Asymptotically free four-fermion interactions and electroweak symmetry breaking

Asymptotically free four-fermion interactions and electroweak symmetry breaking

Markus-Jan Schwindt and Christof Wetterich

Institut fur Theoretische Physik, Universitat Heidelberg, Philosophenweg 16, D-69120 Heidelberg(Received 20 January 2010; published 16 March 2010)

We investigate the fermions of the standard model without a Higgs scalar. Instead, we consider a

nonlocal four-quark interaction in the tensor channel which is characterized by a single dimensionless

coupling f. Quantization leads to a consistent perturbative expansion for small f. The running of f is

asymptotically free and therefore induces a nonperturbative scale �ch, in analogy to the strong

interactions. We argue that spontaneous electroweak symmetry breaking is triggered at a scale where f

grows large and find the top quark mass of the order of�ch. We also present a first estimate of the effective

Yukawa coupling of a composite Higgs scalar to the top quark, as well as the associated mass ratio

between the top quark and the W boson.

DOI: 10.1103/PhysRevD.81.055005 PACS numbers: 12.60.Cn, 14.65.Ha

I. INTRODUCTION

The large hadron collider (LHC) will soon test themechanism of spontaneous electroweak symmetry break-ing. It is widely agreed that this phenomenon is associatedto the expectation value of a scalar field which transformsas a doublet with respect to the weak SU(2)-symmetry. Theorigin and status of this order parameter wait, however, forexperimental clarification. In particular, an effective de-scription in terms of a scalar field does not tell us if thisscalar is ‘‘fundamental’’ in the sense that it constitutes adynamical degree of freedom in a microscopic theorywhich is formulated at momentum scales much largerthan the Fermi scale. Alternatively, no fundamentalHiggs scalar may be present, and the order parameterrather involves an effective ‘‘composite field.’’ In thisnote we investigate the second possibility and thereforeconsider the fermions of the standard model without afundamental Higgs scalar.

As long as we include only the gauge interactions of thestandard model, the microscopic or classical action of sucha theory does not involve any mass scale. An effective massscale �QCD will only be generated by the running of the

strong gauge coupling, which is asymptotically free [1].Confinement removes then the gluons and quarks from themassless spectrum. In addition, the spontaneous chiralsymmetry breaking by quark-antiquark condensates wouldalso imply electroweak symmetry breaking by a compositeorder parameter, in this case the chiral condensate. In thissetting all particle masses would be of the order �QCD or

zero, such that this scenario cannot explain why the topquark mass or the W-, Z-boson masses are much largerthan 1 GeV. Furthermore, all leptons would remain mass-less. One concludes that any realistic model needs furtherinteractions beyond the standard model gauge interactions.Since the strong and electroweak gauge couplings remainactually quite small at the Fermi scale of electroweaksymmetry breaking, they are expected to produce onlysome quantitative corrections to the dominant mechanism

of electroweak symmetry breaking. We will therefore ne-glect the gauge couplings in this paper.Our knowledge about the effective interactions between

the quarks and leptons at some microscopic or ‘‘ultravio-let’’ scale�UV is very limited. Furthermore, it is not knownwhich scale �UV has to be taken. Typically, one mayassociate �UV with the scale where further unificationtakes place, as a grand unified scale or the Planck scalefor the unification with gravity. This would suggest a veryhigh scale, �UV � 1016 GeV and we will have this sce-nario in mind for our discussion. However, much smallervalues of �UV are also possible. In practice, we will onlyassume here that �UV is sufficiently above the Fermi scale(say �UV > 100 TeV), such that an effective descriptioninvolving only the fermions of the standard model becomespossible in the momentum range �QCD � jqj � �UV.

We will formulate our model in terms of effective fer-mion interactions at the scale �UV and restrict the discus-sion to a four-fermion interaction involving only the right-handed top quark and the left-handed top and bottomquarks. This is motivated by the observation that only thetop quark has a mass comparable to the Fermi scale.Interactions with the other quarks and leptons are assumedto be much smaller than the top quark interactions—typi-cally their relative suppression is reflected in the muchsmaller masses of the other fermions. For the discussionin this paper we omit all ‘‘light’’ fermions and the gaugebosons. No other particles besides the top and bottomquarks are introduced in our model.Sincewe do not know the effective degrees of freedom at

the scale �UV, the effective interaction is not necessarilylocal. Nonlocalities involving inverse powers of the ex-changed momenta are typically generated by the propaga-tors of exchanged massless fields. However, we want tokeep the issue open if massless or low mass propagatingparticles are associated to the nonlocalities. This will notbe of primary importance for our discussion.With the inclusion of possibly nonlocal interactions the

limitation to an effective four-fermion interaction poses no

PHYSICAL REVIEW D 81, 055005 (2010)

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http://dx.doi.org/10.1103/PhysRevD.81.055005

severe restriction. Many models with additional degrees offreedom can be effectively described in this way. We stressthat our model is not considered as fundamental for scalesbeyond �UV. Nevertheless, it can be treated with the samemethods as a fundamental theory for scales below �UV.Effects of fluctuations with momenta below �UV can beincluded, and nonlocalities pose no special problem.Modern functional renormalization group methods cangenuinely deal with this situation. In our case even pertur-bative loop computations can be applied as long as thecouplings are small.

Local four-fermion interactions have already been in-vestigated earlier, for example, in the models of ‘‘top quarkcondensation’’ [2]. By simple dimensional analysis a localinteraction involves a coupling �ðmassÞ�2. Such modelstherefore exhibit an explicit mass scale in the microscopicaction. It is indeed possible to obtain spontaneous electro-weak symmetry breaking in this way—the prototype is theNambu-Jona-Lasinio (NJL) model [3]. Without a tuning ofparameters the top quark mass mt turns out, however, to beof the same order as �UV, in contradiction to the assumedseparation of scales. By a tuning of parameters it is pos-sible to obtain mt � �UV, but the issue is now similar tothe ‘‘gauge hierarchy problem’’ in the presence of a fun-damental scalar field. In order to obtain mt � �UV themicroscopic effective action must be close to an ultravioletfixed point. The necessity of tuning arises from a ‘‘relevantparameter’’ in the vicinity of the fixed point (in the sense ofstatistical physics for critical phenomena) which has adimension not much smaller than 1. A rather extensivesearch for possible ultraviolet fixed points for pointlikefour-fermion interactions has been performed in [4].Many fixed points have been found, but all show a relevantdirection with substantial dimension, and therefore theneed for a parameter tuning for mt � �UV.

We will therefore concentrate in this paper on nonlocaleffective interactions. For such interactions the coupling�M�2 is replaced by f2=q2, with q2 the square of someappropriate exchanged momentum and f a dimensionlesscoupling. Interactions of this type do not involve a massscale on the level of the microscopic action—the classicalaction exhibits dilatation symmetry. Still, the quantumfluctuations typically induce an anomaly for the scalesymmetry, associated to the running of the dimensionlesscoupling f. We will discuss a specific model where f turnsout to be asymptotically free. One expects then the gen-eration of a nonperturbative ‘‘chiral scale’’ �ch, in analogyto the ‘‘confinement scale’’ �QCD for QCD. We argue that

the Fermi scale and mt are proportional to �ch. Since therunning of dimensionless couplings is only logarithmic,this offers a chance for a large natural hierarchy �UV �mt, without tuning of parameters.

