Ring

LBL-6199 MULTIPERIPHERAL RING DYNAMICS AND ADEFINITION OF THE COMPLETE TWISTED REGGEON LOOP Philip Harrison Lucht Lawrence Berkeley Laboratory University ofCalifornia Berkeley, California 94720 Ph.D. Thesis November 1977 N\ MULTIPERIPHERAL RING DYNAMICS AND ADEFINITION OF THE COMPLETE TWISTED REGGEON LOOP TABLE OF CONTENTS Abstract 0 ee eeeee eee 0 (1)Introduction. 66ee eee eee 2 (2) Multi-Regge Production Amplitudes... +--+ -.----- 5 (3) The Vertex: Helicity and Parity Conditions ......... UW 1. Helicity Conservation .2... -eee eee eee ee U 2. Parity Invariance...-2-2eeeeeeeeeeeee12 3. Parity with Reggeons .. 2... eee eee ee eee 1B4.Caveats, andtheVertexV .2... --.-+e +--+ 16 (4)TheUnitarity Product... 2... eeeeeeeeeeee WT 2.TheRung... eee ee ee eeeeee223.TheCentral LevelFrames... ..2+++++++ees244.ManyRungs 2eeee ee eeeeee7 (6) The Helicity Pole Expansion»... 2. eee eee ee ee 29 (7) Naturality Condition for the Kernel... -+--+ ++ +++ 34 (8) The Multiperipheral Chain and Phase Space ........- 37 (9) The Diagonalization ofAngular Momentum ......... +41 (10)ThePlanar Bootstrap... .eeeeeeeeeeeeeeeeMM 1. Form ofthe Integral Equation .... +--+ --- +++ 44 2.The Projected Helicity Pole Propagator P, ....... 46 3.The Projected Kernel and ItsThreshold Behavior -.~~ 47 4.The Naturality Diagonalization- -+--+-+--+++-+ 505.TheBootstrap Problem -+--+ +++ee+eeeess546.Counting. 2... ee ee ee eeeeee5S (11)TheCylinder... ee eeeee eeeee 86 1. Diagonalization ofthe Charge Conjugation ...... 58 2.TheCylinder inRapidity. .2.2... eee eee 59 3. The One-Twist Term ofthe Cylinder asaHelicity PoleExpansion. 2... eeeeeeeeeeeeee63 4. Angular Momentum Versus Helicity. .......... 65 5. Regge Cuts and Nonsense Zeros .....-.+-+-++ 66 6. The Complete Twisted Reggeon Loop ....-+..+-++ 67 7.TheFullCylinder .2... eeeee eee eeeee 70 (12) Fixed Poles, Nonsense Zeros, and the Helicity ContourProblem... eeeee ee eeeeeeeeee TI Appendix A:Some Useful Functions .....-.+-+--+++-++ 76 Appendix B:Toller M-Functions .. 1... ee ee eee eee 80 Appendix C:The Helicity Pole Expansion Formula ........ 84 Appendix D:Threshold Kinematics... .-. 2. ee eee es 87 Appendix E:The Cross-Channel Continuation. .......... 89 Appendix F:Reattachment ofthe End-Rungs ....-.----. 95 Acknowledgments. 6... 0cee eee eeee 8B References... ee ee ee Table ee ee 128 <1. ‘ABSTRACT Thet<O multiperipheral formalism ofCiafaloni, DeTar, Misheloff, Mueller, Muzinich andYesian isreviewed, extended, and applied totheordered S-matrix whose ring amplitudes comprise the zeroth level ofthe topological expansion. Toller M-function notation isused throughout. Thebootstrap andcylinder problems areformulated interms ofawell-defined helicity pole propagator; adefinition ofthecomplete twisted Reggeon loop, which appears intheone-twist term ofthecylinder, isgiven asahelicity pole expansion. Some consideration isgiven tothefollowing subjects: diagonalization, naturality, threshold behavior, Regge cuts, and complex helicity. 22. (1) INTRODUCTION During theyear 1969-1970, after aperiod ofvigorous activity inthe field ofmultiperipheral dynamics, Ciafaloni, DeTar, Misheloff, Mueller, Muzinich andYesian presented, infive heavily overlapping Papers, theexact kinematic analysis ofthemultiperipheral mode1.1~5 These papers were, inouropinion, extremely complicated inpart due tothenature ofthe subject, and inpart due tothe fact that they incorporated mathematical ideas which were simultaneously being invented bythemathematicians, notably Mukunda.© Possibly, the relative obscurity ofthese papers has discouraged people from attempting anexact multiperipheral calculation, leading them instead torely upon the approximate Mellin analysis andthereby torelinquish the capability ofhandling the true angular momentum which iscentral toRegge physics. Since the invention ofthe S-matrix topological expansion in 1973-74 byVeneziano, there has been some renewed interest inmulti- peripheral calculations, inparticular asthey pertain toplanar amplitudes. Inarecent review,” ChewandRosenzweig havepartially reformulated these planar ideas interms ofthe so-called Ordered S-Matrix, the connected parts ofwhich are called ring functions. Although the concept ofordered ring amplitudes has not yet been convincingly extended tothebaryonic sector, itseems likely that efforts nowinprogress will soon succeed.® Inthis paper wehave attempted toreview, elaborate upon, and consolidate the ideas ofCiafaloni etal, and toadapt these ideas to the ordered S-matrix framework. -3- Areader familiar with the above-mentioned multiperipheral papers would find, upon comparison ofour descriptions with theirs, many differences inpresentation, some ofwhich wenow enumerate. First ofall, wefeel wehave greatly simplified the group-theoretic aspect ofthe multiperipheral analysis byidentifying, asthe agent which performs the diagonalization ofthe multiperipheral equations, an alnost trivial addition theorem involving thesameLegendre Q,-type functions which appear inthe Froissart-Gribov projection ofRegge theory. These Legendre functions are generalized inthat they carry complex helicity indices whose role we continually stress. The reader is referred to Ref. 9for an extensive discussion of this group theoretic business. Another difference one will notice isour attempt toisolate and identify anobject called the helicity pole propagator which connects cluster discontinuities along the multiperipheral chain. Strangely enough, this propagator owes its existence toafactorization condition which results from the same Legendre addition theorem mentioned above. Obviously spin isanimportant concept inamultiperipheral analysis which purports tocompute Regge trajectories. Wehave attempted toinclude spin infull generality (i.e., onexternal particles aswell asinternal poles) bymaking use ofthe Toller M-function formalism. Toour knowledge, noone has written unitarity equations inthis formalism which seems sowell suited tothe presen- tation ofmultiperipheral kinematics. Interlaced with the discussion onthe following pages one will find asort ofrunning commentary onparity and naturality, leading toanaturality diagonalization ofthe planar bootstrap which is, -4- wefeel, animprovement onthe original discussion byCiafaloni and Yesian. Generally speaking, the exact kinematic analysis allows one to think about things which simply donot exist intherapidity framework which more orless ignores helicity. Wehave extracted the threshold behavior ofthering functions andhave made astart atexamining the Ronsense zeros which are presumed toremove Regge cuts. InSection (5) wedescribe inarather different manner than that ofRefs. 2and 4the construction ofthe standard frames ofthe multiperipheral ladder. Bycontinuing the ladder kinematics tothe center-of-mass cross channel, weshow inAppendix (E)how thepeculiar boost parameters which link the standard frames are the continuations ofvariables familiar from center-of-mass kinematics. ‘The "planar" bootstrap and cylinder problems areboth setup— the cylinder inmore detail because itlacks the counting problem —but nodetailed calculation isattempted because weare stymied byaproblem involving the correct method ofshifting the helicity contour. Me have isolated this problem inthe last section ofthepaper; itmust besolved before the machinery described herein can beput towork. Nevertheless, wedoobtain anexact formal expression for the complete twisted Reggeon loop k(t) which controls the cylinder shifts oftheplanar trajectories inthephenomenology ofChewandRosenzweig? For adetailed outline ofthe paper werefer tothe Table of Contents preceding this Introduction. Ingeneral, the first eight sections describe the multiperipheral construction, Section (9)gives the angular momentum diagonalization, and Sections (10) and (11) apply the analysis tothe bootstrap and cylindér problems. os. (2) MULTI-REGGE PRODUCTION AMPLITUDES Tomotivate the specific form weuse for the multi-particle production amplitudes, weappeal tothe notion ofaparticle pole in the S-matrix. Figure 1shows aparticle pole term known tobepresent inthe 6-point function (repeated indices are implicitly summed), Mp,m,m,mmgm, (@1*82°%37840%5+8—) Daas) ; Mammy,(27785794786) |ee [nmin (Oa7%2787 780)»2. This pole has aresidue which factorizes into two pieces, each piece being a4-point function normalized inthe same way asthe original 6-point function. Each xinFig. 1marks aparticular standard rest frame for the particle onwhose line the *appears. (When the pole isreggeized below, some x's must denote spacelike rest frames.) The notation is approximately thatofToller:'! thes;arethespinsofvarious particles, m,arehelicities (component ofspin along thez-axis in thestandard frane marked byanx). Thesame symbols s;andm;are also used todenote certain Mandelstam invariants and masses of particles; the usage should beclear from the context. The meaning ofadot under ahelicity index isexplained inAppendix (B). Thea,appearing inEq.(2-1) andFig. 1are, foreach particle, theparameters ofa(possibly complex) Lorentz transformation which connects the particle standard rest frame toanarbitrary "lab" frame asindicated inthefigure. Thevariable gappearing inDyas(6) denotes therotation g=a;'a) ;thestandard D-function [see Appendix (A)] isgenerated bycovariation from theM-function on -6- the left according tothe simple rule given inEq. (B.3). s,is the spin oftheparticle pole, andm,,m! are thehelicities ofthat particle intwo different reference frames. Itisperhaps worth noting that, although they carry spin and helicity indices, the M-functions appearing inEq. (2.1) are Lorentz scalars, unlike themomentum space M-functions ofStapp andTaylor.!> Secondly, wehave been careful toproperly order the particles consistently around the connected parts sothat all our equations apply equally well tothe ordered amplitudes (ring functions) inthe ordered S-matrix framework associated withthetopological expansion.” Thefactor (s,-m,+ie)"! in(2.1) isofcourse theactual pole; the numerical constant ¢isdiscussed below inSection (4), and can bearranged toequal unity. Equation (2.1) is, for the pole term, anexact statement. We now assume that this particle pole isinfact one ofmany poles which occur onaRegge trajectory a,. Thecontribution ofa,tothe6-point function shown inFig. 1should begiven bythe above expression with s,continued toa,andwith thevarious group arguments andinvariants continued sothat the equation isinauseful Regge region. Accounting forsignature, theusual Regge machinery” maybeimplemented togive “Regge theory forn-point fynctions withn>4ismuchmorecomplicatedthan wemake itsound.2/»38 Rigorously,28 both thephysical and ordered S-matrix n-point functions must bedecomposed into asum of"spectral components" bymeansofan(n~-3)-variable dispersionrelation (Bargmann-Weil). Each spectral term contains only Steinmann- allowed multiple discontinuities, afact which implies the existence ofaLehmann ellipse ofconvergence for each 2;variable inan appropriate physical cross channel (hexagraph). Asaresult, the infinite angular momentum and helicity sums are convergent atleast somewhere, and this allows the Sommerfeld-Watson continuations tobe defined. So, rigorously one does aRegge analysis oneach spectral component and then adds the results, orone sticks with asingle component and diagonalizes unitarity onto the spectral components. We feel that the form ofour results will bethe same ineither the rigorous Regge theory orRef. 38, orthe naive Regge theory presented in Section (2). -7- the following result (see Fig. 2): Sye+Se. yltyS35455 OT,ySySo0ySyMeese ©MESES Utacconigtty MEE (2.2) where wehave suppressed thea;arguments, andwhere [factor}&t, =eviMenim ocaryham a m+e go ahez)+1-1)"*©a®(2) x)memEE (2.3) 2sint (a+m) The variables g=($, z=cos, $') which appear inFig. 1are now 0(2,1) variables ($,2=coshE, $'). The signature ofReggeon a,is 1,(aspin-¥s particle haspositive signature), and¢=0 or4depending onwhether a,isaboson orfermion trajectory. Using Eq. (A.8) one may show bytaking a,+s, that the Regge form (2.2) duplicates the particle pole term ofEq. (2.1). The final step inobtaining the Regge form weshall use isto perform a"Mandelstam trick" operation!’ which causes thefirst-kind functions in[factor],,, tobereplaced bysecond-kind functions which have simpler asymptotic behavior. Performing this operation we find or -a-1[factor]ig: =Y&Fame (8)> (2.4) where -t-1¢0) 2 enim -a-1 -im'g!Em! (8) e Cm! (2) 2 (2.5) y =-ca'tann(a-e) , (2.6) and _fei™@-9 4, - &= |rsmre- 9J- (2.7) 8 Thefunction esos(2)»defined inEq.(A.10),hastheexpected Regge behavior 2"forlarge z,—isastandard signature factor, and Ycontains the leftover factors. Inparticular, Ycontains a'and therefore hasdimensions E?, InEq.(2.1) these dimensions are generated bythe pole itself. Realizing that the n-point Toller M-function hasdimensions E*~", onemayverify thedimensional correctness of (2.1) or (2.2). The Regge residues appearing in(2.2) are three-particle/one- Reggeon amplitudes normalized inthe correct way soastobecome physical four-particle amplitudes when the Reggeon istaken tothe appropriate value ofmass and spin (and signature, ifMisnot an ordered amplitude). Since the physical helicity amplitudes must vanish when the helicity isout ofrange (has anonsense value), the residues must contain factors toknock out the unphysical poles, since this ghost-killing function isnotbeing performed by[factor] .,- For example, onemight take* 05535455 Dot. ry Majnm.m, 17s [r(a,+1+m,)r(a, +1-,))?. (2.8) Sofar wehave considered the Reggeization ofasingle-pole term inunitarity. Had westarted with the appropriate multiple pole term, wecould have obtained amultiple Regge residue orfour-Reggeon amplitude which, were all Reggeons continued toparticle points, would benormalized soastoyield aphysical four-particle helicity amplitude. Wefeel that this isauseful way tonormalize Regge “In the sense-nonsense region, additional square-root zeros are provided bythe d-functions, See, for example, Fig. 8eofRef. 9. -9- residues, and isultimately necessary ifone attempts acomplete bootstrap of, say, the triple-Regge vertex. Weshall mention this later inSection (10) (but will not attempt such abootstrap). Although three-particle scattering amplitudes have noplace in astable-particle S-matrix theory, unstable particles may becalled upon togive meaning tothefollowing equations. Figure 3shows a particle pole term inthe4-point function. Inanalogy to(2.1) we write Memate tts) De,(e) MeostetCasoay28,)[9] Meera -(2.9) Reggeization inthesame wayasbefore yields this expression forthe Regge pole term shown inFig. 4: Again, the Regge residues (pieces ofthe factorizing residue ofthe Regge pole intheFroissart-Gribov projection) appearing inEq. (2.10) arenormalized sothat, asa,+s,,these two-particle/one-Reggeon amplitudes approach thestandard three-particle Toller M-functions appearing inEq. (2.9). Thehelicity nonsense-zero structure of these standardized "Regge couplings" ispresumably similar to(2.8) above. Notice from therule E* that these Regge couplings have the dimensions of energy. Byreggeizing adouble pole unitarity term, onemay obtain the following Regge contribution forthetwo-to-three production amplitude shown inFig. 