technicolour edward farhi and leonard susskindshufang/phys579c_2006/termpaper/... · e. farhi and...

PHYSICSREPORTS(Review Sectionof PhysicsLetters)74, No. 3 (1981)277—321. North-HollandPublishingCompany

TECHNICOLOUR

Edward FARHICERN, Geneva, Switzerland

and

Leonard SUSSKINDPhysicsDepartment, Stanford University, Stanford. U.S.A.

Received12 March1981

Contents:

1. Introduction 279 4.2. Four Fermi interactionsandquark masses 3012. TechnIcolour 284 4.3. Four FermiinteractionsandGoldstonebosons 303

2.1. A simplemodel without scalars 284 5. The largerpicture 3042.2. Technicolour 286 5.1. Extendedtechnicolourorsideways 3052.3. Technispectroscopy 288 5.2. Grandunified theoriesand technicolour 3102.4. Models andalternatives 290 5.3. Groupsbreakingthemsdves 311

3. Pseudo-Goldstonebosons 291 5.4. Vacuumalignment 3123.1. Goldstonebosonsandpseudo-Goldstonebosons 292 6. Compositefermions 3153.2. Pseudosin theonefamily model 292 6.1. Light compositefermions 3153.3. The Goldstonespectrum 293 6.2. ‘t Hooft’s conditions 316

4. Effective four Fermiinteractions 300 6.3. Technifermionsaspreons 3174.1. Other interactionsandscales 300 References 319

Abstract:The ideasof TechnicolourorDynamical Symmetry Breakingfor theweakinteractionsarereviewed.The needfor technicolouris established

becauseof thegaugehierarchyproblem.Theobviousphenomenologicalconsequencesareexploredwith emphasison thepossibleexistenceof light(lessthan 100 GeV) scalarparticles.The extendedtechnicolouror sidewaysgeneralizationsof technicolourarediscussed.The relationsbetweentechnicolourandgrand unified theories, groupsbreakingthemselvesand vacuum alignment are also explored.Finally thereis a discussionoftechnicolourandthepossiblecompositestructureof quarksand leptons.

Single orders for this issue

PHYSICS REPORTS(Review Sectionof PhysicsLetters)74, No. 3 (1981) 277—321.

Copies of this issuemay be obtainedat the price given below. All ordersshould besentdirectly to the Publisher.Ordersmust beaccompaniedby check.

Single issuepriceDfi. 21.00, postageincluded.

TECHNICOLOUR

Edward FARM!

CERN,Geneva,Switzerland

and

Leonard SUSSK1ND

PhysicsDepartment,StanfordUniversity, Stanford, U.S.A.

t931 . 1981

(PNORTH~HOLLANDPUBUSHING COMPANY -AMSTERDAM

E. Farhi and L. Susskind, Technicolour 279

1. Introduction

According to current thinking, the spectrumof massesof particles originatesin a hierarchy ofsymmetry breakingscales [1]. At the lowest energiesthe world appearsexactly symmetric under asymmetry group SU(3)X U(1), i.e., the strong interaction colour symmetry and the symmetry of

• electromagnetism.Various additional approximatesymmetriessuch as isospin are also manifestbuttheseappearto be accidentalconsequencesof the massspectrum.For exampleit is believedthat theorigin of isospinandchiral symmetryis thefact that the u andd quarkshappento bea gooddeallighterthanthe scaleA~ of quantumchromodynamics[2].

Neverthelessthe real symmetryof the laws of natureis probablymuchbiggerthanSU(5)x U(1) butmost of this symmetry is hidden by spontaneousbreakdown.The spontaneousbreakdownof asymmetry is a phenomenonwhich occurs at a characteristicscaleand it is thesesymmetry breakingscaleswhich appearto control the particle spectrum.Typically, the spontaneousbreakdownof asymmetryis a low energyphenomenonin the sensethat amplitudesinvolving momentalargerthanthecharacteristicbreakdownscaleappearalmostsymmetricwhile below this scaleamplitudesaregrosslyasymmetric.Thus Naturecan be viewedin termsof ahierarchyof spontaneouslybrokensymmetries.As the energyincreasesthe size of the apparentsymmetrygroupalwaysincreases[3].

For example, at energiesin excess of a few hundred GeV the standardWeinberg—Salam[4]electroweak theory predicts that the manifest symmetry expands from SU(3)QCDX U(l%~toSU(3)QCDx {SU(2) x U(1)}Ew. Speculationsabout a further increaseto SU(5) at —. i0’~GeV are verypopular[5].

Typically the breakdownof a symmetryis signalledby an orderparameter.An orderparameteris anon-vanishingvacuumexpectationvalueof some local field which transformsnon-trivially under thesymmetrygroup.For examplein masslessQCD with two flavours u, d the orderparameteris

(Uu+dd)~0 (1.1)

which breakschiral SU(2)x SU(2). In the standardSU(2)LEFTx U(l)Hyp electroweaktheory [4] the

symmetrybreakingorderparameteris (so),the expectationvalueof the Higgs doublet. —

Theorderparameteris generallyadimensionalquantity.Thus in QCD the condensate(Uu + dd) hasthedimensionsof masscubedwhile theHiggsfield in theWeinberg—Salamtheoryhasdimensionsof rnassThevalue of the orderparameteris determinedby the symmetrybreakingscale.In QCD (Uu + dd)(250MeV)3 andin theWeinberg—Salamtheory(q’) -~ (250GeV).

An exact unbrokensymmetry often implies certaindefinite propertiesor relationshipsin atheory.For examplethe equalityof masseswithin amultipletor, in the caseof chiral symmetries,the vanishingof certainmassescan result from unbrokensymmetries.However,the spontaneousbreakdownof thesymmetry will usually causeviolationsof such relationships.Nevertheless,the fact that the symmetrybreaking is a low energyphenomenonpreventsthe violations of symmetry relations from beingarbitrarilylarge.Generallythe massesor massdifferencesresultingfrom symmetrybreakdownwill beon the orderof the breakingscaleor smaller.

In this light it is interestingto realizethat all the ordinaryfundamentalparticles,quarks,leptonsandintermediatevectorbosonswould havevanishingmassesif the electroweaksymmetrywereunbroken.The fermionsform left-handedSU(2)~..~doubletsandright-handedsinglets.Fermionmasstermsareproductsof left- andright-handedfields andcannotbe SU(2)~-~’invariant.Fermion massescan onlyariseout of the SU(2)~F-rbreakdown.Similarly the intermediatevector bosonsW~,Z being gauge

280 E. Farhi and L. Sueskind, Technicolour

bosonsmust be masslessin the absenceof symmetry breaking. It follows that the massesof theseparticles should be on the order of or smaller than the symmetry breakingscale —250 GeV. Anyparticlewhosemassis in excessof 250 GeVmust havean SU(2)i..~vrX U(l)~ypallowedmass.We willsayit is “unprotected”by SU(2)~F.rx U(l)j~y~.In fact, it is a reasonableexpectationthat anyparticlewhich is unprotectedwill havea massdeterminedby somelarger scaleandaccordinglywill be veryheavy. This ideasuggeststhat thepresentlyobservablespectrumis merelythe “tip of the iceberg”withmanyunprotectedmassesatmuchhigher scales.

Look carefully atthe breakingof SU(2)~..~X U(1)~ypandhowit gives massesto the Wth, Z andthefermions. According to the standardtheory [41in addition to quark fields q = (u,d), lepton fields� = (i’, e) and SU(2)~,~.x U(1)~ypgaugefields W~,B there exists a scalarSU(2)~.,~doublet ç =

(ç°,ç~).The SU(2)~..~x U(l)~ypgaugesymmetry is presumedto be spontaneouslybroken by thenon-vanishingvacuumexpectationvalue(~°)250GeV. This orderparametersets the scalefor allsymmetry breakingphenomenaincluding fermion andW~,Z masses.Fermionmassesare,of course,excludedfrom the Lagrangianby SU(2)L~.rx U(l)~~invariance.However,Yukawacouplingsof theform:

yUqL = yu(ULuRç° +dLuR4~)

and

YCJ4L QdR= ~ — üLdRc) (1.2)

areallowed.(~,is definedas ~uc7.) The breakingof SU(2)~..~x U(1)~ypby (ç°)implies thatquarks

movethroughthe vacuumwith effectivemasses:m~= y~,4o°), md = Yd(q’°). (1.3)

Herewe havean exampleof protectedparticlesreceivingmasseson theorderof symmetrybreakingscale.

Next considerthegenerationof massof the electroweakgaugebosons.Theseparticles,beinggaugebosons,are also protectedin thesymmetry limit. To see howtheygainmass,first considertheHiggssectorwhenthe gaugecouplingsareturnedoff. The Lagrangianof theHiggs field,

(1.4)

is chosen so that the vacuum expectationvalue of ~° is —250GeV. The Lagrangian 2’ has afour-dimensionalrotation invariancewhich is mademanifestby writing:

iç=ç1+iq’2, i~°ç3+iç4. (1.5)

So

2= ö ço18~’ç~+V(ç1tp,). (1.6)

The expectationvalueof ç may be takenas:


(~~)= (c2) = (~~)= 0

(~~)= F =250GeV (1.7)

which breaks0(4) down to 0(3).This breakingrequiresthreemasslessGoldstonebosonsof charge±1,0which transformas a three-vectorundertheunbroken0(3).Let uscall theseGoldstonebosons17~andJ70•

TheseGoldstonebosonshavenon-vanishingcouplingsto theweakcurrentswhich coupleto theW5,W°andB. In particular

(0IJ~°I11~’°)= Fq~.

and

(0IJ~]Hb)= Fq. (1.8)

whereJ~’°arethe threeSU(2)~.-~currentsandJ~is the hyperchargecurrent.The W’s coupleto thecurrentsJ~’°with strengthg

212 andthe B couplesto J~with strengthgiI2 so thediagramsin whichgaugebosonsturn into GoldstonebosonshavethevaluesgiF/2 andg2F/2 and areshownin fig. 1.

Now considerthe propagatorof the gaugebosonsW~.As in electrodynamicsit can be expressedas

~ _g~,,q~qJq2 19

— q2(1+ 11(q2))where11(q2) is the vacuumpolarization.The interestingterm in 11(q2) comesfrom intermediatestates

involving a single Goldstoneboson.Accordinglywewrite:

11(q2)= g~F2I4q2 (1.10)

wherethe pole in q2 is dueto the masslessintermediateGoldstoneboson.Evidently the occurrenceof a

masslesspole in 11(q2) cancelsthe masslesspole in 4k,. andshifts it to [6]:m2~=~g~F2. (1.11)

Thuswe againseehowthe symmetrybreakingscaleF controlsthe massof a protectedparticle.The neutralgaugebosonsectoris slightly morecomplicateddueto mixing betweentheW°andB.

The problemis solved in termsof a massmatrix (seesection 2), the entriesof which comefrom thegraphsin fig. 2. Theresult is the familiar superpositionsof B andW°,the Z andphoton.An importantconsequenceof the standardHiggsmechanismis the empiricallyestablishedrelation:

-~cosOw=1+O(a). (1.12)mw

W± nt W0 fl° B 11°

2 2 2Fig. 1.

282 E. Fwhi and L. Susskind, Technicolour

B n° B W° 11° W° w° n° B

g2FZ _

4q2 4q2 4q2

Fig. 2.

Certainaspectsof the abovescenarioareunquestionablycorrect.Theneedfor spontaneousbreakingof SU(2)~~~X U(1)~ypat a scale—250GeVis established.What is lessclear is that the driving forcewhich breaksthe symmetrymustbe acanonicalscalarA~type theory. Fundamentalscalarsmayonlybepart of aprovisionallow energytheoryanalogousto the o model for low energypion physics[7]. Inboth casesasymmetryis spontaneouslybrokenandthelow energybehaviouris adequatelydescribedby aphenomenologicalLagrangian.Perhaps,as in pion physics,the fundamentalRiggs bosonsmusteventually be replacedby much richer compositestructures.This is the main assumptionof thetechnicolourtheoryof dynamicalsymmetrybreakdown.

In fact thereis a very seriousdifficulty with the existenceof fundamentalscalarfields which is calledthegaugehierarchyproblem[8]. To explain thisproblemwemustsupposethat thereexistsasymmetrybreakingscaleatsomevery high energysuchas theone that occurs in grandunified theories[1].Theproblemwhich arisesis how to arrangetheparametersof the theory to ensurea gapof manyordersofmagnitudebetweenthe low andhigh energysymmetrybreakingscales.Generallythereis a very strongtendencyfor quantumeffects to pull the two scalestogetherunlessthe parametersof the theory areadjustedto absurdprecision.

We shall illustrate this situationin thecontextof theSU(5)theoryof Georgi andGlashow[5] but thereadershouldrealizethat it is very general.In theSU(5) theorytwo stagesof symmetrybreakingmustoccur, oneatenergies~~~1015GeVwhich breaksSU(5)down to SU(3)x SU(2)x U(1). This breakdownisusually accomplishedwith a 24-dimensionalRiggs field ~ which must obtain a vacuumexpectationvalue i0’~GeV.Thesecondstageof breakdownfrom SU(3)x SU(2)x U(1) downto SU(3)x U(1)EM isinitiated by afive-dimensionalRiggs field ~owith vacuumexpectationvalue _~102GeV. Thus wemustaccountfor aratio of scales— iO’~.How can this be done?

The obviousproposalis to constructa field potential V(~,ç) with minimumat

(P)=10~(~). (1.13)

For example

V(~,‘p) = A1(~

2—(~)2)2+ A2(ç

2— (~)2)2 (1.14)

The difficulty is that quantumradiativecorrectionsrenormalizeV andintroducecouplingsbetween~oand~. Thesimplestsourceof suchcorrectionsin SU(5) comesfrom thefact that both p and 1’ interactwith the SU(5)gaugebosons.For examplegraphslike the onein fig. 3 give correctionsto V of theform:

~L..ç2~2 (1.15)

The effect is devastating.Theminimum of thepotentialIs shiftedfrom (~),(ç) to —(~), (g2/4irA~/2) x

E.Farhi and L. Susskind,Technicolour 283

Fig. 3.

(P). Thusinsteadof the desiredratio of 10’swe get

(ço)/(~) g2/4irA~’2 (1.16)

which is —a.To compensatefor theseradiativeeffects it is necessary,in each order, to go back and delicately

retunetheparametersin V. In thepresentcasethe requiredprecisionis onepart in 10~.In contrastto this situationconsiderthescalefor chiral symmetrybreakdownin theQCD sectorof

the SU(5) theory.Accordingto our currentunderstandingof quantumfield theory, the QCD couplingconstanta

0is a function of the momentumscalesatisfying[9]

8 ~~q2= I3(a~)= ~ (ll~3)a~+.... (1.17)

When SU(5)breaksdown to SU(3)x SU(2)x U(1) at 1015GeV thecoupling is small so theQCD sectorexperiencesno non-perturbativeeffects at this scale. As the energy scale decreasesa

0 increasesaccordingto eq. (1.17)until it becomeslargeat the scaleA~D.At this point non-perturbativeeffectspresumablytriggerchiral breakdown.The scaleA0~can befound by integratingeq. (1.17)andfindingthevalueof q

2 for which a0 1. Theresult is

M exp(—11/8~rao) (1.18)

whereM is the scale(-‘- 10’sGeV) at which SU(5)—* SU(3)X SU(2)X U(1) anda0is the gaugecouplingat the scale.The ideais shown in fig. 4 which depicts the evolution of the QCD coupling from theunification scaleM down to thescaleAQCD.

