General Field Theory -Weak Interactions

General Field Theory Weak Interactions Math Commins Old Field Theory Notes New Field Theory Notes Phil Lucht notes Feb 1978 -May 1979 MATH Sowa. Fayronam ideaihiew aeeee -—--- +_—— PLease iaOg oeaiakannSarees)1-,--=\uano ot .— AB8.Qo. TAqsBef BecBea] (ean quvinaeye bee ; \ =dt\Bada SGrae) . Vo—.—— Awa Ani 8 —--.-VA ArdeautAnde ©WinBaus Dequacdigdomeis Aseen a 8 as ae - : =ThessSanin80-089 ee__...__A B®. rere) 82 \eAseRyrr ake Thatoan Sayagaaetue (A) 6 a SO 6oYeitt yA 8Gente,meme) (/6n3.199.3 Say mB) 1A) © Qdak: dle =4 Oeees+y(vert) mane §4 6 Oninitiating, aoarntonJomction im" in o — ie ;r+jak£(®) WeWebb ae Mewanucttoheciteto.dortesNougusan” vikaqeodYinsk+Can. 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SS _e AP NBRhey ERE Pe A es Ce Nw a oeAT AAS SSR SE Ee. Gos &eOne(-oak) =TEaexe||a dag wathogete ls —anctandWideRetabion 9 oOOYAYiwomctytia.buvodowle, TyaudJnoarvanishss ionqpsakcisle Unbem ple yee, idASetaSia wee B Read Le 2. .Sato=Jae=ilk @dk ky=-ike kye-ke. dks idk, 2” Jao=-e@k=-%Rot HO QDkys+0 oOSERYG =EQSACKD-saRaLCR- a ait)RAC -_-- - adatSxfGdan i — Totanatas =OMeRoRE StC=“s=‘“=*s*s‘“‘“‘SSSSSSS -GeVa@=OSatk(e - -LS & - . QO | - ~ 4qWere a eeTF 7Sora Lea4 a! woe ++4 or a rn ae ee ee he etme Se ettn a eeOI ee A Se weed et ee, a reee re tt TE 2ea 4er ee oe _-- - : . - ke + - + -‘ t . : yer, roy . Soe Bory ‘ tte . he 7 . a "4 4-4 . eosft,Real. Br, ALL. Lelgoge boooeon- er ° toon - oe ee hK- . . ae : 4 erp eee . se eee - 4 pe ee a wt He f we ts yb Rea ae eee tanh eee . 3 Pert t+tthe sda ote arte eo ee 444 + tr be toneoa _ pee a I - ee a ee ee -phe ee eee ee oe ee gee ee ee t - ee cee ee a ae ee ' + me =- 2 ~ Dong - a + + toa He re - ae HH -- 4 |Bisons Y~space, aSO(s)wbsqeota.A: . . om ONKGye) =4carrer coadviale. Yat ; 6 Rakeyeaewh=GabigRoDelmar anghee (we Reve fhade -t= Rema /¢=Gaweh3-08. BE [email protected] ee cenee ee GEIMG FLRewesme 2 2 ee x=gues =Ramismeesé | cea, wag smg = Rswae sme suf. Lo. ”Oyu, weSasOewere Lo - z= Rsweose wae Gnuw SsRevd sme_cosd . a ne fa Rom sue Sud coe . wa DSSaarcomqubah DerURE yaadouiinnal on nn adadin =dk=RARsikda-swededh| On akeRARMQ A A= Shida. swedeAQ, , OGoogite dothstage Tg aeeef—- ;eo oe on ee| ee ee ameee =--oe- .we oeMaCh=2)=ansad)4sphatsoduueR..| -. Ayo. = -- : : oe a | {8 |Bulge4)spaidhSous oS|ASTER)SSLaeQa$8,he) Abasdecd,ous'sBEADanChews9% fo) _ak4.ue Ce, / VoreN-=“OMT awl . ee QininohOrsJaws: Sabky PCR.gkeg)al_NMnkrashi auebne Sg, Keee, : oe a =a -3q . i :weeee oehela eee - a ©Yaa,daaRyeatte.Ohws a-RR SRa Fr raeFhe a9g— ———___ __oe . _ . Keg=~Gey ==qerhes .WeDamSrousaaron, wee ee ee +i\dhkeACE chee) wc SteUE) =eRete LO wd2veDRRDneice, aaaarnne ambation. ve.OK.cee . Sa : Loe /AaEKaye) =sian)GinSaaee FERHg,—Ry)- JaeAW@)=savyKaeGR) re en OWeosde ngOokodighES 2 ;op=VeSoa\ae3ee) oe e , page 1 Atechnique for proving the Finkman Identity. fe) 1.Hereiswhatwewanttoproveforanypositive integern: a on3 =mn!\Ga (S,)" ayaa ° where we have defined: GNv 7 o~ Yegae TT(Ste)82-1) Sa=Zia 2.Iproposed aproof byinduction. Assume itistrue for n-l, then wmrikyt verify that itistrue for n. We start inthis way: “on mL ow ew at od!VQ,(STEGen!TW(Yass)an)SBsitxty-) ° aseVST]ASay ONT at -hn x\Zat4QaKG Sofar Ihave done nothing more or less than rename the last two integration variables, Iwish to examine the dxdy integrations, holding all the other z's (o) fixed. Iwanttodefinenewvariables toreplacexandylikeso: kexty dud=$fdadk *eECs)=% yeECs)S=X % OHO)= SHEM)SG49) =OCE+S) OC~)BCE) a 8 2s 2S was tes .aSTS Yad ayFQ)=Eyasyak $8, 2). hosas Nowramen wowahaksweoh woe- 2 ot catJEQae)syaktreet) SCZe+k-1) wv \S eS ast hw »TZara tis)+aztts)| Baek the dtoff tothe left, and put twothetas into endpoints: et ee am tan s* =Gey!W(Qaa)\akS(t). Nas|=ardentlyinti] re) wrSe4S‘ et=r so page 2 Next, explicitly execute thedsintegration toget: cet mast fe)zso.pe |Bawta(rssia @sy\/Em) L(derIn) i -t ‘ =nAl mt -n4l <.m4 i|Baas+4,-[Zaxat] =) n>An) m ‘ Wowtake thevariable tandrename it2, 60answer becomes: ° L <nay> aanahpa’ toJoot Sao,[Baal =[goaaraan] HarAo (Aran) 8 vay 2. Now what should Ido next? So far Ilike it because Ihave caused the once-lowered (dz),_,integration toappear. Ihaveyettoprovethatthis isequal tothe thing onthe left. But now lets use the fact that this thing is supposed tobetrue for n-l. Then: ° owl <42)!a2), [aaa] 4,8 unt ry 6° Py ~~ \ Jie,sensoeabet 2BAIae\= (x)!\@aen [Axe ‘aaeeeAneIn PABA np94% ° ¢ 74 os\Aaenan CayFant)lianGuanAmetMALAmtAm a <G@u-\ |\ 4.= . =| oe aeoAGMae Anan \G>4e) Thus itisproved. Iftrue for nel, them mst betrue for n=2, and soon. But is this proof valid for lown? Ishowed that (n-1) implies n. Must show true for ne2 explicitly’: oe 2 -u act “Sydadeflate) [aecraczeny od a u OL =d4vddy SGy-)) fare] % o 8 ‘ - os a 46 -s)\4=t\akSas.1-1) -[atGesyress >) os +t v2 \ = page3 -\f a. _i\ L(tas) +b4(\~ os\adMas)ab2Q-8)] =4 1 fat(iss) +b(1-3) =tNas 2( Een) F(a) -t ~\ =\[=\11i\Y1...3{al-Sf=KG: \=a A> &rvb boa ¢ab Well the case ne? issame asany higher n,soIalready proved n=2 when Idid the general induction proof. Now lets show for n=l: eS AayHNa SQ) Cah =wl 4 ° And socompletes this induction proof ofthe finkman identity. There was no need tosumover permutations orapything fancy like that. Eyaluation ofaFeynman Integral #i. (e) 1,Inthestudyofmassrénormalization inscalar-boson QED,yousoonericounter this integral byconsidering the one photon loop onthe electron propagator: T=Yka (A =Ig,%,hw) —KEX eK)=my Way Hereyouseeaphoton propwithamassinserted, theelectron propatmyanda factor put intoremove the logarithmic UVdivergence. For small k,itdoes not appear tomethat you get anIRdivergence. But lets just treat this ananintegral topractive on. 2.First, use Feynman parameters toget: \ \ \ 2T=2NSaag dayB(lesedees) -Qaheco)4¥(Gak we2p)a so ‘. . . es -3 ~FER aT? \ tog_ 2 @=anaaa \vka° ° s ku . 3[itsaacpee=East} dened =e)N3] Nowcomplete thesquare, then shift ktonewKs 4 ER. =ataedyosSK 7 + xe rec os.(ays fated (rtp eaatert) KY eee. =A ‘ NowIcanlookupthisintegral inmytable: Noe 1 7= cay (E-A] an \ lee orEsrkVaasa \e 8 ON ae Lea(end9°)~NT >seeoo =6 Noa Wenowdothetrivial logintegral: Ade. . : (e)Sa, — tg, [Oa 8]H ° Ob) B Oe) 8 vane B=aeed(Mewe~ pS“Kh ~ . “Nu=Pore Gea) ros a \ Bay ° PHONE 3.Nowdefine asub-integral likeso: Butfirst redefine themeaning ofallvariables bydividing them byp.So: \ c+) =NardaCatt(Sag-x-N) =TER) Jon . .a/K . 2 T=+h(ea){te-T(+*)| 4.Tt remains then only tocompute theintegral 1): \ . ae T=Dax(omer)=BaGer0)] whe deasia)+ \ =2\dxInoa) ©}(Raresa)eQ°wr Wat Oy,=£(Kev) +5]AGN) Qe .? -2- ) ( RxSseGener=\Gebers ~trae)||$ 0 =(leaLaGaa)~ adaa -\ QSTeey= Pe[(read)sucise)=aha;-1| = vt aye=k(mb ERD)&EACaw-A*) 5.Therefore, hereisacomplete andexactresult: . IN4 Te4intEan)A[Gea’)aCe)=a;Sana= iiat hast Ay=EGA) Elan a3: day=E@C-I-N) Bh)20a, ND 6.Nowletslookatsomeiinteresting limits: first, takeLAMBDA captoinfinity, toget the UVbehavior: ‘ cs ie ws yt a=“V8Te a=New- TE=Ah sagt$0 ae A) an =Te4h}Ean(ME)—OREO)\/* . . 2 e 2 ©7 ORBAN) tf Ogre = Saaledt)opcua)Jou[rsmeke |=| ee +ait)¥. arta=BESn)© Dn(isSRR)*aif Thisisthecontrolling termforlarge|\..Thenextlowerterm iaisagonstant whichI-couldfindifIwanted.Ihavereplaced Toxpintdn(4) thep*scdlaingfactor.Youseethelogdivergence asr expected atthestart, andyouseethatnothing veryspecial happens asyou gotothe mass shell, Finally,letsholdJLfixedandgototheIRlimitwhereXO.Thenifindtht:° VaanA=LOWS ah(eraety «oY* =(ews ty2awe Latarws) +or) =(\-wt)=Gea +0(s") dy>[MEANAVee+(5)| =naiepeEE=H] Qi&=~X(Fa)xO Qy=he-\+) ° 3L=4it [vai- (25)mn) [pang +(oben) (8"(Be)) (weena)” a Thia\says thdtfoxtixea/N fn, . /\ TL=cwranyatn allNe .7 —3— 7.Suppose wetake ourgeneral result first tothemass shell, holding |\.and\ finite.Notethetmassshellmeansm2=1,Thenwegets cua=kite ~ we - a) \Rega =8,-\0@) Os=-EXet}sope ="ae a mat) Recompute these with care please: aL A= Ehee Chan’) ASV =Jae » yt a Ss a=£LN=Aan) |=EAT Or)] . athhi-dire Ge)tte G4] = ~& 1 a Va,= -\4ce=ah] Rte -itte a\h\+% OBa=ePDO |=eto lar] : Rot . akJK=~adi~2e=arinXY=a Ssjaxth)5 \Ba\=C@)LACE) suosigtoQa) Joy=24| eVdal—Cinta(id) Soeven ifyou gotothe mass shell first, this integral isstill not IRsingular. Somehow itdifferes from the true spinor-electron integral. Le sev SoaksTitel. (200noe) Ot\e*3RO URT L@eeySh - : ©Pidworquae&combingdamemvente wee wo \ < ok See =Ds Ce@oa LEYLaas} Hoare) Y[atecage yr =©CougPda Quraquenr! coe 2 WA ARAAD =Knacsadleg+0-2) Co .=(k-weal ~Ose+Cir8)_—“URTstee)|—_ i OWE SebMWogoo Cee See ey ee - B=LL \eeegeet.Ad re)SEH)y_yLotecosy oo.QE .=— - @Mtr emunsim Sedogh[Leiaon'kwow ceeee ee : = ants.\fae ae ance PeePaes . “Me fede, aeyent atoso670.Ghee2kdKmata dksotetetar [retary] =Cfer ete)==feerec-s] yas. me ae bah Pt \pae. Leweens . 7©Sa~o=232-2) soLeesan aESseG-esea =&(-e)Esaq a -.Owe. Ro oe aaepr © -To Wigay cey7P" [eae Bore}: Jesus. oe eee Ges yRE . ~ <a ay —— Ste eh.Teais-@) Peay CA).BUZ,846+2-2) =in®(gteG-eil 8Gkgh a a _Bee Sot axGfFSay Bang) (a)seaeany yon nd = 2 DLL. som peep ok ‘ Boxe=iwQD. petere) poweed Yreee) Tess Ken yrEn) 8 a =.joDaag) aye Cae ae Nn ICONS) eS age Se: . .ysGes)gt- ©Deemedgaa (Ba)aBaltladeMEpapa BAL ; —..Te Yaux®GexS* =_BL)-#,\-2] |5 $ - T=ADEse)POexdPi-6) (3) . _@ pase Pa-ep AVL LL ©Qemmouy! ee ee i : oo ey (ety Po Pe sae eadpeg),Plas @* ee_ _ POs POLE) PQ-8-p) a Le —.—Eiapenson Entered # a Ops SiO.MhiniaabldigAnayaSeallgeWAcopes Be ~ey NeSven)\ewtiovaa Loa Tle QaaOR) EMA MEN) \ \-K Sy aapLas GOLmSGT 2 Bk RSRE aage aaa So) —Ghaue Nak =-itVA O—OCSS @_TW2cm)SaesesGR TTSakme fe ee a Reya (Ed TUN =ixsax.. x-asy\G-an ne “i en A,a £E(CV=UOe)Vo wee OANa Ged) etn) im) pp pe ee eee) ea(A ne Nuon mrboian a4omdanan“oowsank' vinA,Graane:A=SongerDoomollStanoraniab er.QasanLenteckmle. cee a A © | ™st -2- Oa tyLeste yo -eT wan aonn ee wo.2|Sie CGE) Saeco 7 ©OeeeeRbeEEG) =aeRO .voThadatansa)asde~)~fino) 280 —@Qustiaadk ©Now,wodameSwomOsk ~ i Mo — ee Tew\=Cin®)eo 8 a oe tertngee al Qe oD Bgtbs=ewe) Ea gy) —lomaounsladeean AT/aqhchean. tena Bustase: ~NRwet)=ATASaatAa OO) TS Tate, ©ake daesRaat ee Qt (YA) a =WES ee Qual ag aD. AGH) Bamye tt. . oe ~3q* a Nes—(fiesBetae#8GS) a kehPGR).eae )ee —sta re in ee AT) =City, dxQa Cefon) =(-it')\dx_t |~dary .7 —3~ Neofra. ee ee Ee) ~4-218 “3 oe. .Salemi)=Bane)tmWES)“AKeSSere — ni) extn(iBiNew-du(@) seireideuo ok=Bf)=ig-= ee re}STdetedeoeten Oa ae OPPS aisve] Sab eeog sy Le SO AT) (tt) dla)BanCet |oeseaoo) “~“age.eeyOMA 36S): 2Rey=eBORE]Eee ST] oe Ltt) WOR gteeTAS oN aa) —oo(os =0(wh)ee Sa ee HewtaMwAso —_— CP) dgBSE Mm ee Miksa.Qnak,alderdecnacoumlencotaybtuuu, Adancenuitdepend DadoOrdeordir.dvGEDDuinQuaaysa feLAS Commins Stasve farricues /padoraten oclat 8 7roanyonoS P ° ~~ y 2 a =e -a weak B -BBs wak a_iKX 5\ms Sr,oh) wal, r 26ms yey ‘ @) & Kt \Qms yey,ww " A 260ps peen? ‘ 8 = {90¢s Aw ‘ raat 150ps on * Ks BAps 20 “ s 80¢s Po " v 80pigs we EN Gy6 ui 5pps. oT eM FM 6 — Paremuteras——— Hoaad+prnaes - ®PEsaye : -2 oR- 2 pve ve Ga a an . Se wie Weloosne. --. - _— eee et ae>v5 Le Cee eee @ ao Lyla ae .. Jae Gotearags Wwe. WIPERS ee.oo.2Koy age eee ee fae ee ee . Wowey.Aods,-=-Wa§so._ . Ro.8Ov) ALA ggg UG,Annmdeateat cums ye - @® mage, Gye Le Menage . .esas :2.OnaTREairtelLp omy .ene rari an . © perm. sehen : D Rasear v.\ssigr ne eweer lasl=o, lens. - oe. ee . 3.Wn. load, dsad@. . . - : Yoo Yale Gee dL ee SUB) paar . oe . Loe - Qabitebe _eunretkodde lyygatsee +Aatabim 08 -- _ Dagion's. Sg*SUL . oe - (@AddnWewugn atekion (GodeueAT) a~BEBO) _decay Carasteobedasa). DFOLe. \e)wocspian,spun GamsretA 2.~=rerdonunsncr andgheeSugawara a Seen ee —---@O_ Ko)eseee eeeeeeeeeea ees oybuntyeae KATY ee J So —--—feaatseil= GalleTastee Fa=Vt8Q) .. . . Kxety) “Ka - oewiuhalowed) ~ooo @KS¥e Kyks —E- oeteh waeHK. Datedronpoiin” eine peancnalt _7 . .J _a : 7| Comming WeakInteractions |Y73, Reread Feb1978 - o/ 1.Introduction. Briefhistory:-in 1932thebetadecayspectrum wasseento‘be continuous; this seemed toviolate energy conservation since reaction was supposed toben-Ppe. Pauli suggested aneutrino, then Fermi made atheory using thisideain1934. Fermi's lagrangian wasjustki,where J"issimple vcurrent,for nucleon andj,isthesimple 4,current forelectron-neutrino. Ofcourse this does not show V-A form, but itworked for beta decay. There are many unstable particles. The ones which are stable against both strong andEMdecay live 1071°seconds which isalong time; ie,awidth of 10ev.ifthere issomeEMdecay, likePI-2%{ thenexpect shorter life10716 sec,orwidth of1eV.Strong decay particles are107° secorwidth 100MeV. In1958 Feynman and GellMann came upwith our present form ofthe lagrangian; ‘ recall bythewaythat itwas1953 when Gelllann Nishijima posulated thestrange ness quantum number toexplain the low production rate ofsingle strange particles. Thus the Feynman GellMann theory had strangness changing ourrents etc etc. Infact, theweak current issum offour currents: electronic and muonio .Surrents, anddS=0 anddS=1 hadronic currents. Each ofthese hadronic currents has aVand Apiece, whereas the leptonic currents are each VA, Separate electron @) 224-monconservation ignoted, endthehelicity story isnoted, Figure 1.4on‘Tyraecay makesthecamplete PalldCviolation ofweakinteractions veryclear. Neutrinos are ofhelicity -Eonly; electrons are also, tothe extent you can neglect their mass. Ie, left-handed neutrinos. Bythe way, semileptonic decay means bothhadrons andleptons in-silvitheretate. Alsobytheway,the0% paradox was that the Kdecays both into 2and 3pions which seems tobreak parity conservation. Thus Lee and Yang. This was 1956. Feyn Gell was 1958. Commins has huge table showing all process types. Blectron neutrino elastic scattering has now been done Ithink. Muon decay isthe only .all-leptonic decay process. Semi~leptanic includes beta decay, neutrino absorption (inverse beta decay), pion decay, the pp process. Then you goto semi-leptonic stuff with strange particles: kaon, sigma and lambda decays here. Also, the direct interaction of the hadronic weak currents should effect npelastic reactions. The dS=4Qrule applies toreactions which are strangeness-violeting and which are semileptonic. The changes apply tothe hadron involved. This rule isbuilt into the Cabibbo form, _Theax\=#rule.Thisrulesaysthatwhenyouhaveareactionwitkahadron ©) —schanging/as{= 1,itsisospin mst change by+$unit. Again theCabibbo form will incorporate this rule. . Since K(long) goes into PI-PI, wehave GPviolation, hence Tvidolation. This still amystery,, Gh2: Muom Decay. Theamplitude Miswritten in(2.6). After squaring, you have the produce oftwo traces Tland 12all ofwhich boils down to(2.26). Notice theneutréno projection operator (2.22) isjust¢since nohelicity option, eo whereas for‘lepton like electron you have mass and spin. Next thing you want to doisintegrate over thé di-neutrino phase space weighted bysome momenta as in(2.28) .Byfooling around youdeduce thetensor I,which iethiephase space and you are left with (2.37). Finally you gotomuon rest frame with the mon ‘polarized along the plus 2,décay rate is(2.40). In(2,40) quantity€ isthe fraction ofmax ofelectron energy. You see that _the degree 6ffront-back assynetry (confficient ofcosd) isenergy dependent as plotted onnextpage.Sincethisformula ‘neglectes electron mass,youseethat *pate is zero into electrons with positive helicity. ~ . : Next section shows how Lisnon Cinvariant, Then comes the piece about trying toconstruct avery general amplitude, more general than V-A. You now . have many constants Casin(2.449) which issame asconstants aandbof(2.45). Théfinal decay rate ingeneral schewe is(2.53). The4parameters called 9,7, Yana§aretheoretically givenfromtheconstants asshown. Thefunctions g(x) and h(x), explicitly stated innext section, are the radiative :corrections. What aré these Michel parameters? Clearly §controls thescaleofthe fore-aft assymetry term,andis-1intheV-A-theory. Parameter ¢controls thee *shape ofthe igh end ofthe electron energy spectrum inunpolarized decay (where thecdsterm averages away andx=1roughly.) Then 4doesthe lowenergy end, andfinally $idtheassymmetry shape parameter. Experiment shows that these 4parameters are all consistent with the V-A theory, see page 49. Next subject isthe magnetic moment anomoly 3(g-2) ofthe muon. The first order théoretical preidetion isjust{/2# “the same asfor thé electron, but the order 4”ana4?values aredifferent forelectron andmon. The Garwin-Lederman method isthisf you stop mons end make them rotate attheir Larmor frequency like a‘compass needle rotating. Using one ‘fixed detector and noting the muon arrival time, the time distribution ofcounts ‘in the detector tells you a)the gyromagnetic‘g, ie, the anomoly a,from the observed frequancy ofthe decay distribution; b)the magnitude ofthe assymetry. Obviously ifthere were nofront-back assymetry this would not work, This experiment has been used tomeasure theassymetry shape factor §mentioned dove. AG-Lexpéiiemtm can also beused togetBy, themagnetic moment ofthemon, Butthis type exper is not good enough toget the anomoly because you need accurate muon mass beyond ability tomeaure. Whathoppens. infactisthattheanonoly “ismeasured inanoth() experiment very accurately, then thejust~mentioned fameasurement gives aacourate massforthemuon, Theexperiment which getsap,isthefamous onethatuses muons rotating inaring.. On a : -2- Finally, Gene notes that youozhmake long-lived’ muonium atoms ‘te. Aswith hygrogen, there! issomehyperfine splitting Which youcancaloulaté‘in‘GED.Unlike fe) hydvogen, there isnbfhing unknown here(1ike préton shayie andsizeeffects) Ofcourse you get the theoretical résiflt very accurately, and this isused as ameasurement of%.Another measurement of4comes fromJosephson experiment. Now{ismmeasuréd two ways and they agree to1ppm. Bythe way, Josephson isthis: you put avoltage VonaJjunction, and you should see ‘anac signak atfrequency W=26V/K . . .. 