Multiperipheral Ring Dynamics and a Definition of the Complete Twisted Reggeon Loop

MULTIPERIPHERALRING DYNAMICS AND ADEFINITION OFTHECOMPLETETWISTEDREGGEON LOOP PhilipHarrison Lucht Lawrence Berkeley Laboratory University ofCalifornia Berkeley,California 94720 Ph.D.Thesis November 1977LBL-6199 MULTIPERIPHERAL RINGDYNAMICS AND ADEFINITION OFTHECOMPLETE TWISTED REGGEON LOOP TABLEOFCONTENTS Abstract.. (1)Introduction. (2)MUlti-Regge Production Amplitudes (3)TheVertex: Helicity andParityConditions.•.1 2 5 11 1. 2. 3. 4.Helicity Conservation ParityInvariance ParitywithReggeons . . . . . . Caveats, andtheVertexV11 12 13 16 (4)TheUnitarity Product (5)Frames. . . . . 1.TheVertex . .... 2.TheRung.. 3.TheCentral LevelFrames. 4.ManyRungs. . . . . . . (6)TheHelicity PoleExpansion (7)Naturality Condition fortheKernel. (8)TheMultiperipheral ChainandPhaseSpace (9)TheDiagonalization ofAngular Momentum (10)ThePlanarBootstrap... ...• •..17 21 21 22 24 27 29 34 37 41 44 1.FormoftheIntegral Equation . . . . 44 2.TheProjected HelicityPolePropagator P~ 46 3.TheProjected KernelandItsThreshold Behavior 47 4.TheNaturality Diagonalization . . . . . . • . . . . 50 5.TheBootstrap Problem ....54 6.Counting 55 (11)TheCylinder.. .~... 56 1. 2. 3. 4. 5. 6. 7.Diagonalization oftheChargeConjugation • . . . TheCylinder inRapidity .... .... . . . . . .. TheOne-Twist Term.of theCylinder asaHelicity PoleExpansion ...• . . . •.... . . . .. Angular Momentum VersusHelicity.... ReggeCutsandNonsense Zeros . • ... TheComplete Twisted Reggeon Loop TheFullCylinder58 59 63 65 66 67 70 (12)FixedPoles,Nonsense Zeros,andtheHelicity Contour Problem ....... .•. . . . 71 Appendix A:SomeUsefulFunctions Appendix B:ToIlerM-Functions .... Appendix C:TheHelicity PoleExpansion Formula Appendix D:Threshold Kinematics•.... AppendixE: TheCross-Channel Continuation Appendix F:Reattachment oftheEnd-Rungs Acknowledgments .. References . Figures Table76 80 84 87 89 95 98 99. 102 124 -1- ABSTRACT Thet<0multiperipheral fonnalism ofCiafaloni JDeTar J Misheloff, Mueller, Muzinichand Yesianisreviewed, extended, and applied totheorderedS-matrix whoseringamplitudes comprise the zerothlevelofthetopological expansion. ToIlerM-function notation isusedthroughout. Thebootstrap andcylinder problems areformulated intennsofawell-defined helicity polepropagator; adefinition ofthecomplete twisted Reggeon loop,whichappears intheone-twist termofthecylinder, isgivenasahelicitypole expansion. Someconsideration isgiventothefollowing subjects: diagonalization, naturality, threshold behavior, Reggecuts,and complexhelicity. -2- (1)INTRODUCTION Duringtheyear1969-1970, afteraperiodofvigorOllSactivity inthefieldofmultip~ripheral dynamics, Ciafaloni, DeTar,Misheloff, Mueller, Muzinich andYesianpresented, infiveheavily overlapping papers, theexactkinematic analysis ofthemultiperipheral model.1-S Thesepaperswere,inouropinion, extremely complicated inpartdue tothenatureofthesubject, andinpartduetothefactthatthey incorporated mathematical ideaswhichweresimultaneously being invented bythemathematicians, notably Mukunda.6Possibly, the relative obscurity ofthesepapershasdiscouraged peoplefrom attempting anexactmultiperipheral calculation, leading theminstead torelyupontheapproxirnateMellin analysis andthereby torelinquish thecapability ofhandling thetrueangular momentum whichiscentral toReggephysics. Sincetheinvention oftheS-matrix topological expansion· in 1973-74 byVeneziano, therehasbeensomerenewedinterest inmulti­ peripheral calculations, inparticular astheypertain toplanar amplitudes. Inarecentreview,7 ChewandRosenzweig havepartially reformulated theseplanarideasintennsoftheso-called Ordered S-Matrix, theconnected partsofwhicharecalledringflDlctions. Although theconcept ofordered ringamplitudes hasnotyetbeen convincingly extended tothebaryonic sector,itseemslikelythat efforts nowinprogress willsoonsucceed.8 Inthispaperwehaveattempted toreview, elaborate upon,and consolidate theideasofCiafaloni etaI,andtoadapttheseideasto theordered S-matrix framework. -3- Areaderfamiliar withtheabove-mentioned multiperipheral papers wouldfind,uponcomparison ofourdescriptions withtheirs, many differences inpresentation, someofwhichwenowenumerate. First ofall,wefeelwehavegreatly simplified thegroup-theoretic aspect ofthemultiperipheralanalysis byidentifying, astheagentwhich performs thediagonalization ofthemultiperipheral equations, an almosttrivial addition theorem involving thesameLegendre ·Q.-type J functions whichappearintheFroissart-Gribovprojection ofRegge theory. TheseLegendre functions aregeneralized inthattheycarry complexhelicity indices whoserolewecontinually stress. The readerisreferred toRef.9foranextensive discussion ofthis grouptheoretic business. Another difference onewillnoticeisourattempt toisolate andidentify anobjectcalledthehelicity polepropagator which connects cluster discontinuities alongthemultiperipheral chain. Strangely enough, thispropagator owesitsexistence toafactorization condition whichresults fromthesameLegendre addition theorem mentioned above. Obviously spinisanimportant concept inamultiperipheral analysis whichpurports tocompute Reggetrajectories. Wehave attempted toinclude spininfullgenerality (i.e.,onexternal particles aswellasinternal poles)bymakinguseoftheToIler M-functionfonnalism. ToourknOWledge, noonehaswritten unitarity equations inthisformalism whichseemssowellsuitedtothepresen- tationof rnultiperipheral kinematics. Interlaced withthediscussion onthefollowing pagesonewill findasortofrunning commentary onparityandnaturality, leading toanaturality diagonalization oftheplanarbootstrap whichis, -4- wefeel,animprovement ontheoriginal discussion byCiafaloniand Yesian. Generally speaking, theexactkinematic analysis allowsoneto thinkaboutthingswhichsimplydonotexistinthe·rapidity framework whichmoreorlessignoreshelicity. Wehaveextracted thethreshold behavior oftheringfunctions andhavemadeastartatexamining the nonsense zeroswhicharepresumed toremoveReggecuts. InSection (5)wedescribeinaratherdifferent mannerthan thatofRefs.2and4theconstruction ofthestandard framesofthe multiperipheral ladder. Bycontinuing theladderkinematics tothe center-of-mass crosschannel, weshowinAppendix (E)howthepeculiar boostparameters whichlinkthestandard framesarethecontinuations ofvariables familiar fromcenter-of-mass kinematics. The"planar" bootstr~p andcylinder problems arebothsetup­ thecylinder inmoredetailbecauseitlacksthecounting problem-but nodetailed calculation isattempted because wearestymied byaproblem involving thecorrect methodofshifting thehelicity contour. We haveisolated thisproblem inthelastsectionofthepaper;itmust besolvedbeforethemachinery described hereincanbeputtowork. Nevertheless, wedoobtainanexactformalexpression forthe cornplete twisted Reggeon loopk(t)whichcontrols thecylinder shifts oftheplanartrajectories inthephenomenology ofChewandRosenzweig.10 Foradetailed outline ofthepaperwerefertotheTableof Contents preceding thisIntroduction. Ingeneral, thefirsteight sections describe themultiperipheral construction, Section (9)gives theangular momentum diagonalization, andSections (10)and(ll)apply theanalysis tothebootstrap andcylinder problems. -5- (2)MULTI-REGGE PRODUCTION AMPLITUDES Tomotivate thespecific formweuseforthemulti-particle production amplitudes, we·appeal tothenotionofaparticle polein theS-rnatrix. Figure1showsaparticle poletermknowntobepresent inthe.6-point function (repeated indices areimplicitly summed), (2.1) Thispolehasaresidue whichfactorizes intotwopieces, eachpiece beinga4-point function normalized inthesamewayastheoriginal 6-point function. EachxinFig.1marksaparticular standard restframeforthe particle onwhoselinethexappears. (Whenthepoleisreggeized below,somex'smustdenotespacelike restframes.) Thenotation is 11approximately thatofToIler: thes.arethespinsofvarious 1 particles, m.arehelicities (component ofspinalongthez-axisin 1 thestandard framemarkedbyanx).Thesamesymbolss.andm.are 1 1 alsousedtodenotecertain Mandelstam invariants andmassesof particles; theusageshouldbeclearfromthecontext. Themeaning ofadotunderahelicity indexisexplained inAppendix (B). Thea.appearing inEq.(2.1)andFig.1are,foreachparticle, 1 theparameters ofa(possibly complex) Lorentz transformation which connects theparticle standard restframetoanarbitrary "lab"frame s asindicated inthefigure. Thevariable gappearing inD7,(g)m7m7 denotes therotation g=a;la~;thestandard D-function [see Appendix (A)]isgenerated bycovariation fromtheM-function on -6- theleftaccording tothesimplerulegiveninEq.(B.3). S7is thespinoftheparticle pole,andm7,m;arethehelicities ofthat particle intwodifferent reference frames. Itisperhaps worthnotingthat,although theycarryspinand helicity indices,theM-fWlctions appearing inEq.(2.1)areLorentz 12 .13scalars, unlikethemomentum spaceM-functions ofStapp andTaylor. Secondly, wehavebeencareful toproperly ordertheparticles consistently aroWldtheconnected partssothatallourequations applyequally welltotheordered amplitudes (ringfunctions) inthe ordered S-matrix framework associated withthetopological expansion.7 Thefactor2-1(S7-m7+ie) in(2.1)isofcoursetheactual pole;thenumerical constant cisdiscussed belowinSection (4), andcanbearranged toequalunity. Equation (2.1)is,forthepoletenn,anexactstatement. We nowassumethatthisparticle poleisinfactoneofmanypoleswhich occuronaReggetrajectory a7•Thecontribution ofQ7tothe6-point function showninFig.1shouldbegivenbytheaboveexpression with S7continued toex,andwiththevarious grouparguments andinvariants continued sothattheequation isinausefulReggeregion. Accounting forsignature, theusualReggemachinery* maybeimplemented togive *Reggetheoryforn-point functions withn>4ismuchmorecomplicated thanwemakeitsound.37,38Rigorously,38 boththephysical and orderedS-matrix n-point functions mustbedecomposed intoasumof "spectral components"· bymeansofan(n-3)-variable dispersion relation (Bargmann-Weil). Eachspectral termcontains onlySteinmann­ allowed multiple discontinuities, afactwhichimplies theexistence ofaLehmann ellipse ofconvergence foreachzivariable inan appropriate physical crosschannel (hexagraph). Asaresult, the infinite angular momentum andhelicitysumsare convergent atleast somewhere, andthisallowstheSommerfeld-Watson continuations tobe defined. So,rigorously onedoesaReggeanalysis oneachspectral component andthenaddstheresults, oronestickswithasingle component anddiagonalizes unitarity ontothespectral components. Wefeelthattheformofourresults willbethesameineitherthe rigorous Reggetheory orRef.38,orthenaiveReggetheorypresented inSection {2}. thefollowing result{seeFig..2): Ma.7SgS4SS .rn,IDgm4mS(2.2) wherewehavesuppressed thea.arguments, andwhere 1 [factor]aT,nun=-irn4>-imf4>'e e x.{d~,-rn'(-z)+T(_l)rn+€d~,(z) ~. 2sin1T(a.+rn) ~ Thevariables g=(4),z=cosS, 4>t)whichappearinFig.1arenow 0(2,1)variables (4),z=cosh~, cP').Thesignature ofReggeon0"is(2.3) T7(aspin-~particle haspositive signature), and(;7=0or·~.depending onwhether 0."isabosonorfennion trajectory. UsingEq.(A.B)one mayshowbytaking 0,7~s,thattheReggeform(2.2)duplicates the particle poletermofEq.(2.1). Thefinalstepinobtaining theReggeformweshalluseisto perfonn a"Mandelstarn trick"operation14whichcausesthefirst-kind functions in[factor] ttobereplaced bysecond-kind functionsnun whichhavesimpler asymptotic behavior. Performing thisoperation wefind where[factor]aL,mm=Y.t'E-a-1C)s.,gnun(2.4) -a-.l( )E.,gmm=e-irn<j>e-a.;l(z) mm-irn'4>'e (2.5) andy=-ca'tantr(a-e) (2.6) (2.7) -8- -a-lTheflUlction emmt(z),defined inEq.(A.IO), hastheexpected Reggebehavior zaforlargez,Eisastandard signature factor, and ycontains theleftover factors. Inparticular, ycontains atand therefore hasdimensions E-2•InEq.(2.1)thesedimensions are generated bythepoleitself. Realizing thatthen-point ToIler M·f · hd··..E4-n ·fhd·· 1 -unctlon asImenSI0ns ,onemayverIyteImenslona correctness of(2.1)or(2.2). TheReggeresidues appearing in(2.2)arethree-particle/one- Reggeonamplitudes normalized inthecorrect waysoastobecome physical four-particle amplitudes whentheReggeonistakentothe appropriate valueofmassandspin(andsignature, ifMisnotan ordered amplitude). Sincethephysical helicity amplitudes must vanishwhenthehelicity isoutofrange(hasanonsense value), theresidues mustcontain factors toknockouttheunphysical poles, sincethisghost-killing flDlction isnotbeingperformed by[factor] t.mm Forexarnple,one mighttake* =SJ:J:' 43=Mt 1[f(a,+1+m,)f(a, +1 -m,)]~ (2.8) Sofarwehaveconsidered theReggeization ofasingle-pole t.erminunitarity. Hadwestarted withtheappropriate multiple pole term,we·could haveobtained amUltiple Reggeresidue orfour-Reggeon amplitude which,wereallReggeons continued toparticle points, wouldbenormalized soastoyieldaphysical four-particle helicity amplitude. Wefeelthatthisisausefulwaytonormalize Regge *Inthesense-nonsense region, additional square-root zerosare provided bythed-functions. See,for example, Fig.BeofRef.9. -9- residues, andisultimately necessary ifoneattempts acomplete bootstrap of,say,thetriple-Regge vertex. Weshallmentionthis laterinSection (10)(butwillnotattempt suchabootstrap). Although three-particle scattering amplitudes havenoplacein astable-particle S-matrix theory, unstable particles maybecalled upontogivemeaning tothefollowing equations. Figure 3showsa particle poleterminthe4-point function. Inanalogy to(2.1) wewrite (2.9) Reggeization inthesamewayasbeforeyieldsthisexpression forthe ReggepoletermshowninFig.4: -cxs-l ]MS152(15Emm'(g)mmm'5 5 12.5(2.10) Again,theReggeresidues (piecesofthefactorizing residue ofthe ReggepoleintheFroissart-Gribov projection) appearing inEq.(2.10) arenormalized sothat,as(15-+ss'thesetwo-particle/one-Reggeon amplitudes approach thestandard three-particle ToIlerM-functions appearing inEq.