EM & Mass & Shifts

EM Mass Shifts Feynman &Speisman (1954) Cini, Ferrari &Gatto (1958) Cottingham (1963) Coleman &Glashnow (1963) Coleman &Schnitzer (1964) Suzuki &Zachariasen (1966) Harar (1966) Ball &Zachariasen (1968) Phil Lucht notes 1977 ra)Weidg)'sSife,PRLSh,MAN oa .- FPWatton 13;G4WomPOthwerwren Awd28.Gwe GoakWAGT'Sweveanygrnttotic,Spastiam»Aa I. : fidedondn4?Garsuan Urendat ~ powaat Te,vutnah genet Ree om “bern era's’ j j QeUskneaWeronanWoyhincanShimsTeemagtached 2" i Fraluchieeishae?—SHH)GOSH OGY |< -eSeanqogekpophewfxsve“f=2 Quarks=spin}hades ;® commStrongondynraleecla. TGs,6.)=o. i -TeaSTcommbychamp eheyta. wede re) Sagem=hon,ontswighit(nesta) ?it,FrS88) Wagslosrataatad wid.26)40>rnscatafraecdnsandSept: ' EM wkdBike38 ‘ eadow on goawod| dudstoware AFL Senoetn)ove cepe4verwrren . ee —— ,Sing=cna ae-—Foe~tWaerpe.:&amar[pnyoepokoneetriny J] fonean-cikBe=\end“amednimaah see eo) Qammchrend -Yodgeten ? . resaae po a pro |QO©Srasyinsonmpating,Duargeass i . to Tt { Wey:dodein?Givesyou192omgistnay. Wasdeo Upsentayocieep thao gage! . |. j PORK ©ietegn MCA te rete! .See Gud .Nes ar ° We=eleieTRIS*¢ <+ .. | FA. Too pete we mm re)CemagartGearsumsiwpedadglaslm|WakinLaqangiar | REAn. 0aged SeAer, 7 iee xeG)isosguadonttit ,feaun- a J. He(S)cehydote.a ; \Somh—seahor frmis eLeUYAy 7 YNemai,comackdreye, haysmae “| | Mabe tndWeddoSumareOnJynckglace. Feynman &Speisman (1954) 500 LETTERS TOTHEEDITOR . Proton-Neutron MassDifference manner,thecompleafvirtualmesonspresumedtobeasociaed spin-orbitiS>,Pomona eSrna withnucleons mayhavetheelfctthatsteuficently highenergy Polson. . cotseeieteSr theelectromagnet couplingofneutronandprotonmaybenealy norephase= ‘Receivedebruny 29,180) thesame,s0thattheintegral representing thedllerence ofthet pepe “my. massesmayconvergewithoutmodieatonofelectrodynamics In polaiedpo GPEOSE alldeviation frmisotopicsinsymmetry aeduethisway,thepresumed convergence afUtemacsdierences might solely toelectromagnetic eects. Then such thingsa8themasseH]ussomethingaboutthecharacterofcouplingwiththeelecteo Thiswor Aiferenceofcharged andneutralxmesons, andtheneutton: Sauce haeokney coq protonmassdiferencewouldhavetobejusttlectrodynamie. Weconcludethatallothedeviationsfromisoopieapin agatete Wehaveinvestigated thispointandhavefoundthatitis& maureasonablepossibility.Forparticlesofzerospin,lke+mesons,SATE : : ieassumedtobeelementaryparticles,thesellenergyisquadraticaly, antag LA.Sivergent.Ifthephotonpropagationfunction’1/4iscatollbya282Feynman,Phys,Rev7,19(94),Wethetainthi convergence actor C)=[4/(Q8—BYP,thereultngenergy STSegyanfar(RPEahtert! Cone ‘SE about _setA2/Sem, where Aisthecut-off energy andmisthe '=Theoretical Physic, Paria, 1980(unpublished). srimeson mass,ahdwe,ssnumed A3>m.Thisgivestheobserved GF saifernc ofaboutIfdestronmusseeeeee —_ |TOprotonmasses? | Tisusually assimed thatthenegative value oftheproton- i . | neutronmassdiferecespeaksagainstanultimateclectromagnetic Polarization ofElastically Scattered Nucleons explanation,‘Thefllowingcalclation ‘shows.thatthiefsan fromNuclei im | Saran esnytontsparsarenesnpaeee ireneaBeeene wr uncertaintyastothecorrectlwofcouplingtotheelectromagnetic RadiosSESEG§NGEROTNE wrJamarV,Lapone te 5pp®Sek.Butforlowenergy,theprotoncanberepresentabythe “ecivelPeacy381580) ProtonfiileaequationwihanatonalPallermtorarestheINUCEEONS ofowormoderateenergywhichareeatcly ‘Theproduct Snomsfousmoment,SincewedonotknowtohowRighanenergyscatteredfromnucle!shouldbepartiallypolarized!bytheandbyTae Thismaybeareasonable approximation,wehavetriedproviding trongsinonbiepotentialunderlyingthepredictionsoftheshell hed.I ‘Sicmomentcouplingtermwithacut-offfactorofitsown.WeIyodelofthenucleusThisspin-ovitpotentialinasonseqoresa aisapron dhemomentcountingtem. rodeofthenucleus.Thisspin-orbitpotentialsaconsequence of aldapreatheA ‘Thusforincidentmaceonswhosewavelengthfsgreaterdanthe ‘oe Fearn70fnGiok—brG)](p—R—ay ghruagacng(2230hes),wouldbeexpestelthathesine ieee |¥ obitpotential oftheshellmodelwouldmakeItselffe.For toonyteal |xf++finlrak—hyocun rac(k)progressively higherenergiestheincidentnucleonbeginstosee processestre itaf eonlyonenucleonat&ineandwhilesspidopeocsoaks {nthenotationofreference1,WesedG(A)=238) toat_elasiccateringcanailbeexpected,fewouldbenoreelection themoment coupling ofatenergies about 2,andC(t) oftheindividual nucleon-nucleon interactions thanofthespin, =—MWAtocatoffthephotonpropagationfunctiona¢orbitpotentialofthehelmodel.TtwilbesupposedthatevenakThenucle [@)energyA.Theexpression fortheneutronisthesame,exceptthatthesehigherenergiesthespindependence hastheformoftheusual symmetric p theyacouplingtermsare omiftedandthevalueof,theanomalous spin-orbitpotential.Ineithercasethisspindependence ofthe ‘oftheincics {‘montntinnudear magnets,g'<LITisleoFLTDbeteeatcnalterngcanbevenga.poonooercopcaly. te targetpartic.Droton,Wstheseomass Stingtheintrofthemolesieteorae ‘ThecrossFortheproton thetermforn=0,representing coupling ofcomplex indesofreraction™an obvious genersfeaton ofthe wherethefi currenwithcurrentispositive, aisaltthetermintButtheoptiealmodelofthencleas# 4theintial ]ossterm,lineain,isnegativeandquitelargeifthemoment isForlowormoderateenergiesthreisnosuitableapproximate |okenergy'? otctoftosoon inthe prtor-netron mas ferencecanmethodforweatingtheclastesetteng-ra phaseshitanaiyaais, |MRRERY2 ' ‘asily turnoutnegative. Forexample,ifAand\arebothtakenat_necessary. Also,athighenergiesanypolarization calculations fowestorder 1.44,theexperimental valueof=2.electronmasoesresultsfor’singconventional approximation methodsaremadeuncertain symmetrieal {WiedterenceGntiecausTacTrantor SUTETSOEMTS forbythedetdependence ofterolesoeGear 've theproton —1.0electron masses). Nosmalldiference oflarge scattered wave,Aphase-shift analysis forvarious energies is 7numbersisinvolved.IfthecutofNisreducedbelowabout0.753,thereforebeingundertakenontheUnivacattheUniversityof2resultin« 2negative mass diference cannot beobtained. Yorhex1OM, California Radiation Laboratory atLivermore incollaboration’ ‘Hae B=24Astigetheepeere “oe ee tengestseeing, oughgration, REA SEedeeis mmomentimpliestha estimatefrsmallanglesofscattering, thoughroughatbest, ‘The(2) | ceeeae fofheAnomalous moment impliesthatthe berenlygbtuned tymakingsoeeaeisaae ct principe,siSRGanal frterorent mustbespreadoveronly8TonsTemagafthepuaretionsvety |stateofzeru Smalldistance(oforderB/G)Thisisaosugestedbythe sivenby |siteaze relativelysmallchangesthatthenucleonmomentsundergowhen pu(ABA)cg, co|preductn nucleos frmmuck e/a|thereareag ‘ThecuteforthepropagationfunctionmaybeinterpretediHere{andBrepresentthesatteringamplitudescorresponding to possiof {§tugwusaHinuy,eltctrodymamics mayfallatNighenergies,thethesin-independent andspindepenent parsofthetersctony ates.Thee failrebeingrepresentedinacrudewaybythecutolITthsi0respectively.