5: ~-10- sy. tigWs «KE fr,Eo] -a,-1 eS yd7The object M isthe two-Reggeon/one-particle amplitude which continues totheToller 3-point function when a,+s, anda,>s,. Again, this "double Regge vertex" has dimensions ofenergy, asdoes the triple Regge vertex which wehave not shown. These vertices differ considerably from thephenomenological Regge couplings (dimensionless) andtriple-Regge couplings (GeV™*). The form ofthe general multi-Regge production amplitude should beclear from Eq. (2.11). Each Reggeon gets abracketed "propagator" factor with linking helicity sums onboth sides. All vertices are standard Toller 3-point functions continued inthe appropriate way. Weconclude this section byobserving that, inthe ordered S-matrix framework where the N-functions in(2.11) are replaced by ordered ring amplitudes, the multi-Regge-pole expansion should be, intheperipheral region, avery good approximation since there are (presumably) noRegge cuts inthering functions. Thetheoretical accuracy of(2.11), when summed ona,anda,,isthus limited only by peripherality andtheconvergence rate oftheRegge asymptotic series, ie., duality. -11- (3) THE VERTEX: HELICITY AND PARITY CONDITIONS WehavebeenwritingthetriplevertexintheformMam,m,(od) tostress the fact that the vertex islike any other n-point Toller M-function. Aswenow show, however, this notation isextremely redundant. Using the freedom allowed bythe Toller invariance condi- tion [see Eq.(B.3)], one can choose tosuperpose the external "lab" reference frame —with respect towhich thevarious a;aredefined, asinFig. 1—onto one ofthe standard reference frames associated with thevertex. Since thevertex standard frames areconnected by certain z-boosts 0,0,, and qwhich are functions only ofthe invariants entering the vertex [see Eq. (5.1)], one may conclude that M,nm,(4198;085) isitself afunction onlyofthese invariants. This mM, situation isillustrated inFig. 6awhere wehave placed the reference frame onto thestandard frame ofparticle 1togetM (e,q7?,077),m,m,m, 1 where eisthe identity transformation. 1, Helicity Conservation Consider now this series ofoperations inwhich represents the zrotation R,(¢): (a,,a,,a,) : m1 ga=Mamam,(Or1s95") et=Mmm, O°»a7",605") =e go .(eo,a7",05°6) Ma mm, : = eime imo ,-im,o mgae e e Mam. (24's05)12M =eid(m,+m, +m)=e Mammy (81>%29%) + (3.1) -12- Inline 1the reference frame isidentified with the rest frame of particle 1,asalready noted. Inline 2the$'saremade toappear via the invariance condition ofEq. (B.2). Inline 3these rotations arecommuted through thezboosts q™’=B,(-q) andoj",andthenin line 4the $'s areseparately covariated tothe right according to Eq.(B.3). Comparison ofthelast line with thefirst then shows that mtmtm =0, (3.2) ie., helicity isconserved atthevertex.” One does not find such acondition for the higher n-point functions because the rotation $does not commute through all the a,nomatter howthey arechosen. 2. Parity Invariance Ifparity isaninvariance ofthetheory, wemayuseanargument similar tothat ofSection 3.1tostate parity invariance interms of thevertex. Since theparity operation, which Toller!® calls s,is anelement ofthelittle group H,ofthe4-vector (*,0,0,0), therotational covariance conditions [shown inEq. (B.3)] maybeextended toread, e.g., sll, (3.3) where Il,istheintrinsic parity ofparticle 3.Since theoperation s fails tocommute through the zboosts [see Eq.(7.6)], itismore convenient touseToller's parity operator s'defined by st=RCs (3.4) equations ofthis section. -13- forwhich the covariance condition onparticle 2reads, according toEqs. (3.3) and (A.8), Maymgm,(219829858 d=[nen Ma,»-m,(@1°82>83) »(3.5) Operator s' brings out the intrinsic parity and negates the helicity oftheaffected particle. Since s'doee commute with the zboosts, one may repeat the argument (3.1) tofind this vertex parity condition: 3 $4-m) am Mamom(@27t2783)=Mom, (8208)Tec» ):(3.6) Asacorollary toEq. (3.6) one has either Ten )zl, orMoco(81282083) =0.(3.7) For example, ifs,=s,=0 then all three helicities must vanish and oneconcludes from theabove that thevertex vanishes ifIIIT, # s (-1)*,asonewould expect fromamoreconventional angular momentum argument. 3. Parity with Reggeons Another convenient property ofthe parity operator s' isthat s',unlike s,belongs also tothe little group H_ofthespacelike Test vector (0,0,0,*), asdoes R,($). ‘Therefore, ifoneormore of the particles atavertex isreplaced byaReggeon —which may be spacelike sothat H_isthe appropriate little group —one shall find that the helicity and parity conditions still exist. The helicity conservation condition ofEq. (3.2) isunchanged, except asnoted below. The Reggeon parity covariance condition is 51520 -imm] ss,Me agsas') =[o, a (a,,a,.4,) (3.8) mm,mym, (21>2>45 3 In,m,,-m,(91>8293. o14- where ; o,=the t (3.9)i i with s,aphysical point ona,,andIl,theintrinsic parity ofthat physical point. Thequantity 0;appearing in(3.9) iswhat weshall call the Reggeon naturality, and isaconstant along anordered Regge trajectory. Another way tosay this isthat the exchange degenerate partners which together compose anordered (planar) Regge trajectory have the same naturality o,,even though theintrinsic parity Il,andspinparity (-1)*4atterate atthephysical points. Oneseesthat,asa,>55, Eq. (3.8) reproduces (3.5). Forfermions, thephysical point parities Il,andnaturalities ©,are, according toEq. (3.9), outofphase by90°, IntheM-function formalism onecanprove!® fromcrossing andTCPthat hm =ent (3.10)ra Apurist, allowing for the possible existence ofself-conjugate fermions, would have toaccept imaginary parities for those fermions. Asemphasized byStapp!” themostreasonable convention istogive all fermions imaginary intrinsic parities. (Toller too uses this convention.!1) Inthiscase, naturality o=+1forfermions aswell as bosons. Weleave tothe reader acomparison ofEq. (3.9) with the more common definition ofnaturality 54-4 n= PG * (3.11) where intrinsic parity P,=+1forboth bosons andfermions, and €,+0forbosons andone-half forfermions. Certainly forbosons, ny=oy -15- once Eq. (3.8) has been established, the argument ofEq. (3.1) may beapplied togive aparity condition for the single Regge vertex shown inFig. 6b: 545,01 5; 8; @,Ma aay =M2? ae, ime,t28)=Mg!onyBase) tan 5,-m, s,-m, -imray 27M «ben ]fen] Lo, - This says, e.g., that two pions cannot couple toann trajectory, even though such acoupling isallowed byG-parity. For the two-Reggon/one-particle vertex acondition similar to Eq. (3.12) results (see Fig. 6c): aja,s -im, -im, sj-m1%5s .1 2 ar) BPeaeeg =Le, ]fo,oe]far] ms a, a, 5 xMe8(aap). (3.13 =m,-m,-m,(91742943 ) The triple-Regge vertex ismore complicated because one cannot always link the three standard frames with 2-boosts. Inparticular, when A(t,,t,.t,) isnegative, thethree frames areconnected by y-rotations’® (seeFig.16). Conveniently, theparity operator s' also commutes with y-rotations; the parity argument then goes through toyield OOO, -im, -im, -im. 123 1 2 s MMs 1,G2Os Mp omyem, 829%)» (3.14) sothat negating the Reggeon helicities isequivalent, for4>0, tomultiplying bytheproduct oftheReggeon naturalities, since thehelicities cancel byEq. (3.2). However, since y-and z-rotations -16- donot commute, the helicity condition ofEq. (3.2) isbroken for the spacelike triple-Regge vertex, A<0. 4. Caveats, and the Vertex V Wemust nowaddtwoimportant qualifications tothepreceding equations ofthis section. Aswritten, they apply toToller 3-point M-functions with allparticles andReggeons being intheinitial "state" andwith allspinor indices oftheundotted upper type (see Appendix B). Tobeconsistent, certain particles andReggeons must beput into the final "state" ofeach vertex. Wechoose toletthis convention bedetermined bythedirection ofthearrows in,say, Fig. 15. Whenever aparticle orReggeon isinthe final state, the relevant bracketed factor inEqs. (3.5)+(3.8)and(3.12)+(3.14) mustbecomplex- conjugated. Secondly, wemust face the fact that inevitably some ofthe helicity indices wearedealing withareoftheundotted lower type. These indices, marked underneath bydots asinEq.(2.1), arenecessarily lower inorder topreserve the spinor covariance ofthe equations. When anamplitude with alower undotted helicity index iscovariated asinEq. (B.3), the Dfunction must bereplaced byD*. Thenet result isthat helicities inEq. (3.2) corresponding tolowered indices will enter with minus signs. However, theparity conditions are the same, regardless ofwh=ther indices areupper orlower. ‘The vertices inwhich weare mainly interested have the form ofthecentral vertex ofEq. (2.11). Inthenotation ofAppendix B andwith the conventions made above (and, asalways, maintaining the cyclic ring ordering) wewrite this vertex as -17- e840, om, *m, a,5,0, : =M, ‘(@,: aba) =Ve . (3.15) ‘The helicity and parity conditions for this vertex are then found from Eqs. (3.2) and (3.13) and the above conventions : momo +mos 0, (3.16) 4,542 -imn!]* sy-m]* -im,] a,5,0We. foe‘|[ren |[o,¢"viene : nim,m, -m}-m,-m, (3.17) Once again itshould bestressed that this vertex Vhas the standard normalization ofaToller M-function, has dimensions of energy, and (in addition tothe labels shown) isafunction only of the invariants entering the vertex. (4) THE UNITARITY PRODUCT Evenwhen allparticles carry spin,!> theunitarity equations for the monentum-space M-functions are completely characterized by theusual bubble diagrams!? together with asetof"Olive's rules," internal line =-2mic 6°(p? -m°) independent loop =d'p/(-2micf) pole =c/(s-m'+ie) . One needs also the relation between the M-function bubbles and the raw connected parts: (S.)=(-2nicf) 6*ext) MO (St)=(-amic)® 6°(ext)MO Inthese relations, the constant £determines the normalization of the single-particle states, -18- ia) e a oe(pim[p'm') =2E66°(P-P'6, gr> and cgives the pole residue, asinEq. (2.1). Authors naturally differ intheir conventions, e.g., Exop:1° c= f=(2n)> stapp:!? cri f=(2n)> taylor:33 =c=-1 f-=1/2. Wefavor the convention ofELOP, but shall always give results in terms of cand f. Once aunitarity equation isexpressed interms ofthe Stapp- TaylorM-functions Mp...(Py,P)-+++)» itmaybeconverted toToller M-functions via the inverse ofEq. (B.4). Details ofthis conversion process with attention paid tothe spinor indices will begiven elsewhere. 1 Before tackling the general multiperipheral unitarity product, we first illustrate the form unitarity takes interms ofthe Toller M-functions bywriting down elastic unitarity assketched inFig. 7. The formula is HyaiseM, (a,,a,,a,,a,)a ampiigih, (1782289285 =vetf4Q,CM,mmm,(297207%6-85)] Ss Se Xdyrs) dygr(Xe) tnt(81992945586) > mm‘smeme6:Mammim! eens 6 (4.1) where 5 + ‘ a'p,8*(pe-me) ap,8(pe-m) 4Q,=6°,+p,-p,-P,) —-—+— —*— 4. 7 1*P27Ps"Pe 73 ? -19- Asusual, wearemaintaining thering function ordering conventions.” Thedots over them,andm,indices onthe left side ofEq. (4.1) are necessary tomaintain the spinor covariance. The rotation functions arise inthe same way asthe D(g) inEq. (2.1), namely, from the Toller covariance condition shown inEq. (B.3). Weare anticipating asystem ofstandard reference frames tobereviewed shortly inwhich itwill turn outthat these rotations will bepure y-rotations, X;, whose presence wasfirst noticed byMisheloff.* Att=0 therotations all vanish, but for t<0 they donot vanish and are determined upto asign bytheperipheral invariants t,[see Section (5)]. From Eq. (4.1) itshould beclear how the general n-body unitarity product appears. Each intermediate particle gets aMisheloff rotation, and the helicity indices are summed over systematically. The n-body phase space is a fats. 6°? m2ag,=é*(ext)TT(ee?) (4.2) isl £ where, asinEq.(B.1), pj=L(a,)B,- Sometimes itisuseful to visualize each produced particle asacluster ofvariable mass and spin, inwhich case Eq. (4.2) can beadjusted byreplacing 8*(pj-m})+6°(p}-5,)ds, andadding spinsuns2, Weare now ready toinsert into the general n-body unitarity product amodel for the production amplitudes, namely, the multi- Regge production amplitudes developed inSection 2,which wenow write as Sa08452-+-SnSb SaSo0 -0,-1 Welptee *[EsEarsea]one=VM 1PiM2Te Pn -20- Thisamplitude isshown inFig.8;theV'sarethestandard vertices described inSection (3), andwearenowusing m,r,p ashelicity labels. Itisperhaps useful toobserve thatthebracketed factors in Eq.(4.3) havethree sources ofphase whenaisreal: i)theazimuthal phase exp[-i(m,¢; +7,6))] fromE(g,)3 ii)thephase (#i)"i'"1fromthee-functions atz>1 arising fron thekinematic spincuts (half angle factors) inthe amplitude; iii) theRegge phase ofthesignature factor &;. Ofthese three phases, onlytheRegge phase willbeincorporated into thehelicity pole propagator tobedefined below. Suppressing theToller a,argunents, wenowstate then-body multiperipheral unitarity product as 1ac.pbsasasb Sh8n2-51508a] ”FrdiscMoaata=-etnfea,-[Mtpt.ept| malta, MPa’? PyPoMa xT(3«)[reste |(4.4) derVPEPE? aPoPassPaM>J 7 where each M-function ontheright hasaform asinEq.(4.3), and wheredQ,isgivenbyEq.(4.2). In(4.4)theonlyvariables not summed over arethose with subscripts aandb.Thespins and helicities appearing in(4.4)arelabeled inFig.9whichshowsthe n-body unitarity product withthemulti-Regge amplitudes inserted. ournotational plan isalways touseprimed variables fortheupper sideoftheladder andunprimed forthelower side. Thereader is again cautioned about ourmultiple usage ofthesymbols s;(spin, invariants), p,(momentum, helicity), andm,(helicity, mass). -21- The next step inthe program istoactually insert the production amplitudes ofEq. (4.3) into (4.4) and make some sense out ofthe resultant expression. Wewish toshow that Regge poles inthe upper and lower amplitudes are converted into helicity poles inthe central kinematic level, and that itisthese helicity poles which determine the Reggeon loop which lies atthe heart ofall bootstrap and cylinder calculations. Before wedothis, however, wemust make some comments about the frames in the various kinematic levels. (5) FRAMES The study ofthe reference frames associated with the multi- peripheral ladder isatbest atedious and unpleasant business. Wepropose only tooutline the development ofthese frames and to provide afew interpretations where useful. The ends ofthe multi- peripheral ladder, where the frames are slightly different, will be completely ignored. Usually inmultiperipheral analysis the end-rungs (oratleast oneend-rung) areamputated, thephysics isdone, and then later the end-rungs are reattached (see Appendix F); Regge physics does not require the end-rungs and this isour justification for ignoring them. Inthe description which follows wehave for noparticular reason adopted thenotation ofCiafaloni, DeTar andMisheloff® rather than that ofMueller andNuzinich.” 