(2.9). Thehelicity nonsense..;.zero structure of thesestandardized "Reggecouplings" ispresumably similar to(2.8) 4-nabove. NoticefromtheruleEthattheseReggecouplings havethe dimensions ofenergy. Byreggeizing adoublepoleunitarity term,onemayobtainthe following Reggecontribution forthetwo-ta-three production amplitude showninFig.5: -10- x (2.11) (1SS4(17 TheobjectM isthetwo-Reggeon/one-particle amplitude which continues totheToIler3-point function whenas~Ssand(1,-+-S7. Again,this"double Reggevertex" hasdimensions ofenergy, asdoes thetripleReggevertexwhichwehavenotshown. Thesevertices differ considerably fromthephenomenological Reggecouplings (dimensionless) andtriple-Regge couplings (GeV-2). Thefonnofthegeneral multi-Regge production amplitude should beclearfromEq.(2.11). EachReggeongetsa bracketed "propagator" factorwithlinkinghelicity sumsonbothsides. Allvertices are standard Toller3-point functions continued intheappropriate way. Weconclude thissection byobserving that,intheordered S-matrix framework wheretheM-functions in(2.11)arereplaced by ordered ringamplitudes, themulti-Regge~pole expansion shouldbe, intheperipheral region, averygoodapproximation sincethereare (presumably) noReggecutsintheringfunctions. Thetheoretical accuracy of(2.11), whensummedon(l6and(l7'isthuslimited onlyby peripherality andtheconvergence rateoftheReggeasymptotic series, i.e.,duality. -11- (3)THEVERTEX: HELICITY ANDPARITY CONDITIONS Wehavebeenwritingthetriplevertexintheform~(al'a2,a3) Im2m3 tostressthefactthatthevertexislikeanyothern-point Taller M-function. Aswenowshow,however, thisnotation isextremely redundant. Usingthefreedom allowed bytheTallerinvariance condi- tion[seeEq.(B.3)],onecanchoosetosuperpose theexternal "lab" reference frame-withrespect towhichthevarious a.aredefined, 1 asinFig.1-ontooneofthestandard reference framesassociated withthevertex. Sincethevertex standard framesareconnected by certain z-boosts 0'1'°2,andqwhicharefunctions onlyofthe invariants entering thevertex[seeEq.(5.1)], onemayconclude that M(a1,a2,a3)isitselfafunction onlyoftheseinvariants. Thism 1m2m3 situation isillustrated inFig.6awherewehaveplacedthereference frameontothestandard frameofparticle 1togetMCe,q-l,cr;l),m1m2mg whereeistheidentity transformation. 1.Helicity Conservation Consider nowthisseriesofoperations inwhich ~represents thezrotation R(<1»:z =M (~~ -1 ~0'-11)mm m.~e,~q,~ 1 2g = = (3.1) -12- Inline1thereference frameisidentified withtherestframeof particle 1,asalready noted. Inline2the4>'5aremadetoappear viatheinvariance condition ofEq.(B.2). Inline3theserotations -1 -1arecommuted through thezboosts q=Bz(-q)and(J1 'andthenin line4the~'sareseparately covariated totherightaccording to Eq.(B.3). Comparison ofthelastlinewiththefirstthenshowsthat (3.2) i.e.,helicity isconserved "atthevertex.* Onedoesnotfindsuchacondition forthehighern-point functions because therotation ~doesnotcommute throughallthe a.nomatterhowtheyarechosen.1 2.ParityInvariance Ifparityisaninvariance ofthetheory, wemayuseanargument similar tothatofSection 3.1tostateparityinvariance intermsof thevertex- Sincetheparityoperation, whichToller15callss.is anelement ofthelittlegroupHofthe4-vector (*,0,0,0), therotational+ covariance conditions [showninEq.(B.3)]maybeextended toread,e.g., =IT3M (a1,a2,a3),m1m2m3 (3.3) whereII3istheintrinsic parityofparticle 3.Sincetheoperation s faiIstocommute throughthezboost5lsee Eq.(7.6)],itismore convenient touseToIler's parityoperator s'definedby Sf (3.4) *ButseeSection 3.4forqualifications onthisandsubsequent equations ofthissection. -13... forwhichthecovariance condition onparticle 2reads,according toEqs.(3.3)andCA.8), (3.5) Operator s'bringsouttheintrinsic parityandnegates thehelicity oftheaffected particle. Sinces 'doescommute withthezboosts, onemayrepeattheargument (3.1)tofindthisvertexparitycondition: =M_m-m-m(a 1,a2,a3)f!(ni(-1)si-mi\. l'2'31-1\ ~3.6) Asacorollary toEq.(3.6)onehaseither 3 1T(n.(-1)Si)=1 , i=1 1or Forexample,ifSI=S2=0thenallthreehelicities mustvanishand oneconcludes fromtheabovethatthevertex-vanishes ifIT1II2II3'f S3(-1),asonewouldexpectfromamoreconventional angular momentum argument. 3.ParitywithReggeons Another convenient property oftheparityoperator Sfisthat s',unlikes,belongs alsotothelittlegroupH_ofthespacelike restvector(0,0,0,*), asdoesR($).Therefore, ifoneormoreofz theparticles atavertexisreplaced byaReggeon-which maybe spacelike sothatHistheappropriate littlegroup-oneshallfind thatthehelicity' andparityconditions stillexist. Thehelicity conservation condition ofEq.(3.2)isunchanged, exceptasnoted below. TheReggeonparitycovariance condition is = (3.8) -14- where cr. 1=TI. 1+i1fs·1e (3.9) (3.10)withs.aphysical pointona.,andTI.theintrinsic parityofthat 111 physical point. Thequantityo. appea~ing in(3.9)iswhatweshallcallthe1 Reggeonnaturality, andisaconstant alonganordered Reggetrajectory. Another waytosaythisisthattheexchange degenerate partners which together compose anordered (planar) Reggetrajectory havethesame naturality 0.,eventhoughtheintrinsic parity TI.andspinparity 1 1s·-e· (-1)1 1alternate atthephysical points. Oneseesthat,ascx3-+S3' Eq.(3.8)reproduces (3.5). Forfermions, thephysical pointparities TI.andnaturalities 1 o.are,according toEq.(3.9),outofphaseby90°.IntheM-function 1 f1· 16f ·dTeph ormalsmonecanprove romcross1ng an tat 2s· TI.n-:-=(-1) 1 1 1 Apurist, allowing forthepossible existence ofself-conjugate fermions, wouldhavetoacceptimaginary parities forthosefermions. 17Asemphasized byStapp themostreasonable convention istogive allfermions imaginary intrinsic parities. (ToIler toousesthis ..11)convent10n .. asbosons.Inthiscase,naturality 0=±1forfermions aswell Weleavetothereaderacomparison ofEq.(3.9)withthemore commondefinition ofnaturality n·=1S·-E·P.(-I)11 1(3.11) whereintrinsic parityP.=±1forbothbosonsandfermions, and 1 €.= 0forbosonsand one-half forfermions. Certainly forbosons, 1 n.=cr••1 1 -15- OnceEq.(3.8)hasbeenestablished, theargument ofEq.(3.1) maybeapplied togiveaparitycondition forthesingleReggevertex showninFig.6b: (3.12) Thissays,e.g.,thattwopionscannot couple toanntrajectory, eventhoughsuchacoupling isallowed byG-parity. Forthetwo-Reggon/one-particle vertexacondition sirnilarto Eq.(3.12)results (seeFig.6c): =[01e-inIDl][02e-~mm2] [ s -rn] . TI 3(~1)3 3 (3.13) Thetriple-Regge vertexismorecomplicated because onecannot alwayslinkthethreestandard frameswithz-boosts. Inparticular, when ~(tl,t2,t3) isnegative, thethreeframesareconnected by ·18(F·16)C · 1h· , y-rotatl0ns see1.g..·onvenlent y,t eparltyoperator s alsoconnnutes withy-rotations; theparityargument thengoesthrough toyield = (3.14) sothatnegating theReggeonhelicities isequivalent, for~>O, tomultiplying bytheproduct oftheReggeonnaturalities, since thehelicitiescancelbyEq.(3.2).However, si,ncey- andz-rotations -16- donotcommute, thehelicityconditionof Eq.(3.2)isbrokenfor thespacelike triple-Regge vertex, .6<o. 4.Caveats, andtheVertexV Wemustnowaddtwoimportant qualifications tothepreceding equations ofthissection. Aswritten, theyapplytoToIler3-point M-ftDlctions withallparticles andReggeons beingintheinitial "state" andwithallspinorindicesofthe1.D1dotted uppertype(see AppendixB). Tobeconsistent, certainparticles andReggeons mustbeput intothefinal"state"ofeachvertex. Wechoosetoletthisconvention bedetermined bythedirection ofthear~wsin,say,Fig.15.Whenever aparticle orReggeonisinthefinalstate,therelevant bracketed factorinEqs.(3.5)-+(3.8)and(3.12) -+-(3.14)mustbecomplex­ conjugated. Secondly, wemustfacethefactthatinevitably someofthe helicity indices wearedealing withareoftheundotted lower!ype. Theseindices, markedunderneath bydotsasinEq.(2.1),arenecessarily lowerinordertopreserve thespinorcovarianceof theequations. 'Whenanamplitude withalowerundotted helicity indexiscovariated asinEq.(B.3),theDfunction mustbereplaced byD*.Thenet resultisthathelicities inEq.(3.2)corresponding tolowered indiceswillenterwithminussigns. However, theparityconditions arethesame,regardless ofwh"~::her indices areupperorlower. Thevertices inwhichwearemainlyinterested havetheform ofthecentral vertexofEq.(2.11). Inthenotation ofAppendix B andwiththeconventions madeabove(and,asalways, maintaining the cyclicringordering) wewritethisvertexas =-17- (3.15) Thehelicity andparitycondi tionsforthisvertexarethenfound fromEqs.(3.2)and(3.13)andtheaboveconventions: +m4=o (3.16) =[-imn~]* [ S4-ID4]*[-imn7·]CX6S4CX7 >cr6e II4(-1) >cr7eV,·-m6-ID4-ID7 (3.17) Onceagainitshouldbestressed thatthisvertexVhasthe standard normalization ofaToIlerM-function, hasdimensions of energy, and(inaddition tothelabelsshown)isafunction onlyof theinvariants entering thevertex. (4)TIlEUNITARITY PRODUCT Eh11 · 1 ·13h·" .. yenwenapartlcescarryspln, t eunltarlty equat10ns forthemomentum-space M-functions arecompletely characterized by h 1bbbld·19 h · h f"01'" 1 " t eusua·u e1agrams (togeterWltaset0lvesTUes, internal line -27Tic0+(p22= -In) independent loop=d4p/(-21Ticf) pole2+iE) =c/(s- m Oneneedsalsotherelation between theM-function bubbles andthe rawconnected parts: (S)c= =C-2wicf) 64Cext)MC+) C~2wicf)* 54Cext)MC-) Inthesere1ations, theconstant fdetermines thenormalization of thesingle-particle states, -18- <p,mIpt.,ni')= andcgivesthepoleresidue, asinEq.(2.1). Authors naturally differintheirconventions, e.g., ELOP:19c=1 f=(271")3 Stapp:12c=i f(27T)3= Taylor:13c=-1 f1/2= Wefavor theconvention ofELOP ~but,shallalwaysgiveresults in tennsofcandf. Onceaunitarity equation isexpressed intermsoftheStapp- TaylorM-functions ~....(Pl'P2•.••)'itmaybeconverted toToIler 1 M-functions viatheinverseofEq.(B.4). Details ofthisconversion process withattention paidtothespinorindices willbegiven elsewhere. 16 Beforetackling thegeneral multiperipheral unitarityproduct, wefirstillustrate theformunitarity takesintermsof~heToIler M-functions bywriting downelasticunitarity assketched inFig.7. Theformulais x (4.1) where = -19- Asusual, wearemaintaining theringf1.lllction ordering conventions.7 Thedotsoverthemgandm4indices ontheleftslideofEq.(4.1)are necessary tomaintain thespinorcovariance. Thero'tation functions ariseinthesamewayastheD(g)inEq.(2.1),namely, fromthe ToIlercovariance condition showninEq.(B.3). Weareanticipating asystemofstandard reference framestobereviewed shortly inwhich itwillturnoutthattheserotations willbepurey-rotations, X., 1 whosepresence wasfirstnoticed byMisheloff. 4Att=0therotations allvanish, butfort<0theydonotvanishandaredetermined upto asignbytheperipheral invariants t.[seeSection (5)]. 1 FromEq.(4.1)itshouldbeclearhowthegeneral n-bodyunitarity product appears. Eachintermediate particle getsaMishe10ff rotation, andthehelicity indices aresummedoversystematically. Then-body phasespaceis =n(.4 +2 2dp.IS(p,.-m.)). 04(ext)'IT 1 1 1 i=l f(4.2) where,asinEq.(B.1),p.=L(a.)p..Sometimes itisusefulto111 visualize eachproduced particle asacluster ofvariable massand spin,inwhichcaseEq.(4.2)canbeadjusted byreplacing +2 2 +2 ~o(p.-m.)-+0(p.-s.)ds. andaddingspinsums""s.• 1 1 1 1 1 1 Wearenowreadytoinsert·into thegeneral n-bodyunitarity product amodelfortheproduction amplitudes, namely, thernulti- Reggeproduction amplitudes developed inSection 2,whichwenow writeas (4.3) -20- Thisamplitude isshowninFig.8;1:heV'sarethestandard vertices described inSection (3),and'wearenow usingm,T,pas helicity labels. Itisperhaps usefultoobservethatthebracketed factors in Eq.(4.3)havethreesourcesofphasewhenaisreal: i) if)theazimuthal phase m--n·thephase(±i)11exp[-i(m_ep_ +r.<I>~)] fromE(g.);111 1 1 from'thee-functions atz>1 iii)arising fromthekinematic spincuts(halfanglefactors) intheamplitude; theReggephaseofthesignature factor ~.• 1 Ofthesethreephases, onlytheReggephasewillbeincorporated into thehelicity polepropagator tobedefined below. Suppressing theToIlera-arguments, wenowstatethen-body 1 multiperipheral unitarity product as 1d­2i1SC =[,'.f..SbSn°005150..Sa]-Cf1TdO·M'nm'p'...p'p'm'bnloa (4.4) whereeachM-function ontherighthasafonnasinEq.(4.3),and where d~isgivenbyEq.(4.2). In(4.4)theonlyvariables not summedoverarethosewithsubscripts aandb.Thespinsand helicities appearing in(4.4)arelabeledinFig.9whichshowsthe n-bodyunitarity product withthemulti-Regge amplitudes inserted. Ournotational planisalwaystouseprimedvariables fortheupper sideoftheladderandunprimed forthelowerside.Thereaderis again~autioned aboutourmultiple usageofthesymbols s.(spin,1 invariants), p.(momentum, helicity), andm.(helicity, mass). 1 1 -21- Thenextstepintheprogramistoactually inserttheproduction amplitudes ofEq.(4.3)into(4.4)andmakesomesenseoutofthe resultant expression. WewishtoshowthatReggepoles intheupper andloweramplitudes areconverted intohelicity polesinthecentral kinematic level,andthatitisthesehelicity poleswhichdetermine theReggeon loopwhichliesattheheartofallbootstrapand cylinder calculations. Before wedothis,however, wemustmakesomecomments abouttheframesinthevarious kinematic levels. (5)FRAMES Thestudyofthereference framesassociated withthemulti­ peripheral ladderisatbestatedious andunpleasant business. Wepropose onlytooutline thedevelopment oftheseframesandto provide afewinterpretations whereuseful. Theendsofthemulti­ peripheral ladder, wheretheframesareslightly different, willbe completely ignored. Usually inmultiperipheral analysis theend-rungs (oratleastoneend-rung) areamputated, thephysicsisdone,and thenlatertheend-rungs arereattached (seeAppendix F);Regge physics doesnotrequire theend-rungs andthisisourjustification forignoring them. Inthedescription whichfollows wehavefornoparticular reasonadopted thenotation ofCiafaloni, DeTarandMisheloff3rather thanthatofMueller and~fuzinich.2 1.TheVertex Theframeanalysis beginswiththesimplevertex sho\~in Fig.10,wheretwospacelike momenta k1andk2meetafuturetirnelike momentum PI.FramecisarestframeofPIinwhichthe3-mornentum -22- -+-... k1=k2pointsinthepositivezdirection. Obviously ,frame cis onlydefined uptoaz-rotation,a £actweshallmakeuseoflater. -1Framebed)isobtained fromframecby az-boostV1(02)which brings k1(k2)tospacelike rest[ki:;:(O,O,O,v'-ti)].Clearly, frames bandclarelinkedbythez-boost qI=v1+(12.Frommomentum conservation itiseasytocompute theseboostsintermsofthe invariants t1 't2andSI: = = (5.1) chq=(s-t-t )I2~-'=t1 1 1 2 1V-1.2 2Thevariable qmaybeinterpreted assensing themasssflowing 1 1 upthecluster PI.Bycomputing (k2-kI)inframeb,onefindsthat qlispositive .because Pisfuturetimelike.1 Theframesb,c,ddefined abovearetheusualBCPfrarnes20 associated withaproduction vertex. 