C by tuning the model between the two limits describedabove.

We expect the dynamics of such a model to be similar to the fine-tuned NJL modeldiscussed in the previous section.Actually, since we are ultimately interested in obtaining a condensate for quarks car-rying only color indices, we will consider models where the technicolor group is broken inthe patternSU(N)T C × SU(N)C −→SU(N)C′,(9)where SU(N)C is weakly coupled at the symmetry-breaking scale. In this case, the sur-viving SU(N)C′ group is weakly coupled, and we will later take N = 3 and identify it withQCD.To make these ideas specific, consider a theory with a gauge groupSU(N)T C ×SU(K) × SU(K)I × SU(N)C,(10)and fermion contentψLj, ψRj ∼(N, 1, 1, 1),j = 1, .

. ., F,χL ∼(N, K, 1, 1),χR ∼(1, K, 1, N),ξR ∼(N, 1, K, 1),ξL ∼(1, 1, K, N).

(11)

For simplicity, we will impose a discrete symmetry to set the coupling constants of the twoSU(K) groups equal. The first three group factors in eq.

(10) are “strong” gauge groupswhich will form fermion condensates, while SU(N)C is weakly coupled in the energy regimein which we are interested. Note that the fermion content of each gauge group consistsof vector-like fermions in the fundamental representation.

Thus, all gauge anomalies arecancelled in this theory. More importantly, this means that as long as only one of thegauge couplings is strong, we can analyze the symmetry breaking pattern by appealing toour understanding of QCD-like theories.The coupling strength of the gauge groups SU(N)T C and [SU(K) × SU(K)]I can becharacterized by the values of the gauge coupling constants gT C and gI evaluated at somefixed scale µ0.

Alternatively, we can characterize the gauge couplings by mass scales ΛT Cand ΛI at which the respective gauge couplings become strong enough to trigger chiralsymmetry breaking. There are several possibilities for the low-energy dynamics of thismodel, depending on the relative magnitudes of these parameters.• Suppose first that gI ≫gT C at the scale µ0, so that ΛI ≫ΛT C. Then SU(N)T C isweakly coupled at the scale ΛI, and the SU(K)I′ interactions give rise to the condensates⟨χLχR⟩, ⟨ξLξR⟩∼Λ3I.

(12)This condensate results in the chiral symmetry breaking patternSU(N)χL × SU(N)χR −→SU(N)χL+χR,SU(N)ξL × SU(N)ξR −→SU(N)ξL+ξR,(13)giving rise to 2(N 2 −1) potential NGB’s. (Each of the SU(K) factors acts like a copy ofQCD.) The condensate eq.

(12) results in the gauge symmetry breaking patternSU(N)T C × SU(N)C −→SU(N)C′. (14)N 2 −1 potential NGB’s are therefore eaten by the broken gauge bosons, which acquiremasses of order gT CΛI.

The remaining N 2 −1 potential NGB’s transform in the adjointrepresentation of the unbroken SU(N)C′ gauge group, and therefore acquire masses oforder gCΛI.Through all of this, the ψ fermions remain massless, and ⟨ψψ⟩= 0. The effectivetheory at scales µ ≪gCΛI therefore consists of the massless ψ fermions subject to variouscontact interactions suppressed by inverse powers of ΛI.

This is qualitatively similar tothe NJL model far into the unbroken phase.

• Now suppose that gT C ≫gI at the scale µ0, so that ΛT C ≫ΛI. In this case, theSU(N)T C gauge coupling becomes strong at a scale where the [SU(K) × SU(K)]I gaugecoupling is weak.

This results in the formation of the condensate⟨ψLψR + χLξR⟩∼Λ3T C.(15)(There are other possibilities for the alignment of the condensate, but a calculation alongthe lines of ref. [9] which treats the SU(K) and electroweak gauge groups perturbativelyshows that this vacuum alignment is chosen for sufficiently weak SU(K) coupling.

This isconsistent with the lore that condensates will align so as to preserve the maximal gaugesymmetry.) This condensate results in the chiral symmetry-breaking patternSU(F + K)L × SU(F + K)R −→SU(F + K)L+R,(16)giving rise to (F +K)2−1 potential NGB’s.