1, The Vertex The frame analysis begins with the simple vertex shown in Fig. 10,where twospacelike momenta k,andk,meet afuture timelike momentum p,. Frane ¢isarest frame ofp,inwhich the3-monentum -22- K,=K,points inthepositive 2direction. Obviously, frame cis only defined uptoaz-rotation, afact weshall make use oflater. Frame b(d)isobtained from frame cbyaz-boost v,(0,1) which brings k,(k,)tospacelike rest [k;=(0,0,0,V-t, )].Clearly, frames banddarelinked bythez-boost q,=v,+0,. From momentum conservation itiseasy tocompute these boosts interms ofthe invariants t,,t, ands,: shy, =(s,+t,-t,)/2V5, Ve, sho, =(s,-t,+t,)/2Vs; Vt, (S.1) cha, =(s,-t,-t,)/2V-t, Vt, . Thevariable q,maybeinterpreted assensing themass”8,flowing upthecluster p,. Bycomputing (k,-k,) inframe b,onefinds that q,ispositive because p,isfuture timelike. Theframes b,c,d defined above aretheusual BCPframes”? associated with aproduction vertex. 2. The Rung Wenow combine two vertices tomake one multiperipheral rung, shown inFig. 11. The triad offrames (b,c,d) just discussed appears onthe lower vertex, and anew triad (b',c',d') appears onthe upper vertex. The primed boosts connecting the upper frames are given by Eq.(5.1) with t,>t}. Frames ¢andc'areboth rest franes ofp,andmust therefore beconnected bysomerotation g=R,($,)Ry(X,)R,(¢1)- Wenowuseup the 2-rotation degree offreedom indefining each vertex frame triad toset¢,=$1=0 sothat theframes candc'arelinked byapure -23- y-rotation X,. This isthe Misheloff rotation mentioned inSection (4). ImAppendix Eweinterpret this variable asacross channel (t+©)Regge variable 2=cos(X); anexpression for cos(X) will be given below. The six frames shown inFig. 11are now interlocked, and all 3-momenta are confined tothe x-z plane. Next, four new frames a,a',e,e' are added asshown inFig. 12. For example, frane aisobtained from frame bbyanx-boost hy. Thisboost ofcourse doesnothing tomomentum k{)=(0,0,0,/%,), butischosen sothatk!(®) isx-zlike;i.e.,theboosth,clears outtheenergy component ofk'(), Boosthlischosen similarly so thatk(@")isx-zlike. Thesestatements maybesummarized asfollows: x)= (0,0,0,) KD 2(0,#,0,) ‘ar (5.2) Ks (0,*,0,4) KG 20,0,0,4) . Itshould beclear from Eq. (5.2) and the lack ofy-boosts in the problem (so far) that frames aand a'are connected byay- rotation, which welabel @,,,. Fromthefactthatt=(k,+k!)? one quickly shows this rotation tobegiven inmagnitude by cos8,,, =(t,+t) -t)/2v-t, V-tT .(5.3) Then, from the loop equation onthe left side ofFig. 12, tay XT MG Shy yy» (5.4) one finds the magnitude ofthe Misheloff rotation cosX, =(cos®,, -shvishv,) /chvichv, « (5.5) Reordering the same loop equation one may then compute the boosts hyandhi: -24- chh, =chvisinx,/sin®,,, , (5.6) chhy=chv,sinX,/sin6,,, . Wehave now described the frames aand a’, and the new transformations h,,hand6,,,- Inexact analogy onedefines the frames eande'andtransformations f,,f)and®,,1- Equations similar tothose above are then obtained bycomparing Eq. (5.4) to theright-side loopequation x,=(o!)"! #10,,,£;' 03°. 3. The Central Level Frames Tothe set often frames sofar defined with respect tothis one multiperipheral rung, two final frames £and gare now added, asshown inFig. 13. Weshall refer toframes like a,b,d,e asbeing lower level frames, those like a',b',d',e' asbeing upper level, and fand gasbeing frames inthe central level. These central level frames are infact brick wall systems (bws) orBreit frames. We define abwsframe forthesystem (k;,k}) tobeanyframe inwhich kj+k}=0,where k;represents thefirst three components ofthe 4-vector k;.Weshallrefer tosuch(t,x,y) objects asvereors”) todistinguish themfromthenormal 3-vectors (x,y,z) likek,. Since kj+k} =0inabwsframe, theoverall momentum transfer Q=kj+k}isatspacelike rest, Q=(0,0,0,V-t ).InAppendix E weperform acomplex Lorentz transformation which converts bws frames~ toensframesinwhichk,+k]=0andQ=(Vt,0,0,0). Now, frame £inFig. 13isthat particular bws frame inwhich versor k,points inthepositive xdirection, andversor k,ist-x like. Similarly, frame gisdefined toputversor k,inthepositive xdirection and tomake k,t-x like. These two frames fand gare -25- thus linked byanx-boost v,whose magnitude weshall compute ina moment. Inallbwsframes forthesystem ofmomenta (k,,k;) the 2-components and versor magnitudes are the same, just asinall cms frames the energy components and vector magnitudes are the same. We find 2 2qty2 x,2 yy2 Os)? =aK)? -a)? -a) =A(t,t!,t)/4(-t) =-K? (5.7) 2 tet! =:Kos (t-t,+tp/avt =x, (5.8) yz ' =2!(Kp)? =(t-tl+ty/2ve =zp. (5.9) Because our interest is limited to the interior runs of the t<0 multiperipheral chain where thekinematics requires A(t,t;,t}) <0, wehavedefined -kjasabove. Whenthesymbol k;appears below asa scalar, itrefers tothisversor magnitude (-k2)* andshould notbe confused with the4-vector k,. Wewish tostress the similarity ofEqs. (5.7) through (5.9) tothenormal cmskinematics. Ifk,andk}were future timelike 4-vectors withmasses (t)*and(t{)"%, theninanyemsframewhere Q= (VE, 0,0,0), t>0, one would have 2. 'py? =acystpeysae : (5.10) EQ=(t+t,-tp/2ve, (5.11) Ei=(t+ ty-ty)/2vt . (5.12) sothat theversor magnitude k,istheanalytic continuation ofthe cross-channel cms momentum. -26- Sometimes thevariables z,and2}shown above arewritten in this way: = idz= R(-t)e -wy .(5.13) ay=a(t)" +wy, » where ttOF(4) : (5.14) * 2-t Thevariables k,andw;areuseful replacements fortheReggemass” variables t,andt!, = ~2 +we tt tysatOD+WD=Ce) (5.15) moo 2+2.+ tyethoset+wD+wc. Inparticular, 1dtydty dkv, =2 —HX_. (5.16)comet 2 "ypeL-att,t, tT Applying the above definitions toframe fofFig. 13wehave () . (f) 2KA =(0,k,,0,2,) KO) =(K,shv,, k,chy,, 0,2,) come : 6), :KO 2(0,-k, 0,21) x} =(kyshy,, -kgehv,, 0,21)« (6.17) Conparison ofk{*)tok{)thenshowsthese frames tobelinked by avery simple y-rotation @,: x=(0,0,0,V-%, ) x)=(0,k,,0,2)) (5.18) =sind, =ky/V-ty cos8, =2,/V-t, - -27- Thus, the new frames £and gare interlocked with the previous ten frames togive atotal oftwelve frames associated with this single rungofthemultiperipheral ladder. Computing p?=(k,-k,)” inframe fwefind that theboost v,isgiven by chy, =(kp+kb+pty/2kk, (5.19) where peo= 5,+(wy-wy)? (5.20) and all symbols onthe right ofEq. (5.19) refer toversor magnitudes. Withtandallt,fixed, v,measures themass” s,oftheparticle orcluster p,;inthis sense thevariable v,issimilar totheBCP variables q,andaappearing inFig.11. Thecomplete setoftwelve frames associated with therungp, isshown inFig. 14. 4. Many Rungs Weare now ready tojuxtapose two rungs ofthe multiperipheral ladder, asShown inFig. 15. Inthis figure one sees that the twelve- frane systems associated with each rung are linked byavery important y-boost called £,. This variable measures the separation ofthe two rungs inaquantity which would becalled the gap rapidity inaone- dimensional model. Notice that thesame variable £,appears inthe upper, lower, and central levels. The frames onthe central level are Linked totheupper andlower levels byy-rotations like 8,ofFig. 13. These rotations are given bythe formulas one would guess looking at Eq. (5.18) above, e.g., sind} =k,/V-t; , cos) =zp/V-ty .(5.21) -28- ‘Theonly transformations notshown inFig. 15arethey-rotations like@,,,appearing inFig.14.Obviously Ot 8+er. Wenowmake some remarks concerning theframes ofFig. 15. First ofall, most oftheframes onthelower level aretheusual BCP frames referred toearlier. Sincethetranformation labeled g,connects twoframes inwhich k,isatspacelike rest, g,mustbean0(2,1) transformation. InBCPthis g,iswritten as 8,=RM) BCEIRW,) . This form, known asthediscrete-basis parametrization, goes allthe way back toBargmann, butwehave put atwiddle over the x-boost Parameter inorder nottoconfuse that variable with oury-boost variable ,.Theazimuthal rotations u,andv,areconjugate tothe Reggeon helicities inthesense discussed back inSection (2), andare connected withtheso-called Toller angles w,=u,+v,,,. Variable E,istheReggevariable, i.e.,z-=cosh(E,), andisconjugate to theangular momentum associated withthelinkk,,which istosay, a,(see Fig. 8). Although thesameBCP0(2,1) transformation g,appears inFig.15, itisparametrized differently, namely, 8,=Bf) BY(E,) ByCh,) theso-called continuous-basis® parametrization of0(2,1). Asalready noted, thesame variable £,appears also ing,the0(2,1) transforma- tion appropriate totheupper production amplitude ofFig. 15. Prior toleaving this section onframes, wewish toadd one more observation concerning theframes connected with thesingle rung shown inFigs. 13and14, Ifonewere toimagine themultiperipheral -29- ladder onthe right asgenerating aReggeon inthe central level, one might draw the figure shown inFig. 16, where wehave redrawn the frames a,a' and f, and their connecting y-rotations. Wejust want to remark that these three frames are the usual standard frames one associates with thetriple Regge vertex!® intheconfiguration A<0, and the thetas are the standard y-rotations. Asimilar remark applies tothe frane triad, g,e,e'. Weare now ready toconvert the Regge poles ofthe upper and lower amplitudes into helicity poles inthe central level. (6) THE HELICITY POLE EXPANSION Consider once again Fig. 15. Inorder tomotivate the next technical maneuver, weanticipate adiagonalization procedure which will beexplained inSection (9). The frames onthe central level of Fig. 15arelinked byalternating x-boosts VWandy-boosts ee Itwill turn out that these frames and variables are the relevant ones for the diagonalized (or even undiagonalized) consideration ofthe multi- peripheral ladder, the reason being that these are the bws frames inwhich the overall 4-momentun Qisatspacelike rest. Wewill show that certain groupings ofthe vand &variables form convenient 0(2,1) transformations. For example, the conbination B,(E,)B,(0,By(E,) is an 0(2,1) transformation in the continuous-basis mentioned earlier which inacertain sense surrounds the cluster p,inthecentral level ofFig. 15. Inthe diagonalization process itwill beshown that the variable v,isconjugate toangular momentum jinthe central level, -30- while theboosts &,andE,areconjugate tocomplex helicity variables A,andA,. Helicity poles inthecomplex helicity plane \will correspond topowers ofellsince these variables areFourier conjugates. Itisfor this reason that weshall now expand the upper andlower Regge propagator functions E~“"1(g) andE™'~!(g*) into powers ofel®!, These functions appear inFig.17,which represents aportion ofthe multiperipheral chain, i.e., aportion ofthe unitarity product ofEq. (4.4) with the model amplitudes of(4.3). Weshallrefertotheformel§I@asahelicity-pole terminthe same wayonespeaks of2%asaRegge pole term, with theunderstanding that the actual pole occurs inthe plane ofthe conjugate variable, beithelicity orangular momentum. Also, the square-bracketed expressions inFig. 17will becalled Reggeon propagators. InAppendix Cwegive aderivation ofthe following (convergent) helicity-pole expansion ofthe lower propagator E-function: Eee =Go" PSoes) Kt nys0 (6.1) oe a eenz.Kom, NysK2Tp Recall that g,=(f,,£,sh,), and that £,andh,arex-boost paraneters fixed bythet,[see Eq.(5.6)]. Thequantity [a,-n,] isthehelicity oftheReggeon whose spin isa,. When a,takes some general non- integral value, the Reggeon helicity takes thevalues a,, a,-1, @,-2,.... inaninfinite sequence. Were a,toapproach aphysical value s,(which does nothappen inthemultiperipheral region ofcourse), wewould expect this sequence totruncate athelicity equal to-s,. This truncation isaffected bythe interaction ofthe functions F -s1- appearing inEq. (6.1) with the helicity nonsense-zeros present in the standard Toller vertices discussed inSection (3), the Vof Fig. 17. These functions Fare given inEq. (C.5). The new index K,appearing inEq. (6.1) will beconnected with parity inSection (7). Basically, K,=sign(E,). ‘The important point tobemade about Eq. (6.1) isthat each helicity term factorizes. Itisnot obvious that anexpression like (6.1) had toexist. Asimilar situation isencountered inamuch more complicated mathematical environment with the Regge pole expansion of asingle Toller/Lorentz pole. Regge poles there are the factorizing daughters ofaToller pole, and helicity poles here are the factorizing daughters ofaRegge pole. The fact that each helicity pole factorizes isthe fact which allows ustomomentarily define ahelicity pole propagator. This concept will greatly reduce the bulge ofcomplexity with which we are now confronted. Had the helicity poles not factorized, wewould be in real trouble. When all the Reggeon propagators [... ]inthe unitarity product ofFig. 17are helicity-pole expanded according toEq. (6.1), certain factors may begrouped tothe vertices, leaving avery simple helicity pole propagator. The new rung with these regrouped factors isshown inFig. 18, and the helicity pole propagator isshown in Fig. 19and has the form 1 + ton}P(E.)=Y,¥yBF8(k,8,)ole! [earna)+Girne] (6.2) ‘Thepowertowhichele!israised inEq.(6.2)isthesumofthe helicities ofthe Reggeons inthe (2,2') channel. Notice that each ofthe helicities isingeneral acomplex number, whereas the Reggeon =32- helicities discussed inSection (2)were always integers orhalf- integers. Thereason isthat here theReggeon helicities areeigen- values ofthe(non-Hermitian) y-boost generator K,which isgenerating theboosts By(E). InAppendix Eitisshown that, when thestructure ofFig. 15iscontinued tothe t>0 cms via acomplex Lorentz transformation, thegenerator K,isturned into anormal rotation generator and the helicities become the normal (discrete valued) helicities mentioned inSection (2). Thevariable £,becomes a rotation (,=ig,) which again measures thesumofthehelicities inthe(2,2') channel, namely, m,+m). The other important point tobemade about thehelicity-pole propagator isthat itstill contains the physical (planar) poles in the signature factor denominators, e.g., 2M,6)‘ fe 2sinn(a, -€,): en ‘2 Se These poles generate the normal thresholds inthe cross channel when tis continued to t>0. Turning now tothe rung orkernel ofFig. 18, the helicity summations r,,r!, p,,p,, andm,sm,canbeperformed since theyare now detached from the rest ofthe chain byhelicity-independent (in this sense) helicity-pole propagators. Wemight first sum over the upper and lower (discrete) helicities togofrom Fig. 18to Fig. 20a, renormalizing for the first time our standard vertices V. Thenewvertex Visgiven by ~—bead|oy Vp(omyGeomKyoKQ581)=>G@Fayeeyr, PD ms 45,0, op mw xVepm,Fngokgm, f2)@ Fy. 63) -33- (Group-theoretically, this’ corresponds toaconversion from the discrete tothe continuous helicity basis.) Finally, wesumovertheMisheloff rotation helicities PP, togofrom Fig. 20a toFig. 20b, which shows the final kernel Kam sty5afmftlsa.myst,s almit!; Keys syt) 2 ~ie(81 ~ => Bp” app0)Wp}=m+6 PyP} =-5y Thiskernel isafunction ofthefourReggeon spins a,,helicities @;-n;, andmasses t;. Duetothekappa indices appearing inEq. (6.1), the kernel isalso afunction ofthe kappa label oneach side. This particular kernel isasingle particle kernel and thus depends onthe : spin s,ofthat single particle. Wecould just aswell have defined P,(the produced object) tobeacluster, inwhich case, asnoted earlier, Eq.