The Multiperipheral Model

Ks ' , i H t ' TheMultiperipheral Model : andits { Application 1 tothe i Rising P-P Total Cross Section 1 ' i :f i |Phil Lucht: 2278 { Dec. 7,1973 : { H i I i 1 \ i ' - aaa - 5 ; 1 ' 1,Thebasic question andtheprogrdmj doubts. ‘Thebasicquestion ofelendntary particle physics seemstobe this:whatisthemostelegant andgatisfying waytoexplain TheData. Ifthisquestionistoomuchtooneallatonce,wecouldaskinstead: what isthe cleanest way toexplain particular chunk ofThe Data. How mightweexplainthsgheb-begofndasurenents whichhavebeenmadeon6=30-3000GeV*proton-proton colbisions? Themultiperipheral model isanattempt atsuchanexplanation. . Like al] quantum mechanical! models, the multiperipheral model assumes theexistence ofanamplitude Awhich onesquares, A*A,toget somekindofindirectly measureable probebinsty foraparticular event tooccur. The model isanamplitude! and assuch isalready part ofanother model called quantum mechanics. Ithould becruel indeed if"amplitude" weretoholditsconventional roantnkonlyinthenon-relativistic limit ofthetruephysics. Barringanyoubhconspiracy, thestatement ofthe multiperipheral amplitude clearlydefinesthemodel.Theoldmachinery ‘forcrankingoutcrosssections,muthipttcsticn andsoonstandsready togrindthroughanytheorist'sstp‘Theprogramofthetheoristisdepicted below. {. oye a ae A Tae Figues4° i { ' Bod, fan oyGirt eet edsacs i “> i 7 ' { Itshould benoted when we'speak ofproton-proton céllisions that suchcollisions mayrepresent apoor{choice ofexperiment. Ianreminded ofFeynman'scommentofafewyears‘ageabouttheSwisswatchmaker's apprentice whospent hisevenings smashing together pairs ofexpensive wristwatches. Hewashopeful that, bylooking carefully atthevarious Jewels, stemsandscrews thatflewcutandbydoingappropriate ensemble averages, hecould learn howworks «watch. Along these sanelines, ne mustwonderhowmany"events"wouldberequired tolaybaretheSchrodingerequation inasystematic studyof(kkypton-krypton, electron) inclusive spectra. Imstadmittohavingmyshareofdoubts. i ' 2.Development_of the Multiperipheral Model. Themulti-peripheral model yasbegotten oftheperipheral model whichinturnwasbegotten -inasehse-ofrelativistic quantum field theory.Fieldtheoryclarifiedtheclvartentpropagator associated withaninternalline,althoughsuchpropnenfors werealreadyknowninoldfashioned perturbation theory assecond-order ¢nergy denominators. The covariant propagator offield theory exposes the pole terms ofanamplitude, Such poleterms mustbepresent independent offield theory andFeynman diagrams, however, since afixed mass,one particle intermediate state isbydefinition euchapoleterm. ‘Thereisnoreaso# forsuchintermediate states tobe absent. ! ‘Theperipheral model! exploited thepion pole contribution tothe nucleon-nucleon scattering amplitude butdidsointhecrossedchannel. Theideawassimplythatapionpoteinthet-channelatt=My=.02Gev? lies very close totheregion ofphysical tfornear-forward scattering in thes-channel”. Ifthescattering amstuteweredispersedinthet-plane, “Near forward suggestdscattering at.‘ impact parameter near the peripheryoftheinteraction region. 7 — j 7> : 3 1 theactivity nearest tot=0otherentheabovementioned polewouldarisefromthecutcontribution starting att=4m=.08Gev2. Unless thepole residue happened tobeanomolously small”, orthecutcontribution anomolously i large”)forscattering closetot“pthepoleisfourtimescloser than the branch point sothe pole should Hominate. Thus, this one particle exchange model ought to“explain the data" ofthe peripheral scattering of nucleons intotwobodyfinal states.! Inapplications toreactions like * +.TPP theperipheral modelndshad--afterafewcyclesthrough theprogram ofFigure 1—~some succbss*. 