Coleman & Weinberg -Gauge Theory

Sidney Coleman: Secret Symmetry 1973 Steven Weinberg :Recent Progress inGauge Theories 1974 Phil Lucht Notes 1974 &1979 SidneyColeman {413 ; NoChie € 1 i, f . “hrSECRET SYRMETRY: ** AN INTRODUCTION TO SPONTANEOUS SyEUERRY BREAKDOWN AND GAUGE FrEnDs* Lectures given at tne 1973 gnternational Summexr. School of Physics Ettore Majoxana Sidney-Colemin * Lyman Laboratory of Physics Harvard University -- * Cambridge, Massacilusétts) USA had i \ ‘3 ww s %Moxk supported inpart by the ilafjonal Science Foundation, under Grant Ho, GP30819%, Se) TABLEOFCONTENTS. Ji.INTRODUCTION L Ja.SECRETSYSMETRIES. INCLASSICAL FIELD‘THEORY 3 2.1 The Idea of Spontaneous Symmetry Breakdown 3 2.2, Goldstone Bosons in an Abelian Nodel 6 2.3 Goldstone Bosons in the General Case 7 . 2.4 The Iliggs Phenomenon in the Apelian Model. 9 2.5 Yang-ills Fields andtheMiggs Phenomenon in 2 the General Case 7 2.6 Summary andRemarks 15 =Js. sxenny nesonanzznmary wsSs 3,1TheOrderoftheArguments .16 3.2 Renoxmalization Reviewed au] 3.3 Functional Nethods and the Effective Potential 22 3.4 The Loop Expansion 24 3.5 ASample Computation 25 ” 3,6 The Most Important Part ofThis Lecture 27 : 3.7 The Physica) ieaning of the Bffective Potential 28 nt 3.8 Accidental Symmetry and Related Phenomena 32 3.9 An Alternative Method ofComputation 33 V4. PUNCTIONAL INTEGRATION (vULGARIZED) 34 4,1 Integration Over Infinite-Dimensional Spaces 34 4,2 Functional Integrals and Generating Functionals 37 4.3FeynmanRoles 42 8 v4 re) 4,4Derivative Interactions 43 4,5 Fermi Fields 46 a 4.6 Ghost Fields 47 Js. nuereynvn RULES oRGAUGE FIEZD THEORIES 49 5.1 Troubles with Gauge Invariance 49 5,2 The Faddeev-Popov Ansatz 50 5.3 Application oftheAnsatz 53 . 5.4 Justification of the Ansatz 55 5.5 Concluding Remarks 57 6. ASYMPTOTIC FREEDOM . 59 ‘ 6,1 Operator Products and Deep Inelastic «: Electropxodaction 89 . 6.2 Massless Picld Theories and theRenomalization 62Ke) Group 6.3 Exact andApproximate Solutions ofthe 6a Renormalization Group Equations 6.4 Asymptotic Freedom 67 6.5 No Conclusions 70 2 APPENDIX: ONE-LOOP EFFECTIVE POTENTIAL IN THE GENERAL CASE 71 a REFERENCES AND NOTES 74 Le) Z ms SECRET SYNAETRY: a ANANTRODUCTION TOSPONTANEOUS SYRMETRY BREAKDO: ANDGAUGE FrELDS* : Sidney Colenan : Lyman Laboratory ofPhysics zMaxvardUniversity a) Cambridge, Massachusetts, USN ra . 2,INTRODUCTION Herearesonelong-standing problems inparticle theory: :1)Howcanweunderstand thehierarchal structure ofthefundamentaL interactions? Arethestrong, medium strong (i.e., SU(3)-breaking), elec- tromagnetic, andweak interactions truly independent, oristhere someprin= ciple that establishes connections between then? 2)Howcanweconstruct arenormalizable theory oftheweak inter- but predicts finite higher-order corrections? . 3)Howcanweconstruct atheory ofelectromagnetic interactions in which electromagnetic mass differences within isotopic multipicts are . finite? 4)Howcanwereconcile Bjorken scaling indeepinelastic electrogro- ductionwithquantymfieldtheory?TheSLAC-HITexperimentsseemtobe a)telling usthat[tneLight-cone singularities intheproduct oftwocurrents axecanonical instructure;| ordinary perturbation theory, ontheother hand, fells usthatthecanonical structure isspoiled bylogarithaic factors, which getworse andworse aswegotohigher andhigher orders inthepor- turpation expansion, Arethere anytheories ofthestrong interactions for which wecantame thelogarithas, suathemupandshow theyareharmless? Enomrous progress hasbeen made onallofthese problems inthelast +)fewyears, ‘There nowexists alarge family ofmodels oftheweak andelece "tromagnetic interactions thatsolvethesecondandthirdproblem, andwe +, Mavediscovered asomewhat smaller family ofmodels ofthestrong inter-"actions thatsolvethefourthproien, Asueshallsce,thestructure of these models issuch thatweaxebeginning togetidens about thesolution tethe(very deep) first problem; connections axebeginning toappear in Mnexpected places, andaoptimist might saythat weareontheroad tothe first truly unified theory ofthefundamental interactions, ALLofthese nerelous developments arebased upontheideas ofspontancous symmetry breekdownandgaugefields,thesubjectsoftheseLectures, (e) Honesty conpels metomoderate thesales-pitch ofthe‘lastpiragraph “Worksupported inpartbytheNational Science Foundation underGraat Ro. GP 30a19x, \ any Index for Notes filed under Coleman paper fa) 1)summaries ofall6sections, alltogether. . . 2)proof offeynman rules, (4.35) atid (4.48) eto. 3)proof ofwhy W(J) issum ofconnected graphs 4)notes onfunctional derivatives 5)deriwations ofequations inSection 4(path integrals, Q(x), etc) 6)aboutderivative couplings, (actk)?, (4.56)and(4.61) 1)derivation of(5.4) and(5.26); theFadeev Popov business inQED . 