‘TheKnownexponent!valueforthedifececel predominant 1scouldguessfomourresultsthattheatureacursatenergieserostectiondo/dmaybeused.Theamplitude,B,forspin.1The(mp) Jntheneighborhood ofthenucleonmass.Another possibility iSdependent sattering maybeestimated bytainte,Dornep on,thattheelectrodynamics iscorrectbuttheeutof!represents, proximation. ThenonlytheimaginarypttofAcantbutestos wishroughly,theercorcommitted inassuming thattheparticles areForsmallanglesthisfsapproximately propostonal tothetotal ishro ‘elementary. Forexample, inthecaseofthe meson, wehave crosssection, Problem. assumed the meson invirtual states actsasasimple particle. For300-Mev neutrons incidentoncarbon,forexample,asquare- acest22 Butforenergies ashigh8M,strongly coupled virtualnucleon wellspin-orbit interaction (ReeL.AANC10- oo)ofblessepoh Ae,pairsmayboredTheyratherhaareofelecueynamin, venpaazationa9percentatheedeiesTahee Hie rayprovide theconvergence atenergies oforder MC.Inatke probably anoverestimate, suggests (hatte exlatenceofaall eat ; FeyrnanondSguizmowo .(Caffe1854.) .-oe os gunOuiDasYukgogranaeemplZe tae —-(CdpeanOneFASBasindontanionsd emptiog., Oesareusatiocoe eeHetOheBY_TpJOMfnceawastes. Ghstaneon.rrauth. ve,gosta ie,TWWoven QinWD2 -eae ee Loe a . erm a . us wa BuuTkeaegeokTROK,AKonanUWcbfAMad-FSsean Dok W-A= Lod Sworarrsa donByeran rereSur ee feb -@Vlad. KER?=~4thelysoarsconsDidar9-74sstetion Toanglan=-Ona.ApatBaopam<0,FScdeoy”dypeever,albnsgvitefoe.22 Dine cogtiny, Eade rnaynmmoni dtrwmiaeuskaledDak_gleckee ae)-SWuteYokA.Swisren pytinyOwoursletete gaQaargns <==@Messing gamedwnt,bitDeywnaeGanYonddodoh Cini, Ferrari &Gatto (1958) vse |Vouume 2,Numoer 1 PHYSICAL REVIEW LETTERS January 1,1959_ ———E—e— Eee eS: lO versal byabout 16°C, incontrast totheTe-Se grals ofthesecond-order electromagnetic self- alloy mentioned above, forwhich thereversal, _‘mass afundamental length hasbeen made bytemperature wasapparently loweredabout5°C.as‘Feynman andSpeisman® Theyshowthatthe 1 ' ‘Theimpurity concentration ofthetwosamples proton caninfact turn outlighter than theneu-~O. ofFig.1,asdetermined fromtheHallcoeffi- tron,inspiteofitselectrostatic energy, ifthe er cient Rat77°K and theapproximate formula anomalous moments ofthenucleons areintro- L Ra1/pe (p=carrier density, ¢=1.6x10"" cou- duced intheinteraction andtheintegrals are lomb),is7.2x10"*carriers/cni* forsample1cutoffatsufficiently highenergy,Sincethenit asHand2.2x10%carriers/cm® forsample2.Inhasbeenrealizedthattheelectromagnetic self- gani addition, theupper reversal temperature fora masses ofthenucleons might befinite even ina e H third sample with p=7.2%10" was found tooccur microscopic causal theory iftheelectromagnetic at498°Kand-thisdemonstrates theshiftoftheformfactorsvanishsufficiently rapidlyathigh rentupper reversal temperature with impurity con- momenta, asindicated bycertain consistency centration. Long'®hascomputedthatthehydro- requirements® Withthisassumption Wickand ts. static pressure used inthisexperiment causes Sorensef®have attempted acalculation ofthe res theenergy gap todecrease by0.032 ev,and this nucleon mass difference onthebasis ofaformal-xed '{inturnisresponsible bothforthedecrease in ismdeveloped byLow®Theircalculation yields U' thelower reversal temperature which here- anegative result, giving aproton heavier than e ported andforthedecrease inHall coefficient theneutron. Nodirect comparison can, however, { withpressure intheregion below the“cross- bemade between thetwocalculations. InFSthe _ H over”pointatabout217°Conbothpairsof relativistic Bornapproximation totheself-energy =Pi curves inthepresent experiment. However, the _isused, while WS are essentially ledtoaBorn causeoftheupperreversal shiftstillremains approximation formula withformfactors, in : tunknown. whichonlypositive-energy. statesarekept. | —_ Forthisreason wehavere-examined thepro- :4p,1,Wold, Phys, Rev. 7,169(1916). blem byusing asabasis theexpression ofthe j *Fukurol, Tanuma, andTobisawa, Science Repts, Compton scattering amplitude derived withadis- Eenpollogy,Selone118,Hbse persionrelationapproach.‘Thesingle-nucleon ‘A,Nussbaum,Ph,D.thesis,UniversityofPenn-_--Sontribution isinourcasesimplytheFSexpres~ fe)sylvania, 1984(anpublished), ‘sion with their arbitrary cutoffs replaced bythe “HL,Fritzsche, Selence 115, 671(1952), electromagnetic form factors ofthenucleons. ‘1,B,Callen, J.Chem, Phys. 22,518(1954). Wefind that themass difference that one obtains 'R.Gispdr, Acta Phys. Acad. Sci.Hung..7, 289 byextrapolating athigh momenta theexperimen-and919(1956). ialformlactonevresgliagnahaa is[wronglin signandinmagni- aDresselhaus, Phys.Rev.105,195(1957). tude.Thereasonisthattheradiloftheexperi- .Nussbaum, Phys.Rev.94,337(1954), nent 47,J,Davies,J.Appl.Phys.28,1217(1957). Mentaldistributions correspond tocutoffscon-,.tH,Roth,asreportedattheInternational Conference Siderably lowerthanthoseusedinFS,WealsoonSemiconductors, Rochester, August, 1958(unpub- ‘findthatthecorrect mass difference canbeob-Hehed). tained,fromthesingle-nucleon contribiition and43D, Long, Phys. Rev. 99,388(1955). without contradicting theStanford data, with a "D.Long, Phys. Rev.101, 1256(1956). rather pathological neutron charge distribution,: concentrated atsmalldistances Ttmaywellbe, i onthe other hand, that the main effect comes ; NEUTRON-PROTON MASS DIFFERENCE from themany-particle intermediate states. j i BYDISPERSION THEORY Westart, asinFS,withtheexpression: id i M.Cini,E.Ferrari, andR.Gatto (p'15)1p)=-ibm(2n)*8lp-pLp’ hulp) ; : Istituto diFisicadell"Universit, “ip ~Roma, Italia, and .Fthe IstitutoNazionalediFisicaNuclearo, ntole-o!feesom),)vea: Sezione diRoma, Halla ‘ample fate ef by) 5 (C4me,” | Anattempt toexplain theneutron-proton mass ##=ifa're“OITG,(B) AyD, (@) |differencebyintroducinginthedivergentinte-withj,,(2)theHelsenbergcurrentoperator.In- (e).1 Gini, Ferrari andGatto (1958) 1.Basically, theyrepeatexactlytheFScalculation ofthep-nmassdifference. © kecati that this caloulation isbased onapproximating the proton-photon Compton amplitude bythe nucleon pole term, InQED this isjustified because coupling weak, but maybe not sofor proton, Anyway, the difference inCFG from FS isthat CFG use actual experimental form factors for the neutron and pboton .Ie, they use the forms (8) inthe formula (9). Bg, the neutron has identieally zero charge form factor because one guesses ithas no charge anywhere init. The other three form factors golike 1/q forlerge q,afact Irecall from feynman. Since thedata setthe soale for these form factors, you get adefinite answer. 