3‘Recall that the muon decay /N/* =64G" (peq)(p.q) .Byjust changing names . around alittle yougetthevarious seattering andannihilation. processes, ie, anything youcandowithtwoharrenta. . ‘ +.However, there -isaslight difference inthecioss sections. ForLy-process, youfindthat theDCSis,onlyenergy. dependent, noangular. See(3.19). However, forthe1% process there issoma angle dependence intheDOS. Reason igthat here youaregoing through a.vector resonance andthis’picks outJ=0andJ=1,whereas inthe$yprocess jouareexchanging thevectof andyougetthegontact effect 80 only J=0.. Obviously, with either ofthese elastic cross sections you.areintrouble’ ©)_withunitarity because only1or2partial wavesinvolved andefergyiaupstairs. Ofcourse this isjust first order perturbation theory, you might. hope that higher orders would fix things up. But higher 4-point interactions just make things worse. Evenwhenyouput.intheWboson, youstill violate ‘unitarity: insay theseroth partial wave, although thetotal cross section goes to,a constant instead ofblowing up. Theproblem with theWis that.the k"k” term in,the . + propagator-ruins your ability torenormalize. Sothe Wimproves unitarity a little butdoes notdoitcompletely. Bytheway, TheWthedry givesBgwhich differs alittle from’.75 oftheV-AFermi. theory. : Chapter 4: Hadronio Weak Currents, ‘ Previous chapters were: introduction, mondecay, other purely: leptonic weak processes. Forfirst time wenowlook into hadronic decays. FirstsubjectisthefamousTyr_piondecay.Yourpiongoesintovacuum _through ahadronic current: this current mst be A,type since axial andno strangeness change. Whyaxial? Because only 4-vector around isq"which ispolar. Thus matrix clement ofcurrent mustbepolar, butpion ispseudoscalar, so”current operatormustbeaxialvector.Piondecayconstantf,isdefined.Youcan fe) termine this constant from thedecay rate ofpions through this mode (which,by theway, isessentially 100% efcharged pion decays). * . Next, consider the,,decaymodel.Calculate insamewaywithsame constant! Result differs from the[ys onlybymass factors, hence youknow theratioasin(4.9);branchfraction isamere1074,Although electron has(#)more phase space, this mode issupresséd because youaretrying toforce electron _into wrong helicity state (pion. hasnospin) ~ Nextcomes thedecays Ky,andKey.Compute inexact sameway,except constant nowcalled fy.TheKy, modélxis nowonly 63%, andKpiswaydown forséme reasoh asbefore, even more so.Sofrom rate into theK,,mode youcan compute fy. Theratio offy/fq istand =.27,Cabibbo. later wewill learn vthat you canallways associate sing with non-strahge and cos@ with strange. changing orstrangeness violating reactions. Nowmoveontoother decays. Consider those likeW->-w°E'Ve which is called pion beta decay. Nowyouneed adifferent matrix element between two pion, states. There arenow two4-vectérs available, both’ polar, socurrent chosen will.be V,. For’each 4-Vector youwill have aform factor, usually called £,andf_andusually defined with asin@ showing. Digression: ifweak hadronic vector cirrent israisiig/lowering part of isotopic cirrenttriplet whose third component istheI,piece of.theEMcurrent, +(isovector t#iplet hypothesis), then what éan you say about pion beta decay? -Firstofall,sinceEMcurrent isconserved, thesecond EMformfactor vahishes (>) andaswewell know there isonly pion EMforma factor Called FiIniso-notation “you canwrite EMcurrent mat}rix element asg*1, withPF).Tmstoget thevector, fionstaange ‘element V, youfeplace with I,say. — Beware: itisthe1;“compenents weighted bycos that aréassumed to . bethe isospin partners ofthe EMcurrent. This isCVC, This, the constants £,and£_mentioned above forpionbetadecay are: f,=Mer, addf=0. Thus, itissimply the EMpion form factor that enterw inthe weak pion *peta decay, the linkage being CVC. Bytheway,pionbetadecay is1078 fraction ofcharged piondecays, Allch pion decay modes areweak andinvolve nelithinos. Incontrast, allneutral pion decays arebasically eléctromagnetic.” Thereason ofcourse isthat fora+fdecay youhave’todumpthecharge onsomelighter particle, andthe’only candidates are ieptons which impliés a’weak decay. Imthejanemodeasabove youcanhavext. Yourcurrent isstill . ¥,strangeness clanging, socosisreplaced with asign.” However, thedifference nowisthet’ Idont think CVCapplies toayycurrent, onlytoaVycurrent, so IT’wouldguessthatyoustillhavetwoformfactors todealwithsificeno~ Q“¥nownconserved current. GCofcourse doesnotapplytooeither A,orA), -3- Pofinish offpionbeta decay, Gene goes ontocompute therate. Since theq"transferred offthepionissosmall,[email protected],ie, O nem pion formfactor normalization. hen your pionbeta decay rate involves onlyG*oos*9 asshown in(4.63). Now, wemeasure Gfrommuon decay, and wemeasure @from ratio ofkaon/pion decay constants, sothis result isapure prediction. Agrees with experiment to10%exper error. So far we have done Meson to Meson +Pair, What about Meson to Meson + Meson +Pair? Nowyouhave 3polar vectorm since 34~momenta, andyousuddenly also have anaxial vector oftheformé.,. kp. Thus, ifyouaretalking xsttree yourA,current has3terms, andyourV,ourrent has1term, "so nowfour form factors toplay with, atthestart. Gene does notpursue this decay verymet? since aSe1there is.no CVC,sowillbeamess. Geneomits things that are amess. Fraction isof course way down onthis. Nawwemove tobaryon stuff. Regular beta decay isBtoB+pair. Nowbaryons havespincohavetodealwiththat. TheB/Bmatrix element must always beoftheform U(p)..... u(p"); this isjust analternative tothe M-function formalism. Byplaying with things, youseethat with your Dirac %anayourtwoavailable momenta, youcanmake3polarvectors and3axial 0 asin(4.29), withf,andg,asthe6formfactors.Now comes CVC again. The EMcurrent would have the 3f-type form factors sincecurrent isoftheV,type, ButEMcurrent conservation reulces thisto theusual twoform factors, Aswritten in(4.39), thefirst form factor gives charge plus usual moment, andsecond form factor gives themoment anomoly. If you procedd towrite this EN matrix element iniso-notation, you get (4.45). Whereas thecorresponding pionmatrix element forEMhadonlyanI,,piece, thenucleon EMmatrix element hasan1;pieceandanIs0piece. ThisI=0 Piece arises from the hypercharge yofthe baryon via the GMNish formula, whereas pion had noy. Soexamilte (4.45). There are2form factors ineach term. Only thesecond termwillbeassumed byCVCtobetheI,pieceofatriplet. Sowhathappens? Well, for the vector piece ofthe current matrix element (which isdS=0 sincenuclear betadecay) youcanset‘fy=0bycurrent conservation, andthen Py)=f, andBy=f) asin(4.52). Ie,thwtwo vector couplers getconnected with EMform factors, the third isset tozero. You are ingreat shape, except you dont know the axial form factors. WhatareargumentsforCVC?TheWweakformfactorf,isexperimentally le] found tonormalize outtounity (cy=1),just like theEMform factor. I think this isthe main motivation. . Now, what are second-class currents? One presumes that ake, ofthe 6form factors appearing innuclear beta decay, 4are completely caused bystrong ifiterdcfions (a11butflandgl).Thus,youmightexpectthefourother r?)terms tobeG-parity invariant. This assumption combined with Tinvariance (which forces allformfactors fobéteal) yields result 6=0. Theg, term, ifitisexperimentally present, iscalled asecond class current, Assuming nog,,thebeta decay current nowhasthetwoEMform factors andthestillunknown g,)and&,axialformfactors. Bytheway,@,iscalled "the induced pseudoscalar form factor” ofbeta decay. . Finally, wehave Goldberger Trieman, Consider nucleon beta decay. We havejustargued thatintheexial pieceyouhaveonlytheg,andtheg,terms. Theone-pion exchange model isamodel forthe&,term, Thismodel ofcourse involves the‘pion decay constant fy (ofweakinteractions, appearing inTe) andthehadronic ion-nucleon coupler called g,here. Thus, themodel here says precisely (4.74), where you put inalittle adjustment tokeep inmind that youshould have used ¢(q°) asarunning PI-N constant. Sowenowhave statement (4.76) withg,nowréplaced by(fy‘eye Next,lookbackatregularTyedecayandobserve‘thattheA,current(which was involved tlere) isconserved to the extent that m= 0, Now since weare dealing withthesamecurrerit operator A,(butbetween nucleon states) wewill(*)assume conservation ofthis current, PCAC, This ofcourse makes arelation detween g,‘and (fey) +But@,=-C,=1.23 asmeasured, Youcanmeasure each ofthese three constants separately and the result works! *What does Goldberger-Trieman prove? Itisevidence for(PCAC)A (g.=0)A *(OPEmodel). Another waytosayGTRisthatg,isrelated tog)- *Obviously, after all this preparation onthe form ofthe nuclear beta decay matrix’ element, weshould dobeta decay. Next chapter. Chapter 5:Nuclear Beta Decay. Thisisa‘long andcomplicated subject, butanexcellent exercise inapplying what has been learned inteearlier chapters. Here are the basic facts: you are - dealingwithahadronicweakcurrent.Inprinciplethatmeans,sincemoolear_,aimplies baryons, there could bethe6formfactors £1,f2,f3,g1,82,g3+ Bythe aN .way,theformfactors f2andg2arécalledthe"inducedtensorcoupling” we and, Iguess, the "induced pseudotensor coupling" because these mitiply theoP?ana#45 terms inthebaryon ‘weak current. Ofcourse theg,termis alsocalled thesecond-class current termandissupposed tovanish byG-parity @) argument. Thef3andg3arecalled induced scalar alldinduced pseudoscalar. Theworkinduced means theyarecaused bypresence ofstrong interactions. ~4- Inthe pure lepton world you have only fland gl non-induoed couplings. Any +othercouplings are"induced". OfcourseinCVCtheoryyousetf,=0andyou e *identify flandf2withyourEMformfactors. Ata=0fliscalled Cyand tpiscalied¢,forobvious reasons, .Inaplane-wave theory, the beta decay amplitude isgiven by(5.1). There you see the usual leptonic V-A current hitting onto the 5~term hadronic current (recall £30), But inthis decay you have totreat the nucleans ashaving definite wave-functions, they art not plane waves. Plane waves are only useful when something isnon-localized. Anucleus is*localized andhassome definite wavefunction for neutron distribution in it, So'we can rewrite the amplitude “putting inwavefunctions andreplacing themomenta (which appear inboth imtineed tensor ters andtheinduced pseudoscalar term, ie,ing2,f2,g3 terms) withposition gradients.’ fhesymbol jy(x) isshorthand fortheweakleptonic current. : Nowthekeysimplification inbeta detay isthie: the@values (energy released) are onthe order of 1MeV for all nuclear beta decays, Since’nucleuses are mich heavier thanthis, theydontrecoil much; Thus, youcanflakethemnon-rel and use thelarge-small spinor reduction. Also, obviousiy themomentum transfer +forthenucleus(theq”whichappearsinthevariousformfactors)isneatly0. fe) Thus,ellformfactors reduce toconstants like£,(0)=Cyadd8,(0)=Cys Moreover, onee you write out the various terms inthe: reduced~Pauli language asin(5.8) ,yourealize tateachgradient: appearing is,in momentum space, 2 butthis issmali, Sointhe "allowed approximation” you Justdropallgradient terms,But,theonlynon-gradient termsaretheCy‘andOy *texms; the £2,g1,g2,g3 terms then all goaway (note that (Eiw) terms also go away since this isq°); Therefore, onceyougointo the"allowed approx", you canlearn nothing atallsbout these other. formfactors. Youcanlearn only about Cy,Cy)andGy=theweakcoupling appropriate forbetadecay. So, everything reduces to(5.14) which shows thaamplitude (called L): |88toterms. “Basically, U=(Goos0/fi) (CAIP59(0)=©,A#7-5(0) ),where 41) isthenuelaer waffefunction value ofoperator “Z+( andigthus always some numberlike1,00or{2),and4#)isthesamethingbutwithaPauli@operator inthere, ie,this issort ofthenuclear spin. Forannucleus, yousumover “all nucleons andyouhave little tfoperators whichalléwthepossibilityof "eachmucleon converting. Ie, you get acoherent sum ofamplitudes asthe total amplitude, Theisospin stuff enters just asnotation, ie,wavefunctionXis 8anisotopic vector, doublet forneutron decay eg.” ° ,Sothe big thing welearn isthat there are basically two terms, the CVand the CAterms. These terms are called Fermi aiid Gamow-Teller asweshall see. Sinceweassumedthatleptonwavefunctions areplanewavesovernucleus, eo Geneargues that this means L=0 for leptons. W1l thats not right. Really the variation ofthe plane waves ofthe leptons issosmall over the nucleus that you appréximate j(x) byj(0). Ie, the actual @ecay occurs inasmall localized place; roughly speaking, the leptons originate atthe same spatial point and therefore can have noorbitral L. They can have S=0 orS=l, and hence leptons have J=0 orJel. Ifleptons carry off Jal, then muclear spin must have changed byJ=1,0. What Imean isthie: J;c Jp@ 1. Thisrequires alittle diression. Consider themtrix element 45(¢|%7. ThePaulicurrent operator transforms asaspin-l object. Thestates iaddf also have rotational transformation properties, ie,nuclear spin J,andJp. This matrix element will vanish unless JC J,@1 .Ingeneral, ifyou have atransition J,=1toJ,=1, this matrix element will have some value, albeit unknown, Theother element will alse have avalue, ie,GUWi) where I have omitted thet*inboth cases, Thus, this isa"mixed FandQTtransition". Ifyouhave J=0goes toJ=0, that mst bepure Fermi, because then theKF) . vanishes from rotation group theory. Or, ifyou have Je0 toJ=l, then pure (GT,sincetheFvanishes fromsamegrouptheory. YeSo, depending onhow the nuclear spin changes, you can have F,GTormixed transifitions, huxkeyxiaxtkxt In(5.14) these facts areobvious without any talk about the lepton L, Its just that the L-0 disoussion explains why there arenodJ=2 beta decays, eg+ an Another detail: ‘electron wave function ,since itischarged, is"distorted" bythe nuclear electrostatic field, soyou need F(Z,E) tocorrect for this. This correction isafunction ofnuclear oharge Ze, aiid electron energy EB. 80with this correction weriowhave Jamp/” as'in (5.17) with (5.18). Fact: since leptons always have the(1+%%) factor, thelongitudinal polarization ofany electron emitted inany weak interactiom, such asbeta decay, is<v/c. HowdidInotkriow this’ Intheneutrino limit youof‘course get~l. ‘This fact has been tested experimentally invarious beta decays. OK, now put inyour amplitude, square it, dothe traces eto and get the rate shown in(5.30) which “applies only forJ=$toJef. Discuss this rate: _ You see the obvious phase space and Coulomb correction F(Z,E) already mentioned, Theoveral ratenormalization shows 0”aswellas$=(5.31). Inthebrackets, youseeal,thenallkinds ofangular terms. Thetheoretical coefficients ofe these angular terms aregiven asa,4,B,D; theidea istomeasure these coceficiants -5- experimentally then compare. . 6 Butfirst, ifyouintegrate overeverything togetthetotaldecayrate, none ofthese coefficients matters. Suppose you integrate over everything except the electron energy, The (5.40) gives you the theoretical "electron energy spectrun". Ie,youlook at1,000,000 beta decays andrecord theelectron energy ineachcase,this,givesyou aspectrum, Experimentally, whatyoulearn from this spectrum isthat ithas the right shape. This fact then tells you thattheemitteed neutrino Vemusthavebeen,lighter than60eV,orelse spectrum would have been wrong. Asimilar beta decay experiment which emits apsitron only gives && 4.1Kev. Somch for the electron energy spectrum, Now integrate this toget your total decay rate, The relevwant dEintegratign iscalled the Fermi Integral andiscalled f,,Thus, thelifetime isgiven by(5.43) andinvolves only Ge:B,$. This implies ofcourse acertain" comparative half-life" which isthehalf-1iRe uébhs@ insone sense ahlef life; this thingiscalled"rt" andft=(5.44). Thus,youcan,determine theproduct Gg§bymeasuring lifetimes. Soifyou somehow mew§youcoulddeduceOp.So,golookatsomeJ=0toJ=0 transifionswhichareofcoursepureFermi,thenyouknow4t};butyou a)donotknowC,yet,nordoyouknowL*7, theWigner-Eckart reduced. 4#>. But, ifyou gooffandmeasure thefront-back electron assymetry inanother experiment, youknow Aof(5.33) andtherefore §of(5.47) atidtherefore$. Thus, you can measure @for various decays, result shown inFig. 5.2. Natice thatallG'saresmaller than frommondecay, evidence forcospresence. OnceyouknowGg,yougobacktoregular neutron decay where 47=3(since nomessy wavefunction anymore) andyouconclude that C,=1.23, thefamous eresult. . Bytheway, neutron lifetime is11minutes half-life. _Nowlets goback andlook again atthose angular terms in(5.30). Note that ingeneral J,,J,case youhave toaddinterms onbottom ofpage 110. Look atthe terms, The aterm isascalar under papity, asisthe Dterm. But the Aad Bterms are pseudoscalars. Thus, ifall these terms are preserit, you have aparity violating interaction. Bg, parity invariance would tell you that A=-A, Similarly, ifDisthere, you have Tviolation. Confirm this. bylooking at(5.35). : 80, first angular thing isthe "electron-neutrino angular correlation" as seen in(5.52). Inthetheory, a=asshown interms of¢.Thus, foreach A decayaandG(orx)msthaveacertain relation, ie,theymstlieonthe main straight line ofFig. 5.3. Experimentally youmeasure aamt bymeasuring electron pandrecoil ionpanddeducing py.Asnoted above, youseparately measure §from thefront-back electron assymetry inpolarized decays, the coefficient A, Since points docome out onstriahgt line, you like. :Nextsection dealswith5+(A,B,D terms). Hereofcourseyouneedto () use polarized nuclei inyour experiments Obviously, the fore-aft coeffient Ainvolving only the electron iseasiest to get: Since this ispseudoscalar term, thefactthatA40inCodecay started theparity revolution. (1957) Ifyoumeasure AandB,you'get aseparate detenmination ofC,=-1.26. Dalways comesoutnear0,indicating thatTandhencePCare@®inbeta decay. Ofcourse small uxxaxxxinrxummams inconsistencies can always be attributed to neglected terms in the allowed transition, Now, when you elevate back up fro the Dirac-Pauli level to the Gamma matrix level, you realize that, after dropping the various terms, your true amplitude for the hadronic current isthe same asfor the leptonic, ’except the (1+3$) isreplaced with(14¢,4y ).Butdontforget thatthisisonlyinthe"allowed approximation", SoitisaV,A theory again. Ingeneral you gight have expected P,S,? terms (note: weassumed vector—vector atthe start, way . backwiththose formfactors). Theexperiments odmbined withgeneralforms of course support but donot*prove the V,A weak hadronic current . ‘What the hell is"weak magnetism" ?This isaneffect which involves the fpformfactor "induced tensor" “termwhich iscompletely neglected inthe (*) allowed approx. Ifyou include this term and look atcertain beta decays ‘which tend toisolate it, itcauses alittle correction tothe electron energy spectYud, ‘This correction involves £,(0), andtherefore (recall CVC)‘the *mucleon magnetic moments. The prediction is(5.70). Measurements ofthis shape correction agree well. Nore support for CVC. InEMcurrent, you would call this fytensor term the“magnetic” form factor since itgives the magnetic anomoly. Thus, this f2term iscalled the "weak magnetism" effect. *Sdmich forf,, Another corrective term isthesecond-class gp.Bvidence forthis term isinconclusive todate. Nocomments onthe83term. {aes“Pa.Faw.Kanscheene,camedasacdion.MabeAnadditiontoMonta3,35, *tFayGT)ReaPe.. Ghapter 6:Nuon Capture, Basic reaction here isfa+p~yn+Vp. Ifyoudid this atFNAL, you would get and compute across section. But, this chapter concerns instead this process occuring asfollows: amuon iscaptured byanatom and replaces a18electron, Since muon mass is,larger than eleetrop, the Bohr radiusisverysmall;,ie, thiemuonspendstimeinthenucleus,sothecapturefo) rate issignificant. Mymain difficulty inunderstanding this isthat you get arate, not, across section. Ie, somehow this rate islike across section =6- multiplied byaflux; the flux iecaused by.the presnce ofthe mon right there intheatom,Iamsousedtoplanewavesandcrosssections, butIknowitis fo) possible to.fold imthe various particle wavefunctions (if they are not plane waves) toget the total transition probability, which inthis case is,a capture rate. Infact we had todo some ofthie wavefunction folding back innuclear eta decay also. Notice that this folding gives you avery particular rate: from som initial muon, 18wavefungtion and some nucleon wavefunction, inté some particular final nuclear wavefunction; the final mon Iguess can excape in aplane wave; itisthe only plane wave here among the 4particle states. . So, forma (6.5) really gives you.a rate into that final nuclear state; you need only, to integrate over neutrino phase space toget that rate. Lets take time out to work on the matrix element for awhile. You see in(6.6) that the amplitude isthe spatial integral ofthe imkmx overlap of the four wavefunctions ateach point x;ateach point, however, the amplitude isjust the leptonic current dotted into. the 4-term hadronic current (here we seeonly f1,f2,g1,g3; ‘theg2second-class. term isdropped. )Thekinematics is such that you are looking atthis 2-tor2 process atthreshold inthe initial channel. This fact means that variables arefixed. Thetransfer t= q*offthe nucleonisfixed.as shownin(6.13);the,neutrino energyisfixed,see(6.12). 