8)derivation oftherenormalization group equation. 6 : ce) SidneyColeman Lectures at_1973 Int.SummerSchoolofPhysics EttoreMajorans Oo"Sdcret,Symmetry: anintroduction toSSBandgauge-fields:" (QIntroduction. : 1.Out of-all this stuff comes several renormalizeable theories ofweak interactions. Theproblem isnowtodosecond order experimenté to’decide which theory iscorrect. 2.Certain topics will not bementioned: fermions will beignored; renormalization will bemostly ignored; (subject ofVeltman andt'Hooft). Thegeneralized Ward identities called Slavnov Identities will beglossed over. @Classical FieldTheory: SSBmeetsGaugeTheory. 2.1 The idea that ifyou like inaworld inwhich the vaccum issuperconducting, you might have lots oftrouble realizing that the Hamiltonian infact has anexact symmetry. Little manintheferromagnet. The symmetry oftheHam. isjust hidden (secret) because the vacuum happens not tohave the symmetry. Ingeneral, ground states often do not carry the full symmetry ofthelagrangian, eg, particle ofspin not equal zero atrest. Firsxttheusualparitysymmetryisconsidered withGoldstones humpedcurve.By ©) tting thevaccum ctthe minumm ofthe potential sothat $=a isthe constant field which gives you the lowest value ofH(ground state), you have toshift the fields in order togetperturbation sense. Newlagrangian has lost theparity symmetry, because thevaccum broke itspontaneously. This isexample ofadiscrete symmetry breaking. Next hedoes the usual 0(2) example. Two scalar fields Aand Bare converted tofields RHO and THETA inobvious way. After shifting the fields, massless THETA particle appears, the Goldstone Boson. Obviously this isb because there isnocurvature along the trough. 2.2 Forgeneral Liegroup case, youstart with Ngenerators. Suppose vacuum hasn-m diensionsional flat sufface through your chosen value. Then there will beN-M Goldstones because that many directions togoflat. Vaccum still hasMunbroken symmetries. Theorem was proven: afield theory which iscovariant andhas positive-def metric must generate goldstones when you SSB. Gauge theories, however, donot satisfy these postulates. You either lose covariance oryou ere stuck with ghosts. Either condition violates the assumption ofanormal field theory. QEDisnotanormal field theory, itisaguage theory. 2.3 Higge idea isto“elevate: your global symmetry toalocal symmetry by addingsomevectormesons.Thenbychoosing gaugefunctions f(x)incertainsimpleway, Oyoucongauge away themassloess Goldsontes;p Higgs mechanism! Their life goes into the vectors and makes them massive. SoHiggs.solvestwoproblems atonce.-“Ihguagetheory, -youwereallwaysstuck withmasslessvectormesonstogetalocallygaugeinvarientlagrangian. InSSBtheory[) youwere always getting stuck With massless Goldstones. when youdombirie thetwotheori S? thetwothings combine andremove allmassless vectors except thephton, ifyoudoitright. OK,thenext step istoseewhat allthis hastodowith quantum field theory. . Sofar has been classical field theory. . e)SecretRenormalizability. le)3.1Outineofwhatwearegoingtodohere.Peculiarorderingofarguments, 3.2Renormalization reviewed. Consider asimple g*lagrangian. Incomputing the4-point function tosecond order you have toadd upgraphs like those shown infigure. Inorder toblock divergences, youputonanultraviolet cutoff. Then allgraphs arefinite and you get (3.2). Now goback and insert cutoff-dependent counterterms inlagrangian to exactly cancel thelogterm. Youhavesuccessfully renobmlized your¢*theory, . Ingeneral, arenormalizeable field theory isoneinwhich youaddafinite number ofcutoff dependent counterterms and you are able tobalance the theory in each order inthis way. Sidney would like togofurther and require that the, counter- terms havethesameformasterms already inthelagrangian, sobyadding couhterterms youarereally only changing parameters inthetheory. Atheory 9”withn=5andupisdefinitely notrenormalizeable, ie,youwill keep needing more andmore counterterms ofhigher andhigher order. "disgusting". Vector mesons present aspecial problem inrenormalization theory. Reason is thatthepropagator hasthatextra kK,terminit.Ifmeson ismassless,’ youare OK.Ifmeson iscoupled to@conserved current like thephoton field (which isalso mass- less),againyouareOK.Butamassivevectormesontendstoscrewyouup.However, ‘o}later wewill see that ifitgot its mass bySSB, then does not count asamassive meson from renorm point ofview. 