2,The answer isthe wrong sign! Comparing toFS, you see that the experimental form factor cuts off roughly at which isless that the artificial outoff's used byFS, Itjustdoesnt work! Ifyouinsist that thenucleon-pole term work, you have tomake theform factors have large k? meat tothem, andthat inturn implies small scale strucutre tothe charge and magnetic moment distributions in the nucleons. Authors make alittle model where the neutron charge form factor isdoctored upinsome way tomake things work, but seems artificial. @3.meysoontosuggest thattheanswer liesinnotusingjustthepoletermfor the Compton amplitude. oa} Cottingham (1963) | .. Biebelant! PO a HEseu i ANNALS OFPitysics: 25,424-432 (1963) ‘There aretwogauge tars i plicity they aredefined kere s thefour-veetor g. 3 TheNeutron Proton Mass Difference andElectron Scattering} Dee s Experiments e Dead € W.N.Corrvoua, ~ a ~ Tys(q,q’)canbeexpresses= Department ofMathematical Physics, TheUniversily ofBirmingham, England coefficients being functisas = Tole.)= Inthiswork equationsarederivedthatrelate,tofirstorderin¢*,theneutron =protonmassdiforoncetoinformationdiretlyobtainablefrmelectronseatter- FromEq.(1.2),Tate7 %ingexperiments. Theequations arederived under twoprincipal assumptions. te= ‘Thefrstthatthemagsdiference iselectromagnetic inorigin,Thesecondre- Takegt)=ie[eetEA Jatestotheconvergence properties ofthetheory. ” ‘© Bod I,THEELECTROMAGNETIC SELF-ENERGY OFTHENUCLEON fisLoe, a Inperturbation theory, theelectromagnetic self-energy ofanucleon canbe = formally expressed, tofirstorderin¢,as(1) a = tip Tad)” shore3|«Xeis | am=thf Toledo” 11)&where>|«)e|isasumov EOalga-gaata4Ifwenowmakethewanst aot. Whereg”isthemetrictensor “ . } Ses Telaa?)=ie[cee [feARON r]de(12)08wherePisthetotalfoar-»ae ow : eh)dul overxinEq.(1.6),weobta= 4"andj,(z)istheHeisenberg currentoperator. (n|P(js(), 3,(0))|n)istheex-Eg (2x)ag| w pectation value ofthetimeordered produet forthesingle nucleon state(at Tyla, g)==D wfiy!Ba Ot rest).Formally, »”7,,(q, q°)istheforward Compton amplitude forthe 2 ed seattering ofavirtualphotonwiththree-momentum q,energyq’,andpolariza- Bik=by tionvector n"fromanucleon atrest. - poyae ‘There aretwoindependent scalar invariants associated withthisamplitude. he=e ‘Theyaredefined tobe {where (1c,I)isthefour-ac ed tote Thedeltafunction ofthe= P=lah =oe tegration overthrec-spsred* Be.11 (13) Consider7,,(q,q°)asafi Bed: P=artsyyole 2seentobeananalytic‘ane osand cuts just below theposit Bo where J,isthefour-momentum ofthenucleon andMisthenucleon mass.The Thepolesarethesingleuel.as calculations arcperformed inthe“laboratory” system inwhich =(0,M). ._,Bed “G'isthephoton energyinthissystem. ge ; aa : r Cottingham, 1963. I,Self Mergy ofNucleon led1.CwritestheusualformafordMintermsoftheobjectT,,,-Then7,iswrittenasad4xintegral of(p/5,(x).i,(0)/p) expo, butsymmetrized. Becarse gurrents must deconserved, your candiate tensors, must begauge invariant, and the usual two tensors arewritten in(1.4). thestructure functigns grethencalled t,and.tp.Whenthecomplete, d4x,integration isperformed, youget(1.7)which showssomeenergy poles andthefeynmanlike SUM(n/jy(0)/k)(k/5y(0)/n). Thethingtonotehere isthatnowbothcurrents areevalyated atx=0, Taking theimaginary partof(2.7) _thentells you: Im7,=SUM. 5,dy+delted, which nowlooks likeunitarity. Ihadbeen wondering where thepole came from, nowIgee: itcomes from thedq° integration andthemonentum translation deal. 2.Soifyou contract uonv,(1.7) gives tHe integrand ofthe dMformula, Asafunctio ofqothis thing has poles and cuts; thephoton propagator also makes two poles. But things are such that you can rotate the contour CCfothe vertical, -thus getting (1.8) where atgisnoweuclidean, Nowhereisthe point: in(1.8) youhaveTy(qitdg): s0energy isalways imaginary. Thus, theq°iéexclusively spacelike, Tms, you dont need timelike information (like from ete-) todothis integral. lo)3Nowsimplysubstitute (1.5)into(1.8)togetdMasanintegral overspadelikeq°andover q°oft,andt,, Next, disprese t,and%,anduote their pole contributions (invbving form factors). Note that theoutdisc Imt, iscalled hy. Finally, ifyou shove these disprels into (1.9), you get final result (1.13) through (1.15). Ie,yougetthemass difference interms ofaspaceliBe q”integral of form factors (which welater get from elastic scattezing), and asadouble integral ofthe Imag part ofthe structure functions, which wehopefully get from inelastic (deep) scattering. Itisnice tohave clean formulas atlest. There seems tobe aclear experimental goal now. II.BlectroneNucleon Scattering, Here, informula (2.4), weseethedeep inelastic DCS written interms ofthe familiar lepton trace and the feynmenic current stuff which werealize isjust Im?,,, asnoted above. Thus, youcanwrite theDCSin terms ofhyandhy(itisthese which must connect toWlandW2). This is(2.6). Can be reparametrized interms oftransferse and long polarized off shell photon cross sections ifyou like, asonpage 429. Similarly, but restricting tothe elastic smmpxe events, you can write elastic DCS asin(2.8), from which you can gettheformfactors fland£2.80ifsomeone wouldgodotheseexperiments, you 6 could find all these functions and plug them into (1.13) through (1.15) and get atrue number for the n-p mass difference. This calculation would then include the effect ofall intermediate states,and would therefore bebetter than the purely pole models ofearlier papers. Cobviously eagerly awaits experiments. III. Possible subtractions, Thecatch isofcourse that whenyoudisperse your functions £1and+2,depending onthelarge q°behavior, there mayhave tobe’ subtraction constants also inthe dispersion relation. Apparently there is awaytoshow that tzcannot have such aconstant, butt;could. Sointhe end, data may show that thet,integration fails tdcdnverge. Then you are cooked unless you cai find away'to get that subtraction constant. *Comparison toCFG paper: CFG know that you should “think about inelastic states *anddout subtractions, butintheendthey onlydidtheelastic formfactor thing. Byinserting only the pole term oftheir disprel (5) into aMformula, &nd assuming nosubtractions (polynomials) they got (7), and then they show based onelastic data that you get the wrong sign. Cottingham did much more: heincluded the. .inelastic stuff andwent ontoshow howyoucould gettheinelastic data from + imelastic e-p scattering, Iwonder: isCottingham one ofthe pioneers of this idea ofdoing deep-inelastic experiments and connecting that data toaM, orwas that whole.program well-known atthe time? : . eo . . e a TT gallpakdtsle gulch gleliesewwlensaotouane2aikaunts.