2.TheRung Wenowcombine twovertices tomakeonernultiperipheral rung, showninFig.11.Thetriadofframes(b,c,d)justdiscussed appears onthelowervertex, andanewtriad.(b',c',dt)appears ontheupper vertex. Theprimedboostsconnecting theupperframesaregivenby Eq.(5.1)witht.+t~.11 Frames candctarebothrestframesofpandmusttherefore 1 beconnected bysomerotation g:;:RZ(<l>l)Ry(Xl)Rz(<l>~). Wenowuseup thez-rotation degreeoffreedom indefining eachvertexframetriad toset4>1:;:4>;:;:0sothattheframes candctare1inkedbyapure -23~ y-rotationX1•ThisistheMisheloff rotation mentioned inSection (4).InAppendixE weinterpret thisvariable asacrosschannel (t+(0)Reggevariable z=cos(X);an expression forcos(X)willbe givenbelow. ThesixframesshowninFig.11arenowinterlocked, andall 3-momenta areconfined tothex-zplane. Next,fournewframesa,a',e,e' areaddedasshowninFig.12. Forexample, frameaisobtained fromframebbyanx-boost hI. Thisboostofcoursedoesnothing tomomentum k~b)=(O.O.O,~). butischosensothatk~(a)isx-zlike;Le.,theboosthIclears outtheenergy component ofk'(b).Boosth'ischosensimilarly so1 1 (a')thatkisx-zlike.Thesestatements maybesummarized asfollows: 1 ItshouldbeclearfromEq.(5.2)andthelackofy-boosts in theproblem (sofar)thatframes aanda'areconnected byay­ , 2rotation, whichwelabelS11, •Fromthefactthatt=(k1+k1) onequickly showsthisrotation tobegiveninmagnitude by cosS11'=(t+t'-t)/2v:::t: ~l.1 1 . ..1(5.3) Then,fromtheloopequation ontheleftsideofFig.12, = (5.4) onefindsthemagnitude oftheMisheloff rotation cosX1= (5.5) Reordering thesameloopequation onemaythencompute theboosts -24- Wehavenowdescribed theframes aanda',andthenew transfonnations h,hIand-e,.Inexactanalogy onedefines the11 11 frameseande',andtransformationsf2,£2and622,•Equations similar tothoseabovearethenobtained bycomparing Eq.(5.4)to theright-side loopequation 3.TheCentral LevelFrames Tothesetoftenframessofardefined withrespect tothis onemultiperipheral rung,twofinalframes fandgarenowadded, asshowninFig.13.Weshallrefertoframeslikea,b,d,e asbeing lowerlevelframes, thoselikea',b',d',e' asbeingupperlevel,and fandgasbeingframesinthecentral level. Thesecentral level framesareinfactbrickwallsystems (bws)orBreitframes. We defineabwsframeforthesystem(k.,k~)tobeanyframe-inwhich 1 1 ki+ki=0,wherelj.represents thefirstthreecomponents ofthe "'-J "",." ~_ 4-vector k..Weshallrefertosuch(t,x,y) objects asversori1 1 ~todistinguish themfromthenormal3-vectors (x,y,z) likek.. 1 Sinceki+ki=0inabwsframe,theoverall momentum transfer Q=k.+k~isatspacelikerest, Q=(O,O,O,y-:t"). InAppendix E11 weperform acomplex Lorentz transformation whichconverts bwsframes ~ ~, -~tocmsframesinwhichk.+k.=°andQ=Cvt.,0,0,0).1 1 Now,framefinFig.13isthatparticular bwsframeinwhich versork1pointsinthepositive xdirection, andversork2ist-x "'-J like.Similarly, framegisdefined toputversork2inthepositive "'-J xdirection andtomake~t-xlike.Thesetwoframes fandgare -25- thuslinkedbyanx-boost VIwhosemagnitude weshallcompute ina moment. Inallbwsframesforthesystemofmomenta (k.,k~)the 1 1 z-eomponents andversormagnitudesare thesame,justasinallerns framestheenergycomponents andvectormagnitudes arethesame. Wefind (ki)2=(k!)2-(k::c}2-(k~)2 1 1 1 =ll(t. ,t~,t)/ 4(-t)--k: 1 1 1 k7=(-t-t.+t~)/2v::t-z. 1 1 1 1 (k·~)z(-t t~+t.)/2v-::t,=- -z. 1 1 1 1(5.7) (5.8) (5.9) Because ourinterest islimited totheinterior runsofthet<0 multiperipheral chainwherethekinematics requires ll(t,t.,t~) <0,1 1 2wehavedefined -kiasabove. Whenthesymbolkiappears belowasa scalar,itreferstothisversormagnitude (-kI)~andshouldnotbe confused withthe4-vector k..1 Wewishtostressthesimilarity ofEqs.{S.7)through (5.9) tothenormal ernskinematics. Ifk.andk~werefuturetimelike 1 1 4-vectors withmasses (t.)~and(t~)~,theninanycmsframewhere 1 1 Q=(y"t,o, 0,0),t>0,onewouldhave (~.)2 ].=, !l(t.,t.,t)I4t 1 1(5.10) et, /2y:t E.=+t.t.) 1 1 1 , (t,t.)I2Vt E.=+t. , 1 11(5.11) (5.12) sothattheversormagnitude k.istheanalytic continuation ofthe 1 cross-channel emsmomentum. -26- Sometimes thevariablesz ..and z~shownabovearewritten in1 1 thisway: ~z..=~(_t)2 W.. 1 1 wherez~=~(-t)~ 1+W.. 1(5.13) W.. 1(5.14) 2Thevariables k.andw.areuseful replacements fortheReggemass 1 1,variables t..andt., 1 1 t.=~t-(k~+w~) 111w..(-t)~ 1 ,t. 1 Inparticular,=J..it-(k~+ 12w..)+ 1w.(-t)~ 1(5.15) dk.dw. 1 11=2dt.dt~ 1 1 [-A(t,t . ,t~)]~ 11(5.16) Applying theabovedefinitions toframefofFig.13wehave k~(f)=(-k2shVl,-k2chVl'O,Z;). (5.17) Comparison ofkef)tok{a)thenshowstheseframestobelinkedby1 .1 averysimpley-rotatione1: -27- Thus,thenewframesf andgareinterlocked withtheprevious tenframestogiveatotaloftwelveframesassociated withthis singlerungofthemultiperiphera1 ladder. computingp~ =(k2~ ~ in"framefwefindthattheboostvlisgivenbyk)2 1 "" where(5.19) 2 PI= (5.20) andallsymbols ontherightofEq.(5.19)refertoversormagnitudes. Withtandallt.fixed, VI the2oftheparticle measures masss 1 1 orclusterPI;inthissensethevariable VIissimilar totheBCP variables qlandq~appearing inFig~11. Thecomplete setoftwelveframesassociated withtherungp I isshowninFig.14. 4.ManyRungs Wearenowreadytojuxtapose tworungsofthernultiperipheral ladder,as showninFig.15.Inthisfigureoneseesthatthetwelve- framesystems associated witheachrungarelinkedbyaveryimportant y-boost called ~2.Thisvariable measures theseparation ofthetwo rungsinaquantity whichwouldbecalledthegaprapidity· inaone- dimensional model. Noticethatthesamevariable ~2appears inthe upper,lower,andcentral levels. Theframesonthecentral levelare linkedtotIleupperandlowerlevelsbyy-rotations like81ofFig.13. Theserotations aregivenbytheformulas onewouldguesslookingat Eq.(5.18)above,e.g., · e1=sln·2,cos82= (5.21) -·28- Theonlytransfonnationsnot showninFig.15arethey-rotations likeall'appearing inFig.14.Obviously all'=al+e;. lVenowmakesomeremarks concerning theframesofFig.15. Firstofall,mostoftheframesonthelowerlevelaretheusualBCP framesreferred toearlier. Sincethetranformation labeled gconnects 2 twoframesinwhichk2isatspacelike rest,g2must bean0(2,1) -transformation. InBCPthis&2iswritten as = Thisform,knownasthediscrete-basis parametrization, goesallthe waybacktoBargmann, butwehaveputatwiddle overthex-boost parameter inordernottoconfuse thatvariable withoury-boost variable ~2.Theazimuthal rotations 112and"2areconjugate tothe Reggeonhelicities inthesensediscussed backinSection (2),andare connected withtheso-called ToIlerangles w.=].1.+v...1.Variable1 1 1+ ~2istheReggevariable, i.e.,z/=cosh (~2)'andisconjugate to theangular momentum associated withthelinkk2,whichistosay, Q2(seeFig.8). Although thesameBCP0(2,1)transformation gappears inFig.15, 2 itisparametrized differently, namely, = theso-called continuous-basis6parametrization of0(2,1). Asalready ,noted,thesamevariable E::2appears alsoing2'the0(2,1)transforma- tionappropriate totheuppepproduction amplitude ofFig.15. Priortoleavingthissection onframes, wewishtoaddone moreobservation concerning theframesconnected withthesinglerung showninFigs.13and14.Ifoneweretoimagine themultiperipheral -29- ladderontherightasgenerating aReggeonin thecentral level,one mightdrawthefigure,showninFig.16,wherewehaveredrawn the framesa,a'and f,andtheirconnectingy-rotations .Wejustwant toremarkthatthesethreeframesaretheusualstandard framesone 18associates withthetripleReggevertex intheconfiguration /).<0, andthethetasarethestandard y-rotations. Asimilar remarkapplies totheframetriad,g,e,e'. Wearenowreadytoconvert theReggepolesoftheupperand loweramplitudes into,helicity polesinthecentral level. (6)THEHELICITY POLEEXPANSION Consider onceagainFig.15.Inordertomotivate thenext technical maneuver, weanticipate adiagonalization procedure which willbeexplained inSection (9).Theframesonthecentral levelof Fig.15arelinkedbyalternating x-boosts v.andy-boosts ~..Itwill 1 1 turnoutthattheseframesandvariables aretherelevant onesforthe diagonalized (orevenundiagonalized) consideration ofthemulti- peripheral ladder, thereasonbeingthatthesearethebwsframes inwhichtheoverall4-momentum Qisatspacelikerest. Wewill showthatcertain groupings ofthevand~variables formconvenient O(2,1}transformations. Forexample, thecombination isan0(2,1)transfonnationin thecontinuous-basis mentioned earlier whichinacertain sensesurrounds thecluster P2inthecentral level of.Fig.15.Inthediagonalization processitwillbeshownthatthe variable v2isconjugate toangular momentum jinthecentrallevel, -30- whiletheboosts ~2and~3areconjugate tocomplexhelicity variables A2andAs-Helicity polesinthecomplexhelicityplane Xwill correspond topowersofel~1sincethesevariables areFourier conjugates. Itisforthisreasonthatweshallnowexpandtheupper -0,-1 -a'-landlowerReggepropagator functions E(g)andE (g')into powersofel~l.Thesefunctions appearinFig.17,whichrepresents aportion ofthemultiperipheral chain,i.e.,aportion oftheunitarity product ofEq.(4.4)withthemodelamplitudes of(4.3). Weshallrefertotheformel~laasahelicity-pole terminthe exsamewayonespeaksofzasaReggepole term,withtheunderstanding thattheactualpoleoccursintheplaneoftheconjugate variable, beithelicity orangular momentum. Also,thesquare-bracketed expressions inFig.17willbecalledReggeon propagators. InAppendix Cwegiveaderivation ofthefollowing (convergent) he1icity-po1e expansion ofthelowerpropagator E-function: (6.1) 1~21(a2-n2) e Recallthat&2=(f2'~2,h2)' andthatf2andh2arex-boost parameters fixedbythet.[seeEq.(5.6)]. Thequantity [a-n ]istheheZicity 122 oftheReggeon whosespinis<x2_When cx2takessomegeneral non­ integral value,theReggeonhelicity takesthevaluesa2,a2-1, a2-2,..- -inaninfinite sequence. Werea2toapproach aphysical valueS2(whichdoesnothappeninthemu1tiperiphera1 regionofcourse), wewouldexpectthissequence totruncate athelicity equalto-S2- Thistruncation isaffected bytheinteraction ofthefunctions F -31- appearing inEq.(6.1)withthehelicity nonsense-zeros present in thestandard ToIlervertices discussed inSection (3),theVof Fig.17.Thesefunctions FaregiveninEq.(C.S). Thenewindex K2appearing inEq.(6.1)willbeconnected withparityinSection (7). Basically, K2=sign(~2). Theimportant pointtobemadeaboutEq.(6.1)isthateach helicityterm factorizes. Itisnotobvious thatanexpression like (6.1)hadtoexist. Asimilar situation isencountered inamuchmore complicated mathematical environment withtheReggepoleexpansion of asingleToller/Lorentz pole.Reggepolestherearethefactorizing daughters ofaToIlerpole,andhelicity polesherearethefactorizing daughters ofaReggepole. Thefactthateachhelicity polefactorizes isthefactwhich allowsustomomentarily defineahelicity polepropagator. This concept willgreatly reducethebulgeofcomplexity withwhichwe arenowconfronted. Hadthehelicity polesnotfactorized, wewould beinrealtrouble. WhenalltheReggeon propagators [...]intheunitarity product ofFig.17arehelicity-pole expanded according toEq.(6.1), certain factors maybegrouped to,thevertices, leaving averysimple helicity polepropagator. Thenewrungwiththeseregrouped factors isshowninFig.18,andthehelicity polepropagator isshownin Fig.19andhastheform =1~21[(a2·n;l)+(a~-n~)] e (6.2) h· hI~21.·cl-E (62')· h fh ThepowertoW1ce 1Sra1se 1n"q..,1St esumate he1icitiesoftheReggeons inthe(2,2')channel. Noticetllateach ofthehelicities isingeneral acomplex number, whereas theReggeon -32- helicities discussed inSection (2)werealwaysintegers orhalf- integers. ThereasonisthatheretheReggeonhelicitiesare eigen- valuesofthe(non-Herrnitian) y-boost generator K2whichisgenerating theboosts By(~). InAppendix Eitisshownthat,whenthestructure ofFig.15iscontinued tothet>0emsviaacomplex Lorentz transformation, thegenerator K2isturnedintoanormalrotation generator andthehelicities becomethenormal(discrete valued) helicities mentioned inSection (2).Thevariable ~2becomes a rotation ($2=i~2)whichagainmeasures thesumofthehelicities ,inthe(2,2')channel, namely, m2+m2• Theotherimportant pointtobemadeaboutthehelicity-pole propagator isthatitstillcontains thephysical (planar) polesin thesignature factordenominators, e.g., =[-i1T(a-f) ] 22e +1" •.2simT(a 2_€2)2(2.7) Thesepolesgenerate thenormalthresholds inthecrosschannel when tiscontinued tot>o. Turning nowtotherungorkernelofFig.18,thehelicity . " ,surnmatlons T1,r1,P1,Pl'andm2,m2canbeperformed sincetheyare nowdetached fromtherestofthechainbyhelicity-independent (intnissense)helicity-pole propagators. Wemightfirstsumover theupperandlower(discrete) helicitiesto gofromFig.18to Fig.20a, renormalizin,g forthefirsttimeourstandard verticesv. ThenewvertexVisgivenby (X)-E r m=_00l'2 (6.3) ·-33- (Group-theoretically, this'corresponds toa·conversion fromthe discrete tothecontinuous helicity basis.) Finally, weSlDDovertheMisheloff rotation helicities p,pt 1 1 togofromFig.20atoFig.20b,whichshowsthefinalkernel (6.4) Thiskernelisafunction ofthefourReggeon spinscx.,helicities 1 cx.-n.,andmassest..Duetothekappaindices appearing inEq.(6.l),1 1 1 thekernelisalsoafunction ofthekappalabeloneachside.This particular kernelisasingleparticle kernelandthusdepends onthe spinSIofthatsingleparticle. Wecouldjustaswellhavedefined p(theproduced object) tobeacluster, inwhichcase,asnoted1 earlier, Eq.(6.4)shouldbesummedoverSI. Beforeconcluding thissection wewishtomakeafewadditional remarks aboutthecritical helicity-pole expansion formula (6.1). Thisformula, orsomething closetoit,hasbeenderived byother workers2,4asonlyanasymptotic expansion. Wewishtoemphasize that(6.1)asderived inAppendixC isanexactandveryconvergent 'equality basedonanelementary addition theoremofthesecond-kind Legendre functions. Inotherapproaches, thestepintheargument represented by(6.1)hasbeentosomeextentobscured bycomplicated grouptheoretic arguments. Forexample, (6.1)canbeinterpreted in termsof0(2,1)mixed-basis matrixelements inthecontinuous series, inwhichcasethediscrete indexKhasacertain mathematical meaning. Alternatively, Eq.