33.Functional Mebhods andEffective Potential. (refs given) Writes down theconnected generating function which they call w(J), Abers andLeecall Z(J). Show thefunctiohal relation between this generating function and the theories greens functions, ie, VEVIOP's. Thefirst derivative ofthegenerating function isaVEVofjustonefield. Thisisthecalssical field, andthisisthegreat connection between thequantum and classical field theories. See (3.12).. Fa Thereexists asimilar generating function GAMMAcal&edtheEffective Action. when yyoutakefunctional derivatives ofthisthingwithrespect totheclassical fieldJ, viyougetthe"proper greensfunctions" ofthetheory. (one-article irreducible etc.)Whentheeffectiveactionisexpandedinpowersofexternalmomentaandconverted RAtocoordinate space, the first term isthe "effective potential". (This classical function ofthe classical field isinterpreted later on). So, out ofthe depths ofthe quantum field theory Sidney has extracted aclassical 6field, andaclessical potential. Nowthatyouhaveyourhandsontheseguys,youcanplot Vagainst @andlook for minima and seeifyou are inthe SSB situation. You can then proceed todoeverything you did before inthe classical analysis. Bek Theloop expansion, Nowtheproblem ishowdoyoucoitpite theeffective potential ifyouaregiven alagrangian? Somehow there isawaytocompute it.Thefirst contribution isUitsel?,ie,thepotential whichappearsinthelagrengien. Butthenthereare[}contributions from loops. Ifcoupling isweak, you could presumably add things upto getanexact result for’thiseffective potential V.Then’youcouldtakeitsderivatives togetthe proper vertices, and soon. 3.5Sample calculation. Hewrites downaformula forthefirst correction “term. the correction term diverges UV, soyou putinacutoff. But these are just thekinds ofterms which youcangetridOfbyadding counterterms tothelagrangian, specifically, adding counter terms into’the potential. Inanon-renormalizeable field theory, youare justplain stuck atthispoint. Youcannot comeupwithafinite value forV,thusyou cannever plotVversus f,tolookforasuperconducting vacuum, 3.6Most important part of’lecture. Thepoint issimply this: SSBdoes notaféect the renormalizeablility ofatheory. After SSBtheories may "look" non-renormalizeable because, eg,there aremassive vectors coupling tonon-conserved currents. However this isanillusion. Thereal Lagrangian isthething youhave before youdoSSBwhich has massless vectors coupling toconserved currents. InSidneys words: secret symmetry’ buysyousecretrenormalizeablility. . . .*) 3.7Meaning oftheeffective potential. Inarguement Idont’ quite follow, heshows that V(J,) isprecisely what itisinclassical field theory, ie,theenergy density. Imaginary partis related tochance ofdecay ofthestate inuation. . 3.8 Howbigisloop correction? Incertain theories, expecially gauge theories, theloop contribution toVmaydominate overthetreecontribution, because theloopallows new couplings tobeused. Idont getthepoint oftheSO(5) business here. 3.9Alternative approach. Sidney takes view ofdoing afield theory’ corrections first getting the renorm done, then doing the fields shift for SSB. Most people really dothings inthe other order, but then the logica isnot asclear. e CB)renetionsl Integration. 4.1Integration overinfinitedimensional spaces.Star'tswith‘aone-dimgaussianintegral. 1)Extends the idea tofinite dimensional space where expo isnow diagonal-matrix ‘element of afinite-dim operator. Actually, expo isaquadratic form inthe integration varilables. But you write itasaninner product: Then make the leap toinfinite diiiensions.. The integration and determinant that you get are defined by the limit only. ‘Inthe limit, Awill beanoperator, still must be'real symmetric Ithink. : 44.2Functional Integrals andGenerating Functionals. Yefines eMM(J), aspathintegral (fields only) -oftheexpo'd action. Just -adefinition. Youwant toactually dothis integral using methods learned inlast sections But, asafunction ofthe.fields g,the integrand isnotdecaying sointegral isnot-defined. Thetrick istorotate thek° and x°components togettheEuclidean action shown in(4.30), which isnowpositive definite s0you can dothe integral. Identifying the operator Aasin(4.31), you ‘cam explicitly dotheintegral toget(4.32). Younowhave thegenerating function notasapath integral ofanexpo, but» asasimple expo ofadouble spacetime integral. Ofcouise you Knew the answer. dll the timé was (4.28), soyou have accidehtitally found the determinant<* The ‘point isthatthisisamethod foractually doing:thepathintegrals. Itdoesnottell @~ you howto figure the determinant, however. - 2ie} The‘aboveexamplewe.afree-field scalerexample.Next,‘allowamintegattion asshown in(4.35). Thegoal here.is again toevaluate thepath integral. First, you writeitintheform(4.38) [note"just»interchange-order. andapply‘little: theorem to seethat (4.36) isthesame asLHS], ButtheRHSof(4.38) isalso obtained directly fromtheDysonequation asin(4.36). Thus,ineffect. youhave:donetheintegral without ever doing it. Yow.are making use ofthe free-field case. which you already’ proved. 