Abers & Lee -Bernstein- Gauge Theories

Abers & Lee Gauge Theories 197 3 Jeremy Bernstein 1974 Phil Lucht Notes 1974 & 1977 Part|2 Physics Reports 9C pO es Pec ee eee tra ian a SS a ee Reine CgatoActPack |\a73 uot BERS e ES : { ae GAUGE THEORIES = : = ‘: ae H Bind : . ' Bex? ; Ernest$.ABERSndBenjamin W.LEE Bee .ot Institute forTheoretical Physics,StateUniversity ofNewYork, * a Coll ‘StonyBrook,N.Y.11790,USA Ba Behe Hlas” * ‘ Bife : ' ate ‘! ae tyBy et vot4 etre fo]NORTH-HOLLAND PUBLISHING COMPANY ~AMSTERDAM. . RRS. Part IAbers and Lee notes: E¥" ©)ounnary ofintroduction. 1)gauge invariance inclassical field theory, summary. 2)Spontaneously broken symmetries a)gauge theory review sheet, a)notes onGoldston Theorem 3)Higgs mechanism, summary a)Higgs technique page 4)weak interaction phenomenology, summary 5)more ofsame, summary 6)more ofsame, summary a)page onPCAC 7)Weinberg-Salam model, summary 8)phenom. ofthe W-S model, suimary 8b) inclusion ofhadrons insuch models 9)heavy lepton models toavoid Zvv vertex. 10) model building, fast review toliterature Addon: notesfrommyowntalkonthismubject givenin1975.3 norloceh~, 2WateedeoneablmgaJonArgon avinoweas. & Qed 378 abnAnding, aterbol, Q- i aa Oe aoo OC N=; GausTote—(bessSu) neCn “REO GouekWisiayolMwandgect: OQaeotalukraryuecA __ ow oovta Q ¥ty. SS.8saad—— hominalarmen. 4Seatpopsbabesian19.68lay__ . Dhomy”) Ouak? 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PatIL Physics Reports AC NEE RET Tr RPRT Se re ee eee Me we xPdayricn RepabeAG) PotIAt, ~Yaoj ve ; ja 60 <€.SAbers end BW:Lee.Gauge theories ¥ PARTIL iUI QUANTIZATION ANDRENORMALIZATION OFGAUGETHEORIES a LLPath integral quantization ”- 4 ‘Onefectsa5Cavatieri musthavefeltcalculating the . 4 ‘lume of2prtamid beforetheinvention ofthecalcul. z RP. Feynman 4 (2) _Inthis seetion wedevelop thequantization procedure basedonthenotion ofpathintegration. Ff gose TheFisthintofthisprocedure appeared in4paperbyDiracin1933;themethod wasperfectedif byFeynman in1948,Weshallfirstconsideraquantummechanical systemwithonedegreeof 4aefreedom,andgeneralizetoquantumfieldtheoryinthenextsection i zBie _LetIg,fybetheHeisenberg picturestatevectordescribing astatewhichattimeianeiger- “gstateofthecoordinate Q,,witheigenvalue q: (Hrsttvan npc-asgendenety —- a NAO =aOy-=FE)NAHtY, i " (0)=eQ.e-U, . any 4 ee eo, whereQ,isthetime-independent position operdtor iritheSchroedinger picture; andHintheex- Bponent istheHamiltonian. Thestate “ “ i Igyseg. ty| : isaneigenstate ofQ,witheigenvalue q! 3 0,14) =lq‘ ag .3 and ( ‘i . 4 1g.=e*#iqy. (1.2y | 4 The transformation matrix element : Fd’, =la’#19,Dy=(q'lexp(—iH(t' —O}1g) (1.3) : al 2 playsa fundamental roleinquantum mechanics. Wearegoingtoexpress F(q’,t';q.f'asa path ier integral. Weshall subdivide thetime interval into +1equal segments, and define = Be alett re(n+Dete. aa) # ~ Wemake useofthecompleteness ofthestate vectors Iq,.f,)towrite - a Fd’tq.=Sdaqulenfoals)finfbagltaXa’s taintnetantsbaofg,02.115)4;1 y. Here andinthefollowing, weshall drop thesubscript-H andunderstand thestate Ig,¢)tomean 3 thatintheHeisenberg picture. Forsufficiently-large n,thetimeinterval ¢,—f,_,canbemade as E: i‘ L=Saget [git&><9C), it . ' i TTT TOD) och eT OR TOM PTO Us ci 3528 Index for Notes filed under Abers and Lee GL): original, oldnotes taken omSections 11thra15,notveryuseful. 2)Sectiom 11summary: "path integral quantization" (reguhar QM) a)about thestate /x,t) th)relation of/x,t) toGreens Function =Propagator c)comments about path integral idea a)actual construction ofpath integral rep ofapropagator =Greens e)details ofthe harmonic oscilldor example ofusing path integrals. f)derivations ofmany equations insection 1l. 3)Seotiom 12summary: “path integrals and field theory" 4)Section 13summary: "Yang-Mills inthe Coulomb gauge §)Section 14summary: "Intuitive Approach toQuantizatiom ofGauge theory " a)the ghost loop expansion business b)proof ofrelatior (16.6), ie, connected Greens fctns and the Z(J) object. 6)Section 15summary: "equivalence ofLandau and Coulomb gauge" a)functional derivative facts 7):Section16summary:"properverticesandaffectivepotential, eto.” fe)a)show (16,26), the power counting rule bd)algebra for section 16 8)Section 17summary: renormalization inthe¢~model 9)Section 18summary: the BPHZ renormalization program a)caleulating §,andbasic ideas. 10) Section 19summary: dimensional regularizing Feynman integrals 11)Section 20summary: Feynman Rules andRenorm, inSSBGauge Theories (Landau gauge) 12)Section 21sunmary: theRggauges andvarious kinds ofghosts 13)Section 22summary: proof that S-matrix does notdepend ongauge paramter§ (and therefore noghost poles inS~matrix) 14) Section 23summary: computation ofweak corrections tomuon anom-moment in the Coleman ~Glashow model. send! 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SMejejeHeA$0SanoSsummaOOS aw XR NSE a 9 Thisisjustthefirst13longsections whichcomprise thesecondpartofAbers andLee's review ofgauge theory. Ifound thesection verydifficult, Manythings I cannot derive without muchwork. Thisisthesecond timeIhavereadthis, last’time was long time ago with Hassam Arfai. Ihavedecomposed thesection into’ subsections because somuchinformation ispresent here. Thegeneral domain here isfirst-quantized physics, like afirst quantized harmonic oscillator. There arebras andkets’ andoperators. There areOperators like Hthe hamiltonian. Things are dont mainly inthe Heisenberg-States picture. The “systen" ofinterest isdescribed inthis section byonecoordinate g,operator Q.Thus, . there are nolagrangian densities, nofield theory atall.. Ifwedivide physics intothese areas: classigal physics, quantum physics, classical field theory, quantum @ield theory, then this section falls into the second area. Path integrals are always about amplitudes, and that always means quantum physics. Ipresume that inthenext section wewill govver toquantum field theory bymaking _ these changes: corrdinate operator Qreplaced byfield operator J(x). Lagrangian replacedbysxaxekimexspaceintegralofLagrangiandensity.OtherwiseI’betthings ro)are much the same, but Ihave not read that section yet. : : (a)_Path integral idea. Amplitude togofrom here tothere isdivided into little times. Ateachtime, youintegrate over allpossible positions. Hence (11.5) Since time evolution ofaHeisenberg state maybewritten inthe form (11.2), you canwrite the *amplitude tomove alittle bitintime asin(11.9). Notice that things were converted toScrodinger states andthe Hamiltonian operator was putininterms ofSchrodinger operators Pand Q,although this was never’ stated. Themain point of(11.9) isthat youhave expressed aquantum amplitude interms ofthec-number classical hamiltonian. Since each‘mini-amplitude isapintegration, whenyoustickalltheseinto(11.5), yougetthekeyresult (11.11), Again,this gives the amplitude togofrom here tothere, a‘finite distance, interms ofthe classical c-numbér Hamiltonian H(p,q). Theprice paid togetthis result however isarether complicated calculational structure ,namely thepath intégral. Incertain simple cases where H(p,q) hassimple form, youcanexecute allthe dpintegrations andgetthesame amplitude asa"path integral" [dq] only ofthe @phasorwhosephaseistheclassicalaction,ie,timeintegralofthelagrangian. re)Remember, thisactioniscalledclassical onlywhenyouputintheclassical trajectorya(t). (b)Incases where theHamiltonian doesnothevethesimple formof4p”+V(q), you canstilldothe[dp]integration andcomeupwithaneffective ‘Jagrangian andthereforeaneffective actionunderyourtime[dq]integration. Afamouscalculation alongthis{oy line wasthat ofYang andLeein1962, Itseems obvious that youcanalways find. an effective action, because youJust dothedp.integrals andseewhat results. (c)Herewefind astunning result:. itisvery easy towrite thematrix element between twoatates (here just Heisenberg q-states, butingeneral anystates) ofatime ordered product ofcoprdinate operators. Ususally ithasbeen myexperience that thetime ordering complicated things, but here itmakes the result very simple. You start with thepath-integral representation ofthe(q't'/qt) amplitude. Youspread itappart.ang stickinasmanyQoperators asyouwant,allina[f0P| Inthepathintegral represntation, allyouneed doisinsert regular c-number qoordinate functions like a(t,). Again, you seetherole ofthepath-integral asconverting from quantum mechanics toclassical |mechanics. . (a)Herewediscuss,the idea ofadding a.source term J(t) a(t)tothat Hamiltonian in theexponent. Intheharmonic oscialltor, J(t) isprecisly thedriving function or external force. Ofcourse J(t)isac-number function. justlikeeverything elseupthereintheexponent. [@) . You can goontocompute the amplitude inthe presence ofthis source term.’ As “apathintegral, itis(11,29), bigdeal. ‘Theprecise definition ofW[J]istheamplitude forsystem tostayinground state under presence ofJ.Ground state means loweset energy state ,eigenstate ofhamiltonian H,soJQisaperturbstion ifyoulike. However, then end-twist integrations thatconvert from(0/0)? to(at/q) arein effect multipliciative constants, once you take certain large time limits. Hence you getresult (11.35) which shows W[J] asthe[dq]integral oftheeffective lagrangian dealwiththesource added inthere. :, Theusefulness ofthegenerating function W(J) isthat ifyouteke itsfunctional derivatives withrespect tothesource, yougenerate,after setting J=0,-the matrix elements oftimeordezed products. Infield theory, these willofcourse bethe Greens Functions. SoW(J) istheGreens Functions generating functions. (e) One slightly unpleasant aspect ofthe W(J) thing isthat you end uphaving toput large imaginary time limits on$bur action integral rather than reel limits. This is avoided when you deal instead with the, Buclidean version. Just compare (11,35) and (11.36)toseethe@ifference betweentheregularandeuclidean generating functions.: Ga Theitsarevery important inthedifference. Inthecase ofWzyouhave regulr real time infinite endpoints. (f)Thefinal subsection works outtheharmonic osciallator example. Anyexample is : ofcourse defined byits lagrangian orhamiltonian. Sotake this big machine and CO)itiect theLagrangian (11.38) andturnthecrank, Firstoff,youcancalculate thething (q'/a)" explicitly. Theanswer isgivenin(11.0) and(11.41) andT'11bet this isone bun-buster excercise. Next, youfold