Several extensions of the standard model have beenproposed where the Fermi scale is directly or indirectlyrelated to a growing dimensionless couplings, as techni-

color [5], condensation via fermions in higher color repre-sentations [6], top color [7], top condensates withadditional U(1)-symmetry [8], top quark seesaw [9],Higgsless models [10], or more generally to running di-mensionless couplings as several versions of supersymmet-ric theories. The particularity of our approach is theabsence of additional fermions or gauge symmetries.This will guarantee minimal flavor and CP violation bythe Cabibbo-Kobayashi-Maskawa matrix. The presence ofonly one dimensionless coupling, which will be transmutedto a nonperturbative mass scale in close analogy to QCD,makes our model in principle highly predictive. If thenonperturbative physics at the scale �ch can be understoodquantitatively, the top quark mass and the properties of theHiggs sector are determined without involving any freeparameters.This paper is organized as follows. In Sec. II we present

our model of a nonlocal tensor interaction. We compute therunning of the dimensionless coupling f in one loop-perturbation theory and establish asymptotic freedom ofour model in Sec. III. In consequence, a nonperturbative‘‘chiral scale’’�ch is generated where the running couplingf grows large. Section IV turns to the effective four-fermion vertices in channels different from the tensorchannel. The interactions in the scalar and vector channelsare generated by the fluctuation effects even though theymay not be present at the microscopic scale �UV. In theregion of validity of perturbation theory they are small,�f4=q2. However, the scalar interaction becomes compa-rable in strength to the tensor channel when f grows largeclose to the chiral scale�ch. As a consequence of this largeinteraction spontaneous electroweak symmetry breakingcan be generated similar to the NJL model [3]. In Sec. Vwe concentrate on the interaction in the scalar channel. Wefind a positive coupling, similar to the NJL model. If thiscoupling grows large enough, one expects spontaneouselectroweak symmetry breaking by a top-antitop conden-sate. This aspect is similar to other models. We computethe resulting top quark mass mt, using a Schwinger-Dyson[11] or gap equation. We find mt � �ch.In Sec. VI we turn to the composite field for the Higgs

doublet. We discuss the qualitative features of the runningcouplings including the composite degrees of freedom. Forthis purpose we go beyond a simple Hubbard-Stratonovichtransformation at a given scale, which is often used for thistype of problem. Using the method of ‘‘partial bosoniza-tion’’ motivated by exact functional renormalization groupmethods, we transfer at every scale the fluctuation-generated effective four-fermion interactions to a corre-sponding running of the Yukawa coupling between Higgsscalar and top quark and of the mass term for the Higgsfield. In a sense, this performs a Hubbard-Stratonovichtransformation at every scale of the running. We use thiscomputation for a first, still very rough, estimate of theratio between top andW-boson massmt=mW . Even though

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quantitatively not reliable, this computation demonstratesthat this ratio is, in principle, a computable quantity whichdoes not involve any parameters (except the weak gaugecoupling for the ratio between MW and the Fermi scale).The top quark Yukawa coupling h is directly related to the

mass ratiomW=mt ¼ gW=ðffiffiffi2

phÞ, withmW and gW the mass

and gauge coupling of the W boson. We present our con-clusions in Sec. VII.

II. NON-LOCAL TENSOR INTERACTIONS

Let us first discuss the possible tensor structures for anonlocal four-fermion interaction �ð �cAc Þ2. The basicbuilding block is a color singlet fermion bilinear �cAc ,where the color indices are contracted. The tensor structurewith respect to the Lorentz symmetry is determined by Asuch that �cAc is a scalar or pseudoscalar, a vector orpseudovector, or a second rank antisymmetric tensor.Interactions in the vector or pseudovector channels involvebilinears with two left-handed or two right-handed fermi-ons, �c L�

�c L or �c R��c R. They conserve chiral flavor

symmetries which act separately on c L and c R and there-fore forbid mass terms for the fermions � �c Lc R.Interactions capable of producing masses for the top quarkand W=Z bosons of comparable magnitude must thereforeinvolve the (pseudo)scalar or tensor channels.

A nonlocal scalar interaction �ð �c Lc RÞð �c Rc LÞq�2 �Rx;yð �c Lc RÞðxÞðx� yÞ�2ð �c Rc LÞðyÞ, with q2 the squared

momentum in the scalar exchange channel, has very simi-lar properties as a model with a fundamental Higgs scalarwhich is massless at the scale�UV. We therefore expect theusual necessity of parameter tuning if we want to achieve asmall ratiomt=�UV. A local coupling ð �c Lc RÞð �c Rc LÞm�2

is allowed by the symmetries and will be generated byquantum fluctuations. The interesting remaining candidateis a tensor interaction, with A� ½��; ��. For chiral ten-sors no local interaction in this channel is consistent withthe SUð3Þ � SUð2Þ � Uð1Þ symmetries of the standardmodel as well as Lorentz symmetry. We will thereforeinvestigate a model with a nonlocal interaction of this type.

We define the microscopic or classical action for aLorentz invariant theory of massless interacting fermionsby S ¼ S2 þ S4, defined in momentum space as

� S2 ¼ �Z d4q

ð2�Þ4 ð�tðqÞ��q�tðqÞ þ �bðqÞ��q�bðqÞÞ

(1)

and

�S4 ¼ 4f2Z d4qd4pd4p0

ð2�Þ12PklðqÞq4

f½�tðqþ pÞ�kþtðpÞ�

� ½�tðp0Þ�l�tðp0 þ qÞ� þ ½�tðqþ pÞ�kþbðpÞ�� ½ �bðp0Þ�l�tðp0 þ qÞ�g: (2)

Here t and b are Dirac spinor fields describing the top and

bottom quark, respectively. (The theory can be easily ex-tended to all three generations of quarks and also to lep-tons.) Contracted indices are summed. The 3� 3 matrixPðqÞ involves the spacelike indices k, l and is defined by

PklðqÞ ¼ �ðq20 þ qjqjÞ�kl þ 2qkql � 2i�kljq0qj: (3)

It has the properties

PklPlj ¼ q4�kj; P

klðqÞ ¼ PlkðqÞ: (4)

The nonlocal character of the interaction arises from thefactor 1=q4.In a standard spinor basis in which

c ¼ c L

c R

� �;

�c ¼ c y�0 ¼ ð �c R; �c LÞ, the 4� 4 matrices �k are de-fined in terms of the Pauli matrices �k,

�kþ ¼ �k 00 0

� �; �k� ¼ 0 0

0 �k

� �: (5)

The fermion bilinears �c�þc � �c R�þc L and �c��c ��c L�þc R therefore mix left- and right-handed spinors andviolate the chiral symmetry that would protect the topquark from acquiring a mass. The matrices � correspondto appropriate projections of the commutator ½��; �� onleft- (right-)handed spinors, such that Eq. (2) indeed de-scribes a tensor exchange interaction (cf. Appendix A).The Lorentz-invariance of the interaction can be checkedexplicitly, if we write Eq. (2) in the form

� S4 ¼ f2Zx½�tR��qL�

@�@

@4½ �qL�

�tR�; (6)

with @2 ¼ @�@� and qL ¼ ðtL; bLÞ.Furthermore, the action (1) and (2) has a global SUð2Þ �

Uð1Þ symmetry—the remnant of the electroweak gaugesymmetry in the limit of neglected gauge couplings. Thissymmetry forbids mass terms for the quarks, such that amass term can be generated only by spontaneous symmetrybreaking. Similarly, the interaction is invariant under thecolor symmetry SU(3), with implicitly summed color in-dices in the bilinears. We note that only the left-handedbottom quarks are involved in the interaction and we willtherefore omit the right-handed bottom quark together withthe other light quarks and the leptons.Similar to the photon-exchange description of the non-

local Coulomb interaction we may obtain S4 from theexchange of chiral tensor fields [12], according to theFeynman diagrams in Fig. 1. A summary of the propertiesof the associated tensor fields and a proof of equivalence ofa local theory with massless chiral tensor fields with ournonlocal fermion interaction (2) is provided inAppendix A.In this paper we will not use tensor fields and concen-

trate on the purely fermionic action (1) and (2). In particu-lar, this avoids the delicate issues of consistency of the

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quantization of chiral antisymmetric tensors [13], and thequestion if the nonlocal interactions are associated to prop-agating low mass particles. One can write a consistentfunctional integral and therefore a consistent quantum fieldtheory based on the action (1) and (2). The functionalmeasure only involves the fermions and can be formulatedin a standard way. The presence of a nonlocal four-fermioninteraction does not affect the consistency of theGrassmann functional integral. As usual, the introductionof gauge interactions (not used in this paper) requires carein the presence of chiral fermions. (Anomaly cancellationinvolves the omitted light quarks and leptons.) Here, weonly need a regularized functional measure for the quarkfields, which is straightforward if the SUð3Þ � SUð2Þ �Uð1Þ gauge interactions are neglected.

