Phil Correspondence with Reuben Freeman

i Covves pendence wale Reuben Freeman ! . Feb[01978 DR. Pri LucuT Phopiesdept,lanvensilyofutoh SuetLacd, Q ; uTRH 84//>) ») >,Phil LucktMev time PipesDept[envesity 4telcultsLekeCllet F412 1 i) ww) O @oper, —-feb.gfae___,ThanlesSoe“Hee”—eey2:bpeur,thes»Ienfess_fastThe noturadadPraayheverydetach(neroe likely to__eloSol)putTol haveSome_ eemmerls towhich gowcerant terespon. _eT aworrieddPhat,deonyshatat.ReggeBaty—asnatfemelow“beenglossedover. perfortsteerp ts__pethags_cverle aptmastic teat!swetping.tidaprobeterdineteesartok: Geta uoing._gprdeap:trenvee veriables2ffords = _Ibn!te, __tealyhe~idetic —obzcurtele lm.Posheeden Ivm. cmcerna! |aboutthe:_—fapesivemol eoonly atlawed_|neblele, abuasmarates hesOASfem tliat ropecenloging—Chenusls. 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OpoowerCanta wgicLira)amas awa”Valin, gyrodysisleeeorranQe)#ztyoreyayymeuch dak \=armen, godowih2gSayectnnss (re,mono), Mesa|om ondnnsre }he iatdagonehSrA frectbecner @ Getnqeds,| : 1 i Jones/Freeman papers: . 1.“foward‘aGeneralProof...”.ThinkofA;inthisway:pagent»Itlooks @sceanA,amplitude, andyouknowthenharetwoSteinman terms.Inmynotesfor this, the first .FJ paper, Ihave shown how these two Steinman terms are contained inWeis' forma (2). Ifyou-4re doing some kind ofunitarity calc, youwantaspartofyourintegration this: SawdiscAgiWt) «Ie,woaresort offaking things byconsiering this pictgre: Bynot bothering toReggeize the upper particlés, they keép itsimple. But it isnot clear tomethat you réally want |the integral quoteds Accepting that you dow,nt this quanityt, youcanexpand A,interms oftwo 2/reggeon/2-particle amplitudes, ofie for each Steinman term, |These Q's are each normal analytic in x,butforfixed 5,5, sayaremitiplida bycertéin (-u?)"** factors which cause kinematic cuts intheM@-plane. ‘Thus, inanyintegrals youencounter, theyclaim thatyourM°integrals ofrefgeon amplitudes willalways beweighted bysuch kinematic factors, This perhaps jeads toasymmetric projections. Whg dont Iencounter these things inmymethod? e@2."Gribtria forgoodFNSR..". Heretheydoitagain,consider Agusingslightly @ifferent notation with V)and¥,Steingm terms. Mynotes there summarize it well. Again youwant aFHintegral ofdisc Ag;youwrite outthetwoterms; the left hand cut thing has to go away to ge} cut killing. Remember that when theFMintegralovsomethinggivesyouonly[2powers,theMellinprojectionof that something has nonsense-geros. Still Iamnot very clear on how to cannect thisstufftome. | 3.“Cancellatiom of.." Third paper ofthe sequence. Here they are finally considering the4-reggeonringctcetint ofFig.2.Youareclearlygoing tohave some kind ofKMintegral aspartlof your unitarity phase space. Tmt integration appearss in(2.1). Again, daseethatyour4-reggeon amplitude is weighted by acertain factor. This ig that problematic asymmetrizing factor. Comments: none ofthese 3papers makes mention ofMellin projections. They are doing energy~plane bootstrap, soall|they care about isfinite mass integrals inu°, ButnowIwill golook atF2vanal? papers. ® | 4.Freeman andZarmi: HereIsuddenly gettheideathatthe"weighting" factor apises from the asymmetric NDC. I-can see this arise inmyown rapidity notesattheend.oftheFZpaper.ThatiswhyIwouldsay:assymetric projetion cane from asymmetric NDC. Bythe way, the three FJpapers never actually doa dootetrap, but refer toFZand P2V. Idea: canperhaps identify theweight factor with threshold factor? 