This gives rise to the gauge symmetry breakingpattern[SU(K) × SU(K)]I −→SU(K)I′. (17)K2 −1 NGB’s resulting from the condensate eq.

(15) are therefore eaten by the brokengauge bosons, which acquire masses ∼gIΛT C.In a basis where the first F entries correspond to ψ and the last K entries correspond toχL and ξR, we can classify the potential NGB’s associated with the broken axial generatorsas follows:X000F 2 −1 physical NGB’s,(18)000YK2 −1 eaten NGB’s,(19)0EE†02FK massive pseudo-NGB’s,(20)K · 100−F · 11 physical “axion.”(21)The fate of the NGB’s of eqs. (18) and (19) should be obvious.

The NGB’s of eq. (20) canbe viewed as 2F NGB’s transforming in the fundamental representation of the unbrokenSU(K)I′ gauge group.

These NGB’s therefore acquire masses of order gIΛT C. The “axion”of eq. (21) can be associated with the chargeψL : + K,ψR : −K,χL : −F,χR : −F,ξL : + F,ξR : +F.

(22)

The current associated with this charge has no SU(N)T C ×[SU(K) × SU(K)]I anomalies,but it does have a SU(N)C anomaly. Therefore, the axion will acquire a mass of orderΛ2C/ΛT C ≪ΛC.The χR and ξL fermions remain massless at this stage, and transform in the fun-damental representation of the unbroken SU(K)I′ gauge group.At the scale ΛI′, theSU(K)I′ gauge group becomes strong, resulting in the formation of the condensate⟨ξLχR⟩∼Λ3I′.

(23)This results in the symmetry-breaking patternSU(N)ξL × SU(N)χR −→SU(N)ξL+χR,(24)giving rise to N 2 −1 potential NGB’s. These potential NGB’s transform in the adjointrepresentation of the unbroken SU(N)C gauge group, and therefore acquire masses of ordergCΛI′.

Thus, ⟨ψψ⟩∼Λ3T C and the low-energy theory consists entirely of the F 2 NGB’sresulting from the symmetry breaking at the scale ΛT C. This is qualitatively similar tothe NJL model far into the broken phase.• We now consider the nature of the transition between the two limits just described.If we assume that the order parameter ⟨ψLψR⟩is continuous across the transition (i.e. thatthe transition is second order), then we can tune the [SU(K) × SU(K)]I coupling so that⟨ψLψR⟩≪Λ3, where Λ ∼ΛT C ∼ΛI.

In this case, both SU(N)T C and [SU(K) × SU(K)]Iare becoming strongly coupled near the scale Λ, but we can analyze this model usingcontinuity arguments.In the fine-tuned limit, it seems plausible that the ψ fermions are propagating degreesof freedom with mass m ≪Λ. The reason is that in the unbroken phase, there are poles atzero momentum transfer in Green’s functions coming from an intermediate state consistingof a single ψ fermion.

If the transition is second order, we expect the position of thesepoles to be continuous across the transition, and the ψ fermions to be massive propagatingstates.Also, in the fine-tuned limit, we expect the decay constant f of the NGB’s to bef ∼m. To see this, defineJ µνAB(q) ≡Zd4x e−iq·x⟨0|TJµA(x)JνB(0)|0⟩,(25)whereJµA ≡ψiγµγ5TAψ(26)

are the spontaneously broken axial currents. From Goldstone’s theorem, we know thatJ µνAB(q) has a pole at q2 = 0:J µνAB(q) −→(qµqν −q2gµν)f 2ABq2asq2 →0,(27)where f 2AB is the matrix of NGB decay constants.

From the Jackiw–Johnson sum rule(eq. (54) below), we know that the NGB decay constants vanish if the fermion self-energyvanishes.

Therefore, if we assume that the transition is second order, then we expect thedecay constants will be of order m for m ≪Λ. (This is made explicit in the approximationsto the Jackiw–Johnson sum rule which we will consider below.