43 Feynman Rules. The Feynman rules are already explicit, itseems tone, inthe Dyson-formla (4.35) for the generating-function: <However, there youdohavetotake contractions and worry about Wick's theoreif andsoon.Byusingthelittle identity (4+4H) which Ihave proved, you cancome upwith another Version ofthe Feynman rules implicitly stated in(4.48), whére your:$derivatives: actaspincers and-reach intothe mess ‘antie together fields tomake propbgators., Wick isautomatic here, although normal ordering isnot antomatic. . so For general field theory (inciuding derivatives) ifyou could only write the genezating function inthepathintegral form,thenyoucouldusetheaboverietlidd to 5le)readofftheFeynamrulesofyourtheoryinnaiveandcorrectway.Youdonothaveanyproblems with pushing.time deriavtives through time ordered products and all that mess. However, uptothis point ourformalism has.always assumed that LxL(y only, noderivatives, eg,asin(4.38) or(4.36). Inguage theory youmust know howtodeal * with derivative interactions, solets now godothis innext section Ihope. he Derivative Interactions. Ifthefulllagrangian isatmostquadratic intimederivatives andhasnospatieak derivatives ....well,Iamnotsureofthestatus e ofthese spatial derivatives. Aregular scalar‘ free lagrandian has spttial derivatives inthekinetic energy part. Maybe hemeans this: 'integrate the lagrangian density..? No. Heissimply not saying anything about the spatiel derivatives here, soIdont know what the conclusion is. Inacertain case, Idont know preci’sely what case itis, you can hendle some derivatives inyour lagrangian. The only effect onthe functional integral representation istoadda certain detK factor inthere asshown. : Amusingly, the form ofthis generating function iscoordinate independent. ~ Ago; it‘ispossible towrite the thing asanintegral over both coordinates and momenta, asinAbers and Lee. Then you can see that the detK thing comes from integrating first overthedpsthff. Coleman doesnotlikethefulldqdpformof the path integral, - 1.5 Fermion fields. Very heuristic: section here. Assuming fermion fields enter quardratically, you would like tobe able todoapath integral ofthe form (14.65). The correct answer turns out tobe(4.67), where detA isinverted from what itwas inthe boson case. This inversion has somthing todowith fermi statistics, but Idont foltheheuristicargument. Ithinkeissumofbubbles,notWasheclaims. "e .But Iwill accept (4.67) asthe presumed answer for the pabh integral. Ifyou really want toknow this, have tostudy anticommuting c-number fields. 16Ghost fields. Right away wearenowgoing tomake useofthelast section. When wehad derivative interactions, you are stuck with abig detK sitting inthe integrand. You cannot set upfeyman rules unless you get that detK into the exponenet somehow. A trick for doing that istowrite detK asthe path integral of some fake’ fermi fields' with Kasthe operator inthe exponent. Je, you make updummy variable fermi fields and write (4.58). These are called ghost fields. Your new lagrangian gets modified inthe abvious way shown in(4.69) and (4.70). You have toinclude the ghost fields’ inyour feynman rulesy Iguess, asifthey were true fields. . Asanexample-of ghost fields,. suppose youtake atriviel lagrangian ofgand define anew variable Aasshown in(J4.72). Obviously this cannot change the theory. But, indoing this redefinition, you have introduced derivative couplings, and therefore you will have aKand dbtK floating around. Thus, inorder toget correct Feynman rules intermsofthenewfieldA,youhavetointroduce someghostfieldsBTAtogetthea).into standard form where you can then read ‘ofthe feynman rules. Soeven asimple change 6f‘variable will require you touse ghost fields. Iwonder what BDhad tosay about dealing with deriavtive couplings? oe ie)1.TroubleswithGaugeInvariance. -We-openwithasimpleexample:takeQED.andtrytoreadoffthephoton propagator inthe,usual functional way. Whatthismeansis:lock attheaction §andidentify theoperator which isquadratic inthefields, inthesense of(4s51). ‘The’greens function ofthisoperator isthepropagator ofinterest, asin(4.52). ‘Theproblem withQEDisimmediate: theoperator ink-space isaprojection operator andistherefore non-invertiable.,Thus,, thepropagator issinghlar,, as schematically indicated by(5.5)- Thereason forthisproblem isasfollows: (1)‘because youhaveagaugegroup, Lfyouintegrate overallvalues ofthefield, youhaveovercounted; (2)youshould canonically speaking, fixagauge fromthestart andproceded therefrom. Ie,findout which fields arenottruevariables andelimiatatthem, Infact, both ideas arethe same as is later shown. . 