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PHYSICAL REVIEW D, VOLUME 60, 114035

Effective electroweak chiral Lagrangian: The matter sector

E. Bagan*Grup de Fı´sica Teo`rica and IFAE, Universitat Auto`noma de Barcelona, E-08193 Bellaterra, Spain

D. Espriu† and J. Manzano‡

Departament d’Estructura i Constituents de la Mate`ria and IFAE, Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Spa~Received 28 December 1998; published 12 November 1999!

We parametrize in a model-independent way possible departures from the minimal standard model predic-tions in the matter sector. We only assume the symmetry breaking pattern of the standard model and that newparticles are sufficiently heavy so that the symmetry is nonlinearly realized. Models with dynamical symmetrybreaking are generically of this type. We review in effective theory language to what extent the simplestmodels of dynamical breaking are actually constrained and the assumptions going into the comparison withexperiment. Dynamical symmetry breaking models can be approximated at intermediate energies by four-fermion operators. We present a complete classification of the latter when new particles appear in the usualrepresentations of the SU(2)L3SU(3)c group as well as a partial classification in the general case. We discussthe accuracy of the four-fermion description by matching to a simple ‘‘fundamental’’ theory. The coefficientsof the effective Lagrangian in the matter sector for dynamical symmetry breaking models~expressed in termsof the coefficients of the four-quark operators! are then compared to those of models with elementary scalars~such as the minimal standard model!. Contrary to a somewhat widespread belief, we see that the sign of thevertex corrections is not fixed in dynamical symmetry breaking models. This work provides the theoreticaltools required to analyze, in a rather general setting, constraints on the matter sector of the standard model.@S0556-2821~99!05621-0#

PACS number~s!: 12.39.Fe, 14.80.Bn, 14.80.Cp

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I. INTRODUCTION

The standard model of electroweak interactions hasnow been impressively tested to the 1 part in 1000 lethanks to the formidable experimental work carried out atCERN e1e2 collider LEP and SLAC Linear collider~SLC!.However, when it comes to the symmetry breaking mecnism, clouds remain in this otherwise bright horizon.

In the minimal version of the standard model of eletroweak interactions the same mechanism~a one-doubletcomplex scalar field! gives masses simultaneously to theWandZ gauge bosons and to the fermionic matter fields~otherthan the neutrino!. In the simplest minimal standard modthere is an upper bound onMH dictated by triviality consid-erations, which hint at the fact that at a scale;1 TeV newinteractions should appear if the Higgs particle is not fouby then @1#. On the other hand, in the minimal standamodel it is completely unnatural to have a light Higgs pticle since its mass is not protected by any symmetry.

This contradiction is solved by supersymmetric extesions of the standard model, where essentially the same smetry breaking mechanism is at work, although the scasector becomes much richer in this case. Relatively light slars are preferred. In fact, if supersymmetry is to remaiuseful idea in phenomenology, it is crucial that the Higparticle be found with a massMH<125 GeV, or else thetheoretical problems, for which supersymmetry was invok

*Email address: bagan@ifae.es†Email address: espriu@ecm.ub.es‡Email address: manzano@ecm.ub.es

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in the first place, will reappear@2#. A very recent two-loopcalculation@3# raises this limit somewhat, to about 130 Ge

A third possibility is the one provided by models of dynamical symmetry breaking@such as technicolor~TC! theo-ries@4##. Here there are interactions that become strong, tycally at the scaleLx.4pv (v5250 GeV), breaking theglobal SU~2!L3SU~2!R symmetry to its diagonal subgrouSU(2)V and producing Goldstone bosons which eventuabecome the longitudinal degrees of freedom of theW6 andZ. In order to transmit this symmetry breaking to ordinamatter fields, one usually requires additional interactiocharacterized by a different scaleM. Generally, it is assumedthat M@4pv, to keep possible flavor-changing neutral curent ~FCNC! under control@5#. Thus a distinctive characteristic of these models is that the mechanism giving massethe W6 andZ bosons and to the matter fields is different.

Where do we stand at present? Some will go as farsaying that an elementary Higgs particle~supersymmetric orotherwise! has been ‘‘seen’’ through radiative correctionand that its mass is below 200 GeV. Others dispute this~see, for instance,@6# for a critical review of current claimsof a light Higgs boson!.

The effective Lagrangian approach has proved remaably useful in setting very stringent bounds on some typenew physics, taking as input basically the LEP@7# @andSLAC Large Detector~SLD! @8## experimental results. Theidea is to consider the most general Lagrangian whichscribes the interactions between the gauge sector andGoldstone bosons appearing after the SU~2!L3SU~2!R→SU~2!V breaking takes place. No special mechanismassumed for this breaking, and thus the procedure is cpletely general, assuming of course that particles not exp

©1999 The American Physical Society35-1

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E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

itly included in the effective Lagrangian are much heavthan those appearing in it. The dependence on the spemodel is contained in the coefficients of higher-dimensiooperators. So far only the oblique corrections have beenlyzed in this way.

Our purpose in this work is to extend these techniquethe matter sector of the standard model. We shall writeleading nonuniversal operators, determine how their coecients affect different physical observables, and then demine their value in two very general families of moels: those containing elementary scalars and those withnamical symmetry breaking. Since the latter become nonturbative at theMZ scale, effective Lagrangian techniquare called for anyway. In short, we would like to provide ttheoretical tools required to test—at least in principlewhether the mechanism giving masses to quarks and feons is the same as that which makes the intermediate vebosons massive or not without having to get involved inspecific details of particular models. This is mostly a theretical paper, and we shall leave for a later work a mdetailed comparison with the current data.

II. EFFECTIVE LAGRANGIAN APPROACH

Let us start by briefly recalling the salient features of teffective Lagrangian analysis of the oblique corrections.

Including only those operators which are relevant for olique corrections, the effective Lagrangian reads~see, e.g.,@9,10# for the complete Lagrangian!

Leff5v2

4trDmUDmU†1a0g82

v2

4~ trTDmUU†!2

1a1gg8trUBmnU†Wmn2a8

g2

4~ trTWmn!2, ~1!

whereU5exp(itW•xW /v) contains the three Goldstone bosogenerated after the breaking of the global symmetry SU~2!L3SU~2!R→SU~2!V . The covariant derivative is defined by

DmU5]mU1 igtW

2•WW mU2 ig8U

t3

2Bm . ~2!

Bmn andWmn are the field-strength tensors correspondingthe right and left gauge groups, respectively,

Wmn5tW

2•WW mn , Bmn5

t3

2~]mBn2]nBm!, ~3!

and T5Ut3U†. Only terms up to orderO(p4) have beenincluded. The reason is that dimensional counting argumsuppress, at presently accessible energies, higdimensional terms, under the hypothesis that all undeteparticles are much heavier than those included in the eftive Lagrangian. While the first term on the right-hand si~RHS! of Eq. ~1! is universal~in the unitary gauge it is justhe mass term for theW6 andZ bosons!, the coefficientsa0 ,a1 , anda8 are nonuniversal. In other words, they dependthe specific mechanism responsible for the symmetry breing. @Throughout this paper the term ‘‘universal’’ mean

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‘‘independent of the specific mechanism triggering SU~2!L3SU~2!R→SU~2!V breaking.’’#

Most Z-physics observables relevant for electrowephysics can be parametrized in terms of vector and acouplingsgV and gA . These are, in practice, flavor depedent since they include vertex corrections which dependthe specific final state. Oblique corrections are, however,same for all final states. The nonuniversal~but generation-independent! contributions togV and gA coming from theeffective Lagrangian~1! are

gV5a0g82@ I f312Qf~2cW

2 2sW2 !#12a1Qfg

2sW2

12a8Qfg2cW

2 , ~4!

gA5a0I f3g82. ~5!

They do depend on the specific underlying breaking mecnism through the values of theai . It should be noted thathese coefficients depend logarithmically on some unknoscale. In the minimal standard model the characteristic sis the Higgs boson massMH . In other theories the scaleMHwill be replaced by some other scaleL. A crucial predictionof chiral perturbation theory is that the dependence on thdifferent scales is logarithmic and actually the same. Itthus possible to eliminate this dependence by building sable combinations ofgV and gA @11,12# determined by thecondition of the absence of logarithms. Whether this liintersects or not the experimentally allowed region is a dirtest of the nature of the symmetry breaking sector, indepdently of the precise value of Higgs boson mass~in the mini-mal standard model! or of the scale of new interactions~inother scenarios!.1

One could also try to extract information about the indvidual coefficientsa0 , a1 , anda8 themselves, and not onlyon the combinations canceling the dependence on theknown scale. This necessarily implies assuming a specvalue for the scaleL, and one should be aware that wheconsidering these cutoff-dependent quantities there are fiuncertainties of the order of 1/16p2 associated with the subtraction procedure—an unavoidable consequence of usineffective theory, which is often overlooked.~And recall thatusing an effective theory is almost mandatory in dynamisymmetry breaking models.! Only finite combinations ofcoefficients have a universal meaning. The subtraction suncertainty persists when trying to find estimates ofabove coefficients via dispersion relations and the like@13#.

In the previous analysis it is assumed that the hypothetnew physics contributions from vertex corrections are copletely negligible. But is it so? The way to analyze suvertex corrections in a model-independent way is quite silar to the one outlined for the oblique corrections. We shintroduce in the next section the most general effectivegrangian describing the matter sector. In this sector ther

1Notice that, contrary to a somewhat widespread belief, the liMH→` does not correspond a standard model ‘‘without the Higboson.’’ There are some nontrivial nondecoupling effects.

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EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

one universal operator@playing a role analogous to that othe first operator on the RHS of Eq.~1! in the purely bosonicsector#

Leff52vqLUyfqR1H.c., yf5y11y3t3 . ~6!

It is an operator of dimension 3. In the unitary gaugeU51, it is just the mass term for the matter fields. Forstance, ifqL is the doublet (t ,b),

mt5v~y1y3!5vyt , mb5v~y2y3!5vyb . ~7!

Nonuniversal operators carrying in their coefficients infomation on the mechanism giving masses to leptonsquarks will be of dimension 4 and higher.

We shall later derive the values of the coefficients corsponding to operators in the effective Lagrangian of dimsion 4 within the minimal standard model in the largeMHlimit and see how the effective Lagrangian provides a cvenient way of tracing the Higgs boson mass dependencphysical observables. We shall later argue that nondecpling effects should be the same in other theories involvelementary scalars, such as, e.g., the two-Higgs-doumodel, replacingMH by the appropriate mass.

Large nondecoupling effects appear in theories ofnamical symmetry breaking, and thus they are likely to pduce large contributions to the dimension-4 coefficientsthe scale characteristic of the extended interactions~i.e.,those responsible for the fermion mass generation! is muchlarger than the scale characteristic of the electroweak bring, it makes sense to parametrize the former, at least atenergies, via effective four-fermion operators.2 We shall as-sume here that this clear separation of scales does takeand only in this case are the present techniques really arate. The appearance of pseudo Goldstone bosons~abundantin models of dynamical breaking! may thus jeopardize ouconclusions, as they bring a relatively light scale into tgame~typically even lighter than the Fermi scale!. In fact,for the observables we consider that their contribution istoo important, unless they are extremely light. For instancpseudo Goldstone boson of 100 GeV can be accommodwithout much trouble, as we shall later see.

The four-fermion operators we have just alluded to cinvolve either four ordinary quarks or leptons~but we willsee that dimensional counting suggests that their contribuwill be irrelevant at present energies with the exceptionthose containing the top quark! or two new~heavy! fermionsand two ordinary ones. This scenario is quite natural in seral extended technicolor~ETC! or top condensate~TopC!models@14,15#, in which the underlying dynamics is charaterized by a scaleM. At scalesm,M the dynamics can bemodeled by four-fermion operators~of either technifermions

2While using an effective theory description based on fofermion operators alone frees us from having to appeal to anyticular model, it is obvious that some information is lost. This issturns out to be a rather subtle one and shall be discussed andtified in turn.