(6.4) should besummed over s,. Before concluding this section wewish tomake afew additional remarks about the critical helicity-pole expansion formula (6.1). This formula, orsomething close toit, has been derived byother workers?’ asonlyanasymptotic expansion. Wewishtoemphasize that (6.1) asderived inAppendix Cisanexact andvery convergent “equality based onanelementary addition theorem ofthesecond-kind Legendre functions. Inother approaches, the step inthe argument represented by(6.1) has been tosome extent obscured bycomplicated group theoretic arguments. For example, (6.1) can beinterpreted in terms of0(2,1) mixed-basis matrix elements inthe continuous series, inwhich case thediscrete index xhas acertain mathematical meaning. Altematively, Eq. (6.1) can berelated tothe 0(2,1) analytically -34- continued Clebsch-Gordon coefficients which couple angular momenta between the upper, lower, and central kinematic levels inFig. 15. ‘These approaches are nodoubt correct, but introduce somuch complica- tion that one cannot tell for sure whether or not aformula is correct without expending much effort. Our approach has been toconsolidate this group theory into afeweasily verifiable addition theorens® which are then used to derive various results. (7) NATURALITY CONDITION FOR THE KERNEL InSection (3) itwas shown that, after accounting for the correct Toller N-function notation for the vertex 045504Vag; =May.ngm;(arapa) 5 (7.1) the statement ofparity invariance for the vertex inFig. 18is 4150p ar, Sy-P) * 2m,Yep, 7(C9 1OC 1f,8) .yn 7.2)TyPym ” where 0;istheReggeon naturality ofEq. (3.9) andI,theintrinsic parity ofthe produced particle. Insertion ofthe parity condition (7.2) into the definition (6.3) oftherenormalized vertex Vthen yields ~ — =) SiPh Vp,(192) =G,o,T,(-1) Vip,KyoKa). (7.3) When this result isinturn substituted into the definition (6.4) of the kernel K, one finds K(k, s,) =To! 0,0) K(-Ky,-,) (7.4) -35- which isthedesired naturality condition forthekernel. Wemaynowinterpret Eq.(7.4) assaying: aparity transformation onthekernel isequivalent tomultiplication bytheproduct ofthe Raturalities ofthefourattached Reggeons. Toseewhyaparity transformation negates K,andK,werefer toFig. 21which shows a Segment ofthemultiperipheral chain withitscentral level boost £. Thefigure also shows thesame chain segment inaparity-inverted world where thetwoframes areconnected bysome boost E'. These inverted-world frames areconnected totheir non-inverted-world counterparts byToller's parity transformation s'defined inEq.(3.4). Since yl 's =(51) By(&)s By(-E), (7.5) oneconcludes that £'=-£. This iswhat ismeant bysaying that parity negates allthe&-boosts inthechain, andtherefore the k;=sign(E;). Equation (7.5) isoneentry inthefollowing table which shows howtheparity operators sands'affect thesigns ofrotation and boost parameters: R R RR BR By By s a re (7.6) st- + - + - + Notice thatofallthevariables listed inFig.15andrelating tothe multiperipheral chain, onlythey-boosts £;arenegated byparity s'. “This condition isderived inRef.5,Eq.(2.8), fortheproductionofspinless particles only; seealso Eqs. (2.7) and(2.10) ofthat paper foraToller angle discussion, andEq.(2.16) which relates toour comments atthe end ofSection (6). -36- Ifthere were z-rotations somewhere, these would also benegated by s', asthe table shows, and this fact has abearing onthe Toller angle which wemention here asadigression. Inthe usual BCP analysis ofthe production amplitude shown, e.g., inFig. 8,one uses for the 0(2,1) transformations gthe discrete basis parameters R,(u)B,(E)R,(v), which wementioned at the end ofSection (5), and interms ofwhich the lower Reggeon propagator function may bewritten “ol,2gcim ol ye)gary Er (8) e ear (che) € : (7.7) Ifthe asymptotic limit ofthis Efunction istaken [see Eqs. (A.15) and(A.16)] toget(ch)* times helicity-factorizing factors, andif these factors and the azimuthal exponentials are absorbed into renormalized vertices 8and the helicity sums done, one obtains for the production amplitudes the form ~a, =O,s+BpCarta)COME)?By(yeti)(CHET«5(7-8) where thep;arethehelicities oftheproduced particles. Then from Eq. (7.2), the parity condition for these renormalized vertices B may beshown tobesimilar toEq. (7.3), =30,Wey 7.9 BpWray) =%9, TY-1) BpCrd) +(7.9) Inthe case ofspinless produced particles, the vertex 8isafunction only oftheToller angle w,=v,+u, andEq. (7.9) becomes 8W,) =5,0,T,B(-w,) - (7.10) Finally, slightly renormalizing the vertices once again, weend up with the asymptotic orphenomenological multi-Regge amplitude for the production ofspinless particles along the chain -37- ~ a, a,+++BWw)(s,) >BWw,)(s,) *... (7.1) Multiplying twosuch amplitudes together togettheunitarity product, one would identify the kernel as ~| ~ PeKw,0) =(B@)] Bey , (7.12) andthis kernel would then have anaturality condition Ku) =To; 0,0) K(-w,,-w!) (7.13) This condition is,however, just aspecial case ofEq. (7.4) which was derived without anyapproximations. Therefore, aparity transfor- mation canberegarded either asnegating the£;variables inthe exact kinematic scheme, orasnegating the Toller angles inthe asymptotic production ofspinless particles. (8) THE MULTIPERIPHERAL CHAIN AND PHASE SPACE JmSection (6)thehelicity-pole propagator P,andkernel K,,; were defined. Figure 22shows how these quantities alternate to compose the multiperipheral chain seeKy.(¥,)P,(E,)K,9(¥,)P,(EK,(¥)«2. (8.1) Thefigure also shows thecentral level frames with their connecting boosts. Thevvariables measure the "rapidity width" ofthekernels (clusters orsingle particles), whereas the &boosts measure the rapidity width ofthe helicity-pole propagators. Since these alternating boosts arenotcollinear (v,=B, and =By),thenotion ofadditive rapidities arises only inthe extreme relativistic limit where -38- chv =chy, chv, +shy, shy, ché, (8.2) becomes ve Mtv, + (8.3) The sums which are implicit inthe chain (8.1) will bediscussed in amoment. First, something must besaid about the phase space. Each particle orcluster (here Kwill beregarded asacluster) gets a momentun phase-space factor d"p;, where p,isthemomentum flowing upthecluster K;;,,- Replacing a*p,withd"k;,where k,isthe 4-momentum ofthe lower Reggeon ofthe system (i,i'), and simply evaluating this 4-momentum inone ofthe central level frames afew removed fromtheframes nearest p,,onemayexpress a°k,interms of the group variables appearing inFig. 22. Recalling the meaning of the central level frames, wehave, e.g., @) _KO) =(0,k,,0,2,) (co).{9 =(kyshv, ,kychv, ,0,2,) x0) =(KyshvgchE,, kychv,, kyshv,shE,, 24), (8.4) where k,(the versor magnitude) and z,were defined inEqs. (5.7) and (5.13). From the last line ofEq. (8.4) wefind that a’k, =[a’k,Jdz, = [kedk a3° ([a’ky]dz, =[kjdk,d&, d(chv,)]a2, cis 2 2 =ank3akdw,[acem,)] , (8.5) where z,has been replaced bythe w,ofEq. (5.14). The portion dk,dw, ofthephase space isthe so-called transverse integration because itcanbeexpressed intermsofa’ptwhere pSisthe -39- transverse momentum ofthe cluster 3whose parallel momentum component plisrelated tothestandard rapidity variable. Intermsofthe invariants t,andt,onecanshow, asinEq.(5.16), that dtjdt! akjaw,=}——*2. (8.6)VBE ty) The second factor inthe last line ofEq. (8.5) shows the 0(2,1) equivalent ofthe d@=d$d(cos®) one finds incms kinematics, e.g., elastic unitarity. Thefact that a,d(chv,)/2m =dg, isapiece ofthe 0(2,1) invariant measure (in continuous-basis parameters) is what allows the exact diagonalization ofthe multiperipheral chain onto central level angular momentum, asisdone inthe next section. Thereader familiar with theChew-Goldberger-Low approximation?2 tothe multiperipheral phase space will recognize the expression in Eq. (8.6) asaportion ofthe asymptotic form ofthe quasi-cms phase space oftwo clusters, as,ds, dt,dt} a'p,a'p, 8(P-p,-p,) ~—— |}—==], @.7 v-A(t,t, .t2) where (s,)%and(s,)* arethemasses flowing upthetwoadjacent clusters. Inorder tocompare Eq. (8.5) with (8.7) wewrite, shifting tothe left one rung, a, a(chy, )d(chv,)@(v-v,-v,) af,=Sbeacwwy =2 gynt a[k(chy,chy,,chv,)] where k(x,y,z) =x?+y?+#2?-2xyz-1. IfitweretruethatV>>V.V, throughout the entire phase space, one could approximate K(chy, chy,, chv,) *(chv)? . -40- Then from formulas like Eq. (5.19), 2 2 2 s,+k?+k2+(w,=) chy,——————— ’ 2k,k, one finds that ds, ds. “4+ , 2mk)s and then ahsak,dw,dsds.[:dt,dt}|ds,ds,2 —s |2SSS] SO”v-AGt, th) which isthe CGL approximation (8.7). Since the approximation s>>s,,8, isnot particularly valid except inspecial cases like double diffractive dissociation, one would expect amore accurate result tobeobtained inany related calculation (like the cylinder) byusing the exact phase space. Natura}ly, anexact angular momentum diagonalization only works ifthis correct group phase space is retained. Wenow consider the sums implicit in(8.1) and Fig. 22. For each segment orpropagator ofthe multiperipheral chain there isa sum ofthe form (e.g., for segment 2,2') Jcy2 where Lefe.>>> (8.9)2 ketaya) nanh=0 and -41- 2n 2 f*=Ffo,f%,eo, (8.10) =o 0 with £being thenormalization factor ofSection (4). Foreach fixedvalueoft,andt;[seeEq.(5.15)] andthediscrete indexx,, andforeachpairofReggeons @,,0}, wesumoverallofthehelicity Poles labeled byn,,n}, these being thehelicity daughters ofthe Reggeons. Next, wesumoverallpossible upper andlower Reggeon combinations. Finally, wesumoverx,anddothetransverse integration. Thegroup integrations dg,willberemoved inthe next section. (9) THEDIAGONALIZATION OFANGULAR MOMENTUM Toavoid confusing themathematics withthephysics, webriefly discuss ourdiagonalization procedure; afuller explanation maybe found elsewhere.® Consider thefollowing mathematical relation among fourfunctions A,B,C,andD,each afunction ofthree variables: . fae, f4,AEwvib') =JSEfaccnv.f 52faccnv,)a¢¢,.v,.0) CC,.,,0) a 1 = 1 xD(Es.V5,63) « (9.1) Schematically, thisequation isrepresented inFig.23.Ifthe variables areintherange=<&,<=andO<v,<e, wemay interpret thefunctions A,B,C, Dasbeing defined onacertain sector ofthegroup SU(1,1) ~0(2,1), andwewrite thesaneequation ingroup theoretic notation asfollows: - Ae)=fet,fot,fac,60e-44,2, BE,CU)008), 00.2) -42- where g=(E,v,6'), £,=(E,,¥,,0), etc. Thevariables g,= (E,,¥,,£}) inEq.(9.1) arefunctions oftheother variables according totheSU(1,1) group multiplication g,=g)'g7g. InEq.(9.2) this fact ismade more explicit byuse ofaninvariant delta function. Equation (9.1) or(9.2) can bediagonalized exactly byproject- ing the functions onto the continuous-basis representation functions ofSU(1.1). These functions are the second-kind generalized Legendre functions #2)discussed briefly inAppendix Aandatgreatlength inRefs. 9and 23. The diagonalization of(9.1) isgiven by je Poh Pa Bi gi piAme-faamAaSarPane>(9-3) where j = jAefe(8)ACa) , (9.4) joo. jBy-fag,BC)BE) (9.5) and the projection ofCislike that ofB;Dlike that ofA. The invariant measures are =2 gt ag=Zacm) S (9.6) ae, ag, ==a(chy,) , (9.7) and the function Q(g) isdefined as ji -ue Jj “ute!it) =2@icer) e : (9.8) with Q(z) given inEq. (A.2). Inthe diagonalized equation (9.3), each group integration has been replaced byahelicity contour integration running upthe -43- imaginary helicity axis. The source ofthis contour isthe second- kind addition theorem which, for convenience, wecompare tothe first-kind addition theorem: J = J J 5Pim(8,82) >Pin(81)Pigs(82) 3 (9.9) nee J = ad oi "By(8,8,)if$eGB)Bye)-(9.10) The familiar helicity sum ofthe first-kind theorem (Pfunctions are essentially the rotation Dfunctions) appears asahelicity integration inthe second-kind formula. InRef. 9,Eq. (9.10) is derived from (9.9) and interpreted group-theoretically. When all the functions appearing inEq. (9.1) are independent ofthe£,variables, thediagonalized equation simplifies somewhat, a=po a : (9.11) where fosifA(chy)Q;(chy)FW) (9.12) 1 More generally this isnot the case and the helicity contours appearing inEq. (9.3) are shifted sideways topick uphelicity pole contributions ofthe integrand. The \are the complex helicity variables to which we referred earlier. Inthe particular mathematical example considered above, we diagonalized achain ofthree functions B,C,D. Hopefully itis clear that achain ofany length may be similarly diagonalized. Each projected function contains the diagonal angular momentum projection label jalong with two helicity labels which are system- atically tiedtoneighboring helicities bythe"summations" fa. -44- Weare, bythe way, referring to jasangular momentum because, inthe Regge language, Eq. (9.4) isatrue Foissart-Gribov projection sothat jisthe analytic continuation ofthe true angular momentum. InEq.(F.5) weshow howtorecover afunction A(g) from its projections Buei.e.,wegivetheinversion offormula (9.4). (10) THE PLANAR BOOTSTRAP 1. Form ofthe Integral Equation The basic multiperipheral chain was illustrated inFig. 22, and weshall now bemore specific. The contribution from three particles orclusters tothe4-Reggeon ring discontinuity isgiven by” Oy "yeRyg(Ev.6") =zad,fad,Py(EK,9(Y,)P2(Ep)Kp3(V,) xPs(ES)Ks4(¥,)P4(Es) (20.1) where the notations P,K,dgand ©were defined inEqs. (6.2), (6.4), (9.7), and (8.9). Since (10.1) isofthe form (9.1), the diagon- alization may beread from (9.3) tobe OR : a far 5 jByrd) =xzSr]GePyCDK), (1,2),2K}(2,3) iXPD KyBAP 4)» (10.2) where “Wehave tried very hard tokeep track ofthenormalization ofampli- tudes, but, alas, have lost the battle. Strictly speaking, ifAis adiscontinuity, equations like (10.1) should contain the overall factor —cftshown inEq.(4.4), 2n/f foreach dg,andanextra 1/fbecause 6(4)(ext) removes oned°kj. However, aswemention in Section 10.8, and show inFig. 29, Aisnot Toller-normalized, soweomit these overall factors. There isalways aquestion of how many T's and 2's appear inthe phase space dofthe planar bootstrap orEq. (11.24), and wehave therefore lost track ofthese factors. -a5- j : a0,2) =JacmBy(chv)Ky), (0.3) faeve na-fFe*ry@ . (20.4) Therefore, defining On, by (3935 - (3),3Ryda =pa)Oaa4)P,@) .40.5) Eq. (10.2) may bere-expressed as 43 - afa j jaad =DP EPSE8c2.20P, 2K), 2,3)?,1(DKAys4) 2,5 (10.6) which isschematized inFig. 24. Inthe usual way one may write anintegral equation for the complete 4-Reggeon ring discontinuity which will besolved byasum ofterms ofthe form (10.1). This integral equation reads AysGV.ED =PLOK,;O)P,E +zf46PEK (4IAy5(Ep5,565) (0.7) which may once again bediagonalized by inspection togive, together with the definition (10.5), Jj =*a,3 SX3.(1,2)P, (2)A3,,(2,3 10.8 Nyt23)RyEPSa22)P)(2)A}(2,3)>(10.8) which isrepresented byFig. 25. The problem ofobtaining the 4-particle discontinuity from the 4-Reggeon solution of (10.8) is illustrated inFig. 26and discussed inAppendix F. Near aRegge pole, the projected ring discontinuity Afactorizes (see Fig. 27): -46- 5 0 a; jx Gh(15a) 6°,(35a)Jj 2iu m Ah 1,3, Lt|i22st) 10. J3)| re = (20.9) Taking the residue ofthe pole onboth sides of(10.8) then yields the vertex bootstrap Obey ad . G30) x]7Ki12)P,(265(250)»(10.10) asshown inFig. 