(6.1)canberelated tothe0(2,1)analytically -34- continued Clebsch-Gordon coefficients whichcoupleangular momenta between theupper,lower,andcentral kinematic levelsinFig.15. Theseapproaches arenodoubtcorrect, butintroduce somuchcomplica- tionthatonecannottellforsurewhether ornotafonnulaiscorrect without expending mucheffort. Ourapproach hasbeentoconsolidate thisgrouptheoryintoafeweasilyverifiable addition theorems9 whicharethenusedtoderivevariousresults. (7)NATURALITY CONDITION FORTHEKERNEL .InSection (3)itwasshownthat,afteraccounting forthe correct ToIlerM-function notation forthevertex =M (ak:a.,a.) mk~imj 1J(7.1) thestatement ofparityinvariance forthevertexinFig.18is =2r* [cr(-i)1] 1 (7.2) wherecriistheReggeonnaturality ofEq.(3.9)andTI1theintrinsic parityoftheproduced particle. Insertion oftheparitycondition (7.2)intothedefinition (6.3)oftherenormalized vertexVthenyields = V_p(~Kl'-K2) 1(7.3) Whenthisresultisinturnsubstituted intothedefinition (6.4) ofthekernel K,onefinds = (7.4) -35- •whichisthedesirednaturalityconditionfor thekernel. Wemaynowinterpret Eq.(7.4)assaying: aparitytransfonnation on"thekernelisequivalent tomUltiplication bytheproductofthe naturalitiesofthe fourattached Reggeons. Toseewhyaparity transfonnationnegates "land K2werefertoFig.21whichshowsa segmentofthemultiperipheral chainwithitscentral levelboost;. Thefigurealsoshowsthesamechainsegment inaparity-inverted worldwherethetwofrarnesare connected bysomeboostt'.These inverted-world framesareconnected totheirnon-inverted-world counterparts byToIler's paritytransformation s'defined inEq.(3.4). Since = (7.5) oneconcludes that ~t=-~.Thisiswhatismeantbysayingthat paritynegatesallthet-boosts inthechain,andtherefore the K.=sign(~.). 1 1 Equation (7.5)isoneentryinthefollowing tablewhichshows howtheparityoperators sandSfaffectthesignsofrotation and boostparameters: Rx~RzBxByBz s +++(7.6) SI + + + Noticethatofallthevariables listedinFig.15andrelating tothe mUltiperipheral chain,onlythey-boosts ~iarenegated byparityst. *Thiscondition isderived inRef.5,Eq.(2.8),fortl1eproduction ofspinless particles only;seealsoEqs.(2.7)and(2.10)ofthat paperforaToIleranglediscussion, andEq.(2..16)\4/11icllrelates toourcomments attheendofSection (6). -36- Iftherewerez-rotations-somewhere, thesewouldalsobenegated by s',asthetableshows,andthisfacthasabearing ontheToIler anglewhichwemention hereasadigression. IntheusualBCPanalysis oftheproduction amplitude shown, e.g.,inFig.8,oneusesforthe0(2,1)transformations gthe discrete basisparameters Rz(ll)Bx(~)Rz(V)' whichwementioned at theendofSection (S),andintermsofwhichthelowerReggeon propagator function maybewritten E-cx-l( )mrg=-irve (7.7) Iftheasymptotic limitofthisEfunction istaken[seeEqs.(A.IS) ~cxand(A.l6)]toget (ch~)timeshelicity-factorizing factors, andif thesefactors andtheazimuthal exponentials areabsorbed into renormalizedvertices eandthehelicity sumsdone,oneobtains for theproduction amplitudes theform ,(7.8) wherethep.arethehelicities oftheproduced particles. Thenfrom1 Eq.(7.2),theparitycondition fortheserenorrnalized verticese maybeshowntobesimilar toEq.(7.3), = (7.9) Inthecaseofspinless produced particles, thevertexBisafunction onlyoftheToIleranglew1=VI+1J2and Eq.(7.9)becomes = (7.10) Finally, slightly renormalizing thevertices onceagain, weendup withtheasymptotic orphenomenological mu1ti-Regge amplitude forthe production ofspinless particles alongthechain -37- ,...., .<X26(001)(52) (7.11) MUltiplying twosucharnplitude5together togettheunitarity product, onewouldidentify thekernelas KCW,w') 1 1=[SCW)] [BCW')]*1 1(7.12) andthiskernelwouldthenhaveanaturality condition = (7.13) Thiscondition is,however, justaspecial caseofEq.(7.4)which wasderived without anyapproximations. Therefore, aparitytransfor- mationcanberegarded eitherasnegating the ~.variables inthe 1 exactkinematic scheme, orasnegating theToIleranglesinthe asymptotic production ofspinless particles. (8)TIlEMULTIPERIPHERAL CHAIN A~J}PHASE SPACE InSection (6)thehelicity-pole propagator P.andkernelK.. 1 1) weredefined. Figure22showshowthesequantities alternate to compose themultiperipheral chain (8.1) Thefigurealsoshowsthecentral levelframeswiththeirconnecting boosts. TheVvariables measure the"rapidity width"ofthekernels (clusters orsingleparticles), whereas the~boostsmeasure the rapidity widthofthehelicity-pole propagators. Sincethese alternating boostsarenotcollinear Cv=Band ~1.=By)'thenotioniX ofadditive rapidities arisesonlyintheextremerelativistic limit where -38- (8.2) becomes (8.3) Thesumswhichareimplicit inthechain(8.1)willbediscussed inamoment. First,something mustbesaidaboutthephasespace. Each particle orcluster (hereKwillberegarded.as acluster) getsa .Itmomentum phase-space factor dp.,wherep.isthemomentum flowing 1 1 upthecluster K..1.Replacing d4p.withd4k.,wherek.isthe1,1+ 1 1 1 4-momentum ofthelowerReggeon ofthesystem(i,il),andsimply evaluating this4-momentum inoneofthecentral levelframesafew 4removed fromtheframesnearestp.,onemayexpress dk.intermsof1 1 thegroupvariables appearing inFig.22.Recalling themeaning of thecentral levelframes, wehave,e.g., = = = (8.4) wherekg(theversormagnitude) andZgweredefined inEqs.(5.7)and (5.13). FromthelastlineofEq.(8.4)wefindthat = =221Tkdkdwg332 =[kgdkgd~2 d(ch"2)]dz g •[d 2;2·d(ChV2)] (8.5) where Z3hasbeenreplaced bythewsofEq.(5.14). Theportion dkgdWgofthephasespaceistheso-called transverse integration cl• fd2tht.h becauseitcanbeexpresse· 1nterms 0pwerep1St e9 g -39- transverse momentum ofthecluster 3whoseparallel momentwncomponent pI!isrelated tothestandard rapidity variable. Intenusofthe 3 invariants t3andt~onecanshow,as inEq.(5.16),that =1 2v-6(t,t3,t~)(B.6) ThesecondfactorinthelastlineofEq.(B.5)showsthe0(2,1) equivalent ofthedr2=d<pd(cose) onefindsincmskinematics, e.g., elasticunitarity. Thefactthat d~2d(chv2)/2n=dZ2isapiece ofthe0(2,1)invariant measure (incontinuous-basis parameters) is whatallowstheexactdiagonalization ofthemultiperipheral chain ontocentral levelangular momentum,as isdoneinthenextsection. Thdf-1--hhChGldb L - -22 ...ereaeramIlarWItt eew-0erger- owapproXImatIon tothemultiperipheral phasespacewillrecognize theexpression in Eq.(B.6)asaportion oftheasymptotic formofthequasi-erns phase spaceoftwoclusters, ~ .~where(S1)2and(52)2arethemassesflowing upthetwoadjacent clusters. Inordertocompare Eq.(8.5)with(B.7)wewrite, shifting totheleftonerung, dg 1d(ehv1)d(chV2)8Cv-v1-"2) =~n[k(chv,ch"!,ch"2)] 2(B.8) wherek(x,y,z) =x2+y2+Z2-2xyz-1.Ifitweretruethatv»V,V1 2 throughout theentirephasespace,onecouldapproximate -40- Thenfromformulas likeEq.(5.19), onefindsthatch'V1= 2kk1 2, andthends1ds2 ~ 21fk2s2, dk2dw2dslds2[tdt2dt; ]dslds2d4k v'-6(t,t2,t;)~ =2S S -whichistheCGLapproximation (8.7). Sincetheapproximation s»SI,52isnotparticularly validexceptinspecial caseslike doublediffractive dissociation, onewouldexpectamoreaccurate resulttobeobtained inanyrelated calculation (likethecylinder) byusingtheexactphasespace. Naturally, anexactangular momentum diagonalization onlyworksifthiscorrect groupphasespaceis retained. Wenowconsider thesumsimplicit in(8.1)andFig.22.For eachsegment orpropagator ofthemultiperipheral chainthereisa sumoftheform(e.g.,forsegment2,2') fdt.E 12 where andE 2(8.9) -41- (8.10) withf beingthenormalization factorofSection (4).Foreach fixedvalueoft2andt;[seeEq.(5.15)] andthediscrete index 1(2' andforeachpairofReggeons a2,a;,wesumoverallofthehelicity poleslabeled byn2,n;,thesebeingthehelicity daughters ofthe Reggeons. Next,wesumoverallpossible upperandlowerReggeon combinations. integration. nextsection.Finally, wesumoverK2anddothetransverse Thegroupintegrations dg.willberemoved inthe 1 (9)THEDIAGONALIZATION OFANGULAR MOMENTIJM Toavoidconfusing themathematics withthephysics, webriefly discuss ourdiagonalization procedure; afullerexplanation maybe foundelsewhere.9 Consider thefollowing mathematical relation amongfourfunctions A,B,C,andD,eachafunction ofthreevariables: Schematically, thisequation isrepresented inFig.23.Ifthe variables areintherange_co<(.<coand0·<;v.<00,wemay1 1 interpret thefunctions A,B,C,Dasbeingdefinedonacertain sectorofthegroupSU(l,l) ""'0(2,1), andwewritethesameequation ingrouptheoretic notation asfollows: . (9.2) -42-: whereg=(~,\l,~t), il=·(~l,\ll,O),etc. Thevariables g3= (~3,V3,~~)in Eq.(9.1)arefunctions oftheothervariables according tothe5U(1,1) groupmultiplication g=g-lg_.lg. InEq.(9.2)this3 ..2 1 factismademoreexplicit byuseofaninvariant deltafunction. Equation (9.l)or(9.2)canbediagonaliz~d exactly byproject- ingthefunctions ontothecontinuous-basis representation functions ofSU(l.l). Thesefunctions arethesecond-kind generalized Legendre functions ~\l(Z)discussed briefly inAppendix Aandatgreatlength inRefs.9and23.Thediagonalization of(9.1)isgivenby where(9.3) j , A 1..111'=Jdg~jI(g)A(g) 1--11..1(9.4) andtheprojection ofCislikethatofB;DlikethatofA. Theinvariant measures are d~ d~t dg=27fd(chv)27f andthefunction ~(g)isdefined as(9.5) (9.6) (9.7) ~1.l1(g)=-ll~ Jee ~,(chv) 111..1(9.8) with -~(z)giveninEq. (A.2)~ Inthediagonalized equation (9.3),eachgroupintegration has beenreplaced byahelicity contour integration running upthe -43- imaginary helicity axis.Thesourceofthiscontouristhesecond- kindaddition theorem which,forconvenience, wecomparetothe first-kind addition theorem: 00 p~,(glg2)=LP~(gl) P!un,(g2) n=-oo(9.9) (9.10) Thefamiliar helicity sumofthefirst-kind theorem (Pfunctions areessentially therotation Dfunctions) appears asahelicity integration inthesecond-kind formula. InRef.9,Eq.(9.10)is derived from(9.9)andinterpreted group-theoretically. Whenallthefunctions appearing inEq.(9.1)areindependent ofthe ~.variables, thediagonalized equation simplifies somewhat, 1 = (9.11) where 00 fj=;~d(chv) Qj(chV) F(v) 1(9.12) Moregenerally thisisnotthecaseandthehelicity contours appearing inEq.(9.3)areshifted sideways topickuphelicity pole contributions oftheintegrand. TheAarethecomplex h~licity variables towhichwereferred earlier. Intheparticular mathematical example considered above, we diagonalized achainofthreefunctions B,C,D.Hopefully itis clearthatachainofanylengthmaybesimilarly diagonalized. Eachprojected function contains thediagonal angular momentum projection labeljalongwithtwohelicity labelswhicharesystem­ atically tiedtoneighboring helicities bythe"summations"JdA. -44- Weare,bytheway,referring tojasangular momentum because, intheReggelanguage,Eq. (9.4)isatrueFoissart-Gribov projection sothatjistheanalytic continuation ofthetrueangular momentum. ~InEg.(F.S)weshowhowtorecover afunction A(g)fromits projections Aj"Le.,wegivetheinversion offormula (9.4). 111.l (10)THEPLANAR BOOTSTRAP 1.FormoftheIntegral Equation Thebasicmultiperipheral chainwasillustrated inFig.22,and tieshallnowbemorespecific. Thecontribution fromthreeparticles orclusters tothe4-Reggeon ringdiscontinuity isgivenby* (10.1) wherethenotations P,K,dgand~weredefined inEgs.(6.2),(6.4), (9.7),and(8.9). Since(10.1)isoftheform(9.1),thediagon- alization maybereadfrom(9.3)tobe jdAjdAf j...j=~.•i1T.i1TPll(l)KllA(l,2)PA(2)KAA,(2,3) 2,3 (10.2) where *Wehavetriedveryhardtokeeptrackofthenormalization ofampli­ tudes,but,alas,havelostthebattle. Strictly speaking, ifAis adiscontinuity, equations like(10.1)shouldcontain theoverall factor-efnshowninEg.(4.4)2n/ffor eachdg,andanextra11f because 0(4)(ext)removes oned~ki. However, aswemention in Section 10.8,andshowinFig.29,AisnotToIler-normalized, soweomittheseoverall factors. Thereisalwaysaquestion of howmanyntsand2'5appearinthephasespace def>oftheplanar bootstrap orEq.(11. 24},andwehavetherefore losttrackofthese factors. -45- 00 (10.3) 00 Pll(1)=f~;e-llg p1(g) _00 Therefore, defining (3)Xjby.1111 t(10.4) (3)Aj(14) 1111',=P(1) (3) Aj(14)Pt(4) 11 111.1" II(10.5) Eq.(10.2)may-bere-expressed as (3)Aj(14) 111.l',='.~.t.·~AJ.d.A'£..J 1'11" 1'11" 2,3K~A(1,2)PA(2)~A,(2,3)PA,(3)Kf'1l,{3,4) (10.6) whichisschematized inFig.24. Intheusualwayonemaywriteanintegral equation forthe complete 4-Reggeon ringdiscontinuity whichwillbesolvedbyasum oftermsoftheform(10.1). Thisintegral equation reads whichmayonceagainbediagonalized byinspection togive,together withthedefinition (IO.S), (10.8) whichisrepresented byFig.2S.Theproblem ofobtaining the 4-particle discontinuity fromthe4-Reggeon solution of(10.8)is illustrated inFig.26anddiscussed inAppendix F. NearaReggepole, theprojected ringdiscontinuity Afactorizes (seeFig.27): -46- . j~a AJ(13)I == '1T ~~', rea)a aGll(1;ex)Gll,(3;ex) j-cx(10.9) Takingtheresidue ofthepoleonbothsidesof(10.8)thenyields thevertexbootstrap (10.10) asshowninFig.28.Thenormalization ofthetriple-Regge couplings Gisdescribed inSection 10.5below. 2.TheProjected Helicity-PolePropagatorPA_ Thehelicity-pole propagator wasdefined inEq.(6.2)tobe P1.(~)=2~H(i)e(~K.) exp[h. I~I],1 1 where(10.11) H(i)- h. 1(a.-n.) +(a!-n~) 1 1 1 1(10.12) (10.13) According to(10.4), theprojected propagator takestheform PA(i)=H(i)/(K.A-h.)1 1(10.14) wherewenowseetheactualhelicity poleatA=K.h.• 1 1 Intheordered S-matrix,Regge trajectories mustoccurin strongly exchange degenerate pairs. Whentheupperandlowersignature factors aresummedoversignature takingintoaccount theexchange degeneracy, onefindsfortheregular (untwisted) propagator P, exp{.-i1T[(ex.-E.)-(Ci~-E~)]} 1 1 1 1 . ().(-It)Sln1T<l.-€.Sln1Tcx.-€.. 11 1 1(10.15) -47- Forthetwisted propagator ,xpusedlaterinthecylinder discussion, 1 ~.t~~."'.,(T•~.)(1:"~.~~)11~ 11~1, 1:".T. 1 1= .(lO.16) sin7T(a. -E.)sin1T(a~ -E~) 1 1 1 1 3.TheProjected KernelandItsThreshold Behavior Inthekernel, shownschematically inFig.18,thereareseven quantities eachofwhichdepends onthekernelmassSIandtherefore onthevariable v10£Eq.(5.19),sothatcomputation oftheprojected kernel(10.3), 00 =fd(chv) 0.~A(chv)K12(v) 1(10.3) intennsofthestandardized vertexVisanunpleasant numerical task whichweshallnotattempt. Thistaskis,however, anecessary aspectofthefunctional bootstrap tobementioned below. jLacking ananalytic expression forKllA,wesearchforany potentially usefulinformation buriedinformula (10.3). Onesuch pieceofinformation isthethreshold behavior whichwenowextract. Since K~A(l,2) isaFroissart-Gribov projection, wearereminded thatitshouldbepossible tofinditsthreshold behavior intheusual way.First,however, onemustidentify thethreshold behavior ofthe unprojected kernel. Inexpanded notation onehas = = wherek.is theversormagnitude (continued cmsmomenta) ofthe 1. channel(i,i'), andk.