6 You are inthe oms frame of course. Thus, all form factors are evaluated not atq°=0butatsomespacelike fixed point. Notethat%=4isthetransfer, ;not the mon 4-mementum. So,from your EMtheory andCVyoudeduce what f,andf,.are at.therelevant points. -g)=C,to.999; theonly other constant isthet g,. Youdefine g.=ng, swhere m=mon mass} thenuseGoldberger Trieman togetthis g..=7C,approxy. Nowyouhave agood idea ofallthe constnet involved, assuming nog,term. The mon wavefunction isgiven by(6.18), has only upper component because solow energy. For the neutrino, pisnot negligeable soyou have the usual (6.21). Note that this @isamatrix inthe lepton Pauli space. Meanwhile WAIT; maybe the reason the muon has only anupper component isthat experimentally youareforced toaverage over-monhelicities. 50% neverappears anywhere. In.contrast,, the nucleon before ang after spin.does appear, asdoes the final neutrino spin; ie,youcould propare-your nucleus inpolarized state, before : yon shoot inyour mons, . . -_ . When thedust settles, theamplitude you get isshown in(6.28). ,You, have thatfolding ofthewavefunctions andaspinormatrix,element ofanperator given 8.in(6.29), Theconstants GyandG,ang.6, areallknown asdiscussed above. Youmust remember that %operates inthe,nucleon 2x2space, whereas gin the lepton, OK, sothe amplitude isnow clear, Toget rate, use upthe eldta function and exhibit neutrino direction integral asin(6.35). Bythe way, notice in (6.33) for the general nucleus that Gene has gone back tothe isonotation, so younowhavethreespacestoworryabout.Thenotation of(6.36)ismtquite(*) clear; the nuclear wave functions f(x) now appear asstates, soaspatial integratio is implicit; you even see the xinside the mon wavefunction, Now theToperator does itsusual lowering function, Soyougetanuclear {jyand 4 typematrix elenent, asinbetadecay. Asnoted, these nould beexactly the beta decay objects ifmon were aplane wave; the exiting electron in beta decay isaplane wave, but here the absorbed mon ésnot. Now comes atrick; you want to sum over available. final nuclear states. Ifyoufudge alittle bypulling bracketed factor outofZig,thenthissum goes directly onto matrix elements squared asshown in(6.40). But then you can use unitarity tomxx absorb this sum; your squared matrix elements then contain double-sums over all nucleons inmuoleus, each has adouble spatial integral, sonon-local. However, ifyou execute the neutrino direction integral before doing the.double coordinate space integrals, you reduce the latter to adouble-radial integral asin(6.45). The final rate result isin(6.45), and the double nuclear sum isstill inthere ofcourse. Ifyou can.ignore .the non-diagonal terms inthis sum, you simplify to(6.46). For light small nucleimyourmon-nuclear overlapissmallsoget(6.47).Simplifythissomemoreto_*)approximate endresult (6.48), which shows that Z4rate increase! Comments: for larger Z,your 15wavefunction.is heavier atthe origin, . asshown in(6.20); this gives 2°.The4thZarrives fromthediagonal sun of(6.47). Thecrude result (6,48) tells youthat forappatom (monic +hydrogen gas) the capture rate is1000 times less than the mon decay rate. But for large Zatoms, even Z=10, itseems that the mon capture rate will exceed the mon decay rate. Notice one bad aspect ofthis process; inessence there-is only one final particle, the neutrino, soyou are forced todo the complete Shase-space; thus, the only data you can measure isthe total rate; it is unlike beta decay - where you get assymetries, angule correlations etc. Not avery rich experiment, What does.experiment say? First, you candothe gaseous pt atoms. But the two fermions can have spin S=0 or1,and these two states have different matrix elements, Gen goes back and undoes some ofthe rate, showing Wonce again. Well, the conclusions are not tery exciting. You can dothis muonic hydrogen, orcapture inEe?orinnet, Section closes with "radiative muon capture" and «somehard-to-do experiment withmmmnueleus recoil aggular distribution. Doese : not prove ordisprove agything; iscompatible with the theory. ~T- Chapter7:theGabibboTrick.Multipletsreviewed.Enpkiricalselectionrules fe) arenoted: fora@Sa0decays, /aI/=1ie,bothV,andAbehave asisovectors. foxdS=1dedays, aS=4Q ;other tule? 10(@8u2-o7- morddecays. fordS=1decays,/dI/= +iey.bothyandA,behaveasisospinors, ThesearerulesIshowld keepinmind.Examples: 2-7WKYisblocked byd8=a9 rule; KS"WEY alsoblocked bysame. Notice thatthe/aI/=% rule contains the dS=dQ rule due toGiN forma, However, there issome evidence of/dl/=b rule violation. . ‘Theexample given forvalid /@l/<} rulé isthedecay KZ4¥F~ Since final state mst have L-J=0, isospin state can be only, I=0 or2.But.if current operator isaInd object, only the I=0 amplitude isnon-vahishing. Therefore, using trivial clebsches, you can deduce that K-short should gointo charged pair twice asoften asneutral pair, ie, ratio of branching fractions should be 2. Nytables show this fraction isinfact 69/31 =2.2. Consider nowKf»WF.Again, asabove,yourequire. 1-0bytheI=} +rule; butI=) forfinal state, sorule says this isblocked! However rate shwsupas.30%oftheKy rate. Isthie really aweak decay? It's rate is.«, intheweak time scale ball park. Iamused toconsidering this asaZweig .,”, A)_—forbidden hadronic decay.YaybeZneigprohibition isexactanditonly.leakethrough weakly; then cylinder correction should bezero. Anyway, intheKta-ef youmustconclude thatitssleaking through in . theI2channél, Doingisospin comparisons, ‘youconclude thatA,/A,=$f,a measure ofthedegree of/dl/=$ rule violation. Next Gene goes into SU(2) endSU(3) discussion. Hewrites down theoctet 3x3matrices inusual way. Ihave reviewed this but have notset myown conventions yet. : Now, howdoyouassociate thecurrents V,,V,,4,;4) .with octet currents? Well, inCVCyouhavealready saidthatVV=j,4iJ, andJ,=d3+Te.you have anisovector, sofar. Now, recall that they key SU(3) assumption isthe identification ofFgwithhypereharge y.=B+S. Consider theweekdecayktsFy. (strange version 6fpionbetadecay*) Thisinvolves only,thecurrent V,(seep84). Clarly this current isacting like V-spin raise/lower, .soyou conclude that vy=dgtiy +Bytheway,J,=neutral ourrent soislikeU-spindirection. Perhaps J,,,=mixofj;andjg} theCVConly says: that V,currents arepartners ofthe isovector portion ofthe EMcurrent; as usual, thing GMI. formla, - eo) WhatabouttheU-spinraise/lower operators which’would‘relatetocurrentsJgand7which nooneevermentions? These would beinvolved in8decay say+ofK%9 Wb) ;thecatch isthat this would violate charge conjugation; sothere seems tobe noway to "access" those members of the current octed. That makes me uncomfortable. Atany rate, Inow understand how the various identifications aremade,Similar tothedssocociation ofthecurrent withaniso-spthor in °“beta decay] ie,recall presence ofTin there. Seems solid. >The&.andA)axialvector currents arecomectea withtheirownaxial octet, ofcurrents. Gene calls these g,+ig, =ay etc. Earlier Iasked: does axial d5=0 current look like avector im isospin; answer isyes. Now comes the Cabibbo stuff. Goback first to the leptons. Just asyou didformandpinhadronic beta decay current, you canput ©andYeinadoublet andigagine &group "weak isospin" SU,. Since Weisdefinitely lefthanded, youmight pititonly with e,component of¢,Inthis- viewpoint, theregular Vaiid Aportions of the weak leptonic current look like weak-isospin raising operators, sothecharges connected withthesecurrents aisiaAlepandYepsare called F,andF,? where the5reminds youofaxial. Youcanofcourse imagine diagnoal currents and charges (these would beneutral currents) which havecharge L,and1?+Clearly youhaveyourself analgebra SU(2) xSU(2) where one SU(2} isfor the vector charges and the other for the qxial charges. f Tyis idthe chiral group. That the charges respect chiral algekra isnotBquestionable,youshowitdirectly,Geneisnotsayingyetwhatyoushoulddowiththisgrouprelevant totheweakleptanic currents, Hementions this(*)because henext sfas: presumably the hadronic weak currents (ie, the related charges) form SU(3) xSU(3), where again, oneSU(3) isforthevector octet andtheother fortheaxial octet ofourrents; these arethe.j, andtheg,currents. Obviously this extension of CVC tothe "octet hypothesis ofthe weak hadronic currents" isnot exact norprovebble, sotheSUSxSU¥ things isnot as good as the SU2xSU2 chiral algebra. . The Cabibbo business now arrives. Youknow experimentally that @Sq1 ~ decays are suppressed relative to dS=6, arid there are no dS=2. The Cabibbo Hypotehsis isreally the octet hypotheses; obviously ifweak hadronic Rie currents transform asU,I,V spin raise andlower operators, youcangetonly” : dS=0anddS=1. Suppose therewerenodS=l., ThenyouwouldsaythatTusSytiig : period, and by "universality" you would give this current exactly the same strength astheleptonic: current Sep=51,10p+31ep+Ie,weak-isospin ; ..vaise and lower operators, same G. But then you imagine that somehow the presence ofstrong interactions causes arotation ofthis J(nad) vector imthe 8-dimensional space; the length ofthe vector does not changehowever. Youchoosejusttherightrotation sothatyoupickupa (a)component along thej,+ij, axisinthisspace. Youchoose theroatation angle sothat d5<0 and.dS=1 parts ofthe current have correct experimental stregnth. -8- Moreover, youassume that theaxial current isrotated exactly asmuch, as .oe) thevectorhadronic current. "Thisleadstotheconciseresult(7.79)whichimthereal thing. Obviously tons ofpredictions willfoloow. "What about charm. Gene's book is1973. Iwould guess offhand: put weakcurrents intoa15repofSU(4). DothesameCabibbo rotation toget (7.19), then rotate again tobring insome d@s1 weak currents. Ie,charm-changing weak currents, These will then beinvolved indecays ofDpesons. Not clear, ‘butseems youwill need atleast twoCabibbo angles. Maybe more. Youmight usetheBox fy/t decay constants ratio togettang, where $issecond Cabbibbo angle. . * The Cabibbo business isfinally very clear andstraightforward. *Chapter 8:Baryon Semileptonic Decays. This chapter opens witha review of‘the Adler-Heisberger relation. Lets see ifIcan't penetrate this and see how it really goes. You start with acommitator ofaxial +and -charges put between two protons. Put insame intermediate states and ofcourse you get matrix elements ofthea8O<thcomponent oftheweak, axial, hadronic current. For neutron intermediate state, this isprecsiely the axial part ofbeta decay and )*yourelatetoC,,TheRHSofthis{ese-charges comrelgivesaconstant. ‘Thus, Ithink you get the left hand side ofthe AWrelation. Now the problem iswhat todoabout higher intermediate States? Instead ofwriting thecharge %asaspatial integral ofA,thistimeyoucen introduce a@/dt atthe expense ofenergy denominator asin(8.12), then you canreplace d/dtwith®yAS .Finally, thislastisreplaced with’pionfield via the Goldberger Trieman. . . Pause toreedll howGTRworks. Youhake aOPEmodel toexplain g,interms offyandYyyw-ThomyouusePOACtorelate g,toB32sincesecond-cldbs g,=0.Butg=C,.Thus,youhavesimplerelation betweentheseguys:+,SueCA.Now,ifyoulikeyoucanuse(8.10) version which saysthatJyN* «!pionfield with constant asshown. From all this, you get for_your “higher intermediate state" sum the form (8.15). But since pion field imthere, the only possible contributing intermediate states are PI-N states, Somehow (Gene skips this) you can then relate this term ofthe Adler Weisberger relation tointegral *over PI-N cross section, whichcertainly sounds teasonable. Thenetresult is (8.16). Amazingly, when you numerically integrate the PI-N’ cross section difference, e youdogettheright answer forG,. ‘ What does this verify? First, weused aportion ofthe SU3xSU3 algebra, but =~ only tie non-strange portion. This whofe thing does not involve strangeness atall. Second, weused Goldberger Trieman, but wealreagy know that works. Weonly uses PCAC inasmuch asitgives the GPrelation, Really, the main thing this testsisthealgebra ofcharges, onlyinthenon-strange sector. Ie,weare e testing theconcept that weak hadronic currents have isospin structure. By thewey, strong interaction currents must allbeisoscalars! Just because weak interactions violate isospin does notrule out arole for isospin. The statement that’ the43-0 currents areisovectors simply bursts with predictive : power.. Now wego tothe baryon decays. Obviously the simplest weak baryone decays are those emitting aQY¥ pair. There are 10such decays. The mtrix elements areasshown in(8.19) foraSq0 and(8.20) fordS=1decays. You preselect your baryons asband b*‘. The currents inthere must have various terms involving BB since you have todestroy one baryon and create another. Question: howdoyoucombine thebaryon fields inthie bilinear wayso that what youhave isanoctet? (Recall that Cabivbo Says currents aresupposed totransform asoctet.) Obviously therearetwoways,the8,anathe8.inthe product of8x8. Question: what are the entries inthe 3x3 bilinear BB matrices thatyouconnect withthe8,and8,?Typical entries aregivenim(8.26). Nowasubtle point. Consider your dS=0 ourrents, both vector andaxial.Thevectorarelikej,+ij andaxielarelikeg,+ig,. Thus,the45-0,wectore weak hadronic current looks like isospin-raising operator, and weknow that such anoperator can never change total isospin I. Recall that apure Fermi beta ecay involves only theCyterm, ie,only thedS=0 vector term, andforsuch decays dI=0. Smae idea. Incontrast, your axial dS<0 current looks like anaxial-isospin raiser, andthus cannot change axial-ieospin. Butaxial-isospin isnotthesame as regular isospin, Thus, the dS=0 axial current can change total isospin. Sowhat? Thepoint comes now: when youmake your 8,and8,entries of BEterms, youfind that allyour 8,entries contain atleast oneterm which changes total isospin ofthe baryon. ‘Thus, for the vector current you mst ruleoutall8,typeBB”terns. a - But, according toCabibbo, once you have removed the Symmetric combination from thedS=0 vector current, youshould also remove itfrom thedS-1 current because this thing isjust aninduction from the aS=0 via the Cabibbo rosation. Ontheother hand,adirectdS=1argument wouldsay:well,theseourrents are notI,s$yle currents, soyoucould notaprioiri saym#I-0.