intheground state fortheH.0. togettheexact xW[J] generating function, Answer tekes thestupendously simple form shown in(11.45). No doubt Icould dothis entire calculation indetail. Youcanseethat ifyoutake thesecond derivative ofthisw[J]youeregoing togetthepropagator D,.Idont really know howtointerpret this "time propagator" because wearenotinquantum field theory, wearejust inquantum physics. Theform isvery suggestive, I'msure Feynam has some words onthis thing. Notice that thetrue propagator hasphasing expentntials. Nowthewhole thing isredone intheEuclidean case. Thetrick isused here forthefirst time obobserving that W[J] isessentially theexponentiated classical action, Ie,ifyouputinthecalssical trajectory, thecorrection vanishes tofirst order. AsfarasJ-dependence isconcerned, therest isjust aconstant. Sowe are down to(11.52). Next,howtocomputetheclassicalaction?Theclassicaltrajectoryisfound fe)bysolving thedriven harmonic oscillatro equation, ie,(11.53). Thesolution isgiven in(11.56), andnowyouseethatDgisaGreen's function inthedifferential equation sense, IB,itsolves (11.54). Butthiseuclidean thing hasnophase init,totally dampt. So,havingfoundtheclassical trajectory q(t),youdumpthatbackintoSs, andcomputetheEuclidean classical actionS,(4,)+Thisgives(11.57).Since W(J) isthesame asthis expo'd classieal action, youget(11.58). Either way,yougetthesamebasic formforthegenerating function, andyou seewhat happens ifyoutake asecond functional derivative andset0=0. NowIseethemeaning ofthepropagator D:itisjust thesolution tothe harmonic oscialltor intheabsence ofsource J.Ie,thepropagator isthe“free Propagator" which describes howthesystem moves withnoperturbation. 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HoleMek OrieYateDeweraurtegral sguobine : 6 ctl =SeSate Leithek’) ‘ Nea GED =oe =ELheGtHeel SoG) Pet=+iSak!GOsxt)YOR Yrinuidagoh sguation Jeadil Franyeoafrmncteon|VHD ond WA baa SEcoticlinn, ©Wo, empas : 5 ~ + A 6KM,CRB) =-Cedtt) Git!|Hbgt? owtmantaVond| oo ~a- Mod, KBGOS;at)=Gi)Re(OB)Seepet) le) Set) 7SN8e-x) +CoG) CLR ed) CR =-Alsi? ==SEM)BGLY) 4heG'A) GDh), =akeCO) =~HeGtxt) Qu:[ee-KH]Ceext)=+8Cx-x) Noa,oquains2donssanidgaindoagumaale ©Qalaoteodain+&Gsx)=G&DOE!)GANtD| re)nue. Quonoloow onguannabe. vod wid: ORDVE =tSAGatco) MEE) Fader06,ygVennarntdivedrone )aeAo 4Nytedonq Oo)henayedO: . Qua, red ost [rk3p-92|Gtiat)=+SGX) 0Que 4 Lo. ySOME2Quso+{Ruason&&gutkegts@)|. Sox ‘ 3, —3- * - LF@tyuO]=+6ot)<atlae> =+G(mt;xt!) 5 ~,* *4¥ Qe (eRe -|Et at)=+BG) [LRH CGH) =Sow). WardnanyouoperaoyomJuskangioma. Sueno,+SPE =GH)OODGt0 (+) eLtG-ty) bout)=DVR Osh8)et 8) * ee 4 [ePewane =+GTeee) [242 -%|e*eitha) =+SoH :i) (- HSGyt) =+8Gex) Fetus Eun kitty=its \®s,andwonckow shaw Dyck ait\xty =SCx'-x) so A=\eeVetet) okoyVw2. 6 _._3+PathIntegralQuantization, _comments, aeeee ie) 1.TheGreensfunctionorpropagatorcontainsalltheinformation ="~77 ">dftheHamiltonian, ToFindG,youcaiwriteoutthedifferential:_Green's equation, putting inE(p,x) where p=id/dx. Ie,thefact. ~that(pedo isalready builtintoordinary Schrodinger equation quantum -- -- -- mechanics... we wee nee ee With the Hamidtonian, wehave adifferential equation for wavefunctions, -----+--+ withGwehaveanintegral equation-for —the-wavefunctions, bigdeat. The : physics isthe same. The integral equation ismore useful, however, for | scattering probleiis and therefore Forparticle physics ingeneral. *2,All the above has noconnection with "second-quantization", which is-.-.=. #0say,withtheories whichallowthecreation anddestructéonofparticles. Inregular Schrodinger theory, even ininggral form with Gfunctions, you - have only -wavefunetions, nefield-operaters-in-a-Fock space. -9f course you oan interpret the Sch theory asthe 1particle sector ofthe second~ —-> >-=> >quantized theory. So, the above stuff has only todowith "first-quantizetion", the ~,|factthat(p,»x)40. Second quantizatiom usually proceeds byinterpreting _ thefieldoperator asa"coordinate" andsaying(i,be)#0.Butin _.the present context, coordinate means x,has nothing todowith field ee -operators. = - : 3.TheGreen'sFunctionG(x,t;x',t')isthereforeanobjectintherealm [o)}offirst-quantized physiéwW, ie,Fégular quantum mechanics. “WeGancompute”Gbydealing witha“quantized” Hamiltonian H(p=id/éx, x).However, there _ 7; isatrick for directly computing the propagator Ginterms ofthe ---.— ~{.--lassical Hamiltonian H(p,q). Ofcourse yougetthesameresult. The - 'trick isjust playing with the /x,t) states and has the name "pathwees 1--dntegral techhique’. Since-you calculate quantum mechanical stuffdirectly.from classical H,you are ineffect performing "first-quantization", hence “ooo coo => =the name"path ittegral quantization. - & RetrTrnbagnate Ceushwckionobdott!Laut). 6 ah(Et) ©(ouJdrsacinUedodoGua: ditt) =He \®.“ “x + . waeTHACBR). Yorcawudt compute OnnaDaundiac Tavaoman opscragsapertiwHBeBRP.On,meGredermaCADyonwoLrone pepe sau. @Sours adick towakDueob.Videdineuslavel Ut shoomsquahstainaAwidth€,Branih crmptalameas onSine % COMED =Laka AkedKCottilena)StabalHoyTor) wie Sedan th)Git] Jr)Jakes. OBGenoa, cro, Ree&oadeciraspalyJemme|ont€. ihGto *ieh Gabe =gle Minn=Galen hey =ld nieGultiley +6) — a ‘Vegaipars) aeBEMESRD = a HQBe,Ge><eln) tm) BOR) : . =VeS [WieHG) +0] =Stele=\&engi(GH)~eAG%o| rs)+o) Soadss"a”dogk 2 . ,Sata|\njh)=\aeei[ReGes)-<«a+o). akoF. .—> ~%— 0@draOriefrsade4DuGndfactarw « eine Cha&)(PAR)«T(i]AeRB.)-€ NG, . A A=opi(2ef&GS)WEI} | ®Fo.sankJackn,SokeBonkaemero.Ghar ; ‘ A=angi|‘Yotje238_wCge,.20)% |a —ON LED. 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Q So: + a nk SaJ@x)9aizn= \[Gale * + % ve Harmonic Oscillator asApplication ofPath Integral Technique. fo) 1.Oneproblem youcandoistocompute thepropagator G(x',t':x,t) inregular quantum mechanics taking asyour Hamiltonian aparticle inaharmonic | potential well with anadded driving force term, This isapath integral calculation, ifyou like, and you can dothe whole thing indetail; the -answer isquoted inAbers and Lee. This example shows how you can compute aquantum amplitude (the propagator) interms ofthe classical action. 2.But here we are more interested incomputing the generating functional called W(J).° Roughly speaking, this isthe limit T= +i ofthe propagator. Rigorously itisthe ground state toground state amppitude. Ifwedbnot care about mltiplicative factors, then wecan use Colemans vulgar gaussian path integral idea tosee what we get. 3.The exact result isderived onpage 68ofAL. Ie, take the exact Feynman Hibbs quantum propagator and fold inground states, then take time limits. Result is(11.45) which isalso the result Iwila get below. 4. Myfirst method isto identify the operator &by immediately rotating the contour with novariable change. Then Iapply Sidney's forma add fo) getananswer. Theproblemwiththismethodisthata)the integral isinfact phasing, not adying gaussian, sothe whole thingisquestionable. b) that fact causes you toneed an extra rule to define your propagator. You have to let, be alittle below real aris. 5.The second method isdifferent. Here, Iactually change variables ad get something that looks more like agaussian. The operator Misdifferent, ad there isno longer anambiguity. This isthe Euclidean method. 6,But either way works fine; the Euclidean thing just serves todefine more clearly what you are really doing/ 7. So this example is acombimtion of Abers Lee with Colemans integral trick. The transition tofield theory is made by Coleman, ond he redoes everything with afree Kelin-Gordan field plus source (field theory analog ofharmonic oscillator +source). Ofcourse now you see d4x stuff, and things are covariant. le) - Yorwnie OzSdain Qneuadion ; . ia[ehakeay] 8ok we \wae * wae LETT=beLaqomgron wilactamal souTe, LeeEm blak+Tx 3Aesok BWankx+3 Que,Jayengewobinaete: om=RK+TLonlanalldns,force @AsYardsoggeoada, Sakesay,wasconSrormntacady Aidestaledheenhus bb0,44) woAromge UnNaralate Tndafrine: Suyy=Yakxu 6 sebask WI]=SLAexpfkEma,YDCteDeCIDE Qkvate GRD=-CKKZLygate,thaw wEI= Wages P=<x,Gamde)x>~dy,tke+dapat =SiaongJdsreinge+HED—G39}ea A « b QhiniawadvvChamenia slencard formsweaAxGime2¢-+ik) ond aT, C=O, Qeakk wih Tare, 6 Wi~exeFe<ea'| =“ Nowe,4A=simde+ik=vi[agakl=so]ut], cl\4)welt 7] tue A=gay[aale it ~ualtt'l whe, fle G&N= Tee. ow ~welt Sopoate exp(-\eus Tse):ere\Eewn| ci juakt~t') Cult=t! 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Quay), ‘i a AaSataba. a,Qasumsannpotas"4xphasonwrosa Paria Meashen Qu) DukoeteRepeman Hs. SO ee (_—— Tp toy 4 le) <aany’=SayTA Ged" CathAT? ow Ww: &tlat) =ene ia soured WhVeBbyel 2ktye GQ)heseedoo ne EET data=2.dale)oreODeid) =ZZ.dG)HQ)€;fr Aapllan. VoraSanchin ybendy. 0”NinZandi eT.aagqe* & site_ -‘wieLEaria] =4GEO (us2) QewihdnoSnidebepullotkBowssikSomowdarsun. So . Sum Caran) - , 1daratejer cer]= JedlteBab&(G)Catlad’ acct)Ble =|emaalaeB(4ss‘“S) (33) SoWSL DBs BHCaitGLQAGait se fe) a _SDs Coll=age baleygeDebale Ge gears fact a. Qetuway,TaskeCoyackttme) aGutecory EYAL phBA)DY Guay SEBS©} | c?) Qrafes. youenSoawamabne, Ex, F=Sangepkod $*\aday \.UV BeehO) 3 Wa,Lemadovedalewe Thy # TOa\= )arLeg~wena4PA] aw.st (2) ix AGAR) at QeK=@ |BK e ~e = -St) soBK= K(at)=Aqk. Spntogs.yh "3aaa =Vueve By “pddom QA TESA sosemtluis Celolglance. Wed dommdanvotinna oavin (M30) OnakT>0QRdemg's e so Wy) =Mh7[ethGe)-QUaS]\9). Phe Scondent@" AlaeJpatiesoaesitdpotGat 3?“Wisi astute: GBSHO=Fata=SUBaloeLYCetHeomdee]coy ‘ -\earnene)LewG8)axl22),(23) Le=Ls=g-WayakanHhoetraformsHon)=9+VG) enw yorWowstoSaduce Lgbydowgy [ag]wlrepntinn | wwEFs BndteQu: WL=LagagSgr<trcqtey=cols? “ =Yaadiy[tsone{YE“rah “ey fo)SoWwtewtner Quaansrgygownstade8).RKfrownW-33), wi ~Grelgse t ‘ J Mint Stade [A STMenh a v weSain\YASorpFehat[eGWGATHE]Tva S a Sw =deLES ayoe[8h BUCK \ Dad . syz0 yaoswip\-dinSSAYat.\qtte)exp"9 (28) SIKHSTAe) 6 =Lim<A"Tlamady\ars SoJumcho danuttuer dWakT=0WHleeTor| Swarwatede 157BoGang.nradnae, marian fatesdksdou:: Wt)=Se7bats 5 Raattaaliefi-ata)=SieTpaon{iLely(EteTekeo QinwoeCuts)Woe €=.os 4Ut)opbymwspac NeeS Se Y J.io Tonybocode Ay=weakdoKio=may, =Aix-6eT Wr c'sgoths GAD =iese elese ow i _ iUaU4 yt) |eMeQTSEonFEZLY(%nt ass) ee Minoex€'-nobegk “=ot{getWa,sae|oni2Aor OnYecwcma,ynent: ~ driary =SeqopltdaLyG4) So Wk . \~~ - wis}~Ren<ama) =\tadhone §Vacly(4,i4) Tate ? 434 Juice DgEudidaan worsen w,Cy] >) Yerorvatilny eee worwiakemar.