The importantpoint to noteabout eq. (1.18) is that with a rather ordinaryvalue of a0 the ratioA0~IMcan easilyand naturally be 10_15. It would obviously be desirableif theweak intefactionscaleat 100GeV could be generatedin an equallynaturalway. This is the goalof the technicolourtheory of symmetrybreaking.

AQCD E(GeV) IO~

Fig. 4.

284 E. Farhi and L. Susskind, Technicolour

Nowreturnto the caseof symmetrybreakingby fundamentalscalarsand to theradiatIvecorrectionsto V((P, ~) illustratedin fig. 3. Anotherway to think aboutthe diagramis asa self-energycorrectiontothe ç field. We illustratethe effect in fig. 5 wherethe external(P linesof fig. 3 are absorbedby thevacuumexpectationvalue(P). Evidently the diagramis a correctionto the massof ~ which is of order(4))2~Themaincontributionto thisself-energygraphcomesfromlargeloopmomentaof order((P). If wecansuppressthelargeioopmomentain fig. Ssothatmomentanolargerthanafew TeVcouldflow, thentheinducedcorrectionto the massof ~ is innocuous.The problemis similar to obtainingafinite andsmallelectromagneticpionmassshift [10]whenthesimplestgraphdiverges.Forthepionthesolutionlies in theform factorswhich resultfrom thecompositenatureofhadrons.ThesamesolutioncanworkfortheRiggsfield (p. If it is acompositeobjectboundof fermionpairswith an inverseradiusaboutiO~GeV,thentheradiativecorrectionsfromveryshortdistanceswill notbeimportant.Thisis theessentialmotivationforthetechnicolour[11]ideawhich is describedin the following sectionsof this article.

Our article beginswith the simplesttechnicolourmodel andendswith themost speculativeideas.Someof theclaims madeneartheendof thearticlearebasedmoreon intuition than on calculation.However,we feel that all of theconsequencesof dynamicalsymmetrybreakdownmustbe pursuedto

gainatrue understandingof particlephysics[12].

~2Fig. 5.

2. TechnIcolour

2.1. A simplemodelwithoutscalars

In order to illustratehow the technicolourmechanismworkswewill startwith afamiliar system[13].Our discussionfollows closely thediscussionof the Riggsphenomenonin section1. Consideramasslessdoublet of quarksu andd interactingthrough ordinary QCD and neglectall other interactions.TheLagrangianof this theoryhasa SU(2)~-~x SU(2)RIGHTx U(1) symmetry.The U(1) is associatedwithbaryonconservationandis irrelevantto this discussion.Although it hasnotbeenproven it is universallyassumedthat QCD spontaneouslybreakstheSU(2)~.~x SU(

2)RIGHT symmetrydown to SU(2)~81~~.

Associatedwith this spontaneousbreaking is the non-zerovacuum expectationvalue of the fieldoperators

(üu+dd)�0. (2.1)

Again, theseare often called condensatesin analogywith similar phenomenain many-bodyphysics.Note that the condensateis not invariant under SU(2)~.,.or SU(2)Rjo~but only under the groupmaking equalleft- andright-handedrotations,i.e., SU(2)181081~~.

The vacuumnow haslesssymmetry thantheLagrangian.The Goldstonetheorem[14]tells us thattherewill be a masslessspin 0 bosonfor eachbrokengenerator.In this casethis meansthreemassless


p~Ofl5,Ha, forming an isotriplet. This pictureis the basis for many of the successfullow energypiontheorems[15].

An importantquantity is thepion decayconstantf,. definedby:

(0lJ~aIHb)= fwq~’öai, (2.2)

whereJ~aare the threeaxial isospin currents.Note that f,. defined in this way is a purely stronginteractionquantity independentof weakinteractions.Its magnitudeis controlledby the universalscaleparameterof QCD, AOCD. f,,. hasbeenmeasuredthroughpion decaysand it hasthe value:

f,,. — 93MeV. (2.3)

The reasonthateachpion hasthesamef,,. in eq. (2.2) is theunbrokenisospin symmetry.Let us complicate the picture. Imagine turning on the ordinary SU(2)~~~x U(1)~electroweak

interactionsbutwithout the fundamentalscalarfields which areusually introducedto give massto W~andZ. You might think that thespectrumof this theory includesfourmasslessgaugebosons:W~,W~,W~associatedwith SU(2)~~rand B. associatedwith hypercharge.This is not true [16].In fact themasslesspions of our previous.discussion replace the conventionalscalar fields and appearaslongitudinalcomponentsof massivebosonsW~and Z. The argumentparallelsthe discussionof theprevioussection.Focusattention on the chargedweak gaugebosonpropagator.We areparticularlyinterestedin the hadronic contributionsto the vacuumpolarization diagramsof fig. 6. Such graphsmodify thepropagatorfrom

g~W— qPqV/q2 gMV — q~4qVfq2 ‘24q2 ~‘q2(1+H(q2))•

If 11(q2) is smoothnearq2 = 0 thenthecorrectedpropagatorhasa poleat q2 = 0. However,in thiscase11(q2) developsa pole dueto masslesspions. The bosonW~hasan interaction~.g

2W~J~5whereg2is theSU(2)~~rgaugecoupling constant.ThecurrentJ~±5couplesto theH~with strengthf,,.±asin eq.(2.2). 11(q

2)hasacontribution

11(q2)~ (2.5)

asis seenin fig. 7. The zeroof the inversepropagatorhasshifted to ~ so the W~hasa massof

Fortheneutralgaugebosonsweneeda massmatrix in the 2 x 2 spacesof B andW°.Theentriescanbereadfrom the diagramsin fig. 8. The resultingmatrix is:

~ g1g2”~f~ (26)

~g1g2 g114w~ n~ w±

w± /-~~ w±

2 q2 2

Fig.6. Fig. 7.


!~~W0 ~B ~

Fig. 8.

whereg1 is the U(1) couplingconstant.The eigenvaluesof this matrix are

m~=0 (2.7)

~

The stateswhich diagonalizethis matrix are identifiedastheusual photonandweakneutralbosonZ.Notice that the ratio

g2 k_ 9k— i

2j 2\1/2; cosmz ~ g2, Jir° j,,.O

where9,,, is theusualweakmixing angledefinedin termsof couplingconstants.We havealreadyarguedthat f,,.+ = f,,.o from isospinconservationso we recoverthe empirically successfulrelation

mw/mz= cos9,.,. (2.9)

This relationis a treelevel result in the ordinaryRiggspicture.Onemayquestionits generalvalidity ifthe Riggssectoris strongly interacting.The significanceof theaboveargumentis that therelation (2.9)follows from a symmetryof the strong interactionsand is thereforeonly modified by electroweakradiativecorrections[17].(In ordinary weak interactionmodelswith a Higgs bosonthereis an 0(4)symmetryof the Riggspotentialprotectingthis relation[181.)

We arenot advocatingthis as a pictureof the world. First, this theory hasno true pion sinceit nowonly appearsas onedegreeof freedomof avectorboson.Second,themassesof theW~andZ areinthetensof MeV’s given theknownvaluesof g1, g2 andf,.. We haveillustratedapoint. The spontaneoussymmetrybreakingknownto occur in the strong interactionshasthe sameeffect on the weak gaugebosonsectoras the carefully constructedRiggssectorof the standardtheory.The differenceis roughlyafactorof 2000 in thesymmetrybreakingscales.

2.2. Technicolour

Let us supposethat an electroweakdoublet of fermionsexistswhich also engagesin anew strong

interactioncalled technicolour[19] (TC). We call this doublet

T=(~). (2.10)

We makeno assumptionaboutthe colour,baryonnumberor lepton numberof this doublet.However,Twr is a SU(2)~..~doubletwhile ARIGHT and BmGHT areSU(2)1~~~singlets.

Thesefermionsfeel the techniforceandwecall themtechnifermions.Theinteractionis like QCD in

E. Fa,hj and L. Susskind, Technicolour 287

its essentialaspectsexcept that it becomesstrong at a much higher energy,nearthe scale of weakinteractionsymmetry breaking[20]. The (A, B) doublet is just like the (u, d) doubletof the previoussubsectionexceptthat colour is replacedby technicolour.All of the argumentsof the previoussectionapply, in particularthe W~andZ becomemassivebut their massesare proportionalto the technipiondecayconstantF,,.:

m~=~g2F,,.. (2.11)

Using the measuredvaluesof mw andg2 we get F,,. — 246GeV if we want theseinteractionsto replacetheusualRiggs sector.

Assumingthat technicolourdynamicsQTD is a scaledup versionof OCD weget

ATcJA~— F,,,./f,. — 2600 (2.12)

giving ATC 500GeVif A~ 200MeV. This solvesthe problemof gettingalargemassfor the weakgaugebosons.As before the technipiondisappearsfrom the spectrum.

We have replacedthe usual Riggs sector of the standardelectroweaktheory with a stronglyinteractingsetof fermions.The vacuumexpectationvalueof ç at the minimumof the Riggspotential(ç>= Vhasbeenreplacedby the stronginteractionsymmetrybreakingparameterF,,.. Numericallytheyareequal.

If we hadmorethanonetechnicolourdoublet of fermions,sayr of them,thentherewould bethreetechnipionsfor each doublet. If each technipion hasthe samedecay constantF,,. then eq. (2.2) ismodified to

m2~=~rg~F2,,.. (2.13)

This would lower the technicolour scale by iiV~An example we consider later has r = 4 soA~—1300A~.

Now considera world with both QCD andcolouredquarks as well as 0Th andtechnifermionsinteractingwith the weak gauge bosonsof SU(2)~.~x U(1). With QCD alonewe arguedthat theordinary pions becomethe longitudinal W~and Z anddisappearwhile with 0Th aloneit is thetechnipionsthat havethis role. In the combinedpicturethe linear combination[21]:

~pionabsorbed>= F,,.Itechnipion)+ f,,. IQCD pion) (2.14)VF~,.+f~.

becomesthe longitudinalgaugebosoncomponentswhile the orthogonalcombination:

~physical~ = FI,TIQCD pion)—f~technipion) (2.15)

remainsa masslesspion in the spectrum.SinceF,,. >>f,. thephysicalpion is mostlyQCD pion while theabsorbedpion is mostly technipion.

To seehowtheselinear combinationsarisenotethat:

288 E. Farhj andL. Susskind,Technicolour

(0IJ~J0CDpion) = f,,,t (2.16a)

and

(0IJ~.Itechnipion>= F,,qM (2. 16b)

whereJ~is the full axial vectorcurrent.Theseimply

(0IJ~~pionabsorbed)=VF~.+f~,q” (2.17a)

(0~J~physicalpion) = 0. (2.1Th)

The weak gaugebosonscouple to thepions throughtheseaxial currentsso the physicalpion hasnocouplingwhile the orthogonalcombinationis totally absorbed.In the standardtheorythesamemixingoccursandthe physicalpion hasaminuteadmixtureof elementaryscalar.

We believe that theabovecombinationof QCD andQTD may be an accurateexplanationof thespontaneousbreakdownof electroweaksymmetry. However, thereis physics missing. In particularadditionalmechanismsare requiredto give thequarksandleptonsmasses(and thereforethe physicalpion).Wewill discussthesemechanismsin a later section.

The dynamicsof technicolourclosely mimic the dynamicsof QCD. Therefore the technicolourinteractionsare probably controlled by a non-Abelian gaugetheory with an associatedgroup G.Imaginethat technicolouris unified with QCD at somehigh energybetweentheweak interactionscaleandthePlanckmass.In orderto haveATC biggerthan~ thetechnicolourcouplingconstantmustgrow fasterthantheQCD coupling constantasyou comedown in energyfrom where they areunified.To accomplishthis the group U should be bigger than SU(3) so the associatedfi function is morenegative.[Seeeqs.(1.17)and(1.18).]

Supposethetechnicolourgroupis SU(N) with N>3. We canusethe resultsof the 1/N expansion[22]to gain information about the N dependenceof the parametersin the technicoloursector. Forexamplethe 1/N expansiontells us that F,. goeslike \/~times the fundamentalscaleas N getsverylarge. So eq. (2.12)is replacedby:

IA Itv~,NiITcJ11QcD—F,./J,. 2. 8

whereF,. comesfrom eq. (2.13)

F_2mw_24~~T 219~V~g

2 V

The technicolourscaleATC getssmallerasN grows. We will useotherresultsfrom the 1/N expansionwhenthey areuseful.

2.3. Technispectrocopy

Supposethere exists at leastone technicolouredfermion doublet andthat technicolourbecomesstrong in the hundredsof GeV’s. What are the obvious phenomenologicalimplications of thisassumption?

E. Farhi and L Susskind, Technicolour 289

2.3.1. TechnihadronsThere exists a rich spectrum [23] of particles called technihadronswhich are bound by the

technicolourforce. We are assumingthat the technicolourforce is anon-Abeliangaugeforce whosespectrumresemblesthe usual QCD spectrumrescaledby ATCJAQCD given by eq. (2.18). Thus forexampletheremayexist atechni-p anda techni-wwith masses—1TeV. The spacingsandwidths oftheseparticleswould alsoscaleup andbe — 10~timeslargerthantheir QCD counterpart’svalues.Thiscontrastssharply with thepossibility of having higher massfamilies of ordinaryquarkswhich wouldhave narrow, closely spacedstates. The technicolour spectrum would be easily visible in e~eannihilationand in fig. 9 we showa possiblepictureof R up to 10TeV.

IogA~c~ logATc

logE

Fig. 9.

2.3.2. LongitudinalW~and Z

The W~andZ will have a strongly interacting componentwhich will behavelike a compositetechnihadron.This is becausethe longitudinal degreeof freedomof thesebosonsis the compositetechnipionin disguise.This effect would showup in e~e—* W~W cross-sections.The productionoftransverseWt would be thesameasin the standardtheoryindicating apoint-like W~.However, thelongitudinal productionwould showacomplicatedform factor including resonances.

The ordinaryp likes to decayinto pions.The techni-pwith a massnear1 TeV would like to decaystrongly into W~Wpairs.This strongdecayinto weakbosonswould becharacteristicof technihadrons.