5.2TheFadeev-Popow Ansatz.Justdont.integrate overthenon-badependent, fields!This idea isfirst expressed in(5.7). Since your actign isgauge-invariant, youcanadd theextraconstwained variables withadelta function asshown in.(5,11). Finely, fe)>youcanwritethesurfaceconstraintinthegeneralwayof(5613)s,whergnowyougust dddthat..determinant since youhave changed variables inyour delta functions. Tey, bychoosing aguage,werestrict integration tosomemulti-dimensiqnal surface, F=0.Verysimple. Thevariables y;(uponwhichtheaction doesnotdepend) arethegauge functions. -Shift your fields with agauge transformation, andyour action will dot change at.all, hence (5.6)., Sotheresult issummarized in(5.15 ).Fine. . Colman hasshown thattheintegral issameregemdless ofwhatsurface you restrict to.Each surface represents adifferent choice ofgauge. Thus, thegenerating function isthesame inallgauges. Soyouonly have toprove itcorrect inonegauge. 5.3Application ofFPANsatz. Letsgobackandredo.{GED} Fixthegaugebyrequiring that. F=0asin(5-17). This happens tobeacovariant gauge choice. Then doalittle trick togetridofthedelta function infavor ofanexponential asin(5.22). This causes ashift intheaction, ie,adds acorrection termtothelagrangian asin(5.2h). Then, when youinclude this term, youcangoback andrecompute your photon propagator. You@indthat thething which wasnot-invertible nowisinvertable. Youget(5.26) and QEDismade well again. Ofcourse this same propagator comes outwhenyouproperly fo}quantizeinthissamegauge,althoughBDdontquiteusethisgauge. “$b,"how‘hat’wé"haveddne’QED;‘Ted“sbpihtheSeineaedtoareal,CHon-abeliam gatighthéoty‘Eogetdit"the'Néinant rules,“HieLine’?“wtibarJAPtise‘the‘same’Wind’~“of’covaridht ‘gauge;VieLitt1d"raudev-Popov detertithdit: isnot’indéperident ofthe*”v “"PYeras vedae ‘oftie‘extrif‘ternsyoualwaysget’inthenon-deitdh thiedry.” Sonow© youhave'a ‘idnitrivial Weterntthant GnYour getisrating funetitnal “interal. thus, |* you hist usethe trick ofghost fields to‘exponentiate thething, asin((5.29). This ‘nakesiteowldor'rection toKtheLabrargian;' inadditidn tothe‘othercorrection - arising from getting ridoffiedettls finetidhl. “Sothe’filly corréckea Werrectven’! lagrangiar'is Pver"in'(5.32): “The lesson Here is'‘sitipled': if“y8u waht! te''coriect’ piopagator tlbseSh’ySur'Peyniiian riiilés,' youahhot’ simply’ igtére gaugl’ invariance: One‘way‘of‘another you'mist advcblit! Pirfé>Tnthérahebional“Sctiéiies’ thie’accounting riothiod Wecai'i4 paliBev-Popdv. 87ERE PT EC Ta ye “5.4 Histaticatiol oftheWnsatz. “Take'sahie’nonableieh guagetheory’ aha’goto"the “Bo-coliet“dkial “gauge!where”yolr Justsb‘ali“A})* cofiponients”% 0.Thisis’of‘course!very non-coVariant, butbigdeal. Itturns outvery quickly that inthis gauge, the determinant is‘nét‘fieldSdepentidht so*}ouVibht‘HiVe‘tb«iveFadbev-Popv’ ghosts. . “theFPAprédicts'(5.35).’ ‘Now,“what dUéS-ne’oltcunorlital qualitization methdd tells us? Goto"te ‘sdite‘gauge’ahd write but1igrangiatl. ’‘Identity tileYndependent’ Guantumffetdb:’rheilwitspathudtegkal fdb‘them,dsin(5.41).Thensoupthisuptégetc*) “gorm(5.423, wtlichyoutehfindafrebd with‘theFPA.) He Ta 5,5CStielusins!’ théFPA"'fdsnow beth‘provéd fFon!Ganonteal-methdds. Ttprovides’ an extremely easy’wayto’movefromonegauge toanother! Thatisitsstrength. re eee ©) Section() Asymptotic Freedom. 6.1OperatorProducts.Indeepinelestic, thephoton-hadron processisdescribed ‘@) Spaebacture Fanermetionsofthetwo,invariants calledq°,thephotonmass,andthe Fgynmanxvariablee. Atlarge -q°, these functions becoue independent, ofvariable q”andseem todepend only onx.This isBjorken scaling, howdoyouexplain it?Integrate overxfodefine moments F(q), thenwanttoexplain waythesemonentsare constant inq@. (2)Forlarge q@infield theory there aresome theorems forexpanding products ofoperators, invented byWilson. Apply this todeep inelastic where theproducted operators aretheJYelectromagnetic currents, This gives 6.4, which isnotveryenlightening. Nowwanttoknowwhythe£ABC(q?) things areconstants. Infield theoryyoucancalculate these functions. Infactthey dodepend onq*asshown in(6.5), andareconstants only infree field theory. Ie,Bjorken scalding suggests that the field theory isgetting free atlarge a2. Butisitpossible that ifyousumtheseries shown in(6.5) that youmight stillgetsgnething independent ofq*?Howcanthisseries possible makesensefor lerge a??? And what about large g? 