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in ETC models or ordinary fermions of the third family iTopC models!. We perform a classification3 of these opera-tors. We shall concentrate in the case where technifermappear in ordinary representations of SU~2!L3SU~3!c ~hy-percharge can be arbitrary!. The classification will then beexhaustive. We shall discuss other representations asalthough we shall consider custodially preserving operaonly, and only those operators which are relevant for opurposes.

As a matter of principle, we have tried not to make aassumptions regarding the actual way different generatare embedded in the extended interactions. In practice, wpresenting our numerical plots and figures, we are assumthat the appropriate group-theoretical factors are similarall three generations of physical fermions.

It has been our purpose in this paper to be as generapossible, not advocating or trying to put forward any particlar theory. Thus the analysis may, hopefully, remain usebeyond the models we have just used to motivate the plem. We hope to convey to the reader our belief that a stematic approach based on four-fermion operators andeffective Lagrangian treatment can be very useful.

III. MATTER SECTOR

Appelquist, Bowick, Cohler, and Hauser established sotime ago a list ofd54 operators@17#. These are the operators of lowest dimensionality which are nonuniversal.other words, their coefficients will contain information owhatever mechanism nature has chosen to make quarksleptons massive. Of course, operators of dimensionality 5and so on will be generated at the same time. We shallto these later. We have reanalyzed all possible indepenoperators ofd54 ~see the discussion in Appendix A!, andwe find the following ones:

L415 i qLU~D” U !†qL , ~8!

L425 i qRU†~D” U !qR , ~9!

L435 i qL~D” U !t3U†qL2 i qLUt3~D” U !†qL , ~10!

L445 i qLUt3U†~D” U !t3U†qL , ~11!

L455 i qRt3U†~D” U !qR2 i qR~D” U !†Ut3qR , ~12!

L465 i qRt3U†~D” U !t3qR , ~13!

L475 i qLUt3U†D” qL2 i qLD” †Ut3U†qL , ~14!

L485 i qRt3D” qR2 i qRD” †t3qR , ~15!

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3In the case of ordinary fermions and leptons, four-fermion opetors have been studied in@16#. To our knowledge a complete analysis when additional fields beyond those present in the stanmodel are present has not been presented in the literature bef

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E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

where it is understood that (D” U)†[gm(DmU)†. Each opera-tor is accompanied by a coefficientd8,d1 ,d2 ,...,d7 ; thus, upto O(p4), our effective Lagrangian is4

Leff5d8L481(i 51

7

d iL4i . ~16!

In the above,DmU is defined in Eq.~2!, whereas

DmqL5S ]m1 igtW

2•WW m1 ig8YBmDqL , ~17!

DmqR5S ]m1 ig8t3

2Bm1 ig8YBmDqR , ~18!

whereY5I /6 for quarks andY52I /2 for leptons. This listdiffers from the one in@17# by the presence of the last operator~15!. It will turn out, however, thatd8 does not con-tribute to any observable. All these operators are invarunder local SU~2!L3U~1!Y transformations.

This list includes both custodially preserving operatosuch asL4

1 andL42, and custodially breaking ones, such asL48

and L43 to L4

7. In the purely bosonic part of the effectivLagrangian~1!, the first~universal! operator and the one accompanyinga1 are custodially preserving, while those goinwith a0 and a8 are custodially breaking. E.g.,a0 param-etrizes the contribution of the new physics to theDr param-eter. If the underlying physics is custodially preserving ond1 andd2 will get nonvanishing contributions.5

The operatorL47 deserves some comments. By using t

equations of motion, it can be reduced to the mass term~6!

d7vqLU~yt31y3!qR1H.c. ~19!

However, this procedure is, generally speaking, only justifiif the matter fields appear only as external legs. For the tbeing we shall keepL4

7 as an independent operator, andthe next section we shall determine its value in the minimstandard model after integrating out a heavy Higgs bosWe shall see that, after imposing that physical on-shell fiehave unit residue,d7 does drop from all physical predictions

What is the expected size of thed i coefficients in theminimal standard model? This question is easily answerewe take a look at the diagrams that have to be computeintegrate out the Higgs field~Fig. 2!. Notice that the calcu-lation is carried out in the nonlinear variablesU, hence the

4Although there is only one derivative in Eq.~16! and thus this isa misname, we stick to the same notation here as in the pubosonic effective Lagrangian.

5Of course, hypercharge breaks custodial symmetry, since onsubgroup of SU~2!R is gauged. Therefore,all operators involvingright-handed fields break custodial symmetry. However, therestill a distinction between those operators whose structure ismally custodially invariant~and custodial symmetry is broken onthrough the coupling to the external gauge field! and those whichwould not be custodially preserving even if the full SU~2!R weregauged.

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appearance of the unfamiliar diagram~e!. Diagram ~d! isactually of order 1/MH

2 , which guarantees the gauge indepedence of the effective Lagrangian coefficients. The diagraare obviously proportional toy2, y being a Yukawa cou-pling, and also to 1/16p2, since they originate from a oneloop calculation. Finally, the screening theorem shows tthey may depend on the Higgs boson mass only logarithcally: therefore,

d iSM;

y2

16p2 logMH

2

MZ2 . ~20!

These dimensional considerations show that the vertexrections are only sizable for third generation quarks.

In models of dynamical symmetry breaking, such asor ETC, we shall have new contributions to thed i from thenew physics~which we shall later parametrize with fourfermion operators!. We have several new scales at our dposal. One isM, the mass-normalizing dimension-6 foufermion operators. The other can be eithermb ~negligible,sinceM is large!, mt , or the dynamically generated massthe techniquarksmQ ~typically of orderLTC, the scale associated with the interactions triggering the breaking of telectroweak group!. Thus we can get a contribution of orde

d iQ;

1

16p2

mQ2

M2 logmQ

2

M2 . ~21!

While mQ is, at least naively, expected to be.LTC andtherefore similar for all flavors, there should be a hierarcfor M. As will be discussed in the following sections, thscaleM which is relevant for the mass generation~encodedin the only dimension-3 operator in the effective Lagranian!, via techniquark condensation and ETC interactionchange~Fig. 1!, is the one normalizing chirality-flipping operators. On the contrary, the scale normalizing dimensiooperators in the effective theory is the one that normalichirality-preserving operators. Both scales need not beactly the same, and one may envisage a situation with rtively light scalars present where the former can be mulower. However, it is natural to expect thatM should at anyrate be smallest for the third generation. Consequently,contribution to thed i ’s from the third generation should blargest.

We should also discuss dimension-5 -6, etc., operaand why we need not include them in our analysis. Letwrite some operators of dimension 5:

qLWUqR1H.c., ~22!

qLUBqR1H.c., ~23!

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FIG. 1. Mechanism generating quark masses through thechange of an ETC particle.

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EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

qLsmnD [m† Dn]UqR2qLsmnD [nUDm]qR1H.c., ~24!

qLUD2qR1H.c., ~25!

qLDm† Ut3U†~DmU !qR2qLUt3U†~DmU !DmqR1H.c.,

~26!

¯ ,

where we use the notation W[ igsmnWmn , B[ ig8smnBmn . These are a few of a long list of about 2operators, and this including only the ones contributingthe ffZ vertex. All these operators are, however, chiralflipping, and thus their contribution to the amplitude mustsuppressed by one additional power of the fermion masThis makes their study unnecessary at the present leveprecision. Similar considerations apply to operators ofmensionality 6 or higher.

IV. EFFECTIVE THEORY OF THE STANDARD MODEL

In this section we shall obtain the values of the coecients d i in the minimal standard model. The appropriaeffective coefficients for the oblique correctionsai have beenobtained previously by several authors@11,12,18#. Their val-ues are

a051

16p2

3

8 S 1

e2 log

MH2

m2 15

6D , ~27!

a151

16p2

1

12S 1

e2 log

MH2

m2 15

6D , ~28!

a850, ~29!

where 1/e[1/e2gE1 log 4p. We use dimensional regularization with a spacetime dimension 422e.

We begin by writing the standard model in terms of tnonlinear variablesU. The matrix

M5&~F,F!, ~30!

constructed with the Higgs doublet,F, and its conjugateF[ i t2F* , is rewritten in the form

M5~v1r!U, U215U†, ~31!

wherer describe the ‘‘radial’’ excitations around the vacuuexpectation value~VEV! v. Integrating out the fieldr pro-duces an effective Lagrangian of the form~1! with the valuesof the ai given above~as well as some other pieces nshown there!. This functional integration also generates tvertex corrections~16!.

We shall determine thed i by demanding that the renormalized one-particle irreducible~1PI! Green functionsG arethe same~up to some power in the external momenta amass expansion! in both the minimal standard model~SM!and the effective Lagrangian. In other words, we require t

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where, throughout this section,

DG[GSM2Geff , ~33!

and the caret denotes renormalized quantities. This produre is known as matching. It goes without saying thatdoing so the same renormalization scheme must be usedon-shell scheme is particularly well suited to perform tmatching and will be used throughout this paper.

One only needs to worry about SM diagrams that arepresent in the effective theory, namely, those containingHiggs boson. The rest of the diagrams give exactly the saresult, thus dropping from the matching. In contrast, the dgrams containing a Higgs propagator are described by loterms~such asL4

1 throughL47! in the effective theory: they

involve the coefficientsd i and give rise to the Feynman rulecollected in Appendix B.

Let us first consider the fermion self-energies. Thereonly one 1PI diagram with a Higgs propagator~see Fig. 2!. Astraightforward calculation gives

SSMf 52

yf2

16p2 H p” F1

2

1

e2

1

2log

MH2

m2 11

4G1mfF1

e2 log

MH2

m2 11G J . ~34!

DS f can be computed by subtracting Eqs.~B7!, ~B8! fromEq. ~34!.

Next, we have to renormalize the fermion self-energiWe introduce the following notation:

DZ[ZSM2Zeff5dZSM2dZeff , ~35!

whereZSM (Zeff) stands for any renormalization constantthe SM ~effective theory!. To computeDS f , we simply addto DS f the counterterm diagram~D4! with the replacementsdZV,A

f →DZV,Af anddmf→Dmf . This, of course, amounts to

Eqs. ~D11!, ~D12!, and ~D13! with the same replacementsFrom Eqs.~D14!, ~D15!, and~D16! ~which also hold forDZ,Dm, andDS!, one can expressDZV,A

f andDmf /mf in terms

of the bare fermion self-energies and finally obtainDS f . Theresult is

DSA,V,Sd 50, ~36!

DSAu50, ~37!

DSV,Su 54d72

1

16p2

yu22yd

2

2 F1

e2 log

MH2

m2 11

2G . ~38!

We see from Eq.~38! that the matching conditionsDSV,Su

50 imply

d751

16p2

yu22yd

2

8 F1

e2 log

MH2

m2 11

2G . ~39!

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E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

The other matchings are satisfied automatically and dogive any information.

Let us consider the vertexffZ. The relevant diagrams arshown in Fig. 2@diagrams~b!–~e!#. We shall only collect thecontributions proportional togm andgmg5 . The result is

Gmf f Z52

i

16p2

yf2

2gmH v f S 1

e2 log

MH2

m2 11

2D23afg5S 1

e2 log

MH2

m2 111

6 D J . ~40!

By subtracting the diagrams~B2! and ~B3! from Gmf f Z , one

getsDGmf f Z . Renormalization requires that we add the cou

terterm diagram~D5! where, again,dZ→DZ. One can checkthat bothDZ1

Z2DZ2Z and DZ1

Zg2DZ2Zg are proportional to

DSZg(0), which turns out to be zero. Hence the only reevant renormalization constants areDZV

f and DZAf . These

renormalization constants have already been determiOne obtains forDGm

f f Z the result

DGmddZ52

ie

2sWcWgmH F1

2~d12d42d22d6!1d31d5G

2g5F 1

16p2

yd2

2 S 1

e2 log

MH2

m2 15

2D1

1

2~d12d41d21d6!1d32d5G J , ~41!