Jets of technihadronscould be producedin e~eor p~collisions either through photons,weakbosonsorgluonsif sometechnifermionscarrycolour. Thesejetswould look like ordinaryhadronicjetsbut themeantransversemomentumwould be scaledup by a factorof 10g. Thesetechni-jetscould becopioussourcesof weakbosons.

2.3.3. StabletechnibaryonsAmong the technihadronsthere can exist analoguesof the baryons.If the technigroupis SU(N)

thesewill consistof N technifermions[24]andcarry a conservedtechnibaryonnumberof N if eachtechnifermioncarriesoneunit. Thesetechnibaryonswill be fermionsor bosonsdependingon whetherN is odd or even.The massof theseparticlescanbe estimatedusing scalingandlargeN techniqueswhich tell us that the ratio of the mass of the baryon to the scaleA grows asN. For example,thetechnicolourgroupSU(4)givesa massof

4 A~ 2.8TeV— m~,.

0000— ,— (3A~ yr

to the technibaryoncorrespondingto theproton. Again r is the numberof technidoubletscontributing


to Wt andZ masses.Like the ordinary proton,the lightestof thesetechnibaryonswill not be abletodecay. However, theremayexist technibaryonnumberviolating interactions(discussedin section4)whichwould allow this lightesttechnibaryonto decay.

2.4. Modelsand alternatives

Furtherdetails of the technispectrumrequire specific assumptionsabout the technigroupand therepresentationsof the technifermions.We haveno preciseknowledgeaboutwhat theseshouldbe butwe will explore somereasonablegeneralpossibilities. In specific models the grouppropertiesof thetechnicolouredsectordeterminethepatternof light quarkmassesand this can be usedasaguide.Thesearchfor the correcttechnigroupis an outstandingproblemin thisfield.

Considerthe SU(3)coLouRXSU(2)~..~X U(l)~ypquantumnumbersof the technicolouredfermions.This groupis anomalyfree [25]in eachfamily of ordinary fermions.By a standardfamily wemeana setof particleswith the gaugequantumnumbersof (up-quark,down-quark,electron,neutrino). We willrequire that the technicolouredfermions are in an anomalyfree representationof SU(3)coLouRX

SU(2)rX U(l)Hyp so that the entire theory is anomalyfree.Two possibilitiesshould be consideredimmediately.

2.4.1. TheonedoubletmodelPreviouslyweintroducedanillustrative exampleof two technifermions

T = (~) (2.21)

whichform an SU(2)~.~.doubletbut arecoloursingletsandtransformasanon-trivial representationoftechnicolour. This is the minimal examplewhich correctly breaks SU(2)~FrX U(1)j.jyp~U(1)EM.Supposethereareno othertechnifermions.Thenin order to avoidtheanomaliesin fig. 10 which are

Y T~EFT

Fig. 10.

usuallycancelledby otherparticleswe makethehyperchargeassignments:

0 TimY = ~ ARIGHT (2.22)

—~ BRIGHT.

This leadsto chargesof ~and — ~for A andB. This model hasasimple spectrumof technihadrons.Itdoesnot haveanyof the pseudo-Goldstonebosonswhich are abundantin thenext model.

24.2. Theonefamily modelOrdinary fermionsappearto comein families. It is possiblethat the technicolouredfermions also


come in a standard family [26]. This would guaranteethe cancellation of the SU(3)coLo~X

SU(2)LEvrX U(l)~.~anomalies.If onesuchfamily existswe could label it:IUa ~ / a j a ,~ a~ red~ I blue ‘~ ~ yellow~

‘Da, ~a ~ a\ red! LEFT ‘~ blue! LEFT \ yellow! LEFT LEFT

/T T~ \ Ii Ta \ IT T~ \ jM~\~U red)RIGHT ~—‘ blue/RIGHT i.’-’ yellow/RIGHT ~ )RIGHT

(D,.Od)RIGHT (DbIUC)RIGHT (D~enow)1uo}rr (E~)RIGHT.

The a is a technicolourindex andwe are assumingthat all particlesin this family havethe sametechnitransformationproperties.We have included a right-handedtechnineutrinowhich is a singletunder SU(3)COLOURx SU(2)~.rX U(l)

1.~yp.As will be seen this particle is necessaryto ensure therelationmw/mz= cos0,.,. Wewill refer to 0= (U, D) astechniquarksandL = (N, E) astechnileptons.

As far as the weak interaction symmetrybreaking is concerned,this model is like four-doubletmodelswhere the doubletsare distinguishedby the label red, blue, yellow, lepton. It is natural toassumethat thesedoubletscondensewith equalstrengthso weget:

(Ured Ured> = (i5reci Dred)= (Ublue Ublue> = (Dblue Dblue>

= (Uyeiiow Uyeiiow) = (~yeuowD~~110~>= (EE> = (RN) ~ 0. (2.24)

Thefactor of r in eq. (2.19) is 4 so thetechnicolourscaleis —1300V3/N if technicolouris SU(N). Iftherewere no right-handedtechnineutrinothen in the technileptonsectortherewould not be theSU(2)LEFT x SU(

2)RIGHT symmetrywhich protectsthe relation mw/mz= cos0,.,. Since this is a goodsymmetryin thetechniquarksectorthe relationwould be violated butnot grossly.

This model hasa rich spectrumof technihadronswith massesnear1 TeV. The techniforcewouldbind technifennionsto anti-technifermionsto maketechnimesonsof spin 0, 1, 2 etc. Someof thesewould be coloured,for exampletheremight be a spin 1 UE which would havecharge— ~and be acolour triplet. Presumablyit would be confinedandform atomswith light quarks.If the technigroupisSU(2n+ 1) wewould havefermionicbaryons.Thesecould haveunusualcolour andchargesaswell. Thephysicsatthis scaleis rich andworth exploring.However,webelievethat the spin 0 mesonsmight havemassesmuch below 1 TeV, somepossiblybelow 100GeV. Someof theseparticlesmight be visible inthenext generationof accelerators.For this reasonwe devotethenext sectionsto thespin 0 sectorofthis model.

2.4.3. OthermodelsWe havediscussedtwo simple models.The list of possibilitiesis endless.We urge the readerto

exploreall the variationsuntil he or shefinds theone that doesit all.

3. Pseudo-Goldstonebosons

The existenceof more than one technidoubletcan lead to masslessor light (on the technicolourscale)spin 0 particles[27].In thissectionwe illustratehowthis occursin theonefamily model.Thereisa rich spectrumof particlessomeof which mayhavemassesin the tensof GeV’s. New interactionsjust

292 E. Farhi and L. Susskind,Technicolour

abovethe technicolourscalecould raisethesemassesto thehundredsof GeV’s. Theseparticleshaveunusualproductionanddecaycharacteristics.

Our discussionof the spectrumof pseudosis almostcompletelydependenton themodel we havechosento analyze.Thereis no compelling reasonto believe in this model except that it is a naturalextensionof low energyideas.

3.1. Goldstonebosonsandpseudo-Goldstonebosons

Supposeyou havea theory which at the Lagrangianlevel is invariant undersomeinternal globalsymmetrygroupG. If the groundstatevacuumis invariant undera smaller groupH, then for eachgroupgeneratorof G not in H you haveamasslessspin 0 particle.ThesearecalledGoldstonebosonsand this result is the Goldstonetheorem[14].If G is an approximatesymmetry thenthe Goldstonebosonsareonly approximatelymassless.Themassesgo to zerowith thesymmetrybreakingparameter.Thesenearly masslessparticlesarecalledpseudo-Goldstonebosons.

As an exampleconsiderthemasslessu, d quarksin QCD. This was discussedin section2.1. Thetheory is SU(2)LEFTx SU(

2)RIGHT invariantbut the vacuumis only SU(2)1~0~~~invariant:

(Uu+dd>�0. (3.1)

Correspondingto eachbrokengenerator(the threeaxial SU(2)generators),thereis a masslesspion.In fact, SU(

2)LEFTx SU(2)RIGHT is only an approximatesymmetry of the real world. Besideselectromagnetismtherearequarkmassesof the form

muUu+ mddd (3.2)

which violate SU(2)LEFTx SU(2)RIGHT.Thepions developmasseswhosesquaresareproportionalto m~and md. The realpion is apseudo-Goldstoneboson.

3.2. Pseudosin the onefamilymodel

Considertheonefamily model of section2.4.2 and ignore all forcesexcept technicolour.This is agood approximationnearthe technicolourscalewherecolour andelectroweakforcesare small. Thereare eight left-handedandeight right-handedfields which have identical technicolourtransformationproperties.Theseparticlescanbelabelledup-red,up-blue,up-yellow,down-red,down-blue,down-yellow,electron,neutrinoin both the left- andright-handedlists.Thetechmfermionsaremasslesssothis theory(againneglectingcolour andelectroweakforces)is SU(8)LEFT x SU(8)RIGHT invariant.

Thetechnicolourforcecausesa spontaneoussymmetrybreakdownof thevacuumat the technicolourscale.Weassumethecondensateshavetheform of eq. (2.24).With eachcomponentcondensingequallythe vacuumis still SU(8)~~

0~= SU(8)LEFT + SU(8)RIGHT invariant.The symmetryhasbeenreduced

from SU(8)LEFTx SU(8)RIGHTto SU(8)VECTORsothereare2 x 63—63=63 masslessGoldstonebosons.Most of these63 particlesarenot strictly massless.If you no longerignoreSU(3)coLo~x SU(2)x

U(1) then theoriginal SU(8)LEFT x SU(8)RIGHT symmetryis only approximate.Most of the Goldstonebosonsarepseudo-Goldstonebosonswhosemassesvanishascolour andelectroweakforcesareturnedoff. In fact thereareprobablyotherforceswhich breakthe SU(8)LEFT x SU(8)RIGHT andthesewill bediscussedin the next section.We now explorethe propertiesof the 63 real and pseudo-Goldstonebosons.

E. FWhi and L. Sus~ckind,Technicolour 293

3.3. TheGoldstonespectrum

Neglecting the colour and electroweakforces the vacuumis invariant under the SU(8) vectorsymmetrySU(8)LEpr X SU(8)RIGHT. For eachof the axial SU(8) generatorsthereis aGoldstonebosonir. Thesecoupleto the currents:

J~a= FYTMV5taF (3.3a)

where

Ublue

Uyellow

F= Dred (3.3b)blue

Dyellow

NE

andthe ta arethe 63 tracelessSU(8)generatorsnormalizedsothat

Tr(t~~tb)= ~o~2b. (3.4)

The Hermiteanfields ir coupleto the currentswith strengthF,.

(f.rIJ~bI0)= F,,q”6~. (3.5)

EachH” hasthe sameF,. becauseof the residualSU(8)vectorsymmetry.In orderto distinguishbetweenthepseudosandthe true Goldstonebosonsit is convenientto choose

abasisfor the ta matriceswhich exhibits theSU(3)cowuR,the isospinandthechargeof thestates.Let

QC andL be techniquarkandtechnileptondoubletsgiven by

c= red,blue,yellow QC = (U)C

L=(~). (3.6)

Let r’ be thethreeisospinmatricesacting on techniquarkand technileptondoubletsandlet A” bethe octet of colour matricesacting on colour indices.The 63 linear combinationsof H” ‘s which havedefinitecolour,chargeandisospincanbe written as:

0~~y5AaT~Q, a = 1, 8; i = 1, 3 (3.7a)

Oa ~b’5A4Q, a = 1, 8 (3.Th)

T~-~~cyTIL; EymJQc (3.7c)

294 E. Farhi andL. Susskind,Technicolour

c = red,blue,yellow

Ey5QC (3.7d)

— (~y5r’Q+ Ey5r’L) (3.7e)

P~-~(Oy5r”Q — 3Ly5r~L) (3.7f)

P3-~(~y

5r3Q— 3Ly

5r3L) (3.7g)

P° ((~y~Q- 3Ey5L). (3.7h)

Thestatescoupleto their respectivecurrentswith strengthF,.. We will now discussthepropertiesofthesestatesin turn.

3.3.1. TheoctetpseudosTheseare the particlescalled O’a and 0, where the 00 and 0~are electrically neutral and 0~are

charged.They areall colour octets.When we take into accountcolour SU(3) forces(aswell asweakinteractions)theseparticlesareno longermassless.The main contributionto their masscomesfrom onegluonexchangeasin fig. 11. A simple estimateof themasscanbemadeby scalingargumentsappliedtothe electromagneticmassdifferenceof ordinarypions. In thecaseof the pionssinglephoton exchangegives thedifferencebetweenthe squaresof masses[101m — m~o. The gluonexchangein thecaseoftheoctet pseudois structurally identicalexcept for a different coupling constantand a colour grouptheory factor.Therefore

2 A2 IAm ~ — “TC a,,frOfl~~/ITC —

2 2g2m,.~—m,.° ~ acm

The factor (A~ ocr))2 arisesbecausethebound statestructure,form factorsetc., aredefinedby the

TC scaleinsteadof the QCD scale.The 3 is thecolour Casimir for theoctets.This model hasfourdoubletsso r = 4 in eq. (2.19)and usinga valueof a,,(A.rc)-~0.1 we get

m8 ~j~260 GeV (3.9)

wherethe technicolourgroupis SU(N).Bjorken andothers[28]haveestimatedthis massusing the currentalgebratechniqueswhich work

well for theordinarypionsand their estimatesagreewith ours.Other interactionswill contributeto themassof theoctetbutareprobablysmaller.Equation(3.9)is certainlya lowerboundon theoctetmass.

The colour octets0~,0~form a techni-isospintriplet while the 00 is a techni-isospinsinglet. Thetriplet is called a colour octet technipionwhile thesinglet is called acolour octet techni-eta.Ordinaryquarks, leptons and gluons are techni-isospinsinglets while the electroweakinteractionsviolate

gluon

--~-~-~--

Fig. 11.


techni-isospin.Thus we expectdecayslike

-~ 4, q-+ gluon,gluon-*gluon, y-~gluon,Z-* gluon,gluon, gluonetc. (3.lOa)

0~-+4,q-* gluon, ~‘

-~ gluon,Z etc. (3.lOb)0~-*4,q

—* gluon,W~etc. (3.lOc)

With thephysicsso far introducedthedirect decayof a spin0 bosoninto afermion—antifermionpairis forbidden.To seethis notethat in the Lagrangianthe light fermionsare only coupledthrough theaxial andvectorgaugecouplingsof SU(3)coLo~x SU(2)~.rx U(1)~..Thesecouplingsareinvariantunderthechiral operation:

çb—ly5çfr. (3.11)

Furthermorethis symmetryof light fermionsis not broken by the technicolourcondensate.As in thecaseof theordinarypion, a spin 0 particlecannotdecayinto a fermion pairwithout violating chirality.The samechirality invariancepreventsthe light fermionsfrom getting a massandmustbe violatedbyotherinteractionsnot yet discussed[29].Theseinteractionswill allow the pseudo-Goldstonebosonstodecay into fermion—antifermionpairs and as in the case of the ordinary pion the amplitude isproportionalto the massof the fermion.By dimensionalconsiderationsthecoupling is then

g~-—(mf+ mr)/F,,.. (3.12)

[Actuallyin specificmodelsthecoupling may go like themassdifferencebut we will work with (3.12).]Giventhis couplingyou seethat theoctetpseudospreferto decayinto heavyquarks.(Leptonpairs

are notallowed becausethey cannotmakea colour octet.)The rateis roughly [30]

F(0-÷if’) —~ ~ (3.13)

wherean extra~is includedasacolour grouptheory factor.For examplethe ratefor 0 -* bb would be—30 MeV given eq. (3.9) with N = 4. This may well be the dominantdecaymodeof the colour octetpseudos.