We will find that the nonlocal interaction indeed growslarge at a scale �ch. At this scale we expect the formationof bound states. A scalar top-antitop bound state may playthe role of a composite Higgs doublet. If the correspondingcomposite field acquires a vacuum expectation value, thiscould explain spontaneous electroweak symmetry break-ing. The phenomenology of our model will depend on thedetailed spectrum and interactions of possible bound statesand on the precise mechanism how the long range charac-ter of the nonlocal chiral tensor interactions is cut offeffectively at the scale �ch. Having started from a consis-tent functional integral formulated at some microscopic orultraviolet scale �UV, the remaining issues of consistency,as unitary scattering amplitudes, depend only on the iden-tification of the correct ground state of the model. Wepropose here that the ground state is characterized byelectroweak symmetry breaking through a top-antitop con-densate. Our computations give indeed indications in thisdirection, but substantial work is needed before this can beconsidered as established.

III. ASYMPTOTIC FREEDOM

The dimensionless coupling constant f is the only freeparameter in our model. We will show in this section thatthe corresponding renormalized running coupling isasymptotically free in the ultraviolet. On the other side,there is a characteristic infrared scale �ch where the cou-pling f grows large. This is in complete analogy to QCD,where �QCD corresponds to the scale where the strong

gauge coupling grows large. By ‘‘dimensional transmuta-

tion’’ we may therefore trade f for the mass scale �ch.Besides this mass scale the model has no free dimension-less parameter. In particular, we will argue in Sec. IV thatour model leads to spontaneous breaking of the SUð2Þ �Uð1Þ symmetry. The Fermi scale ’ characterizing theelectroweak symmetry breaking must be proportional to�ch, ’ ¼ cw�ch, with a dimensionless proportionality co-efficient cw that is, in principle, calculable without involv-ing a further free parameter. If the model leads to acomposite Higgs scalar, its mass in units of ’, MH=’, aswell as its effective Yukawa coupling to the top quark, h ¼mt=’, are calculable quantities in our model.In order to investigate the running of f, we wish to

compute the effective action � corresponding to the clas-sical action (1) and (2), to one-loop order. The one-loopcorrection reads

��ð1lÞ ¼ i

2Tr lnSð2Þ; (7)

where the field dependent inverse propagator Sð2Þ is definedas the second functional derivative of the action withrespect to the quark fields,

ðSð2ÞÞcc0AB;�ðp; p0Þ ¼ � �

��c0B�ðp0Þ

�

��cAðpÞ

ðS2 þ S4Þ:

(8)

Here� are the quark fields, with color indices c, c0, flavorindices A, B (taking values t, b, �t, �b), spinor indices , �,and momenta p, p0. The trace Tr involves an integral overmomentum and summation over all kinds of indices. We

write Sð2Þ ¼ P0 þ F, where P0 is the ‘‘free’’ part of thepropagator, derived from S2,

ðP0Þcc0AB;�ðp; p0Þ ¼ ½ð��p�Þ�ð�A�t�Bt þ �A �b�BbÞþ ð��p�Þ�ð�At�B�t þ �Ab�B �bÞ�� cc0�ðp� p0Þ; (9)

and �ðp� p0Þ ¼ ð2�Þ4�4ðp� � p0�Þ. We treat F� f2 as a

perturbative correction due to S4. Then ��ð1lÞ reads, up toan ‘‘infinite constant,’’

��ð1lÞ ¼ i

2TrðP�1

0 FÞ � i

4TrðP�1

0 FP�10 FÞ þ � � � (10)

where the dots stand for neglected six-quark and higherinteractions. We display the explicit expressions for F, aswell as the formal expressions for the first two terms on theright-hand side of Eq. (10), in Appendix B.The nonlocal factors �ðPðqÞ=q4Þ� appearing in the

interaction (2) are attached in different ways to the fermionlines. We represent them as dashed lines in the correspond-ing Feynman diagrams in Figs. 2–4. Our notation recallsthe one-to-one correspondence with the exchange of theassociated chiral tensor fields. The diagrams in Fig. 2

FIG. 1. Feynman diagrams with chiral tensor exchange corre-sponding to the nonlocal four-fermion interaction S4.


055005-4

correspond to the first term �F in Eq. (10), while Figs. 3and 4 reflect the terms �F2.

Our one-loop calculation results in an effective action

� ¼ �2 þ �ðTÞ4 þ �ðVÞ

4 þ �ðSÞ4 ; (11)

with a kinetic term

�2 ¼ �Z d4q

ð2�Þ4 ðZLð�tL��q�tL þ �bL��q�bLÞ

þ ZR �tR��q�tRÞ (12)

and three different types of quartic interactions �4. (Notethe relative minus sign between � and the classical action Swhich is chosen in order to make analytic continuation tothe euclidean effective action straightforward by replacing

q0 ! iq0.) The ‘‘tensorial part,’’ �ðTÞ4 , corresponds to the

exchange of a tensor field and has the form of the classical

interaction S4. The other two parts, �ðVÞ4 and �ðSÞ

4 , corre-

spond to the exchange of vectors and scalars, respectively,and will be further discussed in Sec. III.

The momentum intergals in the loop expansion (10) arelogarithmically divergent, both in the ultraviolet and theinfrared. We regularize our model in the ultraviolet by asuitable momentum cutoff �UV. In order to investigate theflow of effective couplings we also introduce an effectiveinfrared cutoff k. The effective action �k depends then onthe scale k, resulting in an effective coupling fðkÞ. Insteadof the infrared scale induced by nonvanishing externalmomenta for the vertices (as most common for perturbativerenormalization) we introduce the cutoff by modifying thequark propagators q6 �1 ! q6 ðq2 þ k2Þ�1. This is a proce-dure known from functional renormalization. Indeed, �k

may be considered as the ‘‘average action’’ or ‘‘flowingaction’’ [14]. The precise implementation of the infraredcutoff is not important and does not influence the one-loopbeta function for the running coupling fðkÞ that we willderive next.

The fermion anomalous dimensions arise from Eq. (B2)or Fig. 2. Our computation in the purely fermionic modelyields the same result as in Ref. [12], where it was com-puted in the equivalent model with chiral tensors, namely,

�R � �k@

@kZR ¼ � 3

2�2f2;

�L � �k@

@kZL ¼ � 3

4�2f2:

(13)

The opposite sign as compared to a similar diagram for theexchange of the Higgs doublet in the standard model arisesfrom the particular tensor structure of the ‘‘vertices’’ and‘‘propagator’’ of the chiral tensors.Only the terms visualized diagrammatically in Fig. 3,

which are �A2 in Eq. (B3), generate the same tensorstructure as the classical interaction (2). They provide theone-loop correction to the inverse chiron propagator, i.e., tothe matrix Pkl, and one obtains

�ðTÞ4 ¼ �S4ðP

kl ! ZþPklÞ: (14)

Again our fermionic computation reproduces the compu-tation in Ref. [12] with chiral tensors. The correctionresults in an anomalous dimension for the chiron,

�þ � �k@

@kZþ ¼ f2

2�2: (15)

There are no further one-loop corrections in the tensorexchange channel. Among the quartic corrections shownin Fig. 4, the first four terms in Eq. (B3) contribute to aninteraction channel which is equivalent to the exchange ofa vector particle. These diagrams will be evaluated in thenext section. Similarly, the fifth and sixth term in Eq. (B3)

FIG. 3. Feynman diagrams for the one-loop correction to theinteraction in the tensor channel.