DGmuuZ52

ie

2sWcWgmH F1

2~d12d42d22d6!2d32d5G

2g5F21

16p2

yu2

2 S 1

e2 log

MH2

m2 15

2D2

1

2~d12d41d21d6!1d32d5G J , ~42!

where use has been made of Eq.~39!. The matching condi-tion DGm

f f Z50 implies

FIG. 2. The diagrams relevant for the matching of the fermself-energies and vertices~counterterm diagrams are not included!.Double lines represent the Higgs, dashed lines the Goldsbosons, and wiggly lines the gauge bosons.

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2D , ~43!

d21d6521

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yu21yd

2

4 S 1

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MH2

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2D , ~44!

d351

16p2

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2D ,

~45!

d5521

16p2

yu22yd

2

4 S 1

e2 log

MH2

m2 15

2D .

~46!

To determine completely thed i coefficients we need toconsider the vertexudW. The relevant diagrams are analgous to those of Fig. 2. A straightforward calculation give

DGmudW5

ie

4&sW

gmH F yuyd

16p2 S 1

e2 log

MH2

m2 15

2D 12d222d6G3~11g5!2Fyu

21yd2

16p2

1

2 S 1

e2 log

MH2

m2 15

2D12d112d4G~12g5!J . ~47!

The matching conditionDGmudW50 amounts to the following

set of equations:

d22d6521

16p2

yuyd

2 S 1

e2 log

MH2

m2 15

2D , ~48!

d11d4521

16p2

yu21yd

2

4 S 1

e2 log

MH2

m2 15

2D . ~49!

Combining these equations with Eqs.~43!,~44!, we finallyget

d1521

16p2

yu21yd

2

4 S 1

e2 log

MH2

m2 15

2D , ~50!

d2521

16p2

~yu1yd!2

8 S 1

e2 log

MH2

m2 15

2D , ~51!

d450, ~52!

d6521

16p2

~yu2yd!2

8 S 1

e2 log

MH2

m2 15

2D . ~53!

This, along with Eqs.~45!, ~46! and Eq.~39!, is our finalanswer. These results coincide, where the comparison issible, with those obtained in@19# by functional methods. It isinteresting to note that it has not been necessary to consthe matching of the vertexf f g.

We shall show explicitly thatd7 drops from theS-matrixelement corresponding toZ→ f f . It is well known that the

ne

5-6

m

r-en

ly

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un

a

ieh

x-y

u-ac

asintiv

as

a

,msl

nnotak

bydelin

ac-otanatedh as

icle

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

renormalized u-fermion self-energy has residue 11d res,whered res in given in Eq.~D17! of Appendix D. Therefore,in order to evaluateS-matrix elements involving externalulines at one loop, one has to multiply the corresponding aputated Green functions by a factor 11nd res/2, wheren isthe number on externalu lines ~in the case under consideation n52!. One can check that when this factor is takinto account, thed7 appearing in the renormalizedS-matrixvertex are canceled.

We notice thatd1 andd2 indeed correspond to custodialpreserving operators, whiled3 to d6 do not. All these coef-ficients ~just asa0 , a1 , and a8! are ultraviolet divergent.This is so because the Higgs particle is an essential ingrent to guarantee the renormalizability of the standard moOnce this is removed, the usual renormalization proc~e.g., the on-shell scheme! is not enough to render a‘‘renormalized’’ Green functions finite. This is why the bacoefficients of the effective Lagrangian~which contribute tothe renormalized Green functions either directly or via coterterms! have to be proportional to 1/e to cancel the newdivergences. The coefficients of the effective Lagrangianmanifestly gauge invariant.

What is the value of these coefficients in other theorwith elementary scalars and a Higgs-like mechanism? Tissue has been discussed in some detail in@20# in the contextof the two-Higgs-doublet model, but it can actually be etended to supersymmetric theories~provided of course scalars other than theCP-even Higgs can be made heavenough; see, e.g.,@21#!. It was argued there that nondecopling effects are exactly the same as in the minimal standmodel, including the constant nonlogarithmic piece. Sinthe d i coefficients contain all the nondecoupling effectssociated with the Higgs particle at the first nontrivial orderthe momentum or mass expansion, the low-energy effectheory will be exactly the same.

V. OBSERVABLES

The decay width ofZ→ f f is described by

G f[G~Z→ f f !54ncG0@~gVf !2RV

f 1~gAf !2RA

f #, ~54!

wheregVf andgA

f are the effective electroweak couplingsdefined in@22# andnc is the number of colors of fermionf.The radiation factorsRV

f andRAf describe the final state QED

and QCD interactions@23#. For a charged lepton we have

RVl 511

3a

4p1OXa2,S ml

MZD 4C,

RAl 511

3a

4p26S ml

MZD 2

1OXa2,S ml

MZD 4C,

wherea is the electromagnetic coupling constant at the scMZ andml is the final state lepton mass.

The tree-level widthG0 is given by

11403

-

i-l.

ss

-

re

sis

-

rde-

e

le

G05GmMZ

3

24&p. ~55!

If we define

r f[4~gAf !2, ~56!

sW2 [

I f3

2QfS 12

gVf

gAf D , ~57!

we can write

G f5ncG0r f@4~ I f322QfsW

2 !2RVf 1RA

f #. ~58!

Other quantities which are often used areDr f , definedthrough

r f[1

12Dr f, ~59!

the forward-backward asymmetryAFBf ,

AFBf 5

3

4AeAf , ~60!

andRb ,

Rb5Gb

Gh, ~61!

where

Af[2gV

f gAf

~gAf !21~gV

f !2 ,

andGb , Gh are theb partial width and total hadronic widthrespectively~each of them, in turn, can be expressed in terof the appropriate effective couplings!. As we see, nearly alof Z physics can be described in terms ofgA

f and gVf . The

box contributions to the processe1e2→ f f are not includedin the analysis because they are negligible and they cabe incorporated as contributions to effective electroweneutral current couplings anyway.

We shall generically denote these effective couplingsgf . If we express the value they take in the standard moby gf ~SM!, we can write a perturbative expansion for themthe following way:

gf ~SM!5gf ~0!1gf ~2!1gf~aSM!1gf~dSM!, ~62!

wheregf (0) are the tree-level expressions for these form ftors andgf (2) are the one-loop contributions which do ncontain any Higgs particle as internal line in the Feynmgraphs. In effective Lagrangian language they are generby the quantum corrections computed by operators suc~6! or the first operator on the RHS of Eq.~1!. On the otherhand, the Feynman diagrams containing the Higgs partcontribute togf ~SM! in a twofold way. One is via theO(p2)andO(p4) Longhitano effective operators~1! which depend

5-7

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el,

to

ec

e

ncd

incaa

igg

cak

t

aintivrv

t,d

t

o

yse-o

th’’

t itingaliz-the.

theents

ionnals.’’ale

s’’ale.eredel.sheu-

ry

ten

notdi-nlythetheofnifer-ces-

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

on theai coefficients, which are Higgs-boson-mass depdent, and thus give a Higgs-boson-dependent oblique cortion to gf ~SM!, which is denoted bygf . The other one is viagenuine vertex corrections which depend on thed i . Thiscontribution is denoted bygf .

The tree-level value for the form factors are

gVf ~0!5I f

322sW2 Qf , gA

f ~0!5I f3. ~63!

In a theory X, different from the minimal standard modthe effective form factors will take valuesgf ~X!, where

gf ~X!5gf ~0!1gf ~2!1gf~aX!1gf~dX!, ~64!

and theaX anddX are effective coefficients correspondingtheory X.

Within one-loop accuracy in the symmetry-breaking stor ~but with arbitrary precision elsewhere!, gf and gf arelinear functions of their arguments and thus we have

gf ~X!5gf ~SM!1gf~aX2aSM!1gf~dX2dSM!. ~65!

The expression forgf in terms ofai was already given inEqs. ~4! and ~5!. On the other hand, from Appendix B wlearn that

gVf ~d1 ,...,d6!5I f

3~d12d42d22d6!2d32d5 , ~66!

gAf ~d1 ,...,d6!5I f

3~d12d41d21d6!2d31d5 . ~67!

In the minimal standard model all the Higgs dependeat the one-loop level~which is the level of accuracy assumehere! is logarithmic and is contained in theai andd i coeffi-cients. Therefore one can easily construct linear combtions of observable where the leading Higgs dependencecels. These combinations allow for a test of the minimstandard model independent of the actual value of the Hboson mass.

Let us now review the comparison with current eletroweak data for theories with dynamical symmetry breing. Some confusion seems to exist on this point, so let usto analyze this issue critically.

A first difficulty arises from the fact that at theMZ scaleperturbation theory is not valid in theories with dynamicbreaking and the contribution from the symmetry breaksector must be estimated in the framework of the effectheory, which is nonlinear and nonrenormalizable. Obseable will depend on some subtraction scale.~Estimates basedon dispersion relations and resonance saturation amounpractice, to the same, provided that due attention is paithe scale dependence introduced by the subtraction indispersion relation.!

A somewhat related problem is that, when making usethe variablesS, T and U @13#, or e1 , e2 , and e3 @24#, oneoften sees in the literature bounds on possible ‘‘new phics’’ in the symmetry breaking sector without actually rmoving the contribution from the standard model higgs bson that the ‘‘new physics’’ is supposed to replace~this isnot the case, e.g., in@13# where this issue is discussed wisome care!. Unless the contribution from the ‘‘new physics

11403

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-

e

a-n-ls

--ry

lge-

intohe

f

-

-

is enormous, this is a flagrant case of double counting, buis easy to understand why this mistake is made: removthe Higgs boson makes the standard model nonrenormable, and the observable of the standard model withoutHiggs boson depend on some arbitrary subtraction scale

In fact, the two sources of arbitrary subtraction scales~theone originating from the removal of the Higgs boson andone from the effective action treatment! are one and the samand the problem can be dealt with the help of the coefficieof higher-dimensional operators in the effective theory~i.e.,the ai andd i!. The dependence on the unknown subtractscale is absorbed in the coefficients of higher-dimensiooperators and traded by the scale of the ‘‘new physicCombinations of observable can be built where this sc~and the associated renormalization ambiguities! drops.These combinations allow for a test of the ‘‘new physicindependently of the actual value of its characteristic scIn fact, they are the same combinations of observable whthe Higgs dependence drops in the minimal standard mo

A third difficulty in making a fair comparison of modelof dynamical symmetry breaking with experiment lies in tvertex corrections. If we analyze the lepton effective coplingsgA

l andgVl , the minimal standard model predicts ve

FIG. 3. The 1-s experimental region in thegAe-gV

e plane. Thestandard model predictions as a function ofmt (170.6<mt

<180.6 GeV) andMH (70<MH<1000 GeV) are shown~themiddle line corresponds to the central valuemt5175.6 GeV!. Thepredictions of a QCD-like technicolor theory withnTCnD58 anddegenerate technifermion masses are shown as straight lines~onlyoblique corrections are included!. One moves along the straighlines by changing the scaleL. The three lines correspond to thextreme and central values formt . Recall that the precise locatioanywhere on the straight lines~which definitely do intersect the 1-sregion! depends on the renormalization procedure and thus ispredictable within the nonrenormalizable effective theory. In adtion, the technicolor prediction should be considered accurate oat the 15% level due to the theoretical uncertainties discussed intext ~this error is at any rate smaller than the one associated withuncertainty inL!. Notice that the oblique corrections, in the casedegenerate masses, are independent of the value of the techmion mass. Assuming universality of the vertex corrections reduthe error bars by about a factor of1

2 and leaves technicolor predictions outside the 1-s region.

5-8

e-

r-

Vom

-

esnna,d

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

FIG. 4. The effect of isospin breaking in thoblique corrections in QCD-like technicolor theories. The 1-s region for thegA

e-gVe couplings and

the SM prediction~for mt5175.6 GeV and 70<MH<1000 GeV! are shown. The differentstraight lines correspond to setting the technifemion masses in each doublet (m1 ,m2) to thevalue m25250, 300, 350, 400, and 450 Ge~larger masses are the ones deviating more frthe SM predictions! and m151.05m2 ~plot 1!,m151.1m2 ~plot 2!, m151.2m2 ~plot 3!, andm151.3m2 ~plot 4!. The results are invariant under the exchange ofm1 andm2 . As in Fig. 3, theprediction of the effective theory is the wholstraight line and not any particular point on it, awe move along the line by varying the unknowscale L. Clearly, isospin breakings larger tha20% give very poor agreement with the dateven for low values of the dynamically generatemass.

inothilet

dyo

rd

wni-

on

nen

a

areyry

a

theht

ivenson

theeal-

heyion

eal-thealso-en-

ou-

fiedalint

ses

small vertex corrections arising from the symmetry-breaksector anyway and it is consistent to ignore them and ccentrate in the oblique corrections. However, this is notsituation in dynamical symmetry-breaking models. We wsee in the next sections that for the second and third gention vertex corrections can be sizable. Thus, if we wantcompare experiment to oblique corrections in models ofnamical breaking, we have to concentrate on electron cplings only.