The coupling of an octet pseudoto two gluons or two othervector fields can be calculatedin amannersimilar to the calculation of H°-*yy. Various groups [311have calculated the rate for

~ g~+ g0 and they get

~ 2 3

— . a !!!~~r2‘Va ~b~CJ — 384 ir3 F2,. ~‘

296 E. Farhi and L. Surskind, Technicolour

which for technicolourN = 4 andm9 —260GeV givesa rateof 80MeV. If two gluonsareproducedinthis way weexpectto seetwo highly energeticjetswhosetotal invariantmassis that of thepseudo.

The decaysof the octet pseudosinto the other vector pairs listed in eq. (3.10) have also beencalculated[31].They are all smaller thanthe estimateof 80 MeV for two gluonsbecauseonestrongcoupling constantis replacedby a weakcouplingconstantand thegrouptheoryfactorsaresmaller.

Perhapstheeasiestway to producethe 0~is through two gluon annihilation[32],the inverseof eq.(3.14). This equation and estimatesfor gluon distributions inside protons give at tevatronenergies(\/s=2000GeV)

o(pp—*00+X)— 1035cm. (3.15)

If the 00 decaysmostly into hadronsit will be difficult to seeabovethebackground.Perhapsthebesthopeis to look in thegluon+ photonsignal which would haveadifferent angulardistributionthanthebackgroundandmight stick out just enoughto be seen[33].

An octet pseudoalongwith a vectorbosonmight be producedin e~eannihilation at somesuperLEP. For examplefig. 12 shows the productionof an 0~along with a gluon. Unfortunately thesereactionstypically contributeto R only —10~but theywould befairly cleansignalsof individual octets[34].

Fig. 12.

3.3.2. The colour tripletsThe particles called T~and T~and their antiparticlesin eq. (3.7) are colour triplets and carry

technibaryonand technileptonnumbers.The techni-isotriplet(F, T~,T~)hascharges(1/3, —2/3, —5/3)and the isosingletT~hascharge—2/3. The main contributionto the massesof thesestatesis throughsinglegluonexchangeasfor the octets.The estimateof the massfollows theoctetestimateexcept thatthegrouptheoryfactorof 3 is replacedby 4/3 in eq. (3.8) so the massof the triplet is

m~=~m0 (3.16)

which is =170GeV for N =4.The T~,T~,T and T~all receive different contributions to their massesfrom electroweak

interactions.Thesemassdifferencessquaredareon theorderof a few GeV2(seethesectionon charged

axions)so the massdifferencesarequite small.We do expect

mmT~=mT,~>mTt. (3.17)

Theseparticleswill be producedby thestrongandelectroweakinteractionsof ordinaryparticlesandsincetheseforcesconservetechnileptonandtechnibaryonnumbersthe triplets mustbe pair produced.Sincetheseparticlesarerelatively stablewe can imaginethemseparatingafterproductionwith a lightquark—antiquarkpair appearingfor colour neutralization.This is illustratedin fig. 13.


Fig. 13.

Thedominantdecayof thecolourtripletswill beinto ordinaryfermionpairs.Asfor thecolouroctets

weexpectthecoupling to go as

(mt+mr)/F,. (3.18)

so the triplets preferheavyfermions.The interactionsresponsiblefor thedecaymust respectcolour andbe SU(

2)LEETx U(1) invariantsoweexpectdecayslike:

T—~’b~+v (3.19a)

T~~t~+v~b~+T (3.19b)

—~ t~+ v, i~+ r (3.19c)

T~~t~+r (3.19d)

whereagainc is a colour index.The rate for T~-~br would be roughly

1 (mb+mT)2 — 20 MeV. (3.20)

If thesedecaysarefoundit will be interestingto study them in detail andseeif thereis informationaboutfamily numberconservation.Forexample,will theT~decaysinto br, be,b~all be allowed?Willtherebe complicatedmixing angles?

Again thedecaysof the triplets into ordinaryfermionpairs requirenew interactionswhich violatey~invariance.If theseinteractionswere turnedoff the heaviertriplets could still f3 decayinto the lighterones.Forexamplewe expect

TT~+j~+v (3.21)

just as ir~-4 ~ + e~+ v. The ratesfor individual channelswould be extremelysmall —10_11eV. Thelightest triplet would then be absolutelystablebecauseof conservationof technileptonand tech-nibaryonnumber.

3.3.3. The eatenpionsThesearethethreeparticles,H1, which arecolour neutralandcouplemaximally to the SU(2)LEFTX

U(l)Hyp interactions.They disappearfrom the spectrumvia the Higgs phenomenonas discussedinsection2. Eachcanbe thoughtof asthe sum of fourequallyweightedfermionpairs labelledred, blue,yellow andlepton. Sincethey vanishfrom the spectrumwe haveonly 60 pseudos.

298 E. Farhi and L. Su&ekind, Technicolour

3.3.4. ThechargedaxionsTheP~in eq. (3.7f) are thechargestateswhich arecolour neutral andorthogonalto theH~.This

orthogonalityaccoUntsfor the factor of 3 in their definition. Sincethey arecolourlesstheir massesdonot come from gluon exchangebut from other interactions.Electroweakcontributions have beenestimatedby variousgroupsusingthecurrentalgebraformalismwhich workswell for m — m~o•Theyall get [351

(m ;~P)2 ~ m~iog~~) (3.22)

wheremz is themassof theZ bosonandmTc is themassof a technimesonprobablynear1 TeV. Thus.the electroweakcontribution is (5—10 GëV)2. In section4 we will discussand estimatethe possiblecontributionsto the masssquaredfrom effective four Fermi interactionsamongstthe technifermions.ThesecontributionscOuld be as large as (100GeV)2. Other authorsget lower numbers.We feel thatthereis greatuncertaintyin the massof theseparticlesandthey could vary from 5GeVto 100GeV.

ThesechargedaxiOnswill decay through the additional interactionsresponsiblefor breakingthechiral invarianceof ordinaryfermions.Again using thecoupling of eq. (3.18)wegetpreferentialdecaysinto heavyquarksor leptons.For exampleP~-i c + b would havearateof 3 MeV for a chargedaxionmassof 30 GeV. If theadditionalinteractionscoupletechniquarksto ordinaryquarksandtechnileptonsto ordinaryleptonswith equalstrengththen therateinto lepton pairsmaybeenhancedby afactorof 3relativeto the rateinto quark pairs [36].[Seeeq. (3.7f).] Thedecayinto a fermion—antifermionwouldprobablybeparity‘non-conserving,like ir-~~tv, but notnecessarilymaximally parityviolating.

If a p~p-pair is produced,in e~eannihilationit would contribute1/4 unit to R.Theywould haveathresholdeffect of Ø~where fi is the velocity. Another intriguing source of chargedaxions is intoponium decays. If mp~<m~— m~tomthen you would expect t -~P~+ b again with a couplingproportionalto thefermionmasses.This shouldcomparefavourablywith theordinaryweakdecaysof atop quark. A toponiumstate’would then decayasfollows

(it)—* P+ + i+ b

(3.23)

This might be thebestsourceof chargedaxions[37].

3.3.5. The neutralaxionsTheparticlesdefinedin eqs.~3.7g)and(3.Th) arechargezeroandcolourless[38J.In fact they get no

mass[39] at all from electroweakinteractionsmuch like the ir°. First considerthe axion P3 whichcouplesto

(0JGy”y5-r~Q— 3L’y55r

3L1P3)= F,q”. (3.24)

Thecurrentis not conservedbecaliseof electroweakinteractions.But the P3 alsocouplesto this currentplus avectorcurrent:

E. Fa,hi and L. Susskind, Technicolour 299

(0~y”(1+ ~‘5)-r3Q— 3Ey~(1+ y

5)’r3LIP3) = F,.q14. (3.25)

Since this current is made of right-handedfields it commuteswith the SU(2)1~part of the

Hamiltonian. It is easyto checkthat it also commuteswith the U(l),1~~so it is a conservedcurrent.Takingthedivergenceof both sidesof (3.25)gives

0=F,.m~ (3.26)

so theaxion is massless.The addition of other interactionscan makethis non-zero.In the next sectionwe estimatesuch

effectsand,as for theP~,weget

0sm~~s(100GeV)2. (3.27)

Similar argumentsapply to theP°.In theabsenceof additionalinteractions,thecurrentit couplesto

is conservedso:

0= DM (0I~y~y~Q— 3Ey”-y5L~P°)= F,,m~ (3.28)

Again it getsmassfrom otherinteractionsandwe estimate:

0~m~o~(100GeV)2. (3.29)

Theseparticlesshouldbe ableto decayinto fermionpairs just like theotherpseudos.For a lightaxion, say 5 GeV, decayinginto an s, ~pair we expect‘a rate[30]of around1 keV. The decayswillviolateparity sincethe amplitudegenerallycontainsa scalarandpseudoscalarcoupling[37].This is incontrastto the fermionic decayof theneutralHiggs boson[40]which in the standardmodel is a scalarparticlewith only scalarcouplingsto fermions[41].Thefermioncouplingsof theneutral Higgsparticlearecompletelydeterminedby the fermionmassmatrix sowe expect:

F(Higgs-+4q)/F(Higgs-.k)= 3m~/m~. (3.30)

If theHiggs is replacedby aP°or P3we expectthemasssquaredin eq. (3.30)to be thesamebut the 3will changeperhapsto a27 or 1/3 dependingon themodel [36].[In a modelwith morethanoneRiggsdoubleteq. (3.30)neednothold.]

The P°will also decayinto two gluons whereasthe P3 will not. This is due to the techni-isospinsymmetry.This decayhasbeencomputedandthe rateis [30]

F(P°~gg) 30 eV (mpo/2GeV)3(N/4)2. (3.31)

For a heavyP°this would be thedominantmode.The inverseof this process,gg -~ P°,could leadto theproductionof a P°in proton—protonor proton—antiprotoncollisions.

300 E. Fo,hi and L. Susskind, Technicolour

Perhapsthebestway to produceaP°or P3 is in e4~eannihilationvia a resonantquarkoniumstate[42].For examplewe expect

e~e-+1t--*P°,P~+y. (3.32)

Thedecayrateof a quarkoniumstateinto a pseudoplusphotonis comparableto theleptonicrate,i.e.,(tt) -4 e~eandthe exactratiowould be different if thepseudowasaRiggs.

Anothercharacteristicof pseudosin e~eannihilationis thesmallnessof processeslike [42]:

e~e-~P°,P3+Z

e~e-~P°,P3+ y

(3.33)

relativeto the ratesif the spin 0 particle is theneutral Riggs.The reasonis that theprocessesin eq.(3.33)proceedthroughtechnifermionloops while thecomparableRiggs diagramsareatthetreelevel.The absenceof processeslike thosein eq. (3.33)could signal technicolour.

TheP°andP3 arethelightestof thepossiblepseudo-Goldstonebosonsin this onefamily model.Thediscovery of theseor other pseudoswould indicatethat many of the technicolourideasare correct.Theirabsencefrom thespectrumwould revealthat oursimple extrapolationsfrom low energiesaretoonaive.

4. Effective four Fermi Interactions

Thephysicswehaveintroducedso far hasincludedfermionsandgaugebosons,withoutfundamentalscalars,transformingunderthegaugegroupU(1)~.x SU(2)~,.x SU(3)co~~X TECHNICOLOUR.This cannotbe a completedescription. In particular it fails to accountfor the massesof the lightfermions[43].Recall thediscussionof ~ invariancein section3.3.1 andeq. (3.11).A light fermionmassterm is not y~invariant whereasthegaugeinteraction of the light fermions is 15 invariant. The y~invarianceof light fermions must be violated for the light fermions to get a mass. Evidently newinteractionsarerequired.Thesenew interactionsshould alsocontributeto themassesof thepseudo-Goldstonebosonsandshouldeliminateany empirically unacceptablemasslessGoldstonebosons.

The observedfermions have a wide variety of massesall well below the technicolourscale.Toaccountfor this you might imaginethat the new interactionsinducefermionmasseswhich areof ordera, a2, a3, etc.,times the technicolourscale,wherea is someweak coupling constant[44].Ordera°masseswould bepreventedby specialsymmetries.This possibilitywasfirst consideredby Weinbergandwe exploreit in adifferentcontextin thediscussionof nearlymasslesscompositefermionsin section6.

Anotherpossibility is that thenew interactionsoccurat a variety of scalesall abovethe technicolourscale.In this sectionwe will showhow this can giverise to a complexpatternof fermionmassesbelowthe technicolourscale.

4.1. Otherinteractionsandscales

Let us assumethat at an energyscaleevenhigherthan ATC a new scaleof interactions,AE, exists


whoseprecisenaturewe will not discussnow. (Thesenew interactionsmay comefrom theexchangeofheavyparticles.)At low energiestheseinteractionstake the form of non-renormalizablemultiparticleverticeswith dimensionfulcouplingconstants.This is illustratedby two familiar examples:

(i) TherenormalizableWeinberg—Salamtheory of electroweakinteractionsinvolvestheexchangeofheavyvectorbosons.At low energiesthis theoryis describedby aneffectivefourFermi interactionwitha coupling constantof dimension(mass)2.

(ii) Proton decayin grand unified theoriesis mediatedby extremely heavy bosons,and at lowenergiestheseinteractionsare alsodescribedby afour Fermi effectiveinteractionwith adimensionfulcoupling.

More generally a long or infinite hierarchyof non-renormalizableinteractionsmay be neededtodescribelow energy physics [45]. In particular an n Fermi interaction will require a coupling ofdimension (mass)43’~2to give the Lagrangiandimension of (mass)4.It is usually assumedthat thecoupling constant

g — 0(1)/M3”24 (4.1)

whereM is thescaleof the interaction.Thus thehigherthedimensionof theoperatorthesmallertheinducedeffect.

At thescaleAE aboveATC we expectnewinteractionsproducingeffectivelow energymanyfermioncoupling.Theseinteractionsat the~caleAE must respectthe unbrokensymmetriesat this scale.Theinducedlow energyeffective operatorswill then also respectthe unbrokensymmetriesat AE. Thismeans that the low energy effective interactionsmust be U(l)Hyp X SU(2)~~~X SU(3)COLOUR XTECHNICOLOURinvariant.