(aGupvman! a_ Sn Spans =SahBS ps Th a rR Basinbewaect =APSEeaCD) OnugLitade share sukos+Cealegy Aampetios, Saooelidas. ee. 9SokckBDgOSdeSommemee eo ee ee ep ee ee --aeae=-GVaTaleeRyaaKunis oo. -more 4oea vAoe. peleghtans tos ee Se MeoplGothQuotes-oo 4oe - ee ee ween ae TARR EG Kea=~Keesquealpssyae =CNakake — &_€_. - . ==es ~ '=> a .&--. -—--~AG) Ma) HershAeGe AyMesTo we Tea dn =9¢.-t) Abs)Ape)+OC) ALA lee). COWedeotrueapyT=@aill-(2aSz)? =RaatQuaDAsjnomgranny,QaoseOaalesyorgh . ~ = .-phatHiQeKe7won xa.:_ me SandeCre Sane(Gs)<a\thICSHAHgD.. (GK eta s@)-B emt) =aeses 7 2 BSS BG) BS.eWGA segecdeugadenenMaso: qn -viquse- gy oo 4|Wakecs" ader’as GAnoteArdetoeseArlty. Coleman &Glashnow (1963) > a FS ae %“% But ‘“ ogous“fattus 4 <j a =f,lel—2e'6* A onthemofoan,forallr.(A6) Ad*=florBee". eS severali (kr). db 3 ‘MfIn38Now,fromEq:(1.14)wewriteS=ne intermsof sImagt=—BIme 3 vgthisMe™f-Fife ae _ asoa se¢ata), an =—6[from(A4)].(AO) Bree:4 Saro—Bo" ‘Thus,from(A9)and(A10), q ~tilftheretheprimedenotesdifferentiation withrespectto ; Pa: ‘andTo,Boraand¢,#,o%,g©areevaluated athr=B. joi,|AP+Yélslt426/4 3 ougalt ayAERPs 5 icationOoh. 1,ifandonlyif,Imfu<0;-(A11) ‘a ofthe’ a icand2ptereA=f.p—89". weknowthat]S|*<1,ifandonlyif,fisHermitian. ESusing34H [A+/2lel*—2fImage Thus,fisHermitian, ifandonlyif, ES, aand |S (ag :ampliaa [A|*+feld|?+2f;ImAg* ImfurS0. (A12) strong beThis —— 4 cutofl,aFp § veneral QHPIVSTCAL REVIEW VOLUME 194, NUMBER 3B 11May 1964 ey 4 Departures fromtheEightfold Way:Theory ofStrongInteraction aay Symmetry Breakdown* a i SweyConzuan aah ao 4ae S.L,Grasowt a(An)ea LymanLaboratoryofPhysics,HarsordUnisersity,Combridge, Messachuselis évation Eg (Received14November1968) . ies ‘Weconsiderthethreekindsofdepartuce fromexactunitarysymmetry: midum-trong interactions f(02)gS whichleaveonlyisospinandhypercharge asgoodsymmetries, electromagnetism, andweakinteractions, 4“A Wepostulate theexistence ofanoctetofscalarmesons thatgivethepossiblity ofsyrmmetry-breaking tad aBa: (3 polediagrams, Ourfundamental dynamical assumption-—that symmetry-violating processes aredaninated ttance is bysymmetry-breaking tadpole diagrams—gives animmediate explanation ofthesuccessoftwoempirical oe BogashGellarOksmasformulasadhenoneA=}rl,Moreover,neldingtadpole 4 “aiagrims and someotherelectromagnetic corrections, wecalculatethesxelectromagnetic massspit a ‘then"s ofmesonsandbaryonsintermsofasingleunknownparametercoretlytowithinO8MeV. “e a a. 4 ayi 1.INTRODUCTION schemeofGell-Mannt andNe’eman? basedonthe .aBIE assumethatthefundamental interactions of&°04SUG). inginteractions. * *sagVV‘elementary particles fallintothefollowing ,:Medium-strong_symmetry-breaking interactions, :4 y all invariant under only theisospin-hypercharge subgroup +Miltasses,arranged inorderofdiminishing strength (we SU Thesei. ey (A)Bhigravity): ofSU(S),Whether theseinteractions areintroduced at aYk thebeginning, orwhether theyarisebysomekindof o«,3agél.Very-strong interactions, invariant under the spontancous symmetry breakdown isimmaterial toour fsubatk, Siansformations of“theeightfold way”—the symmetry discussion, 4 ich9729 3.Electromagnetism.oY7hytheU.S,OfficeofNavalRewaren, 2Weakinteractions. ha #Supported in part, .8, OficeofNavalResear 5; Ae inactNONK-3686(0), andbytheU8Ateeoe)——— (AS)Zeeeatiic Research, undercontractnumberA.P.49(638)$89. 4M,Gell-Mann, California InstituteofTechnology Synchrotron i«FAS|Aled P.Sloan Foundation Fellow.OnlavefromthePhysicsReportCTSL20,1961 (unpublished}; Phas,eve158,10) ae tobepegppsctment| oftheUniversity ofCalifornia at.Berkaley, (1802), 410beiloaoenia, #Y,Neteman,Nucl.Phys.26,222(1961). WyMea aeae BH 4 afl a Goleman’ andGlashow: MassSplittings intermsofTadpoles (1963) @)_ ts"Theaoolofthispaper istounify various schemes which havebeeninvented to explain mass splittings. Bg, theusual scheme ofanL, mediumstrong synmetry breaking lagrangian which transforms asFgiscalled octet domaitance andleads tothe Gell*mann Okubo mass formas, There isasimilar rule which leads tocertain electromagnetic splitting sumrules. Also, there arecertain weak interaction rules. Iamignoring the parts ofthis paper which pertain toweak interactions, atleast for now., 2.Theproposal isthis: assume there éxist anoctet ofscalar mesons (bytheway, none oftheusual mesons arescalars). Forsuch ameson, youcanhave "tadpole" feynan diagrang whichgivenon-zero contributions. Eg,in@macgcalculation you would include ‘along with the standard diegrams like —7y-. Thepoint isthat @scalar meson hasvacuum quantum numbers asfarasparity andspin areconcerned, soyou canhave such ameson with momentum q=(0,0,0,0) dissolve into nothing. Ofcourse, ifyoupostulate asCGdothat these mesons form anocte@, then thefact that they candissolve into vacuum represents anSU(3) breakage. Ie,these scalar + mesons are the means by which unitary symmetry breaking ocours. Ie, you assumé a non-zeroVEVforsuchmesons,Obviously, bythéway,thiswholeideashouldbeand © 2 doubt nas been rewooked interms of SSB guage theories. 3.Now, although giving non-zero VEY’s breaks SU3, the Yukawa couplings (interaction lagrangian involving these scalars) areauumed tobesu3 scalars, asin(2). It turns out asusual that there are two scalar couplings, with constants dand f,when you couple tobaryons. So, the information contained in(1) and repeated in(2) leads tothe formulas (3) for self-masses inthe one-tadpole approximation. Note that m,and ug are masses inlimit ofnotadpole contributions, There are 13masses given interms of6parameters, hence there will be7sum rules, asseen onp675. 4,Results: 11 known rules are duplicated inaunified way, and some new rules are found. All rules are reasonably obeyed. Similarly, the decuplet rule comes out, (11). 4.InSection (4)the focus isonthe emmass splittings. Bg, the n-p splitting. 