In Fig. 3 we see the prediction of the minimal standamodel for 170.6,mt,180.6 GeV and 70,MH,1000GeV, including the leading two-loop corrections@23#, fallingnicely within the experimental 1-s region for the electroneffective couplings. In this and in subsequent plotspresent the data from the combined four LEP experimeonly. What is the actual prediction for a theory with dynamcal symmetry breaking? The straight solid lines correspto the prediction of a QCD-like technicolor model withnTC52 andnD54 ~a one-generation model! in the case whereall technifermion masses are assumed to be equal~we follow@9#: see@25# for related work!, allowing the same variationfor the top quark mass as in the standard model. We dotake into account here the contribution of potentially prespseudo Goldstone bosons, assuming that they can be mheavy enough. The corresponding values for theai coeffi-cients in such a model are given in Appendix E andderived using chiral quark model techniques and chiral pturbation theory. They are scale dependent in such a wato make observables finite and unambiguous, but of couobservables depend in general on the scale of ‘‘new phics’’ L.

We move along the straight lines by changing the scaleL.It would appear at first sight that one needs to go to un

11403

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ets

d

ottde

er-asses-

c-

ceptably low values of the new scale to actually penetrate1-s region, something which looks unpleasant at first sig~we have plotted the part of the line for 100<L<1500 GeV!, as one expectsL;Lx . In fact, this is notnecessarily so. There is no real prediction of the effecttheory along the straight lines, because only combinatiowhich areL-independent are predictable. As for the locatinot along the line, butof the line itself, it is in principlecalculable in the effective theory, but of course subject touncertainties of the model one relies upon, since we are ding with a strongly coupled theory.~We shall use chiralquark model estimates in this paper as we believe that tare quite reliable for QCD-like theories: see the discussbelow.!

If we allow for a splitting in the technifermion masses, thcomparison with experiment improves very slightly. The vues of the effective Lagrangian coefficients relevant foroblique corrections in the case of unequal masses aregiven in Appendix E. Sincea1 is independent of the technifermion dynamically generated masses anyway, the depdence is fully contained ina0 ~the parameterT of Peskin andTakeuchi@13#! and a8 ~the parameterU!. This is shown inFig. 4. We assume that the splitting is the same for all dblets, which is not necessarily true.6

If other representations of the SU~2!L3SU~3!c gaugegroup are used, the oblique corrections have to be modiin the form prescribed in Sec. VIII. Larger group-theoreticfactors lead to larger oblique corrections and, from this po

6In fact, it can be argued that QCD corrections may, in some ca@30#, enhance techniquark masses.

5-9

o

ravevin

gl

anisu

ci

satre

nt

es

onat

hatopsisas

loe

ga

mnofn

c-ns,e anto

orlap-

the

n-he

illres-that.the

TC

ueer,

da-ent.

ro-at,enyWes is

githare

,

-

ay

t

e

there--

ldsnsto

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

of view, the restriction to weak doublets and color singletstriplets is natural.

Let us close this section by justifying the use of chiquark model techniques, trying to assess the errors involand at the same time emphasizing the importance of hathe scale dependence under control. A parameter likea1 ~orS in the notation of Peskin and Takeuchi@13#! contains in-formation about the long-distance properties of a stroncoupled theory. In fact,a1 is nothing but the familiarL10parameter of the strong chiral Lagrangian of GasserLeutwyler @26# translated to the electroweak sector. Thstrong interaction parameter can be measured, and it is foto beL105(25.660.3)31023 ~at them5Mh scale, whichis just the conventional reference value and plays no sperole in the standard model!. This is almost twice the valuepredicted by the chiral quark model@27,28# (L10521/32p2), which is the estimate plotted in Fig. 3. Doethis mean that the chiral quark model grossly underestimthis observable? Not at all. Chiral perturbation theory pdicts the running ofL10. It is given by

L10~m!5L10~Mh!11

128p2 logm2

Mh2 . ~68!

According to our current understanding~see, e.g.,@29#!, thechiral quark model gives the value of the chiral coefficieat the chiral symmetry breaking scale~4p f p in QCD, Lx inthe electroweak theory!. Then the coefficientL10 ~or a1 forthat matter! predicted within the chiral quark model agrewith QCD at the 10% level.

Let us now turn to the issue of vertex corrections in theries with dynamical symmetry breaking and the determition of the coefficientsd i , which are, after all, the focal poinof this work.

VI. NEW PHYSICS AND FOUR-FERMION OPERATORS

In order to have a picture in our mind, let us assume tat sufficiently high energies the symmetry-breaking seccan be described by some renormalizable theory, perhanon-Abelian gauge theory. By some unspecified mechansome of the carriers of the new interaction acquire a mLet us generically denote this mass byM. One type of modelthat comes immediately to mind is the extended technicoscenario.M would then be the mass of the ETC bosons. Lus try, however, not to adhere to any specific mechanismmodel.

Below the scaleM we shall describe our underlyintheory by four-fermion operators. This is a convenient wof parametrizing the new physics belowM without needingto commit oneself to a particular model. Of course, the nuber of all possible four-fermion operators is enormous aone may think that any predictive power is lost. This is nso because of two reasons:~a! The size of the coefficients othe four-fermion operators is not arbitrary. They are costrained by the fact that at scaleM they are given by

2jCG

G2

M2 , ~69!

11403

r

ld,g

y

d

nd

fic

es-

s

--

tra

ms.

rtor

y

-dt

-

where jCG is built out of Clebsch-Gordan factors andG agauge-coupling constant, assumed perturbative ofO~1! at thescaleM. The jCG, being essentially group-theoretical fators, are probably of similar size for all three generatioalthough not necessarily identical as this would assumparticular style of embedding the different generations ithe large ETC~for instance! group. Notice that for four-fermion operators of the formJ•J†, whereJ is some fermionbilinear, jCG has a well-defined sign, but this is not so fother operators.~b! It turns out that only a relatively smalnumber of combinations of these coefficients do actuallypear in physical observables at low energies.

Matching to the fundamental physical theory atm5Mfixes the value of the coupling constants accompanyingfour-fermion operators to the value~69!. In addition, contactterms, i.e., nonzero values for the effective coupling costantsd i , are generally speaking required in order for tfundamental and four-fermion theories to match. These wlater evolve under the renormalization group due to the pence of the four-fermion interactions. Because we expectM@Lx , the d i will be typically logarithmically enhancedNotice that there is no guarantee that this is the case forthird generation, as we will later discuss. In this case theand ETC dynamics would be tangled up~which for mostmodels is strongly disfavored by the constraints on obliqcorrections!. For the first and second generations, howevthe logarithmic enhancement of thed i is a potentially largecorrection and it actually makes the treatment of a funmental theory via four-fermion operators largely independof the particular details of specific models, as we will see

Let us now get back to four-fermion operators and pceed to a general classification. A first observation is thwhile in the bosonic sector custodial symmetry is just brokby the small U~1!Y gauge interactions, which is relativelsmall, in the matter sector the breaking is not that small.thus have to assume that whatever underlying new physicpresent at scaleM it gives rise both to custodially preservinand custodially nonpreserving four-fermion operators wcoefficients of similar strength. Obvious requirementsHermiticity, Lorentz invariance, and SU~3!c3SU~2!L3U~1!Y symmetry. NeitherC nor P invariance are imposedbut invariance underCP is assumed.

We are interested ind56 four-fermion operators constructed with two ordinary fermions~either leptons orquarks!, denoted byqL , qR , and two fermionsQL

A , QRA .

Typically, A will be the technicolor index and theQL , QRwill therefore be techniquarks and technileptons, but we mbe as well interested in the case where theQ may be ordinaryfermions. In this case the indexA drops~in our subsequenformulas this will correspond to takingnTC51!. We shallnot write the indexA hereafter for simplicity, but this degreof freedom is explicitly taken into account in our results.

As we already mentioned, we shall discuss in detailcase where the additional fermions fall into ordinary repsentations of SU~2!L3SU~3!c and will discuss other representations later. The fieldsQL will therefore transform asSU~2!L doublets, and we shall group the right-handed fieQR into doublets as well, but then include suitable insertioof t3 to consider custodially breaking operators. In order

5-10

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

TABLE I. Four-fermion operators which do not change the fermion chirality. The first~second! columncontains the custodially preserving~breaking! operators.

L25(QLgmQL)(qLgmqL)

R25(QRgmQR)(qRgmqR) R3R5(QRgmt3QR)(qRgmqR)

RR35(QRgmQR)(qRgmt3qR)

R325(QRgmt3QR)(qRgmt3qR)

RL5(QRgmQR)(qLgmqL) R3L5(QRgmt3QR)(qLgmqL)

LR5(QLgmQL)(qRgmqR) LR35(QLgmQ3)(qRgmt3qR)

rl 5(QRgmlW QR)•(qLgmlW qL) r 3l 5(QRgmlW t3QR)•(qLgmlW qL)

lr 5(QLgmlW QL)•(qRgmlW qR) lr 35(QLgmlW t3QL)•(qRgmlW t3qR)

(QLgmqL)(qLgmQL)

(QRgmqR)(qRgmQR) (QRgmt3qR)(qRgmQR)1(QRgmqR)(qRgmt3QR)

(QRgmt3qR)(qRgmQR)

(QLi gmQL

j )(qLj gmqL

i )

(QRi gmQR

j )(qRj gmqR

i )

(QLi gmqL

j )(qLj gmQL

i )

(QRi gmqR

j )(qRj gmQR

i ) (QRi gmqR

j )(qRj gm@t3QR# i)

io

ir,

nsibrs

a

-oliclam

hethelore-nd

ra-

ly,

ofns-ace

hb

g

determine the low-energy remnants of all these four-fermoperators~i.e., the coefficientsd i!, it is enough to know theircouplings to SU~2!L and no further assumptions about theelectric charges~or hypercharges! are needed. Of coursesince theQL , QR couple to the electroweak gauge bosothey must not lead to new anomalies. The simplest possity is to assume that they reproduce the quantum numbeone family of quarks and leptons~that is, a total of fourdoubletsnD54!, but other possibilities exist@for instance,nD51 is also possible@31#, although this model presentsglobal SU~2!L anomaly#.

We shall first be concerned with theQL , QR fields be-longing to the representation3 of SU~3!c and, afterwards,focus in the simpler case where theQL , QR are color singlet~technileptons!. ColoredQL , QR fermions can couple to ordinary quarks and leptons either via the exchange of a csinglet or of a color octet. In addition, the exchanged partcan be either an SU~2!L triplet or a singlet, thus leading tolarge number of possible four-fermion operators. More i

TABLE II. Chirality-changing four-fermion operators. To eacentry, the corresponding Hermitian conjugate operator shouldadded. The left~right! column contains custodially preservin~breaking! operators.

(QLgmqL)(qRgmQR) (QLgmqL)(qRgmt3QR)

(qLi qR

j )(QLkQR

l )e ike j l (qLi @t3qR# j )(QL

kQRl )e ike j l

(qLi QR

j )(QLkqR

l )e ike j l (qLi QR

j )(QLk@t3qR# l)e ike j l

(QLgmlW qL)•(qRgmlW QR) (QLgmlW qL)•(qRgmlW t3QR)

(qLi lW qR

j )•(QLklW QR

l )e ike j l (qLi lW @t3qR# j )•(QL

klW QRl )e ike j l

(qLi lW QR

j )•(QLklW qR

l )e ike j l (qLi lW QR

j )•(QLklW @t3qR# l)e ike j l

11403

n

,il-of

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-

portant for our purposes will be whether they flip or not tchirality. We use Fierz rearrangements in order to writefour-fermion operators as the product of either two cosinglet or two color octet currents. A complete list is prsented in Tables I and II for the chirality-preserving achirality-flipping operators, respectively.