Let us look at the operatorwith the lowest dimensionswhich by our previousargumenthasthelargestcoupling constant.Our low energytheory only hasfermions and gaugebosonsso the lowestdimensionLorentz invariantoperatoris thetwo Fermi Lorentzscalar~ This aloneis a masstermsoit is preventedby SU(2)~vrinvariant.

4.2. FourFermi interactionsand quark masses

The nextlowestdimensionoperatorsof interestarethe fourFermi operatorsof dimensionsix. Theseoperatorscanbe SU(2)~.~.invariantandgive massto the light fermions.Forexample,considertheonedoublet model of section2.4.1 with a technidoublet(A, B) along with a doubletof ordinary quarks(u, d). Colour is an irrelevant complicationand can be ignored. Define a 2 x 2 technicoloursingletmatrix

MT = 1(fl) + i~rTy5rT (4.2)

whereT is the technidoublet

T = (a). (4.3)

UnderSU(2)w.r

302 E. Fa,hi and L Susykind, Technicolour

MT-+UMT (4.4)

whereU is amemberof SU(2)1~-~andunderU(1)Hyp

MT -4 MT e~°’2. (4.5)

Then anSU(2)~,.x U(l)~.invariantcouplingbetweentechnifermionsandordinaryquarksis:

b_-~-~qLMTqR + -~-~ qLMTT

3qR + h.c. (4.6)lIE

When the technifermionscondenseso that

çT~r)= (AA + ~B) (4.7)

theseinteractionswill give massto the light quarksthroughdiagramslike thosein fig. 14.Thepropagatorsfor the fermionshavemassesdue to thecondensate[eq.(4.7)] andtheevaluationof

thediagramsgives:

(AA + BB)A2 (a+b) (4.8a)

(AA + ~B)md— A

2 (a—b). (4.8b)E

To estimateAE we assumethat:

(TI’) /3,_ (F1.\

3 49

2 VN”~kf,1.) (.)

where (4q) is the one flavour QCD condensateand we have the N dependencefor an SU(N)technicolour.Standard[46]QCD analysisgives (4q)/f~. 17. In theonedoubletmodel F,1~ 250 GeVand taking N= 4 gives (TT)/2 (600GeV)

3. In the one family model F7. is reducedby ~ and

(T1’)/2 (300Ge\’)3. If a andb are of order unity andwe want quark massesof 1 GeV then AE is

around20 TeV for theonedoubletmodel and around7TeV for theonefamily model.This is a newscaleof interactions.

In the realworld we must accountfor the vast differencein massesbetweenthe generations,themost extremeratio being mJm~ttomor ~ Onepossibility is a variety of scalesAE governingdifferent quartic interactions.Alternatively therecould be a variety of scalesof condensateswhich

B

ULUR ULUR

Fig. 14.


separatelycontributeto ordinary masses.A third possibility is that the lightest observedfermionscannotget massesfrom four Fermi interactionsbecauseof specialsymmetriesand only receivetheirmassesfrom six or higher Fermi interactionswhich give rise to extrapowersof (ATdAE). In thestandardweak interactiontheorywith fundamentalscalarsthequarkandlepton massesaredeterminedby arbitraryYukawacouplingsandno explanationis offeredfor their wide rangeof values.

Now considertheone family modelof section2.4.2andalso includeonefamily of ordinaryfermions:

q = (~), ~= (:)• (4.10)

Defining

M0 = 1(~Q)+ ir

ML l(LL)+ ir~Ly5rL (4.11)

we canhaveat leasteight independentcouplingsof the form:

qLMQqR, qLMQr3qR, qLMLqR, qLMLT3qR,

�LMQ�R, ILMQT3eR, eLML4, �LMLT3�R (4.12)

with respectU(1)~.x SU(2)~.~x SU(3)~0~~x TECHNICOLOUR and which contribute to lightfermionmasses.(Herethe labelsL andR on q and� meanleft andright.) Thustheordinaryquarksgetmass from techniquarksand technileptons.These types of mixed operatorscan eliminateunwantedmasslessGoldstonebosons.

In modelswith morethan onelight or technifermionfamily, the couplings(4.12)canbe generalizedto matrices.This gives rise to massmatriceswhich mix families andproduceCabibboangles.

4.3. FourFermi interactionsand Goldstonebosons

In addition to giving light fermions their masses,effective four Fermi interactionscan eliminateunwantedmasslessGoldstonebosons[47].ForexampleconsidertheneutralaxionsP

3 andP°of section3.3.5. In a theory in which the only interactionsare U(1)H-~.X SU(2)~-rX SU(3)cowul(X TECH-NICOLOURthesetwo particlesarestrictly masslessbecausethey coupleto conservedcurrents.Thesecurrentsarelinearcombinationsof techniquarkchiral currentsandtechnileptonchiral currents.

Considera possibleinteractionof the form

I = ~ [~QEL— 075?QL15?L]. (4.13)

This new interactionrespectsall of the gaugesymmetriesbut it violatestheseparatetechniquarkandtechnileptonchiralities. It gives the neutral axions P°and P

3, their masses.We can estimatethesemassesby using Dashen’sformula:

m2= ~ (0I[J°, [J°, H’]]lO) (4.14)

E. Farhi and L. SuEkind,Technicolour

whereJ°is the chargeassociatedwith the currentthat couplesto the pseudo-GoldstonebosonwithstrengthF. H’ is thepart of theHamiltonianthatviolatestheconservationof thecurrent.In thecaseofthe P°weusethecurrentfrom eq. (3.Th)

= ~(Q~isQ — 3L~y5L) (4.15)

andthe interactionwhich violatesthis currentis the onejust introducedin eq. (4.13).Evaluatingthecommutatorsgives

m = ~- ~ ~ (OI~QLLIO). (4.16)

Theinteractionof eq. (4.13)will give thesamecontribution to themassof the P3 if we usethecurrent

definedin eq. (3.7g). -

Now 00 is the sum of threecoloureddoubletswhile LL is onedoubletso

(OIQQLLIO) — 3(’Ti’)2 (4.17)

where(‘Ti’) is thevacuumexpectationvalueof onecoloureddoubletasin eq. (4.7). Usingthevaluesforthe onefamily model of (TT)/2 (300GeV)3, AE =7TeV and F,,. 125GeV we get m2 100GeV.More generallyweexpect

R2(1T)2/F~7.A~ (4.18)

where R is a number of order unity. (In the casejust consideredR is V5/2.) So the lightest

pseudo-Goldstonebosons,who get all their massfrom thesekinds of interactionswill havemasses

m=R6OGeV. (4.19)

Otherauthorshaveobtainedlowerestimatesandone is discussedin thenextsection.Theyassumeadifferent interactionthanthe onewe haveusedin eq. (4.13)but morecrucially they assumea largerscaleAE for the interactions.To makea preciseestimateyou needaspecific model in which all of theinteractionsareknown. In conclusionif thelight pseudosexistthey shouldhavemassesbelow 100GeV.The searchfor pseudosat all energiesbelow the technicolourscaleshouldbe conducted.

5. The largerpicture

We haveshownthat the introductionof a newsetof fermions,technifermions,interactingthrough anewforce, technicolour,cansuccessfullyreplacetheRiggs sectorof thestandardelectroweaktheory.Thetechnicolourandordinaryforces aremediatedby gaugebosonexchange.Ordinaryfermionmassescan be generatedthrough effective four Fermi interactionsbetweenordinary and technicolouredparticles.In this sectionwe showthat the effectivefour Fermi couplingsmight also comefrom gaugebosonexchange.Technicolouredandordinary particleswould then sit in the samerepresentationof

E. FQJhI and L. Surskind, Technicolour 305

somegroupandtheconsequencesof this must beexplored.In particularproblemscanariserelatedtoneutralstrangenesschangingcurrents.

If the Lagrangianof theworld only hasfermionsandgaugebosonsthenit is natural to look for onebig gaugegroupwhich containsall of the forces including technicolour.Thedifficulties of this attemptarealsoexaminedin this section.

Gaugetheorieswith fermionsbutwithout fundamentalscalarsmustdynamicallybreakthemselvesata variety of scalesin order to correctly describethe world. The dynamicsof spontaneouslybrokengaugetheoriesis a difficult subjectbut recentlysomeprogresshasbeenmade.We will examinethequestionsof gaugegroupbreakingthemselvesandthealignmentof thevacuumin dynamicaltheories.

In section 6 on masslesscompositeswe stress a different point of view. The ordinary andtechnicolouredfermions are viewed as compositesand they interactthrough their constituents.Thenatureof their effective interaction is different andit might be necessaryto enlargetheweak gaugegroupto give light fermionstheir masses.

5.1. Extendedtechnicolouror sideways

In a technicolourschemeeffectivefour Fermi interactionsmay arise from the exchangeof gaugebosonswhich mediatetransitionsfrom ordinary to technifermions[29]as in fig. 15. Theseordinaryfermion—technifermion—gaugeboson vertices do not preservethe ys invarianceof the ordinary fer-mions.Diagramslike thosein fig. 16 areresponsiblefor light fermionmasses.The constituentmassofthe technifermionis fed downby bosonexchangeto thecurrent algebra massof the light fermions.If

Fig. 15. Fig. 16.

the mass,mB, of the bosonexchangedis greaterthan~ and the coupling at the verticesis g thenthesediagramsgive light fermionmasses:

(5.1)

Thus mB/g correspondsto the scaleAE of section4. This bosonexchangehasbeencalledsidewaystodistinguishit from horizontal which takesordinaryfamily to ordinaryfamily.

The transitionsfrom techni to ordinary fermions may arise in the following way. For simplicityimagine that the technicolourgroup is SU(N) and some chiral technifermions transform in thefundamental~I dimensionalrepresentation.Now supposethis SU(N) is embeddedin a largergroup,extendedtechnicolour,saySU(N+ 1) andthis SU(N+ 1) somehowbreaksdown to SU(N) at ascaleA E~ The fermions in the fundamentalN+ 1 of SU(N+ 1) decomposeinto a technicolour~ and atechnicoloursinglet,an ordinaryfermion.Thebrokengeneratorscoupleto massivegaugebosonswhichmediatetransitionsfrom technicolouredto ordinaryparticles.


The fundamental(N+ 1) of extendedtechnicolourlooks like

T1T2

technicolour

extendedtechnicolour . (5.2)

TN

q +— technicoloursinglet.

This scenariocanbe generalized.Supposeextendedtechnicolouris SU(N+ m) which breaksdown to

SU(N) technicolour. The fundamentalN+ m representationof SU(N+ m) would break into atechnicolourN andm technicoloursinglets.Thesem singletsmight be m generationsof a particularflavour.

Another possibility is SU(N+3) for extended technicolourwhich breaks down into SU(N)technicolourx SU(3) colour. Thena fundamentalN+3 of extendedtechnicolourbreaksinto a colourneutraltechnifermionanda quark(technicoloursinglet).

Theextendedtechnicolourideacanbe summarizedasfollows. The technicolourgroup is containedin a largergroupwhich at somescaleAE breaksdown into technicolourand otherseither broken orunbrokengroups. The representationsof the extendedtechnicolourgroup decomposeinto tech-nicoloured objectsand technicoloursinglets (ordinary fermions). The broken generatorsmediatetransitionsfrom ordinaryto technifermions.The scaleAE is larger(seesection4) thanthe scaleATC.

Thus theextendedtechnicolourgroupwhich is a good symmetryat energiesaboveAE mustalso respectx SU(2)~ x SU(3)coLo~x TECHNICOLOUR.

It is alsopossiblethat at thescaleAE thereexist broken generatorswhich mediatetransitionsfromtechnifermiontype to technifermiontype. For exampletheremight be gaugebosonsof mass —AEwhich connecttechniquarksto technileptons.Thesemight be responsiblefor the effectivefour Fermiinteractionof eq. (4.13)which could give massto the light axions.Therecould also exist interactionsatthescaleAE amongstordinaryfermions.This could leadto smallbut non-zeroprobabilitiesfor unusualdecay modesof stableparticles, e.g., K-. jse, IT —~e~e~e,etc. The experimentaldiscovery of anunexpecteddecaymodewould be a sourceof information aboutphysicsbetweentheweak interactionscaleandthegrandunification scale.

Letus illustratetheseideaswith someexamples.First considertheone family model of section2.4.2along with onefamily of ordinaryparticles(u, d, e, v). Let the extendedtechnicolourgroupbe SU(5)which breaksdown to technicolourSU(4) leaving onetechnicolour4 andone technicoloursinglet ineach~ of SU(5).The multiplets look like:

fU U U U u\~ (U U U U u)~0~~DDDDd)~ (DDDDd)~fGHT

fUUUUu\~ (UUUUu)~~(D D D D Al ..AbIue

ui L~r U)RIGHT (5.3)IU U U U U\Y&~0W (U U U U u)~ID D D D ‘~ A\Y~HOW(1/ ~FF UJpJQ}fl~

• fNNNNv\ NNNNv)~0~‘E E E E e)~r (E E E E e)~0~

E.Farhi andL. Susskind,Technicolour 307

Assumingthecondensationof eq. (2.24) we getequalmassesfor the technifermionsU, D, E andN.Thesefeedmassesdown to u, d, e andv so we also getm~= md= me= m~.Ignoringthis problemwecan look at the spectrumof Goldstonebosonsassociatedwith technicolour.The spectrumis thatdiscussedin section3. If thereare no additional interactionsthe two neutralaxions aremassless.Toavoid this, imaginethat colour andlepton numberarepart of a vector SU(4) Pati—Salamgroup.ThisSU(4) breaksdown to SU(3)colour andsix of thebrokengeneratorsmediatequarklepton transitions.Thecoupling to heavygaugebosons~ is of the form

gX~[U5’1LN+DCY$E+ ucy~&v+acy~e1 (5.4)

wherec is a colour index to be summedover.This interactionwill violate theconservationof thecurrentswhich kept theneutralaxionsmassless.

If mB is the massof gaugebosonthenthecontributionto themassof theaxion is estimated[48]as:

~ (5.5)mBF7.

If thereis morethanone light generationthengaugebosonsof this kind canmediateKL -. ~ e with theaid of theadditionalcoupling

gX~[~yM~]. (5.6)

Usingexperimentalboundson KL -. j~e theauthorsof ref. [48]estimatem~ s (2.5GeV)2. In models

of this kind theaxion-likeobjectscould bevery light andobservablewith presentmachines.Of coursethis is not inevitableandonly occursin this very specific type of model.