FIG. 2. Feynman diagrams generating the quark anomalousdimension.

FIG. 4. One-loop Feynman diagrams for the four-fermion ver-tex which correspond to the effective exchange of scalars andvectors. The second diagram contributes twice, since we maysubstitute ðtL; Bþ0Þ ! ðbL; BþþÞ.


055005-5

contribute to an interaction with a tensor structure differentfrom the classical action (2). It can be interpreted as aneffective scalar exchange and will also be discussed in thenext section.

The running of the renormalized coupling f2 (to one-loop order) is therefore given by the anomalous dimensionsof the fermions and the correction to Pkl,

k@

@kf2 ¼ ð�R þ �L þ �þÞf2 ¼ � 7

4�2f4: (16)

This implies that the interaction is asymptotically free. Thesolution to the renormalization group Eq. (16) is

f2ðkÞ ¼ 4�2

7 lnðk=�chÞ ; (17)

where the ‘‘chiral scale’’ �ch is the scale at which f2=4�becomes much larger than 1. This is completely equivalentto the result of Ref. [12]. The asymptotic freedom of thechiral fermion-tensor interaction is simply taken over tothe nonlocal four-fermion interaction.

IV. INDUCED SCALAR AND VECTORINTERACTIONS

In this section we evaluate the diagrams in Fig. 4, rep-resenting the terms �A1 in Eq. (B3). We begin with thefifth diagram with all external momenta set to zero. (Thecontribution of the sixth diagram is equivalent, with tLsubstituted by bL.) The contribution to � is

��ð1Þ ¼ 16if4Z d4q

ð2�Þ4PklðqÞq4

PmnðqÞq4

��t�mþ

ð�q6 Þq2

�l�t�

��t�n�

q6q2

�kþt�: (18)

This may be rewritten in terms of Weyl spinors

��ð1Þ ¼ 16if4Z d4q

ð2�Þ4PklðqÞq4

PmnðqÞq4

�tyR�m

ð�q6 Þq2

�ltR

�

��tyL�n

�6qq2

�ktL

�; (19)

where now

q6 ¼ q0 þ qi�i; �6q ¼ q0 � qi�

i: (20)

With the identity

PklðqÞP

mnðqÞ½�mq6 �l��½�n �6q�k� �¼ 5q4½q6 ��½ �6q� � þ 4q6�� (21)

this simplifies to

��ð1Þ ¼ 16if4Z d4q

ð2�Þ41

q8ð�5½tyRq6 tR�½tyL �6qtL�

þ 4q2½tcyR tc0L �½tc

0yL tcR�Þ: (22)

The relative minus sign is due to an exchange ofGrassmann variables. Note the different color structure ofthe second term. The momentum integral can be evaluatedby analytic continuation to Euclidean space, q0 ¼ iqE0.Introducing the infrared cutoff in the quark propagators[i.e., substituting q�4 ! ðq2 þ k2Þ�2], one obtains

��ð1Þ ¼ f4

4�2k2ð5g��½tyR��tR�½tyL ��tL� � 16½tcyR tc

0L �½tc

0yL tcR�Þ:(23)

Using the identities

ð��Þ�ð ��Þ � ¼ �2�� (24)

and

�cd0�c0d ¼ 13�cd�c0d0 þ 1

2TzcdT

zc0d0 (25)

(where Tz are the eight Gell-Mann matrices) this can befurther simplified to

��ð1Þ ¼ f4

2�2k2½�tRtL�½�tLtR� þ 3f4

4�2k2½�tcRTz

cdtdL�½�tc0LTz

c0d0 td0R �:(26)

The total contribution to � from the fifth and sixth diagramin Fig. 4 reads

�ðSÞ4 ¼ ��ð1Þ þ��ð1ÞðtL ! bLÞ: (27)

The first term in Eq. (26) is equivalent to the four-fermion interaction generated at tree level by a Yukawainteraction with a scalar field �, which has a mass k and aYukawa coupling �h given by

�h 2 ¼ f4

2�2: (28)

This scalar transforms as a singlet under color and adoublet with respect to the electroweak interactions. Ittherefore has the same quantum numbers as a (composite)Higgs doublet. The second term in Eq. (26) corresponds tothe exchange of a second scalar which is an octet withrespect to color.Finally we evaluate the first four diagrams of Fig. 4,

which generate an interaction equivalent to the exchange ofa vector particle. The expression for the first diagram is

��ð2Þ ¼ 8if4Z d4q

ð2�Þ4PklðqÞq4

PmnðqÞq4

��t�l�

q6q2

�mþt�

��t�n�

q6q2

�kþt�: (29)

This can again be rewritten in terms of Weyl spinors


ð2�Þ4PklðqÞq4

PmnðqÞq4

�tyL�l

�6qq2

�mtL

�

��tyL�n

�6qq2

�ktL

�: (30)


055005-6

With the identities

PklðqÞP

mnðqÞ½�l �6q�m��½�n �6q�k� �¼ 5q4½ �6q��½ �6q� � þ 4q6ð�� Þ (31)

and

ð ��Þ�ð ��Þ � ¼ �2ð�� Þ (32)

the expression (30) simplifies to


ð2�Þ41

q8ð5½tyL �6qtL�½tyL �6qtL�

� 2q2½tyL ��tL�½tyL ��tL�Þ: (33)

With an infrared cutoff k in the quark propagator themomentum integral yields

��ð2Þ ¼ 3f4

8�2k2½�tL��tL�½�tL��tL�: (34)

This is equivalent to the four-fermion interaction generatedat tree level by the exchange of a vector field with mass kand coupling ~g given by

~g 2 ¼ 3f4

8�2: (35)

The evaluation of diagrams 3 and 4 is equivalent, with twoor four external tL fields substituted by bL.

The expression for the second diagram, in terms of Weylspinors, is

��ð3Þ ¼ 16if4Z d4q

ð2�Þ4PklðqÞq4

PmnðqÞq4

�tyR�m

q6q2

�ltR

�

��tyR�k

q6q2

�ntR

�: (36)

Identities similar to Eqs. (31) and (32) result in

��ð3Þ ¼ 3f4

4�2k2½�tR��tR�½�tR��tR�: (37)

The ‘‘vector exchange’’ interaction generated at one-looplevel can be summarized as

�ðVÞ4 ¼ 3f4

8�2k2ð½�tL��tL�½�tL��tL� þ 2½ �bL��tL�½�tL��bL�

þ ½ �bL��bL�½ �bL��bL� þ 2½�tR��tR�½�tR��tR�Þ:(38)

V. ELECTROWEAK SYMMETRY BREAKING

The presence of the effective Yukawa interaction inEq. (26) indicates the possibility of spontaneous symmetrybreaking and an analogue of the Higgs mechanism. In asense, this interaction can be understood as reflecting theexchange of a scalar top-antitop bound state. If we neglectfor k close to �ch all interactions except for the scalar

singlet exchange channel in the first term of Eq. (26), wemay characterize the effective action by a ‘‘scalar four-fermion coupling’’ ðkÞ,

�ðSÞk ¼

2

Zd4x½ð �c c Þ2 � ð �c�5c Þ2�

¼ 2 ð �c Lc RÞð �c Rc LÞ (39)

with

ðkÞ ¼ f4ðkÞ4�2k2

: (40)

