•The fourth order loop amplitude

mesonstop = (CreateCTTopologies[2, 1 -> 2, Adjacencies -> {3, 4, 5}, ExcludeTopologies -> {SelfEnergyCTs, WFCorrectionCTs, TadpoleCTs}] // DiscardCT // Union[#, AddExternalLegs[#, ExternalPropagators -> 1, SelfEnergies -> False], AddExternalLegs[#, ExternalPropagators -> 2, SelfEnergies -> False]] & // Flatten) ;

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

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

[Graphics:../HTMLFiles/index_36.gif]

[Graphics:../HTMLFiles/index_37.gif]

[Graphics:../HTMLFiles/index_38.gif]

ampFC = CreateFCAmp[loopinsert] ;

LeafCount /@ ampFC

{172, 205, 172, 205, 213, 5446}

amplFC = (tmp = DoSumOver[(# /. {p3 + p4 -> -p1, -p3 - p4 -> p1} // MomentumExpand // ExpandScalarProduct // MomentumCombine) //. (SUNDelta[ExplicitSUNIndex[a_], SUNIndex[b_]] * c__ * d_SumOver) :> (Times[c, d] /. {SUNIndex[b] -> ExplicitSUNIndex[a], b -> a} /. SumOver[_Integer, __] -> 1)] ; Print["Length: ", Length[tmp]] ; WriteString["stdout", "\n"] ; (WriteString["stdout", "."] ; SUNReduce[#, FullReduce -> True] /. subpar /. udrules // Simplify) & /@ tmp) & /@ ampFC ;

Length: 8


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Length: 64

Length: 8

Length: 64

Length: 8

Length: 64

LeafCount /@ amplFC

{133, 136, 133, 136, 127, 376}

amplFC

{(5 i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(3 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ K^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(12 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ η^(ó  ))^2)), (5 i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(3 π^4 (q _ 1^2 - (m _ K^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ K^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ η^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ η^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(3 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ K^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(12 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ η^(ó  ))^2)), (5 i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(3 π^4 (q _ 1^2 - (m _ K^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ K^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ η^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ η^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)), (5 i (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(3 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ K^(ó  ))^2)) + (i (!, _ 0^( ))^3)/(12 π^4 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2) (q _ 1^2 - (m _ η^(ó  ))^2)), (i (m _ π^(ó  ))^2 (!, _ 0^( ))^3)/(4 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (m _ π^(ó  ))^2 (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ η^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ η^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i ((m _ π^(ó  ))^2 + 2 p _ 3 · p _ 4 - p _ 3 · q _ 1 + p _ 3 · ( p _ 3 + p _ 4 + q _ 1 ) - p _ 4 · q _ 1 + p _ 4 · ( p _ 3 + p _ 4 + q _ 1 ) - 2 q _ 1 · ( p _ 3 + p _ 4 + q _ 1 )) (!, _ 0^( ))^3)/(6 π^4 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2)) + (i (2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 2 p _ 3 · p _ 4 - p _ 3 · q _ 1 + p _ 3 · ( p _ 3 + p _ 4 + q _ 1 ) - p _ 4 · q _ 1 + p _ 4 · ( p _ 3 + p _ 4 + q _ 1 ) - 2 q _ 1 · ( p _ 3 + p _ 4 + q _ 1 )) (!, _ 0^( ))^3)/(12 π^4 (q _ 1^2 - (m _ K^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ K^(ó  ))^2) (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 4^2 - (m _ π^(ó  ))^2))}

ampreduced = (WriteString["stdout", "."] ; OneLoop[q1, #]) & /@ amplFC ;

......

LeafCount /@ ampreduced

{176, 200, 176, 200, 176, 898}

ampsimple = (Simplify /@ Collect[# // Expand, {_B0 | _A0}]) & /@ ampreduced ;

ampinfinitiesfull = VeltmanExpand[#, ExplicitLeutwylerJ0 -> True, ExplicitLeutwylerSigma -> True] & /@ ampsimple // Simplify ;

Converted by Mathematica  (July 10, 2003)