1ot 1Gc Theonepionexchenge « model. Note: sometimes k— wt the picture in(a) is % $parpharcd takentoimplyafac~‘ torizeable residue. (a)8 (by eR i 1 t Asthedata onhigh energy jultiparticle reactions grew, itwas discovered that there wasasteep (ekponential) cutoff inthemomentum transfer variables forsuchreactions aswellasfortheelasticreactions. Particles ejected inhotp-pcollisigns stream outalong theline offire, thatis,withsmall p,.Thefollowing peripheral mechanisms weresuggested: t t ! “Unfortunately, thepionpoleroxioforNNscatteringisanomolously small neartO.Infact, itvanished att=0!**The cut isinfinitely long, after 411. +thesepicturesare,Ibelieve,deodribedas"highmassdiffractivedissociation,” especially ifthe pion isreplaced byamodern and stylishpomeron. | ‘ 7 too 4 Asbefore, the rung inthese diagrams represents a(possibly factorized) pion pole. Support forsuch ascattering mechanism existed atleast. fifteen yearsagointhecosmic raylata.Giuseppe Cocconi wasanalyzing thisdatabackin1958andhadneedconclusions tomake: Asafinal comment, thehost important feature ofthe modelisthepossiblePresenge ofthetwofireballs de~ tached from the original nucleons and from which the secondaries, allwith about the same momentum, emerge. Itsuggests that the final products ofNNinteractions are rather insensitive tothe primary energy, once the ultrarelativistic region isreached, asthehigh frequency components ofthe interaction never succeed inemergingfromtheregionofinteraction.® Thistwo-fireball picturewas,vowels,unabletoexplaininacontortion- free manner 1)the lack ofstrong cgrrelations between reaction products inthecentral repidity region, and2)thelogsmultiplicity growth t athigh energy. Ifafireball, ornoya, were todecay asablob of resonance jelly, the multiplicity ofthe resultantspions ought tobe proportional toJs,not logs. | Theseproblemswereresolved bythemilti-peripheral extension . ‘ oftheperipheralmodel,namely,|sank? .Vy ra P AP | ' Indrawing thispicture, wearemaking several implicit assumptions: “IcannotresistfinishingCocconi'sfpovstinAconclusion ofthis kind could have the potential consequence ofremoving some urgency tothe building ofaccelerators ofalways higher energies. Itwould in fact make itplausable that thepresent panorama ofthe . elementary particles isalready rather complete, ifby particles wemeanaproperty brmatter thatlasts long enoughtobesingled out.| 1 ' to‘5 } 1)the pion poles have residues which factorize sothat each blob isa function only ofits own local momentum variables, end 2)the picture is meanttodepictan"average"ofrent\nwhichtheblobsareevenlydis- tributed inrapidity. ‘Theseassumptions correct forthetwodefects of thetwo-fireball model Listed above: factorization kills offmost correlations, andtheevenrapidity} distribution leadstoalog» multiplicity. Themainideahere43}thatthelargeincident energyis shareddemocratically bymanyequivslent blobs.Anyblobhas@ relatively lowsubenergy and,ifthpLinksareinfactpions, theblob isaknown quantity, namely the“TY elastic amplitude. Here theword "Known" isemployed inaloose senst, since off-shell W- amplitudes arenotreallyknownwithoutmanycliveassumptions offactorization andcontinuity. (eg,ChewLow)! \ Asinterblob Links, theploné arejustified inthesaneway asintheperipheral model.Ifthejexternal linesareproperlycrossed, thepionisexposed asapoleinsobecrossed amplitude which is essentially thesameastheunerosse amplitude. Again,thepoleis