)We now turn to the low-energy effective lagrangian for this model. At sufficiently lowenergies µ ≪f, the effective lagrangian describes the interactions among the NGB’s (andpossibly the ψ fermions if their masses are smaller than f).

In this effective lagrangian,the SU(N)F symmetry is realized nonlinearly on the NGB fields. As a result, this effectivelagrangian breaks down for processes involving momentum transfers p2 ≫f 2.

One signalfor this fact is that an infinite number of operators are important for such processes. Wetherefore conclude that the theory must contain new particles and interactions at scalesµ <∼f ≪ΛT C. The simplest possibility is that the symmetry is realized linearly in theeffective lagrangian at scales µ ∼f, implying that the theory contains light scalars withthe quantum numbers of the Φ field in eq.

(6). (It is difficult to imagine a reasonablealternative if one accepts the assumption that the ψ condensate can be fine-tuned to besmall.

)In fact, we may conjecture that the effective lagrangian at scales below ΛT C is in thesame universality class as the NJL model. To make this statement precise, we write theeffective lagrangian asLeff= LNJL + δL(28)where LNJL is the NJL lagrangian of eq.

(1) and δL contains all remaining terms. Weconjecture that the four-fermion coupling in LNJL is close to its critical value, and thatthe coefficients of the operators in δL are sufficiently small that the theory is in the sameuniversality class as the NJL model.

Similar assumptions have been discussed elsewherefor the case of top-condensate models [12] and “strong extended technicolor” models [13].4. Schwinger–Dyson AnalysisWe can flesh out our picture of the NJL limit by considering the Schwinger–Dysonequations for the theory we have described.The full Schwinger–Dyson equations areclearly intractable, and in order to make progress we must truncate these equations and

hope that what remains captures the essential physics we wish to discuss.We do notpretend that this analysis justifies the dynamical assumptions made in the last section,but we will see that these assumptions are incorporated in a simple and natural way inour analysis. Also, we can use the truncated Schwinger–Dyson equations to make crudequantitative estimates for various quantities of interest.

This section can be skimmed bya reader interested mainly in the application of the model discussed above to electroweaksymmetry breaking.4.1. Fermion Self-EnergyIf we defineΨ ≡ψχξ,(29)the full Schwinger–Dyson equation for the Ψ propagator isiS−1(p) =Xr,r′Z d4k(2π)4grγµTrAS(k)Γνr′B(k, p)Tr′BGABµν (k −p),(30)where r, r′ run over the fermion representations of the gauge group, ΓµrA is the gaugeboson vertex function, and GABµν is the gauge boson propagator.

We restrict the fermionpropagator to have the block-diagonal formS =SψSχξ. (31)This is consistent with the full Schwinger–Dyson equations, and is also the correct form inboth of the limits considered in the previous section.

The Schwinger–Dyson equation forSψ then involves only the exchange of SU(N)T C gauge bosons. We will assume in whatfollows that the [SU(K) × SU(K)]I coupling can be chosen so that the mass M of theSU(N)T C gauge bosons has any desired value.We will follow a venerable tradition and approximate the full gauge boson vertex func-tion by its tree-level value (with a running coupling evaluated at an appropiate momentumscale), approximate the gauge boson propagators by their asymptotic forms, and neglectfermion wave-function renormalization.

(See [10] for more details.) The resulting integralequation for the ψ fermion self-energy is thenΣ(p2) =Z dk24π2 G(max{k2, p2, M 2})k2 Σ(k2)k2 + Σ2(k2) ,(32)

where M is the mass of the SU(N)T C gauge bosons,G(p2) ≡3g2(p2)C2p2(33)can be viewed as the strength of the one-gauge-boson-exchange interaction, and C2 is theCasimir of the fermion representation.For p2 ≤M 2, the right-hand side of eq. (32) is independent of p2, and we haveΣ(p2) = mforp2 ≤M 2.