(1.39) aa o= ~Ad(t-2) Awlo-r\Ores:SarSarsire)[pgoe)&me45or)©( e o ~hw(t-o) 2 r aiw(t-* QC —_ QO =SacYee =2Janidwot. (uns)UY). Sdorhareabrns mrendh wh\=Coda oe)=ont|Suite:Te)Dye-yBe)§, WordhevnJyepenenee: Le)Vd wah a - Subs dolTloay esis) TIGA) =2 L w=exe]SSsys} We ot. .ae=oxpSTD) wd2 ae=exp}JdYost.AWVd<O4exeDear),2». Shvro Bw sO fe) Fans SSeasews LATOR adaly) ~d4(-b) Compu. Qias)seeing) (v.38)“ mag ¢ Loa Craving onciath laatewadeMetoneunbeal £ aah yeIGA=ae§Caesret) onve(AUN), anlarapWackoecwnnsclighpkYwinsiiminbeprrdack| VowcamedeQuehad. Gowdoaprotec. &uchncarghMe ON, ati= |gios| on9 e wnQERncmaenssnrntioing herJpinetnan T. Nadas" Letemginekg?+04 3Aa &on,Alena=“eg BS omykg =Dv. SoVWjaSileeeeanion,Wanmrnie oseilaba Jor: os,eryomysewisBa. ; cog . MELe EIST=ongfeargc3)Loewen SAD 2 gmzhw(Fee)=expyi\aoaeYer)\iec—| ver). -- Now.25,ANY ‘Section 12:PathIntegral Formulation forFieldTheory (eo) ‘Thebasicpathintegral formulaforthegenerating function isgivenin(12.2). Itshows integration over both the fields and the canonical field momentte. The argument ofthe action here isreally the Lagrangian written interms ofthe pand a. Incertain cases, you can dothe pintegrations toget the more standard form (12.9) which Coleman uses. ‘The Euclidean stuff ismentioned here. You have toregularize these integrals byWick rotation and soon. The two kinds ofgenerators W(J) and Z(J) are defined, and their relation to "connected" feynman diagarams isstated. Greens functions are written asjderivatives inthe usual way. Onpage 74the field integrations are attempted. The characteristic determinant appears out front, and you get the standard form JxProp xJinthe exponent. In fact, thepropagator isevenwritten asK"'which Ilike. Thiswasafreefield case. For the interacting case, they dothe standard trick ofanoperator onthe free action asin(12.16). Iwasonce very confused astowhythis istheFeynman rules, but studying Colamna notes straightened this out. Ithink the rules are moreobviousinthed/dfnotation. ThefactthatthereisnoneedforaWicktheorem fo)ismentioned, and now Iunderstnad the remark. Wewill not gointo anti-commuting c-numbers. Apparantly just aminor technical detail. le) Section 13: Yang-Mills inthe Coulomb gauge. . a) Standard Quantization, :. 6 T.Pirst, theYMLagrangian iswritten intheusualway,onlyYMfields oresent. However,the Fand Aave treated asindependent objects. Ofcourse this means ‘that the Euler equation ontheF-will setthem equal toexpected object asin(13.3). Under local gauge trensformations, F-transforms asasimple isovector asin(13.2), whereas A goes asan-isovector with thelittle extra piece. Theu(x) arethegauge functions. 2,Consider all the Euler equations. Separate out-those which have time derivative, see (13.5) and(13.6). Computing themomenta, youfind that F,,aremomenta ofA;,butthemomentum ofAyvanishes, thusA,isadependent variable; justasinQED. 3.Nowlook attehconstraint equations. (13.8) clearly tells youthat F,,aredependentvariables determined completely bytheAj.Then(13.9) putsacondition aflthePoy variables sothat not allthree ofthem areindependent. IftheF,; arenot allindependent, then their corresponding fields A,cannot beindépendént. So‘you impose some Kind ofgauge condition onA,here just theusual Coulomb gauge that divA=0, (13.10). ~heSoonlythetranaverse pieceofAyistruevariables. Ithinkyoucattjust.seta20 aswasdoneinBD.”So,breakF,;intoTandLpiecesasin(3.113. Thequantity F.§L)=F; hasnocurl, andmaytheréfore bewritten asagradient ofsomefasin(13.18).Sothethingfisliketheelectrestatic potential. AndthenFpiscalled E.* 5.Whstgoesonhere? Thetruecanonical variables are#4E,andA,=A," .Soyouwanttosolve equation (13.13) forthepotential f.Yougaahead andsolve thisbythe Green's method togetfasin(13.15). Then once weknow f,wegetF,from (13.12). Finally,youwritedown(13.17)andsolveforA,ageinbyagreensmethod.-So-I-was. 0 wrong_in-seying-thet-Aq—=-0;-but-maybe-you-eould still_gauge-it-aways hte Ay. 6.Summary tothis point: fortheYang Mills Lagrangian, wehave identified the fields which ereIndependent andwhich wewafit toquanktize. Allthe ‘other fields aredependent: >?or ate =o AQ) AL=AcondApsosofukYeAr=O,Cotesgang XSuefad, 2 aT > Ct E:=B= coupagshe mcemanben beAy. EL OlverJiode aa: ota(N82) anh(1305) ivjoe (3:8) Rvia3.4) 7.Next, they compute theHamiltonian andget(13.20) which looks right. 68.Thevrocedure uptothispointwasthestandard quantization procedure. Ie,youwouldgoahead andimpose equal time commutators onyour independent objects andgo’from there. (b)Path integral Method. 1,Now, write down thefull path integtal-generating function: only the‘dynamical fieldsenter,henceyouget(tx82x}(13.21). Makessensetome. .°2.With very little work, rewrite this generating path integral asin(13.2h). Now nominally youintegrate over allE,andallA,spatial components, butreally the two delta functions*restrict you totransverse fields only. ,The determinant turns outtobefield~independent spissimply ignored. SoIamhappy with (13.24). Of course itisstill not covariant because you areonly integrating over spatial cpmponents ofPand Q,sotospeak. : . 3.Next, write 1infancy way (13.25). Here fisinisolation. Rewrite the delta as theequation involving f,andcompenstate byjacobien determinant det, where cstandstoremind you weare incoulomb gaugé.. The result ofthis step is(1328) which looks just fine. There are-now three deltas. 4,The rest of.the manipulations are very artifical, noreason tocheck them out. Wehave : three deltas. Dothe[df] integration toremove onedélta, hence get(13.31). Nowyou see all spatial field components present. The goal now istoaddthe time components togetza,covariant formulation. Thetime components aredummied inin(13.32) and.(13.33) and the final result.is very simple, see (13.34). . 5.The final result says: -integrate over all the fields, but dont forget the Popov determinant det, andalso dont forget thegauge delta. Then theexponent issimply your lagrangian. . : 6.Thelast step istoexponentiate thePopov déterminant. Agood method fordealing with such determinants isgiven using the old trace theorem. But the effective lagrangien isnotyetdiscussed. ‘Youcansee‘thattheexponentitated determinant isjustgoingtoe correct the lagrangian from (13.38). - 7Apparantly, the material ofthis ‘gection was the way Fadeev and Popov originally presented some oftheir work. Started from just the dynamical fields, and then dummied inall the extra stuff toarrive atthe simple form (13.34).' Obviously there must besome simpler and more direct way toget this result. Must besome easy wap toexplain the determinant. Section 14:Intuitive Approach toQuantization 61,FirstwecopydownW(J)fromlastsection. Canintegrate trivially overtheF.,so that only [4A] integration remains. Plusyouhave thegauge deita function and the FPdeterminant. Question: why does this W(J) not look like itdoes ina"normal" field theory? This isthe question that Fadeev and Popov answered. 2.Intuitive comments: weknow that inagauge theory, thex operator K(Coleman called A)which appears inthe action isaprojection operator and istherefore non-invertible. The exponent is"immune" tovariations ofthe fields inthe “longitudinal” direction. Usually, asyou let some field gotoinfinity, the exponent isdamped. This was the requirement that the "matrix" Abepositive definite inColeman notes. Here itseems that ineffect some ofour diagonal matrix elements vanish! Ie, the exponent isconstant asyougooffinA,direction. Obviously, thelongitudinal integration then gives adivergence. Torepair the problem, you should ineffect divide out the orbit volume. Another way tosey this isthat you should restrict the integration toahypersurface orcross section f(A) =0which intersects each orbit only once. a)3.Toillustaate thesituation, AbersandLeetellustodefineadeterminant which isessentially the jacobian from the foff(A) tothe gauge parameter functions called u. Byplaying with the"normal form (14.8), youcanexpose theorbit volume asin(14.10). You just divide this out toget your answer! 4.What isthe determinant? You first choose ageuge, this determines £(A+) =0. You then vary thegauge functions causing avariation inf.Then det isdet(df/du). For exemple, inCoulomb gauge youcanread offM,from (14.14). Notice that Mhasthe fordel”times (1+something), where something isfield dependent. Thus, youcannot just ignore the determinant! Mfor Landau gauge isalso A-dependent, ,(14.15). 5.Finelly, onpage 86itisshown howthe determinant canbeexponentiated by introducing fermion scalar ghost fields called c.Theeffect istoaddS,tothe action you already have. Finally, you trivially exponentiate the gauge delta function togetanother Lagrangian correction oftheform(d,A,)* incaseofLandau. Once you get everything into the exponent, you can then read off the feynman rules! These rules are shown inthe table onpage 88, and Ithink Iunderstand now, atlast. @)_%&everturbation theory with@gaugetheory, youwillingeneral needghosts! Tey you have toinclude the ghosts asasymbolic part ofyour feynman diagrams. InQED which isabelian, itturns outthat thedeterminant isA-indep, sonoghosts needed. Howtounderstand theghost loop expansion business. . ‘@xteghosts arecreated tosimilate adeterminant. Ifthedeterminant isdet,then theghostaction willbesomething likedxctMcwhereMissomeoperator. For theghosts, thisisasortof£field theory. Ghosts start outbeing charged scalars, soitisf°theory with arrows onthelines. Itturns outthatthenonA-dependent part ofthe operator Mis just BOK, sothe ghost propagator issame asthat of2 massless Klein Gordan field, ie,1/k* 2.Soconsider forthemonent what aQ*field theory looks like. First obvious fact is that nunber ofexternal particles onaGreens function must beeven, just aswith usual electron case. Here, the total incoming charge must bezero and all particles have charge +or-1,hence even. But the object ofmypresent interest isthe vacuum bubbles expansion. Iknow that w[0] isthe sum ofall vacuum bubbles. Lets show afew terms: aS aS. . Ri 4 wel= \aqdje “=Gle Wy duaSe=Megad: zatgmOnno”: lBeid)={ |tal Qss ColBe2 <> |x2 Omens ChReels 5);}QOo era nden9 ewes =), n A, Fr. \: an Onurad=Loraiagagl ge)=[<a\» @y ~~SYS 4 - (On Q)e +prrmutechait >ans: (LOA+( AD9\+... 