nearty=0,whereft; isthemomentiim transfer ofthepionLink, |pdeat£20 Before investigating themitiperipheral amplitude, wemust wonder whycertain other"pictures" jareleftout,withthe‘assumption thattheycontribute onlyslightly tothecomplete multiparticle amplitude.Forexample,whatveratothédiagramshownbelow? |i | j i 7 a 1 >6 i 1 This diagram isanalogous tothe two pion exthange diagram inthe } peripheral model, anditisnotclear whyyit should besmall. Possibly the picture makes nosense atall since the pion lines cannot be isolated asinthepreceeding sketch. Itiscertainly aviolationi ofprinciple toregard thepicture asahigher order graph,as if ‘ itwere aperturbative Feynman diagram. i Let mequietly brush aside these confusions and use the } accepted ASFdiagram (the multiperipheral model) since itis 1 topologically tractable and has the,pion pole justification just mentioned. i H t 3.What_can bedonewiththeMultiperipheral Model? ‘The multiperipheral amplitude, inits simplest form, isan estimateofthetrueamplitude toheducenpions(letusneglectall chainendeffects). Thus, | fe Then a3=&dueC= y This amplitude, Tr», canbefedinto theprogram ofFigure 1anda fittothetopological orpartial cposs sections canbecarried out. i T=4Aux480)Toon These partial cross sections canthbnbesummed toybéld aprediction forthetotalcrosssection andthe!forward elastic absorptive amplitude, a= Ze= (-AGS) kZzhe4Slum . Alternatively, direct channel unitarity canbepliedtoobtain thenon~ forward,butsmallt,elasticopenamplitude, { ‘ | 1sayy=ye=2|:* . Arsi(st) =-4Zia9TeleThre!. ' b hsOutline oftheASPprogram.3 'TheASFversionofthemathyerigered modelistheoriginal version. Thelinksarepions,andhwopionsareproduced ateachblob. Then=3contribution totheabsorptive elasticamplitude isshown below. Thesame picture, ofcourse, models then=3partial cross section, according totheoptical theorem. The picture suggests the recursion telation, | i | IntheASFnotation (their equation! 3.4), . , Ape)=SasASSYAna(Bt)8[C@-p"s=50]Jere . . ; whereA"(s.)istheresonance-domindtéd, absorptive TTamplitude ~vSoe. Byareasonably straightforward probeedure, ASFshowthat,inthelarge slimit,eachiteration oftherecupsion relation 3.4picksupanother power oflog s,sothat, | ‘| { | Tt LG 1 gy Aw~x ' a i Aa~SOdmys) : i ii ww, Osos | ~ Xd. An~&aap€ic Here, >represents thesizeofthéTTintegrated cross section. , These partial cross sections areinqualitative agreement with theprongdatat HfoArAr As i 39 mn(Ln ' Thefunction $(s)=Gus)starts,outsmall,growstoamaximum on(Genes)valueof©(Gen) ats =e",{then diesout.Foraspecific s, I thetotal cross section isdominated bythepartial cross section . which peaks atn=In(s), remindinglus ofoneofthemultiperipheral model's major assets. } Thetotalcrosssectionasesaidcanbeobtained by formally adding the partial cross s¢ctions, although ASF donot do this.Formally wehave, \ ant 2NHS A-lua, Ams Xx 2 = XKQos Fe"=Ys" Se=Ze %ZGay OS a‘ThissummationisdoneinthemeaddngInapproximation discussed in Horn“p 154. Asemphasized there, {tisnotatallobvious that the series sums inthis simple way, sinde ineach order lower terms have been omitted. AsASF show, however, the net effect ofthese neglected aterms isarenormalization of) Xx. t t | ASFshow this inthefollowing way. Therecursion relation 3-4 istransformed intoanexactBothe-Saltpstor-Like integralequation forthefullabsorptive amplitude f=ZAne moweupst | . Byexamining the symmetry ofthe kernel ofthis equation, ASF conclude thatafactored solution isadmitted, viz, A(syothervariables)=s°Batevariables)whereotherveriables aretheoffshellsquared monenta