° _Soyimagine putting intheBBlinear combinations andletting thefields nit@ onto thestates tomake u(b) spinors. Youthenget(8.30) and(8.31). The ~9- symbols aandsinthere arejust thecoefficients like 2/{ appearing in(8.26). ,o)YouseethatthedS=0anddS#1matrixelementseachhaveonlyonevectortermwhichistoway,the8,term. Notealsothatthesetermseachhave‘overall coefficient: 1,asside from thecosandsin, This 1is£,(0) ofregular nuclear . Reta decay. Recall that this was the experimental fact that suggested CVC, ic, thatf,=D, ie,thatthebetadecay Gissameasmuondecay @aside fromcosine. However, for axial currents, the isospin argument given above fails and youarestuck with both 8,and8,type terms, theusual FandDterms, Again, looking etreguler ndecay, you seethat F+D=C,. . So, after alotofyak, Inow see what (8.30) and (8.31) are saying. All 10allowed baryon simple (semileptonic) decays can befitted with just two constants, DandFf . Table 8.1infactliwts theeffective CyandC,forallthedecays. Bythe way, notice that allthis business ofcombining BBinto8,and8,implicitly means weare assuming exact SU(3) invariance. Inessence weare applying SU3 Wigner Eckhart theorem tothe matpix elements here. However, because there are two octets inthe 8x8, you cannot predict everything from SU3, ‘ever’ exact SU#3. Notice that some times Cy=0 for some processes.' This issimply because [e) thisprocess’ isSU(3)-absent fromthe8,term.Sinceaxialhasmoretermsthanvector, ieaxial has8,+8.,ikisofcourse morelikely thataneffective cyorfyshould vanish than fortheeytovanish. * Bythe way, itshould also benoted that weare treating akl “these baryon decays jnthe same approximation aswe treated the beta decay, ie, weassume small momentum transfer, evaluate allform factors atq2=0, drop allsmall terms such as'thef,weak-magnetism. Ie,inthis limit thedecay theory is just af),g, theory. Now, lets doafit. Choose Fand Dasshown in(8.58),’ use the usual Cabibbo angle. This then enables ustocompute all these decays. Ie, wecan get all the decay rates. Most total rates fit well. +But are there other things you can measure experimentally totest this Theory ofBaryon Decays? Sure, everything you measured inbeta decay you can @lsomeasure here, Bg,the@9correlation, whichisafunction osfheangle ©between Fe,andy -(thiswasthecoefficient ainbetadecay) issomedefinite function ofyour fland gl. Bymeasureing correlation, you can check tosee if f1=0 inthose decays where itissupposed tobe. You cam-also look atthe fo)leptonandrecoilbaryonenergyspectra; thefore-aftleptonassymetry; the recoil polarization; (these latter two inpolarized initial baryon decays). These peripheral experiments oftenyield information onthec,/Cypation asf. |, Apparantly nothing drastically wrong has yet been found, Now afew more topics are inthis chapter. Why does SU3 invariance work sowellhereifmazses aresoseverly broken? Somepeople wetouttofind [*) out, leading to the Ademollo~Gatto Theorem, This theorem isdezived later in this book butsays this: theSU3-breaking term inHaniltonien must beFg=¥ _=hypercharge (octet dominance), This term has relative coefficient .1if superstrong Hhasweight 1.0. ‘Theorem claims this: theFgcorrection has noeffect onour analysis tofirst order inparameter .1, 0expect corrections tobeamere 1%, _Another subject concerns generalized GTrelations. For regular beta decay werelated ¥y,Jpn, C{. Starting withanyother baryon decay youcanmake asimilar theorem. Eg, inastrangeness violating defay, the meson-pole term which gives g,willbeaKsoyouwillinvolve fy,Eng MEgy=Cyfor this decay. Youcangetsome formulas like (8.35) which relate ratio ofC, for two different processes tothe strong coupling constant ratio. How suddenly Gene writes down the two strong coupling legrandian terms asin(8.3%). The two relative couplings are here called fand d.By justexamining theterms, youcangetthestrong-SU3 prediction forstrong coupling constant ratios, Interestingly when you are alld done, you find thataand£ofthehadronic lagrangian arethesaneacDandPofthevaryon weak @ decay legrangian. Not too surprising wince itisall su3 8x8 business. ‘The final punch line then isthis: you fit all your baryon decays via certain value ofFand D, These values agree with hadronic process measurements of fand a, . Sothis chapter has been: generalized beta decay. Roughly, the generalization isSU2toSU} forbehavior ofweak currents (flavor groups), with, Cabibbo relative weights. . Ghapter 9:Nonleptonic baryon decags. Consider nowthepossibility ofheavy stable baryons weak decaying into lighter ones with noleptonic emission. Since initial baryons are"steble", youknow youaretalking weak orEMdecays. Also dont care about decays with photons. So, any purely nonleptonic weak decay of @baryon into abaryon must be atmost dS=1 since weak. But the only mass that will fit inthere isapion mass. Soall decays inthis chapter are hyperon into baryon +pion, (All are dSql. This isthe first time inthis book wehave looked atweak decays with noleptons atall.GobacktoadecaylikeAw@(4,thereyouwereabletoQe ~lo- essentially "factorize" the final state, and group the leptons off bythemselves @_—_—s24Lertonic ourrent, andgroupthetwobaryonstogether inahadronic current.Inthereaction N*M QW with no‘leptons what areyou going todo?‘ Gene suggests youwrite ew] J,J,\N>. Ie,nowboth currents arehadronic andgot inthe hadronic space, soIguess the point isyou cannot factorize this thing into,sayyGe\%|AYo|Talwy. thedifference mustbethatthepandqfinthefinal state interact strongly, unlike the interaction oftHe pand &¥ofaleptonic decay, This must bewhat precludes this factorization, Related toperturbation theory question. . . +Sothink about oneofthese decays, sayRew .In-partial wave terms, initial state has J=$, final state has S=}, soyou ‘must have L=0 or1.Ie, for all these pion decays ofbaryons into baryons, there isonly ans-wave and a powave amplitude, - . Now, independently, you may goontoshow that the amplitude has amost general form (9.3) with amplitudes AandB,Not immediately obvious that Ais the s-wave and Bisthe p-wave, Where are the 4other terms you usually start With? Answer: because only three momenta heré, you can always write pion p interms ofthe fermion p's, and these then kill extra temms asillustrated with ro) example onpage199. . Sogreat, for each decay there are just two complex amplitudes tofind. Now lets gothrough the Pauli reduction. Your initial and final baryon are polarized, say,along directions 4;andawhichyouarefreetoshoose. The ‘amplitudes Aand Bare converted tosand p. Then from (935) "you see very * clearly why sand pare the s-wave and p-wave amplitudes. Wearenot even going totrytopredict total rdtes; there isnoway to ‘connect sand ptothe constant G. So, weonly care bout the relative rate ‘which isgiven by(9.7). 2g,forTinvariance, youexpecth=0.Youshould beable tomeasure the various coefficients toget ratio s/p type information. *Notice aproblem, however. After the ‘and-€ are created “at apoint", they are certainly going todosome PI-N scattering, why not, for awhile they are within afermi, This effect is called "final-state interaction" and Thave never leamned much about it. But Iamnot atall surprised toLearn that wewill getinvolved withonlythesandpwaverphaseshifts, sincethosearetheonly ‘waves there are. . : Inthedecay ofunpolarized A.,youexpect theproton tohave noaverage @ _—sttensverse polarization because theinitial state hasnopolarization, Thisis atrivial electrostatic symmetry argument.‘ Bymeasuring the longitudinal “polarization, you can deduce 4=.65. Somhow you can measure the 3parameters iP,¥%whichisineffect3ofthe4realnumbers present insandpsThe4th number isthe total rate. . Now, there mist besome waytorelate these various decays. Since allaredS=1,youknowthattheweakcurrent oreffective hamiltonian hereshould e transform asInj, Ifitdid, youwould "see" thedl=} rule being repected in these interactions, So for the moment lets just assume this isso. Then you install thespinor behavior ofthehamiltonian byadding a“spurion" tothe initial state; trivial. . Once the spurion isinthere, isospin isnow conserved, soyou use standerd isospin techniques to get ratios of things. Inthe end, these isospin ratios _give you 6conditions onthe 2x7=14amplitudes involved inthe 7decays. In particular there are twobranshing fractions that should come outbeing 2, butwhich areexperimentally only close (to10%.) Soisthereanything elseyoucansqybeyond thisashruleimplicafions? Ie, ifyou assume acertain SU(2) flavor behavior ofthe effective hamiltonian, yougetthésedl=}-rule predictions. WhynotmakesomeSU(3)~behavior assumption forthehamiltonian? Andbytheway, since theeffective Aissupposed tobe Jody) smbyouwould expect toseesomedl=3/2 anddl}; butyoudont seothe 41=3/2, Whyisthis? Well wealready assumed intheabove paragraph that no aI=3/2 whenweputU~spurion, . e . Perhaps youaresupposed totake acertain combination oftheJ,andJ, currents for Hwhich combination goes asanoctet member? Obviously you want tochooge anoctet member that hasIs}. Byindisting that #have properties @Q+0, dSal xmbxix, youconclude that only menbers 6and7arepossible. Later itwill beshown that can beonly one ortother; both would yield CPviolation, So tryHasa7-menber or6-menber ofanoctet. Writeoutallthe SU3scalar combinations ofB,B, M,and h=spurion looking like a7-matrix. UseCPtoreduce number ofamplitudes to3instead of9(among thes-waves). ,Then examine expanded H to get ratios onpage 209. Most are duplicates otthe QI=} rule spyrion method above, but clearly you should get something more. The oneextra relation forthes-waves (forthep-waves ifyouchose member~7) is (9.47) found byLee-Sugawara. . Goback: wehave 7processes ofinterest. ThedI~$ rule gave 3relations among the7s-wave amplitudes, Lee-Suzawara isa4th relation, Arethese relations trueornot? Figure 9.1plote for§ofthe7processes (amv andSA” arenotthere because these arethetwowith nocharged particles, experiment nearlyimpossible). Sothefigureshows5arrows,eachonerepresenting the[o}sand pamplitudes, assumed real. One arrow isrepeated twice because ofsome experimental ambiguity. All relations geem tohold within experimental error. -l- Notice thatthesimple (Lrules arenotopthispicture because, they inyolve thenon-charged decays. Thispicture isatestoftherule9.21from,dI=}, and 6theLee-Sugawara 9.47rule, forbothsandp,infact. Thesimple {2rules are born out by compaming brafiching fractions and they look OK. Chapter 10:Kaonsemileptonic decays. Westart offwiththeoldstory ofKandK,. Fromtheviewpoint ofstrong interactions, andem,thestates K,andK,are,distinct states; ie,strong andemdonotatallmixsuch states (eg, youcould notcouple K,toR,because this would bedS=2). Butofcourse theweak interection canmix . these twostates via anintermediate state ofS=0, say PI~PI. Note thet this mixing mst besecond order inG,soisvery small. .Now when you speak ofastate having adefinite. lifetime, what.do you mean? ,lifetime means lifetime with respect todecay. Since kaons decay only byweak interactions, only these interactions areinvolved inthequestion., A"state" ,Should bediagonal inallquantum nuybers whichcommute withtherelevant hamiltonian. Thus, youwant neutral kaon states that arediagonal inCP;you dont care about diagonal inSbecause (8,H,)40 +Infact, CPisallyouhave inthewayofquantuam numbers conserved inweak decays. This iswhyyou make theKLatidKScombintions. Sowhichoneislongandwhichisshort? ThestatewithCP=+l candeéay 6into PI-PI, sothis will beK,with lifetime 100picosec. Skmxmiharxstate This K,state virtually 100 into charged andneutralwe’, with 107? rate towa¥. Obviously anydecay canaddbremstraullung andbedown order « TheCPe-1 state isblocked intoww andasaresult lives 600times longer, .dey60,000 psec=60nanosec. Theobvious decays hereareKiw(ey).which together give 66%ofthedecays. Theremaining 34%goes intoW¢ which isCP=-1. There aresoemdecays intoPI-PI indicaténg small CPviolation. Alsodecay into YN.Notice thattheK,allousmorevision intovarious decayproducts because . experiment hasmore time, not soswamped . . . : So,onceyoudecide that kKeidKyarerelevant, thenext topic iswhat is meantbyregeneration, Verysimples Youproduce sayK,inWfscattering. Notebythewaythat youcannot produce Kebecause there arenoSat] baryons. Soyou makeyourself anice Kebeam. The“wavefunction" forthisbeamisKetKyas acoherent amplitude sum, Butofcourse your Kydecay quickly leaving relatively moreK,downthebean, After awhile youhaveonlyK,.Butnowrunthisbeam ofK,into some crud andofcourse theKecomponent canreact with|protons whereastheKocannot. So,ineffect,theordremoves moreoftheKecompanent éofthebeam, thus unbalancing themtrture equal mixture ofKe-+Ke®KL. Soyou ineffect causesomeKs4.4reappear, theyare"regenerates". Clearly the KI-KS mass difference can be roughly explained by the CP selection of intermediate states, and result should be quadratic inG.Thus, expect masadifference tobesaneorderofmagnitude asanyweakdecay rate. GyInfact, aM=.44%5~-(S'asZ .OFcourse thisisveryemall compared tothe absolute kaen mass which isroughly 10° sec, ie,typical hadronic size. What can wededuce about decays ofthe Gorm K--WRP? The first fact . isthat ,Sor all such decays, there are only 3amplitudes involved, called f,g, andA*onpage 219. Ofthese, gviolates thedS=aQ rule, Isthere any¢ present? Comment: when you write the hadronic current matrix element, you want tobeusing isospin anddSetc, so-you putinK,,notK.,asin(10.19). Nowconsider K7-9WEV .This decay isreally your only source ofdecay electrons. Youcancompute therate ofe*toe”production asafunction ofdistance down the beam, asin(10.23), Then doexperiment toget exper measure ofg/f. You find .04, soseems tobe4%violation ofthedS=dQ rule. Buterrors arelarge. So,presumably g=0andweareleft with amplitudes fandA*, These are then trivially related bythedI~} rule, ie,byxisospin, soA*=£/{%. Thus, you ean equate some rates ‘and mekem afew predictions. Itworks. Wehave nowboiled things down tothestudy onjust oneanplitude, namely .At=GP\K'D. Wenowwritethisintermsofthetwoformfactors f,and f_.IfthiswereaV,currentwewouldreplace f,ial(heENformatactor}eyusing CVC, and wewould also say things about vaiues atq°=0. Ie, wecould . useforV,current allthepower ofCVC. Now, canweapply CVCtoV,current? This isnotaxial, sowearenottalking PCAC. Inperfect SU(3), since Vv,and V1aveinsaneactet according toCabibbo Hypothesis, snswer is’yes. theconcept ofperfect SU(3) justification wasthat order N Ademollo-Gatto theorem, - Soletsanalyze Kvrowev. Westartwiththetwoformfactors f*andf butbecause eissolight, thef_termgoesaway(notet2#/fr).Theamplitudetheninvolves onlycosinet4¢(a°). Now,smallbytfinite aisinvolved because there islarge Qvalue, butset£,(0) =1byCVGapplied toV,asdiscussed above, andinclude @small empirical q°linear terminf,(q°). Compute total rateand .compare to exper to get still another measure ofCabibbo angle. Oruse old Cangle andconclude that theory andexper agree forKY9WEY process. Once ggain, basically only one constant eneters here and you set itto 1by CVC ,this allowing total rate computation. By looking only atthe pion-energy spectrum, you can evaluate thatsmall linear termin£,(a°). Thenlookatelectron energy spectrum atfixed pion energy, This gives confirmation ofveotor type theory and conflicts violently witheither SorTtheory asseeninfigure -p231. (*)+Whatabout theK+ PHYaocay? Wold, ithasthatextra f_or§ term 80obviously what youwant todoismeasure €,This isdone bycomputing the : -l- muonpolarization intermsof¥andthenmeasuringit.Resultsareshown le) imtable10.25Roughly ¢=£7/f*=-1,dimensionless, althoggh youhaveto, account forq°dependence somehow. Then, using this{informl (10.79) for them/@ ratio inK*decays .(orK,decays, since ratio) youcompute .53for K*and.50forK,(keepinmindthatsuchratios always involve massdifferences andsometimes radiative corrections). Mydata book says 32/48 forK*which is.7, and27/39 forK,which isagain «7, Strangly results areboth thesame butare bothoff, Noreason given yetforthisdiscrépancy! (rate onwita?) Sowhere are wé? Gene discussed the simple Ky, decays back inchapter 4 interms ofthesimple constant fg. Inthis chapter hehas sofandone theKy decays which involve onepion. Thetrick here wasaversion ofCVCtomy f,=1. Thelast semileptonic decay iatheKyyinvolving twopions. 4snoted back in chapter 4,there will be 4form factors here, and both Vand Acurrents operate. Ithink theuseful fact, here isthis: your finals state ofT#eN has200MeV ofkinetic energy, but myintuition says that momenta will beequipartitioned, sopions may only typically, have 10MeV ofenergy, ie, they are relatively soft. Ingeneral, all momenta pare relavively small for both pions? Anyway, Iamtryingtojustifytwofacts:(1)thethrowingawayofthea,termin(10.98) fe)since bilinear inpion momenta; (2) the near constancy ofthe other form factors . intheir arguments. . : Now, wehave three form factors with noknowledge ofthem yet. Byteking . the two different soft-pion limits afd using the soft-piop thearem, you conclude thata,=0anda)=a5,atleastwhenallareevaluated atthatsoftpionpoint. «Fromtheotherlimitweconclude thata)+ay=48Jo+»floodgrief, wheredid these old form factors come from? ,How did they get into soft pion theorem? Well the SPT has fq sitting initintrinsically bepaiise the SFT contains Goldberger : Trieman/PCAG stuff inits derivation. Onthe other hand, the SPT removes apion from final state, andthis brings inty. Soresult nosurprise. Soofthefour caonstants a,through a4,youseta,qa,a0 ineffect, and yousetaj=ay= et,/te»numbers whichyouknow,andthenyoucancompute the total rate! compare with exper: itworks, agreat victory for the Soft Pion Theorem! ' . So lets take alook atthis Soft Pion Theorem, Stated onpage 234. You pullyour softpionoutofthefinal state andyougot18UFeStILES,oflliy - Howdoes itwork? Obvieusly bypulling thefinal state pion, yougenerate apion @_—e148whichcanbeputintoaconmtator Zorfree,see(10.85). Thenyou+replace thepionfield with Se'-3%, ie,thedivergence of:theaxial current; thisisjustGoldberger trieman andyouseeSgappear. Butinthe Limit ofqo (softpion) youcaneffectively replace Yai withg,integrated overspace asin(10.88), which yields thecharge F?,ThusQED, Onlyproblem isthat Idont think the oosine should ‘te inthere. Later when he applies the rule, Gene does not show the cosine. : ‘what about Ademollo-Gatto which Iskipped. This is another one ofthose things which start with acurrent algebra statement with charges F,.Veyy simiar + to Adler Weisberger: you sandwich inbetween pion states, not proton states, put inintermediate state set. When the intermediate stete@ isaK,you get term involving t,andfofcourse, Ifthere were noother terms atall, you could conclude that2,(0) =1 (thisstepinvolves going toinfinite momentum frame). Now consider those extra terms The only available states are single . particle states out ofthe }xx 0”octet, and many particle states (recall in aWitisthe PI-P states that are important.) Look at(10.44). For lerge : energy difference, itisclear that the “other terms" will all beorder where X=«1isthe SU3 breaking Hamiltonian scale. Thus wehave proved that £,(0) =1+order(%). Harlier Iconjectured thisbyarguing thatCVC should apply totheAycurrents, based onSU(#3). Here weseethat even if there isSU(3) breaking, youstill havef £,=1 tolf» : . This isthe general theorem Ithink which says couplings obey SU(3) muchbetter thanmassdifferences. Ie,massdifferences areorder =10% e whereas couplings are SU(3) invariant tooraer =1%. Good argument toknow. Chew used italot, but never used phrase Ademollo-Gatto. Final chapter comments are onneutral currents. Processes ofinterest -hereareKPyay orGE. Clearly these decays violate strangeness andmst therefore beweak, and you can see that the current involved must beneutral. Bxperimentally ,Gene says less than 1079, Butmy1976 data book says these events doocour atrate107%,Ie,di~mon events inK,decays isgoodevidence that wenowhave neutral currents. Maybe this nowagrees with theKy» WY.rate. Some experimenters mst have made amistake in thés business. Ghapter 11: Keg and Ker3 decays. FactL: two pions can only beinstate I=<0 or2for the KW2 decays. Exper seems toindicate a5% I=2 amplitude, which isinviolation oftheai=}rule. Hg,theratio ofKshort intoWatAW’should be.2.0 but seems to be .2.2 . How should you analyze things? For Ky, you cannot doaourrent=current thing with Gbecause this is nonleptonic. Similar to the hyperon decays where youhavetajustmakeupsomeparatmerts andseehowfaryouget.Obviously efor the Kyq you have L=O soonly s-wave amplitude. But asnoted you can have I+0,2, sothere aretwoamplitudes called 4)anaao, -13- Nowobviouslyyouaregoingtowanttomaecomparisons hetweehallthe (o)Kqy decays, half ofwhich involve the neutral kaons. Thus, you immendiately . have, the additional, complication ofthe k*=K¥ system. You have to.understand that "system" before yon can make phenomonologicai analysis. Yuch of this chapter is;about that. system. We