In the limit where the momentum dependence of the scalarfour-fermion interactions can be neglected this can beinterpreted as an effective NJL model. Here the role ofthe UV cutoff in the effective NJL model is played by the

scale k, since the computation of �ðSÞk has involved only

fluctuations with momenta q2 > k2 (due to the infraredcutoff) such that the remaining fluctuations with q2 < k2

still have to be included. It is well known that for k2

exceeding a critical value the NJL model leads to sponta-neous symmetry breaking.For a rough estimate of the top quark mass mt induced

by spontaneous electroweak symmetry breaking we con-sider the Schwinger-Dyson equation

��q� þmk�5 ¼ ��q� þ 2i ðkÞ

Zp2<k2

d4p

ð2�Þ4

� ��p� � 6mk�5

p2 þm2k

: (41)

Setting q ¼ 0 and performing the momentum integralyields the gap equation for the top quark mass mk

mk ¼ 3 ðkÞ4�2

�k2 �m2

k lnk2 þm2

k

m2k

�mk: (42)

We have indicated the scale k for mk in order to recall thatthe solution of the gap equation will depend on the choiceof the scale k which we use as an UV cutoff for theeffective NJL model. Of course, for an exact treatmentthe physical top quark mass should no longer depend on k.Let us next discuss the solution of Eq. (42). Obviously,

mk ¼ 0 is always one possible solution. The expression inbrackets is always k2. If we write

ðkÞ ¼ ðkÞk2; (43)

we see that nonzero solutions for mk (indicating sponta-neous symmetry breaking) occur for

ðkÞ> 4�2

3: (44)

Once the condition (44) is obeyed, one finds indeed anonzero mk obeying


055005-7

m2k

k2ln

�1þ k2

m2k

�¼ 1� 4�2

3ðkÞ : (45)

Since ðkÞ � f4ðkÞ grows arbitrarily large as k approaches�ch and fðkÞ diverges, the condition (44) is always fulfilledfor k sufficiently close to �ch.

For a qualitative investigation we first use the one-loop

result ¼ f4=4�2 and replace in Eq. (17) k !ðk2 þ c2m2

kÞ1=2 with c of order 1. This is motivated by

the effective infrared cutoff �mk which stops or slowsdown the running of f2. In Fig. 5 we show mk=�ch as afunction of k=�ch for c ¼ 1. After the onset of a nonzeromk for k � �ch we find first a very rapid increase until mk

settles at mk ¼ �ch=c for k ! 0. It is obvious that�ch setsthe scale for the top quark mass. This coincides with theresult of a two-loop Schwinger-Dyson equation in a for-mulation with chiral tensor fields in [12]. Indeed, since thegeneration of is a one-loop effect, and the gap equa-tion (41) involves a further loop, the generation of the topquark mass consists of two nested one-loop integrals,which are equivalent to a two-loop integral. The solutionof Eq. (45) for k ! 0, mk � 0 corresponds to an asymp-totic value which is obtained from the conditionðmk; k !0Þ ! 1.

As an alternative to the ad hoc insertion of the quarkmass cutoff in Eq. (17) we may take into account theadditional infrared cutoff due to mk by replacing in thequark propagator q6 �1 ! q6 ðq2 þ k2 þm2

kÞ�1. As a conse-

quence, the anomalous dimensions involve a thresholdfunction sðm2

k=k2Þ,

�L ¼ � 3

4�2f2sðm2

k=k2Þ; �R ¼ � 3

2�2f2sðm2

k=k2Þ;

�þ ¼ 1

2�2f2sðm2

k=k2Þ; (46)

given by

sðm2k=k

2Þ ¼ k2

k2 þm2k

: (47)

The one-loop expression for (40) can be replaced by aflow equation

k@

@k ¼ � f4

2�2

k2

ðk2 þm2kÞ2

þ ð�L þ �RÞ : (48)

[For mk ¼ 0, �L;R ¼ 0, and constant f this reproduces

Eq. (40).] Nonzero mk results in a threshold function forthe running of ,

k@

@k ¼ ð2þ �L þ �RÞ� f4

2�2~sðm2

k=k2Þ;

~sðm2k=k

2Þ ¼ k4

ðk2 þm2kÞ2

: (49)

We show the running of f2 and in Fig. 6, for differentvalues of m2

k=�2ch. Again we stop the flow at some value of

k ¼ �SDEUV and solve the Schwinger-Dyson equation (SDE)

with this UV cutoff. The value of k for which the SDEyields the givenm2

k=�2ch is indicated in Fig. 6 by a dot. The

dots in Fig. 5 show the corresponding k dependence ofmk=�ch.

0.2 0.4 0.6 0.8 1.0

k

ch

0.2

0.4

0.6

0.8

1.0

mk

ch

FIG. 5 (color online). Top quark mass as a function of the UVcutoff k in the NJL approximation.

0.5 1.0 1.5 2.0

k

ch

50

100

150

200

250

300

350f2

0.5 1.0 1.5 2.0

k

ch

20

40

60

80

100

120

FIG. 6 (color online). Running of f2 (red/upper curves) and (blue/lower curves). Solid (dashed, dotted) lines correspond tom ¼ 0:3 (0.6, 0.9) �ch.


055005-8

We are aware that our treatment of the infrared cutoff isonly qualitative. While the one-loop form of the flowequation can be maintained if we interpret the flow as anapproximation to the exact functional renormalizationgroup equation for the average action [14], the approxima-tion of the exact inverse quark and chiron propagators byZL;Rq6 , and ZþPklðqÞ with momentum independent Z fac-

tors becomes questionable in the presence of large anoma-lous dimensions.

VI. COMPOSITE HIGGS SCALAR

An elegant method for the description of compositeparticles in the context of functional renormalization usespartial bosonization [15]. It is based on the observation thatan interaction of the type (39) can be described by theexchange of a composite scalar field. Indeed, one may addto the flowing action �k a scalar piece, with �’ a complexdoublet scalar field

�ðSÞk ¼

ZxfZ’@

� �’y@� �’þ �m2’ �’y �’

þ �hð �c R �’yc L � �c L �’c RÞg: (50)

‘‘Integrating out’’ the scalar field by solving its field equa-tion as a functional of c and reinserting the solution intoEq. (50) yields Eq. (39), with dependent on the squaredexchanged momentum in the scalar channel

ðqÞ ¼�h2

2ðZ’q2 þ �m2

’Þ: (51)

As long as the momentum dependence of the effectivescalar exchange vertex is not resolved (as in our computa-tion where the vertex is only evaluated for q2 ¼ 0), one

may take arbitrary Z’. We will therefore replace �ðSÞk in

Eq. (39) by Eq. (50), and replace the flow of in Eq. (48)by

k@

@k�h2 ¼ � f4

�2~sðm2

t =k2Þ �m2

’

k2þ 2 �h2: (52)

At this stage our reformulation precisely reproduces theresults in Sec. IV. The rule for the replacement of the flowof ðqÞ by a running of the renormalized Yukawa couplingand dimensionless mass term

h2 ¼�h2

Z’

; ~m2’ ¼ �m2

’=ðZ’k2Þ (53)

is to adjust the flow of h2 and ~m2’ such that the flow of ðqÞ

is reproduced, with

ðqÞ ¼ h2

2k2

�q2

k2þ ~m2

’

��1: (54)

For the approximately scale invariant situation for smallcoupling one expects that the relative split intoq2-dependent and q2-independent parts does not depend

much on k. We therefore make the approximation that theflow of ~m2

’ receives no contribution from bosonization,

such that

@k ¼ @kðh2=k2Þ=ð2 ~m2’Þ (55)

or

k@kh2 � 2h2 ¼ 2 ~m2

’k3@k

¼ � f4

�2~sðm2

t =k2Þ ~m2

’ þ ð�L þ �RÞh2: (56)

The effective initial value ~m2’ð�Þ can be computed by

evaluating the momentum dependence of the four-fermioninteraction in the scalar channel. At present, it remains afree parameter. In a more complete calculation one shouldalso evaluate the diagrams in Fig. 4 for nonzero externalmomenta and choose the flow of h2 and ~m2

’ such that the

flow of the vertex ðqÞ is well approximated by Eq. (54).We expect that the resulting flow for ~m2

’ will be attracted to

an approximate fixed point. We note that for a fixed (point)value of ~m2

’ the mass term �m2’ decreases roughly �k2.