6.2Massless Field ‘Theory andRenormalization Group. Inhigh q”limit, mass does notmatter,seemseryTeasonible SowaynotFirstwoworkwithmassless field theories and seewhat you cando. Youneed ascale parameter called M,say, which sets thescale oftheinverse propagator. Ie,yousetthething atp*=M*orwomething. Obviously nothing Physical candepend onthevalue ofM@. This statement leads totheequation (6.8) which says ineffect that &f=0under achange inN*. Noticewhatthevariousparameters are: fahSom =B=m4 =peOR ee oieee 6.3 Exact and Rough Solution tothe RGE. The equation isfirst order, two-variablepartial‘ferential equation, solution isnotobvious. Generalsolution is(6.13)where Fisanyfunction ofasingle variable, andwhere g'is.solution of6.12. This function g'(g,t) has two arguments. The first isthe value ofg*when t=0. Thesecond argument tis1n(Q/M). Thus, gisthespecial value that g'takes when Q=M. Ithink g'iscalled the running coupling constant, orthe invariant charge orsomething like that. Itseems that g’isthe coupling constant atsome particular Q. Now assume you know BETA forsmall gasin(6.14). Then for small g'you can solve forgtasin(6.15). Solution isshown in(6.17). Ifbisnegative, asit was discovered tobeingauge theories, then g'issmall asQgoes large! Te, field theory approaches afree field theory. 6.4Asymptotic Freedom. This simply means that g'goes tozero atlarge Q.Inprinciple you eancompute from lagrangian what field theory does inthis limit bycomputing BETA. Argument isextended toclaim thet result holds even ifgisnot small. So, what doyou predict for Bjorken scaling inanAFtheory? Look backatsolution givenin(6.13). Yougetmainly (6.25) Things arenotconstants, but are slow logs, which isconsistent with deta. Facts: all gauge theories with only gauge mesons (andnoabelian factor groups) are asymptotically free. Fermions added tothe theory tend tomake BETA more positive, putyoucanaccomodatesomefermions.ScalarparticleslikeHiggsbosonsarehard fe)toevaluate astotheir effect onAF. Also: ifyou have nogauge fields, theory cannot beAF. Conclusion: ifafield theory explains stronig interactions and if AFexplains Bjscalaing, then strong intereactions must beanon-abelian gauge theory. yeVex oO wAwlerciclony Goon(455)phe - = _6(428)adosisa,\ecunar Shim AduistinAC i pet(3s)aefalls: Ne ee Ske -enPadme 1=MTClbhentasS——bey a . a ~yosiamAadhJerpeamonsgraghes oe=Nx&, _=\GoousbuureSs) ©©.MadAueds“(4:5)asguadinca(56)?(lawesOsan— _ a eea (Gsm) © =Fla) _ Dyin aagatnBove —— ee -~OK 2% _._ — ebbe =Fe ©RadDadaagee eangone foes — -_ ex— — ——__ ~.. ihe7 —._FO\=2ReEar(boeee _ oo Steet :SS. wssShe =Ene ee Ay S88) daghone 8 i He) 0ee Chouno 2naanuae soaine.oslepllsfofur, sn adendora, (V.39S. 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So Ae some _ . coeMe re) a A AC YS LL ee TX we NSdonna dlfaA.xygtaging RA ROMee) MOOe) Coughing gyaatsinnSsdajinahAMancl),abe coe Qragek be—~— ee ® BnaaDaichaonsge WoWel an ae +sols 5SineN=?dté.. SokAsh aweeeeteaJuwackein 4-S9008_ pancammler ee —aQ-; . -- 4Se =SadUA.. do a anomoe BKCoyNo. wo wMam adesoale hong,Faeat latimvamack,2.22BSSO,SwetoseBrame. terGea SES DRAGTe+QLAMSE +BLDDNVe a 8 ER TOR OT en bk SSaeS=Ge2eT a—-. Te RBS pay +eda aGM =o as oieJasedeunvainameny (ODS pedaeno saan.wn oa+ee Olde F=STai) aMewey &eekeGlen -—-gy Qu,—B=Bs).=O) Sreven Wenberg LAT Apa1974 ‘ecent progress ingauge theories ofthe weak, electromagnetic and strong interactions*t . Steven Weinberg 0LymanLaboratory ofPhysics,Harvard University,Cambridge, Massachusetts 02138 “4 ‘Ateviewispresentedofprogresssnesmid-1972inthedevelopmentofgaugetheoriesoftheweak, electromagnetic, and strong interactions. Abrie introduction tothe history and thefundamentals , ofthesubject isalso provided. CONTENTS discovery ofachangeinnuclearspininbetadecaysof1 Introduction 255 such nuclei asHe’, F",and Al* made itclear that the Tl,Modelé andExperiments 259betadecay interaction wasnotpurevector, andformany { IIL Derivations ofGaugeInvariance fromHigh-Energy yearsthefavored combination wasS,T,P,which ofeen otGangsTheories 283coursewouldquitedestroy theanalogy withelectromag- Y.Dynamical Breakdown ofGaugeSymmetries 26gnetism.Vi.Neutral Currents andAstrophysics 265 Theideaofaunified theory wasrevived inthelateVII. Natural Symmetries andSymmetry Breaking 266 —1950’s, after theform ofthebeta decay interaction finally VID. Strong Interactions andHadronie Symmetries 268 settled down toacombination ofvector andaxial vector. 