(34)This situation is rather reminiscent of the NJL model. In fact, we can writem = m G(M 2)4π2Z M20dk2k2k2 + m2 + δm= m G(M 2)4π2M 2 −m2 ln M 2m2+ δm,(35)whereδm ≡Z ∞M2dk24π2 G(k2)k2 Σ(k2)k2 + Σ2(k2)(36)depends only on the behavior of Σ(p2) with p2 > M 2.

If we naively identify G(M 2) withthe NJL coupling G and indentify M with the NJL cutoffΛ, then eq. (35) is similar tothe NJL gap equation with the addition a “counterterm” δm.

To understand the relationbetween eq. (35) and the NJL gap equation, we note that the NJL coupling G includesthe effects of integrating out all field models with p2 > Λ2, while G(M 2) has no suchinterpretation.

The role of the term δm is to include the effects of the high-energy gaugebosons.For p2 > M 2, eq. (32) is the same as the gap equation for massless gauge bosons,which has been studied by many authors.

The integral equation can be converted to adifferential equation:Σ′′(p2) −G′′(p2)G′(p2) Σ′(p2) −G′(p2)4π2Σ(p2) = 0,(37)where the prime denotes differentiation with respect to p2. This equation has been lin-earized in Σ, which is completely justified in the present case, since we are interested inthe regime where p2 ≥M 2 ≫Σ2(p2).

Using the known asymptotically-free behavior ofG(p2) for large p2, we can write the ultraviolet boundary conditionΣ(p2) −→σ3G(p2)asp2 →∞,(38)

for some constant σ. Eq. (37) has an infrared boundary conditionΣ′(M 2) = m G′(M 2)Z M20dk24π2k2k2 + m2= m G′(M 2)4π2M 2 −m2 ln M 2m2.

(39)It is easily verified by direct substitution that eq. (37) with these boundary conditions isequivalent to eq.

(32).In this formalism, we can explicitly see the fine-tuning necessary to obtain m ≪M.Because eq. (37) is linear in Σ, once we impose the ultraviolet boundary condition eq.

(38),Σ(p2) is completely determined up to an overall constant. We can therefore writeΣ(p2) = m F(p2)F(M 2) ,(40)where F(p2) is the solution to eq.

(37) satisfying the boundary conditionF(p2) −→G(p2)asp2 →∞. (41)The infrared boundary condition can then be writtenM 2R(M 2)4π2≡M 2G′(M 2)F(M 2)4π2 F ′(M 2)= 1 + O(m2/M 2).

(42)The left-hand side depends only on M and is of order 1 for M ∼ΛT C. We see that Mmust be equal to a critical value Mcrit to an accuracy O(m2/M 2) if we want m ≪M.This in turn means that the [SU(K) × SU(K)]I gauge coupling (evaluated at the scaleµ0) must be fine-tuned to an accuracy O(m2/M 2).To show that eq. (42) has a solution for some M 2, we note that the quantity R(p2)defined above satisfies the differential equationR′(p2) = G′(p2) −R2(p2),(43)with ultraviolet boundary conditionR(p2) −→G(p2)asp2 →∞.

(44)Thus, the left-hand side of eq. (42) is small for M ≪ΛT C. Because R′(p2) < G′(p2), wehave R(p2) < G(p2), and thusM 2R(M 2) > 3g2(M 2)C2.

(45)

Therefore, as long as g2(M 2) is sufficiently large, the left-hand side of eq. (42) will begreater than unity and the differential equation has a solution.

This is just the conditionthat the gauge coupling becomes sufficiently strong to trigger chiral symmetry breaking.Note also that Σ(0) is continuous as a function of M in this formalism.4.2. Light ScalarsWe now address the question of the existence of light scalars in this model.Ourstrategy is to compute ψ–ψ scattering in the color-singlet scalar and pseudo-scalar channels.The light scalars will manifest themselves as poles in the s channel.

We will compute thescattering amplitude in the ladder approximation shown in fig. 1.We will make several further simplifications.

Since the phase transition is second orderin this formalism, it is sufficient to establish the existence of the scalars in the unbrokenphase where Σ ≡0. Also, we will compute the amplitude in the kinematic regime shownin fig.