Note that the two upper blocks of Table I contain opetors of the formJ• j , where (J) j stands for a~heavy! fermioncurrent with well-defined color and flavor numbers, namebelonging to an irreducible representation of SU~3!c andSU~2!L . In contrast, those in the two lower blocks are notthis form. In order to make their physical content more traparent, we can perform a Fierz transformation and replthe last nine operators~two lower blocks! in Table I by thosein Table III. These two bases are related by

~QLgmqL!~ qLgmQL!51

4l 21

1

6L21

1

4lW 21

1

6LW 2, ~70!

~QLj gmQL

i !~ qLi gmqL

j !51

2L21

1

2LW 2, ~71!

~QLj gmqL

i !~ qLi gmQL

j !51

2l 21

1

3L2, ~72!

~QRgmqR!~ qRgmQR!51

4r 21

1

6R21

1

4rW 21

1

6RW 2, ~73!

~QRgmqR!~ qRgmt3QR!1~QRgmt3qR!~ qRgmQR!

51

2rr 31

1

3RR31

1

2r 3r 1

1

3R3R, ~74!

e

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E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

TABLE III. New four-fermion operators of the formJ• j obtained after Fierzing. The left~right! columncontains custodially preserving~breaking! operators. In addition, those written in the two upper blocksTable I should also be considered. Together with the above they form a complete set of chirality-presoperators.

l 25(QLgmlW QL)•(qLgmlW qL)

r 25(QRgmlW QR)•(qRgmlW qR) r 3r 5(QRgmlW t3QR)•(qRgmlW qR)

rr 35(QRgmlW QR)•(qRgmlW t3qR)

r 325(QRgmlW t3QR)•(qRgmlW t3qR)

LW 25(QLgmtWQL)•(qLgmtWqL)

RW 25(QRgmtWQR)•(qRgmtWqR)

lW25(QLgmlW tWQL)•(qLgmlW tWqL)

rW25(QRgmlW tWQR)•(qRgmlW tWqR)

msrser

eadtece

inse

res

-wf

oleb

ws.

asin

ur-is

e an-

sen-nottorseentto

for

x Cas

ht-ncers,

~QRgmt3qR!~ qRgmt3QR!51

4r 21

1

6R22

1

4rW 22

1

6RW 2

11

2r 3

211

3R3

2, ~75!

~QRj gmQR

i !~ qRi gmqR

j !51

2R21

1

2RW 2, ~76!

~QRj gmqR

i !~ qRi gmQR

j !51

2r 21

1

3R2, ~77!

~QRi gmqR

i !~ qRi gm@r 3QR# j !5

1

2r 31

1

3R3R, ~78!

for colored techniquarks. Notice the appearance of somenus signs due to the Fierzing and that operators such aL2

~for instance! get contributions from four-fermion operatowhich do have a well-defined sign as well as from othwhich do not.

The use of this basis simplifies the calculations considably as the Dirac structure is simpler. Another obviousvantage of this basis, which will become apparent only lais that it will make it easier to consider the long-distancontributions to thed i , from the region of momentam,Lx .

The classification of the chirality-preserving operatorvolving technileptons is of course simpler. Again, we uFierz rearrangements to write the operators asJ• j . However,in this case only a color singletJ ~and, thus, also a colosinglet j ! can occur. Hence the complete list can be obtainby crossing out from Table III and from the first eight rowof Table I the operators involvinglW . Namely, those designated by lower case letters. We are then left with the toperatorsLW 2, RW 2 from Table III and with the first six rows oTable I: L2, R2, R3R, RR3 , R3

2, RL, R3L, LR, andLR3 . Ifwe choose to work instead with the original basischirality-preserving operators in Table I, we have to suppment these nine operators in the first six rows of the tawith (QLgmqL)(qLgmQL) and (QRgmqR)(qRgmQR), which

11403

i-

s

r--r,

-

d

o

f-

le

are the only independent ones from the last seven roThese two basis are related by

~QLgmqL!~ qLgmQL!51

2L21

1

2LW 2, ~79!

~QRgmqR!~ qRgmQR!51

2R21

1

2RW 2, ~80!

for technileptons.It should be borne in mind that Fierz transformations,

presented in the above discussion, are strictly valid onlyfour dimensions. In 422e dimensions, for the identities tohold we need ‘‘evanescent’’ operators@32#, which vanish infour dimensions. However, the replacement of some fofermion operators in terms of others via the Fierz identitiesactually made inside a loop of technifermions and thereforfinite contribution is generated. Thus the two basis will evetually be equivalent up to terms of order

1

16p2

G2

M2 mQ2 , ~81!

wheremQ is the mass of the technifermion~this estimate willbe obvious only after the discussion in the next sections!. Inparticular, no logarithms can appear in Eq.~81!.

Let us now discuss how the appearance of other repretations might enlarge the above classification. We shallbe completely general here, but consider only those operathat may actually contribute to the observables we have bdiscussing~such asgV andgA!. Furthermore, for reasons thashall be obvious in a moment, we shall restrict ourselvesoperators which are SU~2!L3SU~2!R invariant.

The construction of the chirality-conserving operatorsfermions in higher-dimensional representations of SU~2! fol-lows essentially the same pattern presented in Appendifor doublet fields, except for the fact that operators such

~QLgmqL!~ qLgmQL!, ~QLi gmQL

j !~ qLj gmqL

i !, ~82!

and their right-handed versions, which appear on the righand side of Table I, are now obviously not acceptable siQL andqL are in different representations. Those operato

5-12

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idep

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eat

aturenpl

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ingen-yf-

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

restricting ourselves to color singlet bilinears~the only onesgiving a nonzero contribution to our observables!, can bereplaced in the fundamental representation by

~QLgmQL!~ qLgmqL!, ~QLgmtWQL!~ qLgmtWqL!, ~83!

when we move to theJ• j basis. Now it is clear how tomodify the above when using higher representations forQ fields. The first one is already included in our set of cutodially preserving operators, while the second one has tomodified to

LW [~QLgmTW QL!~ qLgmtWqL!, ~84!

whereTW are the SU~2! generators in the relevant represention. In addition, we have the right-handed counterpart,course. We could in principle now proceed to construct ctodially violating operators by introducing suitableT3 andt3

matrices. Unfortunately, it is not possible to present a closet of operators of this type, as the number of independoperators does obviously depend on the dimensionalitythe representation. For this reason we shall only conscustodially preserving operators when moving to higher rresentations, namely,L2, R2, RL, LR, LW 2, andRW 2.

If we examine Tables I, II, and III we will notice that botchirality-violating and chirality-preserving operators appeIt is clear that at the leading order in an expansion in extefermion masses only the chirality-preserving operat~Tables I and III! are important; those operators containiboth aqL and aqR field will be further suppressed by addtional powers of the masses of the fermions and thus sleading. Furthermore, if we limit our analysis to the studythe effectiveW6 andZ couplings, such asgV andgA , as wedo here, chirality-flipping operators can contribute onthrough a two-loop effect. Thus the contribution from tchirality-flipping operators contained in Table II is supressed both by an additional 1/16p2 loop factor and by amQ

2 /M2 chirality factor. If for the sake of the argument wtake mQ to be 400 GeV, the correction will be below orthe 10% level for values ofM as low as 100 GeV. Thisautomatically eliminates from the game operators generthrough the exchange of a heavy scalar particle, but of cothe presence of light scalars, below the mentioned limit, rders their neglection unjustified. It is not clear where simETC models violate this limit~see, e.g.,@33#!. We just as-sume that all scalar particles can be made heavy enoug

Additional light scalars may also appear as pseudo Gstone bosons at the moment the electroweak symmbreaking occurs due toQQ condensation. We had to assumsomehow that their contribution to the oblique correctiwas small~e.g., by avoiding their proliferation and makinthem sufficiently heavy!. They also contribute to vertex corrections~and thus to thed i!, but here their contribution isnaturally suppressed. The coupling of a pseudo Goldstbosonv to ordinary fermions is of the form

1

4p

mQ2

M2 vqLqR ; ~85!

11403

e-be

-f-

dntofer-

.als

b-f

edse-

e

-ry

ne

thus, their contribution to thed i will be of order

gG4

~16p2!2 S mQ2

M2D 2

logLx

2

mv2 . ~86!

Using the same reference values as above, a pseudo Gstone boson of 100 GeV can be neglected.

If the operators contained in Table II are not relevantthe W6 andZ couplings, what are they important for? Afteelectroweak breaking~due to the strong technicolor forces oany other mechanism! a condensateQQ& emerges. Thechirality-flipping operators are then responsible for geneing a mass term for ordinary quarks and leptons. Their loenergy effects are contained in the onlyd53 operator ap-pearing in the matter sector, discussed in Sec. II. We thusthat the four-fermion approach allows for a nice separatbetween the operators responsible for mass generationthose that may eventually lead to observable consequencthe W6 andZ couplings. One may even entertain the posbility that the relevant scale is, for some reason, differentboth sets of operators~or, at least, for some of them!. Itcould, at least in principle, be the case that scalar exchaenhances the effect of chirality-flipping operators, allowifor large masses for the third generation, without giving uacceptably large contributions to theZ effective coupling.Whether one is able to find a satisfactory fundamental thewhere this is the case is another matter, but the four-fermapproach allows one, at least, to pose the problem.

We shall now proceed to determine the constantsd i ap-pearing in the effective Lagrangian after integration of theavy degrees of freedom. For the sake of the discussionshall assume hereafter that technifermions are degeneramass and set their masses equal tomQ . The general case isdiscussed in Appendix E.

VII. MATCHING TO A FUNDAMENTAL THEORY

At the scalem5M we integrate out the heavier degreesfreedom by matching the renormalized Green functions coputed in the underlying fundamental theory to a four-fermiinteraction. This matching leads to the values~69! for thecoefficients of the four-fermion operators as well as topurely short-distance contribution for thed i , which shall bedenoted byd i . The matching procedure is indicated in Fi5. It is perhaps useful to think of thed i as the value that thecoefficients of the effective Lagrangian take at the matchscale, as they contain the information on modes of frequcies m.M . The d i will be, in general, divergent; i.e., thewill have a pole in 1/e. Let us see how to obtain these coe

FIG. 5. The matching at the scalem5M .

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our-esses

u-

ex-sonfor

rlica-

onte-

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

ficients d i in a particular case.As discussed in the previous section, we understand

at very high energies our theory is described by a gatheory. Therefore we have to add to the standard modelgrangian~already extended with technifermions! the follow-ing pieces:

21

4EmnEmn2

1

2M2EmEm1GQgmEmq1H.c. ~87!

TheEm vector boson~of massM! acts in a large flavor groupspace which mixes ordinary fermions with heavy ones.@Thenotation in Eq.~87! is somewhat symbolic as we are nimplying that the theory is vector like; in fact, we do nassume anything at all about it.#

At energiesm,M we can describe the contribution fromthis sector to the effective Lagrangian coefficients eithering the degrees of freedom present in Eq.~87! or via thecorresponding four-quark operator and a nonzero valuethe d i coefficients. Demanding that both descriptions repduce the same renormalizedffW vertex fixes the value of thed i .

Let us see this explicitly in the case where the intermeate vector bosonEm is a SU~3!c3SU~2!L singlet. For thesake of simplicity, we take the third term in Eq.~87! to be

GQLgmEmqL . ~88!

At energies belowM, the relevant four-quark operator is the

2G2

M2 ~QLgmqL!~ qLgmQL!. ~89!

In the limit of degenerate techniquark masses, it is quite cthat only d1 can be different from zero. Thus one does nneed to worry about matching quark self-energies. Concing the vertex~Fig. 5!, we have to impose Eq.~32!, wherenow

DG[GE2G4Q . ~90!

