2005年直线对撞机国际研讨会

ThresholdResummationforTop-QuarkPairProductionatILCJ.P.MaITP,CAS,Beijing2005年直线对撞机国际研讨会,TsinghuaUniv.Content:1:WhyIsResummationNeeded?2:QCDInteractioninCoulombRegionandEffectiveTheory3:Resummation4:BriefSummaryRef:A.H.Hoang,M.Beneke,………hep-ph/0001286and…..1:WhyIsResummationNeeded?Abouttop:1995:FoundatFermilab,Mass:~180GeVOneof6quarkspredictedbySM,themostheavyoffundamentalparticles!Becauseitisheavy:Heaviestknownelementaryparticle:180GeVmeasurepropertiesofleastknownquarktopquarkmassconstrainsHiggsmasssensitivetonewphysicsshortlifetime:probebarequarkComparingwiththehadronizationscale~150MeVandthetimeforformingatoponium:1/mt!!atop,onceproduced,itwilldecayquicklysothatthereisnotimetoformanyhadroncontainingthetop!!!Andtheinformationfromtheproductiontothedecaycanbestudiedwithperturbativetheory!!But…………..Acleanwaytostudytop-quarkisatane+e-collider(ILC)Measuringthedependence:Scanningarangeofs,onecandeterminethemass.Interestingis:Atthethreshold……..Attree-level:InthelimitTheamplitudeisfinite.Butthephasespaceisproportionaltov!Therefore,attree-levelthecrosssectioniszero!Determiningthezero,themasscanbedetermined!Isittrue???One-loopQCDcorrection:ThecorrectionissingularasThesingularityiswell-knownasCoulombsingularity,itreflectsthattheforcebetweenthetopandanti-topisalongrangeforcebecausethegluon’smassiszero!.SameasinQED,knownasCoulombpotential…….Addingthecorrection:Athigherorders:Eachgluon-exchangegivesafactor:Thecross-sectiontakesthegeneralforminperturbativeQCD:Thecross-sectionisdivergentinthelimitIfweresumthesedivergentterms,wemayhavefiniteanswer……..(?)AhintfromQED-studyinolddays:AresumofthoseCoulombsingularitiesisequivalenttosolvetheSchroedingerequation,itleadstothatae+e-pairisformedintoaboundstate,i.e.,finite!Butnow,wemustdoitinabetter,modernway…….A.Identifywhichgluonsareresponsibleforthesethesesingularities,momentumregionB.Useaneffectivetheoryfortheinteractionwiththosegluons,NRQCD…….C.Resum……..2:QCDInteractioninCoulombRegionandEffectiveTheoryInthelimitWeneglectalltermsproportionaltovinnumerators….MomentumRegion:Hard:Soft:Coulomb:ThecontributioninthelimitisproportionaltoIntheCoulombregion:Neglecting(q0)2ThesingularityisgeneratedbytheinteractionbetweenaquarkwithasmallvandagluonwithmomentaintheCoulombregion.(thatiswhythename…….)Now,thereisastandardwaytodescribeinteractionsindifferentmomentumregion,oratdifferentenergyscales.FromthepathintegraloffullQCD,onecanintegrateoutthosedynamicalfreedomswithmomentalargerthan:Hard:Theintegrationcanbedoneperturbatively.ItleadstoaneffectivetheoryNon-RelativisticQCD,NRQCDThiseffectivetheorydescribetheinteractionwithmomentaaround::two-componentfieldfortopquark:Anti-topCoulombregion:ToobtainaneffectivetheoryfortheCoulombregion,thedynamicalfreedomswithmomentaareintegratedQuarks,gluonsGluonsOneobtainsfromNRQCDanothereffectivetheory,calledpotentialNRQCD,PNRQCDItdescribestheinteractionintheCoulombregionwiththeCoulombpotential.IfonesolvesPNRQCDperturbatively,onehasallthoseCoulombsingularities.Ifsolvedinanonperturbativeway,theyareresummed!ItreallylikesQuantumMechanics!!!3:ResummationThetotalcross-section:InthelimitonecandefinetheCoulombGreenfunction:Witharathercomplicatedmanipulationonehas:Thecorrectionissuppressedbypowersofv!Alsosuppressedbypowersofv!ThepropertiesoftheGreenfunction:+NLOCorrectionAgain,ifonesolvestheequationwithperturbativetheory,onehasallthoseCoulombsingularities.Butonecansolvetheequation“exactly”andobtainfiniteresultsfortheGreenfunction!Includingthewidth:Regularizingthesoftphysics.Sofar,theresummationisdoneforNNLO:SolvingtheequationwiththepotentialV,oneobtainstheresult.Itisfinite.4:BriefSummary•Theproductionratenearthethresholdcanbepredictedpreciselywiththeresummation.“Naïve”perturbativetheorygivesdivergentresults.•Detailedstudyshows:100/fb,includingexperimentaluncertainties,thetopmasscanbedeterminedwitherror<50MeV