1. The Mandlestam variables are then s = 0, t = u = p2.

A light scalar with massms ≪M in the s channel will result in the behaviorT(p2) ∼1m2s+ · · ·. (46)In this kinematic regime, only a single form-factor contributes to the amplitude, and wecan write the following simple integral equation for the scattering amplitude T(p2):T(p2) = G(max{p2, M 2}) +Z dk24π2 G(max{k2, p2, M 2}) T(k2).

(47)Since we are in the unbroken phase, this equation holds for all flavor channels and for boththe scalar and pseudoscalar channels.For p2 ≤M 2, the right-hand side of eq. (47) is independent of p2, and we haveT(p2) = Tforp2 ≤M 2.

(48)Again, this situation is very similar to the NJL model, where the scattering amplitudeconsidered here is independent of t and u.For p2 > M 2, the amplitude satisfies the differential equationT ′′(p2) −G′′(p2)G′(p2) T ′(p2) −G′(p2)4π2T(p2) = 0,(49)where the prime again denoted differentiation with respect to p2. This is the same differ-ential equation satisfied by Σ above, but with different boundary conditions and with the

crucial difference that it holds for arbitrarily large T. Eq. (49) has the ultraviolet boundaryconditionT(p2) −→cG(p2)asp2 →∞,(50)where c is a dimensionless constant, and an infrared boundary conditionT ′(M 2) = G′(M 2)1 + T(M 2)M 24π2.

(51)Once we impose the ultraviolet boundary condition, we can write the solution asT(p2) = T F(p2)F(M 2) ,(52)where F(p2) is the function introduced in eq. (40) above.

The infrared boundary conditioncan then be written1T =F ′(M 2)G′(M 2)F(M 2) −M 24π2=1R(M 2) −M 24π2(53)where R(p2) is the function defined in eq. (42).

Eq. (53) shows explicitly that a solutionexists.

Of more interest is the fact that eq. (53) shows that as we approach the fine-tuningcondition eq.

(42), T becomes singular.† This shows that there are light scalars in thetheory near the critical point in the ladder approximation.4.3. Decay ConstantsWe now discuss the decay constants which result from the ψ fermion condensation.They are given by the Jackiw–Johnson sum rule [11]f 2AB =Z d4k(2π)4 trhγµγ5(ZJTA)S(k)eΓµB(k, k)S(k)i+ δf 2,(54)where eΓµA is the current vertex function with the NGB pole removed, S(k) is the propagatorof the fermions in the current, ZJ is the renormalization constant of the current operator,and δf 2 is a counterterm.† In an exact treatment, we certainly expect the same critical value of M for the chiralsymmetry-breaking transition and the appearance of light scalars, but ´a priori we have noright to expect the critical values of M to concide exactly in our approximation, since itis not clear that the gap equation eq.

(32) and the partial summation we have performedto compute T are part of a single consistent approximation.

If we make the same approximations made in deriving eq. (32) we obtain f 2AB = f 2δAB,withf 2 ≃N16π2Zdk2 Σ2(k2)2k2 + Σ2(k2)[k2 + Σ2(k2)]2(55)=N16π2m2 ln M 2m2 +Z ∞M2 dk2 Σ2(k2)k2.

(56)The integral in eq. (56) is of order m2, so the first term dominates for m ≪M.

If weapproximate Σ(p2) for p2 > M 2 by its asymptotic form eq. (38), we obtainf 2 ≃Nm216π2ln M 2m2 + 12.

(57)To summarize, we see that the gap-equation analysis incorporates many of the fea-tures we expected on the basis of qualitative arguments of section 3: the transition issecond order, the fine-tuning is manifest, the decay constants are small compared to thecompositeness scale, and the light scalars are present.5. Application to Electroweak Symmetry Breaking5.1.

“Top-Mode” ModelIn order to apply the model described above to electroweak symmetry breaking, wetake N = 3 and identify SU(3)C with ordinary color. We also take F = 2 and identifyψ =tb.