5.F2V: they show onpage 8that, when you write aMellin projection of Diso&g (with noweight), andwhen youhold sands,fixed, that weight factor maginally appears. Itisofcourse caused bytheform ofthelimit ofAg interms ofAL. Ie,thelimit given in(2.5). There ,youseethat x A(amplitude with noprime) isanalytic inN°butyoupick upextra or) factor from theregge behaviors. This seems tobetrue without a@list of Steinamn terms, Then Iguess the question becomes: whg doyou want to integrate Agrather than App directly?. . . e ! =JoserDigdad)WugltBechieS--h ee coe ee a2k Vaasados‘make whealwerAg|Sed}Daeaedneiva oaweeaegs re . eens ||Geer —\ isdsidawothQW, 2\° OU ReSG Atl Rae Ante basoa0waCHG,Res——|Agiaanalytic. us28hwuanions. eeeee ee ee i oS an a | TSR gages yee ree CDed cathgute ©) SER AgSO Ae). ne —24 rs —_ i eeDurliga Lo2ee -.-|‘eee Be woe ALNaan3CareLyte-9 ih)=2QD0(4~¥)._ ee oo O(yts-gy -.=-.ani LORI EAEHyer|. ee oe ed weee ee SOG EXD KOH- _Skee oSHKBEDArey~The 9-3)Ge)Oat)|TeMe»roe:Naneo“eal —!weee8 +ne weeee LFwae de oe eee ee -- I31¥Se ey i a nin dy -——- 'cree eeneninenietatts.- LAF AC)Mesh») =ROQAGH3) eee iegeoeee—[Axto~tne! Sa te|--- \ 2wee ee ~~ Be ee. ay ae-oeI:AG=.Yan8)gros et‘NexAwe’ eoeSanbepegadeAG) —_______ pak A-*%=FuSe| siee |—YaeSERGD9G)—— a —+7+ sae |ee ———(FA=AR_toae . yy le Oellien - __|owLadd...LN -wee= +_AQ= SaxAQRE=8) eG-A) -nga. a|-oeSane Jogo) moe ee a Se ete) >napaeen le=.e(R-a).e ft.YerAce! -cere ci“ wee [ModAegean. bo.oe.—<—-- ——-Cee ee ree ener ane-iAy.VaqeSog-pe™Sawe™ eG. |oeSacaenedShyeMagnsege-@~-|-. a«er>|anally,Leahigeanegexsie™om Augeaes _fF)hk ay ~~Poa a -oeba“3XKeea -~\yrsPtee-_—got taedando AF adeage € - aoe ee woo eeapiae-Sea a |TTS Jenne oPNagSeong | { 7 Get2hSatisbsRodten| a RSE KD£eBSacestn@ meth —pectSiqanet ceeeee. - OoFaasSaga — .=f70-8+ROOee SyeSjaagt_ a 60MS onde es a.apiandaa.a mg Viva ladelasebaddtyWTgo eSANBe ae-|ekeey=arprweeenoene -_1Alaa=4hEAQ .|y=osteseAge) Wa a hveGS A eat eseA =r =H GIS AY alDa eo a ee 6 | SoaNawayslouselebeaeesayessvothownsee ReSRRSA |s) a) ~~ak. ILVIV Cree3dasach\LmpplysaAgeeee TL ASSRGARG) eo SY SeaeadShwearwmmckiat (eulet) ee Owe ee a e» Bae dyoddx,antl)sledabreY= and Xreo Vice At beet +hee igiwgropbcte AAGG) xzo Ge,S29). AaesMaslt] >anytin |ee(Wate dtsenedl, onmeit e. : eoGlaim:whenIstartwiththeClL.inReubenspaper,Iamabple tosuccessfully diagonalize jthething atthecost ofhavingtouse"asymmetric" Mellinreson Theseprojectionsthenhavenonsense zeros which correspond + Reutens "Good FMSR". Thus, I amable tocast theClequation into theusual diagonal-j form, but at aprice. However, inmyexact kinematic analysis, Iamagain able toexactly diagonalize onto j;morover, the rojections involved here are . perfectly symmetric. Ihave made nt“high energy approximation". Reuben would say, regarding hisanblysis: look, ifyouwant toproperly take account ofthe low-energy behavior ofamplitudes, you cannot use thehigh-energy Mellin approximatibn, identifying 8,withsf.Ifyou dothis andthereby avoid using my|FNSR, youwill getthewrong andwer for your Cl tern, Reuben wouldgoontosay:look,myPHSRarebasedonstrict observation , ofanagyticity; when Iexamine analyticity, Ifind noconstant terms ine theFMSRandtherefore Iwillhavejcut-killing. ® CereeXnas! | bepet pa ert minal --iia rae ~“epee~ope eatRadius (ou n+ ee ed esthet battsteat Pilincboncaing _vonibles __eT ernenhy lene2_pentenlte ep aes |Erawraeedk >Creche_. ee is oe ew |—Hetshing filaus totofFPNledsPlyGiza(072)4pm a Ve { "> 3re oe NC=oSa aaaaQ> ;-- a | Go oe Ee et) SS 0en 2 EGS es Rb ee a ee Wie elt heaxkenal packet legs TOGA rc ar a aneYet)feAAass) cc Se hs SH AdidineFion _Sehvter. $2wakedsitatuatbewade__th otdey, te.__get__fue—phgarthUdcemfinusly S23 uitaot wea—faaicigedey, veotte Of ptr + -_Ae Sh “BE khte lovioh becmfirule ginarY nek tek tee pheoh becomtimadiggoer2 |(6)HeyyHe2 Be1pAlogSbc=)=Pby)9.)gg). 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HI)FeePO 4)ec) ~~takales “Bimaatbebe.sidesofG)repuuresFal fl pplSen)Aglo) A ;*(6)Loa 0) Hinteaktepicblemei_solyed mtmety6vedenio LA 2 oeWat!L(3-4}e.‘Qf jMea-t bie___egertG4)dseraeaefMs Tus[db agReet ==GYOYfgAA eo..-—SG er =7 — ~a BergeYatsacl, <=).-Ge wes!dng! todturrnaybeveel,hired teteOf . . | aiidjhe,42) oe |pMecgCjHaat)8) en ONBad, Agere chaningtotopLighgues]eblisinwhele_elirnig65 teAftpicksgetefetes9AATof[lov2 _Kew_pestme haMeTeLbes(Sag) (fetsrteer,.)=AUkXe tog Depsista) 2ApeGerke:fan[¥SqTp tg eg AEoeNp ysien).TN glfltiaety)—en 6)ae goeeSe ne — btety tetevighthgles OY 7a) cwhich isLyewhet onegets teenaivevelRvchenge wefelyTimin__mee,aPrejaeforamidy ep.GO) VeonenagtetsAreduohinclie “he Standtyjenscan oeee onhe Frpumabvige Heinysesom.of)pyPaMMe(Sta) at88 2G) PTds53s AL Usaths)eho,teHe be oo ee oo eetke elAA imate)=1ea)Geta)Az)F952 BR GrykyGy" y TS Mw weechatbig + eeee enbo,) IWp) HY ati ens neae—fd, oodtoeETE aTGnahy)4,4, —_—-— ot a ResponsertoReubenFreeman'sLetterofMarch| 6.Hesaysthatifyougothrogehacertainaesandthenshiftthej'contourtothe right, you get aresult which isequivalent fothe dlagonalized-equation result. Whereas ifyou shift the contour tothe left, you get adifferent result. Hethen gives alittle example and shows hbw the two shifts give different results. Myresponse isthis: itiscorrect toshift to|the right. Reason: the projection intheJ*integrand isexpo decaying intherigh} half j'-plane, sooneyoupick upthe pole atj'=J, there are nomore singularities| out there and you the residual contour then vanishes. Thus, the diagonalized equation 4scorrect. Ifyou shift the contour tothe left, and|pick uponly the first regge pole and thenignorethebackgroundcontour,[email protected]:justlookatyour ownexample, Youassumed aspecific projection withanonsense zero ina atJ=\y.Inyourresult shifting totheright, thannonsofse-zero isfeithfully reproduced on both sides ofthe equation; ifyou shift tothe/left, thab nonsense zero only appears onthe left but not onthe right. Ifyou were tocontinued toshift the contéur tothe left and pick upall the lower regge pole contributions, you would eventuglly regenerate that nonsense zero. @ Ofcourseinpractice eithermethodisOK.