The formulation in terms of a composite scalar fieldallows an inclusion of the scalar field fluctuations aswell. This adds new diagrams, which can be viewed as apartial resummation of the box diagrams in Fig. 4. Inparticular, the quark wave function renormalizations ZL;R

get additional contributions from the scalar exchange dia-grams in Fig. 7 (with scalars represented as double lines),as well known from computations in the standard model. Inconsequence, the anomalous dimensions acquire a scalarcontribution with opposite sign to the tensor contribution.

�L ¼ � 3f2

4�2sð ~m2

t Þ þ h2

16�2s’ð ~m2

’; ~m2t Þ;

�R ¼ � 3f2

2�2sð ~m2

t Þ þ h2

8�2s’ð ~m2

’; ~m2t Þ;

(57)

with ~m2t ¼ m2

t =k2 and

s’ð ~m2’; ~m

2t Þ ¼

1þ ln½ð1þ ~m2t þ ~m2

’Þ=ð1þ ~m2t Þ�

1þ ~m2t þ ~m2

’

: (58)

FIG. 7. Contribution of the Yukawa coupling to the quarkanomalous dimensions. The corresponding diagram in terms ofchiral tensor exchange is shown in (b). It is obtained from thebox diagram in Fig. 4 by contracting the tR line, in analogy toFig. 7(a).


055005-9

For ~mt ¼ 0 the scalar correction to �L;R becomes approxi-

mately ��L;R � h2= ~m2’ � f4. As long as f2 remains

small, the effective two-loop contribution correspondingto scalar exchange [cf. Figure 7(b)] remains subleading.However, for large f2 the scalar contribution to �L;R may

become important and modify the anomalous dimensionstowards positive values. This could contribute to a finalstop of the increase of f2 which obeys now

k@

@kf2 ¼ ð�R þ �L þ �þÞf2; (59)

with �L;R given by Eq. (57).

We next turn to the scalar contributions to the running of�m2’ and Z’. The fermion loop correction to the inverse

scalar propagator, as depicted in Fig. 8, results in thestandard result for the anomalous dimension of the scalarfield,

k@

@kZ’ ¼ � 3

8�2�h2sð ~m2

t Þ;

�’ ¼ �k@

@klnZ’ ¼ 3

8�2h2sð ~m2

t Þ:(60)

Because of the Yukawa coupling �h2 a positive nonvanish-ing Z’ is generated, even if it is absent at somemicroscopic

scale. This produces a pole in the scalar propagator forq2 ¼ �m2

’ ¼ � �m2’=Z’, such that the scalar behaves as a

dynamical particle. The contribution of the quark loopshown in Fig. 8 to the flow of the scalar mass term is�h2. Within functional renormalization it has been inves-tigated in [16] and one finds with our cutoff in the fermionpropagator

k@

@k~m2’ ¼ 3

2�2h2sð ~m2

t Þ;

sð ~m2t Þ ¼ ln

~�2

k2� 1� lnð1þ ~m2

t Þ:(61)

The momentum integral for the contribution of Fig. 8 to the

flow of �m2’ has been cut at some effective scale ~�. Here ~�

is a characteristic scale below which the description of theflow in terms of scalar fluctuations becomes a reasonableapproximation. This should be somewhat above �ch, butthe precise value remains somewhat arbitrary without addi-tional computations. (In any case the use of an improved

infrared cutoff within functional renormalization wouldremove this spurious dependence on a scale.)For sufficiently large h2 the flow (61) drives ~m2

’ to

negative values, indicating the onset of spontaneous sym-metry breaking. This is the same physics that is responsiblefor the nontrivial solutions of the Schwinger-Dyson equa-tion for mt in the preceding section. Indeed, for a qualita-tive picture we can take for k < k0 a constant �h

2 and s, andneglect �’ as well as the contribution from bosonization.

This replaces Eq. (61) by

k@

@k�m2’ ¼ 3s �h2

2�2k2: (62)

For the solution one finds the critical value �h2c ¼ð4�2=3sÞ �m2

’ðk0Þ=k20 for which �m2’ reaches zero for k !

0. Inserting s ¼ 12 and using

¼�h2k20

2 �m2’ðk0Þ

; (63)

this indeed corresponds to c ¼ 4�2=3. The vacuum ex-

pectation value ’0 ¼ Z1=2’ �’0, with �’0 the location of the

minimum of the scalar effective potential, differs from zeroif �m2

’ gets negative. For the computation of its value, which

should be ’0 ¼ 175 GeV in a realistic model, one furtherneeds to compute the quartic scalar self-interaction ’. We

can adjust the value of �ch [or the ultraviolet valuef2ð�UVÞ] such that the Fermi scale ’0 has the correctvalue.A particularly interesting quantity is the renormalized

Yukawa coupling h (53). Its value for k ¼ 0 determines thetop quark mass in terms of the Fermi scale

mt ¼ hðk ¼ 0Þ’0 ¼ ht’0: (64)

In other words, the knowledge of ht ¼ hðk ¼ 0Þ is equiva-lent to a determination of the mass ratio mt=mW , where we

use mW ¼ ðgW=ffiffiffi2

p Þ’0, with gW the known weak gaugecoupling (g2W=4� � 0:033). The observational value isht ¼ 0:98. While the scale ’0 is set by dimensional trans-mutation and therefore a free parameter, a computation ofht is equivalent to a parameter-free ‘‘pre’’-diction for themass ratio mt=mW .In our approximation we can infer the flow equation for

h2 from Eq. (52),

k@

@kh2 ¼ ð2þ�L þ�R þ�’Þh2 � f4

�2~m2’~sð ~m2

t Þ

¼ 2h2 � 9

4�2f2h2sð ~m2

t Þ

þ 3

16�2h4½2sð ~m2

t Þþ s’ð ~m2’; ~m

2t Þ�� f4

�2~m2’~sð ~m2

t Þ:(65)

We have solved the flow equations numerically until theonset of spontaneous symmetry breaking at kSSB where�m2’ðkSSBÞ ¼ 0. For k > kSSB one has ~mt ¼ 0 such that

FIG. 8. Quark loop generating the anomalous dimension of thescalar field and contributing to the flow of its mass term.


055005-10

many threshold functions equal one. We display the run-ning of h and f in Fig. 9. For k < kSSB one has to continuethe flow in the regime with spontaneous symmetry break-ing (SSB) and nonzero ’0ðkÞ, as well known from manystudies of functional flow equations [16,17]. Wewill not doso here.

In the present approximation to the flow equations weobserve an increase of f and h to very large values as m2

’

approaches zero. We do not expect our approximations toremain valid for large couplings, even though the one-loopform of the functional flow equations is exact [14]. One ofthe main shortcomings is the inaccurate truncation of thegeneral form of momentum dependence for the fermionpropagators and nonlocal interactions in a region where theanomalous dimensions �L;R;þ are of the order 1. For

example, the quartic interactions for the bare quarks inthe tensor channel replaces in our approximation P

kl=q4 !