6oust (1+ [OY+ [QD+ food). Theinteresting factaboutf°theory isthetthemostcomplicated vacuum graphis Justaproductofloops.Itisnothardtoimaginethatyoucouldwriteallsuch Ps)graphs inthis way: [orsOr e =|e[edesees)+[tatetee]a. This looks reasonable, but Ihave never read about such athing. Maybe, though, Iam now onthe way tomaking sense out ofthe series inthe exponent. . 3.Letsnowdothefunctional integraltogettheFpdeterminant, andthenexpandthe[*) determinant using thetrace theorem: BS <Fe(Ban =BLDme] wa=Ware “=kn) =("= RAMETAL ham axon a ~rotLad : F Pek 3 €=eng[hfBect-k+ el a wyoa s .=engAUFERC) “FACS ]=WO) ; Ihavethedesire toidentify trate)L) withaloopwith2dots, trace(L*) withthe three-dotted loop andsoon. Then youmight sort ofseewhyyouehould putinaminus sign for the fermions. 4,Also,yougetthefeelingthatZ()shouldthenbethesumofclosedloops,justas~] Coleman said. Still, there arepieces missing. Ie,Idontreally knowhowtointerpret theobject called L,andsoon. Also, Imade abasic mistake. Theghost acltion is notaninteraction hamittonian likethe£Iwastalking about. Theghost thing is like anon-interacting theory kinetic energy term, andthere isnoperturbation series because there are no interactions. Somany loose screws need tobelocated, defined, andtightened tomake this stuff go. iy RossteeSALT (QQ _ _ __ an BH. TI 8 _ _ Te Sine sTRpack4QsWario Won,cama (/6-6)_ oo Jlatimae Jorn,leadWutdoCoal. —- ee ae ak _ retw adend dy20arg.by LT Po nN) ——- ~- SalersSee xyes ealet=bash: Sug dd Le w — ss {oohSwag. EideeDy++Daag, wee mH eS _ oo ~wot2SaadHeFo ——- --5 = a BS—=toolNand, TipeiFSaadhefe — =weyPwRbBRCelio oe TG Ey OSoe ee _ Le A@SuaJaceonl22OS 2=AWayoo, OH OID Oe _ =ORV aDyanesthet_ a atta at tatiana thailand =2 -—— = oe- _Seefene —_— oo yd i en Wl ice : = : =wed-— —02a.dalgla-$—-WES4ay a2e--ee ~ ~ 322 Tso7,7 .an =a a2 = = Kolaly = ew2a=3}=Spl=—taliatelyacs v as - ay ee oe ~~—-40-2—(22).= 2-\wiaes Noe: paz fwiae] - as=3\ Dye sat Ds2D3 a ee BY —-=wo|iSe~2a,3s.|= - 5——-BortnitesBdedsesetsesnginetasda! -- _ |gull-2326) (BayeDyay ; Wass aySy dS.)SV, oo a ope ee a a — Swsheidiees _ a an ee ee __ 2 aah ea— @Race aba ste Gaendfos: 9SVSANE)ae a a,2% AD ~IMR =<olr(G-—Gi-~NOr\_ = -oe _ Zoyey Zam —— =¢ QV =teleVV, quran —a,vv — ot Qe,|GH =<oirG BroWi . _ fo P20) Gal); i s“Serame .SarGdo Tiel a Ze) Zale) O.Sawiswhole&jets.Sinuesds a )—_- yea maa -~~ 32h a=GYGTQ — — ~~=| --a —ye aio —C.Aapiily, Maisiearenasnmtcdogseraod saath, —_— soe SQ HK — . 77~~ Sk\=eatGE EI ~ eS 4 _ a ——- Mesamasea ,Oarsfoshang baler. Bikcomma _— ~—any comwst “unde! =odo alee : _ a _ _ ee eee » a VU. ©: Gerack to SL. Soe a er oe Section 15:Equivalence oftheLandauandCoulomb gauges. _ 61.Ithought thatColeman hadmoreorlessconvinced methatthepathintegral was independent ofwhich surface youtook, andtherefore itwasobvious that all gauge choices give the same generating function, Buthere, they explicitly compute therelation betwene Wi,andW,andtheresult isnotthattheyareequal, but almost equal according to(15.6). Infact they arenotequal, andtheunremormalized S-matrices arealsonotequal. After renormalization things become equal again. This isalot more complicated thah Ithought. Right now Iknow nothing about renormalization theory. 6 6 O° 8kFI= Sake) ow F=xAumstion ¥ FmaRanches 4§. Yor BE =4rG SF ae Thefunctional derivative ofafunctional isafunction, notadistribution. 2.However, thefunctional derivative ofafunction isadistribution: FIM =Sa[hG@)Se-x)) =B(FG9),YoruhdeneRoa YorREE a.aRTex) -BREWS ARwe a8ats SQ dene§4. fe) fuller « oat?BAe) Ber). see SAW) “\ 6 Nov:2? Section 16:Generating Functionals forGreen's functions endpropervertices. _ 61.Thisisaverymeatandpotatoes section, crampackedwithusefulideas. 2.Theorem: Thederivatives oftheZ[J]functional givetheconnected greens functions ofthe shifted fields J. This fact resolves earlier conflict Ihad. The first derivative ofZdefines the classical field ginpresence ofsource J.Set J=0toget classical field =v,, thevacuum expectation value ofthe quantum field. 3.The first Zderivative gies the classical field ®x5inpresence ofJ.Ineffect, this defines J(J). Toinvert this equation, yop usethe Legendre transform trick. The dual function iscalled GAMMA, dual toZ. The first derivative ofGamma is-J. Thehigher #derivatives ofgamma yield theproper vertices o(") withexternal props amputated. This isstill aslightly amazing fact? Since the variaus derivatives : starting with m@n=2ofG[f] give theproper vertices, youcanrepresent Gasin(16.20), Gamma iscalled “the proper vertex generating function". Notice that inthe derivatives youset$=v just like yousetJ=Oingetting the regular greens functions. 0. 4.Thesuperpotential isalsoagenerating function ofsorts, defined in(16.23). The coefficient ofeach term inthe sum isthe momentum-space proper vertex greens function (1egs amputated) evaluated atzero momentum. Whyzero momentum? Because then ,asin (16.25), thecurvature ofthis function gets identified with mass! Ie,theinverse renormalized propagator eva,uated atzero momen tum. Notice thet the superpotential isreally almost the same asthe generating proper vertex functional. Socurvature equals mass, andslope =0when =v. Therefore, this isagood candidate toapply the classical field theory analysis onSSB! Ineffect, wehave proved Goldstones to allorders ofperturbation theory, fora“regular” field theory (ie, nogauge group). 5.Obviously, one would like toknow, given some lagrangian, what the superpotential V(#) looks like, soyoucanfind itsminima andthen seeifthere isasuperconducting vacuum ornot. There isaperturbative method tocalculate this V,called the "loop expansion". The first term isjust the regular potential ofthe lagrangian. Tothis same lowest order, the Gamma generator isjust the action, soproper vertices are just the coupling constants inthe lagrangian! Also, tothis order the classicel field is thesameasthesolution fieldoftheclassical fieldequations. whenyougotothe 6 one loop level, the Ziscorrected byafancy trace. This causes Gamma tobecorrected bythesame trace, asshown in(16.38). This trace correction then finds itswayinto the superpotential, 6.The point ismade that ,although itappears that the first fewterms.inthe potential are divergentg infact they can beabsorbed into renormalized constants. . 7.From (16.44), itappears that tothe extent that lambda issmall, the regular poetntial isagood estimate for the superpotential. : : q i . Seng (16.26 a ee ae aSe] eer aS) Sak TP ad: . _ ~ eters ~ yas uO,” OO fe ASODQaex - = aa 2eo UU a | “ON &Ideesk aed: a . Soginal algeGatcsakanal gage 2SSS ne a SC «SO ©CO i Mary ualthpeaseyorkie - Te kt el Le, fp-~— Uoke T= =S-34\ =\=HDew.SE a | fe Pan 4 : coe ‘, _ Cee She ak Wis dg:aBO __ wo = TS = —_— —~ —+— — : _ - _. 2 Lyn kad Use _ SS Sanaa ee Nootate S/T ondoelack? 7oat Sa 5SOPOOTOES be “SESIay oy Lax] *an . Quad gh 4M Yu3 =o ee [ Son Lake — ee. AdL dd Lad: aA Set os Seen ~\y-\. j$ ae&:“hoar Ty.(Mb =O _ as Dye Og= CEDTygon QE)9ve a a we ij ae a. — ww - a °)ce Matngsktadaltme Ree Did=OO)GOVT is | 2 EE ee ” OTE XD a oe —_— ss ae es _ [Soak Seo bg a ie aa a _ PP fA es —aremand ee re ry .= iiOT ooo o>?= eeee . 3.16.78 Section 17: Renormalization inthe o-model. A 1.First, thepunchline. Ifyouknowhowtorenormalize the"symmetric ¢~model"(ie, theU(1) or0(4) invariant lagrangian without that linear term which gives PCAC), inthe regular mode, then you also know how todorenormalization in the Goldstone mode, orinthe motel with the linear term added and either mode activated. The fact that Idont know how torenormalize anything should not block mefrom getting thegist ofthis section. . . 2.Notice the trick ofhaving one instead ofthree pions for didactic purposes. You thereby avoid all the pain ofchirality and 0(4) etc eto. 3.Sohere ishow itgoes. You write down your potential with orwithout the linear term, Ifnolinear term, then the theory has the usual reguler and 2goldstone mode, depending onthe sign of p™. When you turn onthé linear term&sothatof0,itappears from(17.5) thatPsareforced toagoldstone mode o ietoatheory with superconducting vacuum, ie,sigma field hasnonzero VEV. Yy gh4:Sonsider thegenerating function forthesymmetric theory. IfyoutakedeivativesandthensetJ=0,ie,J=(0,0),yougettheéonnected Greensfunctions ofthe pesymmetric theory. Onthe other hand, byevaluating the same derivatives at f 45=(u,0), yougetthegreens functions ofthetheory including thelinear term! 5.Now several other rules get slightly modified. Look at(17.15). Usually we identifythisderivative astheslopeofthepotentialandweexpectittobe fo)zero when $=u, ie,atthe minimal point. Thereason itisnot zero in(17.15) is that |isthevertex generating function ofthe symmetric theory, not ofthe theory-with-linear-term, The idea istosolve (17+15) for u.Ie, given that constant cinthe linear term, what number ucauses (17.15) tobetrue? Then evaluate higher derivatives ofNatthis point feu andyouwill Green vertices ofthe theory—with-linear-term. 6,Restate this last point, Ifyou were doing the symmetric (model without the linear term, you would set o=0 and solve (17.15) for some u.Ie, ingeneral there issome function u(c). Ie,u=u(c). u=u(o) isthe correct ufor the symmetric theory and evaluating atthis uwould give the symmetric greens functions in (17.16). Butevaluating (17.16) atu=u(c) with of0 gives vertices forthe o-model. 7.Ts, ifyou like, (17.16) gives the general vertex asafunction ofu=u(c). Byputting indifferent u's, you get the vertices ofdifferent theories. 8,Look atthe equation above (17-19). This relates f"(u) to{'(o) via simple Taylor series. Ifyour theory isregulr inabsence ofoterm, then regular theory isdescribed bythis [e\. Adding thectorm gives (u). Thus, ifyou know the regular theory, you also know the sigma model. 9.The point being made here isalso tHe point made byColeman: ifyour symmetric lagrangian isrenormaliscable, the fact that itmay have amodef with SSB does not affect renormalizeabliity,. Here, wehave shown this fact byshowing that theregulartheoryvertices(toallorders)determine thetheoryatuf0.Also,| AtheufOmaybecaused either bySSB(internal breaking) orbyexternal linear) * term, Ineither case, renormalization follows from renormalization ofthe w=0 theory. This inturn is well known, 10. Say itonce more: you can drive uaway from 0byeither internal or external symmetry breaking. Regardless of‘the cause, the theory isstill renormalizeable aslong asthe u=0 theory isrenroamlizable. Ll,Ward identities: Ifyourotate your source functions Jatthesame tine asyour fields, the term Jfstays invariant. This simple fact atonce makes arelation between two first order derivatives off', Higher derivatives then give you the more familiar Wara identities. 