On a generalised bootstrap principle

arXiv:hep-th/9304065v116Apr1993DTP-93/19YITP/U-93-09ONAGENERALISEDBOOTSTRAPPRINCIPLEE.Corrigan1,P.E.Dorey2,R.Sasaki1,31DepartmentofMathematicalSciencesUniversityofDurham,DurhamDH13LE,England2TheoryDivisionCERN1211Geneva23,Switzerland3UjiResearchCenterYukawaInstituteforTheoreticalPhysicsKyotoUniversity,Uji611,JapanAbstractTheS-matricesfornon-simply-lacedaffineTodafieldtheoriesareconsideredinthecontextofageneralisedbootstrapprinciple.TheS-matrices,andinparticulartheirpoles,dependonaparameterwhoserangeliesbetweentheCoxeternumbersofdualpairsofthecorrespondingnon-simply-lacedalgebras.Itisproposedthatonlyoddorderpolesinthephysicalstripwithpositivecoefficientsthroughoutthisrangeshouldparticipateinthebootstrap.AllothersingularitieshaveanexplanationinprincipleintermsofageneralisedColeman-Thunmechanism.BesidestheS-matricesintroducedbyDelius,GrisaruandZanon,themissingcase(f(1)4,e(2)6),isalsoconsideredandprovidesmanyinterestingexamplesofpolegeneration.11.IntroductionAffineTodafieldtheory[1,2]isatheoryofrscalarfieldsintwo-dimensionalMinkowskispace-time,whereristherankofacompactsemi-simpleLiealgebrag.TheclassicalfieldtheoryisdeterminedbythelagrangiandensityL=12∂μφa∂μφa−V(φ)(1.1)whereV(φ)=m2β2rX0nieβαi·φ.(1.2)In(1.2),mandβarereal,classicallyunimportantconstants,αii=1,...,rarethesimplerootsoftheLiealgebrag,andα0=−Pr1niαiisanintegerlinearcombinationofthesimpleroots;itcorrespondstotheextraspotonanextendedDynkin-Kacdiagramforˆg.Thecoefficientn0istakentobeone.Ifthetermi=0isomittedfrom(1.2)inthelagrangian(1.1),thenthetheory,bothclassicallyandafterquantisationisconformal;withthetermi=0,theconformalsymmetryisbrokenbutthetheoryremainsclassicallyintegrable,inthesensethatthereareinfinitelymanyindependentconservedchargesininvolution[3].Inthe‘realcoupling’Todatheory,thefieldsaresupposedtobereal.However,therehavealsobeenrecentstudies[4]oftheclassicalsolitonsolutionstotheequationsofmotionfollowingfrom(1.1);forthese,thefieldsarecomplex.Thediscussioninthisarticlewillberestrictedtothereal-couplingtheories.Asquantumfieldtheories,thereal-couplingaffineTodafieldtheoriesfallintotwoclasses.Therearethosebasedonthesimply-lacedrootsystemscorrespondingtothediagramsfora(1)n,d(1)n,e(1)n,andtheothersbasedonthenon-simplylacedrootsystems.Thesefallintodualpairs(dualinthiscontextmeaningthereplacementαi→ˇαi=2αi/α2i),namely,(b(1)n,a(2)2n−1),(c(1)n,d(2)n+1),(g(1)2,d(3)4),(f(1)4,e(2)6),exceptfora(2)2nwhichisself-dual.Thesimply-lacedrootsystemsarealsoself-dualsinceα2i=2.Basedonabootstrapprincipleandanumberofcheckswithinperturbationtheory,ithasprovedpossibletoconjecture[1,5,6,7,8,9]theexactS-matricesforeachaffineTodafieldtheoryassociatedwithaself-dualrootsystem.Thesehavemanyinterestingpropertieswhichhavebeenextensivelyreviewedelsewhere.Inthecontextofthispaperitisintendedtoconcentrateonjustacoupleofthem.2Thefirstimportantpropertyconcernsthebootstrap[10,11].TheS-matrixelementSab(θa−θb)correspondingtotheelasticscatteringofapairofparticlesa,b,isamero-morphicfunctionoftherapiditydifferenceΘ=θa−θb.Forrealrapidity,Sabisaphasegivenbyaproductofelementaryblocks{x}definedby[7]{x}=(x−1)(x+1)(x−1+B)(x+1−B)(x)=sinhΘ2+xπi2h
sinhΘ2−xπi2h
(1.3)where0≤B≤2,inthesimply-lacedcases,thereisevidence[1,5,7,8,12]forB(β)=12πβ21+β2/4π,(1.4)andh=Pr0niistheCoxeternumberforthechosenrootsystem.Intheself-dualtheories,Sabhasfixedpositionpolesandmoving(couplingdependent)zeroesinthephysicalstrip(0≤ImΘ≤π),andmovingpolesandfixedpositionzeroesoutsidethephysicalstrip.Theoddorderpoles,withacoefficientequaltoitimesafunctionofBwhichispositivethroughouttherangeofB,0≤B≤2participateinthebootstrap.Thepolesindicatingaboundstate‘fusing’ab→coccuratpreciselytherapiditynecessaryforenergy-momentumconservationgiventhattheparticleshavemassratiosidenticalwiththosederivedfromtheclassicallagrangian.Moreover,suchfusingsoccurifandonlyifthereisacorrespondingthree-pointcouplingbetweenthethreemasseigenstates(a,b,c)intheclassicallagrangian.Themagnitudeofathree-pointcouplingisalwaysclassicallyproportionaltotheareaofthetrianglewhosesideshavelengthsequaltothemassesoftheparticlesparticipatinginthecoupling.Theprecisenatureoftheodd-orderpolesinaparticularscatteringmatrixelement,andtheexistenceofevenorderfixedpolesonthephysicalstripareexplicablewithinperturbationtheory,intermsofLandausingularities.Detailedcheckshavebeenmade[13]forsecondandthirdorderpolesbutnotfortheothers(uptoordertwelveinthetheoryassociatedwithe(1)8).However,eveninthelattercasestheoriginofthepolesisknowninprincipleinthesensethatsomeFeynmandiagramsintheperturbationexpansionhavebeenidentifiedwhichcontributetoeachofthem.Besidesthebootstrap,themassesandtheeigenvaluesoftheconservedquantities[6,14]areknowntobecomponentsoftheeigenvectorsoftheadjacencymatrixofthesimplerootsαi,i=1...r,afactdiscoveredclassicallyforthemasses[6,7,15]andwhichispreservedinthequantumfieldtheory.Thepossiblecouplingshavebeencharacterisedsuccinctlyin[16]whereitisnotedthattheparticlesofanaffineTodafieldtheoryareeachassociated3withanorbitofasimplerootundertheactionoftheCoxeterelementwintheWeylgroupoftheselectedrootlattice.TheS-matr