4syou see inthe chart page 676, the usual calculation with form factors gives thewrong sign (seenontadpole or-S% colum). Thetadpole graph isdifferent for pand ninthe right way togive anoverriding right-sign contribution, Infact, all 6splittings listed come out with right sign and tomaybe 30% orbetter. Notice thatforthepionthereisnotadpolecontribution §seeresult(8)),sothisexplains © sng the conventional calculation gives the right answer inthis case. Asimple interpretation ofthis tadpole stuff isshown infigure 3(c). Ascalar meson cancouple directly toaphoton (over) andthus yougett-channel pole contributions totheCompton amplitude .These things contribiite tothe cut part ofthe dispersion relation, ormaybe tothe subtraction constant, andsuchtermsareusuallyignoredinthe"conventional" [*) Caclaultion, They dont saymoh about Regge theory, but obviously these exahnges can beRegge-interprested. 5.Comment: itisclaimed that because youaredoing acomplete d4qintegration . toget self mass, only aspin-zero exchange int-channel cancontribute. Iwill have to think about that. 6. Section Von weaks Ihave not read. 7.Remaining remarks: why isGMO forma soaccurate? You would expect that themedium-strong tadpole y'single~tadpole approx isnotsogood, andyouwould expect maybe multi-tadpole contributions would besignificant. But perhaps somehow these have already been included byusing experimental masses... Anohher points Iglossed over isthe fact that the scalar meson coupling has tobeuniversal, ie, coupling ofscalars tothe0“,0+and1mesons isthesame. This fact wasneeded to get various sum rules. 8.Pignotti isquoted privately asconjecturing about scalars =ghost regge effect. This idea later carried byFred and Jim. . +) . e Coleman &Schnitzer (1964) eo? 4 ; He MIYSICAL REVIEW VOLUME 136,NUMBER In 12OCTONER 1964 me totheses (a)and (8) Departures from theEightfold Way. II. Baryon Electromagnetic Masses‘ ( gl y.ry agn vse, inthelimit of4interactions). a. StoxeyCourant“Pyiinentn “He LaymanLeboraoryofPhysic,HarcardUniversity,Cambridge,Mossachusls Intractions that 7, ax vumed, anattempt, jg? Howano J.ScunsrzentfeetianetA DepartmentofPhysics,BrandeisUniversity,Weltham,MostachusetssymmetrybreakaE (Received18May1964)°S(andtheirlarge ‘Theleadingnontadpole contributions totheelectromagnetic masseplitings ofthebaryonsareealewat Inte. Best fitstoexperimentareusedforthemucleonfocmfactorsthestrangebaryonforfactorste +decay Lagrangian -O} tained fromthesebyunitarysymmetry.AccountiakenofthebreakdownofunitarysymmetrybyWing ndtheed se experimental masses inintermediate states andbyincluding theeects ofOwmixing. Then cleatsMey aeaR andtheassumptionoftadpoledominanceareutedtotthesxmasssplittingsofbaryonsandpaeuosealst =0)cannotifs mesonswithonefreeparameter,Allofthesplittings arengoodagrezrantvithexperimentexceptforthe “aafterinclusionof," aonsplittingwhichbastherightsignbutonly#ofthepropermagnitue.Theresultsofapreliminaryvers iden,Nextconsider "4ys ‘sionofthiscalculation werereported inthefirstpaperinthisseries.includes only the,&be otthe symmetry. SFRa ‘ingsofD’which I.INTRODUCTION (2)Equation(1)iscriticallydependentonthebe- sind64,thewhole™iGEVERALyearsago,Cini,Ferrari,andGatto!ob»horoftheforfactorsathighmomentumtarsi validand“allthey,@§Dtainedaformulafortheelectromagnetic selimmassOfwhichweknownothing.Indeed,ifweuseformFactors “Thisisbecausethet-Pef thenucleonintermsofitselectromagnetic formWithhardcores,theformuladivergesquadratically, In hecharacter of‘affpiittors? Theformula isrepresented diagrammatically dettoobtainconvergent results,wemustuseacutoff; iant,However,thet;WYFig.1,wheretheblobsrepresentformfactors,andWidelydifferentmassdifferencesmay'beobtained, Ibelongs tosnadfaalytically bytheexpression depending onthechoiceofcutofl.+vartnerFon‘CPs::° (3)Equation(1)neglectsthecontributions fromand@)arestil siefpde is possiblescalarmesontadpolediagrams.* Thisisnot,tutthisnolonger)caeGaymaFu)+Fale‘\euoky ‘asmightappear,aspecialcaseoftheintermediate-state theK*—+2n and sae. *e criticism statedabove;thescalartadpoles simplyaddaonly ifsuch an ME) pencthxnctmy i constant totheunphysical photon-nucleon scatteringtaeteeAxePapreeMOL, (1)‘amplitude,andmakenocontributiontotheabsorptiveOMeaganee B+2p-btie In part. ineRtDrdecaysabermisthemassofthenucleon,esiseh Thisthirdviewpointwasproposedinthefrstpaperout byCabibbod?3Bandaretheformfactors,nomadsue’fane{8ttaoistheoneveadheretohere.The tcteroftheelectro 8aFO)isoneandF,(0)istheanomalous momenti,22bodyofthispaper(Sec.11)isacalculation oftheofthisF(whichfdtaromsnceens leadingnontadpolecontributions totheelectromagnetic +importance ofthe, JHEquation (1)isobtained byweitingtheselimass infussSPlittings, notonlyofthenucleon, butofalltheLagrangian. Also,-emsofnucleon-photon scattering(withunphysical plvons"singEq.(1).Experimental best-fitform ‘sisofnonleptonie* potons),andthenapproximating thescatteringampli-jeer)teobrednakeoeenuntensume stive.{"itebythecontributionfromone-nucleonintermediateTheseAvsoeatTere oson, ~hisgratitudetoci/@'sates (poleterm).Ithasbeenoftenappliedtocalcu.“Symotrybyusiogmieuhoaloe enunitaryandDr.B.Sakita‘i)btionsoftheheutron-proton_mass difference, withasymmetry byusingphysicalmassesforintermediateF.E.ThrowforseGrrotorious lackofsuccess."* SeveralreasonshavebeenStatesandbycorrecting theformfactorsforromixing?ayfudvancedforthisfailure: warlantandbehaveagaBe'@) Equation(1)neglectsthecontributions from aa vasPhesyaA@htermediatestatesheavierthanonenucleop,)Perhaps DDDD aiind)vs=Hfiesecontributionsarelarge heresultantefecieel Seworkcepportedi , Fic,1.Theleadingnontadpolecontributiontothedlectro- osepre eRettsU,S,ASeeObesofpasaapigWfune laee rapmrannn tadsfPartaliyaupporedbytheNationalScenesFoundation, "=o™aGREformaeons. tetheholargu)gopyit> ©:FathandR.Gatto,Phys.Rev.Letters2,7menosofthe3/2"deeuplettootherbaryonslitngsdonot ‘anexamplewheree2%#4}*Acloselyrelatedformula beenobtainedearlierfromaerated aTenibeastFeSee,ciputations indicate‘stdthe£25magFoyeRev.94500(19847 "aLBande,NuovoCimento26,1068(1962)GSSaharadKFst,PaRevSeaaa.geGlm8Ee wa anes“FRATOYN.Cottinghars, Ann.Phys,(N.¥.)25,424(1969).oF”OL, D.Milig,andR.Wilson,Rev.’Mod.Phys.35, PoPeine,thecontribution fromthenextstate,the3-3resonance, 335(1963)SbeqanTS" 3 "1Eelsesateofteapinaymmetrys vtstigheracre, BleOOCesnow,Phys:Rey.Latins6,423cg),FF {Efoymakenonvanishing contivitions Thecontbutonsf oteOtCalmanandFe.SeanitesBayaRew18WoeUSSoF ro) ey 3223|anit Pe imou.aa= -O fasN_-Aets =eGtayoR ae ~OA 8) _. =t(anb) +(Aveaf ep ae --Saw mFeaafe.dadeok(Le) .: . bela Qype CQ os ope ~ 7 mG Soe 2 oO 2s en -— —_ —_eeaMoakaepeefronFarpranam(29)@IITSa ~ Don onanapn 1% ardeticSay, _ . Ge ~Gu~ \ :~~. ome embed ao... ws Yeo7. Sombodenny) Why : : Coleman and Schnitzer: emmass difference calculations, nontadj O 1.mniesortpaperisafollowup oftetadpole paperofRef.6,Herethey concentrate onthe conventienal form-factor calculation ofthe (nontadpole) one-pole piece. They usetheoldformula, here stated as(1)forbaryons, and thegame is(a)tousetheexperimental form factor data andintegrate and .see what you get ;(b) touse SU3 toguess what the corresponding form factors are for strange baryons. 