For an alternateexample. considera model with SU(4) technicolourwith one doublet of tech-nifermions as in section2.4.1. Consideralso two light families (u, d, e,v) and (c, s,~t, v). Imagineextendedtechnicolouris SU(12)and therearefour fundamental12’sof SU(12)

AB A BAB . A BA B technicolour A BAB A Bu d u d (5.7)u d colour u dud u dv e eC s c sc s colour c scs c s

LEFT 1’ RIGHT RIGHT

and also suppose that the SU(12) extended technicolour breaks at the scale AE down toSU(4>c~jcoLo~x SU(3)coLo~.The SU(3)coLo,~is the diagonalsum of two SU(3)’s which sitdirectly in SU(12). Now thereis only one doublet of technifermions.There are no extraGoldstonebosonsat all. Thereis no axion andno paraxion.


Of coursethis model is also in troublebecausethe u andc quarksare treatedsymmetricallyso youexpectthemto havethesamemass.Thesetwo modelsillustrate thedifficulties of giving massesof lightfermionsin modelswith extendedtechnicolour.The symmetriesof theunbrokenextendedtechnicolourLagrangianare ultimately reflected in degeneraciesin the light fermion masses.To accountfor theobservedpatternof quark and lepton massesyou needa complicatedsymmetry breakingpattern.Ifextendedtechnibosonswith differentmassesconnectto differentlight fermions,theselight fermionswillno longer be degenerate.In general,an extendedtechnicolourmodelwill havea variety of differentmass extendedtechnicolourbosonsbeing exchangedbetweenvarious types of ordinary and tech-nicolour fermions. Considerthe extendedtechnicolourinteractionsof the down quarksd’ (d’ = d,

d2 = s, d3 = b, etc.) with technifermionsT~where a is the set of all the indices carried by atechnifermion.The mostgeneralgaugeinteractionwith gaugebosonsEa is of the form

g~d1y,LE~(1— y5)r +gdiyi4E~(1+ 75)T’

1 (5.8)

where a runs over the possibletypes of extendedtechnibosons.Of coursethis interaction must beU(1)}~.x SU(2)LEFTx SU(3)coLo~x TECHNICOLOURinvariant.This type of interactiongives riseto a down quark massmatrix through the diagramin fig. 17. If the fields d’ aremasseigenstatesthenthe massmatrix is diagonalin the indices i andj.

The generalinteractionof eq. (5.8) can leadto trouble with neutral flavour changingcurrents[49].Thediagramsin fig. 18 arethesourceof the problem.Unlessthesediagramshavethe structureÔI!ÔK1 or8U8K1 therewill be neutralflavour changingprocesses.A part of the diagramin fig. 18, involving left-andright-handedquarks,can berelatedto the quarkmassmatrix of fig. 17 andthis partwill be flavourdiagonal.But thereis no reasonto expectthat the partsinvolving only left- or right-handedquarksareflavour diagonalor suppressed.

d’ Ta d~

~,ALEFT d~Io~N ;l TB

Fig. 17. Fig. 18.

Considerthe contributionof such a diagramin fig. 19 to the KL, K~massdifference.This is verysimilar to the original calculationof this massdifference[50]involving the GIM [51] mechanismbutherean extendedtechnibosonreplacesthe weak bosonand a techniquarkreplacesthe up-quarks.Inour casethereis no cancellationof the leadingpieceandthe effectivefour Fermi operator,leadingto a

~LEFT T dL.EFT

E E

Fig. 19.

E. Farhi and L. Susskind, Technicolour

KLKS massdifference,canbe written [52]as

2 ~7TM(1 — yS)d~yTM(1 — y5)d (5.9)32ir m~

wheremB is the mass of the exchangedtechniboson.To be consistentwith the observedKLKS massdifference,the coefficient [50] in eq. (5.9) must be less than 5 x iO’~GeV

2 so mB/g is greater than80 TeV. SincemB gAE this becomes

AE/g>80TeV. (5.10)

In theone family model thescaleof interactionsAE requiredto give a oneGeV quarkmass is around7 TeV. [Seesection4.2.] For thestrangequarkit would be closerto 20 TeV which is still inconsistentwith the limit of eq. (5.10)if g 1.

An extendedtechnicolourmodelcanalsohaveneutralstrangenesschangingprocessesthroughsinglebosonexchange[52]. (Theseareanalogousto ordinary flavour changingprocessesthrough Z exchangewhich againvanishbecauseof the GIM mechanism.)Considera toy modelwith extendedtechnicolourbeing SU(N+2) which breaksdown to SU(N)technicolour.A fundamentalN+2 might break into atechnicolouredfermion D and two technicolour singlets d’ and s’ where the prime meansweakinteractioneigenstates

D2

technicolourSU(N)SU(N+ 2)etc. - (5.11)} technicoloursinglets.

Thebrokengeneratorconnectingthe last two componentswould mediated’ to s’ transitions.If wewritethe weak interactioneigenstatesin termsof masseigenstates

dL = dL cos°L + SL sin °L

sL =5L cos9L — dL sin 0L

d1~= dR cos

0R + 5R sin OR (5.12)

Si~= 5Rcos9R — dE S~fl9R

whereL, R meanleft andright, then the diagram in fig. 20a would leadto the neutralstrangenesschangingdiagramin fig. 20b with anglefactorslike cos2°L sin2 °Rdependingon thehandednessof theparticlesinvolved. Theeffectiveneutralstrangenesschangingoperatorwould havethe form

~ cos20 sin20 ~7TM(1±y5)d~y

TM(1±y5)d. (5.13)

310 E. FazI,i and L Suyskind, Technicolour

Fig. 20.

Again, to be consistentwith the KLKS massdifferencethe coefficient mustbe less than5 x 10_13GeV2so

AE-- mB/g ~ 1500TeVcos20 sin20. (5.14)

Evenfor 0L 0~Cabibbo 13°the scaleAE usedfor fermionmassgenerationviolatesthe inequality.Thesituationcan besummarizedasfollows: two generatorsof extendedtechnicolourconnectingthe

sametechnifermion to two different ordinaryfermionscan havea commutatorthat connectsthetwoordinaryfermions.The bosonassociatedwith this generatormediateshorizontalcurrentsandthe limitson horizontalgenerators[53] areoften higherthanthescaleAE weexpectfor extendedtechnicolour.This problemalongwith theproblemof neutralflavourchangingcurrentsassociatedwith doublebosonexchange,might be eliminated if there are GIM [51] type mechanismsat work in the extendedtechnicolourmodels. Unlessthe neededsuppressionfactors can be discovered,the extendedtech-nicolourmechanismis beingseverelychallengedby low energyphenomenology.

5.2. Grand unified theoriesand technicolour

Grandunified theories[54]unify theelectroweakandstronginteractionsby making SU(3)coLo~,SU(2)LEFT and U(1)HTp all subgroupsof one group. Quarks andleptons sit in the sameirreduciblerepresentationsof the grand unifying group. These theoriesare elegantand have certain strikingphenomenologicalsuccesses.

The extendedtechnicolourschemes[43] put technifermionsand ordinary fermions in the samerepresentationsof theextendedtechnicolourgroup. It is naturalto ask if technifermions,quarksandleptonscan all be put in the samerepresentationof a super-unifyinggroup.This groupwould containtheknowngaugegroupsas well asthe technicolourgroupassubgroups.

Oneof the simplestways to implementthis ideais to generalizethe SU(5) Georgi—Glashowgrandunification scheme.Supposetechnicolouris SU(N) and the super-unifyinggroup is SU(5+ N). ThisSU(5+ N) containsthe SU(N)TECH x SU(3)COLOUR x SU(2)~..~embeddedin theobviousway

SU(N)

SU(3) . (5.15)

SU(2)

The U(l)HTp would be a diagonalgeneratoras it is in SU(5)without technicolour.Various represen-tations of SU(5+ N) would have technicoloursinglets(ordinary fermions) as well as technicolourednon-singlets(technicolourfermions).

The super-unifiedgroupwould be a good symmetryonly abovesomevery high energy.Then thesuper-unified group would breakdown, perhapsthrough a sequenceof breakings,until at a scaleAE

E.Fathi and L. Susskind, Technicolour 311

aboveATC the symmetry is (TECHNICOLOUR) X SU(3)~~0~X SU(2)~.rx U(1)~yp.Thenat ATCthe technicolourforce becomesstrongand the SU(

2)LEFTx U(l)~. groupbreaksdown to U(l)EM.Using SU(7) as the super-unifyinggroup [55], the authorsof this papertried to implement these

ideas.Other models[56]havebeendiscussedin the literatureandwe will not discussany model indetailhere.However,we will list the typical successesandproblemswith theseschemes.

Successes:(i) The low energyRiggs sectorresponsiblefor the breaking of the electroweakgroup can be

replacedwith fermions.Thesefermionsaresuper-unifiedwith ordinaryfermions.(ii) More thanone family of light fermionscan be naturallyincorporatedin the largerepresentation

of the super-unifiedgroup.

Problems:(i) It is difficult to implement the requiredsymmetry breakingbecauseit requiresmany stages.

Perhapsa tumbling gaugetheory [571 is the solution.Therearealsodifficulties with vacuumalignment[58] for certainchoicesof the technicolourgroup(seesection5.4).

(ii) There are often a large number of technifermionswhich carry SU(3)coLo~,SU(2)LEFT and~ quantumnumbers.Thesefermionscontributeto the f3 functionswhich determinethe couplingconstantrenormalizations.In the grandunified picturethe couplingconstantrenormalizationsetsthescale for proton decay and determinesthe low energyweak mixing angle Ow [1]. The addition oftechnifermions(especiallymany of them)canruin thesepredictions.A super-unifiedmodel with manytechnifermionsmay also havea techniforcewhich is not asymptoticallyfree [59].Then it is hard toimaginethe techniforcebecomingstrongas you comedown in energyfrom the unificationpoint.

(iii) The massrelationsamongstthe light fermionsareusuallywrong.This is similar to the standardSU(5)modelwith aRiggswheremClCCfrOfl = mdowfl.

The ideaof super-unifyingtechnicolourwith the, otherknownforce is soattractivethat it shouldbepursueddespitethe difficulties.

5.3. Groupsbreakingthemselves

In the previoustwo subsectionsweexploredthe possibility of placingthe technicolourgroupinsideof alargergroupwhich breaksdown to technicolourat somescaleAE perhapson the orderof 10 TeV.Thenat somescaleATC the technicolourforce breaksSU(2)LEFTx U(1)HTp. Theelectroweakgroupandthe colour grouparebelievedto breakapartat the grandunification scale=1015GeV.

All of theseideasrequire a sequenceof symmetry breakingscalesbetweenthe grand unificationpoint andpresentenergies.The technicolourphilosophyprohibitsthe existenceof fundamentalscalars.The questionarisesof whetheror not a gaugetheorywith only fermionsandgaugebosonscan exhibitthis kind of sequentialbreaking.Recentideasindicatethat this is possible[60].

Startwith a gaugegroup anda setof fermionsin variousrepresentations.The numberof fermionsshould be small enough to keep the theory asymptoticallyfree. At the tree level the fermionswillexchangea gaugeboson andbe attractedor repelleddependingon the sign of a group theory factor(seefig. 21). This factordependson the representationof the initial fermionsandthe representationofthe combinedchannel.Forexamplein colour SU(3)a quark(~)andantiquark(~*)areattractedif theyscatterin a singlet but repulsedif they scatterin the octet (s). The strengthof the attraction (orrepulsion)for fermionsf

1, f2 to scatterinto astateof representationr is

312 E. Farhi andL. Surskind, Technicolour

Fig. 21.

c(fi, f2, r) g2(E) (5.16)

where c is the group theory factor (positive for repulsion, negativefor attraction)and g(E) is therunningcouplingconstantof the theory.

It is possiblethat if the combinationin eq. (5.16) is big enoughand the final stateis a Lorentzinvariant then condensationwill occur. For example in QCD at low enoughenergiesthe couplingconstantis big enoughso that thereis a condensationin the quark—antiquarksinglet channel.(Seesection2.1.)This causesa spontaneousbreakdownof SU(2)LEp.rX SU(2)RIQHT. Sincetheattractionis inacolour singletchannel,colour is still agood symmetry.

Givenaset of fermionrepresentationsof a grouptherewill beamostattractivechannel[57], MAC,that is to say a choiceof two fermionsfor theinitial stateanda representationfor the final stateforwhich eq. (5.16) is most negative.As you comedown in energythe couplingconstantgrows andthischannelwill condensefirst. If thecondensateis not asinglet of the group,then thegroupwill breakitself. For exampleif ordinarycolouris the forceof attractionbetweena set of fermionswhich are allcolour3’s (no ~ thentheMAC is two ~‘sto attractinto a ~ state.If condensationoccursthe groupcolourSU(3) would breakitself down into colour SU(2).

Imaginea non-AbeliangaugegroupG anda setof fermionrepresentationswith the gaugegroupunbrokenat somehigh scaleE

0. If thegroupis asymptoticallyfree, thenthecouplingconstantincreasesasyou lower theenergy.In fact hg

2goesas logE.At somescaleE1 condensationwill occurin themost

attractivechannel and 0 will break to a subgroupG1. Now the fermionsshould be describedundertheir representationof G1. If thereareno symmetriesto forbid themfermionmasstermswill appearatthescaleE1 so somesubsetof fermionswill becomemassivewith masses—E1 and thesefermionswillonly weakly influencephysicsat energiesbelow E1. Now track thecouplingsassociatedwith the gaugegroupG1. If oneor moreof themis asymptoticallyfree therewill be ascaleE2 at which condensationoccursin a newMAC andG1 will breakto G2. Newfermionmasstermsappear,etc., andthe processcontinues.In this way a gaugegroupG canbreak in a seriesof steps03 G~3G2. . .3 GN at scalesE1 > E2-~. > EN. The logs,of thesescalesareseparatedby numbersof orderunity so the scalesdifferby factorsof —10 or ~~102or —10~,etc.At the endof the sequencethe group GN cannotbreakitselffurther.This pictureis calleda tumbling gaugetheory.

The tumblingideaimplies a sequenceof scalesof symmetrybreakingbetweenthe low energyworldandthegrand unificationpoint. No fundamentalscalarsareneededto implementsymmetrybreaking.Models which tumble havebeenconstructedandwe urge the readerto consult ref. [57] for specificexamples.

5.4. Vacuumalignment

In technicolourmodelsaspontaneoussymmetrybreakingis inducedby the technicolourforce whichbreakstheweak interactionsymmetrygroup.The dynamicsof thesesystemsis extremelycomplicated


andit is usuallyonly an assumptionthat thedesiredsymmetry breakingpatternoccurs.However,it isoften possibleto comparetheenergiesof thedifferentvacua(or groundstates)which resultfor differentsymmetry breakingpatternsandto choose the correctsymmetry breakingpatternby minimizing theenergy.