12, Recall inQED how the Ward identity was helpful inthe renormalization process. Same thing will happen here, Again, all these Ward identities are functions ofu, Atus0 they apply tothe regular theory, atsome other u ‘they apply tothe broken theory, spontaneously ornot. The next task istolearn something about how todorenormalization! a Section 18, TheBPHZ renormalization program, 1.References hereare:the1959bookofBogoliubov andShirkov;a1965paper fo)‘byHepp; some 1970 lectures by Zimmermann .Also Symanzik has afoot inthis somehwer. 2.Many things are defined: most are OK: proper diagram superficial degree of divergence D the processed integrand Rand its corresponding finite integral J primitively divergent diagram operation (1-+) toeffect subtractions renormalization part the contraction notation Bogoliubov's Roperation: (pre-process integrand, then subtract with (1-t). Zimmermann's forest theorem propagator regularization 3.Whenisatheory renormalizable? Ie,whatisthepunchlineo ofrsthis high technology? Answer: look back atthe definition ofthe index %.Recall Coleman's remark that this thing isdefined incomplete disregard ofthe dimensions ofthecoupling constant. Thus, ifLy=af>,ie,PHI-FIFTH theory, dim(L) =5and $=+1. Ofcourse dim(L) =4ifyou include dimensions ofthe couplings constant thatisnoteepoint. The point isthis: ifall %{ofthe various interaction terms are£0, the theory isrenormalizeable. Asimple result toremember! Technically this issobecause then only afinitenumber ofcounterterms arenecessary toimpletent theR-operatjom andachieve renormalization. Obviously O3.—#?theoryisnotrenormalizeable, butaand3theories are.QEDhasPY,ty*which is3/2+3/2 +1=4,hence yes, QED isrenormalizeable, Same goes for the sigma model since allterms arelike #4, Dont forget this very simphe and useful result! 4. Thex rest of the section is to technical for me to follow at this time. co” 3.16.98 VoteonBAYQrrmadigabin Denia ta aMhadhy.heHeZL, 6 bomaWomSueAo R=Afonmim Jase d&=+dnwalia 9, . Node: dimti= bet+Reds dxdt, node dbund, f=We ,Lawdgt DXoan.Xe&Ve 1,“iad Ebad Maud fodeedumtn USodiy 2%=dim(QV—4 Baath Joramtrenchdemnamecig Eg= Galera Losnss Ee=4Erle PouDsOPEgpede Asmasot Tye ATdeushbamLerVsgregh Tesoo: Ne=HHvere UAwh S=4 mb=AB\csm Liner sinduds a=) omverte - m4 8 b=3 Que, rosEg+ate A, Qn, (B23) roeLior. Grea . bdaver)Eeae>€ daie >Ka E*! Jog >Bk~VAL >et Qe, MEAD Te=coment Ademogense, =Dipnae, QkgaTarte-V+\5‘ (ley). DO)=S[tete-vei) ad,te=aeeteWey Joponwhseotqradenls, oddinAnde. © Wo:0SSud:+]-GeeEnh+3[-&eZ]~Yury =Fon\vb,+2)-eZee WN. +SH, “Y =On,—G-2G+EENeo=(ita)gee&7s me. Th=agen)|YepemengangDP Oe=poe Fro=Guhepnh P=veche, 7 Bppocscrd wba) (o5,subinal) Weesettdedk bagel, (hc), ce) ~ -2~ se Piapa‘davange”eosa)yeheres,1¢,OFF. : gd)beams: emmepal Yop buaaLin, Firsideayagiie[ishamanbolatolmaitcamragech), Quaachetoog” ‘ Rp=Ci)Ep : :.\eysbbosrdlermaling SowaJovousa,Ga.surhashrng, 7 ndBoarb>omanddocorm doube|rower C3"dinpiddrag” WireadLWA =¢ . F=P2448 cowaene+ahagedf+Stake2adsSiaSutsudodogra Yooprck‘goghawdAgia. =Pe.a~% = : Beyaeee “Zin OF — Begedintoar sagesATeeDO,MamdoRp=CYR won he ayZsWoe : Te,QukpomoadsomMadewa begownakg), Denyr Nevdog! ——samearfnDrowned, Section 19: Dimensional Regularizatiom. 1.Theideaistocontinue inthenumberofspace-time dimensions fromn=4 ©) tocomplex njdoyour integrals which can then converge, then take limit as nQ4. The divergences will then appear aspoles, These poles are precisely removed by the R-prescription. 2.Technically, here iswhat you do. 2axae Take aparticular Feynman graph. Put all propagators into usual exponential parametrization. Dothe usual completion ofthe square trick orwhatever (recall ELOP )sothat you candotheaeintegrals. Onlyremaining integrals arethedd‘s,TheUVdivergence now appears asend-point singularity caused bydenominator shown. This is the usual procedure. 3.Hereisthedimensional procedure. write aeattax wereygu,extendtolarger space. Setupthealpha parameters asbefore. Thendothea”“K integral directly. Your result then ms thosep potential endpoint siggularities. ‘Then choose nsmall enough sothat these endpoint sings goaway. Then you have desired integral as afunction ofn. 4,4reason this method isnice isthat itpreserves ateveyy step your various symmetries, since these dont depend ondimension ofspacetime, Recall inBDvol 1 how they kept doing things insuch away astopreserve gauge invariance. I guess that isdesireable. 5.Big footnote onthe Adler anomolies. Apparently even the fancy dimensional regularization cannot handle these guys. You really want toavoid anomolies inaguagetheory.Somelittletheorems: re)a)iftriangle anomoly absent, then all anomolies are absent. )ifthe triple-gluon fermion loop isanamoly-free, your NAGT isOK. Infact you can write down anexplicit condition for the xrtatexr loop of interest. You can gause anomoly cancellation bychoosing particles carefully. Ithink charmed particle does this inGeorgi Glashow model. 6.Anexample ofthis t*hooft Veltman regularization isdone: the two graphs which add togive lowest order vacuum polarization inscalar EDtheory. You have two terms. Bygoing tocomplex n,you find that the second term =0?I thought you were supposed toget apole? Well, Ireally dont have apy details here. The idea isclear though. Section20:FeynmanRulesandrenormalization ofSSGgauketheoreis, Landaugauge, © 121Weconsider «specific model ofagauge theory. Model has 0(3) as“gauge group, so non-abelian. The only carrier particles are some scaler pions with isospin 1,ie, SU(2) group. Add inthe gauge fields and the ghosts and figure out the Feynman rules. For noSSB this israther steaightforward. 2.Atrick tolocate and identify all the necessary counterterms isto scale all the bare fields andcouplings inacertain way. Then (20.6) and (20.7) gives you allthe counterterms. Then you renormalize the theory inthe "usual way", ie, you show how the counterterms fix things uptoany order. For more details see Lee and Zinn-Justin. 3.Now, what happens ifyou.are inthe Goldstone mode, ie, there isssb. Inthe usual way goahead and define your proper vertex generating functional; wenow have aclassical field for the isovector ofscalars and for the isovector ofgauge fields. Suppose there were alinear $+) term inthelagrangian. Here¥isaconstant source and plays the role ofconstant cinthe sigma model. Choose {inz-direction imisospace. Ascume that this causes thefield §tohave anonzer VEVcalled y,. Soyousolve(20.15) sothat,given aX,youcanfindu/.upoints alsointhezdirection.Sinceyouhavetherefore brokentwoPeugegenerators TyandI,byassumingthat has aVEV pointing inthe I,direction, you expect toget two magsless goldstones. ‘There are referred toasXKffelds.Look’at(20.18).Itsays:supposethatconstant ¥=0sothere really wasnoexplicit breaking inthelagrangian. Butsuppose u,40still. Youareforced toconclude that m,= 0;just statement ofthegoldstone theorem.So,whatdoyoudotogetfeynman ruiés andsoon?Asusual youshiftthe zpiece ofthe¥field tosomething called, 60nowyoureplace thethree fields @withthethreefields¢(2)and-{1). Rewritethelagrangian intermsofthese ©) snitted fields. only problem nowisthat, even though these areall physical fields " after the shift, you seem tostill have a-Ytadpole term inthere. Somhow you make this thing goaway. Then once this tadpole term iskilled off... you renormalize the theory in exactly the same way asbefore for the theory without SSB, Notice that there are no massive vectors here because weare not inthe unitary gauge. Weare inthe Landau gauge where the field has some mass and isstill inthere. Ie, Iknow that this fields ..nowait. Look carefully at(20.25). Youdoseeamass term for some of the vector gauge fields. But you also see the. -fields asmassless goldstones. In the U-gauge these goldstones would have been gauged away. Here they are still present. Somehow they must decouple. From this lagrangian you read off the feynman rules for this theory inthe goldstone mode. You see that two ofthe gauge fields are now massive, asjust noted; one isstillmassless (photon); youalsohaveamassive ¥*propagator. Inaddition youhavemasslessghost cfield and that goldstone fields X. 4.But, when yourewrite themassiye vector boson propagator, youseethat ittoohas some kind ofpiece with poles atk“=0. These are some kind ofawful negative probablitity scalars, ie, ghosts in another sense. Asyou would expect, when you gotothe mxS-matrix, you get aglorious cancellation ofthe three kinds ofmassless bosons sitting inthis theory: the ghosts c,the ghosts just mentioned, and the goldstones %,. The goldstones "decouple" asexpected. Summary ofthis section: Given agauge theory, youhave toconsider both possiblities ofno-SSBandSSB,InthefirstcaseyoureadoffyourFeynmanrulesintheusual CDwey and you show theory isrenormalizeable inusual way. But, inSSB case you have toshift field with nonzero VEV. This creates goldstones. You then have toshow that theory isstill renormalizeable, and that the goldstones decouple. . aneye 4 Section 21;TheRe-Gauges. 61.Themodelissetupinrathergeneral terms.Let9besomegaugegroupvectorofmesons, say ofdimension m, ‘These are the Higgs scalars because one will break the symmetry. Couple inyour gauge fields tothese scalars intheusual way; addtheF, gauge term. Assume that the Higgs potential issuch that ithas aminimum soyou have SSB. The vacuum vector visstill invariant under Mrotations, but breaks N-M ofthem. Recall that this means that the vector visannihilated bythe N-M generators inthegoldtone grouping. Soshift thefield vector gtog*and observe thatthereisnowagauge-field mass-matrix called Wr,Forthem-(N-M) Higgs fields thatarenotconverted togoldstones, there willbeamassmatrix M°.The fancy projection operator isinthere merely toseparate the goldstones from the other Higgs scalars. Now, notice in(21.14) that after the shift for SSB, there isadirect coupling of thegauge fields A,totheHiggs scalars dg. Theyaregoing tochoose afancy gauge that kills off this coupling. The gauge surface isstated in(21.23) and isalmost theLandau gauge, but ithasapiece added onto itinvolving the'field. Iguess there isnothing wrong with choosing Finthis way; recall that itjust had tobe agaussian tobereasonable. So, exponentiation-of-the-delta creates aterm which @)cancels that,just-mentioned 4-$*coupling; italsoaddsalittle constant termto thegaugepropagator sothatisjustliketheythingealier, nowcalled§.