Namely,DG is the difference between the vertex computusing Eq.~87! and the same quantity computed using tfour-quark operators as well as nonzerod i coefficients@recallthat the caret in Eq.~32! denotes renormalized quantities#. Acalculation analogous to that of Sec. IV~now the leadingterms in 1/M2 are retained! leads to

d152G2

8p2

mQ2

M2

1

e. ~91!

VIII. INTEGRATING OUT HEAVY FERMIONS

As we move down in energies, we can integrate lower alower frequencies with the help of the four-fermion operat~which do accurately describe physics belowM!. This modi-fies the value of thed i :

d i~m!5 d i1Dd i~m/M !, m,M . ~92!

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ate

a-

-

or-

i-

artn-

ds

The quantityDd i(m/M ) can be computed in perturbatiotheory down to the scaleLx where the residual interactionlabeled by the indexA become strong and confine the tecnifermions. The leading contribution is given by a looptechnifermions.

To determine such contribution it is necessary to demthat the renormalized Green functions match when compuusing explicitly the degrees of freedomQL , QR and whentheir effect is described via the effective Lagrangian coecients d i . The matching procedure is illustrated in Fig.The scalem of the matching must be such thatm,M , butsuch thatm.Lx , where perturbation theory in the techncolor coupling constant starts being questionable.

The result of the calculation in the case of degenermasses is

Dd i~m/M !52 d i S 12 e logm2

M2D , ~93!

where we have kept the logarithmically enhanced contrition only and have neglected any other possible conspieces.d i is the singular part ofd i . The finite parts ofd i areclearly very model dependent~cf., for instance, the previousdiscussion on evanescent operators! and we cannot possiblytake them into account in a general analysis. Accordingwe ignore all other terms in Eq.~93! as well as those finitepieces generated through the Fierzing procedure~see discus-sion in the previous section!. Keeping the logarithmicallyenhanced terms therefore sets the level of accuracy ofcalculation. We will call Eq.~92! the short-distance contribution to the coefficientd i . General formulas for the caswhere the two technifermions are not degenerate in macan be found in Appendix E.

Notice that the final short-distance contribution to thed i

is ultraviolet finite, as it should be. The divergences ind i areexactly matched by those inDd i . The pole ind i combinedwith singularity inDd i provides a finite contribution.

There is another potential source of corrections to thed istemming from the renormalization of the four-fermion copling constantG2/M2 ~similar to the renormalization of theFermi constant in the electroweak theory due to gluonchange!. This effect is, however, subleading here. The reais that we are considering technigluon exchange onlyfour-fermion operators of the formJ• j , where, again,j ~J!stands for a~heavy! fermion current~which gives the leadingcontribution, as discussed!. The fields carrying technicolohave the same handedness, and thus there is no multiptive renormalization and the effect is absent.

Of course, in addition to the short-distance contributithere is a long-distance contribution from the region of in

FIG. 6. Matching at the scalem5Lx .

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llys

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arges

heng

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EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

gration of momentam,Lx . Perturbation theory in the technicolor coupling constant is questionable, and we haveresort to other methods to determine the value of thed i at theZ mass.

There are two possible ways of doing so. One is simplymimic the constituent chiral quark model of QCD. There oloop of chiral quarks with momentum running between tscale of chiral symmetry breaking and the scale of the cstituent mass of the quark, which acts as infrared cutprovides the bulk of the contribution@28,29# to f p , which isthe equivalent ofv. Making the necessary translations, wcan write, for QCD-like theories,

v2.nTCnD

mQ2

4p2 logLx

2

mQ2 . ~94!

Alternatively, we can use chiral Lagrangian techniqu@34# to write a low-energy bosonized version of the techfermion bilinearsQLGQL and QRGQR using the chiral cur-rentsJL andJR . The translation is

QLgmQL→v2

2trU†iD mU, ~95!

QLgmt iQL→v2

2trU†t i iD mU, ~96!

QRgmQR→ v2

2trUiD mU†, ~97!

QRgmt iQR→ v2

2trUt i iD mU†. ~98!

Other currents do not contribute to the effective coefficienBoth methods agree.

Finally, we collect all contributions to the coefficientsd iof the effective Lagrangian. For fields in the usual represtations of the gauge group,

d15aLW 2G2

M2 S v21nTCnD

mQ2

4p2 logM2

Lx2 D

21

16p2

yu21yd

2

4 S 1

e2 log

L2

m2D , ~99!

d25S aRW 211

2aR

32D G2

M2 S v21nTCnD

mQ2

4p2 logM2

Lx2 D

21

16p2

~yu1yd!2

8 S 1

e2 log

L2

m2D , ~100!

11403

to

oe

-f,

s-

.

-

d351

2aR3L

G2

M2 S v21nTCnD

mQ2

4p2 logM2

Lx2 D

11

16p2

yu22yd

2

4 S 1

e2 log

L2

m2D , ~101!

d450, ~102!

d551

2aR3R

G2

M2 S v21nTCnD

mQ2

4p2 logM2

Lx2 D

21

16p2

yu22yd

2

4 S 1

e2 log

L2

m2D , ~103!

d651

2aR

32G2

M2 S v21nTCnD

mQ2

4p2 logM2

Lx2 D

21

16p2

~yu2yd!2

4 S 1

e2 log

L2

m2D , ~104!

while in the case of higher representations, where only ctodially preserving operators have been considered, onlyd1andd2 get nonzero values~throughaLW 2 andaRW 2!. The long-distance contribution is, obviously, universal~see Sec. II!,while we have to modify the short-distance contributionreplacing the Casimir of the fundamental representationSU~2! for the appropriate one@1/2→c(R)#, the number ofdoublets by the multiplicity of the given representation, anc by the appropriate dimensionality of the SU~3!c represen-tation to which theQ fields belong.

These expressions require several comments. First ofthey contain the same~universal! divergences as their counterparts in the minimal standard model~MSM!. The scaleLshould, in principle, correspond to the matching scaleLx ,where the low-energy nonlinear effective theory takes ovHowever, we write an arbitrary scale just to remind us ththe finite part accompanying the logarithm is regulator dpendent and cannot be determined within the effecttheory. Recall that the leadingO(nTCnD) term is finite andunambiguous, and that the ambiguity lies in the formasubleading term~which, however, due to the logarithm inumerically quite important!. Furthermore, only logarithmi-cally enhanced terms are included in the above expressiFinally, one should bear in mind that the chiral quark modtechniques that we have used are accurate only in the lnTC expansion~actually nTCnD here!. The same commentapply, of course, to the oblique coefficientsai presented inAppendix E.

The quantitiesaLW 2, aRW 2, aR32, andaR3L and aR3R are the

coefficients of the four-fermion operators indicated by tsubindex ~a combination of Clebsch-Gordan and Fierzifactors!. They depend on the specific model. As discussedprevious sections, these coefficients can be of either sThis observation is important because it shows that the ctribution to the effective coefficients has no definite sign@35#indeed. It is nice that there is almost a one-to-one corresp

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thio

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r-u-

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, as

encecalthe

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

dence between the effective Lagrangian coefficients~all ofthem measurable, at least in principle! and four-fermion co-efficients.

Apart from these four-fermion coefficients, thed i dependon a number of quantities~v, mQ , Lx , G, andM!. Let usfirst discuss those related to the electroweak symmbreaking~mQ andLx! and postpone the considerations onMto the next section@G will be assumed to be ofO~1!#. v isof course the Fermi scale and hence not an unknown a(v.250 GeV). The value ofmQ can be estimated from Eq~94! sincev2 is known andLx , for QCD-like technicolortheories, is;4pv. Solving for mQ , one finds that ifnD54, mQ.v, while if nD51, mQ.2.5v. Notice thatmQ andv depend differently onnTC, so it is not correct to simplyassumemQ.v. In theories where the technicolorb functionis small ~and it is pretty small ifnD54 and nTC52!, thecharacteristic scale of the breaking is pushed upwards, soexpectLx@4pv. This bringsmQ somewhat downwards, buthe decrease is only logarithmic. We shall therefore takemQto be in the range 250–450 GeV. We shall allow for a msplitting within the doublets too. The splitting within eacdoublet cannot be too large, as Fig. 4 shows. For simplicwe shall assume an equal splitting of masses for all doub

IX. RESULTS AND DISCUSSION

Let us first summarize our results so far. The values ofeffective Lagrangian coefficients encode the informatabout the symmetry breaking sector that is~and will be in thenear future! experimentally accessible. Thed i are thereforethe counterpart of the oblique corrections coefficientsai ,and they have to be taken together in precision analysithe standard model, even if they are numerically less signcant.

These effective coefficients apply toZ physics at LEP, topquark production at the Next Linear Collider, measuremeof the top decay at the Collider Detector at Fermilab~CDF!,or indeed any other process involving the third generat~where their effect is largest!, provided the energy involvedis below 4pv, the limit of applicability of chiral techniques~Of course, chiral effective Lagrangian techniques fail wbelow 4pv if a resonance is present in a given channel:also @36#.!

In the standard model thed i are useful to keep track of thlogMH dependence in all processes involving either neuor charged currents. They also provide an economicalscription of the symmetry-breaking sector, in the sensethey contain the relevant information in the low-energygime, the only one testable at present. Beyond the stanmodel the new physics contributions is parametrized by fofermion operators. By choosing the number of doublets,mQ ,M, andLx suitably, we are in fact describing in a single sha variety of theories: extended technicolor~commuting andnoncommuting!, walking technicolor @37# or top-quark-assisted technicolor, provided that all remaining scalarspseudo Goldstone bosons are sufficiently heavy.

The accuracy of the calculation is limited by a numberapproximations we have been forced to make and whhave been discussed at length in previous sections. In p

11403

ry

all

we

s

,ts.

en

of-

ts

n

le

le-at-rd

r-

t

d

fhc-

tice, we retain only terms which are logarithmically ehanced when running fromM to mQ , including the long-distance part, below Lx . The effective Lagrangiancoefficientsd i are all finite at the scaleLx , the lower limitof applicability of perturbation theory. Below that scale thrun following the renormalization group equations of tnonlinear theory and new divergences have to be subtrac7

These coefficients contain finally the contribution froscalesM.m.mQ , the dynamically generated mass of thtechnifermion@expected to be ofO(LTC).# In view of thetheoretical uncertainties, to restrict oneself to logarithmicaenhanced terms is a very reasonable approximation wshould capture the bulk of the contribution.

Let us now proceed to a more detailed discussion ofimplications of our analysis. Let us begin by discussingvalue that we should take forM, the mass scale normalizinfour-fermion operators. Fermion condensation gives a mto ordinary fermions via chirality-flipping operators of ord

mf.G2

M2 ^QQ&, ~105!

through the operators listed in Table II. A chiral quark modcalculation shows that

^QQ&.v2mQ . ~106!

Thus, while^QQ& is universal, there is an inverse relatiobetweenM2 andmf . In QCD-like theories this leads to thfollowing rough estimates for the massM @the subindex re-fers to the fermion which has been used in the left-hand s~LHS! of Eq. ~105!#

Me;105 TeV, Mm;10 TeV, Mb;3 TeV. ~107!

If taken at face value, the scale forMb is too low: even theone for Mm may already conflict with current bounds oFCNC, unless they are suppressed by some other mechain a natural way. Worse, the top quark mass cannot besonably reproduced by this mechanism. This well-knoproblem can be partly alleviated in theories where techcolor walks or invoking top quark color or a similar mechnism@38#. ThenM can be made larger andmQ , as discussedsomewhat smaller. For theories which are not vector like,above estimates become a lot less reliable.

However, one should not forget that none of the foufermion operators playing a role in the vertex effective coplings participates at all in the fermion mass determinatiIn principle, we can then entertain the possibility that trelevant mass scale for the latter should be lower~perhapsbecause they get a contribution through scalar exchange

7The divergent contribution coming from the standard modeld i ’shas to be removed, though, as discussed in Sec. V, so the differis finite and would be fully predictable had we good theoreticontrol on the subleading corrections. At present onlyO(nTC ,nD) contribution is under reasonable control.