start off with amass mtrix which would bediagonalized bythestates K,andRjwere there-noH, .The weak interaction, asitviolated strangenesa conservation, connects these two .States caugeing offdiagonal entries inthemass matrix intheK,,KR,basis, see(11.29). Soobviously, youre-diagonalize andcall your newstates K,and Kg.Howaretheseexpressed interms ofK,"4K, 7,Tygeneral form(11.37) and CP? then leads you tothe most general form (11,46), showing the one parameter @.Clearly, if€$0 you have some CP-violation. From (11.50) you.can sebt that. CKs\Kur#O when EO. Inother words, these two states can couple inpresence ofCPviolating interaction; ifthere were no OPviolation, each would be aGP eigenstate. Actually, Ithink they are CPeigenstates anyway, CP-violation: justallows themtocouple. -Ifthere were nocoupling,.no CPviolation, then you.know that KyandKy wouldhavetohavesamemassto2partsin109,AssumingCPT,theyhavetolwe (oe)exactly the same mass, Thus, the fact that there even.is aK-long/K-short mass difference. isevidence ofCP-violation. . Gene throws inalittle analysis of regeneration, though not used: later. Ie, what happens when you run aneutral kaon beam down through some material? Invacuum youknow that mass eigenstates willbe K,andK,. But, insone material thenew,eigenstates will bedifferent, namely K,'andK,". Inthe . end, these new eigenstates are given.as in(11.67) and (11.68) .Phere you see the regeneration parameter r, This arises because, thescattering .onprotons * ornuclei ofK,differs from xfor..reasons that areclear. Ie,'you rediagonali,e your mass.matrix once again and get these new eignstates. . Besides the regeneration complication which you cannot avoid inanexper, therearealsostrong interaction s-wave placeshifts intheI=O'and I=2 channels. “Iheffect, these phasé shifts’ change thephase ofthe2=pidn final statebyei8E,soturnsouttobaverysimplecorréctioh %6‘actount ‘for.I dontknow thetheory ofthis, however. Sd,whenyouinoludes tHeoé phase shifts atidtheusual isospin Clebéch's, your final possible pion states areasin(11.70) and(11.71).Combinethiswith’yourearlierconstiuctioh oftheKyandKy le]states and yéu end upwith main results (11.77) through (11.80). Asnoted, . everything isnowdescribed byAp,4yanaE. Recall that A,ismeasure ofviolation ofthedIw} rule, and@ measures the CP violation. - What canbeexperimentally measured ?Consider K,/K, ratios. Obviously such ratios are ‘small because inexact OPconservation, K,cannot gotoPI-PI. Tworatios are.called hae andne. See(11.81). IfdI=}isOK,thenno A, andthen these ratios should both equal €. Experiments say\9'~| =2x1077, although somerecent controversy in1976 particle data book, ‘The mass difference ismeasured bythe very clever gap~ +regenerator method toaccuracy of1%. Notice from from (11.21), bythe way, that intoorder tobeable tocompute dM,youneed tocompute M,,, that -off diagonal strength. That inturn requires you to compute quadratic terms like that shown in.(11.21) (first term vanishes since dS=2), Since, well, theresult will beorder G?andprobably propertional to€,.but youcannot really compute itbecause wnkrown intermediate states, ie, a.long calo isneeded totry to compute dM. No trivial relation between dMand€ . Thephese ofM*" iscalled ye andismeasured bysimply looking atthePI-PI production rate versus time inavacuum regenerator. Youget45° with error of4°, Youcanalso trytoget.\Nee| andphase by ;also people can measure Re({) to10%. Theresults whowclearly thatthereis.CPviolation, butnotmuchelse(*) can besaid, Obviously there isalso some Tviolation going on. What isthe - cause of this CPviolation? Isitjust that the weak interaction violates CP? Orisitmaybe that strong orem are violating GPand itshows uphere? Or is there some superweak interaction which isdoing the CPviolation? Strong and EM. are obsérved torespect CPP and Ptohigh degree, soonly possibility issome small C/T violation there? Noone really knows. InWolfensteins simple “superweak” thregry, you just assume there issome extra dS=2 interaction which letsK.~yKgwhich thengoestoPI-PI, Thislittle theorylet predicts *~=A” which looks asthough itmay betrue. But dis} aso predicts this. So,tosummarize theMw story, there arethree parameters inthe phenomological “theory” called4),4,and©.Whatyoucanmeasure areA"PS” nN”, q”, Reb, QM. There isno"test" ofanythreory here, youjustdefine some parameters and measure them. You, observe tix both from mess difference and _factthat MLS WWthatthere isCPviolation. SoCPviolation isreally the main point ofinterest intheKy, Story. , WhataboutKyg?Obviously evenlesswillbedoable, ‘Thereareseveral) such decays ofkaons, obviously try tomake some predictions using isospin etc. -u- Sincecondarethree-body decays with(néar) equalmasses, youcdnplotevents inaverysimpleDalitaplot,seefigure11.8.Asshownbg(11.163), the 6 density ofpoints atsome place inthe’diagram (sayT,,7, orr,®) tells you the square of the amplitudé. Notice that there isno Dalitz diagram for a2-body decay since omskinematics fixes T,and7,bymasses. Ie,forKye decay, Tl=T2= kndwn number. _ Sowhat canyoudowith amplitudes A,(7,9) where i=process label? First there issymmetry 9to-@ because two pions are usually the same. Second isthi: pions havé atmost the entire Q=75MeV, usually mich less, typically 25KeV which means pions are slow. Ie, not much phase spac eavailable for these decays, soyou might say this: the amplitude atsome point inthe Dalitz diagram isthe amplitude atthe celter r=0 plus asmall correotion. Hence (11.166) which incorporates this statement with the 9symmetry. The correction turns out tobe afunction ofyercos®, the size ofthe correction is measured byparameter g” called “slope parameter". * Isospin and centrifugal barriér also have somthing to say here. Infact, considering pion plus dipion, you can correlate GP, Iaid L‘of final state; the netresult isshownintable11.4‘which alsoshawsthedI-$rule.Result isthatKYandK,caifbothgohealthily intoIeltripionstate,whereas loaeK, cant goinany I-spin final state, Ofcourse there will besome bleedthru because dix} isviolated, CPisviolated, and barrier isviolated. *Asusual, the dis} rule byitself makes various ratio predictions, see Table 11.5 page 283. Obvious small violations ofthe rule, asusual. * Finally, soft-pion theorem has some stuff tosay here, because ifallows * youtoextract pions from final state torelate Ky toKg decays. And pions are definitely soft. Results seem towork OK. Tosunmarize, theKyyamplitudes areparametrized byAatcenter of diagram, plus aslope parameter. There isno theory to compute either parameter (foreachdecay). Relations between decays intheKyg class arise from isospin, dI=$ rule, barrier effect, 2~pion symmetry, softness ofpions. Obviously apainful subject. Youhave 4theory which works nicely forsemi leptonic and pure leptonic sectors; but itsays little about nonleptonic dewayse Chapter 12: Parity Violation imNuclear Forces. The point here isvery simple: Thereshouldbeaweakinteractionoftheformkne|Sods+3,5ls92.>ieon ©_topofthehugeparityconserving stronginteractiom weshouldexpectaparity violating weak interaction. Recall that EMHamiltonian also conserves parity. How are you going tofind this effect? Certainly not looking atthe nbcross section, Clearly, what you doissay that the effective nuclear potential nowhas @small, parity violeting term. This then admixes states ofopposite parity, whichwereformerly unmimed. Thus,thisadmixing causesphoto-decays [*)which youwould normally think notpessible. _Soonetrick istolookforsomestrictly forbidden transition under strong +BM, and see ifinfact itleaks through alittle. Bythe way, the wavefunction admixing effect isabout F=107’, soyour leakthru transition Willbedownbyhugefactor of10¢ fromaregular allowed transition. Iel, this requires good experimntal technique tosee. Asexample, Gene uses a certain forbidden decay Go» C'*4 4where the0! isprepared inacertain excited state. Ifyouseealphas ofthischaracteristic energy, thenyouare ‘seeing the forbidden transition. This has been seen and measured, and result seems about what you would expect. " Besides looking forforbidéen events, youcanlookatallowed transitions which mayhave mixtures offorbidden events inthem, Ie,itisprobably hard tofind things thet are reall forbidden, eoyou have toaccept amixture. But thatisenough. There aretwoClassic experinénts. Inone,youlookatasimple photon emitted inanuclear transition, Ie, agamma ray, Ifthe ‘states involved aregetting parity mixedbypresence ofparity violating terminhamiltonian, @thenyouexpect toseecircular polarization ofemitted photan (ie,emitted fromunpolarized nucleus, andfinalnuclear polarization isnotmeasured). Just draw picutre and you can see thet there ehoutd be nonet longitudinal polarization ofaphton in‘reaction W9W°X. Circular pol=longitudinal polarization. So set upyour polarized iron target and look for this gamma ray polarization, ‘Typical netcircular polarizations are107! effect, soyouneedmanymany events togotstatistics onthis, Often your process isstarted bythermal neutron hitting some nucleus, getting "captured" into ‘an excited state which them dumps “agamma ray. . The other obvious experiment isto look at fore-aft gamma asymmetry when ‘initial nucleaus ispolarized (captures apolarized neutron). ‘This effect is sometimes relatively large, 1074, andhasbeen ‘seem. Chapter ends with quick discussion ofthe "theory" involved, not’ very exciting atpresent. Bythe way, what about parity violation inatomic transitions? Now you areasking aboutCEP\WwiEG7 .Since lepton factors off,youseethatthis ieamoutral current effect. Soparity violation inatomic transitions has to dowithneutral weakcurrents, andisreally adifferent subject altogether (nog -inour theory sofar); ofcourse experiments are. very similar. Ithink this is what Commins isnow up towith Steve Chu. : -15- Ghapter13:meutrino scattering. Obviously youdontwanttolookforweak 8 scattering inaprocess that isdominated byother forces, like ep. That is whythereal thing todoisYp4e%. (Standard charged-current deal), The Kinematics here isthe same.as depp inelastic epscattering, more orless. Ie, ‘ interesting variables areq”andJ. First consider elastic Vp? Fe. Obviously youputinyourusual BtoBweak hadronic current with flthrough g3form factors. Now however youwill needthelarge q°behavior ofthese form factors. Analogously tothe form usually taken and/or measured for the BMformfactors (vector dominance etc) youmighttakeformsasshonin(13.19). Youmightassume that£32220 (though not obvious that CVC still works at"high energy"). Ifprecision experiments could bedone, you could test theory: Atthe time ofwriting, only apoor statistics CERN 1967 experiment had been done with neutrinos. What about inelastic stuff, like nucleon throws off apion. Adler has some current algebra relation about composition of products indeep inelastic. Inthe deep inelastic, ingeneral you donot imow the tensor that getts mltiplied *bythe(weak orEM)lepton trace; hence theusual structure functions Wyand W,forBM;andalsoW,forneturinos, whicharisesfrompresence ofaxial © sicce not present inEi, This isthe sane old hadronic stroy: when you have too many hadrons, you cannot make upform factors, soyou set upsome pheonom. parameters, Aand Binhyperon decay, structure functions here. All structure functions are expected toshow Bjorken scaling. Chapter 14: Neutrino Astrophysics. You can compute solar neutrino flux; Davis etalmeasured itin cave inSouth Dakota, number was not inagreement with solar theory. (the famous cleaning fluid experiment). Gene goes ontodothe neutrinos out ofhot stars, and cosmological topics. Iamnot reading this chapter now. Excellent book! However, most recent references are about 1971; itisnow 1978. Much has happened inbetween, Neutral currents found, sothe 1958 Feynman-Gellman theory now obselete; must use something with weak currents, like Weinberg-Salam. Ofcourse any new theory must duplicate all the steble-particle decay results described inthis book! Charm also found, sothe whole theory has tobe e generalized tounclude bothstrangeness andcharm changing currents. Then there isQCD for strong interactions, quarks looking like leptons, and soon andsoforth. Areview oflast 7years would behelpful atthis point, rather than detailed reading ofspecial papers. 4 Vouume: 4,Number 7 PHYSLCAL REVIEW LETTERS Arne1,1960Fyouustn ‘- — i Watanabe.” Itshouldbenoted,however,. that “Thepossibility thatweakintergetions aremediated onth ;somevariantsoftheirtheorymaybefound byabosonhasbeencongideredbymanyauthors:H.calcur (oe)which-cannot bedisproved byonenegativeex- Yukawa,Proe.Phys.=Math, Soc.Japan17,48(1938;|PUh periment alone. Therefore, itisdesirable that J+Schwinger, Ann.Phys.2,407(1957);T.D.Lee Helatk4 othoftheseexperiments beperformed. It Rae eee taeeeasdfi . 4toberatherdifficulttoInventa.theoryof|_evnmanandM-Gell-Mann, Phys.Ree etsealan | seems if Y¥.Tantkawa, Progr.Theoret.Phys.(Kyoto)3,338{|sealar, : theTantkawa typewhichdoesnotgivearesonance (1545);Y,TantkawaandS,Watanabe, Phys.Rev.113,eleona } inanyoftheprocésses v+p,v+n,e+p,and 1244(1959). Wew |1 +n, where»andedenotecittierparticles or $5,L,Glashow, Phys.Rey.(to’bepublished). astricti. antiparticles. LeeandYang(reference2)showedthattheproducfF-f,2050| ‘Weconcludethatitisprobablyworthwhile to_#08erosasectionofsuchabosonintheneutrino tateq search forpossible anomalies inelectron and ‘nucleus collision isabout10-%om!foranIncident bei: Saree erecta oucrtheenergy rangewhich "outsin0 energymuchgreater than2Bev.Thisis myays m ey morepractical thantheresonance scattering method {poseth4 inay becovered with thepresent accelerators ‘asadirect testoftheintermediate boson hypothesis. beta de before looking into weak processes athigher Irshould benoted thattheabsence of1~e+isnotproper! I.energies. conclusiveevidenceagainsttheintermediate-boson 4h Ishould liketothank C.R.Schumacher for hypothesis. SeeG.Feinberg, Phys. Rev. 110,1482 rie checking someofthecalculations. (ag56). fateii Ty,Tantkawa andS.Watanabe, Phys.Rev.113, ay a 1944(1958). aye. “ThetheoryofTanikawaandWatanabeleadstothe Hi ‘Supported inpartbythejoint program oftheOffice V-A theory onlyafter spinors arerearranged bythe iy ofNaval Research andtheU.8.Atomic Energy Com- Fiera transformation. Inthiatheory, therefore, it {re mission, 15noteasy (ifnotimpoesible) tofindanatural ex- ti 1K,P.Feynman andM.Gell-Mann, Phys. Rev. 109, planation fortheconservation ofthevector current iy 193(1956); E.C,G,Sudarshan andR.-E, Marshak, partoftheweak interstion, iftisInfactconserved4 Phys.Rev.109,1860(1958). (R.P.Feynman andM,Gell-Mann, reference 1).If q he 2M,Schwartz, Phys.Rev.Letters4,360(1960); thepresenceofthewealmagnetism[ff,Gell-Mann, Ye ‘T.D,LeeandC.N.Yangy Phys. Rev.Letters 4, Phys. Rev.111,362(1958)] 1sestablished exper!~ ‘Thepi 367-(1960).- Similar considerations haverecently been mentally, itmayagain behardtoexplain by‘Fanikawa's} tumm4s radebyY.Yamaguchi (fobepublished), andN,Cabbibo theory.Theexperimental resultisnotconclusive yet.|thedis| H feiausuallyaseumedthatneutrinos emittedinthe _slightlytotakeaccountofanadditional decaymodeof|garmni 11-4decay andBdecay areofthesame kind. Thereis theBparticle. i jopositive proot fordhie, however. Obviously, itis ""R,Holetadter, F,Bumiller, andM,R,Yearian, Fl@l i desirablethatthisbeexaminedexperimentally. Reva.ModernPhys.30,482(1958). iiti ji AXIALVECTORCURRENT CONSERVATION INWEAKINTERACTIONS*| Yoichiro Nambu ji EnricoFermiInstitute forNuclear StudiesandDepariment ofPhystes 1aii University ofChicago, Chicago, Ilinois where i - (Received February 23,1960) + low m| q for 14 Inanalogy totheconserved vector current in- momenta. Suchanattempt hassome appeal in thatF] Jkr téractioninthebetadecaysuggested byFeynman viewoftheapparently modestrenormalization thenw 4 ‘andGell-Mann, some speculations havebeen effect ontheaxial vector betadecay cons weald Aye current." Oneeanformally construct anaxial_point,’namely, thepossible forbidding ofne+y,} imme Me vector nucleon current, which satisfies acon- hasnowlostitsrelevance. piondWd tinuity equation, (xe,1sConsents) ‘Theexpression (1),unfortunately, cgmbe pion-atN A easilyruledoutexperimentally, as-#aspointedi}ph = Av pl-ivgy 2Mgt,/a',api, (1)outbyGoldberger andTreimgnr*sinceitintro- iy # # i ducesalargeadmixture ofseudoscalar interac} wy,4 wherepandofseethesandfinalnucleon|tion. wid lie 2LMjoo CACurnct)\\afock fla.- i 380 Byaad ih p88 AOO) tonGTR &Dede YO 160°|Vorume4,NUMveR 7 PHYSICAL REVIEW LETTERS ‘APM1,1960 eS _ ie aOntheotherhand,Bq.(1)arousestheoreti- secascompared withtheobserved value2.56 iBjy;|calcuriosity astotheoriginofthesecondterm x10"sec. Hee“Witreallyexists; according toourconventional Goldberger andTreiman’® havearrived atthe (Sea. . fieldtheory, wewouldhavetointerpret thede- same'relation Eq.(4)(inthelimitoftheirself-" By. “95077nominator q?asimplying amassless, pseudo- ‘energyintegral J=«)fromanentirely different ‘athmyscalar,andchargedquantumbridgingthenu- approach. Inouropinion,thisisnotacoinci- 2 18|cleoandleptoncurrents. dence,aswillbeexplginedelgewhere, On+Wewould liketosuggest thatthere may notbe Wearetempted toéxterd thisapproximate BY .that.wemayhaveanapproximate conservation alsothevectorcurrent) tothestras 4‘=‘whichbecomesrigorousinthelimitq™>>m.7, conservingbetadecays.Wetake,forexample, iiizm,belngthepionmass.Specifically, wepro-“theANaxialvectorintheform R= {posethattheaxialyector partofthenucleon Ot+m) iebetadecayvertexhasthefollowing formand r4p, Pe NS 6) OHSproperties: wen Pg Gm hs= Bete: afAlpp) andattributethesecondtermtothepseudoscalar. ifee #Kmeson.*Thedegreeofaccuracyoftherela- A 2My-4tion,(5)willbepoorerthanintheprevious case i =eylivgyFle) a Ra), inviewoftheA-Nmass difference (which de- 6) VS LO) am" a Bstroys vectcr conservation) andthelargeK- iesmesonmass.Atanyrate,weobtainananalogGR: HAM FO), /ay=F(0); ofBq.(4): fi : +My) B4'=G, a a| F(Q)-FA@) forgt>>m_2. @) Og+My)Bg"=Cbg (6) ie= which relates theAbetadecay-axial vector et ‘Thepionis.then theanalog ofthemassless quan- aes*,theANK Gx,andtheK, oCaaa vs{tummentioned above.‘Thisisconsistent with teenythmecouplingGxandtheKy Aiei & ion rel :: i B ieeee Faas ectedoren WiththeobservedKlifetime2.1%10°*sec CHG, Aamely,F,andF,shouldhave ingeneraltheandatentativevaluedy/4x=46,2/42, weget ne . Lege. 7) Pres FF (-m2) ayy ¢ “net iS Thisisnotinconsistent withtheobserved beta. ie ©p.m)dm? decayofAwhichseems‘anorderofmagnitude a =(qtemafwas lessthanpredictedfromauniversalcoupling aied TSne" 7 scheme gy’=g4'=ay." 2a WecanStillgofurther,thoughtheargument a= { (i=1,2), (8) becomes morearbitrary. Letusassume thata Rsfundamental weakcoupling(WNW: riset ie wherem3m,unless therearenewparticles of"damental weakcoupling (NNNA) gives riseto i foe SmsI aneffective V-Ainteraction (oratleast partof alene lowmass.ThustheF'swillbeslowlyvarying it)oftheform ahs forIq*i<cm3. Theconditions inEq.