(58)The low energy theory contains two Higgs doublets; in the notation of eq. (6), we haveΦ =H0tH+bH−tH0b,(59)where Ht and Hb are Higgs doublet fields.This theory may seem to be a phenomenological disaster, since it appears that custo-dial symmetry of the technicolor interactions implies mt = mb, and the theory contains anaxion (see eq.

(22)) with a decay constant near the weak scale. Such axions are ruled out bya combination of laboratory experiments and astrophysical considerations [16].

However,the model actually does not suffer from these problems, as we will describe below.

In the standard model, we know that the custodial SU(2) symmetry is broken by theweak hypercharge. In fact, the hypercharge assignments areY (ψL) = 13 ,Y (tR) = 43 ,Y (bR) = −23 .

(60)Thus, U(1)Y gauge boson exchange will mediate an attractive force between ψL and tRand a repulsive force between ψL and bR. Although this force is weak, when the theory istuned near the critical point, the contribution from U(1)Y gauge boson exchange can resultin ⟨bb⟩= 0 while ⟨tt⟩̸= 0.

The phenomenon of the amplification of small perturbationsnear a critical point has been discussed in the context of technicolor theories [14] and intop-condensate models [15], where it has been dubbed “critical instability.”To see how this phenomenon emerges from the gap equation, writeY = YLPL + YRPR(61)= 13 12PL +13 12 + τ3PR,where PL,R are the left- and right-handed helicity projection operators. The ψ self-energyis a diagonal 2 × 2 matrixΣ =Σt00Σb.

(62)The gap equation has the same form as eq. (32) with the replacementG(p2) =34π2p2g2T C(p2)C2 + g2Y (p2)YLYR.

(63)In the approximations we are making, the Schwinger–Dyson equations for the fermionmasses split into separate equations for the top- and bottom-quark self-energies. Supposenow that the theory is tuned so that mt ≪M.

ThenGt(M 2) = Gcrit + O(m2t/M 2),Gb(M 2) = Gcrit −2g2Y (M 2)M 2+ O(m2t/M 2). (64)Thus, Σb = 0 as long asgY >∼mtM .

(65)(Note that we can consider the possibility that the scale M is chosen so that the hierarchymt/mb emerges entirely as a result of U(1)Y custodial symmetry breaking. However, thisrequires M ∼1 TeV, and results in a top mass mt ∼500 GeV, as we will see below.)

We now consider the low-energy theory for the case where Σb = 0.Taking intoaccount the breaking of SU(2)ψR due to hypercharge, the symmetry breaking pattern canbe writtenSU(2)W × U(1)Y × U(1)bR −→U(1)EM × U(1)bR,(66)resulting in 3 potential NGB’s which are eaten by the massive electroweak gauge bosons.The effective theory below the scale ΛT C contains two Higgs doublets with⟨Ht⟩= v,⟨Hb⟩= 0. (67)Hb is an unbroken doublet with mass ∼gY M, which can be far above the electroweakscale.

In this model as discussed so far, all fermions other than the top quark are exactlymassless.The axion of eq. (21) is not present, but this is only because there is an unbroken axialU(1) symmetry in the low-energy theory as a result of the massless b quark.

Clearly, thissymmetry, as well as the flavor symmetries of the remaining quarks and leptons, must bebroken somehow if the theory is to account for the observed fermion masses and mixings.This is exactly the situation in technicolor theories, where the technifermion condensatedoes not give mass to ordinary quarks and leptons in the absence of additional “extendedtechnicolor” interactions which connect the technifermions with the quarks and leptons.Thus, as in technicolor, we assume that the effective theory at the scale ΛT C containsfour-fermion “extended technicolor” (ETC) interactions of the form1M 2E(ψψ)(ff)(68)where the ψ’s represent third-generation quark fields and the f’s represent quark or leptonfields which are SU(N)T C singlets. We will not be concerned here with the dynamicswhich gives rise to these operators, or with exhibiting a completely realistic theory.