5-16

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rd

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

FIG. 7. Oblique and vertex corrections for thelectron effective couplings. The ellipses indicathe 1-s experimental region. Three values of theffective massm2 are considered: 250 GeV~a!,350 GeV ~b!, and 450 GeV~c! and two split-tings: 10% ~right! and 20%~left!. The dottedlines correspond to including oblique correctiononly. The coefficients of the four-fermion operators vary in the range@22, 2#, and this spans theregion between the two solid lines. The standamodel prediction~thick solid line! is shown formt5175.6 GeV and 70<MH<1500 GeV.

-

fa

-a

ite

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onb

nl

s

frw

arsdsu

anffi

tec-o

th-

erin

theex--iton

er

geal-

o-

is-

ses,

of

ic-

GeV

doesntalue

ffi-

some of them can be generated this way!. Even in this case itseems just natural thatMb ~the scale-normalizing chiralitypreserving operators for the third generation, that is! is lowand not too different fromLx . Thus the logarithmic en-hancement is pretty much absent in this case and some oapproximations made become quite questionable in this c~Although even for theb couplings there is still a relativelylarge contribution to thed i ’s coming from long-distance contributions.! Put in other words, unless an additional mechnism is invoked, it is not really possible to make definestimates for theb effective couplings without getting intothe details of the underlying theory. The flavor dynamics aelectroweak breaking are completely entangled in this cIf one only retains the long-distance part~which is what wehave done in practice!, we can, at best, make order-omagnitude estimates. However, what is remarkable in ais that this does not happen for the first and second gention vertex corrections. The effect of flavor dynamics cthen be encoded in a small number of coefficients.

We shall now discuss in some detail the numerical csequences of our assumptions. We shall assume the avalues for the mass scaleM; in other words, we shall placeourselves in the most disfavorable situation. We shall opresent results for QCD-like theories andnD54 exclusively.For other theories the appropriate results can be very eaobtained from our formulas. For the coefficientsaLW 2, aR3R ,

aR3L , etc., we shall use the range of variation@22,2# @sincethey are expected to be ofO~1!#. Of course, larger values othe scaleM would simply translate into smaller values fothose coefficients, so the results can be easily scaled do

Figure 7 shows thegAe , gV

e electron effective couplingswhen vertex corrections are included and allowed to vwithin the stated limits. To avoid clutter, the top quark mais taken to the central value 175.6 GeV. The standard moprediction is shown as a function of the Higgs boson maThe dotted lines in Fig. 7 correspond to considering obliqcorrections only. Vertex corrections change these resultsdepending on the values of the four-fermion operator coecients, the prediction can take any value in the strip limiby the two solid lines~as usual, we have no specific predition in the direction along the strip due to the dependenceL, inherited from the non renormalizable character ofeffective theory!. A generic modification of the electron couplings is of O(1025), small, but much larger than in thstandard model and, depending on its sign, may help to b

11403

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-

de.

yra-

-ove

y

ily

n.

ysels.ed,-

d

ne

g

a better agreement with the central value.The modifications are more dramatic in the case of

second generation, for the muon, for instance. Now, wepect changes in thed i ’s and, eventually, in the effective couplings of O(1023). These modifications are just at the limof being observable. They could even modify the relatibetweenMW andGm ~i.e., Dr !.

Figure 8 shows a similar plot for the bottom effectivcouplingsgA

b , gVb . It is obvious that taking generic values fo

the four-fermion operators@of O~1!# leads to enormousmodifications in the effective couplings, unacceptably larin fact. The corrections become more manageable if welow for a smaller variation of the four-fermion operator cefficients ~in the range@20.1,0.1#!. This suggests that thenatural order of magnitude for the massMb is ; 10 TeV, atleast for chirality preserving operators. As we have dcussed, the corrections can be of either sign.

One could, at least in the case of degenerate mastranslate the experimental constraints on thed i ~recall thattheir experimental determination requires a combination

FIG. 8. Bottom effective couplings compared to the SM predtion for mt5175.6 as a function of the Higgs boson mass~in therange@70,1500# GeV!. The ellipses indicate 1-, 2-, and 3-s experi-mental regions. The dynamically generated masses are 250~a!, 350 GeV~b!, and 400 GeV~c!, and we show a 20% splittingbetween the masses in the heavy doublet. The degenerate casenot present quantitative differences if we consider the experimeerrors. The central lines correspond to including only the obliqcorrections. When we include the vertex corrections~depending onthe size of the four-fermion coefficients!, we predict the regionsbetween lines indicated by the arrows. The four-fermion coecients in this case take values in the range@20.1,0.1#.

5-17

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E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

charged and neutral processes, since there are six of the! tothe coefficients of the four-fermion operators. Doingwould provide us with a four-fermion effective theory thwould exactly reproduce all the available data. It is obviohowever, that the result would not be very satisfactoWhile the outcome would, most likely, be coefficientsO~1! for the electron couplings, they would have to beO(1021), perhaps smaller for the bottom. Worse, the samasses we have used lead to unacceptably low values fotop quark mass~105!. Allowing for a different scale in thechirality-flipping operators would permit a large top quamass without affecting the effective couplings. Taking thisa tentative possibility, we can pose the following problemeasure the effective couplingsd i for all three generationsand determine the values of the four-fermion operator coficients and the characteristic mass scale that fits thebest. In the degenerate mass limit we have a total of 8knowns@5 of them coefficients, expected to be ofO~1!# and18 experimental values~three sets of thed i!. A similar exer-cise could be attempted in the chirality-flipping sector. If tsolution to this exercise turned out to be mathematically csistent ~within the experimental errors!, it would be ex-tremely interesting. A negative result would certainly ruout this approach. Notice that dynamical symmetry breakpredicts the patternd i;mf , while in the standard modeld i

;mf2.

We should end with some words of self-criticism. It maseem that the previous discussion is not too conclusivethat we have managed only to rephrase some of the lostanding problems in the symmetry breaking sector. Hoever, theraison d’etre of the present paper is not really tpropose a solution to these problems, but rather to establtheoretical framework to treat them systematically. Expeence from the past shows that often the effects of new phics are magnified and thus models are ruled out on this baonly to find out that a careful and rigorous analysis leasome room for them. We believe that this may be the casdynamical symmetry-breaking models, and we believethat only through a detailed and careful comparison withexperimental data will progress take place.

The effective Lagrangian provides the tools to look for‘‘existence proof’’ ~or otherwise! of a phenomenologicallyviable, mathematically consistent dynamical symmetbreaking model. We hope that there will be any time sosufficient experimental data to attempt to determine the fofermion coefficients, at least approximately.

ACKNOWLEDGMENTS

We would like to thank M. J. Herrero, M. Martinez,Matias, S. Peris, J. Taron, and F. Teubert for discussionsE. wishes to thank the hospitality of the SLAC Theory Growhere this work was finished. J.M. acknowledges financsupport from Generalitat de Catalunya, grant 1998FI-006This work has been partially supported by CICYT graAEN950590-0695 and CIRIT contract GRQ93-1047.

APPENDIX A: d54 OPERATORS

The procedure we have followed to obtain Eqs.~8!–~15!is very simple. We have to look for operators of the for

11403

,.

fethe

s:

f-ta

n-

-

g

ndg--

ai-s-is,sinoe

-nr-

D.

l4.t

cGc, wherec5qL ,qR andG contains a covariant derivativDm and an arbitrary number ofU matrices. These operatormust be gauge invariant so not any form ofG is possible.Moreover, we can drop total derivatives and, sinceU is uni-tary, we have the following relation:

DmU52U~DmU !†U. ~A1!

Apart from the obvious structureDmU which transform asUdoes, we immediately realize that the particular form ofGRimplies the following simple transformations for the combnationsUt3U† and (DmU)t3U†:

Ut3U†U†°GLUt3U†GL† , ~A2!

~DmU !t3U†°GL~DmU !t3U†GL† . ~A3!

Keeping all these relations in mind, we simply write dowall the possibilities forcGc and find the list of operators~8!–~15!. It is worth mentioning that there appears toanother family of four operators in which theU matrices alsooccur within a trace:cGc tr G8. One can check, howeverthat these are not independent. More precisely,

i qLgmqLtr~DmU !t3U†5L43, ~A4!

i qLgmUt3U†qLtr~DmU !t3U†52L411L4

4, ~A5!

i qRgmqRtr~DmU !t3U†5L45, ~A6!

i qRgmt3qRtr~DmU !t3U†5L421L4

6 . ~A7!

Note thatL47 ~as well asLR8 discussed above! can be re-

duced by equations of motion to operators of lower dimesion which do not contribute to the physical processes weinterested in. We have checked that its contribution indedrops from the relevantS-matrix elements.

APPENDIX B: FEYNMAN RULES

We write the effectived54 Lagrangian as

Leff5d8LR81 (k51

7

dkL4k , ~B1!

where dk are real coefficients that we have to determithrough the matching. We need to match the effective thedescribed byLeff to both the MSM and the underlying theorparametrized by the four-fermion operators. It has provmore convenient to work with the physical fieldsW6, Z, andg in the former case, whereas the use of the Lagrangian fiW1, W2, W3, andB is clearly more straightforward for thelatter. Thus we give the Feynman rules in terms of bothphysical and unphysical basis:

5-18

EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

~B2!

~B3!

~B4!

~B5!

~B6!

The operatorsL47 andL48 contribute to the two-point function. The relevant Feynman rules are

~B7!

~B8!

unthit

o-e

g ofow

Rather than giving the actual Feynman rules in thephysical basis, we collect the various tensor structurescan result from the calculation of the relevant diagramsTable IV. We include only those that can be matchedinsertions of the operatorsL4

1,...,L46 ~the contributions toL4

7

11403

-atno

and L48 can be determined from the matching of the twpoint functions!. The corresponding contributions of thesstructures tod1 ,...,d6 are also given in Table IV. Onced7has been replaced by its value, obtained in the matchinthe two-point functions, only the listed structures can sh

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alar

tan-

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t but

e

-

on-r

he

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

up in the matching of the vertex: otherwise, the SU~2!3U~1!symmetry would not be preserved.

APPENDIX C: FOUR-FERMION OPERATORS

The complete list of four-fermion operators relevant fthe present discussion is in Tables I and II in Sec. VI. Italso explained in Sec. VI the convenience of Fierzingoperators in the last seven rows of Table I in order to wthem in the formJ• j . Here we just give the list that comeout naturally from our analysis, Tables I and II, without futher physical interpretation. The list is given for fermiobelonging to the representation3 of SU~3!c ~techniquarks!.By using Fierz transformations one can easily find out retions among some of these operators when the fermionscolor singlet~technileptons!, which is telling us that some othese operators are not independent in this case. A lisindependent operators for technileptons is also given in SVI.

Let us outline the procedure we have followed to obtthis basis in the~more involved! case of colored fermions.

There are only two color singlet structures one can buout of four fermions, namely,

~ cc!~c8c8![cacacb8cb8 ~C1!

~ clW c!•~ c8lW c8![ca~lW !abcb•cg8~lW !gdcd8 , ~C2!

wherec stands for any field belonging to the representat3 of SU~3!c ~c will be either q or Q!, a,b, . . . , arecolorindices, and the primes~8! remind us thatc andc carry sameadditional indices@Dirac, SU~2!, . . . #.

Next, we clasify the Dirac structures. Sincec is eithercL@it belongs to the representation~1

2,0! of the Lorentz group#or cR @representation~0,1

2!#, we have five sets of fields toanalyze, namely,

$cL ,cL ,cL8cL8% @R↔L#, $cL ,cL ,cR ,cR%, ~C3!

$cL ,cR ,cL8 ,cR8 % @R↔L#. ~C4!

There is only an independent scalar we can build with eof the three sets in Eq.~C3!. Our choice is

TABLE IV. Various structures appearing in the matching of tvertex and the corresponding contributions tod1 ,¯,d6 .