(2)imply Pla thatF,/F,=1 forallg,Ii'm,=0andF,/F,=1, eneerAeYepAL ts ha thenwerestoreexactcurrentconservation,? and BRONNSwpOPIAT yie: wealso-expect F\(0) =g4/gy=1. ‘ v a)1 Here ©, =iy,, which isapproximately con- heA iY PP: y nsatiadopt Ba.(2),thesecondtermofTA servedbyitself,and7,4standsforE49.(2)or tt4= .‘loatelyciesaiter)eotants,fareth (5).WeseeeasilythatEq.(8)containsinfor- MigBiondecay(Pgeudovector) constantgp,andtherationabouttheA~N+zdecaymatrixelement: is pion-nucleon (pseudoscalar) coupling Gy: fips vA ag =-m_#)= . (21NE (a - 9) ome 2Mg,=2MgyF(-m2)v8Gg..(4)(2Mals264tahynae(9) ,iai 7Y)Withgq=1.25gy=1.75x10-" ergom’,G,2/40 Combinedwiththeassumption ofAT=}selec- ktfy:<13.5, thisgivesar-udecaylifeof2.7x10"* tionrule,thisgivesalifetime of2.5%107 sec PU . 381 the alk = . : |Votume4,Numere7 PHYSICAL REVIEW LETTERS Apri.1,i9@You iki - a Micefor8"»gyascompared withtheobserved value fieldisnotrigorous, possibly becauseofasmi tayt2.810" sec, baremassoftheorder.of thepiohmass. = FARIt‘spossibletoapplythiskindofconsidera- Theabove-mentioned modelofelementary pe iC)tiontootherhyperons. Moreover, iftheFeyn- particleswillbestudiedinaseparatepaper. at iman~Gell-Mann couplingschemesuchas = by Hie(nné)isformallyextendedto(Krev),etc.as ThisworkwapsupportedpysheU.S.Atomic ing Aulihasbeentriedbysomepeople,alltheobserved: EnergyCommisajop. fore 44 decay,processes maybecovered. Herewe $3:G.Taylor,Phys.Rev,110,1216(1958): eren| wouldliketopointoutthatifallbaryonsshould j1..C-Polkinghorne, Nuovoelmento8,179and761bpub,ig satistyEqs.(4)and(6),theratiosg4/G,and Oegolbergerant6.B,TrPhys.Rrays £4'/Gx mustbeapproximately common constants. 410,1478(asa) “eSB: Treiman,Phys.Rev, at.otfinalremarkconcernsthetheoreticalbasis, aaInternational Con|NUC (|!|g__fortheassumptionsmadehere.Ifthebaryons ferenceonHigh-EnergyPhysics,Kiev,1989(unMA: TAT!,yrage278derivedfromsomefundamental fieldywhich ubiiches) StucFiar‘gh’vossessesaninvarianceunderatransformation 5M.L.Goldbergerand’s,B,Trelman,Phys,Rev.|TllintH{|Stigofthetypey-explia-7y,)y," thentherewillbea.«110,1178(1958);M.L.Goldbérger,Neva,nadeborHIT!42conservation ofthepseudovector change-current. PAYS.2,7971950. Jan)IAIigd Afiniteobserved masscanbecompatioic with withtheiFacerareaculateslatmebonighA tyi theconservation iftheparticleiscoupledwitha ytaxialvectorunaccounted for7 resi BeitHI bosonaswas-noted inEq.(1).. "AgainEq.(5)andthesubsequent conclusions are The! sa)if—7ThissituationmaybeunderstoodbymakinganessentiallythesamesetreeC.H,Albright,Phys.toieiyi SnenGeytothetheoryofsupercondctivity origi-av.114,1648195)sodBySabla,Piys,hesteeebati, natedbyBardeen,Coopér,andSchrieffer,* and1650(1959),whicharebasedontheGoldberger Thebay refinedbyBogoliubov.!® Theregaugeinvariance, TFeimanmethod.FortheA-dueaysavebelow,oeteft theenergygap,andthecollectiveexcitationsare,Tenaglla,Nuovocimento14,499(1959). La'F.Giirsey(privatecommunication) hasrecently F iAtf logicallyrelatedtoeachotheraswasshownbyobtainedsimilarreaults.onthexdecaybaseg'onthisWeis -theauthor." InthepresenteaseweRaveonlytoy, invariance, Wenseonteshecifythe'interactionAEN replacethemby'y,invariance, baryonmass, ofthe#field,which'may.be ofthenonlinear Heisen- peandthemesons. Infact,themathematical me- bergtype,ofduetoanintermaiate boventard om ry: thodused insuperconductivity may betalten over fromame Ang"tostudytheself-energy problemofelementary 9,RationUN.Cooper,andFRSobran) |R1 eae particles. itisinteresting thatpseudoscalar Phys:Rev.106,162(1957). EYE.meson snomateny epee ane, Comaht meme fon ghtP Shirkov, ANewMethod intheTheory ofSuporeon. Weyboundstatesofbaryonpairs.‘Thenonzero ductivity(AcademyofSelencesofUSSR,Mascon ots ‘alimeson-massesandbaryonmasssplittingwould1958) ofa eyfi“ndizatethatthey,invarianceoftheparebarjor_ "ty.nambe,Phys.Rev.11,648seo, late cae -the| alWaataenguin aatouly Queonidtonpin gougeuvoromer iSlaroken,)te] HH\ ~ SoyAyPr- tion cea OO of| BSii ERRATUM hoch«oteb a4ene taeroilasclastprckayofene BR;iHELICITYOFNEGATIVEMUONSFROMPIONmeasurements haveyieldedanatuendeaovatconsan) DECAY. W.A.Love,8,Marder, I.Nadethaft, fortheasymmetry parameter a,as.defined in‘|toa|IandR.T.Siegel(Phys.Rev.Letters2,107thigLetter,of@=(0.64+0.58)%. Ttthusappearsave] geeft(1959). thatinpentanetheB’?isdepolarized’ byoneofmat SeHi thevarious.interactions (quadrupole coupling, conigal| JpthisLetterwepresented preliminary results, multipleelectroncapturewnaloss,etc.)which-fconaflHofanéxperimentdesignedtomeasuretheforward-mightcausespinreorientation. Therefore,ajfoteal|| backward asymmetry of#raysemittedfromBY,definiteconclasion aboutthehelicityoftheneg-“ifSan]gen [ishfasbeenproducedbyabsorption ofpolar~ ativemuoncannotbedrawnfromcorcennieto“fslidMaul! izedmuonsincarbon(inC,H,,).Continuing date. ofles:Shed 3a2 Sault it OldFieldTheoryNotes Btn f088SshDharDodeanOlokn Ande ke eae a -©Raga.shoankeasond _-sinumehiea _&,OZ_woe =ORipatiecedee a —--® Somdorsd Tuwetio 8 ee -—-—-f Tpananenass chonere: -ee Iii Rely Gontents ofthis section: re)1,Variouspageinvolvingdiscretesymmetries likeP,C,T,G: a)how the field operators transofmr b)howthecreation operators transform 2c)whattodoabout squares ofoperators likeP”,7”etc. d) time reversal e)G-parity f)parity andeffect onlagrangian and current; pairs like BBandFF . g)charge conjugation and pairs 2.Reviews offield theory method a)the two methods offield theory b) the U-matrix and the S-matrix c)another review ofthe two methods, including Wick d)eescattering asanexample 3.Feynman rules for QED. 1,,‘The Reduction Formulas 5.Propagator functions 6,Pield theory Norms (like C,etc.) 7.More commutators and Ppopazators8.Thephoton spinsumbisiness9.The transverse photon propagator 10, The old pion nucleon page. 11, Some detail on the Dirac current. 12.Dimensions infieldtheory. fo)13. Unitarity infield theory. 14. Poincare versus Lorentz groups. 15. Crossing infield theory. 16, Noether Theorem for internal and external symmetries (various verions) 17, Feynman graph combinatorics, loops, internal lines etc . 18,Feynman rule forvertex inspinor QEDandscalar QED. PeProofbyinduction ofFeynmanIdentity. Field Operator Transformations S 1.Parity: eat6's+4%) OXG4)&=to(-%,t)PAGHF =~Al-x,t) 2.Charge Conjugation: i"Bamc’ =go _BHO =CVMH= ivtth= CV GA@C =-A®) 3.Time Reversal: S4caad =29%-t) SVGNS =THe -t)=i8887K-t) SAGNS' =-AG-b) —heTCP: 2 Cees)8)(83S==a*-*) . (©63) Yq)(63 =4b¥FV(-x) (PG3) A®(ees)! =-ACX) 5.Poincare: UWA) 9OSAA =O(Ax+a) VAsa)WG)UICA)=SCA) (Anta) Us) AG) Ua) =Cry AYAx+e) Additional information: . -\ - CHC =-H WaoHiVVeM-THTesHaye “oy yon Savsay=Nv a=Av% : a pe=ieBy S=Peleg mcahaGlaTaHi? ; ateTete -f Ny= Sy+40% =-¥ Pam G=+aGk Recen=wGRs) GB vaevt. +doy Raceag =—ows Savy =aw gba S= AQ 4 GawHC =ao cata T= lps) Be UK 4 Ublesd K=—b(-¢-8) Wdips)«=—dtpas) ES ro)o_See=6SgeNCseseat ee _@SxS"=ve=2BQ ee A ———j,9 NON = GO)2NEA CRA ee a ____attee),alae Ep SCONE AGgeese eaCoeeto —— ye Ln gy-220. Ge\BS230f GB_SH1O= 2wit=ae Fe Sap iat ftaesGames [Ra aen A =a\ Jospsaye soda] ze eee zo 8 “0Friahtye,DME tensSean ceaaghh eee ee eee aees——-ANGE ENsed =Cay Whee=eV Me os 8kTer Re)2ASe eS EEL tS EOS RT een een aseAO!(issue) 2“Se. Gee=eae OT aEe. - pote sGs atyp eee LLa ESS : . ®Vasey" nn ee oo. .Ot Te SS . ee eee Pee a ¢ rt * oe . : _Ss__\R@= -GP! ee _. GAMES TORNS ee a?| SCRYERS ERSTE) AOE FL Ste gate este of) _. der 8B LCL TS bake LL. oo. 0eet [eee _ . _SSeS le Pt “Q@eRRS HSH TIWI.L ==EPs) _ “CWS SEE LET eosie=¥eres) LS (eS 9iteeaar eri@esy= WERS | gersVEVvE S| ayes Ssh_ aSY Ad2Se Ot 2Se aeAGE S385 ESAU |e=SSERD =-l2ETS PO EP CQ 1) a i aa eo * ; °oy @d==ceegh=R=-1%* > Sel @s}+-\ Cee eBke) a a. G=*l Geet - . @R=eRh)y .RS= -BK Vv - - 2 Che-ac | Re=-EK v ~ 8ts-ce RG =+OK ; ‘Sone ofthese equa ionsagree, withLeeandWick, somedonot. Eg,(2.18) through _-tose (2.21)-0fLee-Wick-agree-with-the-abover -I-think-we vanBetrid‘Ufthediscagreement by_redefining theToperator sothat.newT=GR,orsomething like.that.OK,but. - Tdont want todothat now. What tokeep the above relations strictly for the BD ‘conventions+ .Time Reversal. = 1.Weknow theeffect oftheoperator Son‘Yand bandd: =\ >| sys =TY Sb =bEP HS) de. 2.Now consider a"process" and the defined "time reversed process" + ' es re! We~<e8'| GIp.S> ewe@QO< hs , ' ~P3 ~33 mia es MW~BS)%\-253> ~eOF Notice that electrons are not turned into positrons. It's just that ¢the final electron reverses its momentum and becomes the intial electron. ‘The energy does not change sign. 3.Now dothe usual trick toobtain: Ss ro} Slesr =sBENT ID=BEeD0> =|-2-D -\ 1. aa m= 88) S%3 13> Well, notice that script T,as Ihave it,does not change anINstate toanOUTstate. Itjustchanges themonentum andspinofaplane wave state and the state remains inthe same picture itstarted in. 4.Now assume QEDsowrite HasJ,AU where Jisthedirac current. What can be said? 1 nn Vays molMM~LES)LIS \PPPBA'S By ae ~Ss|TAs? Aw (ol y\X*. mM~uPSYE™n(ps) - I ~ teCc min &C0S)YauC) . Zoe “S/o hanaw TAC, G1,8,0,¥ TWdtsnowallhyyouosaumeoTonkOringine?. NondQarechviors panirQB. ToastCalfapos,anmachtwopogo. Wo nee Tr= Q-L,sometimeouTHOcam LoxAndadark.Kreg, Q=\andY=Z, youSus Mok @=0Wancl allowedstolemnMe AX,soT=O imporsiare. Swmor horgrd srgeusteled, c- 0ore WHE!P=FaoOcamdhauyIyercharge.CPTcommat hasugeSnyperdaaageAimeeHaLoletia wil ansreepaak. DuryGparityondagood4[prosene Sos cvotaar& Q=0 Come 4sausten ohaysteura A,dabmhe G.ManG=GxG% oe QosZ:febos~closoS-o8A,onleuyaryaoandB=O~ GQWRconnAmy GeCryPS4O Spamdevoer. ©.faery face Mote+spinisgamudovactinsixsamalfached lyspace UAbustaw, Yoke: pardorcator onraony (B=0)Sramaagain panciyAadadiue JoMe anu Heied:dheparity((urmsnd) ofaloanyon comnst loxdalownurad doting awkiloouyone ust las—. Gilde —8k(O8 =LCnt)CineO=Cx) PQAHKa&QCX) CNRDE=CK) oe,©ale=+aCe) <\ io) PLANE=ER) Part =—dces) P=e, C1)" “aks Coosa=(anboee) e(88)=-(-i N(Sua =-%Consttars) ecm) =+Cy S ChargeConjuopiioy olPanay . ALbesBes, VES) =LPRIR SCR OGs,s)BRIA) Lod5 BS) goats: $Cke)=COSO48) .48)=CrASS) Garunkagspunsondy!) Quer C=tS 2.deat. “3(na)=SFePL'SCeeg(ss)BCes)aes}[o> gs)~Oe (drhosidspin\*SMA -Swiar Wd @ e=cs 3, besepours ;ats VE)=os aF(es)huis)(oy ates@=I), asBudnivoyvunideonCAY™sineShiateover Sige3mustlo+)Srlooser, *Choseinshedou:(Sucker) geusd =[Sit=Z,<smserts(S>|sym>|sem> iat=CIPS Ga) Aop326o\PDG, CFO Fadcratlinad tadonibd&oxype_coesonsiwomy) he unita8uspadaeanoedijeraata. OkeolajickGu.bur Tia,coDed a"tousJuco” fheSistda aseomnostda)inikal fsauss owokimen Onin” 8jactesn!! ——-@ Mad,necmaidino Weomadiecowmnatig TsTessas prikore do__——Lhs—drr_prokine +$0xt)=Boxe CEU) MiaB00mu Oi Ceciansbebenllenae:—=55.-— oo ASHO) . 2G xe)=LolTGoadtaa o>=__<olTuna) E™Vlo> ——ho9h=We8,3Gus), sadQuodloronmamal ordoud a0. ein aZoltGay Gna) 2LO)ly Fao 6?thom, Dueusndd lor©.QenoxtSonaio pm ent)=SEVaGalEEGailey) My)KalVBS Be ae ooi.‘ ar _—___Gombackio uwmdd be 9 eee eee . (hwcmmpudains meldody?Oheonenmal pespagadne. Wace ansdroh” *- @_A_dull rouckmronvions olporkurloatem (hoo)ompurater ih a Meigete OS ©DitaDatadolOitaattaQeddofleorgan aula ao a ON: oO-jhUosmctaix.omgFieldSuvory ———. 2oeANS ‘Rteaenee: pp.Asi>18Sobeonas Oo {>Riana.Weanewardfooudingopwalarsandstaleveshona.ontneSehatdiga,~fidtarn,Fa,Osamopueden, $8.50starauctor.Wetwau Ouatt0)—pinedaprardanee izivethsstoleoectr,mowed5iG9=HOA;dare BAWSE.Deongese HO=HPTHS. Veohonafoumnde tu.situation -~- Koon aaSvllowos: rye - —i a=eeecute i CW= SHE Sth Mow Jind Sree: _- meee ee®te adHrgo®. O=ilte,d] —- -NetwteatHOHte,Soi.Oiemune pichune,theopratons. oat.aiven" thyHosDl.Heme O,rierLootbeMeVerrenlarng) gichuw, Osoin-- Qemerspicks Qaslaleamchan, oxdius”why:bugWay.se.Saary) .o) Gaomrge aSewdig® UAWinn, - - ee -»Also nuobion Chat,akk=O,amuntuachinn pictureoperectn”eyate) weoimeddedwilh.tswiles Souddingen eowmrpaib ——.--- ©hsV-ograten, Wearsaud”deinbuackioupidune’, DaasWaele,Qe- jopoata wlialy pingagater Ohiunlaactun pictstelevadis (lowes): - SO VEELGE) ABWGDaHeWie)-—Lpom0. o Manat) HPWasmoderns cytootHedate.Vanesheclagelines,MewvinbearcioTShoye“ke!Cworalete,.)feo. (RLU AUG =DS Ueto wee NaoattAadandaeAGGIE KGun,BE : & : - .aluaesy-1] =RADU a oo 4 xne)Qaww:Wiad=14SHa)URa -eee “DreaDemonyJomoud Ulupelaqation tideinotOreWut e -o- OF—-lthicctgpetin. quadabegal uquadionvaitle. bara,aeetuerdty Newman - ~-Enid, COMMEREMESHEEa,tayTKpasdnoaoe -= .oeSiMe,AlfkmghsSatunatasideINapdoorSlakee,DpSse-- Ea, - ee —~— Le. ~-- + de=DUbs- 2) - .Dow Pot=plt=+e) 2.eeeeeee TT Spm Sanlu =LoeaIdesp TTT -.fox,wnO(te)= Ue2)dCs)_: :OS'irdot=UGS,-©) bu .coon . : =Coin U'Gss)(diod 7 - © 8SE UG) =tagecankeropal f= TEnp(ase . . -|. — ®Commuter JShans.lason/ bakheidea.vebo dea, . howdtasSen pide’) she,ikgdanepe: oe oO FieldTheorySession April24through264\QQ5” Motivation: Iwas trying to_read p139ofFeynman onparton calculations ‘and was unalbe torecall clearly the trace business. Aiso, Ihave been ~ wanting tosee howfield thery looks inmyall—norm notation. Finally, I have been interested inthe various pseudoféeldtheoretic cdlodlations ~ ,ofthe massive vector mesons, All these things justified this long session. Applied Field Theory. Ireviewed thebare bones offield theory, skipped over “'postderivations anddetails, otherwise I-Would ‘néver havefinished. 1found~that ‘there are basically two methods ofdoing calculations:- Rist Hephog: inthiswethow (1SZ)youuse’thereduction-formikas-to -~|oxpreas esEnatriz elenent interasofagiantvacuum expectation valueof -«>4-9tineordered product oflotsoffivtdss Prior-to~perturbation-theoryy - -~ithere isafield for each external particle, and these are all “full” fields, 7 —{One then goos-to aperturbation-theory- by-transforming-these full- --~ |Heisenberg Piture fields toINficldé viatheoperator U(t,+').. This~~~ |-caunes-all the-external -particle fielde-to become“in“fields (free. fields). -:andyou are left with U(~inf,+inf) insidé theT-product with these éxternalpofielas: ‘This:operator-U-is-then replaced-with-the. usual-exponentiated .—.|interaction Hamiltonian offree fields, and each ocourrence ofHisnormal — {ordered to-prevenit the possibility -of-tadpole type graphieto.... _._ - You are now ready todo“contractions”. Choose an"order" and insert 1the corresponding number..of H's-and d4x integrals. Then pair offthe. |_ fieldsinallpossible waysusingWickstheorem. Inthecaseofeeelastic @),seattering insecondorder, éachcontraction isaproduct of5propagators. |After doing all contractions, you put the huge mess back into your LSZ;Teduction whereupon alltheexternal_propagators are"amputated". Then _ you canconvert thewhole thing tok-space andgetoutthe feynman rules. - ! "What égn besaid about nopns for this first method? First ofall, the“square root factor" or“external wayefunction” owwhatever youwant tocall|dtmustbethesameasthefactor occurring in(§;a)field/creator commutation|Felation, This"external wavefunotion"is détermined ineffect bythésizes ~~ |ofsingle particle states andbythesize ofthe.spinors forDiracions.|Remember that fields arethesame"size™ foreveryone. This, ingeneral |"external wavefunction factor" ¥C,. Nevertheless, itigetill Cythat|Sours inthedefinition ofT, 7° “= aNowinmost norm systems C4=extersial wavefunction factor, butGas,|aeWeialisanotable exception. (seeKinewheetdaggers). ----— ‘ When all issaid and done, wehave amatrix clement Mpeyn given ~ |exattly by“the‘foynnan ruleswith-no external leg-factors+-the connection --- |between Mpand the all-norm Tp; ist tTe=+469) Gel (“Gas)~Mean .| - vs cx HInanyofthestandard (daggar) normsystems thisreduces to: i :oO- Th=HCG)Wee _ |For Gasiorowies the result isinstead, | -—-— | wee eee ee | Tks +¢ Tlam:|ket ffA@atMine, a ‘—— re 4ae -e2- oO ForGasiorowics itisasifhehadanextrafeynmanrulewhichreads:"Fora ~GS0hi ‘exterrial”lifie iiGlude afactor of (~)2r - ForKallen theresult is: ~--) Ske=CY)Mhe, —Geate -—~ ——~,,~,¥e should noteinpassing thatTo,isnotalways C¢independent. ‘hisisbecause thephasespace“knows” aboutCyinGertin novesyste, —— - . like Bjorken Drell_and Gasiorowids.Inthisfirst method (1SZ) eachtimeyou“pull” aparticle outorthe~~~ state and into afield, you get afactor ofi. This issimply the ithat ~ occurs inthe inversion formula “(1279)page 27resulting fromatine——— derivative onafree field, The renormalization factor ofZIalways set ~~" "-““gqGal tootis,atidIuséVedormulized (physical) masses-amt couptings-in— ~— any. caloulation. Second Method. This technique ismuch more efficient for doing acalculation.- -eeToate“wayINcreaton-and annihiletion-operators-for yourexternal - particles, and you replace Sdirectly with the.exponentiated interaction - ~-Hamtltonkany-Then“atonceyowdothe-contractionsyEach timeyou-contract-—— afield with acreation operator, you pick upone ofthose “external wavefungtion-factore,-The-nuiibet:of-spacetime integrationsequalsthe- —— order of the calculation. It is this method that allows afast derivation ad -of-the feynman-rules—f: theory. This isthe method used-by-Sakurai, — fe) ‘peepages130etc.The factorial factor, Ineither perturbation method, there isalways an—~-—____—_n inthedenominator_for_an_nth order_caloulation, butsomehow thisnever__appears inthe result. The reason isthat there are n!"contractions" which ——--- —~belong tothesametopologically distinot_feynman graph, so.the factorial __ isalways cancelled. Wicks Theorem, This isoneofthose combinatoric things that isbest proven .