|Ie,thetwoexpressions yougive agreewhenjisnear4. i | | " —|— fae. Tt. FuserimdecwymrLylanieedusbeeyprefrsin deedYack: Ss det) deg oara Ase)=Yas, ce)(8)fa) V@)AG) —)x 1 whaTada asi-B selsSal melaDogueik ere caquoten. View, enggor Sia la : Yana*Aey 0 () Oy =W., P\nt,Dameuntcmerophnr PD>7Y, =-_Oe |& I eemateyHes» nit AGs>=SsneGey" VG)sa) ©)dssGS)ACs). @Ses Neo comune Onsk ehadkyoleg=2 AG)=BS(2) >) JSecaSouge,bee . Act Anelka Xeg-% ot. dnd~| (San Cr)(3) hye 4geeVaxx aGu) =Go 5SOME oa) Gace) Ou. yt AyskCaleg ath <aAgd=S38) SHE) VE).96)4.50Abode) « |MW.QuonsodwRwily | q=WyG9"VEDge (8) e [a~ve} QeiaDaunlercrmen Oamartin leootodeg. Creeding aaddayne Ero umd)JdraccderallCiaSaNode?JKRWateOwaaey/DadDebt,WHtobegoagen: Yokamesak VG) =StsUG\VOr8). Qosqretbion ae e Aisa=SitSas." {eadecsh \ *SassosAs)‘Ges)Po (SasSSa) \ AG=4-452) pM Mae AG=te-tes-t) pry oles ~T=y-t Auli)= Sissi“A,Co.s2)| (5) e * ~jree onaslgrourda-e) =Sis,[ENA Ce) ee | e ; SD rr eS j =\~%-1. SS ~\7v eet] ha(raed)=Si5,(55ipArs(ss)( > agit |\ . {ry AwGd\ =am WayGs)Aes) Windateaotagen withator | ages @sO)=Yas,c5°" ForTaf Que aca Qa bwth DA v=dat) osar \x;Vota fy sry oy Aas(i~ae-) QuinOrenaaaDewarledsdeOx.VoyWeapomstia wheasymoan! j —_ 4 i AGe)=SonSd.wey'}(oarsqesyy . e °; hed roksst ekeakan ~ \ ( >xSaskGe)&Ga~wbaGs)Res(Meet) : | Qaea Bam doGes3: »ass©(Seg~SaSs) (ss)“e eh deni . Jdantune:(ree e AGe)SWJaca osie-snt C>)Pawdems jindee RokQubevanQamonarneryercdunfstOho e Aba)=aTYass PMOL (sayPt Sdss Sus, we| \ \'oeea (SsjeeSedhugh \ds,-(s3) =(Sef) Gee) jio%e Uw +f “T er -3- | - 4 1 ° 1 iSa ~v-\(oot 6as S)= SyGS Ss 3m Sy ®23) DASGs) a4 > "3 .at we OMIRalyeden) =345m SSabepo Sp Ark a po Gd 7 ot Ok,=aheftDee!embomSaftfogh ; 2 AGa)=Sar Qase {C3 ¥ ah hnA~ tardye An!shGufr) re)A Gt Fades Kege% a= 9.fsGs) p 11 What has been done tothis point? Reuben has tektn abootstrap equation with the left cluster deltad atsome fixed mass, hence nocounting problem. Hethen takes the largeSpqlimitofbothsidesandrequireses thattheypeconsistent. Assuming thenonsensezétoorFSRorwhatever youwanttocallit,he4abletogetconsistency withacertainboostrap condition. Fine. Then, hetries toredothesamething byfirst expanding theobject ACs) ontoitsMellin projections, and then heexecutes the s,i#tegral. Finally, heclose§°the Jcontourandtakesthelarge 85,limit. Thisprecudee léaddtothesamelittle bootstrap condition.Thus, hehasshofm thatifyouaredoing sodething asymptotic,you canfreely expand onto Mellin projections, doyour energy integral, jthen close your contour and regain the same little vertex bootstrap. Sofarsogood.Nocontradictions claimed ofnbticed byme.ItisOKforhimto @etine nis om Neliin projection however hewished. { { ~ BerAHe~ AGesy= SarVG)GS Gag Yas (sy?"AG,2) Se Anygoeleue Sdsa(seyee -\ iss BRSSss(32)ere Gu) WecadeVeRCyeeeseg2)dehcewrensrodenndwily Diesodinardeben Nerhile »§ndlgeA4bah AQ). Se aus=7OWLvadoadhaespudbong). e “Bwsiet. Qtek det “ \oaee AGs)=San,GO"V6) SSagaGeoGre) 4Ses)Mereotes