=]. w.=].-48- 1~ ..~[-A(t,t.,t.)]/2(-t)].1 (t.-t~)12(-t)~ 1 1 Weshalldefine"threshold behavior inthe(i,i1)channel" tobeany approach tothekinematic boundaryA(t,t.,t~) =0asshown,e.g.,in 1J. Fig.40,sothatatthe(1,1')threshold k1-+0. t~herearenegative.) J.(Variables t,t., 1 Todetermine, then,thebehavior ofK12ask1ork2vanishes, weexamine thefunctional andkinematic structures ofKasshownin Figs.18and14.Asdemonstrated inAppendix D,ask1-+0,onehas butwhenk2-+0thesituation isreversed -1ch(f)-+constx(k)•2 2 InEq.(C.5)thefunction F(-h1),whichappears aspartofthekernel inFig.18,isgivenroughly as -ex,(k) 1 1 Similarly, Collecting similar factors fromtheuppervertexVofFig.18,one mayconclude that =-49- (10.17) whereK'isareduced amplitude, realontheuncutportion ofthe realtaxis. Thethreshold behavior oftheprojected kernel K~Amaynowbe fOWldfromtheFroissart-Gribov projection (10.3). Equation (5.19) whichexpresses ch(v1)intermsofS1showsthat aseitherk1ork24O.Therefore, usingonceagainthelargez · - 1behavior of~J(z)~z-J-andremembering thattheintegration inllV Eq.(10.3)actually beginsaboveZ=1atthelowestproduction threshold ofthekernel, wepickuptheusualextrafactor(k1k2)j, sothatthecomplete threshold behavior oftheprojected kernelis givenby = (10.18) Whenthiskerneliscontinued tothephysical crosschannel t>0andthefourReggeons takentotheirphysical points,we regaintheusualthreshold behavior given,e.g.,byJackson and H·-t241e, L1(min) (k1) whereL.(min)=J -S.(max)andS.(max)1 1·····1, *=s.+s..11 *Inderiving thisthreshold condition wehaveignoredparitywhich causesthedistinction between threshold andpseudothreshold and whichmayraisesomeL(min)byoneunit. -50- Sincetheaboveanalysis usedonlythekinematic structure ofFig.IS,onemayconclude thatthisthreshold behavior isequally applicable tothesingle-particle orclusterizedkernelsas wellas tothefullamplitude. 4.TheNaturality Diagonalization InSection (7)itwasshownthatinaparity-conserving theory thekernelK12(V)hastheparitycondition (10.19) wheretheo.arethenaturalities -asdefined inEq.(3.9)-ofthe1 Reggeons attached tothekernel. Theproperty (10.19) passesimmed­ iatelytotheprojectedkernel K~Avia(10.3). Atthisjunctureitisconvenient toconvertallFroissart projections likeK~Aof(10.3)tolower-case projections k~A defined by 00 k~A(1,2) -fd(ChV)q~A(ChV) K12(V) 1(10.20) whereqissimplyrelated to~asinEq.(A.13). Thereasonfor thischangeisthatqhasasimpler helicity-negation symmetry thanQ,asymmetry whichofcourseiscarried overintok~A' = (10.21) Combining (10.19) through (10.21) wefind (10.22) Onemaynowstudytheeffectofthissymmetry onthering -51- discontinuity components. Converting the(2)Aequation tolower- caseprojections asinEq.(10.20), onefinds (2)aj(13) lllJ' ,='""'r·d.AHjkj(l2)P,\(2)kj(23)(10.23)L.J2·L.J17T AllA' 1\ .All', ,K2 wheretheK2surnhas beenremoved from~2 andexplicitly displayed, andwhere Hl-r(j+1+A)r(j+1 -A) (10.24) Fromthesymmetry of(10.22), andtheobvious fact[seeEq. (10.14)] that = (10.25) onemayeasilyshowfrom(10.23) -usingthesymmetry oftheA contour -thatthesymmetry of(10.22) propagates into(2)a, = (10.26) andsimilarly intoall(n)aandthefulla.Thepersistence of thissymmetry meansthatallourprojected equations canbediagon- alizedinthe2x2spaceofthekappaindices; thisisthenaturality diagonalization discussed byCiafaloni andYesian.5 Wenowperformthisdiagonalization onthefollowing proto- typeequation =Lt-?A K~.I1T 2 wherepisanyfunction, anda,b,careanyfunctions havingthe symmetry ofEq.(10.26). Define aj.(K··K )aKK-K'fK 'I ,l'3.•1 3 ItA'3to"(10.28) -52- andsimilarly forbandc,andnoticethat (10.29) (10.30) Intermsoftheprojections ofdefinite naturality* aa1--[aff ++t +0°1°1a ]-+(10.31) Eq.(10.30) takesthediagonal form Thus,thenaturality diagonalization of(10.27)isgivenby(10.32) =Y~AJ17f where(10.33) jaallll, (10.34) Intermsofupper-case projections like(10.3), Eq.(10.27) becomes (10.35) *Poradiscussion ofwhyaisidentified withnaturalitYJ the readerisreferred topage438,equation (9.59)ofthetextbook ofMartinandSpearman,Ref.32. -53- anditsnaturality diagonalization isgivenby (10.36) jcra,(1,3)llllwithprojections oftheform j01[..j tr(j+1+ll)j ]. AlJlJ,=ffAlJlJ'(+,+)+GalaIr(j+1-lJ')A_lJ,ll'(-,+)· (10.37) Now,since(104!8)isoftheform(10.35), thenaturality- diagonal bootstrap equation canbereadfrom(10.33), =k~~,(1,3)+Lfd~2.r~; a2a.~ andxH(2) A-h2(10.38) (10.39) where±referstotheKvalues, andsimilarly foraj,(1,-;2,+)]..'-ll,+ll (10.40) Left-shifting thehelicity contour assuggested inEq.(12.12) yieldsthisfinalformfortheplanarbootstrap equation: jcrat(1,3)lll.l=k~~,(1,3)+2~fd~2H(2)~2k~~/1,2) (l2CX2 tn2n2 (10.41) -54- A Awherekisaprojection ontoqof CA.20).Allthatremainsisthe transverse integration dcP2and thesumoverallupperandlower·loop Reggeonsand theirassociated helicity poledaughters. WithII=h1 andl..lt=hg,Eq.(10.41)isamatrix(lattice) equation inthespace ofthehelicity indices. Inpassing, wenotethattheapparent Reggecutsin(10.41) duetopolesof~shouldbecancelled bythenonsense zerosof 2 A"theproduct k a . 5.TheBootstrap Problem Equation (10.41)statestheintegral equation whichisthe planarbootstrap forthefour-Reggeon ringdiscontinuity. Assuming theexistence ofafamilyofReggetrajectories {Cl.},andgivena 1 knowledge ofthestandard vertexVijk,onecartinprinciple compute thesingle-particle kernelKanditsprojection K~A.Sincethe propagator istrivially knownasinEq.(10.14), onecanthen searchforsolutions Aoftheintegral equation. Theexistence of asolution depends functionally ontheReggeon set{a.}andtheform 1 ofthevertexV. Theresidue ofthebootstrapequationat anyReggepolej=a, whereae{a.},yieldsthevertexbootstrap (10.10) whichisperhaps 1 moreinteresting thantheoriginal equation becauseitcontains only oneunknown function V(tpt2.tg).giventheReggeon set{ail. Wereturntothevertexbootstrap inamoment. Firstwemustnoteacertain inconvenient property ofthe functions AofEq.(lO.8)andthevertexGof(10.9). Asideeffect ofdoingthehelicity poleexpansion isthatthesefunctions arenot -55- normalized inthesenseofthestandard ToIlerM-function discussed inSection (2).Figures 29and30showschematically howAandG arerelatedto'thenonnalized ringamplitudes (ordered M-functions). Thefunctions FandF'areliketheF'sappearing inFig.18and Eq.(C.S). Asnotedearlier, theapproximate roleofthesefunctions istoconvert theReggeonhelicity fromthediscrete valuesm,T, p,...asinFig.18tothecomplex values(a-n). Accounting forthesenonnalization factors, wenowrewrite the vertexbootstrap intheextremely schematic formofFig.31which showsthebootstrap asanonlinear functional integral equation of the3-point ringamplitudeandtheReggeon set{ai}.Inprinciple, thisequation shouldallowthecomputation oftheorderedtriple- Reggevertexasafunction ofallthreearguments. Toourknowledge, thiscalculation hasneverbeendone. 6.Counting Approximate bootstrap calculations usingaverysmallleading Reggeon set{a.}haveoftenindicated thatthesingle-particle kernel, 1 withexperimentally determined couplings, doesnothavethestrength necessary toelevate thegenerated outputtrajectories totheir experimentally observed intercepts. Assuming thatthisresultis notanartifact oftheapproximations made,onemustconclude that theperipherality and/ortheRegge-expansion convergence assumptions whichgointothemUltiperipheral modelaresimplynotviablefor single-particle production, andoneturnsinsteadtocluster production. Onereplaces thesingle-particle kernelwithacluster oflimited maximum width,butsufficiently broadsoastoapproximately -56- Regge-factorize,eventhough thereisnoReggepoleinsuchakernel. Thisistheconcept ofthedottedReggeon, andtheprogram ofFig. 31isthenreplaced withthatofFig.32which,whentheleft coupling isboldlycancelled onbothsides,givesthefamousequation n1=gNg"ofRef.25,whereN =N·(flavor) . Ingoingtothemulti-particle kernel, however, oneencounters certain counting problems whichinvalidate thediagonalization procedure whichledtothesimpleequation (10.8). Thenecessary alterations involve pre-convoluting thecluster/kernel witha -d26,27 T-dh- ·bl propagator onone51e. 0aV01t1Scount1ng proem,we havechosentoconcentrate instead onthecylinder calculation where thereisnocounting difficulty. (11)THECYLINDER Theordered orplanarbootstrap discussed inthepreceding section consists ofsewingtogether twoordered amplitudes (zero handles, oneboundary) inanordered mannersoastoobtainthe discontinuity ofanother ordered amplitude. Bysewingtogether ordered amplitudes (h=O,b=l)withacertain well-defined disorder, onemayconstruct thecylinder component (h=O,b=2)ofthephysical 4-point function. Figure33showspartsofthiscylinder component inseveral different notations. Figure33adepicts, inquarkdiagram notation, aparticUlar contribution tothetwo-twist-pair piece C(2)ofthecylinder resulting fro~theunitarity product oftwo 9-point ordered amplitudes. Figure 33bshowsthecomplete C(2), butthefigureonlyhasmeaning intermsofdiscontinuities after -57- theupperandlowerReggeexpansions havebeeninserted. Theseare showninFig.33cwhichisnowdrawnintheringnotation. Finally, Figs.33dand33edisplay thetopological meaning ofthetwistsin theabsence ofquarknotation. Toconform withthekinematic diagrams likeFig.15,weshallcontinue tousethenotation of Fig.33c. Thefullcylinder isdefined asthesumofallitstwist-pair components, ex> C=L:C(n)(11.1) n=l WhentheC(2)component showninFig.33cisdiagonalized onto angular momentum j,chargeconjugation* L,andnaturality CJ,one obtains thetriplepoleconfiguration showninFig.34(whenthe simplest assumptions aremadeforthejandflavordependence of thevarious elements, andwhenonlytheleadinghelicity poleofthe leading ReggeonpairiskeptineachReggeonloop) where(2)CL111gj _a(Lk)j_a.(Lk)j_a.g (11.2) k=k(t)=k(j,t) :=fdcj>l·g(t,tl,t~)2 x[otherfactors]. (11.3) Inthephenomenology ofChewandRosenzweig10thecylinder- shiftsofthef,ft,w,and~trajectories aresimplefunctions of k,whichissometimes approximated bysetting j=0,.Roughly, the *Wearerelieving Tfromitstradition dutyofrepresenting signature sincesignature seemstoplaysuchasmallroleintheordered S-rnatrix, andalsobecause therearealready toomanyCtsfloating aroundinSection (11). -58- shifted f=pomeron hastheintercept [inSU(l) ] ap(O)-a(O)+k(a,O) Wewishtodiscuss thetechnique usedtoarriveatthe(11.4) expression (11.3)forkandtosuggest howkmightmoreaccurately becalculated asahelicity-pole expansion. Afterfirstdiagonalizing thechargeconjugation, wereviewaone-dimensional cylinder calcu- lationandthenproceedtothethree-dimensional helicity pole analysis. 1.Diagonalization oftheChargeConjugation Sincethecylinder termsCen)carryzeroad~itive quantum numbers, itisdesirable todiagonalize thechargeconjugation in addition tothenaturality sothatcylinder polescanbeidentified withphysical particles. Thisprocedure isverysimple, aswenow show. Theordered ringdiscontinuitiescarry orientation indices whichhavebeensuppressed throughout thispaper. Onemightwrite (1,0'1IA12,0'2)where 0'i=±ldepending onwhether theordered channel 7iliesintheclockwise orcOlD1terclockwise Hilbert space. Asa 2x2matrixinthisorientation space,Aisdiagonal withequal diagonal elements, (1,01IAI2,02)=AOcra.Changing tothecharge-l'2 conjugation basis Il,T}=[11'°1=+) +TI1,01=-)] 1-.12",onefindsthat (1,1"1IAlz;r2)=A01"1,1"2' sothereisnoneedforAitselftocarry aLlabel. Incontrast, thetwisted Reggeon propagator xpalwaysconnects statesofopposite orientation, <l'0'1IxpI2'0'2} =00'-0'xp.Inthel'2 Tbasisxpisagaindiagonal buttheelements haveopposite sign,so -59- Xp · ·lIb 1 mustcarryatr1vlaTae, 'where (11.5) Therefore, theonlyeffectofchargeconjugation diagonaliza­ tionistoaddaTlabeltothec(n)andtoreplacexp-+TXpevery- where. \ Bycomparison, theuntwisted Reggeon propagator Pwhichappears intheplanarbootstrap doesnotmixorientations, soallcontribu­ tionsA(n)totheringdiscontinuity Aarediagonal withequal diagonal elements intheorientation spaceandtherefore alsoin theTbasis, assuming thespecial caseofzeroadditive quantum numbers alongthechain. 2.TheCylinder inRapidity Forcomparison withthekinematica11yaccurate (thoughstill physically slippery) cylinder calculation presented inthenext -sections, wereviewherea"typical" rapidity analysis ofthecylinder. Forsirnp~icity, onlyoneflavorisassumed instead ofthethreeflavors (with1=2'f3symmetry breaking) usedbyChewandRosenzweig.10 I fh 1 -d- ChP- .28 -bI ntenns 0teusuarapl·.1.tyor· ew-19nott1 varlaes, andwiththeCGLphase-space approximation discussed earlier in Section CB),onewritesintheenergyplanetheone-twist termof thecylinder asfollows (seeFig.35}: f++XT + + +x.d<l>2A(t;,x1,t;> P(g,t;)A(t;,x2,t;), (11.6) -60- withd<P2asgiveninEq.(8.10). Here,Aistheabsorptive partof afour-Reggeon ringamplitude ofrapidity widthx.,andxplisthe 1 twisted (nocosine) Reggeon propagator ofgapwidthgtobegiven below. Thelabellindicates thattheequation hasbeendiagonalized inthechargeconjugation, l=±1. Incorporating theNeveu-Schwarz shifta-+a-1,wenormalize ourtriple-Regge couplings intheusualway, ++ a R(t~,s,t;) -glg2r(l-a)(-s) (11.7) ++ 7T a A(t~,s,t;) =fCa)glg2(+s) (11.8) andtakefortheMellin-projected ringdiscontinuity aformexhibiting '\ symmetric nonsense zeros, + + A(t~,j,t;) (11.9) Thepresence ofnonsense-zeros inaMellinprojection isequivalent totheabsenceoffixed-poles intheFroissart-Gribov projection; wewantsuchfixed-poles tobeabsentbecause weassumetheretobe nofixedpowersintheringamplitude R(tf,s,t;). Theassumption ofthefirstnonsense-zero inaMellinprojection corresponds totheabsenceofaconstant tennontheright-hand side ofaFMSRoverA(ti,s,t;). Byattempting torespect theanalytic - 29structure ofmulti-Regge amplitudes, several authors haveused somewhat controversial asymmetric FMSRtoarguethat,ineffect, the amplitude showninEq.