—nlybyexamples, because thegeneral proofistoohardtounderstnad for more than one day. First, you prove Wicks theorem for two fields. Todo ~~___._this,youchoose aparticular timesequence soyoucanremove theT-product.Thenyouobserve thatthetimeordered thing differs fromthenormal ordered __ _____._thing only bysome field commutators, and these are always o-numbers, 80 . a the TorderandWorder canonlydiffer byac-itiiber. “Thenyoutakea-VEV~ toshow that that c-number isjust the VEVoftheTOP. ~“Next, YoumoveonWothreefields. Agaid; ‘picks definite tiveorders =~The game istoget "from" this particular order, "to" the normal order._-—"~"~ “This Tuvelves shifting“ all-plus fields to~the-R'LEFT where-they-can-do ~no harm. Each such "shift" causes aterm with apropagator for the two — ---— interchanged"fielde; and-alt-the-other-fietds~just sit -theres—Eventuallyy—-- all fields that donot appear inpropagators end upnormal ordered. Ishould _- ~~~“ do-a better proof-some-day-on-thisy—— —— Note that any VEVofaTOPofanodd number offields must vanish, due—--—-- +directly to-Wieke—theorem—————— = =--——- —___-- —-. a Oo * a e -3- BisBNdlea quoerrcnser+ ie)Braces. Whenever there isaDirac particle inafeynman amplitude, tracesHIToccurwhentheamplitude iesquared, Theseariséfromtheprojectionoperators and the Inversion theorems. tt; - te SS [+%$ z =m) (1+8H ~ ef - ~~ letye- [x@X sO] =AQXa X=KY Rather than first write te amplitudes ,then squere them, then convert |totraees, ste, "itis eaéiesttousethe("trave dtaprans'} wsillustrated énthe back ofpage 12, there for electron electron scattering insecond-“order QED;Theideaisto“form allthe-"unitarity- like"digrams-with =—~-appropriate statistics minus signs. Then each "line" may beclosed onto |ttself-ané stands fora trace. Zachtime-you-hit-an external particley— insert aprojector. Each time you hit avertex, insert the feynman rule - +vertex: factor-(egy ~ie{*-)+—The-trace-cleses-when-you-return-to-starte -- ‘Sometimes there isonly one trace, sometimes two orthore. Don't forget~ +‘the-propagators apropo. henyou:do--all~this, -you-have -at-once _/M/2and .|youarereadytogoto&crosssectionorwhatever. a - = | Compton provides another-example -ofthe.technique.—(See. page-15). -Again|you use vertex factors inthe various traces, except here ithappens|thatpolarization 4-vectors-are. tied_to eachvertex.. Also,in.this.example. |therearerealfermionpropagators andthesetwomustbeinsidethetraces, (@!: Ingeneral, any quantity from the "top". unitarity graph_should_be. |complex conjugated, eg, propagators with ie's,polarization vectors, |coupling. constants. Usually..we donot worry_about such things. LoL | In actually exeouting the traces, there are very important tricks to|beobserved. SeeBDonthis,Ihavenotgonethrough allthesethings here. | — |SpinSuge:ZonDinac_clentnonas Spinsummingistrivial, Aspinsummerély_ {clears the.projector ofits spin portion. Ifyou are averaging over initial. .spins, donotforget toputinaPoreachinitial fermion. Forfinal. fermions, summing isdonebysimply ignoring thespinpartoftheprojector. . Porphotons sndmassive vector mesons thespinsumisnotsotrivial.Forexternal photons, thereabeOnlytwotransvérse polarisation states ~~although you could combine them téget circulars. Itisbest todophoton polarizationsumsasthéVeryénd-of-a calculation (asis’not~thecase=~ for electrons). Basically you have todothe -sum explicitly. Ihave done anexample on&separate sheet (ie;—the Compton-case)s "Note that-the three-1aafourdotproducts6fapolarization vectorwithitsmomentumvanish :-Cthe‘three-dot vaniswing isnotrelativistio- statement «+.-we-adjust -our-|frame byarranging forthistobetrue, thisistheradiation gauge) -+ For massive: vector mesons; there-are three polarizations toworry--about~ .|The four-dot condition still holds. The parallel polarisation vector gets - !stretched; seedack-of page20:Againy-you~have-to.do-spin sumsexplicitly. i|par, Howdoyoucomputefieldcomiiitateredtiméqiat‘tiie?You"~~~|Girfteyourlagrangian andcomputeyourcanonicalmomenta.Thenimpose oO |theequdltinecémPélé. Makemomentum éxpansionsandinversions; andther |Compute the creator comrels bylooking atequal tine. Then, using these comrels, the théqual time fiéld Conimitators folaow, See pags 20. ~~~ ~ 1 - \a.: H - i 2- “ ~ ad-4- . O ‘The REO decay. Now weare getting into the realm ofphenomonological= “field theory calculations, asin-wéakinteractions, For-the tho ~ _interaction with two pions, wepostulate aldgrangian, . 4— SevigderdBsh4sshown onpage 24, this contains two terms involving the neutral rho.-——~- —-Thenyusing~the “"second method" we-oan-get-the-feynman-rutes-and—in- ==particular the decay matrix element, Inadaggar standard norm wefind: ~~ ~ Ty Here, pisthe4-momentum ofeither of__[Tal =aype tne tinal'pions, Thefactor of2arises“because therearetwocontributing terms~ Using this amplitude, we inthe lagrangian.°GanqaLckIy compatetie— ~ns width ofthe RHO )into these two pions: eeeg WE(|—Sm - an Tgete =spegtte oe _ -From_theLagrangian youcanshowthatgisdimensionless. Here,Ihave — xumumixarerxtiaxt trivially averaged over initial polariaations ofthe RHO, a+ —.—+Theresult isindependent ofpolarisation anyway. 9_ Usingthefirstmethod, weoanshowthat: —— ee Scale wee a - OCF. ©AONFOSE ND=_AMLA(S)- AigREGAL _ Except for normalization, this isthe result quoted inGas page 455 top. --=~ —“We-eould havearrived atthisformfrouLorentz—invariance-without— —— ever using alagrangian and perturbation theory. Thus, wecan justify — - that phenom, lagrengiar-in this-ways——————— ———-———- —- ~~ ———-———Rhe-production. Itried doing-the-i#ttie-perturbation-calc-thet-Gas—shows——~ but Idid not get the right thing exactly. Atthat point Istarted this summary "~~ -- oe = cocoadetA"mutate404 ee u's o_unamupexerise So.pastorAmsig..d waddekelorodine Han . — Laganqisn, rasaltop — -——— . OL OPS Ge ERENT eeDo © [iatmittenadtoiinaldebateJainabe S i k ecunnandarnd)AA; --- aOwnSada.cysiclsiy— fotLSBcaesar, ulesarSanna———. —— Nio_panhustastiion Sunny.ee a= |Mea,UB.torauger doloraKlamalpnmatvon adabiong Queusdeancbing —| ——__—-{pane "ienlSde” Waoualabdosonsbuact acilbewstand — TGR etna dpDeneue)Mowmee2Scnaden,SODIUM, saben es eee ee ~ gee Ue eee ee COfoves=7(effete Gull oe ALS Zaman ey ARSEaWitmerSemele) HIG — _oOSock abanVedTo am ron_ -SLs SATCOG)Range).PEG)UGE)UGS oO; YooSAVE FUE) =OMUEH AGA)GeO, wee ee ee HUH Gt) , pegey- -s“ CepaSe 4 222 insnaBowate MopmdDaWiat SaSAENATCCE) ERIC NCEACCESACNAC|ILS ETE DARE 4aTS asSerer7ae LETS IY ShAe ee CECA CNaCe de S Fexgrernam Quates(ae) =\= R=-CC2n)' &(oa)| 3) S-\=R Cx)caTHi)mM “Rules GonWhe (tres) Ly ier" cack wuden J4Swckermal photes .Tuete.Bractermoh Lommian, Jymre7FCAmocksatesfeminine Cpatinganes) Ya ge =LDPCWecodLdewah photo Dire . qtte e d ./s)\scookUdewol Loup J) 4Sapitermad Jermncon Love YD -Anatipowrchours actadeed Alum/enccforn. 2)+idwe coumberterm. atZ as , 2. &(all,40=[GBHerhomeyTet Conta* . \apdHeomeneComan. AD .) Lo) ;S+\=eeScoCBnk _* Ji)-ie* c&sucha Varker pee RD : vSWESpoMntefdine.Ceeave~2e) a. Reaction Formulas (forremoving oneparticle fromtheS-matrix) Oe OE) [EME B[cosy] e| oA fe ORSo — O(A[eem®. fa9 e| -)(SS Je th etre & 2a (RJ Teaap a ce)0©(We 4WlPues,et] —_ —aye EL3lp¢(Fe)le‘Taes)Bleww °| ec (Ele rend1ABl el Sh Gt)(Fe)le “|¢rPataNOs)otRET Remarks: Time increases tothe left. All momenta are defined asbeing tothe left. Incoming particles always have aminus inthe exponential. The spinor indices donot closerightontothefieldoperator shown,but,thrutheTeyienaThus,theDiracorderingcannot,bechanged.Statebracketisalways(out/in).AlphaandBetarepre- ey T(ADBY) =RCAM BE) +BH)AG) . be=eGynLAM,BO], ScalarandSpinorInvariant Propagator &Conmutator Functions (exact.Bjorken-Drell notation) j @ zip S Sipe|Se Sacemel?| A= Sue’? |iSr=6@)S? +as” BAe=OB)O074OCR)A? =<oT(4 #)op =<oT(&g)o> iGS)= SPs? ps4 =BO =LoUYFop =(440 ‘ . ao | a) 1Com|8=O|cowl|=O! -S a . ~) ~) *ls(H-*)=—LOLPE)AKOD\0D Rex)=+LolIH)QH[O> | |@F-m)se =+8@ (a+mt) Ae=-8@) a Ln |= wo+Mhenwiseapecasieg.mune™ues®* ' ||Sf=Gam 2°) Mite gape aei CS) a& due SB (nF 2E B|ese 4S? A=Oe 1 o™~ | Cy|FiasthenyNowsSinehon, = a - ©Aveeno QRGALbkyuuDangBiiationyeenone|OYacasitinnall uso.ae ee ~Gammne 45BkoecDinwpasitan Selpuddhieseeinverte. =2Wee hsADombie wadeOokAB=ABOAB, Ohiobade © Queme amoneanakdwt.=G/CM Wniemamne tame.- withbenue sachomssus exluwer QoqSochors ofanMawDeruinallCypullediyTandafrsichtnn ATie vehMananmuewmd &OKoweeDatayoud?Chnoseadh.tasARfoo ol.<pteyh = -nee.Gs=[ONO|"=Qt, - _ -. 4. - ae Ce—@Wefutonpamaiend. Onprmememine APAOM (| sealanotal fats aweley[ewes das | _|duagdsedanSal:§)=SEEGeHMEsHa] jDiins,asd: “gel=BEEceLethamagesdiyEY :-44S43 li,sy+file: themSpan Ko286[atonetnhe wet‘|patie seinSad:|FRE oFfauneranes cher. Fe © Matsa’ tanmabeiees io2 | Bedal=ae [aadel ee- | asa7 nip Ta, 6@)= ote5TGOLBELES Cywacesn et aaa oO|Lees,GolsCyEee ee 2g TRO, dex=Cetery Oo Oo1 | O- [FiskCouumutodoud. woo.p- [HseCeamte ~ _ ‘Seton.jb: [gadal =iAG» —— .. (IMO, BGS Ciprmle Ades) =-iSeley) - a UtaRog]=cig?AG) . —_ Ware ovacnSactd ._ -eeee~ . Lates,ey]S=2(geag38")AG-y) -0MaletatAG)=AGeyjoe) daamte --ee oppo pas OMA - —_a _ . -: 1 7 -—- : -~ oO ,Tromeanogs, ee eea ee ©|Sestan pata dete). =LIT3@FY : ede SEO Lp” ---ne eve Lee} - . Dirne anh oo . :” RSCedea =<AT(MO BEI—- ~ .tak hoop ge TT“oT “SerS[aaa - ce TROT PhatYd _- 0- ERG) =ColT(AnedAG\\lo> _ i mo TS age SHO) Lig yo “ho 7 a To , -- ~2th®p oo <iDpagpe =<alT(4.04(alo)- BeSEEM[ig+4Rule bsetle ~ 2 StU Myo | iS | CO PidSinSewe, -2lee Loe ©©Seas ieBDp8ie enmenstn -oe. wn Wek,Aemene Bacher apiedummy, Soaacoumanole! wk SATB ee] =ans28:Aeeef AD akga0Dee ne : ~le~~on at :an oo Qn Bete ee.,Z&eso, Beso Gaal | Wa Ziteela [HoaSeMSJ= C4Eade) Lm BAe Kisco) SALAreRGsedS]| osapdepeasn(sedef2afbabesuey| omDrsfullyontengad Cregionarettai(asst) YQ O- (TeStonyonOuramos PhotonPropagat, 2nas : 1G SEF Tasleases Fase] OT 9WEE aatigoaaatheb edChasa” a SATACAI =TLORC LL Qh Voneeane cents GasesAPoaetySelweyhavent DRG aa=StySEPTY! S,Stan)cha) 0.GO DHsangdostesGad a ---1)Sheqe -yoBl=Rok - lechCyae es ' _=RAK =ROCAef40)=ae\Gtay_<o. BMS IECee[>a Oa apeeesPeper§ ywdeLapphonaomabin SeSAR), ree ASA, >Ahatan(At A)=PEM Seni . hiaonli,doedietsiemuswahifiet .Q-_waeamtqelgeCnunanceomOanpiskploneBacallv NO“ - . ou -eae ‘sRee|theteaSe¢7. aadxUnsestig ackGasoRandre ditapwagek - aeOTAY geeDEP ark"2gaywhtera anmesand~ywbebeep2SYRgee aeraent sm1Bhe.c = =YM weoePEMD 2 SRS kn a S .2Z6ge + St Ly —— . me me akeerent” *PCatale - oete oho’ BPS mM cavesManewachirreporarytingow sdLouse'pA AYeyeaes _ nr \.aCS(Rx) aor DRONEN aeYererePAR aide ot~8UeIROa) NEES veeks : ~oye SAD 4.5. rs aa LD pnosoFt ‘RD ja! ah ‘isseaeoyre-OKC. iyso . GRO) =eVOIve a4 x. 2RA sawed Roe va Dy d .oveTandy sorts -Vadnees,; .Pern“ Sgvithdain |ARH,=iste, | a ee Fe oe Peanl ange le RAL URmS cedkera ahte ~ DR ceat leet POU ©ake &=£(qria) SS blow_wucison pace Q=£(4%+4) t=(83) BEG) Ee(8) =~UgoTAGE? ¥=-g.5iGem +oFterGig,Bisgnte . ~WageRitspT Tocarnpult amyormalsuds ,uae’ vesKoBAEXE) Feedv6[She <sey | ao wm Nc} ;° o= [oe B= (459) - 2,30 .Tee*2lf°| OT=GM+.2O4+R 2.4. 23=[4] Rade:)K=()&pag: GY&vom. :.. 2)vac0,4Sapode|VivGyankigarde. Erowvsle: yf _ .Sah (IER9Toru aa & BLECOW =QPPTegeiVam they]Peete +Ae—t] . Hermiticity, adjointing, normal ordering, andthedirac current. oS 1.Adjointing operators. +ae ABle> =\q(AS)=BA Aine v+ Whenyouadjointaproductoftwo <elSe=<aloperators, the order reverses and =thereare'nominussignsevenif Salat=<4these are anticonmuting dirac fields. Also, this reversal ofthe operator, the adjoint operation order has nothing whatsoever todo has nothing todowith transposing with matrices. When you adjoint an and complex conjugating anything. 2.Adjointing matrix representaions ofoperators.. Square matrices have the above property which isone reason that matrices can beused torepresent operators inthe first place. With matrices, the conventional adjoint operation applies: twiddle and *. 3.Playing with the dirac current operator. P Inwither the first orsecond quantized theories, the dirac current jyis& hermitian. Theonlypossible pointofconfusion isthatinthesecond quantized theory you donot throw inaminus sign because atnopoint are the fields anticommuted. See 1above. +at oentAg yee A=LAG =CDs MV it. + A“fe LTH =de Notice that the adjoint ofanumber isjust its *. Asshown inlecture 22 of230A, the hermiticity ofjisunaffected bynormal ordering. hsWhen does that anticonmutation minus sign come into play? when you charge conjugate the'direc current! .5 4-\ a\ tv. +CyG= CYC CYGWG =YECHICHY Trx . =VOGT =Gy =Ay ce Bytheway,thefirstquantizedtheorydoesnotgivetheaboveminussign. a ©mist bechanged to-ebyhand. = :. a os 2“= GENE =ov Drinonsions weFidoThess E=t L=7 Seoton£02:pmE atv Ee” -3ivield vitee%ato©4 Uy=dumausindend OD Cuda: 0)VOFwWy =Se=E/— GQ-entdte EeE(w @pred E=E- ()-eVHA =eere/Lv ()q¥G2Ev= Serv Waalra :psBovcap nw£B a bxyrs<ehoood 4.22.76 Unitarity inField Theory - - Ce 1.Because youaredealing withahermitian lagrangian, unitarity inafield theory isautonatically repected: ‘So 18Lérénts invariance and cotiger'ved quantum numbers; Wealways make’suchabigdealabouthowhardit“isto satiefy unitarity, but here in-field-theory-you dont-even give-it a-thoughty --‘Formally, unitarity follows from$=el,Butyoucanshowunitarity. onder ee byorder in,say,QED.Herearesomegraphs: . .- - ae eee -OreTy=YadTeTey - : dort =0 ~Lordsr &.. - dmFA=ade (ed)x(1) 7]ondana 2.Infact, unitarity istrivial inQD. ItHolds for each graph. #11 you ao isputtheintermediate lines-on massshell€utkosky-like. -—~ — 3.Ofcourse theothersymmetries follow frominvariances of.theLagrangian.6 Suchas:lorents invariance, isospin invariance, andsoon._ : 4.Well, what ever happened topion-nucleom field theory? Igather that the _ large dimensionless coupling ruins perturbation theory, but what about theaxiomatics? Andthenthere isthéproblem offundamental entitities which appear inthe lagrangian. . : Comments onthePoincare groupversus Lorentz groupproblem, .‘Theproblemhereisthis:whyistheRHOsometimesavector(4,3)oftheLG,bat fo}other times itisa(0,1) ofthe same group, and other times amass=m, spinal rep ofthe Poincare group. How are these ideas all related? Idont have afull answer yet. Here are some ideas though. 1,IkmowthatIcanconstruct spinor states oftheform\p™*) whichtransform as the vector rep ofIG. Insuch acase, neither ofthese indices isahelicity. T canthenmakeavector wavefunction bychanging basis, sothat wehave|p*) and @Xp) asourmomentum-space wavefunction. This transforms inobvious way u(t)Ite) =2%ite) orsomething like that. This thing isjust afunction. Obviously, you also know how the state transforms: uJ u(L)/*)=LiAp’) Next, you should beable tofind out how the creation operator transforms. Then finally you find out how £he field operator goes. ‘The conclusion isroughly this: the field operator (x) offield theory transforms just like the wavefunction (spinor) @(p), except perhaps under translations. e Sincethesetwothings areroughly related bysimplefourier transform, thatisjust what you expect. So, both wavefunctions and fields should belong torepresentations ofthe Lorent group rather than the Poincare group. Atleast wecan see the Lorentz group inboth pand xspace. 2.Onthe other hand, ifyou take any massive particle and put itatrest, itmst have some spin sdue tothe Poicare group. Ie, itmust belong tosome rep ofthe Poincare. The RHO has s=1 for sure. The spinor state Iwould normally make for the RHO would transform with (0,s)= (0,1), not (3,4). This latter hasthe wrong number ofstates (4). Nevertheless, intheories ofthe RHO, the three states are often embedded into a vector field. Ithink this iswhy you need guage invariance, tocompensate for this erreonsous, embedding ofthree objects into a4-plet. InS-matrix theory, Idont think there isany reason atoll tothink ofthe RHO as a4-vector. lo] NewFieldTheory Notes Sadow$nNous Jd.Momoles) ee OoSoran)oa;EstiGovaney .. © (node c\_dauaa, 2 y Nytnnn _@eeVsanahstaneSGseomaFahne) oe -==®Distekias Dowd summoner shi__PVRiiyhDuaaGasme@ GTSwunchnaea. |BauaiorTasafmnetes —- B® anges Sedu$Scum SO iain fo oN Grossing infieldtheory. ° duly7,1977 ra)1.Considerthemainreductionformila(16.81)ofBjorkenDrell:thefieldsinsidethe Toperator have adefinite order. Now lets goatonce toperturbation theory where this T-function isexpanded asin(17.22). The INfields in(17.22) are ordered inthe same way asthe full fields in(16.81). Now, the interaction hamiltonian musk contain aneven number offermion fields (as, eg, inQED) toconserve fermion superselection. Thus, you can move afield through thesymbolic exponential in(17.22) topline. Thefact that Hy isnormal ordered makes nodifferen to,this argument. Thepoint isthat ithas even number of fermion fields. 2.Now wehave to say something about how weshall write states. Notice that: aati =Wy = + SB sett =Auey = Sola This isimportant. Wehave defined astate /1,2) andnever deal with astate /2,1), atleast not here. When you conjugate the state, the ordering ofthe labels inthe Ce)finalstateisthereverseoftheorderoftheannihilation operators! Oftenpeopledonot order things inthis wy, but itseems right tome. 3.Therefore, itwould bemyinclination toreverse the label order for the final state in(16.81). Then this equation combined with the other equation would give: poe7CytanSeas) me ~ — <— x =GY SeeTSagQed(Ot*)4sew) SeGy) BrSey rR ¢SAT(ASMHEY-BYR)QO)+QGm))A) BAAS Pm pes —viK =o) (Qala) Malye) =MACgn) QabG)-- PsdOn)S ‘Yt Now you can see what crossing will do. Without changing any orders, you can cross particle y,,=P, from thefinal state toaposition totheleft ofa,=x;inthe initiel state. The total 4-momentum gets negated because fgoes to£8.Any 6 spinindex.. well,lookatself-reduction page.Au(p,s)infbecomes u(~p,s)which isv(p,s) orsomething, soessentially spin stays put. This crossing is inaccord with Taylor's crossing rule, except they have nophase "over the top" |(Modanfretwote) BS DASouadorsYaRecitation [DomatvarEidersuntanvibedningNoha| Qloata: crnihe,onywantin,Te4ymadal. YorbennoOwak SL=ab©Tp3Gp8) BukfunnSadergorHoweDek,SHAg=A.W"so Bh=(QWYTB+TF.HB) =[ISR SwiewHemennstum uwhwadobe Baral \roraneLrswrd BuskVS=HO anhOrsalae adhwordbins Shasssymmaliee Roodusbanc WePaguingon LAr) ia.daPinadJoge,Oeact wtJrvomy Re@,WeakfeeS,fheaifelsclutume. UYbes, ydBSao,”OkecheegongatEndyUaMerkplace.‘Whowviaspelol)a“sepwmasas AiC"vsersaaBRsuckMakBH=o aakLootsudMokYhkst=o ,fram : @ 7NoethursMansa, Conumastss @wmitra,SoonSladek 4DueOngDrakwalebobpw Deivadion: (gulesOrssnleusion doomanyJidecodwoiakdocfader, TdBoma.cwhy,onaJad.Weshatvypositing, contac SHO=goredas YStaopersedve Suien SpLBAI= BpodSh+a@noR), weaswartyodSL WeOreSovoudwey+rseamSinafrins SwVUD eeeBh=(SE)faTalsTW[dh [email protected])] cane WsAk/d[pd]. Soavdajuns. aran,dosere Dok: a=GearwontOB _aeBel To} DronwrcourSoy! , Nakdh=LokPSa4LSet@s| 6~ Tana =Sah[aL_9,(20511ae\s[Qa%(2B)hee @ Now, reeanaes Sh&Suppo. a :donenthpiakDoak nSAIMSRondon, 6 VACH=0. Onpariaudon ,sweDecalgougebsLaie|Orond.bosaielyTriaemdchm fralevhorp, paromnsber Suncar a&),we OmetUde Mok arse_9,(del) *Spasmaby EskewBa(Ba a : Rows:cbvisualy Dwia.proia.ettdoflaAtsinadring A)fheGide 33-02\tkeh=Sak|&->.Ah)50 QaeatersSpmroqueckme ounsumbow yoakdonat cmPuen up.Syparkrondar, windastmake noreDe> BelooSii2woucllyou uSooliDomefroSyrahwhaoe=wnacpebaucke dXOebuneeot m5" Sylapt. Oa ondGrow NahBt=Qk(2) =o Ackwasta\mduerggnee, Taiomanareal] . ow Te,SKommatbeposdtetedowdy ©OTS 3Aaa ) prsosemSavIL)JARi) wa.pindOnaDrsSim.BaSledea 2%; 6 “ sQk=3|Ta Que,woeanuine ckowermdomuad, eur : Yo=ShNs HY =o Ak! Js : .Bel-9.dajiueobisskx Cowamai: byOne ja) wor,dAroseaonsongsyne 7Asad dawhre ,pedwous Seo collKooTe(Qal=ofxsportal,watDabeofubial)“ Comoe: suree iacmanalay rvabecbiny yriasmgtVeale oe Ob" ke Oa aa =o. laneDutaundhonedoomronot(43-4)a)—Oxlba,a),2A Spinkongmiskat,SeanSeng, enYaowJodDremrerofioa SLsdteaJumchind prrarnederc, 6 1) Bb=doh-a09=Apo)soWad". 