-Terthieparameter €isafreeparameter inthegaugecondition. ThelastterminF’Ffrom thedelta causes anadditional $masslike term which you seein(21.2h). Finally, you have tocompute the FPdeterminant and then use ghosts toexponentiate it. This gives (21.32). So, nowyou areinaposition toread offallthe Feynman rules ofthe theory. a)gauge propagator is(21.27); recall thatfisamatrix, sothisshows that someofthegauge fields havemass. There issomespurious poleatoy. b)the Higgs particle propagators are given in(21.28). The still massive ones propagate atmasses detbymass matrix M*. TheonesintheGoldstone space seemtohavemass Cptinstead ofbeing really massless goldstones. c)theghosts cnowhavemase¢Jialso,andthereisanewc-c-fcoupling. 2.Whet have wedone? Wehave taken anarbitrary gauge theory with anarbitrary multiplet ofHiggsscalars. WehavedoneSSB,andhaveshosenapeculiar gaugesurface. The gauge surface has aparameter$. The reason for doing this will become clear in thenextsection,Ipresume.As§2®,obviousthatyougototheoriginalLandau fe)gauge. They claim that ifyoutake §%D intheend, you have the unitary gauge. Igthey canshow that theS-matrix isindependent of§,then ineffect they have shown that all those poles ofthe gauge, ghost, and goldstones have somehow managed to cancel. MoweonSidi theRe~gouger.. 6 wag aOSLeEHS) =Jewkesgod$fag,Qaqeeryrects. YawNoySUAjoomahlWundee, . Yb=-1S-G4 . pa igAp). vas(6,6) =LEG,=LCR Ykoie5Oydram Oye~4Ce.YonsanpackatscoleSFandCxJoctosneg- S: HeEMT SAG iSR'8) —vO) ra 1 &ossheoya=O,OKR=BTandriteSean LeECEHL(GOWER iEANON) ~VGay). Siaumeot:bintSOY) ~4épeER)L atCA,Wyecanes oxserkonarated, Qe * ” @=te(GGBAL +N (0(EBC = 1 “ (0) 5oo Conte [EE] GeOArhyp Loa@=a(y,Oa)KA(E(uP)Achy,@ joddsesth +(bi) of=pohqeegvactn,soc,gosuawskOeSree pone Foanogunoy+ombiagm ie (BdGav)=CORA) =(HEY =—(V\ Ge) Gut nm enabler 0+Odog. i(oC,G18) KE e e Section 22.Proofthattherenormalized S-matrix isindependent of$ 9missection hasseveral distinct partswhichcanbeseparated: 1,First there isanelaborate set ofmanipulations which all lead tothe result (22.18). That isonebusiness. Then comes another business ofinterpreting the resultant (22.18). 2.Lets examine this first. Equation (22.18) tells usthat the net effect ofchanging the gauge surface from FtoF#AF istoadd couplings ofthe source Jtothe fields. Ican see that indeed (22.18) does say this. What Idonot understand isthe next conclufia, namely, that when you add terms which are aninteraction offields with the source J,that merely has the effect of shifting your various Z's, These shifts are then compensated when you gotothe renormalized S-matrix, since you divide bythe same Z's. Thus, the conclusion they claim isthis: ifyou change your generating function byadding couplings between Jandthefields J,yousimply donotchange the renormalized S-matrix one iota. Idonot really understand why this isso, but . Idosee the flow pattern ofthe logic here. Itisthis: 1,Show that changing gauge adds only J.fields term toaction 2.Claim that this does not change renormalized S-matrix. 3. Since changing parameter ¢islike changing gauge, the renormalized S-matrixcannotdependon§. 4Therefore thespurious poles must cangel because they were located ata §~dependentposition,iedrat$eye fe) 3.Now, howisthis equation (22.18) derived? Onestarts with thegenerating functional w[J] inenarbitrary gauge F.Onethen shifts thefields gtonewfields g*inarather peculiar way such that the group paratmers which Icall©(andtheycallu)depende on thefields inacertain way, This little trick enables you toderive the Ard-Takahashi identiy forthis theory, ieappears aseither (22.10) er(22.11), Basically, this is astatement ofgauge invariance interms ofthe generating function. Iamsure there isaneasier way tocomprehend this thing. One then computes the change inWifyou change Fbyasmall amount. This isshown in(22.16), TheWITisthen recast intheform (22.17), andyou finally arrive at (22.18) which isthe desired result. 4.Itseems tomethat Coleman convinced methat the generating functional was somehow independent ofthe gauge surface you chose. But now itseems that that statementx istrue only after you renormalize. Section 23:Weakcorrections tomuonmagnetic moment. 1,IntheWeinberg-Salam typemodel,theanamolous magnetic momentdaetotheweak 6 graphs which convert muon into Wboson before higting vertex, these contributionsarealloforder Gm where m=mass ofmuon, But Gm =g%¢@,soroughlytheweakcorrections areoftheordera(m/my), Theregular electromagnetic 5anomoly isoforder {, soforthe muon, theweak corrections aredown by(.1/30)* =10°”. Buttheexperimental muonsnomoly isonlyknow toJ,places, sothiswillnotbetoo relevant. Notice that for the electron, you will bedown anadditional 10-4 becausze electron mass sosmall, thus weak anomoly forelectron inWeinberg Salam isorder 10-9 well beyond any experiment. 2.Inmodels with heavy leptons like the Coleman Glashow, the E-exchange diagrams like those shown onpage 133 arestronger than theWeinberg graphs bypower (mg/m, )whichratiocouldbe(B/-2)=20,somaybethesegraphsinthistheorywillcontribute something oforder 10" tothe muon correction, and you could measure it?? 3.‘Theactual calculation 1sdoneinanRggauge. Youreadoffthenecessary vertices from your set ofFeynman rules, and your final result comes out asshown in(23.47), or(23.51) asanumber. Answer isabout .5(M/m) inlast place, soifratio were 50, you would have acorrection ofabout 25units. The experimental error is32units. 4y.Onething youseeisthis: ,ifM,islarger than 10GeV, thecreated correction will belarge and take you out ofrange oferror. Thus, you have xxkmwmx anupper bound on the Mmass inthis theory. Teremy Bernetein 1974 Spontaneous symmetry breaking, gauge theories, theHiggs mechanism and allthat Jeremy Bernstein Stevens Institute ofTecnology, Hoboken, New Jersey [Amoreorlesself-contained introductory reviewispresentedoftheso-calledHiggsphenomenon. ‘This isthemechanism bywhich, inacertain class ofgauge theories, the“photon” andwould-be Goldstone sealer mesoas conspire together toproduce massive vector mesonsviaa“spontaneous” breaking ofgauge invariance. [tisconceivable thatthisisthe wayiniwhich nature haschosen to rityweadadSecwommgneeinterasioustishopedthaaearofthtreviewwlcometo a |tndersand theeasing oftheistthre sentencesinthisabstractandwilhenbeabletoproceed wo" ‘|toconfront arapidly growing literature inthesubject ofgauge theories. . CONTENTS oforthogonal stateswiththeproperty thatTi,TheGoldstone Theorem io ammieeeLoophole wh is HI0>=0, uy /-TheHiggsMechanism, orWhere where HistheHamiltonian ofthetheory, anfi|0)isonevy,TAateGoldsonesGone! Wrofthevacua.Sinceohasthequantumnumbersofa Vi.Non-Abelian GaugeSymmetries 3gvacuum, itsvacuum expectation valueisngtforced byVii,Weinberg’s 1967Model 32anysymmetry principle tovanish,Le,wemayhave VIII.Conelusions a | Ola(x)|0>=<OJa(0)I0>=A#PO,(1.2)1.INTRODUCTION where\isarealnumberandoisanHerngitianoperator |Elementary particle theory seems toproceed from Withdimensions ofamass.Weassume thft - fashion tofashion inintervals oftwo tothree years. A ipfewfamiliar namesfromtherecentpastwillgivethe a(x)=exp[-i(Px)o(0) exnfi 3) general idea: reriormalizable field theories, dispersion with—note themetric convention )beused relations, conserved and partially conserved currents, throughout— current algebras, Regge poles, etc., etc.Atthemomentwhen thesespecialities areattheheight oftheiractivity, (Px)=Pox=Pot, (14) most practitioners have neither thetime norinclination, ~hime difogobackandreadtheveryearlyliterature intheWherethe&arethegenerators ofspace-time: displace-discipline, sothatagroupofstandard references is™ents.Wealwaysassume that arrived atandthese become oftquoted andrarely read. Rilo) =0. (us)When oneactually does_go back toreadthese early * : .‘Papers.oneisoftenamazedbyhowmuchtheirauthors.yewanttogivethegfieldaparticleinterpretation, we“kneworconjectured, andonecomestotheconclusion areforcedtoredefineitinsuchawaythat1 thatthese papers rearranged andunified areprobably thebestintroduction tothesubject. Thatwillbethespirit @le"(0)|0> =0 (1.6) andmethodology ofthisreview. Anything novel onthe yeh partofthepresent author isunintentional.‘Theplanofattackinthisreviewisasfollows.We Y(x)=o(x)— |begin(inSec,11)in1960whenNambu(Nambu,1960; a)=of3)=> | Nambu andJona-Lasinio 1961) observed thatthenatural otherwisethe“vacuum”andtheone-particlestatewill interpretationofaconservedAS=Oaxialvectorcurrentnot_beorthogonal,Asweshallsee,itispossibleto |inweakinteractionsisinthelimitingcaseofaworldinarrangetheLagrangianoftheomodelinsuchawaythat, &whichthemassofthemesonissetequaltozero.Atattheoutsetthepionandohavea“bare”mass.The |essentially thesametime,Gell-Mann andLévy (1960) nucleons appear tobemassless but,infact,acquire aa produced several fieldtheoretic models inwhich -this massthatisproportional toA,thevacuum expectation eel Phenomenon wasshown tooccurasaconsequenceofthevalueoftheo.Toachieveexactconservationoftheaxial “elfieldequations appropriate tothemodel. Ofthese mo- vector current, A,,inthistheory, itisnecessary togive..J dels,theso-called“‘o”modelis,inthepresentcontext,thepionzerobaremass.This,itturnsout,corresponds to the‘ostinteresting. Initssimplestversion,therearealimitinwhichthetheoryisinvariantwithrespecttothe : threebasicfieldsintheLagrangian: anisotopicvector groupSU(2)xSU(2);exactchiralinvariance. Hence,inpion,anisotopicdoubletnucleon, andanisotopicsinglet,_thiswayofrealizing thesymmetry, zeromassbosons—in| Lorentz scalar,¢meson. Thelatterhasthequantum thiscasethepions—make theirappearance. In1961,>numbersof“avacuumstate”ofthestronginteractions. Goldstone conjectured thatsuchzeromassbosonswould(C7) Werusethephraseavacuumslateadvsedlysinceinthisbeaninevitableconsequence ofa.symmetryrealizationclassoftheories thereareingeneral aninfinite number oftheories Tiketheo-model inwhich theLagrangian —— wouldbefullyinvariant_withrespect_to_a_continuous |7WorkpariallysupportedbyNSFgrantGP-3677, proup,batinwhichthevacuumwouldnotbeinvariant : Reviews ofModem Physics, Vol.46,No.1,January 1974 Copytight ©1974American Physical Society 7 4Prue “highMadd : in . 