(11.9)shouldhaveanonsense-zero onone sideortheother(depending onwhichexternal overlapping invariant isheldfixed), butnotonbothsides;i.e.,thattheformof(11.9) -61- shouldbeasymmetric. Wefeel,however, thatthefour-Reggeon amplitude shouldbeleft/right symmetric, evenifasymmetric NDC isusedinitsgeneration, andthisisourmotivation fortheform (11.9), though wehavenorigorous argument tosupportthisconjecture. Oneofthephysical weaknesses ofthecylinder calculation is thatsmallchanges inthesmooth(i.e.,non-singular) j-dependence oftheprojected planaramplitude, suchasnonsense zeros,cancause violent changes intheoutputpomeron location,3D sonocalculation canbetrusteduntilthelow-energy/smooth-j behavior oftheplanar amplitude hasbeendetermined fromtheplanarbootstrap. Hopefully, suchbehavior mightbecomputed fromthehelicity poleformalism. Meanwhile, weshallusetheform(11.9)onlyasaprototype and continue ourcalculation.* Thetwisted Reggeon propagator appearing in(11.6)is =,.....,+ TH(t;)exp(gac) 2(11.10) InMellinprojection thisbecomes =j-ac2(11.11) with = , (11.12) whichmaybecompared to(10.12) with(10.16). Now,theC(l)equation (11.6)maybetrivially Mellindiagon- alizedtoyield *Lowenergydataare,ofcourse, helpful onthispoint. -62- Inserting theexpression (11.9)wefind (11.14) with (11.15) Byemploying symmetric nonsense zerosin(11.9), wehaveremoved the Regge-cut generating factorofthepropagator (11.11), andhaveadded another factor (j-ac)inthenumerator; k(j)isj-dependent. 2 Fromthediagonalized integral equation forthefullcylinder, orbysimplysumming thegeometric seriesthefirsttermofwhichis givenby(11.14), onefindsthat -(J-.-_-a-)(--j_k__(~"""-')-_-T-k-(j-)~)[g3(:~:::)} (11.17) whichshowsthepomeron (T=+)atthesolution of j=ex+k(j) (11.18) Finally, addingCtotheplanartermAextinguishes theunshifted poleinthemannerofRef.10, A(1,j,3) +CT(1,j,3) (11.19) andthesymmetric nonsense zerosappearalsointhecylindrically corrected amplitude. -63- 3.TheOne-Twist Cylinder TermasaHelicity PoleExpansion Thetypicalmulti-cluster contribution (3)A(~,v,~') tothe four-Reggeon ringdiscontinuity wasgivenin(10.1)andillustrated inFig.22.Thesumofallsuchtermsdefines thecomplete four- Reggeon ringdiscontinuity inthe"energy plane." Ofnecessity, the objectAcontains thepropagators onboth·ends ofthemultiperipheral -."ladder. Itisimportant torealizethatAcontains theseend- propagators inconvoZution, sothat,unlikethekernel, cannotbewritten intheform-.", A(~,v,~ ) Onlyafterdiagonalization cantheend-propagators beremoved asin (10.5). Forthisreason,itisdifficult towritecylinder terms­ inparticular C(l)-intheenergyplane,butveryeasytowrite thesetermsinprojection, aswenowshow. Letusdefineanextremely condensed notation sothat,for example, Eq.(10.1)oritsdiagonalization (10.2)bothread: (3)-'"A=PKPKPKP Similarly, theplanarbootstrap reads A=PKP+PKA A=K+KPA[energyplane,see(10.7)] , (11.20) [j-plane, see(10.8)] Inthisnotation, theC(l)cylinder termmaybewritten inthe energyplaneas -"'(1)C=AKPxKA +PKPxKA +AKPxKP +PKPxKP ,(11.21) wherePxisthetwistedhelicity polepropagator (seebelow). The -64- diagonalization of(11.21)is,inourcondensed notation, again (11.21). Once(11.21) hasbeendiagonalized, wemayuse(10.5)to exposethepropagators sothat "'(1)C=P[APK+K]Px[KPA +K]P Inserting theplanarbootstrap (11.20) twiceyields(j-plane). (11.22) (j-plane) . (11.23) Finally, inanalogy to(10.5), wedefine C(1)intermsofC(l) toget C(l)=APxA (j-plane) , which,infullj-plane no~ation, reads (l)C~l1'(1,3)=~j~;A~A(1,2)xPA(2)A~l1'(2,3) (11.24) Thisequation, illustrated inFig.36,givestheprojected one-twist cylinder termintennsoftheprojected ringdiscontinuity Awhich * solvesthebootstrap (10.8). From(10.14), =H(2) KA- h222(11.25) whereH(2)isgivenby(10.12) withthesignature-factor product replaced thistimeby(10.16). Addingthenaturality andchargeconjugation labels[see Sections (10.4)and(11.1)], (11.24) becomes (1)CjOT(13)=""j.dA jO[T] jO 1111" ~.·hrH(2).\A(1,2)A_h 2AA11'(2,3) *Notice thatEq.(11.24) doesnotsayC(l)=KPxK.(11.26) -65- LjcrA- h 2aAll,(2,3) (11.27)(l)CJOT(13) 1..11..1t ,or,intermsofthelower-case projections of(10.20), =L..j··~AH(2)Hjajcr(12)2.·1U.A·~~., withH~=r(j+1+A)r(j+1 -A)• 4.Angular Momentum vs.Helicity Wepausetomakeafewobservations aboutEq.(11.27). First, itshouldbeclearthatthe"Reggeonpropagator" (11.25)isnot directly related totheangular momentum j,incontrast tothe feeling onegetsfromtherapidity approximation. Thatis,the leadinghelicity-pole propagator hastheform 1/(~- h2),not l/(j-(lc)asinEq.(11.13). Asemphasized inAppendix E,the variable ~whichmeasures theenergydependence oftheobject we looselycallaReggeon propagator istheanalytic continuation ofan azimuthal Eulerangle,notacentral Euleranglelikeeof(<I>,e,<I>'). Therefore, thecorrectly projected propagator isafunction ofthe variable conjugate tothatcontinued azimuth ~,namely, thecontinued helicity ~,andnottheangular momentum j.Therapidity formalism withitscollinear boostsisincapable ofdistinguishing angular momentum fromhelicity, andprojects everything ontoahybridized Mellinprojection index"3".Onefeelsthatasymptotically -i.e., nearsingularities intheprojection index-thishybridization is acceptable. Evenso,itseemsunlikely thatthelow-energy behavior ofaplanardiscontinuity couldbedetermined fromaplanarbootstrap whichusessuchanapproximation, andthesamegoesforthecylinder. Forexample, itisjustthisdistinction between jandthevariable A(whichisforcedtothevalueh2=(le+1)whichgivesrisetothe 2 threshold factorappearing inEq.(11.38) below. -66- 5.ReggeCutsandNonsense Zeros TheA-plane forEq.(11.27)isshowninFig.37.Asjis varied, thehelicitycontour isrepeatedly pinched betweenthehelicity poleatA=h2andthepolesofr(j+1 -A).Eachpinchgenerates a poleinjwhichisinturnconverted toaReggecutbythetransverse integration d<P2•Thesej-plane polesare,ofcourse,explicit when thehelicity contour in(11.27)isleft-shifted asper(12.12)to give (1)CjcrT(13)1111t , (11.28) Keeping onlytheleadinghelicity polesoftheleading Reggeons sothat h2(11.29) andsetting ~=11'=0,(11.28) becomes (11.30) whichmaybecompared totherapidity result(11.13), (11.31) Whereas (11.31) showsonlythefirstReggecut,(11.30)exhibits thecomplete familyofReggecutsassociated withtheReggeonpair Recall, however, thatiftheupper-case projections -67- lackfixedpolesatthenonsense points, thelowercaseaj ~h~e l.lll nonsense zeros. Wepresume, then,thattheseamplitudes infacthave astringofnonsense zeroswhichcancelallthegarnma function poles andthereby eliminate allReggecutsfromthecylinder, justaswe contrived todointherapidity model. Moresignificantly, thesame mechanism shouldremoveReggecutsfromtheplanarbootstrap. Unfortunately, wehavebeenunabletopursuethisquestion duetoa technical difficulty whichwediscuss inSection (12). 6.TheComplete Twisted Reggeon Loop SincetheA'sappearing in(11.26) aretheprojections ofring discontinuities, theirj-plane singularity structure contains, hope- fully,onlyReggepoles. Wethenwriteasanasymptotic series, A~A(1,2) =L: ex[~G~cr(l;a)] [I;fu G1cr(2;a)] (j-ex) (11.32) wheretheGaretriple-Regge couplings, =jcr, ,G(a.,Cl.;t.,t.;Cl,t)lJ1 1 1 1 whosenonnalization wasdiscussed inSection (10.5)andshownin Fig.30.Nearaparticular Regge-pole, Eq.(11.32) reduces tothe form(10.9)givenearlier, butingeneral wewishtomaintain the j-dependence inthetriple-Regge couplings, asdiscussed below. SinceGj(l;a)couples twoReggeons (a1,a;)toathirdReggeona, itisclearthatthecoupling vanishes ifahasthewrongnaturality, sowenowdropthenaturality label,keeping inmindthat(11.32) represents asumonlyovertrajectories ofthepropernaturality. -68- Inserting thepureReggepole expansion (11.32) twiceinto (11.26), wefindfortheprojected one-twist cylinder term (l)Cjl.(13) 1-11-1t , where=:2:[Irra)G~(1;a)]j=a{Lk~a'(t)}j_la' a,at x[/r(~') G~,(3;a')J (11.33) = Whenthehelicity contourisshifted totheleftassuggested in Section (12),theresultant kis =____2'IT__1:H(2)[G(2;a)]~ [G(2;a')]~ v'r(a)r(lt) 2 2 2(11.35) InourReggepoleexpansion (11.32) fortheordered four­ Reggeon discontinuity A~A(1,2), wehaveexhibited thetriple-Regge couplings- asbeingj-dependent, justasintherapidity version (11.9). Usually Reggecouplings arepresumed tobeindependent ofj,e.g., =YaYb (j-a) sowewishtocomment onthispoint. TheReggeexpansion givenin (11.32)issupposed tobeareasonable approximation totheexact jpartial waveamplitude AVA(1,2). However, weknowfrom(10.18)that wh~nk1-+0ork2-+0[k.arethecontinued emsmomenta, seeEq.(5.7)],1 . thepartial waveamplitude A~A(1,2) mustexhibitthecharacteristic threshold behavior, [A(1,2)]~A = (11.36) -69- Therefore, inorderthatthe(finite) Reggeexpansion (11.32) be accurate, wemustassumethatthecouplings alsoexhibitthisthreshold behavior, (11.37) Wemightthenignorethej-dependence oftheresidual coupling G'. Inparticular, wehavealready notedthatG'shouldhavenofixed polesinj. Therefore, amodelforthecomplete twisted Reggeon loopk accounting forthisthreshold behavior andlackoffixedpolesis = (11.38) wherewehaveused2d$2=(2wjf) kdkdwasgivenintq.(8.10), and whereH(2)isEq.(10.12) with(10.16). Inthepast,expressions forkhavenotshownthisthreshold behavior because theprojected triple-Regge coupling hasbeenidentified withthej-independent dual coupling ,g(t,t2,t2)=rea+1)r(a-(le) 2 Thecomplete twisted Reggeon loopanditsrelation toC(1)are showninFig.38.There,theloopiscross-hatched toindicate that itisacompZete twisted Reggeon loopincorporating theeffects ofall thehelicitypoles ofallReggeonpairs. -70- Wemaynowcomparethisprecise Reggeon looptoitsapproximation intherapidity modelasgiveninEq.(11.15). First,sincewehave included morethanoneReggepoleinourapproximation toAj,thek of(11.38)isamatrixinthespace.of theReggeonset {ex.},whereas 1 (11.15) showsonlytheleading diagonal elementofthismatrix. Secondly, theusualnurneratorgammafunctions of(11.12) whichcontain thephysical polesofthepropagator, nowappearassinesinthe denominator ofthefactorH(2),withthejobofghostremoval now incumbant uponthecouplings GinthesenseofEq.(2.8). Wehave retained theE2,E~factors inH(2) toallowforferrnionson thetop and/orbottomoftheReggeon loop.Forexample, theupperandlower Reggeons mustbothbebaryons inthecontribution tothecylinder whichmixesregular mesonswithbaryonium states,31 (seeFig.39). Finally, thekappearing in(11.15) contains onlytheleadingpair ofReggeons (a2,a~), andonlytheleadinghelicity polecorresponding tothatpair,i.e.,n2=n~=o. 7~TheFullCylinder Sofarwehavediscussed thezeroandone-loop contributions to thefullcylinder, Eqs.(11.32) and(11.33), whichwenowrewrite in anabbreviated notation =g(l).P.g(3).111 =g(l).P.K..P.g(3). 1 1 1JJ J (..)-1 .wherenowP.=J -cx..andK..=Tka.a.. 1 1 1J 1 J cylinder including theplanarpart, 00 C(1,3) =L:C(n)(1,3) n=OTocompute thefull -71- weremovetheexternal couplings ontheendstomakematrices ofthe cen),andwereplacetheP.withdiagonal matrices 1 P..1J=0..P.=1J1..-1o...(J-a.)1J.1 ThenCisageometric matrixserieswhichonesumstoget [C(1,3)] ...1J=[P+PKP+ ...]..=[P-1_Kf1= 1J ij[T-1]cof(P-K)·...1J -1det(P-K) Thelocations ofthepolesofthefullcylinder arethendetermined by wheredetD(j,t)=0 [D(j,t)]...=(j-a.et))0..-Tk~.et)1) 1 1J 1)(11.39) (11.40) withk...asgiveninEq.(11.34). Inpractice, onecanrestrict to1J asmallnumberofleading planartrajectories andinclude symmetry breaking. Ifthematrixspaceiscrudely limited toonedimension, Eq.(11.40) showsthatkrecovers itssimplesignificance asthe shiftbetween thepomeron andplanarReggeon intheone-flavor model, asinEq.(11.18). (12)FIXEDPOLES, NONSENSE ZEROS, AND THEHELICITY CONTOUR PROBLEM Whereas thediagonalization procedure described inSection (9) isstraightforward, theproblem ofshifting thehelicity contour in thediagonalized equation isstill, wefeel,anunresolved question. Ratherthanburythisdiscussion inthecylinder calculation above, wethoughtitbesttoexposetheproblem clearly inthehopethat someone willsolveit,andtoshowthedrastic assumption wemake°in theend.Theproblem described hereineffectblocksthecompletion -72- ofthehelicitypole expansion program. Consider asimplerversionofEq.(9.1), oritsdiagonalization A~ll'=t~;~~Acill, withprojections asgiveninEqs.(9.4) and(9.5),whichwenow writeas(12.1) (12.2) j CAll'ex> =fdz~ill'(z)CAll'(v) 1(z=chv) (12.3) andsimilarly forA~ll"butfor~~Awehave(12.4) ~(v) 1100 =f~; _00~\(z)~(v)111\.11(12.5) (12.6) Thefunctions ~~ll'(z),liketheregularQj(z),havepolesinj andtherefore (itturnsout)inAand11',andthiscertainly suggests thattheprojections like C~lltofEq.(12.3)mightalsohavethese "fixedpoles," although thisisnotnecessarily thecase.Neverthe- less,itisusefultocQnvert fromthefunctions~~ll' totheq~ll'of Eq.(A.I3)whichareanalytic inj,A,11'andhavenozeros,atleast forRe(j)>-1.Defining new,lower-case projections asinSection (10), 00 - fdzq~llt(z)CAll'(v) 1(12.7) -73- wecansaythatifC~ll'hasno"fixedpoles," then c~lltmusthave zeros(nonsense zeros), sincecistheresidue ofthepoleinC. (Inthissense, cisclosertotheMellinprojection thanC.) Intermsofthelower-case projections, (12.2)becomes (12.8) where H~=r(j+1+A)r(j+1-A).AsA-+±ioo,H~-+exp(-1T1AI), providing theexcellent apparent convergence forthehelicity inte- gration. Onepaysapricetogetthisexponential damping, however; H{haspolesgoingoffinbothrealdirections intheA-plane (see Fig.37).Suppose ~~Aisanalytic inAandc~ll'hasasimplepoleat, say,A=h=-1+i.Onewouldliketosaythat,whenthecontourisshifted totheleft,thispolemakesacontribution toa~ll"However, thepoles ofH~alsomakecontributions, andtomakematters worse,thepolesof jcAll'atA=hcanpinchthecontour againstallthepolesofr(j+1 -A) causing a~ll'tohavepolesinj("Reggecuts"). SinceReggecutsare unwanted inthecylinder orplanarbootstrap, wewouldliketoclaim thatthepolesofr(j+1 -A)arecancelled bynon5:ense zerosinthe projections like c~ll"whichistosay,theFroissart-Gribov projection C~ll'hasnofixedpoles. Thissoundsreasonable ifC{ll'isthe projection ofanordered (planar) amplitude wherefixedpolesmust beabsentsothattheReggecutdiscontinuity formulas givezero discontinuity, circularly speaking. Granting thattheproduct hasfullnonsense zerosto cancelthepolesofr(j+1 -A)andremoveReggecuts,onemuststill consider theproblem o'fshifting thecontour. Itmustbeimpossible toshifttotherightbecause thenonegets a~ll'=0,certainly not -74- desirable. Shifting totheleftyieldsacontribution fromthepole atA=h,butstillthereareallthepolesoff(j+1+A).Theconjec- turedformofthehelicity nonsense zerosofEq.