6P IPA. aixe44el J. DEX =oh =eb. oe Seyel.wt=%}{gw=¥'[qui] Jha, ApeSoc Cuwak &Dun: ne7h.0')-gp fe)(honed Driecewidhar are=Thebe)gwd=oneeeeyo raga, Gauaroh,Oecomand claongeacnt VeTo=f =bhdamatorton ayoatt aes Gale: B4e)=869-469=ate46)=FHoy-4ew : =ded -RO)=GIO BO=g9LO)asomeway, vO he weGar @ ‘ rita ~ 6SO=4.8)~&&-\¢° \s&@-0)=oth&@) tt YaleQrakDuieiaaTypeDhemaformaction WaBanckahy # C=Ws =aR Py =SyOetoe) +2 a=-98 Yofoy=3h=fee LErsvoo =©hh Te oth Ferd, =Cer woryh - ;aa Wakexhotennoi=dn[ear xe )¥seroea)aiisXg™t=0 fe)=i So,ecmamench cansat P=Wsy-d =TELiga -egxgnst = ye[eva -eal~(pew) Soy S vByeWa, eT wT) =WwW” 3,100 Brrovss) heoryea. ove~ 9, JNe ee ee ee SeaG9-M =a(GO=-BH) 2CA®)O(KAR) =HO] B® dab+oye ~ACAyaX)O) Que, FaLaeyal oo owdofindAWolk.CavatsmutMeedeasiperrnetay” LagIpeya tH,soy=LO) $@)=Lar= wefe*e(eal QnSt=VekftcBbe]. Jsgeekdye. . DrmslaLeMaynatEL), Medat,OlearmswaleeseodeUnvariouk asin’ BikGeady NPUL.Wkadook a"adord=Vek3[eeola*leeGo) goes Seba EBAY 3SeYEW) FPA) =SV Se,0Raspongian Like,LOR=EQAve-Aeussatsuvarah| aw back toMe: Bh=Loo-L6) =aldtexvX =W= Targa lh Prortoasln yawcheeseAaudetudkAueALaNSf TPH [teev]h-PL=coanned AWelablan ourrod wysemascl,ALDgrrumdighTT(AitZIDG:=BH=Teab+%eT_Thrace D=VAS=Decheye=sanke 5 ©Reamsden 9apapainqraghina Tanyon: maWAwWae&wedex THhidend Sawn EBtraclewel Dave LaHQ : Va &varies- “Vowaracequoncinie aetehad7 Dust: fraYeonmuckd” Jupwman qragh[reamaJ2Spteedl, QuininQue: G-y4+r=L PratsstedwithTratere, OfUfcoached.thawrmushbe okSackW-DuleuelMine.4THT),Denyohoe OF&SurqpegeweLerpe,SackoddahWl Laefrenan Conlon FonSnoags,r=. QnomornOe:Ae“yteyets¢,a=alt+s. Quran: imaCainsWideoaaduackin, Lagann DashUrb pushLolde|re\rour:de@rj-Y+ 4 al:Aanetalemd,ogptinaenkadocomeched qeaghe. ods ii offtogetafiniteintegral. re) 1,Consider anintegral ofthefollowing formwherethereissomeparameter inside:a Ie)=ANSe .pxra Itisobvious from inspectiong that this thing islogarithmically divergence. Here are three ways ofputting inacutoff:. a)just stop the integral atafinite value: (e .dx n Ata TG) BGA) =BE=AeG|y =a(S)0 b)doasubtraction: es T@)> Te)-X(s) =de|Xa KER8 axa eth =aa(S)-2- (5)=dn(A). . Bysubtracting theintegrand atsome point (the subtraction point) yougetanew fo] integral whichisconvergent, although hereIevaluated thethingbyshortcut. c)addamultiplying factor: ofa derNGeSdb(coe ) SM. (=) =Ge x » SER)=2G,A)=\xaGr aly SenKAQW Moo. Gb aK ,A). Yan xem en xan)An* : 2.Comment: forfinite LAMBDA each ofthese methods gives aslightly different result. Butforlarge LAMBDA allresults arethesame, basically 1n(I/a), which shows upthelogarithmic divergence. Youcanuseanymethod youwant: d)changethenumber ofdimensions ofspacetime, Idon'tknowhowtodothisnow. ‘TheMinimal Substitution andFeynman rules forvarious kinds ofQED. 1,First,calibratethingsbydoingregularQEDalaBD.Hereisthelagrangina +a)and the minimal substitution: La+Fgm) Ap>AHR—Apsig3potted YS “Tha, te= VK foniw(155). Now get the feynman rule for aQED vertex byb soneidering asimple S-matrix element (even though kinematically blocked): ,€N y.‘we Dome=Qk S-alp) =i<phkl FA pS Lo NN WBley wc) =TG)Piey| wey€”* eLAS LL,this istheFeynman Rule fortheQEDvertex. 2,Nowredo this foracharged scalar field. First, notice that althought there is a44inthelegrangian foraneutral scalar field asin(12.2), there isnone in thecharged case because youadded twoKlein Gordon fields, see (12.54). Also, youmustkeepthedagger alsoond, togettherightresult: a ~Pa an . o= Hg O*R)~ whFY >Brod, v .Tan +A ave + a=Geld HeORVied =ied” [@da- F@e)] e —apne. a yop=2Gelied(@a)4FOND{—) &\¥* fo} Sc) Cit) . woh at . A”=iGiele* Ghtip) =[rie rte . LyFeynman Ruleforscalar QEDvertex. —~-DeakQusypmmadio Adincdomie sloademset. Ss SpgLOcak ote” aSoOS) Ws ete PA)TESeGW =eo. .a gancoset, seamed - . es Qe a —@Noadij PY=Rahg=wksywere )eOe). (RB soma,TP2oman ILEN, adsTT=8/3 Tank, 6YeaemDonfaseweshalloma AN S804 aSoepee Oy 2PyeM22le)=WGN)TG) Sal iat Sa adsfodatas dlQain1ladVpconeocasatits Dersrboah ©NeaadoanSlsensebeg [email protected] Pe OPPakCe a1East otaley ondasypendeaantee 6DDame: Yo? =iatae ey Maer =SE =Eae)=o —@Yanan: BPaspoemadie a Sr >) TSPage )-aE he)¥Ge AoawkGH) cadAne G8iade ie i BeEE 7LOSE ReaEPR) SYR (Gg? SS “ay ye)aeS a eB oy Caer arrSan PLES Dinan akPP eg Vane, ee — a ee eeoO_2Nee yearea ee Tams Wecounaalar Qaa2007LentonATRSram oe(ASD).OED, 1Raaamose +(Colegun SaskJAN) ;ae {ee Oe ee |Radin@ouabe J —.. ee ~LQys.Ostasavers suannaakssano“VPCauiSsagaumahc). ouchfagafar|—onaareonuencsannoa WY Weawea-y eh ae ee_ WP=Ga eT Wea a We . 8 OP2Cer) 53,Craeeweke) \Bask aeso Neer ae ns aN sR (Wer gatee Pesy) >RA. See we ~ |S&B =Se =ee — a Howtodefine theLorentz andTranslation Generators. _ 6 1.First,letsgetstraight thenotionofusingacoordinate transformation Rto define atransformation inthe Q.M. Hilbert Space. You define Rinterms ofRin this way: ot<ARley=CRED Youcould, ifyouwanted, make analternative definition byusing Rinstead ofR7. BUT, ifRinaddition tobeing atransformation isalso representing some group, thenyoumustuseR™sothattheoperators Ralsorepresent thegroup, ie,you want:RyRo=Rytobereflected byRR,=Ryfortheoperators. Notice, bythe way, that the above setup isjust the usual "regular representation": Ralm= g(a 22.Sohere ishowyoudoangular momentum. Define fil” like so: iow show 4cas gy=CM axel Here both Mand@Mwillsatisfy thesameLieAlgebra. (Ifyouweretouseopposite signthatwouldnotbetrue).Gotoinfinitesimal tolearnthat: (nie:M=Meo).ne, 8<A ey=TW"Lysx2QL vga sy . 7 Bae=ALSp SSI IEbE)=LGEI-W3*) GO=HMQO) 3.Since thetranslation group isabelian andtyto=toty, wedont have toworry about. thedistinc£ion made above. This means that the sign ofoperator Pisjust mm aconvention, soIwill doitthe way Ihave always done itand inaway thet duplicates classical definition ofthe relation between angular and linear momentum. Soz i rNwig RASIN Vey=Lavalay ok TINY =Sa) . SRL PALEY =iee®e@ =Pde “Men: WeCPLR eg, BY=RRP =Ls 4. Summ of results: 2 n SAPSY =PRG whew PAshy fe) SOE =MIA wre AY=a(IEX’9") GQawrabupition YoPndudssew. eo) OoVsowsty a=(RSOIR oean c=ceefpguunbin space S& spe agad: we,= Wwe EVAL= eaedo205 Ted=LooaieSoIC . ©Awraranetein witspineGomalrna De905 HGS=LRA =RAR) . > . y a 5)5) z Nae:TRAE =KS,IRAE) =RPERRN(R KO=CDG(fer), WeearwithMioe SreLQ BD=RyRelea waAAHKEARY= CARL KyGlad. OvcBad =<BR ond KEAADE RG ALES . So,RyyissomeFDRofsomegroup, ie,oftheLorentz group (need notbeirreducible, eg, might bethe4x4Dirac spinor rep), andRijactsonlyinthespinspace. Notethatthe useofRj;andnotRyiscorrect intheconsistency sensementioned onearlier page. Ie, now wehave three different representations ofthe Lorentz group acting indifferent. spaces. 3.Nowwecandefine M°Ywhen spin ispresent: —iot™ iow iow” <0)SMVay =LEM CE Tatex|badd ConerDredenanstakionbosey we—a ~-is L= ev =oe am sieln’@ we —e he”=Sowrowel So oeASR=ABI+108 AWww Se\W olllas=HM Cadile ion ~io,=SOMay TSP" Ysdead Ohua walowe: Oweeay =Gere aimtiay =feanay éei\SoVes=jee"iydele =>delwiley=(weldels Cordhanae : SalWelt) =FRY,Beene) ceded =UNI Heil WAR =dre Mer +10171 &d9 =BUHRIARD Fd WYded) 6={Fed HUM }Grraceved =(HM Goalde.dd Qyue: Sele =Wels a waa LMS =AU Ly fo] WoswQunchuon Nenad: 6 ' BO =C= o@) TeWYuaeyak BOO=jo eleacy=~ie<xl tt=~10HL8 BM agt BR) =rigGeilPQ =C1a)(*) Re)=+addi®) CGdeggomarous usmbsyoSleeaspuvies. QunsWY. WW@=-icWLO ro) : - io] _. Introduction ofFields. Global, comrels, variations, and2i”, 01.Wehavealreadyintroducedtheobject.Risasaspinmatrix,Rxasatransformation onx,and asanoperator intheQMHilbert Space. Wenowdefine a"field with spin” as follows: u s =\ RE =KRROR =Ry (Rs) Youhavetodoitjustthisway.If,eg,youweretoswitch theRandR-!which surround the field, you would find agroup multiplication inconsistency. Notice that ifyou take the VEV ofthis field, thus getting the classical field, you would find aclassical field that fransforms consistently with out previous wavefunction transformations. 2.For linear and angular momentum, then, wehave shown that: cot” —iow” ~ieW™, ie <eCgee -le kh&l€ yer) ” te A|sige LieQeBe)© =&(xxa) 3.Now put insmall paramters toget the commutation relations: wey] = . = Wr. seLW”,Resl=SHO =-lLeLiely&@ hap]P",BO)=SR =-ignP*GH) v qe’ AW WD) 5 >(THe,HP]=Tg,BOD=fewwg+WETRie [8@, PA]=Baw ="bo &Examine &O(8) doQooun . wr to6 By=XIN"|g : °) »ys—=—*Vva Son (Taree, w=4(perry +22.)RE Comment: for afree field wecan make this expansion: ‘ Qa By ; a 6 ea oefoapean Ar) +<ppcnled ae,4 where you sum over all momenta and spin states n.Iamallowing here for achange ofspin basis. Then wehave: =Sy omot R&S=ridLR&)AG+gh"alGerd] *, wor) > CABH MOMS =<olROlpH =QE) Thus, byusing @single particle state onone side wecan get aconnection between the regular wavefunctions and the fields; wecould have used this toshow the transformation properties ofthe fields ifweliked. , Dilatakim. Gammatot @©Wine 0wouelmetinn Seat Komakrrmnakin oefallows: [oeCiad) a= 2e(ds) Lenaval, RKaarsrmalhbogasrat ADO =1CA4 BSH) =Bantad B=i(aetd,) @Funitidamepriaim YunMoaDaFranoraadd: iad had da 0ef ee =Sf&& @®Godwn boe-dedmaln dogit +eLD,GH]=BQ =-iBde >(ae, DL=-Baw 6 ©Wao: [BW =[Lean,iGevxH\] =0 LB,&=Licastan ,iat]=-(ia,a4]=atge-0 1B,BL=~em,%|=0 lo] Diototin ondGrae Func. AdGasesDestsadeaon emrevamancr FyyerDaaggeeae LATAWN: KESNQ =ATTHYLor gay... QEeats Sx deee Maw . ey, derdytetde[scotavoor]> Cwm) =QQ &Om,eeAX) ©WowLekAfounsr: Pra e=Yay«deoS SOen) en Vino ait XfAK so dike Malu. Oe on Lala) dyes doe=GeO88"Vad ©‘ OQ) Otel.xh) P Agenda LtA =ae erldsPe/A+Qald)- @®Vrduarma Peer) =DIM* GPO0, 0%) |@ Bad) ~)Brecnmy =pi Eeceay.path) weraonhueiyQaoySrnareodswverauct, ©Romo dann: Layee tay)COMPSRAE) =OYE GOR, Be)CisHoa=) 6a =XB) oy [Sa-4nts] aPepsGray) =OY oR BS) . ge ®awgoseercronSchue BPE br=GPRD EEGD Gp) TOCrates pow)bo,24 \ L v z\ aa eA COMME CTS) U3 SeumegitAcedetdeYRS©1A-2de+CaMO- iea" “ Pay. -ee) nL Yau: =Pm. be): x W-24)Went w= OQ POCaprese geald, ©Sermmany : 6 ‘ hawSO deyde)=OY Gare, es) “ Pereaes SCapurer, +Age)=OY ‘SSeeate, a) & 4-Ba -Pe Meybs) =OD POCasey Pan) @Suy toook dammed (Senay aMeer) we|CPO, xd]=Ba Owe Sows a2alone \ ©. Sage BILE rs Pe) = Ba—Ye 0dawLIroy=Y-Za Sonmony JoPY,"ant Dz 6©Licgeo: (=Map=MCieemnat) wvToe) =a|teeth int5gt =han?| [em] =3{gyP- e*| [IWple ak Ire l=[pol- lpw-o aistt? ~ient” ie io @ €we =ek aew) +ig." ignP* €kw € =dra) iad sia do e ke © =©d&(Cek) @Goose : Lae,me)=Uy RO Ly=Reay4Lady Tao, MY=Baie Bnax \%©O, Pl=Bea Dei(dsotan) Odeo - e HEP=ieta x3) Iw" =BS SMG =a(eroe3"] 84+28) onEber Thomann ohThomann Paaniaait Baath oeCit tySoda 2eSFGete Gat) =AigiagengeedieSETSHES|SET ~Woguatuet andapOudkvuligake Sgh 0=eSMeena VigdeSSE RGge) eae NeSeeSe TG =GE)Se SSE FR TDaatiges ioRG) inMallya polafe Tiss. Fumckiows amdFou. Tromalown.. 6SeCMe) =OAT(om) aS) =Syedgpsaee“Tromslodin vvornmnce Yabbayer!”GOK te) GORA,KAdA) Roowen. Ong(io)8Gox(~ia®)=QGera) @Wine : BMcesraeestey &Yarden SH BMG) oe) Bikmvndhomadodcin ABLEyt ta UrMer (a)=CRSeger toe) Sane dtrGo ©Hy¥en, 0) [a , ‘=BC tyeped Naa: Raiadepumdast wendiitn meE>gameUredowcbadett ouswit, CieSrvusion Smudoa: (b=\/C2*) oy EESagedgSOShCowper ped) Dru Dwate of oo ow Aire) =RaKaaKn)oekeXn)-v.Vagdea&we& xSPCeateyees- Bann) ©Rropogadns oamed . EPcay =NawetGuo) =e) =PH) @ =p,Kr) aBO ewe aa ETMGhee)‘ POM) PCR) Qaanne, wreBDPRIey&POY=Se&), PCY=SG)=e Vrs.PreponVorexFumckin. Bom wondunale space: a” Ge_iPew=Kooy)=LEDGrsi)i 2 Ou Qe gy FGreKa)=SaudyeRew).-BoGocyr)Peng,ts)laa @VeodoJournboait are crete Brerstst) =Sage dyeSanedee xe.. eit Res) ReCmrya) PUyugey ge) fe] ©SaasageSterdge eAeth«SE SheRED.REDSPy,ys) =RUDBGD RA. KOCapp) @NoosxdotOvh8'( ep)dogk Sete Bed=REY BCADDOC,ge,pas)oFota “ee Ray=6a,) fo] ©UijinsDananbasic:byte %BtCoane oe)= = caneplthe, Goum. Te (n) 1ea,oe)=Nex” =poser£07uadler, Yaw = ae .Souree)= o—o% ke @Rona bearer4WtondMt:CatsvwannY). 0, beSBE okTHs LN- (eke) =BY -20).1) Bako y)Se=Ba21S =Cs (=°=PAastee) BIKES SIM) 3)SE=~S50=Meany=Seep (=XAuclee)) SNR 34H) . ay Ke Xe Xx& Dah: .LWS=SanSeee =—398)=-36-x). BIH)ILSee) STH) 0[GeerwrewthcmAkA = ? —-—2— eQheown:%4.yea-(tw) Peak:doinormnakChenin, WateratkLogharr)dogel tytyeiGaysPeg =—26%G2) oR aL a 2 Ldydy, |S SY aien --\sv . \ws(Saal Saal esany\ (Baal aa ann cae 9 les Youn QhieWarn shabuGot 7) (at) oe. SsTGste)=SdFuerst KGa) Qo Ges Sp Ayres Syoe BIW)WOH)++8404]YsSSTAH)FA--39s : SY =2» Qeed:dono BPSCoy) TR fe) a ~3o xs pe % eoDrown:Dow,=wordfo, * Pad u%s = As, Xs oks Xs LDKs0Oar yds,-|-obo)bash ovaabin Severe. Ore. Get: Oluwm: opmatin, <2Uwduabrloutie ontallblader. (‘ And x nOBalleasheemangled)eleeteinght @Non: xa Dp©ok .* +3Bia a’ . Nea Deal:yakdelmat,abnerMowe. es Fldoveale NieLastcath: We ~ t3 o sed‘oe=MB=lieGoesleowxy _ ‘ 2=enara ~2somban rs: . *—O-e “ W2 ( om=6p+tsvindles+fe‘ 3 , .4 je)4 L =wobeodeon atte } x a \ a=pane +Usum+ne \ 2 3 M . 8 baw 3.WAY OoFicrsitonBercondealacsqess, GansFakes Tene) =GILCheKohglo 9 Sea acaege feBE a ee=RayEt Se OM Ras. ‘se 8GeeLe ees TeChachara). Eee. Danes aoe Qe. STGE)Tec 2? sayy ro)TruncatedGreensFunctions. 1.RBcall BDdiscussion ofINfields, and OUT fields. These are supposed tobefree, non-interacting asymptotic states. Wheo you examine anS-matrix element, you dosoin terms ofthese free, asymptotic states, id,Sp=(£,0UT/i,IN) =(f,IN/S/i,IN). Ithink the idea isthat the external lines ofadiagram representing anS-matrix element are supposed torepresent these free states. Such "states" donot self-interact, andthus donot“build up"thefactor Z,duetotheir self-interaction. Theexternal lines are supposed tobepropagating asfree particles without even self-interactions. Sohowwould yourelate J,totheunrenormalized field ?Well, the propagator youwould getbyfourier transforming (,.,,,) would bejust 1/(p-n?), whereastheunrenormed (but"interacting" )fieldswouldyield(9,f,)=2/(p?-n"). Thus, you can imagine how these two kinds offields might berelated byasquare root ofZ,,assuggested inthe"asymptotic condition" (16.20) ofBR. Dont beconfused. Here weare discussing the relation between afree-field (which makes just the bare propagator) andaninteracting field. Inthepast Ihave often compared twokinds ofinteracting fields: theunrenormalized which Icallg., andthe renormalized whichIcall$.Thesetwointeractingfieldsaresimplyrelatedby fe)amiltiplicative factor. 2,Iclaim that the ISZ formula really says this: compute your momentum-space Greens function, then knowk off the contributions which are due toself-energy insertions onexternal lines. The result isthe "truncated oramputated" Greens function and isprecisely the S-matrix element you seek. You knock off oramputated the external Greens Propagators because you are supposed tobecomputigg S-matrix element between asymptotic states. Inother words, anS-matrix element isamilti-pole residue ofaGreens function. Then the object you get issimlar tothe 1PI function (which also has its external self-energy insertions knocked out) but the amputated guy isnot IpI. ‘e) Qseinask blesonoorSiderdogit: vy wv rae(uty ge)=Ue)ew)... Qar’) aCOsbayseQn) ERK Banos ae =@)Nya dy,CS “ee oyes 4+COs) COeat)Shy Ye) - =ZalhQeny---- feo)|B The+choices depend whether particles isingo oroutgo anddepends onbabelling convention, butthesignhasnoeffectonthesignofBox”operator. Sohefeyousee the"amputation" going oninthex-space ibtelf. Thus, wearrive atprecisely BD's LSZformila (16.81) except Idont have any2's. Thereason issimply that Iamusing renormalized fields, whereas they areusing unrenormalized fields. Icould gettheirresultifImaketheusualreplacementthat$=(23)?$,.Then 0O youshould imagine BD's fields as9's. 4.Bytheway, according tothewayIdidit,Ijust pulled offexternal propagator insertions. Thus, the first term inmyS-matrxi element here called iT will bethe 1PIproper vertex. Te,thisistheconnected-part oftheT-matrix.- . - 5.‘TheseZ'ssitting outside theformula inBDaresometimes called“external waye- function renormalization", which terminology relates toparagraph 1above. For me, these Z's arejust the Z'sthat getyou toarenormalized Greens function, soIdont have tocallthen anything. . le 2 3.Implement theabove conments inscalar field theory, say#*orsomething. Then a)letG(pjs-++-p,) betherenormalized andthusfiniten-pointGreensfunction.It contains the overall delta ofcourse. Now, remember that LSZ gives you amatrix element ofS-1rather than 8,soyouarereally getting theT-matrix. Lets saythat. S-1-iT which isastandard form. Then here iswhat wecansay: . . Gy @ @) ‘Duby Pe)=Sigeage:MtCea)wodong 6Tq: i) . ‘ :: anCry an soigt<T Here Iamusing all-script functions inmomentum space because these arethécomplete ones which include the(2PT)* (...) overall. Nowlets expose allthose delta functions and the result ismuch simpler, and wegotonon-script functions: aay OG Cy ()GSConte,teal) =FRYGG SRY EVGates Bary,OD) Here Iputthelast momentum arguement inparentheses because really p,,should beregarded asafunction ofthe other n-1 momenta, since weare now conserving 4-momenta. So invert togett, fe)ce :@= ow=8ITCeyeyees Gad=LEI TE) GCate tes), ., 4eS) Qytewey, LFFB8y=HM.) &) @): 24a= |Hy) e- Lo Qk Sw= 0G) wr B= LL. : poe . =S wot Qua, D= Get) amd: &) woom. 1 + sozt RoKTGuta ts)=P) Gee) Gee) &Cerys few) Ofcourse theimplication isthatyouaretaking alimit hereaspj”goonshell. This ishowyou‘aretakingamiti-poleresiduetogettheS-matrixelement.Ifwewerenow oOotoputthings back into x-space, wewould get p*=-d2soeach factor makes a(-1) which combines with each facotrs's 1/itomake aplus i.Thus: QaQaeda Voge sO, ©acimRadiomanrmratizeMeowwebaad“Oued(cinecle). : ~\arleamgw+QTE:.” Spa@ %8@)°Me ;oy, e* [aveg BIH) @AvDiormole + . _: —Tk<ascain? SGI? . Nae: :-Tin g chu LAVAS =LADGUMMY =LawMGat =LAWCAMDAN : : le):-EWG.” . oa\iisy=LAM WYP : a : Ho, EMM) SoweconSay - =Saredy[2QE)MEAG)+4OTOPT) \aa\© . —_7 eo okMRKPheASE WET -ne[askMeA)| eag 3YTG)) @WiderweWELCengecomeedah rs Yea aQB-sSwhy|=we) WEY SIE) -~ : : ©Weedsil COdogit 1) siYan[2geTHgo©SeoT09]ba © .. — aoe os [3Ronysey] Yakm Y\eewareWIS)Quewevoodhsow . war is a3W =\(ad)(iid) © ryao LEAISIS)©=GYSitGago\uaye’ © 2 s \pne>Gos)=Cbdsent_\o: WSTEN) =15Mes) =Wes), DG@s). a . Seaaust MG)=DGay),Megopagede, ~ Aah)ax]£80)TRAE)4QOOTE) 3LardySTEMS)TC wiSpae? \ |.as ‘s) Takh . «aoS Veo =Cew= OWsewsl | ease +hRG)=RDG). =<\T(GOYD)]s>