1/77 sertein, Int@paution. SVEnah CF1:TeeonlyKindoffieldthatcanhaveanon-zero VEVisafieldwithvacuum quantun no'es 2.The "g-model"wasinvented byGellmanntLevy in1960,Jnthismodel,youhavethe nucleons asiso-doublet andpionasiso-triplet, andsingle field called o.Atthe start, both thepion and signma have mass, butthe nucleons donot. After shifting the sigma field due tohumped potential, nucleons acquire mass. Inprocess, Ithink thesigma loses itmass, ormaybe thepion does. Yes. pion loses mass. This isjust agoldstone example. This theory also has C.A.C. since itischiral invariant. The idea that PCAC results from non-zero pion mass isintroduced here maybe for first time. This Gellmann-Levy thing came beofre Goldstone paper. 3.In1962, itwasthought thattheGoldstone Theorem wasproved andthatyouwerestuck therefore with massless bosons ifyou have asuperconducting vacuum. Higgs found the loophole. This loophole has something todowith asubtlety ofgauge invariance and gauge choice, theidea that A,isnota4-vector ifyousitinradiation gauge which is non-covariant. . 4Higgsmakesasimplemodel: chagged scalarfieldf,and$.andphoton. Bothscalars start outmassless, asdoesthephoton. Butafter doing supercond vacuum shift, Jy gets amass, $,disappears, andthephoton becomes amassive vector meson. Youalways Qs weds solar field1ikeJtogetgoingbecause youneedafieldthatcanhave non-zero VEV, asnoted above. Such fields are Higgs fields. Some ofthem are "eaten" bythe“photons butothers stay around inthetheory. 5.Later Higgs in1966 shows how towrite legrangians directly interms ofthe massive vector fields. . 6.Important point: intheshifted Lagrangian, theoriginal symmetry isnolonger present andisextremely broekn, InNature, thesymmetry maybeundetectible directly. 7.In1961, Englert andBrout doacovariant trip andfind that photon propagator gets shifted. Thisis(Higgs mechanism indisguise.) Nodoubt lotsofmessy field theory here. 8.Idea thet conserved current maynotincertain cases imply aconserved charge! This is new tome. GHK showed this in1964. Kibble worked this upin1967. 9.InWeinberg unified theory, one might besurprised that the photon and Whave such tremendously different mass. But"broekn symmetry" inthespontaneous sense does not imply slight mass differences like conventional H'symmetry breaking term inaHamiltonian. 10,Combined unified theory ofweak andEMisforsome reason renormalizeable, free bonus. Lo] - 7 (2.}he Goldstone Theorem (10ha-pagew) 1,Inagange theory you have currents andcharges. Consider Q/0). Usually this 0 iszero, vacuum hasnocharge ofapykind. But,inSSBanalysis, ifsomescalar field$hasanon-zero VEV,then‘anycharge whichdrives thatfield"through the usual commutator has this property: Q/0) #0. Inever thought of this before. 2.Prrof ofGoldstone theorem. In(2.27} isdefined the FTofcommutator ofJ(x) with 6(0). Insert complete set ofstates, assume positive norm etc. Igather that these intermediate states /n) can beatmost single particle, because J can create atmost one particle. Also, any such states /n) must have J=0 , based on rotation properties. The proof here (Gilbert's of1964) assumes (2.32) which seems obvious tome, and (2.29) seems wrong tomeatthe moment. IfGilbert can show that there is some contribution fo (2.28), then heknows there isaspinless particle floating around. The fact that some field has anonzero VEV allows him to prove that there mst besome term inthat sum, Moreover, ifthis current Jisconserved, then (2.32) tells you atonce that that state mst bemassless. So, tosummarize: ifyour theory has @conserved current, but avacuum which isnot killed off bythe charge ofthat eame current (ie, nan zero VEV ofsome scalar field), then you mst have massless spin-zero particle inyour theory. Maybe thecatch here isthis: (2.32) iscorrect only ifJ,isanhonest 4-vector. This isthe point precisely! 3.The"g~model" ofGellMann-Levy, 1960.Writealagrangion of#’end¥mesonfields which ischiral-invariant, ie, SU(2)xSU(2), sort ofisospin and axial isospin together. This theory hasaconserved current VandA,. Hence charges QandQ;type. Ifthere?is noSSB,thenallmesonshavesamemass;ifthereisSSB, 5: then you have massless pion come out. This is application ofnormal Goldstone Theorem: you here have originally 6conserved currents. From (2.59) you see that ,ifsigma has anonzero VEV, then the three axial charges have property thata,fo)#0.Thus,youexpecttogetthreegoldstones, thethreepions,Notice that, inthis theory, shifted lagrangian still has isospin,invariance; you only get aGoldstone boson for each conserved current you "break", Here you started with 6,and you break down tothree, soyou get three Goldstones. Bach "broken" charge will no longer kill the vacuum. 4. Finally nucleons are added. Ifno SSB, then you have massless nucleons and equal mass pions and sigmas. Experimentally terrible because implies massless nucleons and sca&ar mesons degenerate with pion. Inthe SSB view, nucleons acquire mass and pion ismassless, looks much better. 5.The Gellman-Levy model isanexample ofzm amodel with SSB inwhich the Goldstone Theorem works inthe way you'd expect. You break some symmetries and thereby create massless particles. The theory isnice because you want to think ofthese mass less guys asthe true pions. Ie, add asmall perturbation tothe lagrangian and get very light pion. 6.what weshall really beinterested inisthe following: under what conditions can the naive Goldstone theorem be violated? That isthe Higgs business ofthe next section. 4 Comments onGilbert's 1964 Goldstone Theorem proof. fo) 1.Ihaveworkedowtthedetails ontheattached sheets. 2.What are the main ingredients? a)ifJ,isatrue 4-vector (manifest covariance) then (2.32) iscorrect. Ingauge theories this step will fail ifyou choose anoncovariant gauge. Condition (2.32) leadsatonceto(2.33)withnootherassumptions. d) one assumption skipped over ss that vacuum does not contribute insum over intermediate stetes. Iamnot very clear onthis; they say something about positive definite states only are allowed. Certainly youcannot putthevacuum inaunitarity sum ce)Next, this chosen J, (acurrent corresponding tosome gauge generatorcalled Q)isassumed tobeconserved. This fact leadsto(2.39). At thispoint here iswhat weknow: ifM40, then there must besomescalar and massless particle contributing. The massless part came from the conservation ofthe current. The spinless came from the fact that gisascalar field, Maybe Ishould add that. ad) the field gisatrue scalar field. e)finally, assume that (0/9(0)/o) #0,ie,SSB, Then from (2.46) you getcy#0andyouaredone. Ingredient here isbasically (2.41) which says: the fields transform insome definite way under the charges. Then (2.43) isarestatement ofthis fact. 3.Overview: inatheory there are gany gauge charges and conserved currents. Byexamining theway inwhich the charges "drive" thefields, andbydeciding whichfieldsyouwanttohavenon-zeroVEV,youdecidewhichofhecharges lozQfail tokill vacuum, Ie,you decide forwhich currents (2.42) istrue. 4.What wasnotclear from theproof here isthat each "broken charge" gets its owngoldstone. Weonly proved that ifyou have atleast onebroken charge, then you have atleast one goldtone. fo) sex . 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BesaaWinnsyneeureWokG<Vew)we3sdf.Qe,rw enrntads Wak eens) | weo. Que,gorsender peter comacng, Wall)ocrwoedre| OninPetts FanBd%*S0. OSs y=BLED.Ga!faWCua@ig—arersoro}CE - =Q. ®Wowwarkbocompu, cy.Conaittens Ma= eCedykyBME) +heS(Wh) =YnCer)LaeCes+e\. Qadandale :e Yeown ; \SxMocs\=SaeeStet sec aSol). Sher\th) a aye ae=| i)e2|4CAAA) yeCR)xcae[vettesa+6weIsveCti sz] = EXPEGW)-eCH)] +¥(oe-sy) Qa EX)=8)~8%) eX) =oO ~0(x) e So €@)~EO) =2se \Se«MC)=Sponyvoter| OnnBs.oOlrnonnd g — Nes :Sata =Soe Saye colon, galls3 QeSST algOnly =avs: V8. Lo][UENO] |)S =e@re. "Lg ele=\Ye.WER):VR)AeZaROY. =drCAROL. [0vssboskBxo- SoyWah: Ja=3HOW. |(246) ®BkwyachKoay! T3.@, dO) —FO40. {QWO)=Z..4@) \dedovnafrn tnSomewweey/ Bernstein: (3.) TheHiggs Loophole. : 1.WhenyoudotheusualCoulombgaugequantization with@-Aeo,youineffect fo) force theobject4,totrénsform.not asaregular 4-vector, butassomeweird object with thé extra piece’as shown in(3.16). This extra piece isfound by B.intwoways: first, byrequiring thatthegauge condition betrueinany twoframes; second, byjust-looking atthewayyouquantize with 8object. 2. Another viewpoint is this: since you are choosing this non-covariant gauge condition, youineffect pick outthetime direction asspecial. Thegauge condition is still rotationally covariant, but not bogst covariant. Thus, in effect, youcanimagine thatanything youhavecanalsobeafunction ofthe A-vector p= (1,0,0,0). Ie,yougotoanyframebutthisvector stays asit ie,ie,youareviolating boost covariance, Thissameypshows upinthe Feynman propagator but has no effect on S-matrix, argument repeated from BDvol 2. “3. The presence ofthis extra vector, iethe loww ofboost covariance, ruins the proof ofthegoldstone theorem. From (3.42) inmomentum space youfind: : Ge WAViaKAY+YouseethatJisconserved, butsinceAisnota true4-vector,neitherisJ.ClearlythiselimifiatesakeystepintheGoldstone Oo”proof which newgetsreplaced by:ColTple)(07 =a(KjkVKy +b(K) kx)Ip.another result isthat the charge @corresponding tothis current isnow Q(t) and moves in time, something very new. However, the current isstill conserved. (4.)TheHiggs Mechaniom, _- . 1.The preceding section ofthis paper showed that the usual’ Coulomb gauge “allowed presence ofWaandthisthenallowed an"escape" fromtheGoldstone Theorem; this isthe Higgs Loophole, socalled. Inthis section wewill-examine theHiggs idea in,more detail. Section isalong 15half-pages. 2.First interesting trick istowrite alagrangian without showing second-order derivatives. This isjust aSchwinger trick toeliminate extra work. Of-course second-order derivs reappear when you vary everything. This isthe came asthe idea inAbers Lenn oftreating A,and¥,, asindependent fields forawhile. 34Phe model ofthis section is@charged Klein~Gordon scalar field with photons, is, scelar ED. Charged means that scalars come inadoublet. Hass term isput inwith wrong sign, intentionally. Afinite charge rotation (gauge transform tion) isidentified inits effect onthe scalar doublet; just asimple rotation. S_me gauge function ofcourse appears ingauge trans. forA,.Thecovariant Df ° CX 5SBentiFied asusual, standard gaugetheory. 