(2.8)suggests that thezerosintheA-plane shouldbesymmetric andtherefore thepoles off(j+l+A) arealsokilled(although thedAconvergence isnow jeopardized bytheremovalofHt). Now,presumably, theintegrand ofEq.(12.8)isanalytic inA exceptforthepoleatA=h(the"helicitypole")andwewouldlike tosay:shiftthecontour totheleft,pickupthehelicity pole contribution, andhopethecontour integration vanishes asitis shifted offtoRe(A)=_00. However, intheC{l)cylinder calculation ofSection {It)we foundthat,asidefromthehelicity pole,theintegt'and wassymmetric inA,sothatifwedisallow ashifttotheright, wemustalso disallow ashifttotheleft. Thesituation isanalogous totheproblem oftheSommerfeld Watsonrepresentation whichisresolved bythe"Mandlestam trick"of replacing thepoorlybehaved functions pjwithj-decaying functions likeQ..InRef.3itissuggested thatasimilar procedure beapplied J inthepresent context. Presumably theprojections ct~tarebadly behaved asReCA)-+±oobecause the q~J..llarebadlybehaved. Asshown inAppendix A,onecandecompose = (12.9) Ajwhere qA~'haspolesonlyontheright,thoseofr(j+l-A),andis wellbehaved asRe(A) ~-00,ascanbeshownbyapplying Watson's Lemma Atotheintegral representation, Eq.{A.IS).Defining projections ~ Aandcintheobvious way,(12.8)becomes -75- = ThetwocrosstermseitherNow,sadly,onedoesnotreallyknowthelarge ~behavior ofthe various projections because, lookingat(12.7): (a)infinite range Atthispointonethrowsupone'shandsandmakesaguess. Of thefourtermsontherightsideof(12.10), thefourthterrnmay be harmlessly shiftedofftotherightwheretheprojections~j '\and-lJ,-1\ Ajc,,haveatleastpowerdecay.-1\,-11 cancel, ormayalsobeshiftedofftotheright,alsoyielding no contribution toaj,.ThefirsttermTTlUstbeshifted totheleft,llll Aj Ajinthedirection that Uandcareatworstpowerbehaved.PllA.All' Thistermpicksupthehelicity poleatA=h,givingthefinalresult =2Hj•~jRes[~j.Jh.llh hll'(12.11) or,intermsoftheoriginal equation, (12.12) Inthispaper, wehavemadetheassumption that(12.2)canbe replaced with(12.11)or(12.12) inthefollowing locations: (10.41), (11.28),(11.30),(11.35),and(11.38). -76- APPENDIX A SOMEUSEFUL FUNCTIONS References 9and23describe atlengththeproperties ofthe generalized Legendre functions P~vand~~V'Herewereproduce only theirdefinitions andbasicsymmetry properties: =( 12-Z.) Fj+l+v,-j+v;V-11+1; r(v-II+1) (A.!) ~~v(z) (1)~(1l+V) =~r(j+l+~)r(j+l-v) ~:1 F(j+l+~, j+l+v;2j+2;_2_)l-zx r(2j+2)()-j-1z-1-2- (A.2) pj=pj pj=p-j-l l.lV-v,-ll llV l.lV (A.3) ~~v=~j~~v=Gj ~~~ -V,-l.l llV where Gj=r(j+l+1.1)r{j+1-v)(A.4)l.lVr(j+l-1.1)r(j+l+v) Allvariables aregeneral complex numbers. Sometimes wemakeuseof thefollowing combination: ~~,(g) =_llt~te (A.5) Theusualrotation d-functions32aregiyenby[±forIm(z) ~0] ordj,(z)nun=(±i)m-m' (Gj)~pj(z)m'm mm'(A.6) x-77- ·~em'-m.em'+m (G~tm) (+sin2) (COS2) F(·l' ·t t1.28) J++m,-J+m;m-m+;S1D'2 r(mt-m+1) Whenj,m,m'areallintegers orhalf-integers, onehas dj t(-z)m,-m d~t(a=n)=j+mj(-1) dt(z)mm j-m' (-1) tSm,_m t(A.B) Asusual,thecomplete rotation-group matrixelementisgivenby D~t(g)= =e-im~dj t(a) mm-im'cI>'e (A.9) Theaccompanying second-kind e-functions33aredefined by withej t(z)mm= , (A.IO) E~t(g) =-im~j ~-im've e,(ch~) emm(A.II) Thedandefunctions havethesehelicitysymmetries: (_l)m-m'dj=dj=dj,mm' m'm -m,-m' (_l-)m-m'ej=ei=ej mm' m'm -m,-m'(A.12) Sometirnesitis convenient tousestillanother version ofthe second-kind function, =r(j+1+ll)r(j+1 -V)(A.13) Thisq-function hastheadvantages ofbeinganalytic ini,ll,v,and -78- havingverysimplesymmetries: = (A.14) Theasymptotic behavior isgivenby q~v(z)2j-j-l limz=r(2j+2)z+oo andtherelation between qandeis(A.IS) ej,(z)mm=m-m'(±i) [HjJ~qrnmj,(z)mm'(A.16) where Hj=HjHjandHj=r(j+l+11)r(j+l-11). Anintegralmm'mm' II · f · eb 34 representatl0n orq1Sglven.y where00 =~fdaf(a) _00CA.17) f(a)=e-lla(chv+shvcharj-l[efJ. +th(vI2) ]A 1+eath(vI2) jSinceqllAisanalytic inll,andhasnoidentical zeros,thefunction ontheleft-hand sideofEq.(A.17)haspolesgoingoffinboth directions inthell-plane. Bydecomposing theintegral intotwo parts,itispossible toproduce afunction whichhaspolesonly ontheright, o H~~A(chv) ::;~Jdaf(a) _00(A.18) andiswellbehaved asRe(ll) -+-00.Comparing (A.18)with(A.17), we find j qllA=CA.19) -79- Letting e-cx=llin(A.18)andusingBatemants35formula [5.8.2(5)], qmaybeshowntobeatwo-variable hypergeometricfunction wherexFI(cx,s,al,y;-th~,-cth¥) fey) y=a+l=j+2-11 S=j+l+A a'=j+l-A(A.20) Thepolesmentioned abovearenowevident.AOurfunctions qandq appearinRef.3asd-functions [noconrlection toEq.(A.6)above]: = (A.21) = -80- APPENDIX B TOLLER M-FUNCTIONS Wepresent herethedefinition andsomebasicproperties ofthe ToIlerM-functions usedinSection (2).Ourconventions differsome­ whatfromTOller'sll andwillbepresented indetailelsewhere.16 Asnotedearlier, theM-function formalism applies·equally wellto 7thephysical orordered S-matrix connected parts. AToIlerM-function representing a2-to-3amplitude maybe defined asfollows: where(~rn5t ~m4t ~m3tI-[pslu(as)0[P4lu(a4)~[P3lu(a3) scIU(a1) 1i>:1)~u(a2) Ii>:2)), (B.I) 4 <5(ext) p. 1 p. 1= = =4 <5(p1.+P2 -P3-P4 -ps) L(a.)p. 1 1 (m.,0,0,0) , 1 andtheconstant A=-27Ticfisdiscussed atthestartofSection (4). TheL(a.)arethe4x4Lorentz matrices whichacton4-vectors, whereas 1 theU(a.)areunitary operators whichrepresent theelements (O,a.) 1 1 ofthePoincare groupinthesingle-particle Hilbert space. The statesI )andI ]aredefined anddiscussed inRef.13;basically theyarelinearcombinations oftheusualI)stateswhichare designed totransform asundotted anddottedspinor representations oftheLorentz group. InEq.(B.l)allhelicity (spinor) indices are -81- oftheundotted upper(contravariant) type.Generally therearefour kindsofspinorindices:mmx,x,x,andx.,which canberaisedm m (Gacts ontheleft)orlowered (Gactsontheright)byananti- symmetric metricspinor G1mm=mm'G = G••,mm==sd,(-1T)mm wheresisthespinoftheparticle involved. Exceptonafew occasions weuseonlythexmandxindextypes. However, inorderm toallowroomforexplicit spinlabelslikes ,wehaveadopted the following notation: = Thatis,upperindices arewritten aslower,andlowerindices are alsowritten aslower,butwithadotunderneath, thisdothavingno mrelation tothedotsofxandx..m Theonlyproperties oftheToIlerM-functions statedhereare theinvariance andcovariance conditions. Otherproperties suchas crossing, Tep,Reggization, etc.willbediscussed elsewhere.16 Thestatement ofLorentz invariance intermsofToIlerM-functions isverysimple: (B.2) Thisinvariance condition, inunediately evident fromthedefinition (B.I)sincetheoperators U(a)areunitary, statesthataToIler M- function transforms asaLorentz scalar. Theequation isthesame foralltypesofspinorindices aslongasbothsidesmatch. Inaddition totheaboveoverall invariance condition, the -82- ToIlerM-functions haveacovariance condition oneachparticle, e.g., (B.3) wheregisanyrotation. Thecovariance conditions arealsoobvious fromEq.(B.I),giventhatthereststatesfpm)transform inthesame wayunderrotations astheusualIp,m)states, whilethestates Ipm] transform asn*. ToIlerextendshiscovariance condition toincludeparity, rotations andparitycomprising thecomplete littlegroupH+ofa rest4-vector. Thismatterisdiscussed further inSection (3). Toverifythecounting ofvariables, onefindsforthegeneral n-point ToIleramplitude: nx6 -nx3 -6 -4 3n-10.eacha.=6variables 1 covariances invariance 04(ext) Finally, theToIlerM-functions arerelated tothemomentum- 13 12spaceM-functions (spinorial amplitudes) ofTaylor (Stapp) by M (a aaaa)mmIn Inml'2'3'4'51 2 3 4 5 (B.4) -83- (0S·) .1316whereD· ,1.arecertain spinorrepresentation functlons'of SL(2,C),andp. =L(q.)p.•111 ToIlerhasshownthatthen-pointfunction M(a...)ism••• analytic in[SL(2,C)x SL(2,C)]n ~[complex Lorentz group]n, theonly singularities beingreflections viap.=L(a.Jp. ~ofthepositive-a111 Landausingularities whicharetheonlysingularities inthep.of 1 theStappM-functions (e.g.,nonnalthresholds, poles,triangles); kinematic singularities andconstraints arenotpresent. However, \whentheMCa,...)areconfined tocertain surfaces withinm••• [SL(2,C)2]n, asbyusingthestandard framesofSection (5),these kinematic singularities reappear. Thisisobvious whenonerealizes, e.g.,thattheToller4-point function, whenwritten asafunction of geO(3),isanordinary helicity amplitude. -84- APPENDIXC THEHELICITYPOLEEXPANSIONFORMULA InRef.9wehavederived acertain "alternative" second-kind generalized-Legendre addition theorem andhaveproveditsconvergence. Thisformula, Eq.(2.11)ofRef.9,whenconverted fromthe~tothe qfunctions ofEq.(A.13), becomes ex> =-22: m-j=1C_l)m-j-mcxe x (C.1) Therelation between variables (~,z,;') and(Zl,ex,Z2) isgivenby Eqs.(2.8)and(2.9)ofRef.9.Wenowmakethefollowing setof changes onEq.(C.l): ~-+-ill ~'-+-iv Z-+ishf Z2-+-ishh1 11-+-m 11'-+-r (C.2) ex-+~ m-j-+n+l ""Z-+ch~ Taking j-+-j-lintheequation whichresults fromthesechanges leadsto ex> +2'"(_l)nre-2j+n)L....Jn! n=O whichconverges when Re(~)>O.However, fromthediscussion in Appendix EofRef.9itcanbeshownthat ~+-~iscompensated in -85- Eq.(C.3)by(1-1,v) -+-(-\.1,-v). Aftermakingthesechangesin(C.3), onemaythentake(m,r) +(-m,-r)togetanequation identical to(C.3) exceptthatm,rand~arereplaced bytheirnegatives ontheright side,andthisnewequation converges forRe(~)<o.Bothequations canbewritten simultaneously byintroducing anindexK, 00 n+2L:e(K~)L (-~~T(-2j+n) K=± n=O xe1t1U-n)q:j-l(ishf)q:j-1(-ishh)J-n,Km J-n,Kr (C.4) nowvalidfor _00<F;<00.Asthelaststep,theqfunction onthe leftsideof(C.4)isreplaced withitse-function equivalent [see Eq.(A.16)]sothat,upondefining Fj(f)n,Krn=_[2re-2j+n)n(-!1)nH~j_1]~q:j-1(ishf)J-n,KDl,CC.5) Eq.(C.4)becomes -j-1 '"-E(ll,F;,v)rnr =E-j-1(g) mr=(+i)m-rL:e(K~) K 00xL: n=OFj(f)n,KmeI~IU-n)Fj(-h)n,Kr(C.6) Setting j=cx,thenputting asubscript "2"onallvariables yields theresultquotedinEq.(6.1). FromEqs.(2.8),(2.9) ofRef.9and(C.2)above,therelation between theBCP(Bargmann) variables (ll'~,V) andtheFig.15variables (f,F;,h) maybefound: chF;=shfshh+chfehheh~ (C.7) +ille=-86- [shhchf+shfchhch~ +ichh sh~] ,.,." sh~(C.B) ivandanexpression foregivenbyh...finEq.(C.8). -87- APPENDIX D THRESHOLD KINEMATICS InSection (5)westudied theleft-andright-side loopequations ofFig.14inordertocompute theMisheloff rotation. Hereweexamine instead thelowerloopequation ofFig.14,namely = (D.I) Analysis of(D.l)inthemannerofAppendixE ofRef.9showsthat = = where,asfoundinSection (5),shv1sine1-­shQ1 shv1sin62-­shQl sine1 shQ 1 Therefore, ask1-+0, chf2 chh1= = = = = =k1 ~[n(sl,t1,t2)]2 1 1 2(-t1)~(-t2)~ 1 [ll(-k~ .-k~.p~)P 2k1k2 = =const. -1const. xk1(5.18) (5.21) (5.1) (5.19) -88- andask2-+0, 2 2 chf2(p1+k1)-1=k=const. xk22shq(-t)2k1 I 2 ( 2+k2) ehhlPI 1 =k=eonst. 2shql(-t2)2k1 Theselastfourequations areusedinSection (10)tofindthe threshold behavior ofthekernel. -89- APPENDIX E THECROSS-CHANNEL CONTINUATION Inordertocarrythekinematic structure ofFig.15fromthe multiperipheral regiontothephysical crosschannel wheret>0,one mustanalytically continue intheMandelstarn invariants toappropriate newvaluesandperform acomplex Lorentz transformation. Ourmain purpose indescribing thisprocedure istoshowthatthepeculiar group variables appearing inFig.15aresimplycontinuations ofthefamiliar variables onewouldusetodescribe thelarge-t Reggelimit. Consider, insteadofFig.15,thesingleladderrungshownin Fig.14.Th··'d .elnvarlants t.,t.,tan 1 1saredefined by 1 t'=(k')2 t'=(k')2 1 1 2 2 t=(k)2 t2=(k)2 1 1 2 P1=(k-k)=(k'- k' )2 1 .1 22s=P .1 1 Ourgoalistostartinthemultiperipheral regionoftheReggeon ,,process 1+2-+-1+2 ,where , Q,k.,k.=spacelike1 1 ~(t,t.,t~) =negative1 1 PI=futuretimelike central level=bwsframes and·windupinthecrosschannel physical regionfor2+2'-+1+l', where -90- Q,k.,k~=futuretimelike 1 1 li(t,t. ,t~)=positive 1 1 PI=spacelike central level=ernsframes Bws(erns)meansbrickwallsystem(centerof masssystem). Thefirststepistocontinue allthet's.Figure40showsa "movie" ofthiscontinuation, andTable1describes themovie. Thebranchpointdetours werechoseninthesamewayforallvariables. Whateffectdothesechanges haveontheequations ofSection (s)? Firstofall,Eqs.(5.1)become = = chql= whicharenowthecorrect BCPboostformulas fora2-tirnelike/l-space- likevertex. Moreinterestingly, Eqs.(5.3)and(5.5)become -91- cose22t cose22, Butnowcos(8t)>1whichimplies cos(Xl)>1,sowedefine22 togetCOS822, cosXI chn,22= = = =coshn22, COSh~l Fromtheseexpressions onerecognizes thatn22,=-i622tistheusual rapidity boostparameter connecting therestframesofthe(now) ""-incident particles k2andk~,andthat ~l=-iXIistheReggevariable ofthe0(2,1)link (~l'~I,VI) onthenowspacelike linePI(recall that ~1,Vlweresettozero). Tocomplete theabovedescription, wenowconstruct afigurelike Fig.14inwhichtheparameters n22t=-ie22,and ~l=-iX1appear explicitly asframe-connecting boosts. Theframesinthisnewfigure (whichwedonotdraw)formasortofshadowcabinet fortheframesof Fig.14inexactly thesenseofFig.21,exceptthatthes'of Fig.21isnowreplaced bythiscomplex Lorentz transformation T=Bz(-i;). Theoperator T,giveninthe4-vector spaceas 000-i 0100 T=0010 -i000 -92- turnstimelike vectors intospacelikeand viceversasothat,e.g., k2(inaneofitsspacelike restframes) becomes Tk2=(~,O,O,O) wherek2isthevectorappearing inthenewfigure. According totherulesofEq.(E.1),thebwsversormagnitudes k.ofEq~(5.7) become1 k.= 1,~[A(t,t.,t.) ]21·1 12(t)~ whicharenowtheinitial andfinalchannel cmsvectormagnitudes. The·z.ofEq.(5.13)isnowimaginary 1 z. 1=i(t+t.-t~)1 1 ~2(t)2iE. 1 asdesired, sincek2inoneofitsbwsframes becomes = = +whichisthenormal formofastandard erns4-vector, exceptk2points inthex-direction ins·teadofthez-direction [seeEqs.(5.10)through (5.12)l. Ourimaginary newfigurecanbecompleted verysimplybyexamining theactionofTontheLorentz generators (seeAppendix AofRef.9): sothat,e.g.,-93- G T-1GT Jx-+iKy Jy--+-iKx Jz.