Pkl=ðZþq4Þ in Eq. (2). We may consider an effective

momentum dependence of Zþ given by the anomalous

dimension �þðq2Þ evaluated for k ¼ ffiffiffiffiffiq2

p, i.e.,

Zþ � ðq2Þ�2�þðq2Þ—this would indeed be a valid approxi-mation for small j�þj � 1 and q2 � k2. However, for�þ ¼ 1 the momentum dependence of Zþ would effec-tively cancel the nonlocality �1=q2 of the interaction. Fora quasilocal interaction in the tensor channel we expect

strong modifications of the flow of f—for example, itcould reach a fixed point f, replacing in the interactionf2=q2 ! f2=ðc�2

chÞ. (In the language of chiral tensor fieldsthis would correspond to the generation of a nonlocal massfor the ‘‘chirons’’ [13].)For a realistic model of electroweak interactions the

increase of the Yukawa coupling should stop or be sub-stantially slowed down in the vicinity of its final value atk ¼ 0, say for 1 & h & 2. For the corresponding region inthe scale k, 1:1< k=�ch < 1:4 we find values 0:8<f2=ð2�2Þ ¼ �þ < 2:4. [We use ~m2

’ð�Þ ¼ 0:1 for the quan-

titative estimates.] In view of our discussion, it seems notunreasonable that the nonperturbative infrared physicsstops the further increase of f and h in this range. If thishappens, the ratio mt=mW may come out in a reasonablerange. A typical value of the scale for nonperturbativephysics where the increase of f and h stops, may be knp ¼1:3�ch. At this scale the quantity [cf. Eq. (63) with k0 ¼knp] has reached a value ðknpÞ � 10, not too far from the

critical value in the Schwinger-Dyson approach. It is wellconceivable that the top quark mass is substantially belowthe scale knp, such that the region of strong interactions

may correspond to the multi-TeV scales and not disturb toomuch the LEP precision tests of the electroweak theory.Our computation of the flow of f and the Yukawa

coupling h has further substantial quantitative uncertain-ties. For example, we have neglected effects from theinteractions in Sec. III that correspond to the exchange ofcolor-octet scalars or vectors. This may be motivated forthe region of k where m2

’ is already small, since a reso-

nance type behavior and spontaneous symmetry breakingis only expected in the scalar singlet sector. On the otherhand, these contributions may still play a role in theinteresting region where the increase of f may stop.

VII. CONCLUSIONS

We conclude that models for quarks and leptons with anonlocal four-fermion interaction in the tensor channelappear to be promising candidates for an understandingof the electroweak symmetry breaking. No fundamentalHiggs scalar is needed. The theory is asymptotically freeand generates an exponentially small ‘‘chiral scale’’ �ch

where the dimensionless coupling f grows large.Furthermore, a strong interaction in the scalar channel isgenerated at scales where f is large. A solution ofSchwinger-Dyson equations suggests that this interactiontriggers the spontaneous breaking of the electroweak sym-metry at a scale determined by �ch. This would solve thegauge hierarchy problem.We have also introduced a composite Higgs scalar and

investigated the flow of its mass. We find indeed sponta-neous symmetry breaking with a Fermi scale somewhatbelow the chiral scale. Our first attempt of an estimate ofthe Yukawa coupling of the top quark is encouraging,yielding a reasonable range for the ratio mt=mW .

1.0 1.5 2.0 2.5 3.0 3.5 4.0

k

ch

2

4

6

8

10

1.5 2.0 2.5 3.0 3.5 4.0

k

ch

0.3

0.2

0.1

0.1

0.2

0.3

FIG. 9 (color online). Running of f (red curves), h (bluecurves in the upper picture) and ~m2

’ (lower picture). At large

values of k we started with ~m2’ ¼ 0:1 (solid lines) and ~m2

’ ¼ 0:3

(dashed lines).


055005-11

The model has also other interesting features. It wasshown [12] that the flavor and CP violation is completelydescribed by the Cabibbo-Kobayashi-Maskawa matrix[18]. Masses of the light quarks and leptons arise fromappropriate four-fermion couplings [19]. First phenome-nological constraints from LEP precision tests and theanomalous magnetic moment of the muon have been com-puted [12]. The details of possible experimental conse-quences depend mainly on two issues. The first concernsthe spectrum and interactions of the ‘‘low energy boundstates.’’ One typically expects two strongly interactingcomposite scalar doublets, perhaps with masses in therange around 500 GeV. There could be also other boundstates. The second issue concerns the precise ratio betweenthe Fermi scale h’i or theW-boson mass, and the scale�ct

where the long range character of the interactions in thetensor channel is effectively cut off. Indeed, there seems tobe no reason that the strength of the tensor interactionsdiverges �q�2 as the exchanged squared momentum q2

goes to zero. Because of strong interaction effects werather expect that the tensor interactions reach a maximalstrength��2

ct . Both h’i and �ct are parametrically of theorder�ch, but there may be a substantial factor in between.A first phenomenological investigation suggests that �ct

has to be in the multi-TeV range [12]. The quantitativeestimates depend, however, on other details of the non-perturbative physics of the strong tensor interactions thatare not settled at the present stage.

At the present stage the understanding of the stronginteractions around the scale �ch, which would be a fewTeV in a realistic model, is still in its infancy. For thisreason our estimate of mt=mW is only very crude.Nevertheless, no free parameter enters in the determinationof this ratio in our model. If the strong interactions can beunderstood quantitatively, our model leads to a unique‘‘pre’’-diction of mt=mW . It would also predict the massesand interactions of the composite scalar fields.

APPENDIX A: NONLOCAL FOUR-FERMIONINTERACTION AND CHIRAL TENSOR FIELDS

In this appendix we show the equivalence of the micro-scopic action with a theory of chiral tensor fields[12,13,20–22]. This is similar to the equivalence of quan-tum electrodynamics to a theory with a nonlocal four-fermion interaction, generated by integrating out the pho-ton. We may start from the nonlocal four-fermion interac-tion and obtain the chiral tensor theory by a Hubbard-Stratonovich transformation. We proceed here in the oppo-site way, starting with a chiral tensor theory and integratingout the chiral tensors.

The starting point is the theory of chiral antisymmetrictensor fields investigated in Ref. [12]. We consider a com-plex antisymmetric tensor field �� ¼ �� which is a

doublet of weak isospin and carries hypercharge Y ¼ 1.The field can be decomposed into two parts which corre-

spond to irreducible representations of the Lorentz group.The two parts are the ‘‘chiral’’ components of ��:

�� ¼ 1

2�� i

4��

��: (A1)

These components can be written as 4� 4 matrices actingin the space of Dirac spinors via

� ¼ 12�

��

��; (A2)

where �� ¼ i2 ½��; ��. The matrix �þ (��) acts only on

left-handed (right-handed) fermions, i.e.,

� ¼ �1 �5

2: (A3)

One may introduce an interaction between the fermionsand the chiral tensors:

�Lch ¼ �uRFU~�þqL � �qLF

yU�~�þuR þ �dRFD

��qL

� �qLFyD��dR þ �eRFL

��lL � �lLFyL��eR:

(A4)

Here the chiral couplings FU;D;L are 3� 3 matrices in

generation space, qL are the left-handed quark doublets,uR (dR) are the right-handed up-type (down-type) quarks,lL are the left-handed lepton doublets, eR are the right-handed electron-type leptons, and we defined

�� ¼ 1

2ð�

��Þ�� ¼ ��1� �5

2;

~�þ ¼ �i�Tþ�2;�~�þ ¼ ��2 ��þ;

(A5)

where the transposition �T and the Pauli matrix �2 act inweak isospin space, i.e., on the two components of theweak doublet �þ

��.