Ouraim is simply to show that there is in principle no obstacle to making the theory realistic.Four-fermion interactions of the form of eq. (68) can give rise to masses for all thequarks and leptons, but they necessarily preserve a U(1) symmetry which counts thenumber of quarks from the first two generations.† If this symmetry were exact, elementsof the Cabbibo–Kobayashi–Maskawa matrix involving the third generation would vanishidentically, and so this U(1) symmetry must be broken somehow.

This U(1) symmetrycan be broken by dimension-7 interactions such as1M 3E(QLi/DψL)(χLχR),(69)† I am indebted to R. Sundrum for pointing this out to me.

where QL is a left-handed quark field from the first two generations. Alternatively, wecan imagine that all quarks carry SU(3)T C rather than SU(3)C, and the fact that thetop quark is the only fermion which condenses is explained by critical instability, due forexample to additional higher-dimension operators.

In this case, all masses and mixings canbe obtained from operators of the form of eq. (68).

Thus, there seems to be no obstacle inprinciple to incorporating light fermions into the theory.The scale ME > ΛT C is associated with new interactions, such as massive gaugeboson exchange. It may seem that much of the motivation for writing a renormalizablemodel is lost once we introduce these non-renormalizable terms.

However, unlike the four-fermi coupling which drives the symmetry breaking in the NJL model, these four-fermiinteractions are perturbatively weak, and no exotic dynamics is required to generate them.The present model can be viewed as a valid effective theory for scales up to ∼ME, whichcan be well above the compositeness scale, while the NJL model cannot be extended abovescales Λ ∼1/√G.Through the standard ETC mechanism, the four-fermi interactions of eq. (68) willgive rise to fermion masses of ordermf ∼1M 2EZ M d4k(2π)4 tr 1/k mt1/k ∼M 2mt4π2M 2E.

(70)This can be viewed as due to the presence of a condensate⟨ψψ⟩∼M 2m4π2 . (71)Alternatively, using the NJL model as a guide, we expect that the ETC interactions eq.

(68) will give rise to Yukawa couplings between the ordinary fermions and the compositeHiggs field of ordery ∼M 24π2M 2Emtv(72)This gives rise to the fermion masses eq. (70) when the Higgs field acquires a vacuumexpectation value.We now consider the case where M is far above the weak scale.

In this case, theeffective lagrangian at low scales is indistinguishable from the standard model, and weexpect the renormalization group analysis of ref. [3] to be valid.

This scenario necessitatesa severe fine-tuning of the order m2t/Λ2T C, and results in a top-quark mass of 230 GeV forΛT C ≃1015 GeV (200 GeV for MT C ≃1019 GeV). Such values for the top quark massare disfavored by a global analysis of the radiative effects in the the standard model, whichgives mt < 201 GeV at the 95% confindence level [17].

Note that if M is far above the weak scale, eq. (70) shows that realistic fermionmasses can easily be generated with ME sufficiently large to suppress flavor-changingneutral currents which can arise from four-fermion operators of the form1M 2E(ff)(ff).

(73)Of course, this feature is obtained only at the expense of the fine-tuning of the SU(N)T Cbreaking scale.If M is close to the weak scale, the leading-log effects which form the basis of thepredictions of ref. [3] are not expected to be important.

Using eq. (57) and noting thatv = f/√2 = 246 GeV, we find e.g.mt ≃440 GeVforM ≃10 TeV.

(74)While the approximations used to obtain this result are rather crude, it is clear that thisis well outside the limits on the top mass coming from ρ-parameter constraints.It isof course possible that other sources of custodial symmetry breaking (for example fromhigher-dimensional operators from the ETC sector) can cancel the effects of the top quarkon the ρ parameter, although there is no good reason to expect this to happen.Since the “top-mode” model described above seems to have phenomenological troublesfor all reasonable choices of parameters, we now turn to models in which other fermionsare responsible for the electroweak symmetry breaking.5.2. Technicolor-like ModelsIn this subsection, we consider models in which the ψ fermions are new strongly-interacting fermions.

