•The fourth order tree amplitude

mesonstop = CreateCTTopologies[1, 1 -> 2, Adjacencies -> {3, 4}, ExcludeTopologies -> {SelfEnergyCTs, WFCorrectionCTs, TadpoleCTs}] // DiscardCT // Union[#, AddExternalLegs[#, ExternalPropagators -> 1], AddExternalLegs[#, ExternalPropagators -> 2]] & // Flatten // AddCT ;

Paint[mesonstop, PaintLevel -> {Classes}, AutoEdit -> False, SheetHeader -> False, Numbering -> False, ColumnsXRows -> {2, 1}] ;

mesontreeinsert = InsertFields[mesonstop, {Scalar[2][0, {i1}]} -> {PseudoScalar[0][0, {i2}], PseudoScalar[0][0, {i3}]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

Paint[mesontreeinsert, PaintLevel -> {Classes}, AutoEdit -> False, SheetHeader -> False, Numbering -> False, ColumnsXRows -> {2, 1}] ;

[Graphics:../HTMLFiles/index_84.gif]

[Graphics:../HTMLFiles/index_85.gif]

[Graphics:../HTMLFiles/index_86.gif]

amp4 = (CreateFCAmp[mesontreeinsert, Sum -> True] /. D -> Sequence[] // PropagatorDenominatorExplicit // DoSumOver[#, Drop -> {}] & // SUNReduce[#, FullReduce -> True] &) /. subpar /. udrules // ScalarProductExpand // SUNReduce // Simplify

{-(128 (3 L _ 6^( ) + L _ 8^( )) (!, _ 0^( ))^3)/(p _ 3^2 - (m _ π^(ó  ))^2), -(128 (3 L _ 6^( ) + L _ 8^( )) (!, _ 0^( ))^3)/(p _ 4^2 - (m _ π^(ó  ))^2), (128 (L _ 8^( ) (m _ π^(ó  ))^2 + L _ 6^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^3)/((p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2)), (128 (L _ 8^( ) (m _ π^(ó  ))^2 + L _ 6^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^3)/((p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2)), (64 (8 L _ 6^( ) (m _ π^(ó  ))^2 + 4 L _ 8^( ) (m _ π^(ó  ))^2 + 4 L _ 6^( ) (m _ K^(ó  ))^2 + 3 L _ 4^( ) p _ 3 · p _ 4 + L _ 5^( ) p _ 3 · p _ 4) (!, _ 0^( ))^3)/((p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))}

Converted by Mathematica  (July 10, 2003)