The fields �� can be represented by three-vectors B

k ,

�þjk ¼ �jklB

þl ; �þ

0k ¼ iBþk ;

��jk ¼ �jklB

�l ; ��

0k ¼ �iB�k :

(A6)

Rewriting the kinetic term

�Lkin ¼ 14ðð@��Þ@�� 4ð@��Þ@�

�Þ (A7)

in terms of the B fields and in Fourier space gives

� Skin ¼Z d4q

ð2�Þ4 ½ðBþk ÞðqÞPklðqÞBþ

l ðqÞ þ ðB�k Þ

� ðqÞPklðqÞB�

l ðqÞ�: (A8)

The propagator

Pkl ¼ �ðq20 þ qjqjÞ�kl þ 2qkql � 2i�kljq0qj (A9)

has the properties

PklPlj ¼ q4�kj; P

klðqÞ ¼ PlkðqÞ: (A10)

In the following we ignore for simplicity all couplingsexcept f � ft, i.e., we assume


055005-12

FU ¼f 0 00 0 00 0 0

0@

1A; FD ¼ FL ¼ 0: (A11)

In this case, only the Bþ fields couple to top quarks t andleft-handed bottom quarks b. All other fields are free if weignore gauge couplings. We denote the two weak isospincomponents of Bþ as Bþþ and Bþ0, since they correspondto electric charge þ1 and 0 after electroweak symmetrybreaking. The action for the chiral interactions is then

� Sch ¼ 2fZ

d4x½�Bþ0k

�t�kþtþ Bþþk

�t�kþb

þ ðBþ0k Þ �t�k�t� ðBþþ

k Þ �b�k�t�: (A12)

In a spinor basis in which

�0 ¼ �i0 1212 0

� �; �i ¼ �i

0 �i

��i 0

� �;

�5 ¼ �i�0�1�2�3 ¼ 12 00 �12

� � (A13)

the matrices �k are defined in terms of the Pauli matrices�k as

�kþ ¼ �k 00 0

� �; �k� ¼ 0 0

0 �k

� �: (A14)

In the action�Sch, a summation over quark color is under-

stood. The classical field equations are obtained by varyingSB � Skin þ Sch with respect to the Bþ fields. They deter-mine these fields as functionals of the quark fields. Inmomentum space, these relations are

Bþ0k ðqÞ ¼ �2f

PklðqÞq4

Z d4k

ð2�Þ4 �tðkÞ�l�tðkþ qÞ;

ðBþ0k ÞðqÞ ¼ 2f

PklðqÞq4

Z d4k

ð2�Þ4 �tðkÞ�lþtðk� qÞ;

Bþþk ðqÞ ¼ 2f

PklðqÞq4

Z d4k

ð2�Þ4�bðkÞ�l�tðkþ qÞ;

ðBþþk ÞðqÞ ¼ �2f

PklðqÞq4

Z d4k

ð2�Þ4 �tðkÞ�lþtðk� qÞ:

(A15)

We then insert these relations into SB and obtain the actionS4 of the nonlocal four-fermion interaction given inEq. (2). In a functional integral formulation this procedureis equivalent to performing the Gaussian integral over theBþ fields.

APPENDIX B: ONE-LOOP EXPRESSIONS FORTHE EFFECTIVE ACTION

In this appendix we evaluate the interaction contribution

F to the inverse propagator Sð2Þ in Eq. (8) for the different

quark types separately, (Sð2Þ ¼ P0 þ F). One obtains

Fcc0�tt;�ðp; p0Þ ¼ 4f2

Z d4q

ð2�Þ4��tc

0� ðqþ p0Þtc�ðqþ pÞ½�kþ��

l�� þ �l��kþ��

��P

klðqÞq4

�

þ �cc0 �tc00� ðqþ p0Þtc00� ðqþ pÞ½�kþ��

l�� þ �l��kþ��

Pklðp0 � pÞðp0 � pÞ4

þ �bc0� ðqþ p0Þbc�ðqþ pÞ�kþ��

l��

��P

klðqÞq4

��;

Fcc0�bb;�

ðp; p0Þ ¼ �4f2Z d4q

ð2�Þ4 �tc0� ðqþ p0Þtc�ðqþ pÞ�l��

kþ��

PklðqÞq4

;

Fcc0�tb;�ðp; p0Þ ¼ 4f2

Z d4q

ð2�Þ4 �cc0 ½ �bðqþ p0Þ�l�tðqþ pÞ��kþ�

Pklðp0 � pÞðp0 � pÞ4 ;

Fcc0�bt;�

ðp; p0Þ ¼ 4f2Z d4q

ð2�Þ4 �cc0 ½�tðqþ p0Þ�l�bðqþ pÞ��k��

Pklðp0 � pÞðp0 � pÞ4 ;

Fcc0�t �t;�ðp; p0Þ ¼ 4f2

Z d4q

ð2�Þ4 tc0� ðp0 � qÞtc�ðpþ qÞ½�k

þ��l�� kþ��

l��

PklðqÞq4

;

Fcc0tt;�ðp; p0Þ ¼ 4f2

Z d4q

ð2�Þ4 �tc0� ðp0 � qÞ�tc�ðpþ qÞ½�kþ��

l�� kþ��l��

PklðqÞq4

;

Fcc0�t �b;�

ðp; p0Þ ¼ �4f2Z d4q

ð2�Þ4 tc0� ðp0 � qÞtc�ðpþ qÞ�kþ��

l��

PklðqÞq4

;

Fcc0tb;�ðp; p0Þ ¼ 4f2

Z d4q

ð2�Þ4 �tc0� ðp0 � qÞ�tc�ðpþ qÞ�k

þ��l��

PklðqÞq4

;

Fcc0BA;�ðp; p0Þ ¼ �Fc0c

AB;�ðp0; pÞ: (B1)


055005-13

Inserting this into the expression (10), we find an anomalous dimension term for t and for the left-handed b

i

2Tr

�1

P0

F

�¼ 4if2

Z d4pd4q

ð2�Þ8p�

p

Pklðq� pÞðq� pÞ4 f�tðqÞ�l��kþtðqÞ þ 2�tðqÞ�kþ��l�tðqÞ þ �bðqÞ�l��kþbðqÞg: (B2)

The term �F2 in Eq. (10) produces an effective four-fermion interaction

� i

4Tr

�1

P0

F1

P0

F

�¼ 8if4

Z d4pd4p0d4qd4q0

ð2�Þ16p�

p2

p0�

p02

��PklðqÞq4

Pmnðq0Þq04

A1 � 6Pklðp� p0Þðp� p0Þ4

Pmnðp� p0Þðp� p0Þ4 trð��kþ��n�ÞA2

�; (B3)

with

A1 ¼ ð�tðq0 þ pÞ�n��kþtðqþ pÞÞð�tðqþ p0Þ�l��mþtðq0 þ p0ÞÞ þ 2ð�tðq0 þ pÞ�mþ��l�tðqþ pÞÞ� ð�tðqþ p0Þ�kþ��n�tðq0 þ p0ÞÞ þ 2ð �bðq0 þ pÞ�n��kþtðqþ pÞÞð�tðqþ p0Þ�l��mþbðq0 þ p0ÞÞþ ð �bðq0 þ pÞ�n��kþbðqþ pÞÞð �bðqþ p0Þ�l��mþbðq0 þ p0ÞÞ þ 2ð�tðpþ qÞ�kþ��n�tðpþ q0ÞÞ� ð�tðp0 � qÞ�l��mþtðp0 � q0ÞÞ þ 2ð�tðpþ qÞ�kþ��n�tðpþ q0ÞÞð �bðp0 � qÞ�l��mþbðp0 � q0ÞÞ (B4)

and

A2 ¼ ð�tðqþ p0Þ�l�tðqþ pÞÞð�tðq0 þ pÞ�mþtðq0 þ p0ÞÞ þ ð �bðqþ p0Þ�l�tðqþ pÞÞð�tðq0 þ pÞ�mþbðq0 þ p0ÞÞ: (B5)

Color indices are suppressed, since all pairs of fermions in large brackets are color singlets.

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Asymptotically free four-fermion interactions and electroweak symmetry breaking