2XAsymptotic Freedom BtSpecific models wereproposed byseveral authors, nota: Acknowledgements 274blySchwinger (1957), Bludman (1958), Glashow (1961),andSalamandWard(1964).However, two‘greatobsia~clesstood inthewayofasynthesis. Onewastheobvious |.INTRODUCTION ‘iscrepancy inmass‘betweenthephotonandtheinterme-. diatevectorbosonW;ifthe Wcouplings arecomparable :_Renormatizable ugtheoriesofthenaeeewiththeelectronicchargthenitsmassisdetermined otso,andhavebeenpopular atleastsincemid1971.Pythecondition thate'/imhshouldbeoftheorderoftheThere arebynowanumberof excellent surveys ofthe etm!coupling constant Gr,sothalsy including Ben(Lee's)comprehensive-Fepor mw(137X19m)" =30GEV. (Lal)featNALand themorexecer ‘lesbyBernstein ”(1974),S1973b),Aberscal(1973),andAlessobviousthoughnolessimportantdifficultyhadto LiowelyalSmith119733)‘Tintetid-in-thé bodyofthisdowiththeproblemofhigh-energy behavior.Althoughreview toConcentrate ontheexciting new developments theintroduction ofanintermediate vector boson amelio- which have occurred since thepreparation ofLee's rated thebadasymptotic behavior ofprocesses such as report, some ofwhich are only afew months old, lepton-lepton scattering, there were known tobeothers. However, before getting intothisnewmaterial, which issuchastheprocess »+7W*+W~considered by Atherealsubject ofthisarticle, itmaybewelltotakeaGell-Mann, Goldberger, Kroll,andLow(1969), where {brief look inthisIntroduction atthehistory andthe theBorn approximation amplitude grows sorapidly with i fundamentals ofgauge theories. energy that unitarity forces afailure ofperturbation z ‘Thehistory ofattempts tounify weak andelectromag- theory atenergies above 300GeV. Even worse, the aneticinteractions goesbacktoFermi’s(1934)proposal ofoccurrence ofsuchrapidlygrowing matrixelements as i ®afour-vector lepton-hadron betadecay interaction, anal- pieces ofhigher-order diagrams (asintheexample of a ogous tothevector interaction ofelectrons with the »—Fscattering considered byLow, 1970) invalidates a . electromagnetic field. Shortly thereafter Yukawa (1935) perturbation theory atallenergies. Atwuw— 2 ° suggested thatthebetadecay interaction wascarried by _In1967awayoutofthesedifficulties wassuggested iaspin-one boson, analogous tothephoton, butwitha(Weinberg, 1967b; seealsoSalam, 1968). Itwasproposed a p large mass. (Yukawa intended this vector boson toex- that the photon and the intermediate vector bosons fe plainnuclear forces aswellasbetadecay, sohegaveitshould ariseasquanta oftheYang-Mills vector fields! Fiwhatinmodern terms would becalled astrong interac- associated withsome exact local gauge invariance of ition with hadrons and asuperweak interaction with nature. Inorder toavoid theproblem ofultraviolet i leptons, andamassoforder100MeV.)Thesubsequent divergences, itwasproposed thatthetheoryshouldbe iconstructed toberenormalizable, Finally,incontrastto Ton,earliertheoriesofintermediate vectorbosons,itwas tb ‘WarkgareinpabytheNationalScienceFoundationunderProposedthatthet arise(romasbontay : “breakdown ofthegauge group. 5 i Itwasalready known since1964through theworkof 4 "Note: This eview isbased inpartona talkgiven atthe ItConférence ‘ Inemationale Aix-en-Provence surlesPartcues Elementairs, Septem. <—————— :ber6,1973.Apreliminary versionofthisreviewisincludedimtheThefistexampleofatheorybasedonanon-Abelian gaugegroupwas "conference proceedings, publishedbytheJournaldePhysique,thatofYangandMil(1954).Thistheorywasgeneralizedtoabitary a ‘Comptes-RendusdesColloques no.|(1973).Permission toreprintpartssemisimple LiegroupsbyUtiyama(1956)andGell-Mann andGlashow Ls ofthe preliminary versionisgratefullyacknowledged. (i361), : Weinberg:"KEcentProgressinGaugeTheories_ April197), © 1.imrroduction. Several other reviews are listed, one byI-Suith Ihave not seen. Then aLittle review ofpast attempts tounify EMand Weaks. The problem isthat phpton and Whave such different mass, and that you have high energy UWproblems. Hard tounderstand the severe mass difference iftheory istobeunified insome wey. In196), Higgs mechanism appeared, but people were happy with the pion asa Goldstone and did not want toget rid ofgoldtsones sobad. In1967 cams the Weinberg and Salam models inwhich the Higgs thing was used toget mass onto the vector bosons byusing SSB. Bythus avoiding anexplicit massive vector bosons, there was aleast ahope ofrenormalizeablity. In1967 itsimply was not known whether ornot such SSB theories were renormalizeable. The real problem was the SSB part ofit. In1971 'tHooft did it, showed that SSB theories are renormalizeable. Then tbher people carried onhis start and showed proofs for weak and EMtheoreis (Ben Lee Zim Justin, Veltman 'tHooft. Finally, the XiGauge thing ismentioned asaway todebug calculations. Atpresent there are nodefinitely accepted gauge theories. But the idea that the various conserved weak currents are gauge currents isattractive. And then thereistheneutralcurrents recently found(seebelow). 