Whenthe technicolourforceoranotherstrongforcecausesa fermioncondensateto form, theglobalsymmetrygroupof the theoryis generallyreduced.For exampleQCD reducesthe symmetryof a twoquarkworld from SU(2)LEFTX SU(2)RIQHTdown to SU(2)v~roR.(Seesection2.1.) If theoriginal globalsymmetrygroup is G andtheunbrokengroup is H C G thenthevacuumis left invariantby any groupelementh E H, i.e.,

hlvacuum)= vacuum) (5.17)

whereas

blvacuum)~ ~vacuum) (5.18)

if b E0 but b~H. In the absenceof otherinteractionswe canuseeq. (5.18)to definea newvacuum:

blvacuum)= vacuum)’ (5.19)

and the new vacuum,Ivacuum)’, will be invariant under a new subgroupH’ of 0. In generalH’ isequivalentto H. To be lessabstract,supposeour originalLagrangianis 0(n)symmetricandthevacuumpoints in the n direction spontaneouslybreaking0(n)down to O(n— 1). If we rotatethevacuumtosomeotherdirectionthe theorywill still havea left over O(n — 1) symmetry.As long as all directionsareequivalentthe choiceof vacuumdirection is arbitrary andtheunbrokengroupsareequivalent.

However,theweak interactionsgaugea subgroupG~of 0 so all directionsin thegroupG arenotequivalent.The energyof the vacuumwill be a function of its orientation relative to U,,. The truevacuumis theonewith thelowest energyandthe problemthatmust be solvedis how to find the truevacuum.Weinberg[61]first setup the formalismto solvethis problemin generalandtwo independentpapersby Peskin and Preskill [58]have applied theseideasto systemswhich are quite relevanttotechnicolourmodel building. We will showtwo examplesfrom their papersto seehow importantthisproblemis. The first examplestartswith a technicolourgrouplike SU(N) andtwo technidoubletswithconventionalweak interactionpropertiesbut different charges:

(Ai) = q1 (Al)IUGHT Y = q1+ ~B1 LEFT (B1)iuGi~rr Y — —

fA2\ (A2)IUGHT i’ = q2+ ~ (5.20)= q2 (B2)RIGHT Y = q2—

Neglectingweak interactionsthe world looksSU(4)LEFTx SU(4)RIGHT symmetricandweusuallyassume

thecondensation

(A1AI + ~1B1+ A2A2 + B2B2) � 0 (5.21)

which reducesthe symmetry to SU(4)VECTOR. As far as technicolourinteractionsare concernedany


SU(4)LEFTX SU(4)RIQHTrotationof the vacuumof eq. (5.21)shouldleavethe vacuumenergyinvariant.However,whenweak interactionsareconsideredtheenergyis extremizedby only two directions.Onedirectionis that of eq. (5.21) and theotheris thevacuum:

(AIA1 + ~1A2+ A2B1 + B2B2)~ 0 (5.22)

which is left invariant by a differentSU(4). The physicsof thesetwo vacuais very different. The firstchoicebreaksthe SU(

2)LEFTx U(l)HTp gaugegroupdownto U(l)EM whichis the conventionalbreaking.Thesecondvacuumleavesno unbrokengroupandthe photonbecomesmassive.The energeticsis suchthat if

0<~q1—q2I<1 (5.23)

thesecondis preferred.This result is quite surprising.Two technidoubletsof slightly different chargewill not line up and

leave the photon massless.In building technicolourmodels the charge assignmentsof the tech-nifermions,contributingto theweak interactionsymmetrybreaking,arecrucial.

We havebeengenerallyassumingthat the technifermionstransformundera complexrepresentationof the technicolourgroup.For example,if technicolouris SU(N) technifermionsusuallytransformasN’s while anti-technifermionstransformasN”s. If we assumethat the technifermionstransformundera real representationof the technicolourgroup thentechnifermionsand anti-technifermionshavethesametransformationproperties.Thiscan affect theoriginal global symmetriesof thegroupaswell asthepatternof symmetrybreakdown.Our next exampleillustratesthesepoints.

Considerone doublet of technifermions(A, B) with an arbitrary meanchargeq. We will useanotation where all fields are left-handedso right-handedfields are replacedby left-handedchargeconjugates:

fA\ - A~FT Y=-q-~Y—q ~ (5.24)

\D / LEFF DLEFF i — —q ~.

If the fermionstransformundera complexrepresentationR of the technicolourgroup, thenALEFT and8LEFT will transformastwo R’s andA~FTandB~FFwill transformastwo R*~s.The global symmetryof technicolouris SU(2)x SU(2). However,if the representationis real thenwe havefour R’s andtheglobal symmetryis enlargedto SU(4).

Let us assumethat the technifermionstransformunderareal representationR with the additionalassumptionthat the antisymmetricproductof two R’s makesasinglet. For exampleif technicolourisSU(2) andtechnifermionsaredoubletsweknow that(1/2x iI2)~rIs~ .mic is a singlet. This will alsobe true for fundamentalrepresentationsof SP(N)groups.Now the condensatewe usually assumehasthe form:

IA C A A A C C~ttLEFT/1~LEFT — ‘1~LEFT’~ LEFT + DLEFFDLEFT — 1~LEFT1~LEFT .

The minus signscomefrom:(a) antisymmetrizingin technicolourindicesto makea technicoloursinglet(b) antisymmetrizingin spin indicesto makeaLorentzscalar(c) Fermi statistics.

E. Fathi and L. Sugskind, Technicolour 315

This condensatebreaksSU(4)down to SP(4).Theweak interactionsbreak from SU(2)LEFTx U(h)~.down to U(l)EM. We must comparetheenergyof this condensatewith the energyobtainedby makingall possibleSU(4) rotations on the condensate.The result is that the energyis extremizedby twocondensates:theonein eq. (5.25) andalso by:

/A 1~ A L AC C T~C A C\rtLEFrOLEFr — L~LEFT/ILEFF T .

1~.LEFFD~ — DI..EFrI~tLEPT�

Thiscondensateviolates electricchargebut it is SU(2)LEFTinvariant so the electroweakgaugegroupbreaks from SU(2)LEFFX U(1)F~-+SU(2)LEFT which i•s an unacceptablebreaking pattern. Whichvacuumis preferreddependson the relative strengthof theSU(2)LEFr coupling constantg andtheU(l)HyP coupling constantg’ as well as the meanchargeq of the doublet. If I~I<~ the preferredvacuumis thesecondone leavingSU(2)LEFTunbroken.If I~I> ~ and

cot28,, = g~/g~<~[16q2— 1] (5.27)

then the first vacuum leaving U(l)EM as a good symmetry is preferred.For one doublet and theobservedweak mixing anglethis requires~qI> 0.85 which is not satisfiedby quarks(III = ~)or leptons(IqI=~)-

In modelswith real representationsof the technicolourgroup the correct breakingof the weakinteractiongaugegroupmaybe hardto implement. (For a discussionof modelswherethe symmetricproductof two representationsformsa singlet seeref. [581.)Technicolourgroupsof this kind oftenarisein modelswhere technicolouris unified with other forces and again theseresultscan place severeconstraintson modelbuilding (seesection5.2). However,the true vacuummust also be alignedwithrespectto the interactionswhich give light fermionstheir massesand theseinteractionsmight causethevacuumto align differently.

6. Compositefermions

Physicsat energyscalesup to the weak interactionscalemay well be describedby quarks, leptonsand technifermionsviewed as pointlike fermions interacting through gauge boson exchange.Thesuccessesanddifficulties of this point of view havebeenreviewedin the previoussections.However,there may be advantagesin imagining that the observedfermions and the technifermionsare notpointlike but rathercomposite[621just like theprotonis a compositestateof threequarks.This opensthedooronto a newsetof interactionsbetweenordinaryfermionsandtechnifermionswhich mayhelpcuresomeof our previousproblems.

6.1. Light compositefermions

If quarksandleptonsarecomposite,the forceswhich bind their constituentsmustbe acting on ascale much higher than the ordinary particle massscale of —1 0eV. This must be true becausetheelectronlooks pointlike down to distancesbelow 10_16cm while its inverseComptonwavelength ism 4x 10~”cm. How can a particle havea massso much smaller than the scale of interactionsexperiencedby its constituents?As was emphasizedin section 1, the mass must be protectedby asymmetryandfor fermionsthis will be achiral symmetry.


Supposethat the leptonsarecompositebut their massesareprotectedby achiral symmetry.Therearestill limits which can be imposedon the scaleB of thebinding forcesfrom low energyexperiments.Thegyromagneticratiosg~andg. of the electronandmuon arevery closeto their Diracvaluesof twowith thedeviationsfrom two beingaccountedfor beautifullyby QED. Fortheelectrontheagreementisgoodup toafractionalerrorof—Sx h0’°andforthemuon— 10_8.If afermioniscompositeyouexpectg todiffer from twoby factorson theorderof me/B.However,if themassmtismuchlessthanB becauseof achiralsymmetry,thesamechiralsymmetrywill guarantee[63]thatthedeviationsfromtwo areoftheorderof (m~/B)2.For theelectronthis gives B ~ 25GeVandfor themuonB ~ h TeV. It is possiblethat theobservedfermionsareboundby forces which are strongat orabovethe technicolourscale.

6.2. ‘t Hooft’s conditions

Supposea setof fermions interactthrough a strongforce associatedwith a confining gaugetheoryandtheforcegetsstrongatthescaleB. Thefermionswill formavarietyof boundstateswhicharesingletsunderthe stronggaugegroupand someof thesebound stateswill also be fermions.Which of thesecompositefermionsarelikely to bemasslesson thescaleB? ‘t Hooft hasprovidedapartialanswerto thisquestion[64].

We will call thefermionsinteractingthroughthestrongconfiningforcepreons.Thepreonstransformnon-trivially underthe strong groupG~andtherewill also be someglobal chiral flavour group0F~ (Ifthe strongforce is QCD andthe preonsare like the up anddown quarksthenG~is SU(3)and 0F isSU(2)LEFTX SU(2)RIGHTX Baryon Number.) In generalif the chiral flavour group 0F is not broken,either explicitly or spontaneously,it will protect the preonsfrom getting a mass. (If SU(2)LEFTX

SU(2)RIGHT is unbrokenthe up and down quarksare massless.)Now the theory will haveanomaliesassociatedwith masslesspreonsrunning throughtriangle diagramswherethereis a flavour currentateachcornerof thetriangle.Supposeyou wantedto gaugetheflavourgroupO~.Forthegaugetheoryof theflavour groupto besensibleit mustbe anomalyfree [50]sotheseflavouranomaliesmustbecancelledbyotherfermions.‘t Hoofthascalledtheseotherfermionsspectatorsbecausetheydo not feelthestrongforcebut they do havenon-trivial transformationpropertiesunderGF. (The flavouranomaliesof quarksareexactlycancelledby theflavouranomaliesof leptonswhicharespectatorsherebecausetheydo notfeelthestrongforce.)

Thepreonsandspectatorstogethermakeasensibleanomalyfree theory. However,thestrongforceis confiningandthephysicalstateswill bespectatorsandboundstatesof preonsall of which aresingletsunderG~.It shouldbe possibleto write down a sensiblelow energyanomalyfree theory in termsofphysicalstates.Thiscan beusedas a guide in decidingwhich of thepossiblecompositefermionsshouldbe masslesson the scaleB.

All of the possible fermions which can be made of preons in a G~singlet will have differenttransformationpropertiesunder the flavour group GF and each will contribute differently to theanomalydiagramsassociatedwith 0F• ‘t Hooft arguesthat the set of masslesscompositefermionswhichexists in the spectrumis the set whose 0F anomalycontributionsexactlycancelthoseof the spectators.In otherwords, themasslesscompositefermions will havethe sametotal 0F anomaliesas the preons.

‘t Hooft only consideredthecasewherethestrongforcedoesnot causea spontaneousbreakdownof0F andhe foundonly oneexampleto illustratetheseideas.Again it is QCDwith two quarksu andd.If QCD does not breakSU(2)LEFTx SU(2)RJGHTthen you expect the spin ~colour singlet states,theproton andthe neutron,to be massless.And in fact the GF anomaliesof the proton and the neutronexactly cancelthoseof the leptonswhereasbeforeit was the quark anomaliesthat did the job.

E. Farhi andL. Susskind,Technicolour 317

More generallythestrongforcewill causea condensationthat breaks°F down to anotherflavourgroupG~.Usually G~is containedin GF but it mayalsobe equivalentto 0F- The anomalycancellationshouldbeenforcedonly with respectto G~currents.Examplesof this kind havebeenconstructed.

A usefulaid in finding examplesis thecomplementarityprinciple [65].This statesthat in certaincasesa gaugetheorywhich undergoesspontaneoussymmetrybreakdowncan equivalentlybe viewed as atotally confiningtheorywithout symmetrybreakdown.This principle is very elegantandcan beusefulbut we will not elaborateon it here.

The point we want to emphasizeis that thereis a natural framework in which you would expectfermionic preonsbinding atthe scaleB to producemasslessfermionsor at leastfermionsmuchlighterthan B. This will occur in a theory with a chiral global symmetry group if you require anomalycancellationsof theeffective low energytheory. The symmetrygroupcan then protect themassesofcertaincompositefermions.

6.3. Technifermionsaspreons

Theobservedquarksand leptonscould be bound statesof fermionicpreonswhosebinding scaleisexperimentallyrestrictedto benearor abovethe technicolourscaleATC- As we sawin the last sectionitis evenpossibleto imaginethat therecould be good theoreticalreasonsto expecttheseparticlesto belight on the scale ATC. It is then possible that thesepreonsare the technifermionsor that thetechnifermionsaremadeof thesamepreonswhich makequarksandleptons[66].

Imaginea theory of preonswhich feel astrongforceassociatedwith agaugegroupG~.Thereis aglobal chiral symmetry group 0F which protectsthe preon masses.Also imagine that a subgroup0,,. C 0F is gaugedand is associatedwith theweak interactions.U,, shouldatleastcontain SU(2)LEFTX

U(l)H.~.and at the Lagrangianlevel all of °F is unbroken.Thereareno Riggsbosonsin the theory.At the scale ATC the strong force causesa spontaneoussymmetry breakdown and the global

symmetrygroupis reducedfrom 0F to GLE. Certainpreonswill havemassesprotectedby GLE andwillremain masslesswhile the otherswill get massof the orderof ATC. We can also look at thefermionswhich can be madeof confinedpreonsin 0~singlets.We arguedin the last sectionthat a setof thesewill remainmasslessalso beingprotectedby G~.Theotherswill get masson theorderof ATC. (Thosefermionsmadeof massivepreonsgetmass.)Wewill identify themasslesscompositefermionsasordinaryquarksandleptons.

The spontaneoussymmetrybreakingalso affects the weak gaugebosonsector.The weak bosonsassociatedwith generatorsof 0,,, which arealsoin G~will remainmassless.Theotherswill get amasson theorderof gWATC whereg,, is the weak interactioncouplingconstant.