2.One technical aspect ofthe paper involves the idea of"mixing", Idont know what this means, except itinvoles correction tothe SU3 predictions ofthings. There isref 's8and 9on this ifIcare to learn about it. ‘The results are given inTable Iand II, The numbers differ byorder of1MeV depending onwhether ornot you incorporate "mixing". Their point here issimply that you probably cannot trust these calculations to better than 1MeV since different calculations differ bythis mich. 3.Remember that @“and@t arenotantiparticles, sotheir predicted splits from the2°arenotthesame. (nor aretheir experimental splits). Ie,the9*and37 6 donothavethesamemass! . 4.The meson caloulations were done by Socolov inaHarvard thesis. CShave only done the baryon calcs. 5.Page 224shows theexperimental form factors forPyandP,, ;these seem to disagree inasymptotic form from what feynman gives inhis 1970 book. How to resolve this (see attached sheet for comparison) 6.Pinally, Table III gives the tadpole and non-tadpole stuff and compares toexper. Thie table supercedes that inRef 6because they made some numerical errors. All signs are right, agreement isgood. 7.Now Iknow what mixing means. Remember that vector meson dominance gives atheory of form factors. Soyou get asimple pole for each vector that couples. This the isoscalar formfactor willhaveaR&@xamix@¥anW andpcontribution. Thus, thevector dominanace says asymptotic FPbehavior is1/q2, not1/a4, Feynman says: VDMalways predicts FFgoas1/q°. Exper, thisistrue forthepion ¥F,butFeyn says thenucleon FFgoas1/q+, This isthefact being claimed different1 here. Suzuki &Zachariasen (1966) na966~ Votume17,Numper19 PHYSICAL REVIEW LETTERS ‘7Novempur 1966 ; scattering” asforproton-proton scattering* fm,R=1.82A fm,and1V9i=22 MeV. The atlower energies. Inview ofthe similarity results are shown asdashed curves inFigs. 1 “ ofproton-nucleus and proton-proton scatter- and 2for Be, Cu, and Pb. The agreement with ingathigh energies demonstrated inthepres- experimental results isapproximately asgood ent paper, itisreasonable toassume that the aswith our first model. ratio ReT(0)/Im7(0) isnegative athigh ener- _ gies also forproton-nucleus scattering, which —— gives arepulsive potential V,=+12 MeV. 4G,Bellettini, G.Cocconi, A.N.Diddens, E:Lille ‘Thewayinwhich thepotential scattering has thun, J.P.Scanlon, andA.M.Wetherell, Nucl. Phys. beenintroduced isonlyoneofmanypossible 19,609(966), ways,theessentialpointbeingthatwithout 1.vanHove,NuovoCimento28,798(1963). Yel potential scattering, V,=0,weshouldhave sant Nottingham andRF.Peters, Phys.Rev.. =3(1-(1 137, . ‘ raeayenaom Seatering, hesandhy101 “LJ.Cook,B.M.MoNillan, J.26.Petersen, and ee~p)?], Inordertoinvestigate thispointwe D.G,Bowell: Pigacev.15,1808(19halsomadeanattempt tofittheexperimental SH.A.BetheandP,Morrison, Elementary Nuclearune results using acomplex potential, i.e., weput ‘Theory JohnWiley &Sons, Ine.,NewYork, 1961), i pp.7-11.(9) h=G/2)(1-e 7%), (1) *D,Hartingetal.,NuovoCimento 38,4640(1965). 'P, Bjorklundand8.Fernbach, Phys, Rev,109,1295 wherex=2Vo6-'(R*-6*)" describes potential age ndS:Fernbach, Phys.Rev.108,scattering while7isgivenbyshadowscatter- 4G.Bellettini, G.Cocconi, A.N.Diddens, E.Lille- heingonly,e~27=1-p(b). Thevaluesof4pand thun,J.Pabl,J.P.Scanlon, J.Walters, A.M.Weth- eAareasbefore, butwehadtochoose Ry=1.00 ereli, andP.'Zanella, Phys. Letters 14,164(1965). e . SIGN OFTHE PROTON-NEUTRON MASS DIFFERENCE AND AGHOST SCALAR MESON* MahikoSuzukiandFredrikZachariasen oe): California Institute ofTechnology, Pasadena, California (Received 3October 1966) aAnewinterpretation oftheproton-neutron tionZpwill become Zp+02p, andweniaywrite : massdifference has-been givenindependently 2,(g,m, 4m.); byPagels! andFriedandTruong.’ In-this note, bm, oop Mon bmve itisshownthat,ifthismodelworks, itimplies my= we fem, ?5 ‘@ghostscalarmesonwithT=1andG=-1, P Useismadeofthetechnique ofspontaneous +85,(f) @) ‘ale symmetry-breaking theory.’ Byidentifiying pOvems"‘ theghost with thatoftheRegge-pole model, i wsonecancrudely estimate themagnitude ofthe ‘Thismayberewritten intheform feedback bymeansofalinearextrapolation, CCAR) aa @)Thephysical mass oftheproton, intheab- ’ “™ = sence ofelectromagnetic interactions, may using thefactthattheproton wave-function bewritten renormalization constant is z - 1 »myrmgt2p,"op'"oW=m, aefea |w _- wf.70 whereZpistheunrenormalized properself- , ” ship80 energy part. Itsdependence onthe bare masses 90 oftheproton andneutron areindicated explic- Similar equations can,ofcourse, bewritten 10 ity.Itisalsoafunction ofthebaremasses fortheneutron, andwenotethattolowest or-a0 ofotherstrongly interacting particles andof derinthefine-structure constant wemaytake 8 thebarestrong-interaction coupling constants. Zy=Z,=.a Nowletusintroduce electromagnetism. The ‘Thephysical effect exploited byPagels andae) protonmasswillshifttomp+6émp,thefunc- byFriedandTruongisthe,feedbackontheelec- H 1033, ! eee Harar (1966) ; _o- halist “wei e : +ame ondwb=wy) oeRapesrae onYO \ t e Vorume17,Nummer26 PHYSICAL REVIEW LETTERS 26Dacennen 1966OO SUPERCONVERGENT DISPERSION RELATIONS AND ELECTROMAGNETIC MASS DIFFERENCES* Haim Hararit 5 | Stanford Linear Accelerator Center, Stanford University, Stanford, California i (Received 9November 1966) ' Using dispersion relations andtheconvergence properties oft=channel isospin ampli tudes, weshow that(a)4/=2 electromagnetic mass differences should becorrectly ob- tained bysumming theself-energy contributions ofafewlow-lying states; (b)A=1mass differencescannot beobtained inthis way, andasubtraction termisalwaysnecessary; (6)thesubtraction termhasthecorrectsignforexplainingtheproton-neutron massdif :ference. i ‘Theproblem ofcomputing theelectromag- simple approximation willgivethecorrect or-|neticmassdifferences betweenparticlesin derofmagnitudeoreventhecorrectsignfor |agiven isomultiplet hasalways been oneof theA7=1 mass differences, thegreatest puzzles ofelementary-particle (@)Aconsistent calculation oftheAf=1terms | physics. Itiswell known thatthesimple, na- must include anadditional “subtraction” term, ivecalculations whichincludeonlythecontri- Weshowthatthistermhasthecorrectsign |stbutionsofafewlow-lying statestotheself- and,roughly, thecorrectorderofmagnitude energydiagramleadinmostcasestototally requiredbytheexperimental