-+Jz Kx-+-iJy(E.2) Ky---.iJx Kz~Kz =-i822,[-iKx] -+e lending credence totheaboveremarks concerning thevariableseandX. Finally, consider thecentral levelboostparameters ~.andv. ~ 1 whichweresoimportant forthediagonalization ofSection (9). According to(E.2),thecombination whichsurrounds thecluster pinFig.15,becomes inthenew 1 picture where cl>=i~1 18=-iv1 1(E.3) aretheEulerangleswhichcharacterize thisrunginthephysical cross ""channel (81isthescattering angle,k1•k2=cos(81)),exceptthat,as already noted,theazimuthal rotations happentocomeoutbeing x-rotations instead ofz-rotations. Weconclude withashortcomment abouthelicity. Equations (E.3) showthatthecentral level~. boostsofFig.15arethecontinuations 1 -94- (totheimaginary axis)ofthecrosschannel azimuthal rotations of theprocess 2+2'~1+It•Therefore, thevariable conjugate to~l' namely lJ1,istheanalytic continuation ofthevariable conjugate to 4>1'whichhappens tobethechannel helicity m=rn1+m;.Thisjustifies h··f h· \··b1 1hI··· 36 ourcaracterlzatlon oteIIor1\varlaesascompex.elCltles. -95- APPENDIX F REATTACHMENT OFTHEEND-RUNGS Howdoesoneobtainfromthesolution ofEq.(10.8)thephysical discontinuity forparticles ratherthanReggeons? Onewayisto continue theReggeon discontinuity Ainthemasses, spins,andheli- cities~to thedesired physical points. Unfortunately, the4-Reggeon discontinuity Aappearing inEq.(10.8)isnotastandardized ToIler M-function (seeFig.29),sothatonemayconclude onlythatthe continuation ofAwillbeproportional tothephysical amplitude. Analternative andmoreconventional waytoobtainthephysical amplitude istoaddthe"end-rungs" backontothemultiperipheral ladder. Asthisinvolves thespecial end-rung kinematic configura- tionswhichwehaveomitted fromSection (5),wesimplystatethe answerwithafewcomments. Intheenergyplane,theend-rungs are reattached according to Aisthesumofallcontributions oftheform(10.1), Ka1(n1)isthe leftend-rung, andP(CP)istheconventional helicity propagatora1 associated withthe(a,at)channel, i.e., -i</>1(ma+m~)=e (F.2) Variable CPl·.sensesthechannelhelicity rn=m+rntofthetwo-particle aa system(a,a'). Thevariables n.appearing in(F.!)arelikethev.1 1 appearing in(10.1), butnotquitebecause theend-rungs arealways inamixed-basis configuration9whichcauses z=ch(v)tobetwisted intoz=ish(n). -96- Thediagonalized version of(F.1)is Tj t(a,b)mmPm(a)Kj(a,l)P (l)Aj t(1,2)mll II·llll xP,(2}Kj, (2,b)P,(b)II 1.1ID m(F..3) where P(a)m21T =1.d<p2n o= (F.4) InFig.26weschematize theprocedure forreattaching theend- rungstogetthephysical amplitude. Onceonelras solvedtheintegral equation (10.8)forAj Iandcomputes theTj,asin(F.3),the 1.111 mm absorptive partTt(s,t)intheenergyplanemaybefoundfromthemm usualinversion oftheJacob-Wick expansion. However, onemayreturn directly totheenergyplanewithout reattaching theend-rungs by rnyansofa~expansion formula whichisineffecttheinverse ofthe projection (9.4): d](2j+1) i [j]-1-j-1.[j ]X$l.1'l.1 ~-l.I,-l.1' (g) Pl.I(1)~l.1' (1,2)Pl.1' (2) (F.5) where sin1T(2j) sin1T(j-11')sin1T(j+lJ)(F.6) Thecontours in(F.5)runupvertically totherightofallsingulari- tiesoftheintegrand. However, thejcontour Ccontains, in addition tothisvertical piece,clockwise loopsaroundtheintegers andhalf-integers totheleftofthevertical component. Formula (F.5) -97- canbederived fromacompleteness relation [qdefined inEq. (A.13)] 6(x-y)=+ildj(2j+1)csc1T(2j) q~~(;) ~,,{y) C (X,y>1) (F.7) whichinturncanbederived bythetechniques ofRef.9,Appendix G.l. Finally,itshouldbenotedthattheprojection (9.4)isprecisely thecontinuation oftheusualReggetheoryFroissart-Gribov projection toimaginary andingeneralcomp~ex helicities. Formula (F.S)is(the discontinuity of)theMandelstam-Sommerfeld-Watson transform, the discrete-helicity version ofwhichwasusedtogetEq. (2.2)with(2.4). --98- ACKNOWLEDGMENTS IwouldliketothankEvelynGrant,Deberah Olson,CandyVoelker andDessaBucksbaum forinvaluable technical assistance inproducing thisreport; myparents fortheirconstant moralsupport, theU.S. Government andtheLawrence Berkeley Laboratory forpayingthebills, JaimeMillan, Jean-Pierre Sursock andDr.HenryStappforusefuland interesting discussions, Ms.Georgella Perryforherconstant surveil­ lance, andGeorgeWeissmann forcontinually reminding methatthere areotheraspects oflifeatleastasinteresting asparticle physics. Iwishalsotothanktheprofessors oftheBerkeley Physics Department rortheirexcellent teaching, inparticular Professors GeneCommins andJ.D.Jackson, andalsoMs.TeriDoizaki forbeingsonice. Finally, andmostimportantly, foranexhilarating educational experience Iamindebted beyondmeasuretoProfessor Geoffrey F.Chew whosecivility, sharpinsight, andinfinite patience Iwilltrymy besttoemulate. TheworkcitedinthisThesishasbeendoneundertheauspices oftheD.S.Department ofEnergy. -99- REFERENCES 1.A.H.Mueller andI.J.Muzinich, Ann.Phys.(N.Y.)57,20 (1970). 2.A.H.Mueller andI.J.Muzinich, Ann.Phys.(N.Y.)57,500 (1970). 3.M.Ciafaloni, C.DeTarand M.N.Misheloff,Phys.Rev. 188, 2522(1969). 4.M.Ciafaloni andC.DeTar,Phys. Rev.D1,2917 (1970). 5.M.Ciafaloni andH.J.Yesian, Phys.Rev.D2,2500(1970). 6.N.Mukunda, J.Math.Phys ..!i,2005(1973); J.Pasupathy andB.Radhakrishinan, Ann.Phys.(N.Y.)83, .186(1974). -7.G.F.Chew andC.Rosenzweig, Physics Reports (tobepublished). 8.H.P.Stapp,"Ordered S-Matrix Approach totheTopological Expansion forBaryons andMesons," LBL-6735 (August 16,1977). 9.P.Lucht, "TheGeneralized-Legendre Addition Theorems, SUe!,!), andtheDiagonalization ofConvolution Equations," LBL-S527 (October 11,1976). 10.C.Rosenzweig andG.F.Chew, Phys.Letters 58B,93(1975); G.F.ChewandC.Rosenzweig, Phys.Rev.D12,3907(1975). 11.M.ToIler, Rivista delNuovoCimento ~,403(1969); M.Taller, NuovoCimento 62A,341(1969); M.ToIler, NuovoCimento 53A,671(1968),Section 2·, M.ToIler, NuovoCimento 54A,295(1968),Section 10; c.Cosenza, A.Sciarrinoand IvI.ToIler, NuovoCimento 57A, 253(1968), Section 3. 12.H.P.Stapp, J.Math. Phys.~, 1548(1966); H.P.Stapp, inH.E.Phys. andElem.Part., ICTP,Trieste (196S),p.3; H.P.Stapp, Phys.Rev.125,2139(1962). -100- 13.J.R.Taylor, J.Math.'Phys.7,181(1966). 14.S.Mandelstam, Ann.Phys.(N.Y.)19,254(1962); W.Dreehsler, NuovoCirnento 53A,115(1968); C.Cronstrom, Ann.Phys.(N.Y.)~,340 (1973). 15.M.ToIler, NuovoCimento 54A,295(1968), equation (6.8). 16.P.Lueht,(inpreparation). 17.H.P.Stapp,Phys.Rev.128,1963(1962). 18.M.N.Misheloff, Phys.Rev.184,1732(1969). 19.R.J.Eden,P.V.Landshoff, D.I .OliveandJ.C.Polkinghorne, TheAnalytic S-Matrix (Cambridge University Press,1966). 20.N.F.Bali,G.F.ChewandA.Pignotti, Phys.Rev.163,1572(1967); N.F.Bali,G.F.ChewandA.Pignotti, Phys.Rev.Lett.19, 614(1967). 21.L.Serterio andM.ToIler, NuovoCirnento 33,413(1964),p.418. 22.G.F.Chew,M.L.Goldberger an'dF.E.Low,Phys.Rev.Lett. ~, 208(1969). 23.Ya.I.Azimov, Sov.J.Nuel.Phys.4,469(1967). 24.J.D.Jackson andG.E.Hite,Phys.Rev.169,1248(1968). 25.C.Rosenzweig andG.Veneziano, Phys.Letters 52B,335(1974). 26.J.R.Freeman andY.Zarmi,Nuel.Phys.B112,303(1976). 27.J.Finkelstein andJ.Koplik, Phys.Rev.D14, 1437(1976). 28.G.F.ChewandA.Pignotti,Phys. Rev.176,2112(1968). 29.J.R.Freeman, Y.ZarmiandG.Veneziano,Nucl. Phys.B210, 477(1977); J.R.Freeman andC.E.Jones,"Toward aGeneral Proofofthe PlanarPoleBootstrap," Nebraska preprint (tobepublished in Phys.Rev.D); J.R.Freeman andC.E.Jones,"Criteria forGoodFMSRforReggeon -101- Amplitudes," Nebraska preprint (tobepublished inPhys.Rev.D); J.R.Freeman, "TheCylinder...,1tNebraska preprint (submitted forpublication. 30.Ref.26,page320. 31.B.Nicolescu, "Evidence forBaryonium Exchange inMediumand HighEnergyScattering," tobepublished inNucl.Phys.B, alsopublished asLawrence Berkeley Laboratory ReportLBL-6701. 32.A.D.MartinandT.D.Spearrnan,Elernentary Particle Theory (North-Holland, Amsterdam, 1970). 33.M.Andrews andG.Gunson,J.Math.Phys. ~'1391(1964). 34.Ref.9,AppendixH.16. 35.Bateman Manuscript Project, A.Erdelyietal(McGraw-Hill, NewYork,1953),Vol.1. 36.P.Goddard andA.R.White,NuovoCimento lA,645(1971); C.E.Jones,F.E.LowandJ.E.Young,Phys.Rev.D4,2358(1971); c.E.DeTar, C.E.Jones, F.E.Low,J. H.WeisandJ.E.Young, Phys.Rev.Lett.26,675(1971); C.E.DeTar andJ.H.Weis, Phys.Rev.04,3141 (1971); C.E.Jones, F.E.Low,J.E.Young,Phys.Rev.D6, 640(1972); H.D.I.Abarbanel andA.Schwimmer, Phys.Rev.06,3018 (1972); J.H.Weis,Phys. Rev.D6,2823(1972). _.- ~ 37.R.C.Brower, C.E.DeTar andJ.H.l'ieis, Physics Reports 14C, 257(1974). .38.A.R.White,"TheAnalytic Foundations ofReggeTheory," CERN ReportTH.2136-CERN (February 16,1976); Proceedings ofthe LesHouches Institute ofTheoretical Physics (June1975). -102- 6 5......_..... XBL779·2260 Fig.1.Aparticle poletermcontained inthe6-point function.2 5 4 36 • 4 36 1 XBl779-2257 Fig.2.AReggepolecontribution tothe6-point function. -103- 4 3 XBl779·22612 Fig.3.Aparticle poletermcontained inthe4-point function . 32•-- 3XBl779·22622 Fig.4.AReggepolecontribution tothe4-point function. 3 245 • 234 XBl719·22565 Fig.5.Adouble-Regge contribution tothe5-point function. -104- 23 2 q 323 XBl779·2265 Fig.6.Thestandard TallervertexinnO-Regge, single-Regge, anddouble-Regge configurations. 2 Fig.7.Elastic unitarity; particle 6.XBl179·2253 X6istheMisheloff rotation for o 2-105- n-1 n \a XBL779-2266 Fig.8.MUlti-Regge production amplitude.b --II I I I•mla1r1m2a2r2 I I Po PI So SI Po Pl m1alr1m2a2r2 XBL779-2252 Fig.9.Spinandhelicity labeling forrnultiperipheral unitarity product. Uppervariables areprimedversions oflower variables. -106- v,o,qBz XBL779-2267 Fig.10.TheBCPframetriadforaproduction vertex. XBL779-2268k't'2'2 .-tv~~ c' '\(12 .....-Ao-.....-__--.....k2 't2b'-..Jd qlk't'l'1 Fig.11.Twovertices combined tomakeonerung. a'b'd' el k' k'1 2 )XIh,f=Bx 8 8221 8,x=Ry 11' C f2~ k1b dk2a e XBl719-2250 Fig.12.Fournewframesaddedtotherung. -107- a'bldte' k' k'1 2 \)=Bx9 X e=Ry k} k2ab deXBL779-2271 Fig.13.Central levelframes fandgaddedtotherung. k'2~~~.-......---.oI!,.X_------.JIl --.---4fOo-----_-....---- ......_ ~ t:~.-....-----Il}.F=--------::'I~-- ......-.....I¥--------.;~~---- .....--k2~~k'1 k 1 XBl779-2247 Fig.14.Complete 12-frarne systemdescribing onerung. •~o 00 Ix~,, 93~ ~Al)- ~~ q2 9392, 92•ql~ et 1 ~IIhI .VI~~r- kl~~ 91 ql9' 1~ E;X~ -S1 [x,e=Ry Vtf,h=Bx~=By q,v,o=Bz] Fig.15.MUltiperipheral ladderformedbycombining rungsofFig.14.Rungsareseparated by y-boosts ~i.Legendindicates meaning oflabeled parameters. XBL779-2246 -109- t XBL779·2251 Fig.16.Standard framesfortriple-Regge vertexinits spacelike configuration, ~(t,tl,t;) <o. -110- ](',}-[-a'-l]*(a~S2a'3)*-[al51<X2 - V r,p'rn'y'E;'E2 , -Vr~p~rn; 2 2rn'r'(g2)112 2 2 ISI51d2dp'p(Xl) p'p(X2) 1 1 2 2 I)-[I}-[]_(ValSla2-{X-1]_( a2s2a3 y2~2Ern2r(&2) Vr prnr1P1rn2 22 223 XBL779·2270 Fig.17.Functional structure ofthernultiperipheral ladder,after Mande15tarn trick. ---(,-r (i)1*)-- )---- Fig.18.WhenE-functions ofFig.17arehelicity-pole expanded, residual functions Faregroupedtotherungstoform kernels. ThisisthekernelK12. XBl779·2248 -111- a'n'2'2 P21~2'[(a2-n2)+(a;-n~)] e a2,n2 XBl779-2269 Fig.19.Thehelicity-pole propagator. (a) (b) --- Fig.20.Thekernel. XBl779-2249 --112- ..,--- I IIxs'xI I~I-------.-_._-- XBl779·2264 Fig.21.Dottedfigureshowssegment ofmUltiperipheral ladder inparity-inverted world. Inthatworld,framesshown areconnected byt;'=-E;. ~=B ]y XBl779·2254 Fig.22.The3-particleor3-cluster contribution tothe 4-Reggeon amplitud~. -113- 9 -- 11j11' j jA'j11' XB1779·2259 Fig.23.Functions B,C,Dareconvoluted togivefunction A. Thevariables onthebottomlineareconjugate tothe boostparameters asshown. Thediagonal variable jis angular momentum, variables IIandAarehelicities. K(1,2)P{2)K(2,3)P(3)K(3,4) 11j ~'1.1jAjA'j, 1.1 Fig.24.The3-particle/cluster contribution tothe4-Reggeon amplitude, inbothenergy-plane andj-plane. XBl779·2258 -114- A(1,3) K(l,3) K(l,2)P(2)A(2,3) IIjll' IIjlJ' 11jA jll' ~....+- ~\) ~tE;\)~t ~lV~2"'2~;1 Fig.25.Thebootstrap equation inbothenergy- andj-planes. XBL779-2263 T(a,b) P(a)K(a,1)P(l)A(1,2)P(2)K(2,b)P{b) mjrn' rnj l..ljll'j a' a' --- a b a b <P <P' ep1n1~2\)~3n3ep'2 3 Fig.26.Thereattachment oftheend-rungs (seeAppendix F). XBL779-2255 -115- a'n' a'n' a'n' o:'n'1 1 2 2 1 1 2 2 -L- Cl aIn1a2n2 XBL779-2312Fig.27.TheRegge-pole expansion oftheunnormalized 4-Reggeon ringdiscontinuity. lJaAa G--- Kp G Fig.28.Thevertexbootstrap. XBL779-2313 -116- -- Fig.29.Relation between discontinuity AandtheToller­ normalized 4-Reggeon ringdiscontinuity. The R-notation isthatofRef.? XBL779-2318 Fig.30.Relation between thecutvertexGandtheToller- normalized ringfunction R. XBL779-2319 -117- XBL779·2316 Fig.31.Highlyschematic bootstrap forthe 3-Reggeon ordered amplitude. - XBL779·2317 Fig.32.Vertexbootstrap with"dotted Reggeon" replacing thesingleproduced particle ofFig.31. (0)'-118- J---- ~---[ (c) (d) (e).\ ... e.e. :.......-.-..,'. .~--4-RJ-,I ~........tII••,........~ Fig.33.Thetwotwist-pair contribution tothecylinder invarious notations (seetext). XBL779-2314 I ...... ....... (.!) I 91 j-etk1 j-etk1 j-a9 Fig.34.Thediagonalized twotWist-pair cylinder termnearj=et. XBL779-2320 -120- A A 9y------ ....... ......~....Io--- ....................... ..... Xl Fig.35.TheC(l)cylinder termintherapidity model. XBl779-2321 A(l~2)xp(2)A(2,3) (l)C(1.3) 1Jj 1-1' -- F; \>~'j j Fig.36.TheCCI)cylinder termwithexactkinematics. XBl779-2322 xxxhX xX -j-l-121- j+l xj+2 xxxx XBL779..2315 Fig.37.Thecomplexhelicity planeforEq.(11.26)or(11.27). Ifhelicity polehisintherighthalf"'plane, contour shouldbedeformed totheright. Theotherpolesmay ormaynotbepresent depending onnonsense zeros. -122- a,a' C(1) k XBL779·2323 Fig.38.C(l)maybeexpressed asadoubleReggesuminvolving thecomplete twisted Reggeon loopk.Thecross-hatch indicates thattheloophasbeensummedoverall helicity polesofallpossibl~ Reggeon pairs. Fig.39.Quarklinestructure oftwisted Reggeon loopcoupling rneson tobaryonium. SinceloopReggeons areferrnions, f-factors inpropagator aresettoone-half. XBL779-2324 t'1 t ~<O e Bt1 t'1 Bt1I ~ N ~. I XBt779-2311 Fig.40.Fiveframesofamovieshowing thekinematic continuation of asegmentofthemultiperipheral ladderfromthemultiperipheral regionAwheret<0tothecrosschannel physical regionDwhere t>o.Heavylineisequation 8(t,t1,t;)=o.Framesofmovieare described inTable1(nextpage). -124- TABLE1. A(-t)~ (-t')~ -l~(t,t1't~)~(-t)~ (+s)~ 1 1 1 B(-t)~ (-t')~ +i(+l1)~ (-t)~ (+s)~ 1 1 1 B (-tl)~ (-t')~ +i(+l1)~ +i(t)~ (+Sl)~ 1 C(-t)~ +i(t')~ +i(+l1)~ +i(t)~ (+s)~ 1 11 +i(tl)~ +i(t~)~1 +i(t)~ (+Sl)~D+i(+l1}~ +i(tl)~1 +i(+l1)~ +i(t)~ -i(-s )~D +i(t~)'~1