In this case, the direct connection between the top-quark mass andthe electroweak scale is lost, but raising the compositeness scale may make these theo-ries more phenomenologically attractive than technicolor theories (at the expense of fine-tuning). We will argue that because of the critical instability mechanism discussed above,we cannot have the electroweak symmetry broken by new fermions without additionalfine-tuning.There are several strong constraints on the new fermions, which we will genericallydenote by “T.” We assume that the new fermions occur in weak doublets so that thecondensate of these fermions is an I3 = 12 order parameter.

The masses of the T fermionsare constrained by demanding that they give rise to the correct value of the electroweak

scale v. Using the approximation eq. (57) to the Jackiw–Johnson sum rule, we obtainXjm2j ≃8π2v23ln M 2m2 + 12−1(75)∼(400 GeV)2forM ∼10 TeV,(76)where the sum runs over all T fermions and m is their average mass.

(This formula willhave large corrections if ln(M 2/m2) ≫1.) This shows that in any such model, we expectnew heavy weak-doublet fermions which will be accessible in future experiments.In order that they do not contribute significantly to the ρ-parameter, T fermions inthe same weak doublet must be nearly degenerate in mass.

In particular, we must avoidlarge mass splittings induced by U(1)Y couplings through critical instability, as discussedabove. Of course, we can always add additional interactions to cancel this effect, but thestrength of these interactions must be fine-tuned, and we want to see whether fine-tuningcan be avoided.To avoid the critical instability mechanism described in the last section, we musttake the weak doublet T fermions to carry zero U(1)Y charge.The SU(2)W × U(1)Ysinglet T fermions must carry U(1)Y charge so that the condensate ⟨TLTR⟩preservesU(1)EM.Thus, the only possibility is to take the extra fermions to transform underSU(3)T C × SU(2)W × U(1)Y asTL ∼(3, 2; 0),UR ∼(3, 1; 1),DR ∼(3, 1; −1).

(77)When SU(3)T C becomes strong, they give rise to a condensate⟨U LUR⟩, ⟨DLDR⟩̸= 0,(78)where we have definedTL,R ≡UDL,R. (79)The theory has an approximate discrete custodial symmetryTL,R 7→0110TL,R ≡XTL,R,Bµ 7→−Bµ,Wµ 7→XWµX,(80)

where Bµ and Wµ are the U(1)Y and SU(2)W gauge fields, respectively. If the T fieldswere the only fermions in the theory, this custodial symmetry would be exact, and wewould have mU = mD.

In the full theory, this symmetry is violated by the U(1)Y gaugecouplings to the quarks and leptons and by ETC operators. In particular, there mustbe strong ETC operators to give rise to the large top mass, so that there will in generalbe custodial-symmetry violating four-fermi couplings among the T fermions with strength1/M 2E ∼1/M 2, which will give rise to a large splitting |mU −mD|.

(Note that we cannotexplain the large top-quark mass by taking the third-generation quarks to transform underSU(N)T C, since the effect of U(1)Y would give mt ≫mT . )We have argued that additional fine-tuning is required in “non-minimal” models in-corporating NJL-like dynamics to break electroweak symmetry in order to avoid an unac-ceptably large value for the ρ-parameter.

If we accept this additional fine-tuning, however,there seems to be no obstacle to making such theories fully realistic.6. ConclusionsWe have considered a renormalizable model which we argued can break the electroweaksymmetry via NJL-like dynamics.

The crucial dynamical assumption is that the chiralsymmetry-breaking transition in the model is second order. The fine-tuning needed toobtain a large hierarchy between the weak scale and the compositeness scale is mainfestedin the fine-tuning of a gauge coupling constant.

We have seen that the minimal “top-mode” standard model can be obtained from such a model, as well as models in which newfermions are responsible for electroweak symmetry breaking. In the latter case, we haveargued that additional fine-tuning is required to avoid unacceptably large corrections tothe electroweak ρ-parameter.7.

AcknowledgementsI would like to thank R. Rattazzi and especially R. Sundrum for discussions on thetopic of this paper. This work was supported by the Director, Office of Energy Research,Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S.Department of Energy under Contract DE-AC03-76SF00098.

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Figure CaptionsFig. 1: Ladder approximation to the ψ–ψ scattering amplitude.

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