Group theory factors for Feynman diagrams

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1、UM-TH-98-01NIKHEF-98-004GrouptheoryfactorsforFeynmandiagramsT.vanRitbergena,A.N.Schellekensb,J.A.M.VermaserenbaRandallLaboratoryofPhysics,UniversityofMichigan,AnnArbor,MI48109,USAbNIKHEF,P.O.Box41882,1009DB,Amsterdam,TheNetherlands1February2008AbstractWepresenta

2、lgorithmsforthegroupindependentreductionofgrouptheoryfactorsofFeyn-mandiagrams.Wealsogiveformulasandvaluesforalargenumberofgroupinvariantsinwhichthegrouptheoryfactorsareexpressed.Thisincludesformulasforvariouscontractionsofsymmetricinvarianttensors,formulasandal

3、gorithmsforthecomputationofcharactersandgeneralizedDynkinindicesandtraceidentities.TablesofallDynkinindicesforallexceptionalalgebrasarepresented,aswellasalltraceidentitiestoorderequaltothedualCoxeternumber.Furtherresultsareavailablethroughefficientcomputeralgorith

4、ms.arXiv:hep-ph/9802376v120Feb199811IntroductionAsthenumberofloopstowhichperturbativefieldtheoriesareevaluatedincreases,thegroup1structureoftheindividualdiagramsbecomesmoreandmorecomplicated.Thisproblemhasbeenrecognizedmanyyearsagoandonagroup-by-groupbasissomever

5、ycompactalgorithmswereproposed[1]fortheircomputation.EspeciallyforthedefiningandtheadjointrepresentationsoftheclassicalgroupsSU(N),SO(N)andSp(N)thesealgorithmscanbeimplementedrathereasilyinasymbolicprogramthatwillthengivethecolortraceofadiagramasafunctionofthepar

6、ameterN[2].Thedisadvantageofthesealgorithmsishoweverthattheseresultsgivenoinformationaboutgroupinvariantsandhenceitisonlypossibleforverysimplediagramstogeneralizetheresultssuchthattheyarevalidforarbitrarygroupsandarbitraryrepresentations.Henceadifferenttypeofalgo

7、rithmisneeded,ifonewouldlikeamoregeneralanswer.ThatsuchinformationisusefulcanbeseenfromsomerecentcalculationsinQCD[3]inwhichtherepresentationintermsofinvariantscouldshowimmediatelywhyextrapolationsoflowerordersinperturbationtheorycouldnotbesuccessful.Inadditiont

8、hepresentationintermsofgroupinvariantsismoregeneralandneedshardlyanynewworkwhenoneneedstoapplyitfordifferentgroupsorrepresentations.Theneedforthiskindofgeneralityiscle