masses, ri wrongresults(including thenotorious wrong Wealsodemonstrate thatthestatements (a)- |‘Signsfortheproton-neutron andK*-K°mass (c)arecorrectforallsixelectromagnetic had~ |diferences). Ontheother hand, thesame sim- ronmass differences which areexporimental- pleapproachgivesthecorrectsignandmag- lyknown,andweproposefurtherexperimen- @|_niteinatowotnercases(suchasthex*—7° taltestsofourassumptions.difference), Inthispaper wepropose simple, Inperturbation theory, theelectromagnetic_|Seasorabieascunptins onthesnergydepos, goreneitiontheory,theelsesGenceof¢~channel isospin amplitudes forfor- 2wardCompton scattering and,usingtheseas- 1pera gh! (a)All4/=2 mass differences should becor- “2 rectlyobtained whenweapproximate theself- whereHe"P(92,»)istheforward amplitudeenergy diagram bythecontributions ofafew forCompton scattering ofavirtual photon with \low-lying states. massq",energy g°=, andpolarization ¢#from(0)Thereisnoreasontoexpectthatthesame)ahadronwithmomentum pandmassM(Mveth. Wewrite728 Peale weave 1yates -0,9Jettvive,+Lapot 2 “h wiMats, Noe,a0,}+tgla,IME,+PP+7y(,,+P,f) @) Cottinghan®has shownthatbyrotating theintegration contourin(1)fromtherealtotheimaginaryaxisinthecomplex vplane, onecanexpress AMinterms ofscattering amplitudes forspacelikePhotons, allowing us,inprinciple, touseexperimental electron scattering datainorder tocompute ; theintegral. Substituting iv andintograting over theangular variables ii(1),wefind ag? ¥eaegfSEYanette) (o2lti), ) Wecannowwrite,forf,andf,,fixed-q?dis~|SSP NPSESEEEonDNEreneeepersionrelations invandcomputethef7’5in firstfewlow-lying stateswilldominate the e terms oftheir absorptive parts?*The main expressions forf,,f,andhenceforAM®*The obstacleatthispointis,ofcourse, theques- ‘convergence properties ofthedispersion in- tionofpossible subtractions inthedispersion tegrals aredetermined bytheasymptotic be- relations, since onlyinthecaseofnosubtrac- havior oftheabsorptive parts oftheamplitudes. tions canwehopethatthecontribution ofthe Tolowest order ina,theelectromagnetic 1303 : Tou6,1007 Harari: WassDifferences, 1966 @}_—s}+ ‘Hereistheproblem: sometimes thenaive aMcalculation works (pions), and sometimes itfails sobadly that itgives the wrong sign: Why? Recali that CFG in 1958 put inthe experimental form factors and. showed that p-n mass difference comes out with the wrong sign inthe.polesapproximation tothe aM. Thus, CFG disproved the conjecture ofFSin1956 that form factors. could explain the anomolous sign. 2.Huses Regge theory tofind theanswer. First, heclassifies mass differences interms oft-channel isospin amplitudes, something Ihave not worked out atthis time. Itturns out that mass differences are either Iy=1 orIy=2. Hnotes that itisonly the I,=1 differences that arecausing thewrong answers, Basically hewill show that the reason isthis: for the Iy=1 ampitudes (mass differences) like the p-n difference, the pole approximation derives its validity inthe first place from adispersion relation written for the structure functions, The pole is only agood approx ifthere happens tobenosubtraction constant inthe dispersion relation, The presence ofthis constant isdetermined bythe high energy behavior of the integrand, and for Iy=2 there are nohigh Regge trajectories sothere isno constant. Forthe Iy=1 however there istheApwith intercept .4,sothere isa constant. Thisconstant isthereasonp-ncomesoutwithwrongsign. 63.Now lets gothrough details qhickly. The mass difference isgiven byacertain integral ofthe Compton amplitude for photon-hadron scattering. Happens tobe inforward direction, sot=0. Variable nuplays role ofs,incident energy. This amplitude isT,,. Using uvsym, andconserved current (ie, gauge invariance), you can express Tyintermsof twostructure functions called here t)andtp.Obviously these arerelated tothenowstandard WyandW,functions ofdeep inelastics. Hence equation (2). There isad4qFeynman integral, butangles aredone andthen the energy orvcontour isWick/Cottingham rotated toyield equation (3). The idea isnow toestimate the structure functions and plug inand compute dM. Bythe way, seems tomethat in1978 these functions are now very well known, soyou could just doit, But in1966 Harari must approximate them bypole term ofdispersion relations. Soequation (4)shows thedisprels forthet,,alsoclassified by isospin by supersoript. The term out front isnot the subtraction constant but israther the pole contribution tothe integrand. Recall that pole contribution can bewritten interms ofcoupli gswhich inturn are given interms ofthe various Eand Mform factors for agiven particle, hence (5) and (6). The Regge step is e™ invoked andyouconclude thattheI=1disprel fort}musthaveasubtractionconstant. This isnow included in(10). Thus, thenaive pole computation isreally @computation ofthe 2nd and 3rd term in(11), involving the form factors, Hgoes on toestimate the size of this subtraction constant .You pick out the constant dytaking acertain limit, but this limit also picks out the magnetic moment of neutron and proton, The sign ofthe determined constant now comes out correctlyforthependifference, e 4.What areother tests for this little theory? Regge part of itmakes predictions for the photon-proton crdéss section (now known) .Iwonder how this oame out? .Other mass differences are very hard tomeasure along with magnetic moments. e Ball &Zachariasen (1968) Reprinted from:Tax Purysicat Review, Vol. 177, No. 5,Pant I,2264-2272,25January1969 ined ipUS.A. fe) Mass Splittings.and Ghost Tadpoles* J.8.Bau University ofCalifornie, LosAngles, California 90024 FP.Zacuantises California Institute ofTechnology, Pasodens, California 91109 (Receive! 11-September 1968) xgreelpana antesotefrownNeoyatseile,agieas ofthesinglen=0,If=0(4)amplitudeforthesumeproces.Tarepresentation allowsquitedirectlyn. uabtative undeianding ofthinefthenasdere,edaloallows,wthmoredatesump. ons,anapprosimats mumerical evaluationofthemassdifeencebasedonexperimentally determineparameterfromtheAsReggetrajectoryThisnumerial.etimate isinreasonableagreementwithexper eat,Thespproschiseasilygeneralized tootherelectromagnetic massdifferences, andisshowntoyield I.INTRODUCTION ‘asignchange? Ittherefore became necessary tolook7insk isatothercongequenées ofthestronginteractions. [°2Edigerence insignbetweentheexperimental "Asafirststepinthisdirection, itwasobserved by Protonmass “dilference and’thevalueHdrari? ‘thattheexistence of,theRegge trajectory calculatedontheassumption ofpoint Dirac nucleons ‘ete withthe adeit ademonstrates unequivocally theimportance ofeffectsitythat’the.sim,leFeyuman"Spel&trenelyorf duetothestrong interactions ofthenucleons, ifoneis y " Synmnan-Speisman-tyPeton culation ofthem-pmassdifference couldhivetuned toascribethemasssplittings solelyto.electromag- ‘, (hisisinmarked a netism.ThefrstattempttoestimatesomeofthéseOutto,Becorrect.ThsisIpmaria cccorork effects was byFeynman and Speismanin19563‘They™2SSdifferences,wheresuchacalculation doeswor!