28. The normalization ofthe triple-Regge couplings Gis described in Section 10.5 below. 2.TheProjected Helicity-Pole Propagator P,_ The helicity-pole propagator was defined inEq. (6.2) tobe P5(E) =2mH(i)O(EK,) exp[h,|E|], (10.11) where De “ls yt) © EF!H(i)=(2m) Og¥p) +GEE), (10.12) hy=(,-m\) +@-nj) : (20.13) According to(10.4), the projected propagator takes the form PAG) =HG)/(«,d-hy) , (20.14) where wenowseetheactual helicity pole atX=K,h,. Inthe ordered S-matrix, Regge trajectories must occur in strongly exchange degenerate pairs. When the upper and lower signature factors are summed over signature taking into account the exchange degeneracy, one finds for the regular (untwisted) propagator P, -,_exfinlta,- 4)-@-<I} BEL ee =—t—++42° .«0.15 irifin sinn(a, -e,)sinm(a, -€!)at iS iG -47- Forthetwisted propagator “Pused later inthecylinder discussion, aE? Doespojep -SS ft.16)ad sinn(a, -¢)sinm(@ -1)Fart 3. The Projected Kernel and Its Threshold Behavior Inthe kernel, shown schematically inFig. 18, there are seven quantities each ofwhich depends onthekernel mass s,andtherefore onthevariable v,ofEq. (5.19), sothat computation oftheprojected kernel (10.3), 3,01,2) =facenv) @,(chy)K,,() (10.3)yal q uA 12 ’ . interms ofthe standardized vertex Visanunpleasant numerical task which weshall not attempt. This task is, however, anecessary aspect ofthe functional bootstrap tobementioned below. Lacking ananalytic expression forKywesearch forany potentially useful information buried informula (10.3). One such piece ofinformation isthe threshold behavior which wenow extract. SinceK),(1,2)isaFroissart-Gribov projection, wearereminded that itshould bepossible tofind its threshold behavior inthe usual way. First, however, one must identify the threshold behavior ofthe unprojected kernel. Inexpanded notation one has Kyo) =Kpolty tistyotys vw)=Kyolky myskjowys v) where k,istheversor magnitude (continued cmsmomenta) ofthe channel (i,i'), -48- .1yys 35 k= Lact tply2cn? and =' 35 w= (y=tp)/204) : Weshall define "threshold behavior inthe (i,i') channel" tobeany approach tothekinematic boundary A(t,tytj)=0asshown, e.g., in Fig. 40,sothat atthe(1,1') threshold k,+0. (Variables t,t;, thherearenegative.) Todetermine, then, thebehavior ofK,,ask,ork,vanishes, we examine the functional and kinematic structures of Kas shown in Figs. 18and 14. Asdemonstrated inAppendix D,ask,+0, one has ch(h,) *const *(k,)7? ch(f,) +const , but when k,+0thesituation isreversed ch(h,) +const ch(£,) +const x(k,)7*. InEq.(C.5) thefunction F(-h,), which appears aspart ofthekernel inFig. 18, isgiven roughly as oy woy-1 Fakir (-hy) =oy-n,sx,7,35%hy). Since -ish(h,) *© ask,+0, one finds from the large z behavior 2%ofq7%~1(z)_ that w ay =a, k, +0F“(-h,) ~(k,) : Similarly, a, 0, k, +0°F ‘(+£,) ~(k,) : Collecting similar factors from theupper vertex VofFig. 18,one may conclude that -49- =,+0)) -(@,+0,) Ky.) =(ky) (k,) KM). (0.17) where K'isareduced amplitude, real onthe uncut portion ofthe real t axis. Thethreshold behavior oftheprojected kernel Kamaynowbe found from the Froissart-Gribov projection (10.3). Equation (5.19) which expresses ch(v,) interms ofs,shows that chy, *5,/2k,k, aseither k,ork,+0.Therefore, using once again the large z behavior ofgi,(z)~273-1andremembering thattheintegration in Eq. (10.3) actually begins above z=1 atthe lowest production threshold ofthekernel, wepickuptheusual extra factor (k,k,)?, sothat the complete threshold behavior ofthe projected kernel is given by j 5-3) 5-48) 5KiyG2) =k) (k,) (K)),0,2) «(10.18) ~ When this kernel iscontinued tothe physical cross channel t>0 and the four Reggeons taken totheir physical points, we regain the usual threshold behavior given, e.g., byJackson and Hite,24 L,(min) L,(min) ,) &,) , whereL;(min)=J~S;(max)andS;(max) =s,+5}. “Inderiving this threshold condition wehave ignored parity which causes the distinction between threshold and pseudothreshold and which may raise some L(min) byone unit. -50- Since the above analysis used only the kinematic structure ofFig. 18, one may conclude that this threshold behavior isequally applicable tothe single-particle orclusterized kernels aswell as tothe full amplitude. 4. The Naturality Diagonalization InSection (7) itwas shown that inaparity-conserving theory thekernel K,,(v) hastheparity condition K(k, ,K,) =9,0} 6,0, K(-K,,-K,) > (10.19) where theo;arethenaturalities —as defined inEq.(3.9) —ofthe Reggeons attached tothe kernel. The property (10.19) passes immed- jately totheprojected kernel 8,via(10.3). Atthis juncture itisconvenient toconvert all Froissart projections likeKaof(10.3) tolower-case projections Ka defined by j = i0.2) =fa(ehy)a),(chv)Ky) (10.20) 1 where qissimply related toQasinEq. (A.13). The reason for this change isthat qhas asimpler helicity-negation symmetry thanQ,asymmetry whichofcourse iscarried overintoWy Woaz) =e a2). 10.21unl ) Son ) i ) Combining (10.19) through (10.21) wefind Karke) =90)F292Cesky) (10.22) One may now study the effect ofthis symmetry onthe ring -s1- discontinuity components. Converting the(aequation tolower- case projections asinEq. (10.20), one finds (2),5 = A id jFr =ED LSMHKy.) yOKy(23), (20.25)2 where theK,sumhasbeen removed from 2,andexplicitly displayed, and where Hi}=TG+1+aA)TG+1-a) . (10.24) From the symmetry of(10.22), and the obvious fact [see Eq. (10.14)] that PK) =PACK) 5 (10.25) one may easily show from (10.23) —using the symmetry ofthe contour —thatthesymmetry of(10.22) propagates into 7a, (2), = ' »(2),5aireK,)=0,0,0,0;PahvCxysk,) 5(10.26) andsimilarly intoall[Ja andthefulla.Thepersistence of this symmetry means that allour projected equations canbediagon- alized inthe 2x2 space ofthe kappa indices; this isthe naturality diagonalization discussed byCiafaloni andYesian.> Wenow perform this diagonalization onthe following proto- type equation Jj a5yi Jja),(Ky.K,)-Ef2 Hydiy(KyKz)P(KLA) 4,11(K,0K) HurOK)=DPaeBarra) PONSysaoa) where pisany function, and a,b,c are any functions having the symmetry ofEq. (10.26). Define =al aeEe (9K) (10.28) KK, feworutaos -52- and similarly for band c, and notice that Fk 79,0;0,05Bees (10.29) Taking w> ku, A*KA, Ul+Kyu" inEq. (10.27) yields a yi a. Aw op )¢ (10.30) KKP»Iea»Pek,PKK sincewh=H]. Intemsoftheprojections ofdefinite naturality” &21a soctal , (10.31) Eq. (10.30) takes the diagonal form ° a yi4 a av=IFeRepa?. (10.32) Thus, the naturality diagonalization of(10.27) isgiven by jo 2 fk yipio joaneJeHYbiPA)ey (10.33) where jo 21fj raded ay(++) +00,0) a Gt]. (10.34) Ayvr 171 Fy Interms ofupper-case projections like (10.3), Eq. (10.27) becomes j = a yj JAiurlyoK))=>]SrBua(rok2)PCKA)Cyr(KK) (10.35) 2 *For adiscussion ofwhy oisidentified with naturality, the reader isreferred topage 438, equation (9.59) ofthe textbook ofMartin and Spearman, Ref. 32. -53- and its naturality diagonalization isgiven by jo | [ar ,io joAueIeBPOGi (10.36) with projections ofthe form JF 2DL] Gyscoot PGI iOy AutVE[RueD400FESTayMyytot)] - (10.37) Now, since (10.8) isofthe form (10.35), thenaturality- diagonal bootstrap equation can beread from (10.33), jo =‘Io + 5 {doAy3)=Wy3)>foJGrBAKy(2)2203 nny HQ) jooesacr (10.38) where HQ)=(ny! y,7,«Eq.(10.15) Wo=TG4142)PG41-2)|(20.39) Sado, =Eq.(8.10) and jo at j +i 28anh) =[ue524)+000)a)yo 2,)]. (10. 40) where+refers tothekKvalues, andsimilarly forwe. Left-shifting thehelicity contour assuggested inEq.(12.12) yields this final form fortheplanar bootstrap equation: 5S1,3)=13%(1,3)42 jpio gioaye(153)jutCha3)>GNDHhRin,2)A023), 12083 ayn, (10.41) -54- where isaprojection onto qof(A.20). Allthat remains isthe transverse integration dj,andthesumover allupper andlower loop Reggeons andtheir associated helicity pole daughters. With H=h, andyu"=h,, Bq.(10.41) isamatrix (lattice) equation inthespace ofthe helicity indices. Inpassing, wenote that theapparent Regge cuts in(10.41) duetopolesof4,shouldbecancelled bythenonsense zerosof theproduct ka. 5. The Bootstrap Problem Equation (10.41) states theintegral equation which isthe planar bootstrap forthefour-Reggeon ring discontinuity. Assuming theexistence ofafamily ofRegge trajectories {a,}, andgiven a knowledge ofthestandard vertex V,5,onecaninprinciple compute thesingle-particle kernel Kanditsprojection 8).Sincethe propagator istrivially known asinEq.(10.14), onecanthen search for solutions Aofthe integral equation. The existence of asolution depends functionally ontheReggeon set{a;}andtheform of the vertex V. The residue ofthe bootstrap equation atany Regge pole j=a, where ae(a,}, yields thevertex bootstrap (10.10) which isperhaps more interesting than theoriginal equation because itcontains only oneunknown function V(t,,t,,t,), given theReggeon set{a,}. Wereturn tothe vertex bootstrap inamoment. First wemust note acertain inconvenient property ofthe functions AofEq.(10.8) andthevertex Gof(10.9). Aside effect ofdoing thehelicity pole expansion isthat these functions arenot -55- normalized inthe sense of the standard Toller M-function discussed inSection (2). Figures 29and 30show schematically how Aand G are related tothe normalized ring amplitudes (ordered M-functions). The functions Fand F'are like the F's appearing inFig. 18and Eq. (C.5). Asnoted earlier, the approximate role ofthese functions istoconvert the Reggeon helicity from the discrete values m,r, p,-+. asinFig. 18tothe complex values (a-n). Accounting for these normalization factors, wenow rewrite the vertex bootstrap inthe extremely schematic form ofFig. 31which shows the bootstrap asanonlinear functional integral equation of the3-point ring amplitude andtheReggeon set{a,}. Inprinciple, this equation should allow the computation ofthe ordered triple- Regge vertex asafunction ofall three arguments. Toour knowledge, this calculation has never been done. 6. Counting Approximate bootstrap calculations using avery small leading Reggeon set{a,}have often indicated that thesingle-particle kernel, with experimentally determined couplings, does not have the strength necessary toelevate the generated output trajectories totheir experimentally observed intercepts. Assuming that this result is not anartifact ofthe approximations made, one must conclude that the peripherality and/or the Regge-expansion convergence assumptions which gointo the multiperipheral model are simply not viable for single-particle production, and one turns instead tocluster production. One replaces the single-particle kernel with acluster oflimited maximum width, but sufficiently broad soastoapproximately ~56- Regge-factorize, even though there isnoRegge pole insuch akernel. This isthe concept ofthe dotted Reggeon, and the program ofFig. 31isthen replaced with that ofFig. 32which, when the left coupling isboldly cancelled onboth sides, gives the famous equation "1 =gg" ofRef. 25, where N=N (flavor). Ingoing tothe multi-particle kernel, however, one encounters certain counting problems which invalidate the diagonalization procedure which led tothe simple equation (10.8). The necessary alterations involve pre-convoluting the cluster/kernel with a propagator ononeside.*°*?7 Toavoid thiscounting problem, we have chosen toconcentrate instead onthe cylinder calculation where there isno counting difficulty. (11) THE CYLINDER ‘The ordered orplanar bootstrap discussed inthe preceding section consists ofsewing together two ordered amplitudes (zero handles, one boundary) inanordered manner soastoobtain the discontinuity ofanother ordered amplitude. Bysewing together ordered amplitudes (h=0, b=1) with acertain well-defined disorder, one may construct the cylinder component (h=0, b=2) ofthe physical 4-point function. Figure 33shows parts ofthis cylinder component inseveral different notations. Figure 33a depicts, inquark diagram notation, aparticular contribution tothe two-twist-pair piece cl)ofthecylinder resulting fromtheunitarity product oftwo 9-point ordered amplitudes. Figure 33bshows thecomplete c(2), but the figure only has meaning interms ofdiscontinuities after -57- theupper andlower Regge expansions have been inserted. ‘These are showninFig.33cwhichisnowdrawnintheringnotation. Finally, Figs. 33dand33edisplay thetopological meaning ofthetwists in the absence ofquark notation, Toconform with the kinematic diagrams like Fig. 15,weshall continue tousethenotation of Fig. 33c. The full cylinder isdefined asthe sumofall itstwist-pair components, c=) cM, q1.1) nel whenthec(2) component shown inFig. 33cisdiagonalized onto angular momentum j,charge conjugation” t,andnaturality 0,one obtains thetriple pole configuration shown inFig. 34(when the simplest assumptions aremade forthe jandflavor dependence of the various elements, and when only the leading helicity pole ofthe leading Reggeon pair iskept ineach Reggeon loop) Mrs gti sgmmyige (11.2) where k=k(t)=kG.t) =f46,+e(t.t,.t})? x[otherfactors). (11.3) Inthephenomenology ofChewandRosenzweig’? thecylinder shifts ofthe£,£",w,and>trajectories aresimple functions of k,which issometimes approximated bysetting j=a. Roughly, the *Wearerelieving tfrom itstradition duty ofrepresenting signature since signature seems toplay such asmall role intheordered S-matrix, andalso because there arealready toomany C'sfloating around in Section (11). -58- shifted f=pomeron has the intercept [in SU(1)] ap(0) =a(0) +k(a,0) . (11.4) Wewish todiscuss the technique used toarrive atthe expression (11.3) for kand tosuggest how kmight more accurately becalculated asahelicity-pole expansion. After first diagonalizing the charge conjugation, wereview aone-dimensional cylinder calcu- lation and then proceed tothe three-dimensional helicity pole analysis. 1,Diagonalization ofthe Charge Conjugation Since thecylinder terms C™)carry zeroadditive quantum numbers, itisdesirable todiagonalize the charge conjugation in addition tothe naturality sothat cylinder poles can beidentified with physical particles. This procedure isvery simple, aswenow show. The ordered ring discontinuities carry orientation indices which have been suppressed throughout this paper. One might write (1,0, |A|2,0,) where a,=#1depending onwhether theordered channel iliesintheclockwise orcounterclockwise Hilbert space.” Asa 2x2 matrix inthis orientation space, Aisdiagonal with equal diagonal elements, (1,0, |A|2,0,) =A85,10," Changing tothecharge- conjugation basis |1,7) =[|1,0,=+)+1]1,0,=-]/WV2, onefinds that (1,7, [A]2,7,) =A6,,1,»80there isnoneedforAitself tocarry at label. Incontrast, thetwisted Reggeon propagator *Palways connects states ofopposite orientation, (1,0,|*P|2,0,) =8,,-0)P-Inthe 1basis *Pisagain diagonal buttheelements have opposite sign, so -59- *pmust carry atrivial tlabel, (a,t/*P[2,71) =bnPe where *pT2op. (11.5) Therefore, the only effect ofcharge conjugation diagonaliza- tionistoaddatlabel totheC(")andtoreplace “P+t*Pevery- where. Bycomparison, theuntwisted Reggeon propagator Pwhich appears inthe planar bootstrap does not mix orientations, soall contribu- tions A™ tothering discontinuity Aarediagonal with equal diagonal elements inthe orientation space and therefore also in the tbasis, assuming the special case ofzero additive quantum numbers along the chain. 