ah 37he J . | AKA K+ iret)2S88 ie\ sot eGay yowilee ~)fey2bay i { a) apt N seu!Ko-ordone) =YD VEeeGayhesGio1) » toxf Qi) Ved wo> a Cnasdpoe6 Cet 4ok SoAly-dysdee2) =SMecA){(Ss)+ieh™6* e *SD(R%) SseenttceeFOCBaldpe Myrepatabotiaadstheag,fea, . Aordendgnt) =SA|CE vehey|3k[AsGao =DarenalsaepSoge)anal ° | 1 a, os \ VS s Acondensation ofReubens letter. =I @1.First,rotscotao1=8,5=,andletsgetridofthetransverse integration, putinV=some constant fof thé*potential. Then the Yootstrap equation becomes: Sz=S. Sy=S. a A@= tS. Sts,AC) : ( ' 2.Nowletusassumethattherede@nonsense“asfollows:o oo oe ‘yas,AGO=o.Thaw,Ach-\s®\aAse)(*) Nowassume these asymptotic limits: ! x '. AG)=aqS\ YamaeySoke,==L &—ez . AG)=gq,S. Jamis Oeysn;ka#-\ . 4 C4Xe, Xen >ams- -Vs" \ae.Lands§| e S | ColomFEQa8a. Ss) x!) @&~ee) »yeste, | Ca~ae) { Thisthenistheresultantbootstrapcondition,tsequation(6). :3.Nowletssimulate Reuben's secondmove,whichj8this:expandA(s,)inequation (*)above: ,ek . y= eriPA |Pee=! AG@\= arda’Gd AG‘he-1) ‘ \ To» Waymacklencamp, alo2 Ase)=BaYayv@dAG). eeol Vow Qa:AG)=-¥sSas.3Saves AG) ®s 2 ; a~ =--Ys* aye AG).Vase6) T+isone Cs) . Trl Wak,sighdo|sveee AGL)>AGHbeeped yeas of: : yeah eke RO AG)=~Vs"™AchyAG+4a-!) G)odQe) YorsthSaf.ShadeOok: Pe a) ASG~| ~~)Nal 2 RQta)=Sd.Ge)“K923e) ay Oe x\y Se)2 . voa) > é KAass AG)=3? =4Vs™ pyC (a=e) e=twSang\oohsivg. Now finally wecome toReubens third move. Westart with equation (**) above and project the whole damn thing: what happens? How doyou deal with the two contours Jjandj',ie,where arethese variables relative toeach other? Atthemoment, the J*contour lies tothe right oftheleading singularity inA(s5). Maybe Ishouldbemoreexact about this. Thecontour mustbetotheright of}=alpha, because that isthe leading pote inthe. projection shown inthe equation .Now weare allowed toproject with some very large positive jtogetgood convergence. Thust . o .: . AySés3"AG) =AG)5° ‘ eS ‘ at ; Senn Ag=VAP)gtKatee)Suhoe- Y gees A VWs \Kot *®. >AO=Y¥aha Al\rekee1) @xr¥). Gide) ({!- ~a- Sundlsary agracontrobaesotlasBBD]LNs.So AQ) = ce y 5 & (AY =Voy _? lve WASarleortshe, (e-HeeyeyS) ire) Iwould never close thisthing totheright; Reujen merely points outthatifyoudoso, you get anequation which looks tohim like the diagonalized Mellin equation that Ipropoung 0vehenently. Nemely: AG)=AG8) | G-4e) | Hethen arrives atthis conclusion: since this result disagrees with the result you get bydoing directs integrations, itmust bewrongj} and since this result looks like my iagonalized equation, mydiagonalized equatiors must bewrong. Qrovever, Iwouldclaimthatinthissituation thpdiagonalized equation iswrong. But‘that isonly because his boostrap equation iswrong. Ifhehad the right bootstrap, IthinkitwouldMellindiagonalizeinthiswry.iioks !e Aay=Fey"YasAe Ithink the fact isthat this thing isnot aconvplution equation, and therefore itwill Hot diagonalize inthe “trivial "way. lets see whathappens: 2 aS ABS=VO) --ds,ACs2)882)e INS gy a: ~ ~-\ 2~s-\ . Yas00)=AG=Sas,fdJasO°) \ y <a A Sh (die o 4 a-y -=\) de=TSas,65°M6)«SSANsSS) e ail2) ' —— iad(Sa) oe\ = AG: yao, Corvesfeseneah Reuben Freeman Phil. Lucht |