Tensor structure d1 d2 d3 d4 d5 d6

i qLg@t1W” 11t2W” 2#qL1 1

i qLt3@gW” 32g8B” #qL1 21

i qL@gW” 32g8B” #qL21

i qRg@t1W” 11t2W” 2#qR21 1

i qRt3@gW” 32g8B” #qR21 21

i qR@gW” 32g8B” #qR21

11403

ee

-re

ofc.

d

n

h

cLgmcLcL8gmcL8 @R↔L#, ~C5!

cLgmcLcRgmcR , ~C6!

where the prime is not necessary in the second equabecauseR and L suffice to remind us that the twocand c may carry different @Su~2!, technicolor, . . . # in-dices. There appear to be four other independent scoperators: cLgmcL8cL8gmcL @R↔L#, cLcRcRcL , and

cLsmncRcRsmncL . However, Fierz symmetry implies thathe first three are not independent, and the fourth one vishes, as can be also seen using the identity 2ismng5

5emnrlsrl . For each of the two operators in Eq.~C4!, twoindependent scalars can be constructed. Our choice is

cLcRcL8cR8 @R↔L#, ~C7!

cLcR8cL8cR @R↔L#. ~C8!

Again, there appear to be four other scalar operatocLsmncRcL8smncR8 @R↔L#, cLsmncR8 cL8smncR @R↔L#,which, nevertheless, can be shown not to be independenrelated to Eqs.~C7! and ~C8! by Fierz symmetry. To sum-marize, the independent scalar structures are Eqs.~C5!, ~C6!,~C7!, and~C8!.

Next, we combine the color and the Dirac structures. Wdo this for the different cases~C5!–~C8! separately. For op-erators of the form~C5!, we have the two obvious possibilities ~hereafter, color and Dirac indices will be implicit!

~ cLgmcL!~ cL8gmcL8 ! @R↔L#, ~C9!

~ cLgmcL8 !~ cL8gmcL! @R↔L#, ~C10!

where fields in parentheses have their color indices ctracted as in Eqs.~C1! and ~C2!. Note that the operato(cLgmlW cL)•(cL8gmlW cL8), or its R version, is not indepen-

dent @recall that (lW )ab•(lW )gd52daddbg22/3dabdgd#. Foroperators of the form~C6!, we take

~ cLgmcL!~ cRgmcR!, ~C11!

~ cLgmlW cL!•~ cRgmlW cR!. ~C12!

Finally, for operators of the form~C7! and ~C8!, our choiceis

~ cLcR!~ cL8cR8 ! @R↔L#, ~ cLlW cR!•~ cL8lW cR8 ! @R↔L#,~C13!

~ cLcR8 !~ cL8cR! @R↔L#, ~ cLlW cR8 !•~ cL8lW cR! @R↔L#.~C14!

5-20

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EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

All them are independent unless further symmetries@e.g.,SU~2!L3SU~2!R# are introduced.

To introduce the SU~2!L3SU~2!R symmetry, one just assigns SU~2! indices~i, j, k, . . . ! to each of the fields in Eqs~C9!–~C14!. We can drop the primes hereafter since thereno other symmetry left but technicolor, which for the preseanalysis is trivial~recall that we are only interested in foufermion operators of the formQQqq; thus, technicolor indi-ces must necessarily be matched in the obvious wQAQAqq!. For each of the operators in Eqs.~C9! and~C10!,there are two independent ways of constructing SU~2!L3SU~2!R invariants. Only two of the four resulting operatoturn out to be independent~actually, the other two are exactly equal to the first ones!. The independent operators achosen to be

~ cLi gmcL

i !~ cLj gmcL

j ![~cLgmcL!~ cLgmcL! @R↔L#,~C15!

~ cLi gmcL

j !~ cLj gmcL

i ! @R↔L#. ~C16!

For each of the operators in Eqs.~C11!–~C14!, the samestraightforward group analysis shows that there is onlyway to construct an SU~2!L3SU~2!R invariant. Discardingthe redundant operators and imposing hermiticity andCPinvariance, one finally has, in addition to the operators~C15!and ~C16!, those listed below@from now on, we understandthat fields in parenthesis have their Dirac, color, and aflavor indices contracted as in Eq.~C15!#:

~ cLgmcL!~ cRgmcR!, ~C17!

~ cLgmlW cL!•~ cRgmlW cR!, ~C18!

~ cLi cR

j !~ cLkcR

l !e ike j l 1~ cRi cL

j !~ cRk cL

l !e ike j l , ~C19!

~ cLi lW cR

j !•~ cLklW cR

l !e ike j l 1~ cRi lcL

j !•~ cRk lcL

l !e ike j l .~C20!

We are now in a position to obtain very easily the cusdially preserving operators of Tables I and II. We simpreplacec by q andQ ~a pair of each: a field and its conjugate! in all possible independent ways.

To break the custodial symmetry we simply insertt3 ma-trices in theR sector of the custodially preserving operatowe have just obtained~left columns of Tables I and II!. How-ever, not all the operators obtained this way are indepensince one can prove the following relations:

~ qRi gmQR

j !~QRj gm@t3qR# i !

5~ qRgmt3QR!~QRgmqR!1~ qRgmQR!~QRgmt3qR!

2~ qRi gm@t3QR# j !~QR

j gmqRi ! ~C21!

11403

ist

y:

e

o

-

nt

~ qRi gm@t3QR# j !~QR

j gm@t3qR# i !

5~ qRgmQR!~QRgmqR!1~ qRgmt3QR!~QRgmt3qR!

2~ qRi gmQR

j !~QRj gmqR

i ! ~C22!

~ qRi gm@t3qR# j !~QR

j gm@t3QR# i !

5~ qRgmqR!~QRgmQR!1~ qRgmt3qR!~QRgmt3QR!

2~ qRi gmqR

j !~QRj gmQR

i ! ~C23!

~ qRi gm@t3qR# j !~QR

j gmQRi !1~ qR

i gmqRj !~QR

j gm@t3QR# i !

5~ qRgmqR!~QRgmt3QR!1~ qRgmt3qR!~QRgmQR!.

~C24!

Our final choice of custodially breaking operators is tone in the right columns of Tables I and II.

APPENDIX D: RENORMALIZATION OF THE MATTERSECTOR

Although most of the material in this section is standait is convenient to collect some of the important expressioas the renormalization of the fermion fields is somewhatvolved and also to set up the notation. Let us introduce thwave-function renormalization constants for the fermifields:

S udD

L

→ZL1/2S u

dDL

,

uR→~ZRu !1/2uR , ~D1!

dR→~ZRd !1/2dR,

whereu(d) stands for the field of the up-type~down-type!fermion. We write

Zi511dZi . ~D2!

We also renormalize the fermion masses according to

mf→mf1dmf , ~D3!

wheref 5u,d. These substitutions generate the counterteneeded to cancel the UV divergences. The correspondFeynman rules are

5-21

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

~D4!

~D5!

~D6!(L)

a l

~D7!

dia

zeh

o

eof

slys.pe

onsenta-s

Here we have introduced the notation

dZL5dZVu,d1dZA

u,d , dZRu,d5dZV

u,d2dZAu,d ~D8!

and

v f5I f

322QfsW2

2sWcW, af5

I f3

2sWcW. ~D9!

Note that the Feynman rules for the vertices contain adtional renormalization constants which should be familfrom the oblique corrections.

The fermion self-energies can be decomposed as

S f~p!5p8SVf ~p2!1p8g5SA

f ~p2!1mSSf ~p2!. ~D10!

By adding the conterterms one obtains the renormaliself-energies, which admit the same decomposition. One

SVf ~p2!5SV

f ~p2!2dZVf , ~D11!

SAf ~p2!5SA

f ~p2!1dZAf , ~D12!

SSf ~p2!5SS

f ~p2!1dmf

mf1dZV

f , ~D13!

where the caret denotes renormalized quantities. Theshell renormalization conditions amount to

11403

i-r

das

n-

dmu,d

mu,d52SV

u,d~mu,d2 !2SS

u,d~mu,d2 !, ~D14!

dZVd5SV

d~md2!12md

2@SVd8~md

2!1SSd8~md

2!#, ~D15!

dZAu,d52SA

u,d~mu,d2 !, ~D16!

whereS8(m2)5@]S(p2)/]p2#p25m2. Equation~D14! guar-antees thatmu , md are the physical fermion masses. Thother two equations come from requiring that the residuethe down-type fermion be unity. One cannot simultaneouimpose this condition to both up- and down-type fermionActually, one can easily work out the residue of the up-tyfermions, which turns out to be 11d res with

d res5SVu~mu

2!12mu2@SV

u8~mu2!1SS

u8~mu2!#. ~D17!

APPENDIX E: EFFECTIVE LAGRANGIANCOEFFICIENTS

In this appendix we shall provide the general expressifor the coefficientsai andd i in theories of the type we havbeen considering. The results are for the usual represetions of SU~2!3SU~3!c . Extension to other representationis possible using the prescriptions listed in Sec. VIII:

5-22

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EFFECTIVE ELECTROWEAK CHIRAL LAGRANGIAN: . . . PHYSICAL REVIEW D 60 114035

a05nTCnD

64p2MZ2sW

2 S m221m1

2

21

m12m2

2 ln ~m12/m2

2!

m222m1

2 D1

1

16p2

3

8 S 1

e2 log

L2

m2D , ~E1!

a152nTCnD

96p2 1nTC~nQ23nL!

3396p2 lnm1

2

m22

11

16p2

1

12S 1

e2 log

L2

m2D , ~E2!

a852nTC~nc11!

96p2

1

~m222m1

2!

3H 5

3m1

4222

3m2

2m221

5

3m2

4

1~m2424m2

2m121m1

4!m2

21m12

m222m1

2 lnm1

2

m22J , ~E3!

wherenTC is the number of technicolors~taken equal to 2 inall numerical discussions!, andnD is the number of techni-doublets. It is interesting to note that all effective Lagrangcoefficients~except fora1! depend onnD and are independent of the actual hypercharge~or charge! assignment.nQandnL are the actual number of techniquarks and techniltons. In the one-generation modelnQ53, nL51, and, conse-quently,nD54. Furthermore, in this modela1 is mass inde-pendent. For simplicity, we have writtenm1 for thedynamically generated mass of theu-type technifermion andm2 for the one of thed-type, and assumed that they are tsame for all doublets. This is of course quite questionablea large splitting between the technielectron and tenineutrino seems more likely and they should not necessacoincide with techniquark masses, but the appropriatepressions can be easily inferred from the above formuanyway:

d15nDnTCG2

16p2M2 aLW 2H m121m2

2

22m1

2S 11m1

2

m122m2

2D logm1

2

M2

2m22S 11

m22

m222m1

2D logm2

2

M2J , ~E4!

11403

n

-

as-

ilyx-s

d25nDnTCG2

16p2M2 $~aLR32aRR3

!A21aR32A11aRW 2

B1%,

~E5!

d35nDnTCG2

16p2M2 $~aL22aRL!A21aR3LA1%, ~E6!

d45nDnTCG2

16p2M2 aLW 2H m121m2

2

21m1

2S 12m1

2

m122m2

2D logm1

2

M2

1m22S 12

m22

m222m1

2D logm2

2

M2J , ~E7!

d55nDnTCG2

16p2M2 $~aLR2aR2!A21aR3RA1%, ~E8!

d65nDnTCG2

16p2M2 $~aLR32aRR3

!A21aR32A11aRW 2

B2%,

~E9!

d750, ~E10!

where

A657m12 log

m12

M22m22 log

m22

M2 , ~E11!

B6562m1m22m12S 16

2m1m2

m122m2

2D logm1

2

M2

2m22S 16

2m2m1

m222m1

2D logm2

2

M2 . ~E12!

We have not bothered to write the chiral divergence counterms in the above expressions. They are identical to thosSec. VIII. Although we have written the full expressions otained using chiral quark model methods, one should be waware of the approximations made in the text.

,

g.

.

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v.

i,

v.

eM

s.

yss.

21.

or

.

s.

ry

. B

.

.

E. BAGAN, D. ESPRIU, AND J. MANZANO PHYSICAL REVIEW D60 114035

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A. Longhitano, ibid. 22, 1166 ~1980!; A. Longhitano, Nucl.Phys.B188, 118~1981!; R. Renken and M. Peskin,ibid. B211,93 ~1983!.

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