oftheexternal lines(westillhaveaforwardanjiituse here,sot=0).This factorization isthecrucialsteptntheASFanalysis. Itsaysthat the multiperipheral model yields tHe sane power law behavior manifestedintwo-bodyReggetheory.Power‘behavior Asobtained byASFwithout anymention ofthej~plane. ! Theintegral equation forfbecomes, viathefactorization, an eigenvalue equation; $44 aresdmeeigenfunctions, and44}arethe corresponding eigenvalues. Themjtpersphere modelthereforeprovides aschene forcomputing theexponent, ofthepower behevior ofthetotel cross section for large s! All ond need doisfind the leading eigen- value @.3 InRegge language, themultiperipheral model yields the t=0intercepts oftheappropriatd Regge trajectories. Unfortunately, theeigenvalue equation is@messonaccount of theVolterra aspect resulting from|the oddlyshaped elastic phasespacedonain i { \ L. ;&id=TassAltsaaxxydu!— : smaj) t 1 7 .n i Iftheloweru'endpointisscornilates byu'=0,thenca)isindependent ofu,theequation isebsitysoluble, andtheeigenvalue obtainedfor%isthesameaswageobtainedbythenaiveleadingInapproximation, namely, Yo =~]+integrated WM resonance cross section. ! InamodelsimilartothatL.ASF,Goldberger? touchesonthesameproblemofcalculating theeokingeigenvalue 4%.Inhismodel, whichisslightly simplerthanthatofASF,the273%r blobsare tornapart andtheremaining vertex!functions arereplaced byconstants. Thus, Goldberger's amplitude isthib, | Hisintegrelequationisgivenby,| Yh-Le YX1 Actually, hedefinesZ=A-(poleler)andworkswiththisequation instead, | !' =aris i ‘The question ofthe u'=0endpoint ofcourse comes upagain. The problem ofhandlingthisendpointwasinveshigated byGoldberger's student Saxtonwhodidthefollowingmentation thatJ.D.Jacksonwouldhavecertainly enjoyed. Saxtonkepttheexactphasespaceendpoint.w=u(xweyandtransformed theA(u,s)equationtothel-representation viaaMellintransform (allatersts).Next,heappliedanobscure differential operator tohisa(u,1)lintegral equation, converting it | 1 | j into ahypergeometric differential equation and taking care tomatch the rightboundary conditions. Hethen!identified theleadingpoleofthehypergeometric solutionasq,=-x+)haeandthatsettingthe u’endpoint equal tozero has the spme effect asmaking the "weak couplingapproximation", ie,5c’,80that%%~\tiar.“intsgt term corresponds totheASFintegrated TFT resonance cross section. Getting back totheASFprogramy they proceed tocalculate the multiplicities (1ns),inclusive, spectra(centralplateau), and non-forwardelasticamplitudethatleptiowfromtheirmodel.Their AGE) calculation ofA(s,t) =ImT(s,t) fortoms thesanelinesastheir =cts<procedure forgetting40,08" ‘the$u11complexelesticamplitude is Mo) obtainedfromtheusualfixed~tatoporaton relationandtheresult coincides withthestandard Regge form, 7 +69)=eS"[45Sle]t Finally,attheendofseshpaper,ASFobservethat‘their amplitude Aisnotself-consistent 4threspect todirect, channel unitarity, whtth states that, i , 4Yak=SOC +tag. , |‘theones shouldexistinRand.But,ifBosPra then the elastic unitarity term goes as , Sone =Sash5G)=AGH) Acontinuous superposition ofReggépoles XL=isacut. Itisthis *the function C(t) ispositive definite intheASFmodel, therefore non— vanishing. Itleadstothe1-0ghostdifficulty. H soe '{ ev i ‘ .4 cut inangular momentum that isperhaps considered: the most important i epenentoftheASFpaperendisalwhysreferredtointheliteratureas“thewell-known ASFcut."Theedisreallyindependent ofthe uitéportphersh modelandmerelystatesthattheiterationoftwo tfpolesisacut.ASFthemselvesobservedthattheelastic itarity term has, inthis model, four particles inthe