2. The Cylinder inRapidity For comparison with the kinematically accurate (though still physically slippery) cylinder calculation presented inthe next sections, wereview here a"typical" rapidity analysis ofthecylinder. For simplicity, only one flavor isassumed instead ofthe three flavors (with 1=2#3 symmetry breaking) usedbyChewandRosenzweig. 1° Interms oftheusual rapidity orChew-Pignotti?® variables, and with the CGL phase-space approximation discussed earlier in Section (8), one writes inthe energy plane the one-twist term of the cylinder asfollows (see Fig. 35): MDete*y,e4) =-sl| pyoty. fSfefa,6(y-x,-g-x,) i i) xfa,Lye PTD ACExD (11.6) -60- with dg, asgiven inEq. (8.10). Here, Aistheabsorptive part of afour-Reggeon ringamplitude ofrapidity width x,,and“PTisthe twisted (no cosine) Reggeon propagator ofgap width gtobegiven below. The label tindicates that the equation has been diagonalized inthe charge conjugation, t=+1. Incorporating the Neveu-Schwarz shift a+a-1, wenormalize our triple-Regge couplings inthe usual way, R(thsty) =8,8,T-a)(-s)* qa1.7) A(tt,s,t3) =Foy&8Cs)", (11.8) and take for the Mellin-projected ring discontinuity aform exhibiting symmetric nonsense zeros, +j -a, j-a (1.9) The presence ofnonsense-zeros inaMellin projection isequivalent tothe absence offixed-poles inthe Froissart-Gribov projection; wewant such fixed-poles tobeabsent because weassume there tobe nofixed powers inthering amplitude R(t*,s,t#). The assumption ofthe first nonsense-zero inaMellin projection corresponds tothe absence ofaconstant term onthe right-hand side ofaFMSR over A(t,s,t#). Byattempting torespect theanalytic structure ofmulti-Regge amplitudes, several authors”? have used somewhat controversial asymmetric FMSR toargue that, ineffect, the amplitude shown inEq. (11.9) should have anonsense-zero onone side orthe other (depending onwhich external overlapping invariant isheld fixed), but not onboth sides; i.e., that the form of(11.9) -61- should beasymmetric. Wefeel, however, that thefour-Reggeon amplitude should beleft/right symmetric, even ifasymmetric NDC isused initsgeneration, andthis isourmotivation fortheform (11.9), though wehave norigorous argument tosupport this conjecture. Oneofthephysical weaknesses ofthecylinder calculation is that small changes inthesmooth (i.e., non-singular) j-dependence oftheprojected planar amplitude, such asnonsense zeros, can cause violent changes intheoutput pomeron location, *°sonocalculation canbetrusted until the low-energy/smooth-j behavior oftheplanar amplitude hasbeen determined fromtheplanar bootstrap. Hopefully, such behavior might becomputed from thehelicity pole formalism. Meanwhile, weshall usetheform (11.9) only asaprototype and continue our calculation.” Thetwisted Reggeon propagator appearing in(11.6) is *ptcg,e2) =H(t]) exp(ga,) (11.10)2 InMellin projection this becomes : ahces)Xpt ps 4+ 2PG) =Fran (a1.11) 2 with sat + -H(t]) =TO-a,) Ta-o,) . (11.12) which maybecompared to(10.12) with (10.16). Now,theCO)equation (11.6) maybetrivially Mellin diagon- alized toyield Meteat sty)= +.+PQ) JagatstcT(tyjt.) =fdo,Alt.5.tL))5=e,A(t,,j,t,)- (11.13) *Low energy data are, ofcourse, helpful onthis point. ~62- Inserting the expression (11.9) wefind j- kG] j-4, Motat, ot|ot 5-4.) [u cyCat) =rey|eilara |——z|8s(aca (11.14) i G-@) 3 with G-o,.). a Paes cekG)=ryfesey) (11.15)Tray Jats&BE)Oe 2 Byemploying symmetric nonsense zeros in(11.9), wehave removed the Regge-cut generating factor ofthe propagator (11.11), and have added another factor (j~ae,) inthenumerator; K(j)isj-dependent. From the diagonalized integral equation forthe full cylinder, c¥a,3,3) =Meta,5,3) +fe,ac.3,2)76,2)€2,3,3), (11.16) orbysimply summing the geometric series the first term ofwhich is given by (11.14), one finds that ja) KG 5-0, Tos- 1 tk) 3 c°(1,5,3)Fay[-(ae G-AG-a=TKD [8s\a-ae)p 1 (11.17) which shows the pomeron (t=+) atthe solution of j=a+k(j) . (11.18) Finally, adding Ctothe planar term Aextinguishes the unshifted pole inthe manner ofRef. 10, j=, jo, 53)+clag - 1 cy ACL5,3)+C053)=Tey[)] J-G@+tk)[eat P 1 3 (11.19) andthe symmetric nonsense zeros appear also inthe cylindrically corrected amplitude. -63- 3. The One-Twist Cylinder Term asaHelicity Pole Expansion ‘Thetypical multi-cluster contribution “)A(E,v,£") tothe four-Reggeon ring discontinuity was given in(10.1) and illustrated inFig. 22. The sum ofall such terms defines the complete four- Reggeon ring discontinuity inthe "energy plane." Ofnecessity, the object Acontains thepropagators onboth ends ofthemultiperipheral ladder. Itisimportant torealize that Acontains these end- propagators inconvolution, sothat, unlike thekernel, A(E,v,£") cannot be written in the form REE) =Py) A) PLE - Only after diagonalization can the end-propagators beremoved asin (10.5). For this reason, itisdifficult towrite cylinder terms — inparticular c“)—intheenergy plane, butveryeasytowrite these terms inprojection, aswenow show. Let usdefine anextremely condensed notation sothat, for example, Eq. (10.1) orits diagonalization (10.2) both read: GOR =pxpxprp Similarly, the planar bootstrap reads K=PKP+PKA [energy plane, see(10.7)] , (11.20) A=K+ KPA [j-plane, see (10.8)] . Inthisnotation, theC“!)cylinder termmaybewritten inthe energy plane as GG)=Axp_KK +pKP,KA +AKP,KP +PKPLKP ,(11.21) x! x! x xKPoy(11. where P,isthetwisted helicity pole propagator (see below). The -64- diagonalization of(11.21) is, inour condensed notation, again (11.21). Once (11.21) has been diagonalized, wemay use (10.5) to expose the propagators sothat EO)=plapK+kK]py{kPA+K]P—(-plane). (11.22) Inserting the planar bootstrap (11.20) twice yields zs). .c PAP,AP (-plane). (11.23) Finally, inanalogy to(10.5), wedefine C)interms of€“) to get oO =apa (j-plane), which, infull j-plane notation, reads [ears] = og Ai x iCv DPF ALD HO)Ryd .ar20 This equation, illustrated inFig. 36, gives the projected one-twist cylinder term interms ofthe projected ring discontinuity Awhich solves thebootstrap (10.8)." From (10.14), x HQ)i rn (11.25)a EA, By where H(2) isgiven by(10.12) with the signature-factor product replaced this time by (10.16). Adding the naturality and charge conjugation labels [see Sections (10.4) and (11.1)], (11.24) becomes (Qpjot dd jo T jo1,3)= 5 J J Ci3)xPnea)ASc.2)[xsi]ay(2,3), (11.26) *Notice thatEq.(11.24) doesnotsaych).KPK. -65- or, interms ofthe lower-case projections of(10.20), M33) =rfeH@)Whalga,2)caday.2.3). (11.27) withw=rG+1earge1-2. 4, Angular Momentum vs. Helicity Wepause tomake afew observations about Eq. (11.27). First, itshould beclear that the "Reggeon propagator" (11.25) isnot directly related tothe angular momentum j,incontrast tothe feeling one gets from the rapidity approximation. That is, the leading helicity-pole propagator has the form 1/(X-h,), not 1/G-@,) asinEq.(11.13). Asemphasized inAppendix E,the variable &which measures the energy dependence ofthe object we loosely call aReggeon propagator isthe analytic continuation ofan azimuthal Euler angle, notacentral Euler angle like @of(¢,0,¢'). Therefore, the correctly projected propagator isafunction ofthe variable conjugate tothat continued azimuth €,namely, the continued helicity 2,and not the angular monentun j. The rapidity formalism with its collinear boosts isincapable ofdistinguishing angular nonentun from helicity, and projects everything onto ahybridized Mellin projection index "J". One feels that asymptotically —i.e., near singularities inthe projection index —this hybridization is acceptable. Even so, itseems unlikely that the low-energy behavior ofaplanar discontinuity could bedetermined from aplanar bootstrap which uses such anapproximation, and the same goes for the cylinder. Forexample, itisjust this distinction between jand thevariable A(which isforced tothevalue h,=Ge,+1)which gives risetothe threshold factor appearing inEq. (11.38) below. -66- 5. Regge Cuts and Nonsense Zeros TheA-plane forEq. (11.27) isshown inFig. 37. Asjis varied, thehelicity contour isrepeatedly pinched between thehelicity pole atA=h, andthepoles ofP'(j+1-A). Each pinch generates a pole injwhich isinturn converted toaRegge cutbythetransverse integration d},. These j-plane poles are, ofcourse, explicit when thehelicity contour in(11.27) isleft-shifted asper (12.12) to give MesoC3)==Dfe.H(2)TG#1+h,)TG+1-h,)oatnn) aio aio *Gn) BR23) (11.28) Keeping only theleading helicity poles oftheleading Reggeons so that h,=a,+0, =id1 (11,29) and setting y= y"=0,(11.28) becomes Me3ora,3) =tfao,(2H2)]TG+a,+1)TGag) ‘00 > 2 Cp om aio aio xAa2 (2,3) (11.30)Gon,92) Aho which may becompared tothe rapidity result (11.13), Osta,3) =«fa,(iy)G-a¢,)7 +aJ(1,2) a9(2,3). (11.31) Whereas (11.31) shows only the first Regge cut, (11.30) exhibits thecomplete family ofRegge cuts associated with theReggeon pair @,,0,. Recall, however, thatiftheupper-case projections an -67- lackfixedpolesatthenonsense points, thelowercasea.have nonsense zeros. Wepresume, then, that these amplitudes infact have astring ofnonsense zeros which cancel all the gamma function poles and thereby eliminate all Regge cuts from the cylinder, just aswe contrived todointhe rapidity model. More significantly, the sane mechanism should remove Regge cuts from the planar bootstrap. Unfortunately, wehave been unable topursue this question due toa technical difficulty which wediscuss inSection (12). 6. The Complete Twisted Reggeon Loop Since the A's appearing in(11.26) are the projections ofring discontinuities, their j-plane singularity structure contains, hope- fully, only Regge poles. Wethen write asanasymptotic series, jaa-> (reri039][Vrerse], us© G-@ (11.32) where the Gare triple-Regge couplings, Isa) =Pal; tythsat) whose normalization was discussed in Section (10.5) and shown in Fig. 30, Near aparticular Regge-pole, Eq. (11.32) reduces tothe form (10.9) given earlier, but ingeneral wewish tomaintain the j-dependence inthe triple-Regge couplings, asdiscussed below. Since (130) couples twoReggeons (a,,a,)toathird Reggeon a, itisclear that the coupling vanishes ifahas the wrong naturality, sowenow drop the naturality label, keeping inmind that (11.32) represents asum only over trajectories ofthe proper naturality. ~68- Inserting thepure Regge pole expansion (11.32) twice into (11.26), wefind fortheprojected one-twist cylinder term pit 3)= Tdasa]stefc} 53 C353) =Diray8,059] praHarOfpew a,a" ul J3:0" x[eas6),(30)], (11.33) where j an j 1 iwo)» amy] Bae [sea cre ea] :‘ao Fore 74 aNXe,Od(11.34) When thehelicity contour isshifted tothe left assuggested in Section (12), the resultant kis j 2n esa) [6 jw(t) =—AR—Yu2) [6(230)]}, [6(2s0')], - avr@re) 2 i Be(11.35) Inour Regge pole expansion (11.32) for the ordered four- Reggeon discontinuity AJ,(1,2), wehave exhibited thetriple-Regge couplings asbeing j-dependent, just asintherapidity version (11.9). Usually Regge couplings arepresumed tobeindependent ofj,e.g-, ; Ya" AyG) =Gray > sowewish tocomment onthis point. The Regge expansion given in (11.32) issupposed tobeareasonable approximation totheexact partial waveamplitude AG?) However, weknowfrom(10.18) that whenk,+0ork,+0[k,arethecontinued cmsmomenta, seeEq.(5.7)], thepartial waveamplitude 0,2) mustexhibit thecharacteristic threshold behavior, 5 ja, -1 j-ay-1 .je 1 cr” rat i (AG,21 =(k,) (k,) [a2h: (11.36) -69- Therefore, inorder that the(finite) Regge expansion (11.32) be accurate, wemust assume that thecouplings also exhibit this threshold behavior, ; Sag .Us@im =a)? [o@ah . (11.37) Wemight then ignore thej-dependence oftheresidual coupling 6". Inparticular, wehave already noted that G'should have nofixed poles inj. Therefore, amodel forthecomplete twisted Reggeon loop k accounting forthis threshold behavior andlack offixed poles is 5rf 5-0, Kyi(t)=1 >>Ffofaod 2H(2) T(@)r(a') , o, c]G03 non} to“0 ar Jpay 14) x[6'(2;0)) (6'(250')] > (11.38)hy hy where wehaveused do,=(2n/f) k"dkdw asgiven inEq.(8.10), and where H(2) isEq.(10.12) with (10.16). Inthepast, expressions forkhavenotshown thisthreshold behavior because theprojected triple-Regge coupling hasbeen identified with thej-independent dual coupling "2 T+)g(t,t,,t.) =T@-a,) 2 Thecomplete twisted Reggeon loop anditsrelation toC) are shown inFig. 38. There, theloop iscross-hatched toindicate that itisacomplete twisted Reggeon loop incorporating theeffects ofall thehelicity poles ofallReggeon pairs. -70- Wemay now compare this precise Reggeon loop toits approximation inthe rapidity model asgiven inEq. (11.15). First, since wehave included morethanoneRegge poleinourapproximation toAJ,thek of(11.38) isamatrix inthespace oftheReggeon set{a;}, whereas (11.15) shows only the leading diagonal element ofthis matrix. Secondly, the usual numerator gamma functions of(11.12) which contain the physical poles ofthe propagator, now appear assines inthe denominator ofthe factor H(2), with the job ofghost removal now incumbant upon the couplings Ginthe sense ofEq. (2.8). Wehave retained the€,,€, factors inH(2) toallow forfermions onthetop and/or bottom ofthe Reggeon loop. For example, the upper and lower Reggeons must both bebaryons inthe contribution tothe cylinder which mixes regular mesons withbaryonium states,°!(seeFig.39). Finally, the kappearing in(11.15) contains only the leading pair ofReggeons (a,,a,), andonly theleading helicity pole corresponding tothat pair, i.e., n,=n) =0. 7. The Full Cylinder Sofar wehave discussed the zero and one-loop contributions to the full cylinder, Eqs. (11.32) and (11.33), which wenow rewrite in an abbreviated notation cMa,3) =g(,>,8, cPa,3) =80D, KPa > wherenowP,=(j=a)?andKj;=Tkyja,+ Tocompute thefull cylinder including the planar part, tas) = cMa,s) , n=0 -71- weremove theexternal couplings onthe ends tomake matrices ofthe c), andwereplace theP;withdiagonal matrices 5 -1 Pay=6Pat ByG-4) : Then Cisageometric matrix series which one sums toget _ aoe [cot"(pr?-YO,(ca,3)],; =(p+pKp+...],5 =[PO-k]=———_ .4 J ij det(P™*-K) ‘The locations ofthe poles ofthe full cylinder arethen determined by det D(j,t) =0 (11.39) where G91; =G-a,(8))8;5-TeE;(t) (11-40) withkj;a8giveninEq.(11.54), Inpractice, onecanrestrict to ‘asmall number ofleading planar trajectories and include symmetry breaking. Ifthematrix space iscrudely limited toonedimension, Eq. (11.40) shows that krecovers itssimple significance asthe shift between thepomeron andplanar Reggeon intheone-flavor model, asinEq. (11.18). (12) FIXED POLES, NONSENSE ZEROS, AND ‘THE HELICITY CONTOUR PROBLEM Whereas the diagonalization procedure described inSection (9) isstraightforward, theproblem ofshifting thehelicity contour in thediagonalized equation isstill, wefeel, anunresolved question. Rather than bury this discussion inthecylinder calculation above, wethought itbest toexpose theproblem clearly inthehope that someone will solve it,andtoshow the drastic assumption wemake in theend. Theproblem described here ineffect blocks thecompletion -72- ofthe helicity pole expansion program. Consider asimpler version ofEq. (9.1), A(g) =fg, B(,) Cle) , (12.1) orits diagonalization ij=f2giciAmt=JeBySue ’ (12.2) with projections asgiven inEqs. (9.4) and (9.5), which wenow write as Jj= J =Oye=feBarZ)Cy) (2=chy) (22.3) i fae orefactwet : Cyr)=féeor&C(E,v,6") (12.4) imi j Jj andsimilarly forA),,,, butforBJ,wehave jo. JBy.fe%@)B,() (12.5) 1 =[Se_ve BO)=fseB(E,v,0) (12.6) Thefunctions @j,,(z), liketheregular Q;(2),havepolesinj andtherefore (itturns out) in\andwu", and this certainly suggests thattheprojections likeCj,,,ofEq.