4.Nowweallow breaking, ie,letJ)have nonzero VEV. Several paths arethen floowed from this point. 5.Inthe first path, you examine the equations after linearizing them. This linearization is justified after shifting field because fields are "small" insome sense. You then assume the covariant Lorentz gauge and you find that youdohaveaGoldstone (massless scalar $,') butthisthing solves afree eKGequation andisthus afree Goldstone, te,uncoupled torest oftheory. Meanwhile, ifyoudefine 3,tobeA,plus4,of,',thisGoldstone, youfind that 3,isamassive vector meson. Soyou get the idea that inthis gauge, although Goldstone theorem isvalid .and you doget your Goldstone, itisdecoupled. Moreover, your gauge field has acquired amass. Your other meson also has mass, the fo’ field. The ratio ofthe masses ofvector meson and scalar meson isset bythe twotheory couplings, the$4 self coupling fandthegauge coupling. e,- 6,Now lets redo this inthe Coulomb gauge. Again play with the equations to seewhat youcansee, First, g,' isnowidentified with a/dt ofA,+80$i" isnoteven atrue Lorentz scalar! However, B,constructed asbefore isatrue4-vector (cofariant), The fieldJ)’ has thesame mass asB,, still decoupled.- Thus, Goldstone Theorem isexpected tofail since loss ofcovariance, and Golston particle now has amass, All fits together. : Inthis guage, the physical results are the same: you have amassive scalar and amassive vector with masses ratiod as before. Finally, explicit failure ofGoldstone Theorem isshown inthis gauge. I wont bother tofollow it.Hinges onfact that J," isnotatrue scalar. Hence itcan couple the vector BY tovacuum. . Theexplicit commtator ofthecharge and$,'iscomputed andyouseeexplicit timedependence ofcharge, even though current isconserved. 7.Ideatoremember: thereason youget 440 issurface terms. These surfactermsarelikelytobepresent whenyouhavemaxxivax m,ssless particles in“eo your theory. Gauge fields are massless particles. 8.The third parth isthe U-gmge. You start bychoosing agauge function that rotates yourLees6,"fieldtozero,so itisjustplaingone.Thissametransformation! takes yourA,intoA,’=ByeOnlygp"and3B,areleft.Youthen doSSBandshift f."field YoCHI" afdCHI becomes your massive scalar. This isthe U-gauge where S-matrix interms ofCHI and Bfields ismanifestly unitary. This path iseasy togeneralize proof and analysis toall orders, 9.Last piece here adds the baryons, Global gauge invariance gives you baryon conservation and some "S-number" conservation for meson. You doSSB andbreak theS-number, butmaintain baryon conservation. NowtheB,field is driven by the baryon current. More physical theory. Big deal. 10. This section féllows Higgs own work closely. .. §5.) Toshow how particles acquire mass imperturbation theory. ° "i,Inthelastsectionwestudivdindetailalittlemodelwith’achatgetscalar fo) Klein-Gordon field and some photons. Inthe U-gauge, wemade orfe ofthe mesons "go away" byacarefully chosen gauge rotation. Them weshifted the’ other meson from BotoCHI, andthefield: B,picked uptheremnants of‘theGoldstone.Thequestion hereis:howdoesalEthislookinperturbation theory? Youstart asshow in(5.17) with alegrandian inwhich SSB has aready taken . place andashifted field $'appears. Neither the$'field mortheA,field have explicit mass terms inthis lagrangian's L, part. Ofcourse the mass terms doappear inthe Lz part because wehave already shifted. The game here istotreatthesemasstgrmsasinteractions. Thus,yourbaregpend4are massless and sohave 1/q° propagators. Then the game istowatch how the masses ofboth these particles appear asyou do perturbation theory. 2.The lowest order corrections tothe bare massless photon proagator are shown . infigure 4, You see the "point" coupling which isreally the direct mass term, afd then you see acontribution ofthe massless meson inthere. Notice that you can couple ascalar toavector if-you have gradient couphing,. The meson term causes apole in“(,) and this pole then causes the renormalized photon prop.- (renorm, +0first ofMer) tohaveashifted pole! Thieishowthephoton acquires an its mais. . . 3.Notice that the"mass term AAU itself does notshift thepole, Butofcourseitmstbetheretoyieldgaugevariance ofay. 4.Next, heshows how’ the meson itself acauires mass. Inthe meson cause, the trivialpointinteraction doegcausethemassshift.Thereisadifference ombetween the two: notice theq©factors in(5:26) versus (5.6) forthephton. Thus, you dont need massless photon pole togenerate mass onyour meson. The whole difference isrelated tothe uafd vindices, gauge invariance, conseved tensor, etc. , 5.This section isbased onthe work ofEnglert and Brout, 1964. Recall that these guys independently discovered the Higgs Mechanism inperturbation theory. Ie, they discovered that one ofthe mesons attaches tothe vector and makes it massive. . 6.Further work along the lines ofthis section was done byColeman and Weinberg 1973+ 7. Adetail Iomitted: notice that the "massive photon” has apole atsome mass, but itdoes not have the usual propagator you, would associate witha massive vector meson. Inparticular, ithesayq,/a? rather thanq¥q’/M°, Thus, itwill bemore open torenormalization than atrue massive vector meson! Ie, itwill not have the obvious UV problem. 8.Itisalso noted that, inthis theory, ifyou donot allows SSB, you have massless photons and scalars and theory isanIRnightmare. 6.)Nom-abelian cases. : . 1.Section opens with review ofYang Mills. Start with some fermion fields in isospin doublet asusual(nucleons). Global isospin ratations yieldthe [*)usual conserved isotopic spin current, Local isospin invariance requires addition ofthree gauge fields, theb| is1,2,3 since three SU(2) isospingenerators. Asusual, theextra non-Abelian termisadded toF,,goget atrue isovector F,, and usual gauge fields term. The bfields mst be massless, aithongh nucleons could have mass. @his iswhere Yang Mills stopped, 2.ButBernstein now goes ontodosome SSB, Add aniso-doublet ofHiggs scalar fields. ‘These fields add some terms tothe full lagrangian, called Lye‘ . 3.Atthis point, what are our currents? The usual electric charge type trans- formation (phases oncharged fields, nothing onb,)yields aconserved . electic ofhypercharge current called ¥,(global transformation). Also there isthe, isospin current 1, .Each Current has acharge, and weset up theelectraé charge inusual GHWmanner. *4.Next step istoallow field $tobreak vacuum. WegotoaU-gauge andkilloffeofield.Thenweletthef°fieldbreak,thenshiftbytofield¥. What is the result? a)formerly massless Higgs scalars now dothis: . . al)one Higgs field isgone away a2)theotherHiggefieldis{andpasacquired mass. »)all three gauge fields bhave the seme mgss, nonzero, . . 5.Comments: asshown in(6,47),-we have broken three isospin generators,”hencethreegoidstons, butthesewerealleaten bygaugefields. Thereare~o no"massless gauge fields left inthis model after SSB takes piace. ~ ane (1.) Weinbderg's 1967 Model., (oe) 1,Nultiplets: lefthanded electron andneutrino putintoaweakisospindoublet, both are massless. Right part of:electron put into singlet. The, requirement of locel weak-isospin invariante brings inthe compensating fields ‘By asinYang-ltills. This iswegk-isospin, notisospin however. 2,Sofar lagrangian has only lepton fields Land R,and gauge fields b.There is atonce aweak isostoic spin éurrent which isconserved. This current has two pieces: the lepton obviais piece, and the gauge -field purely bpiece. This latter iscalled Jwith charges %.The lepton piece iscalleé J!with charges . T'~ Total weak-isospin current iscalled 7,Thecharges ofT,,arecalled T,and these ere the conserved charges which generate the SU(2) part ofWeigberg's gauge group. 3.Nowthe"hypercharge-photon” a,isadded; this istheU(1) part oftthe gauge group. Hypercharge of coursé means "weak hypercharge" and isrelated tothe"weak 7," component ofweak~isospin togive thetrue charge & 4.Argument isgiven forwhythis hypercharge photon couples with g'toRy butwith g!/2 toL,Idonotfoliow argument, but could ifIhad to. 5.Now lagrangian has these fields: L,R,bya. Ifyou like, you can regroup things, taking linear combinations, sothe fields 1L,R, band aare replaced with e,), Z,W& A,Zisfound bylooking atwhat couples toVueV. The photon isthen the orthogonal field ,couples only to 6...e. fo}«6.NowcomestheHiggsfields. Theycomeinaweakiso-doublet. ThescalarHiggs lagrangian isthen made local invariant under entire U(1) xSU(2) bycoupling inthe band afields tothe Higgs fields. Higgs fields are massless. 1.Now, thefield g*isgauged away andsomehow thefield #°isrelaced with aHiggs field called R(x). The final Higgs lagrangian interms now ofthis field Rand the A,Z and Wis shown in(7.82). There isnoterm like AAR. Thus, when we later do SSB with R, photon will not acquire mass. 8.However, you dowant your electrons toacquire mass. Thus, couptings 6Re isadded byhand, coupling G,anew free parameter. 9.Sofinally you do SSB and shift field Rinto usual CHI, The results ofthis SSB are enumerated: electron acquires mass, sodoes Higgs CHI field. Sodo the fields Wand Z;Weinberg angle defined; photon stays massless asnoted. 10. Aproblem noted with this theory isthet the Aand Zfields can mix, since oth couple tothe Sesystem. Thus some kind ofmass-matrix diagonalization is needed. ll. Rest ofthis section not very interesting tome. Situation reanalyzed in terms oflinearized equations, look atdecoupled Goldstones inthe Loentz gauge. 12, Last section isabout possibility ofmagnetic moment and minimal coupling. j Incertain cases itseems that minimal prescription does allow ananomolous6 moment term.??? Skipfornow. ae (8.) conolusions. . 1.Bjorkenscommentson"believable" gaugetheories. °e 2.Division oftheoreis into Clas 1with neutral vector carrier like theZoe and Class 2without neutral vecobs, but with heavy leptons. 3.TheZ,implies neutral currents, Bernsteain in1974wasnotsurethere were any such animals, Atthe time this was arguement infavor ofClass 2models _like theSU(2) GeorgiGlashow with itssingle heavy lepton. o 4.Bernsteain showshowtheZzoftheWeinberg modelcurestheunitarity problem which the Wbosons themselves do not cure. : 5.Hegives afew other examples ofhow gauge theories mysteriously "cure" problems ordiseases ofweak interactions. . :.