t-channel | ' intermedjate states. This situation could never arise innon-relativistic i Rotentialscattering and that iswhy nocuts appear inpotential analysis.” H i ‘ i' ' 5.Reggeization ofthelinks. { Sofarwehaveconsideredthoversionsofthemultiperipheral model, ' H tito | end | where thelinks arealways pions. Several authors have suggested replacing thefixed spin pions with variable bpin reggeons. Intheperipheral model,thisreplacemént becomestwolbodyReggetheory, Seken { * i Ss —_—? erHea]gh,FF \1 i At~[Beein| A~|Sa ] ' a cal { i Pits | i ‘“tereissomediscussion inCollinb andSquires® page131thattheASFcutisactually cancelled bysomefisherunitarity graphs,butthen yreappears under consideration ofso-called non-planar diagrams. This work ofMandlestam leaves memystified. *tiere Ihavefosfunwritten theKhuri—Jones formo$fthepartial wave hmplitude asdescribed inC&S. t 'i t \ ' t { soe > TheReggeists, then,wouldLiketodeethefollowing multiperipheral amplitude , H ee 7 There 4s,however, aslight problem: Thewhole point ofthemultiperipheral ‘OGUE decomposition, asexpounded byASF,istohavelowsubenergies ostomakeuseofthedirectresonance,lowenergyWWcrosssectionasakernel.ButhereiswhatthePeaseits arestiggesting: iSi LBsBry SYpds 44 ' H i dhedirect channel TTlowenergy absorptive partsarebeingreplaced ¥theleadingcross.subchannel Regiepolesasifthesubenergies were verylerge. ‘Thisclearcut travesty] isjustified onthebasisofDolen,Horn,Schmidtdualitywhichsaysrethink)thattheleadingcross channelReggepolesdodescribe|lowdirectresonanceamplitudes inanaverage weyandaretherefore} anacceptible substitute forthe direct channel resonances. 8s*+ Wnat,then,aretheimplicationsofthesereggeonLinks?First, hayallow anestimate oftheshort! range correlations. Thestraight chainappearstohavenoeorrelatich, butintheChew-Pignottd modeledeLinkmodel),acorrelation develops (see Chew Goldberger andIox?),— *mat.happenedtothebackground iptegrai? Ithoughtitcontained |theresonanceinformation. | | a' i |1Girt,KAKew LALA—| oe. i Ri:Pint Rest | i i ‘' on(Pi*) | Os. OF tp. -0,.%|G(tes,Pes fern)=BORGAD[CfoPen] j This correlation extends only one link ineither direction along the i chain,‘Thefullamplitude forthechainisthen ' E(ord)(PoPP2)CP.PesPs)fs+--GCfeorsPose) 7 ‘Ofcourse the ASF model also had ohe-link correlations arising from the“WWII blobs. Theslight difference isthatinthereggeon H case, there are two ways toview the chain, sothe correlations go 1 bothways, { ' { ' 1 ' |Thesecondimplication ofrhegeons aslinks,andthedriving dotivation forusing themasLinksfinthefirstplace, isthepossibility of"bootstrapping" these reggeons. ,The"input" reggeons arethelinks, andthe"output" reggeons arethemltiperipheral ladders. Theself- consistency integralequationissomethinglike, a i | { oe | } 6i al ' z|x 7vara H Ofin=oot. | 1 ChewandPignotti attempt suchaboptstrap calculation.? theyallowthe pomeron andoneaymknkie representative meson trajectory intheir model. Long segments ofpure meson ladder are found totelescope down to renormalized rung,often _nw i _tal Sincethepuremesonchaindoescontribute totheresult,andsincethe result must respect the Froissart bound, they find anupper bound on theirvertexcoupling constant, ge£2(\-dm) +whereOmigthe “pare inputtrajectory. ' Sinceallmeson chain segmehts canbereplaced withanM'rung, theentireproblemofthepomeronchntributions teAconsideration of hatswithyalternating PHPMPM... structure. AlthoughChewandPignottd|nsiderthesecontributions, theyfittheppinelasticdatawithoutthem,|TheFroigsartresultga£2(\-ae)ispinnedtotheequality bytherequirement ofconstantornpsr-constant totalcrosssection(ie,dy=1).Afittots intherange10-30GeVgivesthe results @q=43andgq?