(12.3) mightalsohavethese "fixed poles," although this isnot necessarily thecase. Neverthe- less,itisuseful toconvert fromthefunctions @.totheayeof Eq. (A.13) which are analytic inj,4,u"and have nozeros, atleast for Re(j) >-1. Defining new, lower-case projections asinSection (10), oh ff a) cw) (12.7)aut 4aut2)Sue , -73- wecansaythatifQihasno"fixedpoles," thendy.musthave zeros (nonsense zeros), since cisthe residue ofthe pole inC. (Inthis sense, ¢iscloser tothe Mellin projection than C.) Interms ofthe lower-case projections, (12.2) becomes a.=|gWh#3,due: (12.8) where cH=T(j+#1#APG+1-2). AsA>tia,W>exp(-m/a|), providing the excellent apparent convergence for the helicity inte- gration. One pays aprice toget this exponential damping, however; Hjhaspolesgoingoffinbothrealdirections intheA-plane (see Fig.37).Suppose aisanalytic inAandduhasasimplepoleat, say, A=h=-1+4. One would like tosay that, when the contour isshifted totheleft,thispolemakesacontribution toabn However, thepoles of1)alsomakecontributions, andtomakematters worse, thepolesof duatA=hcanpinchthecontour against allthepolesofP'(j+1-) causing atohavepolesinj("Regge cuts"). SinceReggecutsare unwanted inthe cylinder orplanar bootstrap, wewould like toclaim that the poles ofI'(j+1-A) are cancelled bynonsense zeros inthe projections likeuewhichistosay,theFroissart-Gribov projection Gahasnofixedpoles. Thissoundsreasonable ifdu.isthe projection ofanordered (planar) amplitude where fixed poles mst beabsent sothat the Regge cut discontinuity formulas give zero discontinuity, circularly speaking. Granting thattheproduct ®,gj"hasfullnonsense zerosto cancel the poles of[(j+1-.) and remove Regge cuts, one must still consider the problem ofshifting the contour. Itmust beimpossible toshifttotherightbecause thenonegetsas=0,certainly not -74- desirable. Shifting tothe left yields acontribution from the pole atA=h, but still there are all the poles of['(j+l+A). The conjec- tured form ofthe helicity nonsense zeros ofEq. (2.8) suggests that the zeros inthe A-plane should besymmetric and therefore the poles ofT(j+l+A) are also killed (although the dAconvergence isnow jeopardized bytheremovalofHw). Now, presumably, the integrand ofEq. (12.8) isanalytic in2 except for the pole atA=h (the "helicity pole") and wewould like tosay: shift the contour tothe left, pick upthe helicity pole contribution, and hope the contour integration vanishes asitis shifted off to Re(A) =-». However, intheC'!) cylinder calculation ofSection (11)we found that, aside from the helicity pole, the integrand was symmetric inA,sothat ifwedisallow ashift tothe right, wemust also disallow ashift to the left. The situation isanalogous tothe problem ofthe Sonmerfeld Watson representation which isresolved bythe "Mandlestam trick" of replacing thepoorly behaved functions P)withj-decaying functions likeQ,. InRef.3itissuggested thatasimilar procedure beapplied inthepresent context. Presumably theprojections daarebadly behaved asRe(A)+4”because theayearebadlybehaved. Asshown inAppendix A,onecandecompose ay=Gye+Byye (12.9) whereahaspolesonlyontheright,thoseofT(j+1-a), andis well behaved asRe(A) +-«, ascan be shown byapplying Watson's Lemma totheintegral representation, Eq.(A.18). Defining projections p and €intheobvious way, (12.8) becomes -75- 3.fayr .e a4aaIe[RaatBad[Sue*Sy,ye]:2-20 Now, sadly, one does not really know the large 2behavior ofthe various projections because, looking at(12.7): (a) infinite range integrations canchange asymptotic behaviors and(b)C,,,,(v) also depends onAvia the Fourier projection (12.6). Atthis point one throws upone's hands and makes aguess. Of the four terms onthe right side of(12.10), the fourth term may be harmlessly shifted offtotherightwheretheprojections Bi_yand ©,xhaveatleastpower decay. Thetwocrossterms either cancel, ormay also beshifted off tothe right, also yielding no contribution toan Thefirsttermmustbeshifted totheleft, inthedirection thatByandoryareatworstpowerbehaved. This term picks upthe helicity pole atA=h, giving the final result i = ww. aae 2HByResCope] > (12.11) or, interms ofthe original equation, jk afiores FOAe 2B),*ResCoy] : (12.12) Inthis paper, wehave made the assumption that (12.2) can be replaced with (12.11) or(12.12) inthe following locations: (10.41), (11.28), (11.30), (11.35), and (11.38). -76- APPENDIX A SOME USEFUL FUNCTIONS References 9and 23describe atlength the properties ofthe generalized Legendre functions vilyand®..Herewereproduce only their definitions and basic symmetry properties: Bw ye” czy F(jet4y,jv;veer;52)w 2 2 T(v-u+1) (a2) ; 45(u+v) 5-25 ; +1 <1 G2) =%sPGetrGer- (HH) & ’. Py ; 2F(js14u, elev;25425i) xee a?2. (A.2) T(2j +2) jo. pi Joo. pintPo=Py Pv=Puy , (A.3) J.gi j.@giBy=yy > By=SyBy> where j |PG+tewrGet-vGy Tero ae All variables are general complex numbers. Sometimes wemake use of the following combination: J a) 1)= eMgicenote Bye(8) RyEve") erBays (chv)e +(A.5) Theusual rotation d-functions*” aregiven by[+forIm(z) 20] ij =(4iy™™ odytpiaa) =i Ctra) , (0.6) or -77- , . m'-m msmJ J in& 8(8) =(Gin) (*sin$) —(cos$) F(je1em', ~jem';m'-me1; sin®2) xSa, (a.7) T(m' -m+1) When j,m,m' are all integers orhalf-integers, one has i - jmG2 =C1" ae), 5 yi (a.8)dint (8=m)=(-1) n,m! . Asusual, the complete rotation-group matrix element isgiven by i =p 1)= enim Qi -im'g"Dit(8) Dit(¢58,6") eay18) ~(A.9) Theaccompanying second-kind e-functions®® aredefined by J@ =av™™ @ ee, (A-10) with j - BF aEyy = et™ ol cond ecim'y. (AIDEnt(8) =ER,(u,8,v) e Snmt(ChE) € The dand efunctions have these helicity symmetries: m-m' 43 -@ -.@ Caeat ig =Gag * (A.12) m-m' oj eed «2e cyom! Satm ~©m,-m! . Sometimes itisconvenient touse still another version ofthe second-kind function, Jj(2) i: M Uy(2)a aTG?i-w * . (A.13) This q-function has the advantages ofbeing analytic inj,u,v,and -78- having very simple symmetries: j= gi = = qiGy Wy yy Typ * (A.14) The asymptotic behavior isgiven by in @ .D2Binqy@ =Repay? (A.15) and the relation between qand eis Jv@ =eo™Wht dw (a.16) ‘nm3 im) Wnt , - je ya J2pe j i whereHJ, =HH, ,and H)=T(j+i+)T(j+1-y). Anintegral representation forqisgiven by“ Jgi = ufHYayy(chy)=ufda£(a) (A.17) where 5Oth2a £a)=eM(chy+shvchayS-t] 2*th@/2) |1+e*th(v/2) Sinceayisanalytic inul,andhasnoidentical zeros, thefunction onthe left-hand side ofEq. (A.17) has poles going off inboth directions inthe u-plane. Bydecomposing the integral into two parts, itispossible toproduce afunction which has poles only onthe right, ;9 joa 2HY@a(chv) =ofdaf(a), (A.18) and iswell behaved asRe(i!) +-~. Comparing (A.18) with (A.17), we find Jj .gia giay=Ayr Oy: (a.19) -79- Letting e™= in(A.18) andusing Bateman's*> formula [5.8.2(5)], qmaybeshown tobeatwo-variable hypergeometric function Song -B -Bt igi = j ¥ ¥H)@)y(chv) =4%TGe1-u) (ch3)(sn3) iye Y ccth ¥.F,(a,8,8".7; -th3,-ctn3) TY) where Y= ati =j+2-n, B=j#14r , (A.20) Bi =j+1-a : Thepoles mentioned above arenow evident. Our functions qand q appear inRef. 3asd-functions [noconnection toEq. (A.6) above]: Jjgi = nadAyGye=Tye , (A.21) igi = adHyQue Tay . ~80- APPENDIX B TOLLER M-FUNCTIONS Wepresent here the definition and some basic properties ofthe Toller M-functions used inSection (2). Our conventions differ some- whatfromToller's!! andwillbepresented indetail elsewhere. 1° Asnoted earlier, the M-function formalism applies equally well to thephysical orordered S-matrix connected parts.” AToller M-function representing a2-to-3 amplitude may be defined as follows: -m,m,m,m,m MN’7°" Scyays 2.03: 3,053 4a, 5.a,) A6"(ext) an es yt M3) ot=(elu'(a,) @[p,lu’(a,)®Lp,“lu@)| My ~™s_|Uca,15,Ove,IB,); @.1 where 3 gt(ext) =6°(p,+p,-P,-P,-Ps) > Py=Lapp P;=(m,,0,0,0) , and the constant A= -2nicf isdiscussed atthe start ofSection (4). TheL(a;) arethe4x4Lorentz matrices which acton4-vectors, whereas theU(a;) areunitary operators which represent theelements (0,a;) ofthe Poincare group inthe single-particle Hilbert space. The states |)and|]aredefined anddiscussed inRef. 13;basically they are linear combinations oftheusual |)states which are designed totransform asundotted and dotted spinor representations ofthe Lorentz group. InEq. (B.1) all helicity (spinor) indices are -81- ofthe undotted upper (contravariant) type. Generally there are four kindsofspinor indices: x",x,,x",andx,,whichcanberaised (Gacts onthe left) orlowered (Gacts onthe right) byananti- symmetric metric spinor Gagt =Oh =Gia, =Gm =aCm), where sisthe spin ofthe particle involved. Except onafew occasions weuseonlythex™andx,index types. However, inorder toallow room for explicit spin labels like s,wehave adopted the following notation: $15,555,5, _~MM,M, Meese,” Mameae, =Maom That is, upper indices are written aslower, and lower indices are also written aslower, but with adot underneath, this dot having no relation tothedotsofx*andx, The only properties ofthe Toller M-functions stated here are the invariance and covariance conditions. Other properties such as crossing, TCP, Reggization, etc. will bediscussed elsewhere. !® The statement ofLorentz invariance interms ofToller M-functions isvery simple: Masmemny(24°84885020408) +Myynyepn (sitet 8. (B.2) This invariance condition, immediately evident from the definition (B.1) since the operators U(a) are unitary, states that aToller M- function transforms asaLorentz scalar. The equation isthe same for all types ofspinor indices aslong asboth sides match. Inaddition tothe above overall invariance condition, the -82- Toller M-functions have acovariance condition oneach particle, e.g., (a,,a,,a,,a,8,a,) Mamma,(Hes OE, Sy s*My. 1 . auMpM3MM, (2,+8,54;42,35)Dory(8) (B.3) ‘My mt=-s, an where gisany rotation. The covariance conditions are also obvious from Eq.(B-1), given that therest states |p") transform inthesame wayunder rotations astheusual |p,m) states, while thestates |p™] transform asD*. Toller extends his covariance condition toinclude parity, rotations and parity comprising the complete little group H,ofa rest 4-vector. This matter isdiscussed further inSection (3). Toverify the counting ofvariables, one finds for the general n-point Toller amplitude: nx6 each aj=6variables -nx3 covariances -6 invariance -4 6"(ext) 3n-10 . Finally, the Toller M-functions are related tothe momentum- space M-functions (spinorial amplitudes) ofTaylor!® (stapp!*) by M, (a,,a,,4,,4,,a,) In,m,m,m,m, (41742743994 985 5 ' (0,84)=satmtmrmt Py»PyoPyoP,oP,) TTDe(a), (4) Mosmsmgaraz PPP P57)TTDyeecay -83- where DSi) arecertain spinor representation functions!>?1° of SL(2,C), andp;=L(4,)P,- Toller hasshown that then-point function M,(a...) is analytic in[SL(2,C) xSL(2,C)]" ~[complex Lorentz group)", theonly singularities being reflections viap;=L(a,)p, ‘ofthepositive-a Landau singularities which aretheonly singularities inthep,of theStapp M-functions (e.g., normal thresholds, poles, triangles); kinematic singularities andconstraints arenotpresent. However, whentheM,_(a,.-.) areconfined tocertain surfaces within [sL(2,c)?]", asbyusing thestandard frames ofSection (5),these Kinematic singularities reappear. This isobvious when onerealizes, e.g., that theToller 4-point function, when written asafunction of ge0(3), isanordinary helicity amplitude. -84- APPENDIX C THE HELICITY POLE EXPANSION FORMULA InRef. 9we have derived acertain "alternative" second-kind generalized-Legendre addition theorem and have proved its convergence. This formula, Eq. (2.11) ofRef. 9,when converted from the Qtothe qfunctions ofEq. (A.13), becomes eegi(aye BE’=25 (-p™s _-m hu mj=1 xPG+m) 5 jFOgray G21) Ayr(Za)- (c.1) Therelation between variables (€,z,£') and(z,,a,z,) isgiven by Eqs. (2.8) and (2.9) ofRef. 9. Wenow make the following set of changes onEq. (C.1): E> -ip Et >wiv z,>ishf z,>-ishh wen yhosar (c.2) aoe m-j >nel z> che . Taking j+-j-1 inthe equation which results from these changes leads to -imp -j-1, 5)-irv cp" . enimgh(chb)e =>—r(-2j+n) n=0 (€.3) EG-n) Q-5-1 “5-1 0g xe952mmOshf) a52)(ish) which converges when Re(E) >0. However, from the discussion in Appendix EofRef. 9itcan beshown that &+-€iscompensated in -85- Eq. (C.3) by(u,v) +(-u,-v). After making these changes in(C.3), one may then take (m,r) +(-m,-r) toget anequation identical to(C.3) except that m,r and &are replaced bytheir negatives onthe right side, and this new equation converges for Re(E) <0. Both equations can bewritten simultaneously byintroducing anindex k, :© n im0-5-1 eanFyeit 5 (1) 5 emtgi?(chee =42D)o(xe)>-r¢-25+m) ket n=O x¢l6lG- Gem(she)4374 ish) (c.4) now valid for -»<— <, Asthe last step, the qfunction onthe left side of(C.4) isreplaced with its e-function equivalent [see Eq. (A.16)] sothat, upon defining j 2r(-25+n)(1)Ht5ja Fem) =|S —|GPeglish) ,(C.5) Eq. (C.4) becomes lay 2 gv) 2Gam?Ebr(6,¥) =ED(g)=(+i)zx8(KE) SF 1E1G-™) pig n=0 Setting j=a, then putting asubscript "2" onall variables yields the result quoted inEq. (6.1). From Eqs. (2.8),(2.9) ofRef. 9and (C.2) above, the relation between theBCP(Bargmann) variables (y,f,v) andtheFig. 15variables (£,£,h) may befound: ch=shfshh+chfchhchE , (c.7) -86- evin.[shhchf+shfchh_ché +ichhshé] . (c.8) sh& andanexpression fore*”given byh#finEq.(C.8). -87- APPENDIX D THRESHOLD KINEMATICS InSection (5) westudied the left- and right-side loop equations ofFig. 14inorder tocompute the Misheloff rotation. Here weexamine instead the lower loop equation ofFig. 14, namely a1 "ve, =haf, - (D.1) Analysis of(D.1) inthe manner ofAppendix EofRef. 9shows that shy, che,=sin,sa _.shy, ch=sind,agi > where, asfound inSection (5), ky, sin, =—y (5.18)(-t,) k, sind, =—“y (6.21)(-t,) [as,.t,.t,)]¢sq,= (5.1) 2(-t,)7(-t,) 2 42aya [a(-k},-k.Pi)] shy,= —+A4 —e (5.19) 2kyk, Therefore, ask,+0, @)+k) cht, =——2 =const.2(-t,)* shq,k, (+k) -“ chh,=a =const. *ky.2(-t,)* shayk, -88- and ask,70, ae @,+k) “1 chf, =——"——— =const. xkj2shq, (-t,) *k, @p+K) chh, =—+~—*,— =const.2shq, (-t,) *k, These last four equations are used inSection (10) to find the threshold behavior of the kernel. -89- APPENDIX E THE CROSS-CHANNEL CONTINUATION Inorder tocarry the kinematic structure ofFig. 15from the multiperipheral region tothe physical cross channel where t>0, one must analytically continue intheMandelstam invariants toappropriate new values and perform acomplex Lorentz transformation. Our main purpose indescribing this procedure istoshow that thepeculiar group variables appearing inFig. 