The weak interactionscan also give small massesto the masslesspreons.If someof the brokengeneratorsof 0,., connectamasslesspreonto a massiveone thenwe expectthemasslesspreonto pickup a massof the orderof aWATC (a,.,= g~,/4ir).This was the schemeWeinberg[61] originally hadinmind for giving light fermions mass in dynamicaltheories.The sameeffect then givesa massof theorderof aWATC to someof themasslesscompositesby connectingthenthroughweakbosonexchangetomassivecomposites.If we imaginethis to be the massof a realquarkor lepton it is aboutright for theheavygeneration.Perhapstheothergenerationsonly get a massof ordera~ATCor

Many of thesepointscanbe illustratedwith anexample[67].Imaginethestrong forceis SU(3) justlike QCD andtherearethreeflavoursof preonsPA, PB and PCeachtransformingasa left-handedandaright-handed~ of SU(3). The flavour symmetryOF is SU(3)LEFTx SU(3)RIQHTx PreonNumber andallof thepreonsaremassless.Supposethestrongforce causesthe unusualcondensation


(Pcpc)�O. (6.1)

This will break the GF symmetry group down to SU(2)LEFTx SU(2)~0~x PreonNumberx PreonCNumber. The preon PC becomesmassivewith a masson the order of ATC while PA andPB are still

massless.The physicalstatesshould be singletsunder the strong groupSU(3). With threetypes of preons

thereis an octet of possiblespin ~singletsunderSU(3). This octetis analogousto the baryonoctet inQCD with threeflavours.The two stateswhich do not containPc shouldbe masslessandin fact theirmassesare protectedby the unbrokenflavour group. In fact thesetwo masslesscompositeshavethesameflavour anomalyasthepreons,with respectto theunbrokenflavourgroup.Wewill think of thesetwo masslesscompositesasordinary fermions.

Now imaginegauginga subgroup0,,, of OF which will be theweak group.Supposeit is the SU(3)formed as the diagonal sum of. SU(3)LEFT and SU(3)~G~.The condensateof eq. (6.1) breaksSU(3)wE~jc-*SU(2)wE~&jcx U(1) wherethe U(1) is associatedwith the eight generatorAg of SU(3)w~ic.Theunbrokenweakgaugegrouphasfourgeneratorsandthe four generatorsof the brokenweakgroupconnectPA andPEtOPCin aleft-right symmetricmanner.Thebosonscorrespondingto thesegeneratorswill give PA andPB a masson the orderof aWATC. At the sametimethe two masslesscompositestateswill alsoget a masson the orderof ~ Thiscan bethoughtof in two ways.The masslesscompositepicksup asmall massbecauseoneof its constituentpreonsgetsa mass.Alternativelyyou can imaginethe masslesscompositeemits a heavyboson and becomesa massivecompositewhich reabsorbstheboson.Thesedifferent viewsare shownin fig. 22 wherethe masslessstatespick up small massesfrommassivestates.

PA

b () b

massless massive massless mossless ___________________mo.sslesspreon ~ ~ composite PB composite

massless massive masslesscomposite composite composite

Fig. 22.

Of coursethis exampleis totally unrealisticbut it hasmanyinterestingfeatures.In particular it is

possibleto give massto light ferinionswithout introducing a new scaleof interactionsAE- Ordinaryfermion massesmight be a,, or a~. effectson the technicolourscale.The searchfor a truly realisticmodel of this sort is underway.

It is not clear that this type of model will avoidthedifficulties which exist in extendedtechnicolourmodels.The simplest technicolourmodel works beautifully in helping us understandweak interactiondynamicsbut it doesnot providefermionmasses.The complicationsrequiredto give fermion massesleadto severeproblems.In theoriginal weak interactiontheorywith fundamentalscalars,massesarisefrom arbitraryYukawa couplings.The theorycan fit thedatabut no insight is gained.Understandingtheorigin of fermionmassesis oneof thegreatchallengesfacingparticlephysiciststoday.

E. Farhi andL. Susskind,Technicolour 319

References

[1] H. Georgi,H.R. Quinn andS. Weinberg,Phys. Rev.Lett. 33 (1974) 451.[2] We arenot concernedherewith aprecisedefinitionof AQCD.[3] H. Neilson hasa different point of view, for example in: Proc. 17th ScottishUniversitiesSummerSchool in Physics,SI. Andrews,eds.I.M.

BarbourandA.T. Davies(1976).[4] S. Weinberg,Phys.Rev. Lett. 19 (1967)1264;

A. Salam,ElementaryParticleTheory, ed. N.Svarthold(Almquist and Wiksells, Stockholm,1968).[5] H. Georgi andS.L. Glashow,Phys. Rev. Lett. 32 (1974) 433.[6] F. Englertand R.Brout, Phys.Rev. Lett. 13 (1964) 321;

P.W. Higgs, Pisys.Rev.Lett. 13 (1964) 508;G.S. Guralnik,C.R.Hagenand T.W. Kibble, Phys.Rev. Lett. 13 (1964) 585;P.W. Higgs, Phys.Rev. 145 (1966) 1156;T.W.B. Kibble, Phys.Rev.155 (1967) 1554.

[7] M. Gel-MannandM. Levy, NuovoCimento16 (1960)705.[8] For ageneraldiscussionwith referencessee

Ta-PciChengandLing-FangLi, Preprint numberCOO-3066-143(1980).[9] D.J.GrossandF. Wilczek, Phys.Rev.Lett. 30 (1973) 1343;

H.D. Politzer,Phys.Rev.Lett. 30 (1973) 1346.[10]T. Daset al., Phys.Rev. Lett. 18 (1967) 759.[11]This reviewis in thespirit of

L. Susskind,Phys.Rev. D20 (1979) 2619andS. Weinberg,Phys.Rev.D19 (1979)1277.Manyof the ideascan alsobedirectly tracedtoS. Weinberg,Phys.Rev.D13 (1976)974.Earlierwork on dynamicalsymmetrybreakingincludesJ. Schwinger,Phys.Rev. 125 (1962) 397; 128 (1962) 2425;R.Jackiw andK. Johnson,Phys.Rev.D8 (1973) 2386;J.M. Cornwall andR.E. Norton,Phys. Rev.D8(1973) 3338;M.A.B. BegandA. Sirlin, Ann. Rev.Nucl. Sci. 24 (1974)379.

[12] Otherrecentreviews includeK.D. LaneandM.E. Peskin,XV RencontredeNonond,Vol.11, ElectroweakInteractionsandUnified Theories,ed.J.TranThankVan(EditionFrontières,France,1980);P. Sikivie, CERNpreprintTH.2951(1980), to bepublishedin: Proc.VarennaSummerSchool 1980.

[13] L. Susskind,ref. [11].[14] J.Goldstone,NuovoCimento 19 (1961)154;

J.Goldstone,A. Salamand S. Weinberg,Phys.Rev.127 (1962)965.[15] For a reviewsee

S.L. Adler andR.F.Dashen(eds.),CurrentAlgebraandApplicationsto ParticlePhysics(NewYork, Benjamin,1968).[16] T. HagiwaraandB.W. Lee,Phys.Rev.D7 (1973) 459;

M. Weinstein,Phys.Rev. D7 (1973) 1854 andD8 (1973) 2511;L. Susskind,ref. [11].

[17] Ref. [16]andS. Weinberg(1979) in ref. [11].[18] Seethediscussionaftereq. (1.4).[19] Technicolouris alsocalled hypercolour,metacolour,heavy-colour,extra-colour,ultra-colour,super-colourandsometimescrazy-colour.Note

thespellingof colour is by orderof theCEURNSecretariat.WE know how to spell!!![20] L. Susskind(1979) andS. Weinberg(1976)in ref. [11].[21] SeeT. HagiwaraandT.D. Lee(1973) andM. Weinstein(1973) in ref. [16].[22] G. ‘t Hooft, NucI. Phys.B72 (1974)461; B75 (1974) 461.[23] L. Susskind,ref. [11].[24] E.Witten,NucI. Phys.B160 (1979) 57.[251C. Bouchiat,J. IliopoulosandP. Meyer,Phys.Lett. 38B (1972) 519.[26] A model in which onetechnifamily occursnaturally is

E.FarhiandL. Susskind,Phys. Rev.D20 (1979) 3404;seealsoS. Dimopoulos,Nuci.Phys. B168 (1980) 69.

[27] S. Weinberg(1976) in ref. [11];S. DimopoulosandL. Susskind,NucI. Phys.B155(1979) 237;E. EichtenandK.D. Lane,Phys.Lett. 90B (1980) 125;M.A.B. Beg,H.D. PolitzerandP. Ramond,Phys.Rev.Lett. 43 (1979)1701.


[28] J.D.Bjorken,unpublished;S. Dimopoulos(1980) in ref. [26];

M.E. Peskin,Nuci.Phys.B175 (1980) 197;J. Preskill, NucI. Phys.B177 (1981)21.

[29] S. DimopoulosandL. Susskind(1979) andE. EichtenandK.D. Lane(1980)in ref. [27].[30] S. Dimopoulos,S. RabyandG.L.Kane, Nuci.Phys.B182 (1981)77;

J. Ellis, M.K. Gaillard, D.V. NanopoulosandP. Sikivie,Nucl, Phys.B182(1981) 529.[31] Seeref. [30];

F.Hayot and0. Napoly, C.E.N.Saclaypreprint Dph-T-86(1980).[32] S. Dimopoulos(1980) in ref. [26];

seeref. [31].[33] SeeF. Hayot and0. Napoly (1980)in ref. [31].[34] J. Ellis, M.K. Gaillard, D.V. NanopoulosandP. Sikivie (1980) in ref. [30].[35] Reference[28]aswell as

E. EichtenandK.D. Lane(1980)in ref. [27];S. ChadaandM.E. Peskin,CERN preprints3023 and3038 (1981).

[36]J. Ellis, M.K. Gaillard, D.V. NanopoulosandP. Sikivie (1980)in ref. [30].[37]A. All andM.A.B. Beg,DESYprepnnt80/98 (1980);

seeref. [33].[38]Thesearesimilar to theaxionsof:

R. PecceiandH.R. Quinn,Phys.Rev.Lett. 38 (1977) 1440;S. Weinberg,Phys.Rev.Lett.40 (1978) 223;F. Wilczek, Phys.Rev.Lett. 40 (1978) 279.

[39]E. EichtenandK.D. Lane(1980) in ref. [27],aswell asref. [28].[40] J.Ellis, M.K. Gaillard, D.V. Nanopoulos,NucI. Phys. B106(1976) 292.[41] M.A.B. BCg, H.D. PolitzerandP. Ramond(1979)in ref. [27].[42] A. Au andM.A.B. Beg(1980) in ref. [371;

J. Ellis, M.K. Gaillard, D.V. NanopoulosandP. Sikivie (1980) in ref. [30].[43] S. DimopoulosandL. Susskind(1979)in ref. [27];

E. EichtenandK.D. Lane(1980) in ref. [27].[44] S. Weinberg(1976) in ref. [11];

M. Peskin,unpublished.[45] S. Weinberg,HarvardpreprintA023 (1980).[46] S. Weinberg,Theproblemof mass,in: Trans.N.Y. Acad.Sci. SeriesII, 38 (1977)185.[47] This is a generalizationof otie useof theextendedtechnicolouror sidewaysideaintroducedby

S. DimopoulosandL. Susskind(1979) andE.EichtenandK.D. Lane(1980) in ref. [27].[48] S. Dimopoulos,S. RabyandG.L. Kane(1980) in ref. [30].[49] M. Dine, E. FarhiandL. Susskind,unpublished;

S. DimopoulosandJ. Ellis, CERN prepnntTh.2949(1980).[50] M.K. GaillardandB.W. Lee,Phys.Rev.D10 (1974) 897.[51] S.L.Glashow,J. Iliopoulos andL. Maiani, Phys.Rev.D2(1970)1285.[52] S. DimopoulosandJ. Ellis (1980)in ref. [49].[53] R.N. CahnandH. Harari,NucI.Phys.B176 (1981)135.[54] Forexample,H. Georgi andS.L. Glashow(1974) in ref. [5];

J. Pati andA. Salam,Phys.Rev.D8 (1973) 1240;H. FritzschandP. Minkowski, Ann. Phys.93(1975)256plus manymore.

[55]E. FarhiandL. Susskind(1979) in ref. [26].[56] S. Diinopoulos,S. RabyandP. Sikivie, NucI. Phys.B176 (1981)449;

B. Holdom, Harvardpreprint (1980).[57] S. Raby,S. DimopoulosandL. Susskind,Nucl. Phys.B169 (1980)373.[58] M.E. Peskin(1980) andJ. Preskill (1980) in ref. [28].[59] P.H. Frampton,Phys. Rev.Lett. 43 (1979)1912.[601Ref. [571andH. Georgi,unpublished.[61] S. Weinberg(1976) in ref. [11].[62]A partial list of compositeideasincludes:

J.C. PatiandA. Salam,Phys.Rev.D10 (1974) 275;J.C. Pati, A. Salamand J.Strathdee,Phys.Lett. 58B (1975)265;H. Terazawa,Universityof Tokyopreprint INS-Rep35 (1979);O.W. GreenbergandC.A. Nelson, Phys.Rev. D10 (1974) 2567;


0.W. Greenberg,Universityof Maryland report 76-012(1975);J.D.Bjorken,unpublished;G.R. Kalbfleish andB.C. Fowler,NuovoCimento19A (1974)173;E. Novak,J. SucherandC.H. Woo, Phys. Rev.D16(1977) 2874;M. Veltman,Proc.Intern. Symp. on LeptonandPhotonInteractionsat High Energy, FNAL, Batavia(1979);H. Harari,Phys.Lett. 86B(1979) 83;M.A. Sharpe,Phys.Lett. 86B (1979)87;JO.Taylor, Phys.Lett. 88B(1979) 291;R. Raitio, Helsinki preprintHU.TFT.79-39(1979);V. Visnjic-Triantaflllou,Fermilabpreprint Pub-80/15(1980);E.Derman,Universityof ColoradopreprintCOLO-HEP-19(1980);J.C.Pati, Universityof Maryland preprint (1980);S. Dimopoulos,S. RabyandL. Susskind,NucI. Phys.B173 (1980)208.

[63] SI. BrodskyandS.D. Drell, SLAC-PUB-2534(1980);R.Barbieri, CERN preprintTH.2935(1980);L. Susskind,unpublished.

[64] 0. ‘t Hooft,Lecturesat CargèseSummerInstitute (1979).[65]L. Susskind,unpublished;

T. BanksandE. Rabinovici,Nucl.Phys.B160 (1979) 349;E. FradkinandS:H. Schenker,Phys.Rev.D19 (1979) 3682.

0. ‘t Hooft in ref. [64];S. Dimopoulos,S. RabyandL. Susskind(1980)in ref. [62].

[66]Many of theideasin this sectionareduetoM. Peskin,unpublished.[67]This exampleis dueto P. Sikivie.

technicolour edward farhi and leonard susskindshufang/phys579c_2006/termpaper/... · e. farhi and...

Documents