=Lan t Asdiscussed inHorn’pagep12,themultiperipheral equation *other bootstrap efforts haveplaced theponeron atgj)=«974-01 and g@)= +47t.09.(Discussed inreferences 11} | i ‘ 7 a : 1 |I |: i | ifwritten inthestrong-ordering limit withnooff-shell massdependenceonanyofthelines,takesthevoform,* PKR) = .a& Neileet |AGH)=papeAtteBSBAL(EMEK) ACS) thedrivingtermAgisaBorntorn1skearg28(sm).Thisequation nhsthedelightful propertyofbecojingatrivialalgebraic equation undertheMellintransform. Thus,ifthenotation ofreference 12, wehave, |ds!FO)=BG)+53"K(&)FC)\ and \; $@)=SQiRD\ | where fandkaretheMellin transfbrms ofFandK. |Thissimplemultiperipheral!equation isoneingredientnecessaryfortheoscillation prediction. The;otheristhepresence ofNNthresholds within thecentral blotsinthechain. Inamultiperipheral anpiithde, thelinksareusuallyendowed withcutoffsintheirmomentumtrangfere Th€stcutoff,mstactually reside intheadjacent blobs, elthopgh sometimes aspecial cutoff blob isinserteddirectlyintothesaefei, | "|Ihavethefeeling thatthere is&hidden assumption hereabout ApJ(s,t)being afactorizeable kernel inthediagonalized (t-channel partial waves) |integral equation, butIdonotknowwhere itfitsin. i I t .tot ‘ n H \ i bah Sa Sp Ce a ~REA RN-S ietOt : ;~e* (a4) | thiscutoff directly implies aminimum rapidity gap separating two blobs onthechain. Italsoimplies;a minimum value forthevertex boostqaroundablob.Thisminimisgivenby, ' is owi m at| Wscde(Yalan 1oe 1an coe iC;a 2a 1 bE | 1 Ii Thus, amultigeripheral blob is"bbrn" into thechain atthis minimun sitecharacterized bytheboost q.Such abirthisrather naturally interpreted asaproduction threshold ofsomesort. i ChewandSnider propose thatitistheNNthreshold thatis aking itsappearance.1? withthe‘assumption thatthefirst baryon- antibaryon pairoccursinthecenterofthechain(notontheend),and taking arather simple kernel, | K6) Frslentted) —Dus>wCY?tus)= i ' ° i Buse jtheymakethefollowing simple calculation,oe. RG)=Sax0")[¢oCw] : ?60<j. 1 o™.=dVdxgft =sye"l ew i I! + .H .|a 8 \t i !! |: ‘thus,thepartialwaveamplitude sy tt=BW)4| . —Wy47be‘| where ‘ $i)=GS? stew Ta) =YSBO-S) |i !1 ‘Thepoles ofthis amplitude are atvalues of which solve: | =Wa ‘oy Wott| x=rte >aw.tert! ' | Sinceaandwarereal,itisclearthatallcomplex solutions occur asconjugate pairs. ! Inorderthattheleadingrtebeatj=1,wehave 4=tew =tee‘ war)30thespectrumofreggepolesisjtetineaby:jx=elonewandSnider takethevalue w=1.5andfindthefollowing poles: 'H : oo=1 1 ye |KA"F5232i , ate —.2Ebe ‘ ES Theeffectofapairofcomplexpalesonthetotalcrosssectionis, | Sy, & at HRRAS +BLAS +S)ea —"—_-—_.—_—_— 1 Re[teQ-t=pamseosbt)Ws) ‘H |rmuethepoles %and4%,willhdvetheleading influence inthesecond |5 {erasincetheyhavethelargest. realparts.Thistermwilloscillate1 '. i { i —— fi i} 1 1 fi inthevariable Inswith aperiod { i | A(us)= 2| 1) } paUsingtheapproximation Im@«=or]wefind, ‘ xé Relay os)=AS+BSMONY cos([T] mus+P : ' i Thenwiththeo&,obtained fromwp15weeet 1 PAC a a '} ; Thewhole point ofthisanalysis isthattherising total cross dections mayberising onlytemporarily. Thefullpicture issupposed ! :tobethis:| VAS : tu~ 53! 