15are simply continuations ofthe familiar variables one would use todescribe the large-t Regge limit. Consider, instead ofFig. 15, the single ladder rung shown in Fig. 14.Theinvariants t,,t},t ands,aredefined by te ty? ' 12th=(kK) th=(k)) - 2 -2 t,=(k,) t,=(k,) ' ' 2 Q= (ky+k}) =(+k), t=Q toy 2 Py=Cy-ky) =O-RD , sepe Our goal istostart inthemultiperipheral region oftheReggeon process 1+2 +1'+2', where Qk;sk; =spacelike A(t,t,,t{) =negative P, =future timelike central level = bws frames and wind upinthe cross channel physical region for 2+2' +1+1', where -90- Qk;kp=future timelike A(t,t;,t]) =positive P. =spacelike 1 central level = cms frames. Bws (cms) means brick wall system (center ofmass system). The first step istocontinue all the t's. Figure 40shows a "movie" ofthis continuation, and Table 1describes the movie. Including t,andt,onemayconclude that sare yt Ct)?+i) cd *itt 45(tp i(ty) (E.1) (ty! +ace)’ (s,)% +-iGs,)* . The branch point detours were chosen inthe same way for all variables. What effect dothese changes have onthe equations ofSection (5)? First ofall, Eqs. (5.1) become shv,=(5,+t,-t,)/2(-8,) %(t,)? sho, =(s,+t,-t,) /2(-8,) %(t,)%2 atte ty vt, = Bec,5} cha, =(t,#t,-s,)/2(t,)*(t,)? which are now the correct BCP boost formulas for a2-timelike/1-space- like vertex. More interestingly, Eqs. (5.3) and (5.5) become -91- .' Nscetyt cos0,,, =(t-t,-t}) /2(t,)*(e!) cos®,,, =sh, sho) +cho, cho} cosx, « Butnow cos(8,,,)>1 which implies cos(X,)>1, sowedefine c0s8,,, =cosh n,,, cosX, =coshé, to get ' seetyt chnj,, =(t-t,-t3)/2(t,) *(t3) chn,,, =sho, sho} +cho, cho} che, From these expressions onerecognizes that n,,, =-i8,,, istheusual rapidity boost parameter connecting the rest frames ofthe (now) incident particles k,andkj,andthat£,=-iX,istheRegge variable ofthe0(2,1) Link (1,,,,v,) onthenowspacelike linep,(recall that u,,V, were set tozero). Tocomplete the above description, wenow construct afigure like Fig.14inwhich theparameters n,,,=-i0,,, and£,=-ix,appear explicitly asframe-connecting boosts. The frames inthis new figure (which wedonot draw) form asort ofshadow cabinet for the frames of Fig. 14inexactly the sense ofFig. 21, except that the s'of Fig. 21isnow replaced bythis complex Lorentz transformation a T=(45) - The operator T,given inthe 4-vector space as. 0 0 0 -i o 1 0 0 T= 10 0 1of ° -i 0 0 0 -92- turns timelike vectors into spacelike andvice versa sothat, e.g., k,(inoneofitsspacelike rest frames) . k,=(0,0,0,V-t,) =(0,0,0,iV%,) becomes Tk,=(V%,0,0,0) =kK, where k,isthevector appearing inthenewfigure. According totherules ofEq. (E.1), thebwsversor magnitudes k,ofEq. (5.7) become "ytLoe [ace,t, 27 ,7 SC”* 2(t) which arenowtheinitial andfinal channel cmsvector magnitudes. The2;ofEq. (5.13) isnowimaginary i(t+t,-t!) z= op2(t) * asdesired, since k,inone ofitsbws frames ky=(ykp,0,2,) =(0k,,0,4E,) becomes K,=Tk, =(B,,k,,0,0), which isthenormal formofastandard ems4-vector, except k,points inthex-direction instead ofthez-direction [seeEqs. (5.10) through (8.12)]. Ourimaginary newfigure canbecompleted very simply byexamining theaction ofTontheLorentz generators (see Appendix AofRef. 9): -93- rler Jy ky Jy -iK, jp > 3, Ko -iy (.2) ko is, ko K sothat, e.g., RG)=eofeeey;eeeKd=ByC18;54) =BglMegs) lending credence totheabove remarks concerning thevariables 6andX. Finally, consider thecentral levelboostparameters £,andv, which were soimportant forthediagonalization ofSection (9). According to(E.2), the combination By(E,) By(¥,) By(E,) which surrounds thecluster p,inFig.15,becomes inthenew picture Rx(4)Ry(By)Re) where 6,=48, 8,=-iv, $,=48, (B.3) aretheEuler angles which characterize this rung inthephysical cross channel (8,isthescattering angle, k,+k, =cos(8,)), except that, as already noted, theazimuthal rotations happen tocome outbeing x-rotations instead ofz-rotations. Weconclude with ashort comment about helicity. Equations (E.3) showthatthecentral level £;boosts ofFig.15arethecontinuations -94- (to the imaginary axis) ofthe cross channel azimuthal rotations of theprocess 2+2' +141", Therefore, thevariable conjugate to£,, namely u,, isthe analytic continuation ofthevariable conjugate to 4,»which happens tobethechannel helicity m=m,+m!. Thisjustifies ourcharacterization oftheuorAvariables ascomplex helicities.>© -95- APPENDIX F REATTACHMENT OF THE END-RUNGS How does one obtain from the solution ofEq. (10.8) the physical discontinuity for particles rather than Reggeons? One way isto continue the Reggeon discontinuity Ainthe masses, spins, and heli- cities tothe desired physical points. Unfortunately, the 4-Reggeon discontinuity Aappearing inEq. (10.8) isnot astandardized Toller M-function (see Fig. 29), sothat one may conclude only that the continuation ofAwill beproportional tothe physical amplitude. ‘Analternative and more conventional way toobtain the physical amplitude istoadd the "end-rungs" back onto the multiperipheral ladder. Asthis involves the special end-rung kinematic configura- tions which wehave omitted from Section (5), wesimply state the answer with afew comments. Inthe energy plane, the end-rungs are reattached according to Tyg)=Xfoesfoe,PCO,)Ka(1,JAqy(,5¥,8,)Kyp(,)P,,(0).ae (F.1) Aisthesumofallcontributions oftheform(10.1), K,,(n,) isthe left end-rung, andP_($,) istheconventional helicity propagator associated with the (a,a') channel, i.e., P26.)=fiat) : (F.2) Variable $,senses thechannel helicity m=n,+m) ofthetwo-particle system (a,a'). Thevariables n;appearing in(F.1) arelikethevy appearing in(10.1), but not quite because the end-rungs are always inamixed-basis configuration® which causes z=ch(v) tobetwisted into z=ish(n). -96- The diagonalized version of (F.1) is 4 = duffdus j J Ti(ayb)=rfelFarPn@DKh(@,DDP,DAN,(1,2) 1,2 x jPir(2)King(25d)P4(0) (F.3) where an =|oo_-ime = Pf)={3ePi)onmem,.(F.4) 0 InFig. 26weschematize the procedure for reattaching the end- Tungs toget the physical amplitude. Once one has solved the integral equation (10.8) forAueandcomputes theTaeasin(F.3), the absorptive partThm(S,»t)intheenergy planemaybefoundfromthe usual inversion ofthe Jacob-Wick expansion. However, one may return directly tothe energy plane without reattaching the end-rungs by means ofanexpansion formula which isineffect the inverse ofthe projection (9.4): x+e [email protected].") peyey Qin)c J pl git j«(ShT!Oi@[daly 29,29] (F.8) where si, =——Sinn(2j) (F.6)Ld sinn(j -p")sinn(j +) The contours in(F.5) run upvertically tothe right ofall singulari- ties ofthe integrand. However, the jcontour Ccontains, in addition tothis vertical piece, clockwise loops around the integers and half-integers tothe left ofthe vertical component. Formula (F.5) -97- can bederived from acompleteness relation [q defined inEq. (A.13)] =Hifi (25 qt gi 6(x-y)ites(25+Nesem(25) ANG)Gyo) (xy >1) (F.7) which inturn can bederived bythe techniques ofRef. 9,Appendix G.1. Finally, itshould benoted that the projection (9.4) isprecisely the continuation ofthe usual Regge theory Froissart-Gribov projection toimaginary and ingeneral complex helicities. Formula (F.5) is(the discontinuity of) the Mandelstam-Sonmerfeld-Watson transform, the discrete-helicity version ofwhich was used toget Eq. (2.2) with (2.4). -98- ‘ACKNOWLEDGMENTS Iwould liketothank Evelyn Grant, Deberah Olson, Candy Voelker andDessa Bucksbaum forinvaluable technical assistance inproducing this report; myparents fortheir constant moral support, theU.S. Government andtheLawrence Berkeley Laboratory forpaying thebills, Jaime Millan, Jean-Pierre Sursock andDr.Henry Stapp foruseful and interesting discussions, Ms.Georgella Perry forherconstant surveil- lance, andGeorge Weissmann forcontinually reminding methat there areother aspects oflifeatleast asinteresting asparticle physics. Iwish alsotothank theprofessors oftheBerkeley Physics Department fortheir excellent teaching, inparticular Professors Gene Commins andJ.D. Jackson, andalso Ms.Teri Doizaki forbeing sonice. Finally, andmost importantly, foranexhilarating educational experience Iamindebted beyond measure toProfessor Geoffrey F.Chew whose civility, sharp insight, andinfinite patience Iwill trymy best toemulate. Thework cited inthisThesis hasbeen done under theauspices ofthe U.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.DeTarandM.N.Misheloff, Phys.Rev.188, 2522 (1969). 4, M,Ciafaloni andC.DeTar, Phys. Rev. D1, 2917 (1970). 5. M.Ciafaloni and H.J.Yesian, Phys. Rev. D2, 2500 (1970). 6. N.Mukunda, J.Math. Phys. 14, 2005 (1973); J.Pasupathy and B.Radhakrishinan, Ann. Phys. (N.Y.) 83, 186 (1974). 7. G.F.Chew and C,Rosenzweig, Physics Reports (to bepublished). 8. H.P. Stapp, "Ordered S-Matrix Approach tothe Topological Expansion for Baryons andMesons," LBL-6735 (August 16,1977). 9. 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Rev. Lett. 26, 675 (1971); C.E. DeTar and J.H.Weis, Phys. Rev. D4, 3141 (1971); C.E.Jones, F.E. Low, J.E. Young, Phys. Rev. D6, 640 (1972); H.D. I.Abarbanel and A.Schwimmer, Phys. Rev. D6, 3018 (1972); J.H. Weis, Phys. Rev. D6, 2823 (1972). 37. R.C. Brower, C.E. DeTar and J.H. Weis, Physics Reports 14C, 287 (1974). 38. A.R.White, "The Analytic Foundations ofRegge Theory," CERN Report TH.2136-CERN (February 16, 1976); Proceedings ofthe Les Houches Institute ofTheoretical Physics (June 1975). -102- ae 6. 1 ax a a1 6 14 e as z 2 5 2> 5 9 4 3 4 3 XBL778-2260 Fig. 1.Aparticle pole term contained inthe6-point function. 6 7 ~ 5|6 7 5 e. = 077 521% 7 2 a3 } 4 3 XBL.779-2257 Fig. 2.ARegge pole contribution tothe6-point function. -103- + ay ‘ , ance =‘1 5 3 2 wT g«0(3) 3 2 xa.rs.2061 Fig. 3. Aparticle pole term contained inthe 4-point function. 4}T 4 = 5 ge0(2,1) 32 /2 3xeL T9282 Fig. 4. ARegge pole contribution tothe 4-point function. 345 3, 4 5 i/ \2 2 1 1XBL792256 Fig. 5. Adouble-Regge contribution tothe S-point function. -104- 3 2 3 o9, 9, q 3 9 1 ww 9, XQ 2 7 7 1 1 xaL779.2265 Fig.6.Thestandard Toller vertex inno-Regge, single-Regge, and double-Regge configurations. 4 a,3 3 4 a, a, wast aX 2 1 2 7 xa 779.2083 Fig. 7.Elastic unitarity; X,istheMisheloff rotation for particle 6. -108- 0 1 2 n-1 " pow a, on WKw /WwW 3, 9, B 5 in b a XBL719.2266 Fig. 8.Multi-Regge production amplitude. saa i) Sama MOT) mary Tenth MSS ¥ a - = So Sn Po Pn Say ™Sp SM MOT, mayry MOP, MSS XL779.2252 Fig.9.Spinandhelicity labeling formultiperipheral unitarity Product. Upper variables areprimed versions oflower variables. -106- P95, < v0.9 =By vT™ kot, kyst, NA a xeLrs2267 Fig. 10, The BCP frane triad for aproduction vertex. a aaakatt kath v\Ao: xk ~\.<_ ¢ TN kyoty kaste is* rouse % Fig. 11, Two vertices combined tomake one rung. Pner rs Ki kK hoe te hyf=By ho€ t, — k, ke 2 ¢ oe xa7792250 Fig. 12, Four new franes added tothe rung. -107- ab get kt kK a v= x x :y ony 8 k, k, ae aoexeL7782271 Fig.15,Central level franes £andgadded totherung. ky x ke a ° , .1 6 enwe : v ey iu 8,Rr, 22 8 k, kNE, NEA xo.7192207 Fig.14.Complete 12-frane system describing onerung. -109- a 8 4 f t 8 a 8, XBL 779-2251 Fig. 16. Standard frames for triple-Regge vertex inits spacelike configuration, A(t,t,,t}) <0. -110- vias‘pu *(v35:0). —(Vipm vib; Eqeys(8!) WVeipim J)1PM mT. ? 72PaM3 5;| al) dyn,O%) sp% PIP, PP) 18,0, -O,-1 2520 ](tee)[YahEa,GP Vern, xa792210 Fig. 17. Functional structure ofthe multiperipheral ladder, after Mandelstam trick. om vyy's 98 tycyt? TTFFatigPDVerptimtFatnegng(6) araaPyms TMK IM | 1 qepip,O) -r,a, eyPetite ee om ———([ « - 6GFayanyr, ORDVep.m, Fam, fe) Fig. 18. When E-functions ofFig. 17arehelicity-pole expanded, residual functions Fare grouped totherungs toform Kernels. ThisisthekernelK,>. xoL7782008 -111- a},n} Oe P= a 1,[E(a,-n.)+(@j-n})]2 YateG:F,O0F,8) aE o, ny xaL792088 Fig. 19. The helicity-pole propagator. (a) (b) aint©any eaon Sy =4 = K..=Sp, = «, K 12 : ‘, @,m,(,.)a,n, am oP, Fig. 20. The kernel. xeL782249 “112 s —\—- La Xe Xx| x x [WEwe a ne xeL792264 Fig. 21. Dotted figure shows segment ofmultiperipheral ladder inparity-inverted world. Inthat world, frames shoxn are connected by§'=-£. Kyat) Pol6,) Kyglvz) P3(E,) Kaqlv,) uw 2 3 a a b c \d e f Xe x % & Ve &, Ya 1 2 3 4 [v=B, €=By) xa79254 Fig. 22. The S-particle or3-cluster contribution tothe 4-Reggeon amplitude. -113- 8 By f. 95 gov sg BM YM Bs MS BS ujouw uij ajwg ut xeL7792259 Fig. 23. Functions B,C,Dare convoluted togive function A. The variables onthe bottom line are conjugate tothe boost parameters asshown. The diagonal variable jis angular momentum, variables and Aare helicities. aca) K(1,2)P(2)K(2,3)P(3)K(364) wojow wGAG Goa x»}xx=xx]xx]xx]xx eo ove By SM &Yy&y Fig. 24. The 3-particle/cluster contribution tothe 4-Reggeon amplitude, inboth energy-plane and j-plane. xaL779258 -114- A(1,3) K(1,3) K(1,2) P(2) A(2,3) jou wget vgoo4 jo Bapx=xxpooe*Ghee Voks gov sg eve By & v, Bp Fig. 25. The bootstrap equation inboth energy- and j-planes. XBL 779-2263 Tla,b) P(a)K(a,1)P(1)A(12)P(2)K(2,b)P(b) m jm mojow oj wg om a a’ b' a’ a b a b oy 6 a Fig. 26. The reattachment ofthe end-rungs (see Appendix F). XBL 779-2255 -115- ain! ajn} atin! ajny a an an, am OM, Fig. 27. The Regge-pole expansion ofthe unnormalized 4-Reggeon ringdiscontinuity. yg)779931 vo wad a O)- = 6 kK oP 6 Fig. 28. The vertex bootstrap. XBL 779-2313 -116- Fig. 29. Relation between discontinuity Aand theToller- normalized 4-Reggeon ring discontinuity. The R-notation isthat ofRef. 7. XBL 779-2318 Fig. 30. Relation between the cut vertex Gand the Toller- normalized ring function R. XBL 779-2319 -117- XBL778-2316 Fig. 31. Highly schematic bootstrap for the 3-Reggeon ordered amplitude. XL779.2317 Fig.52,Vertex bootstrap with"dotted Reggeon" replacing thesingle produced particle ofFig. 31. (0)SAAraaranr°JuUuL JUL JUL erKR EO ()36gSeeeeen TOU CEONus notations (see text). XBL 779-2314 -119- \ e (x) . :se s PR -| 3 en +s= (x)2 § . a 5 3 et 3 t 3 z 3 2 a g ()g é 3 : = 3 3 3 5 E 3 @(-|: 7q z (2) Co cy -120- () he Oh =soso <> <>< ><> y x, 9 x, Fig.35.Thec‘!)cylinder termintherapidity model. XBL 779-2321 Meq,a) (1,2) *P(2) (2,3) uiju ui a ijut Xx XX = XX XX gov e By & % & Fig.36.TheC!)cy1inder termwithexact kinematics XBL 779-2322 A f+ 542 xX xXXXKX "x xX X KX XK XK Fig. 37. The complex helicity plane for Eq. (11.26) or(11.27). Ifhelicity pole hisinthe right half-plane, contour should bedeformed tothe right. The other poles may -122- “ => 2 ~ a,a" ree) k rouroz Fig.38.co)maybeexpressed asadoubleReggesuminvolving thecomplete twisted Reggeon loop k.Thecross-hatch Indicates that the loop has been sunmed over all helicity poles ofallpossible Reggeon pairs. Fig. 59. Quark Line structure oftwisted Reggeon loop coupling meson inpropagator aresettoone-half. XBL779-2326 -123- 7 by die Be Bo 8“sh eu B22s Bes :- Eas ead es SF dt z. ages a gag¥ BS,et See ° = Baked ebeae BRSee BRen. 2Egeey ° “Bees ease feo an FFued o so eg2 goe8 ween SE wipes w gol8S+ set zo EE i oes iF eeacss gfas3 . Baeis « e¢ veo aa <ry -124- TABLE 1, OE AGe Gey -A(tyt,01)#(-t)# (5,9 BoGt ety +i(a)# (ty (5,9 BoGt eet +i(ea) HO Gs,y Coty sant) (+a) sit) Gs) dDsitty® —sacey* +i(+a) +i(t)? (+s,)% D+i(t,)% +i(et) +i(+a) +i(t)™ -it-s,)% eee