‘ fe andSp %fairlywell,andwherethereisnoanalogofthedsvj deinonstratedthattheinclusion ofelectromagnetic‘ hernot itativelse oy formfactors(presumably presentbecauseofthestrongalectory.Jtwasfurthernoticedthat,qualitatively,” yet ‘Z,interactions) inthecoupling ofphotons tonucleons O2€Might expect the,existence of‘theAstosefféctarts ngofP’ “/unusually stronginteractioninthe J=0+)T=1states 3°couldleadtothecorrectsignforthemassdifference) St eetnandtheetht " dependi behavi foemfacton 5 this . on"srdependingonthedetailed behaviorofthe SCouldproda aeOeeinem * “Yo Asexperimentalinformatignontheformfactopsbecame ONGProducesignchangeinthemassdifference year!”available,however,itbecameclearthatinfacttheypePrycn Brgeesicswhichscam.tobeimportantae formfactorsdidnotbehaveinsuchawayastoproduce 4,0.StonesofenGatinetboundstate(chati,«bound state with zeroresidue') ofnegative mass squared, withJ0%,I=1,whichis’produced bytheverystrongforce.Thisextinctboundstatecanbethoughtofasthe“tadpole” whichhabeeninvokedasamoreorless pheromendlogical way ofobtaining. thecofrect mass + shifts. [Since the4sisameinber ofanSU(3) octet, . (afOnEFes,andR.Gatto,Phys.RevLet2,7 coSHHaraPhysey,Letters17,1303(1966). a H (ME,Suull andPeduchargcen liye, Rewbatters 17,1083,gpeelny 1)(ie a ; cannot be that 4 mn[in Proceedings ofthe pcditesepaaltnnnmn tessieoriw tesvolleeMitznate,teie BOEDEER creep tgeiolgyEe Coa aeae ees duestotheStrongfntractionintheJP=O"channelofNA(56r)outmestratherbethatof.F.Chew[PiyaRev,Lettersscattering. . 16,60(1966)}.Thereasonforthisisthatwerequireazeroof —" . fsenefactionwheethedetrfectoryfeethroughere, +WerksupportedinpartbytheNationalSinceFoundation, roonlyinthenegecnseDfunction,NheChewméckantam, cn. aoabytheUsAttEaegyCommisionsnderContractNo.heotherhan,ateoftestiDton,Fortuna fo)ELofheSenFrancoGpeatiomofcU5,AtsmictheCheymchitnminactstemetbeconsentwithts “HEP,PommeandG.Speisman,Fhye,Rev.94,500(1984),“'ESicolsmay and,Glashow,Phys,Rev.134,BAT(1964)./ ITT 2264 2265 MASS SPLITTING AND GHOST TADPOLES 177 theextinct bound sate, ortadpole, presumablyialso;momenta,¢andgarethephotonfourmomenta (we() therefore, allpredictions ofthetadpole model are take g’=g*), andwhereuand»arethephotonpolar- obtained.] The“tadpole” isthusaconsequence oftheizations.*Thenucleonandantinucleon are,ofcourse,strong-interaction dynamics ratherthananew¢le-onthemassshell.mentary particle. Wemay place Eq.(21) between thespinors Bp Itisourintent here toexplore further andtoelabo- andtlp., sumover spins andsetBequal to—p,Note rate onthisphysical picture ofwhat isimportant for that therelation theI=1 mass shifts. Tothisend, weshould liketo ressthemassdifferencesintermsofthephysics2(r-byTor(Gdi PP)Moe) oftheWNchannelratherthanintermsofvirtual tow‘Compton scattering asisusually done, Ourfirststep, - »then,willbetoderivefromtheusualCottingham Rig enlbri 2.2) formula? anexpression for8Minterms ofthepartial- wave amplitudes forIV—>ry forvittual photons ofholds, where Tiv.sy- are(essentially) helicity ampli- equal mass. This isdone inSec.II.Theprocess evi- tudes forVW—ry,andwhere wedefine dently isofinterest atzero total energy ;O(4) symmetrythusapplies andwealsoexpress 54interms ofthe teatom (Ob n=0, M=0, 0(4) amplitude forNN—yy. and ‘The formula obtained inSec. IIisspecialized in vag: p/M. Seeatateparticiunitarityincludinganynumber_9g4result,wemayrewriteEq,(2.1)intheform NN—yyamplitudes. Withtheassumption thatthese ay1‘amplitudes satisfy Unsubtracted dispersion relations, ai=iiftry; g)\m0, 23) itisshown, qualitatively,howthesignchangeproduced (2m)@ bytheexistence ofthe’As-can.come about. Wealso,‘outline,inthissection,apossiblecalculation basedonWherewedefine tel0(4)symmetry andtheBethe-Salpeter equation which cotaleh :includesthesamebasicphysics,andwhichmaypermit Ty@)=—TEALTomlbrig) 2A)Oothemass shift toberelated more directly toexperi- mentally determined 4,trajectory parimeters. and,ofcourse, |p|= (4-2). [Precisely thesameInSec.IV,wemakesomesimplenumerical esti-result,Eq,(2.3),obtainsforabosonmassshift,wheremates, withtheintent ofshowing thatthephysical T(1p;2)iatheamplitude forBB—>7,summedover ideasweareusingnotonly.canprovidethecorrect(equal)helicitiesofthetwophotons. signbutmayevengivearesultwhichisquantitatively Wemayalsoapplythereasoninginventedby nottoounreasonable, thoughitisimportant toempha- Cottingham’ toEq.(2.3).Thisyields,inastraight- sizethatnumericalestimatesrequirefarmoredrasticforwardmanner,theequationassumptions than arenecessary forunderstanding thecon 1podgpenneFinally,inSec.VwecommentontherelationofourSM*=——~ f>fdo(—g—vyresultstothetadpolemodel,andinSec.VIwegivea BrJawFJain bp: 2.5)briefsummary ofwhatwebelieve tobethebasic XTO ng). 25) conclusions. NowthehelicityamplitudesTayhaveavery11,RELATION. OF8MTOTHE simpleparta-wave expansion, namely JPROCESS NNyy Paral QEQIFDPs@Tami (9),(2.6) Our starting point istheusual formula forthe lectromagnetic mass shift ofanucleon tofirstorderinwheresisthecosineofthecenter-of-mass scattering ,thefinestructure constant, which wewriteas angleforNN—yy.Thuswehave,inthephysical hopai regionfortheprocess,therelation J—ia=faaptoe-8h—D, (4) rays—[(e-40)"(0-49)/4M}s. (2.7) ‘Thesamepartial-wave expansion evidently holds whereTega,p)istheFeynman amplitude forthe .ProcessNN—+v¥,forvirtualphotonsofmass4«ournormalization ssuchthattheSmatriciarelatedtoTyebyere andfarethenucleon andantinucleon four eeeetppitiTolwlee/teaboay QD WH, Cottingham, Ann,Phys, (N.¥.)25,424(1963)./ _otherwiseweusetheFeynmanconventions throughout, aeee oeeeee . wee ee Ng¥-- — =arly ee SeSiLe] Dor =ohYeecometeme A.WSastty ee ee we©GamedromeDeneenh WoVnadlacthsy snamvengmaambuiyluk.rangescanuh yon.aunmange antpeikape —Se - oo. Bs Va) Jo TayPe - oe}tet[esSP ~s, ee ee Nee wee ee Ce ee LMTAh) _couvenchinall morn (o ee aa HN aa Wechad 2Gad) gataglecenT CRATE \5ENEOh a .- ckaOo : ne gadeol neue. - ws_vs -*>BE Ee. any _ ©@S.dwWAonThyeese} gt)oein(26).Anteckaul,sins oon,Bromesenna. Yality, p=papesOsoRearty, Agaws, ©SorshdoColerivoan dogidQs).Quavulapche oeDeaces (ait).08,fone Cea. 2.- -- ---- --D Qus). Duckatheclowwdh patil wooo UeOGANd akTaoMae,LESnlyondgeen