8 t Chew andSniders' paper waswritten before the’p-prise was i discovered. They were thinking more interms ofthe rising Ktp and attptotal cross sections. (HastheNALdata come inonthis? Inotice theseexperiments amongthoselistedintheapproved NALproposal No.t 10h,Kycia etal.) .4 Thereareafewpapers whidhdomention oscillations andnn{ thresholds fmxthep-pdata. Joel‘Koplik3 repeats theChew-Snider argument,downplaysthetriplePensoncontribution totherise,but { 1 i | - ae | | postpones afit toalater publication now inthe works. ; Gaisser andTanlookatthe!ISRinclusive Z*date! :owSe ‘and, being careful toinclude non-imltiperipheral contributions toaf,trytwoseparatefits.tabirquestioniswhethertointerpret thebreakinthedataasthefirstPsecondnfithreshold. Theyseem tiolean toward the former conclusion. Their rather broad fits are shownonthelastpageofthisrept.‘ Asafinalremark,wenotethatcomplexpolepairserefairly domoon,inReggetheory. Balietalshowthatcomplex polepeirsin theleft:halfJ-planedoweryexciting things!” InaseriesoftrulydrotteAllustrations suggestive of@Nyquistdaydreamtheyanalyze theReggetrajectories ofadual*Yoikawapotential invariouscases. Whenthecouplingisweak,thepoleszoomaroundinthelefthalfplane,aincomplex pairs,but;theyneverpenetrate RE(j)=+. Whenthecouplingissufficiently strong,one(ormore)ofthepoles Breaks intotheright halfplane ardbecomes aconventional Regge Pole exhibiting aboundstateatJ.0ehspinless) andathreshold breakaway Ineothefirstquadrant. Hereisanexample takenfromtheirpaper: i. | PoP itA:4 =;2 i *notVenezianodualbutdualinwaetherearetwoYukawapotentials added together, one attractive ‘one repulsive. | i 7 TeeH ey= | |Gann Tomdak | . | ds%eF “a 2, 3 4 Ri 46 i?__siGev") 10 10 ©Ref.24(BNL) | e a +Ref.1(SR) 4aql9Ref,25(Serpukhov) | / : Ret. 26 &Ret2 ; we } 3% &Ref.27PNAL VEL¢Ref.28 ; poe i 38 i. :a . :mo| 10 10°Pyaa(GeV) 10°i z '46 .oPs(Gev*) 110° tot ’‘ , ee 4q| |MF A ‘v/ Le‘34 | - . ig p Was fSa | \l¢1% q Ra: ia 3¢| f |I H ul 10 1 |1PyagiGeV) 10"i FIG,2| | ' ; F | a | i References. H References. lgnew 1958—relate gtoNNscattering amplitude viaOPEmodelgrewLow1959—relatewWAt™ toofre|viaOPE Trieman Yang 1962 —-azimuthal uniformity test ofOPE Ferrari Selleri 1961,1963 --addf¢rm factors togetcutoff Amaldi Selleri 196, '—explicit form factor Gottfried Jackson 1964b —included absorption oflower partial wavesHearnDrell1967—phases i Clegg 1964 —angular momentum barrier, kinematics provide form factor Durr Pilkuhn 1965 -~add some spinistructure Wolf 1969 —review ofperipheral successes Eden,"HighEnergyCollisions ofElfnentary Perticles" (Cambridge ,1967)‘the source ofalltheabove references. 26,cocconiPR111,1699(1958)~cosmicrays,oakvenSt Bamati, Stenghellini ,Fubini, NuovoCimento 26,896 (1962).Thisworkduplicates most ofanearlier paper ofBertocchi, Fubini, ToninNuovo Cimento 25,626 (1962). The,BFT paper expends much efforttryingtosolvetheintegral equhtion,‘Gorn&Zechariason "HadronPhysicsatVeryHighEnergies (Benjamin ,1973)M.L.Goldberger "Multiperipheral Dynamics" in1969Ericelectures(courtesy ofG.I.Ghandour) H+ 6gollins andSquires, "Regge PolesjnParticle Physics", Springer!TractNo.45—anA-1book.a) Tpolen, Horn, Schmidt "FESR andtheir application toWN CEX",PR 166,1768 (1968) 8chew, Rogers, Snider, "Relation bebween MultiRegge model andthe ABFST pion exchange multiperipheral model" PRD2, 765 (1970) Sgnew, Pignotti, "Multiperipheral Bootstrap Modél", PR176,2112(1968)Chew, Goldberger, Low, "Anintegral equation forthescattering amplitude,"PRL22,aoe(1969) im lchew, Snider PRD1,3453 (1970) PRD3,420(1971) —theschizo pomeron l2chew, Snider Phys. Lett. 31B, 75(1970) -oscillations 133, Koplik, LBLReport No. 2175 \st.K. Gaisser and C-I. Tan, "Perturbative treatment ofthreshold contributions © '_toarising pptotal cross sectilon,"preprint July 1973 (BNL No.18070)15pa1i,Chu,Heymaker,Tan “Regge trajectories fortwoYukawa potentials," |PR16h,1450(1969) | : { i : H ! | ' i L