2.Models and Experiments. How doyou test these theories? First, look for the Wbosons, HietecatnatdethatIn1973yet.Next,lookforthevarioustindsofneutralcurrents. ‘ACERN exper in1973 showed oneevent ofthetype¥,eelasticscattering which seems tobeanhonest event. The absence ofother neutral events puts certina limits onthe Weinberg angle Thingsareeasierwhenyougotosemileptonics andlookforneutralcurrentevents. ©) People are doing neutrino-induced deep inelastic events toget high statistics. From CERN itseens that there definitely are these semileptonic dS=0 currents, because they have R=.21 +03, and R=O means noneutral currents with vinduction. The ¥induced R=.453.09, soagain you have definite neutral currents. Youcaninterpret thesehadronic neutral currents byputting sonequarks intotheWeingrg modelinnaivewey.Thenmodelprddicts RandRasfunctions oftheta. Ifsin?0 isintherange of.3to«4,then yougetagreement. Looks good. What are the faults ofthe Weinberg model? First, ithas @S=1 neutral currents at first order, and these predict things that should not happen when you add quarks in the naive way, eg, Kpdecay into muons. Now Weinberg discussed several ways to"fix things up" one ofwhich istoadd more quarks. This review ispre-PSI ;now ofcourse this isclearly the way tofix things up. The other possiblities listed sound alittle obscure. Another problem with the theory isthat itjust isnt basic because ithas two independent couplings, ie, ithas that free parameter the Weinberg angle. Itwould bealot nicer ifyou could use asimple gauge group which then has one parameter. The most famous model that does this isthe Georgi Glashow which uses only BU(2). Inthis model, however, the neutral geuge field isthe photon and there isnothing else, this neutral currents can elimitate this nice model. Another idea istoconsider theSU(2)xU(1) group asbeing asubgroup ofsome larger group. Weinberg gives ang example which then preducts that sin’0 =.25, which matches experiment! Other problems are: Weinberg theory can have the anomolies; itdoes not give theCPviolationinaniceway. re)Generally, the model isnot "natural", This isthe main problem. Itdoes not predict masses say ofthe muon, oranything baisc. -2- .Can youderive gauge invariance from HEconstraints? Itcanberoughly shown that P theonly possible theory ofvector particles which are“perturbatively unitary"@¥ arethegaugetheories. Recall thatinprinciple anyfieldtheory isunitary formally ifyou sumtoall orders, but ifitisnon-renormalizeable you are cooked. Tamconfused astothe relation between renormalizeability and perturbative unitarity? Can atheory berenormalizeable, andyet violate unitarity atsome given order, butbeunitary toallorders? References given forpursuing this interesting subject. ..Regge andgauge theories. Canapraticle beelementary intheitappears in2Lagrangian, butatthesame time lieonaRegge trajectory? Answer isyes. In fact, even theelectron lies onaregge trajectory. Refs given forpursuing this also interesting path. 5.Dynamical Synmetry Breakdadwn? Usually youdoyour SSBbyadding some Higgs scalars‘nd shifting some ofthe figlds. Oneproblem isthat you arealways left over with some scalars that dont seem tomatch any physically observbed particles. Isthere some way togetSSB without introducting these explicit scalars into thelagrangian? Appranatly there isaweycalled "dynamical SSB" andsome references aregiven « Somehow you aregetting Goldstones outwithout putting anybosons in. Idont follow this section, but the subject matter isclear enough. 6.Neutral Currents andSstrophysics. Ifthere areinfact neutral currents, this changes thedecoupling times foryversus 4intheearly cosmology. However, this does not seem tomake much difference inearly universe history because the basic facts dont depend onwhether things are coupled inthermal equilibrium, ordecoupled. Next,whatabout“neutrinocooling"ofstartsbyobviousneutrinoloss.Clearly eaany such calculation should beaffected byneutral currents. Calculations show that neutral currents can change the neutrino cooling rate byafactor of10. Finally, supernova theory isaffected. One theory isthis: you have astar with aniron core and anenvelope. The core islarge enough tobegin shtinking dom tothe neutron ster limit; asitshirinks, neutrinos are emitted which are then captured inthe envelope, causing the envelope tofly off, leaving aneutron star. Such envelope-blow-off theories are obviously affected byneutral currents. ‘7.Natural Symmetries andSymBreaking. skip 8»Strong Interactions and Hadron Symmetries.