•The fourth order loop amplitude

mesonstop = CreateTopologies[1, 1 -> 2, Adjacencies -> {3, 4, 5}, ExcludeTopologies -> {SelfEnergies, WFCorrections, Tadpoles}] ;

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

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

[Graphics:../HTMLFiles/index_45.gif]

ampFC = CreateFCAmp[loopinsert] ;

LeafCount /@ ampFC

{586, 5390}

amplFC = (tmp = DoSumOver[(# /. channel /. {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


........
................................................................

Length: 64

LeafCount /@ amplFC

{88, 324}

amplFC

{(5 i !, _ 0^( ))/(48 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2)) + (i !, _ 0^( ))/(12 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ K^(ó  ))^2)) + (i !, _ 0^( ))/(48 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ η^(ó  ))^2)), (i (m _ π^(ó  ))^2 !, _ 0^( ))/(16 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ π^(ó  ))^2)) + (i (m _ π^(ó  ))^2 !, _ 0^( ))/(48 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ η^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ η^(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^( ))/(24 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^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^( ))/(48 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ K^(ó  ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ K^(ó  ))^2))}

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

..

LeafCount /@ ampreduced

{89, 581}

ampsimple = (Simplify /@ Collect[# // Expand, {A0 | _B0}]) & /@ ampreduced

{-(5 A _ 0 ( (m _ π^(ó  ))^2 ) !, _ 0^( ))/(48 π^2 (f _ ϕ^(ó  ))^2) - (A _ 0 ( (m _ K^(ó  ))^2 ) !, _ 0^( ))/(12 π^2 (f _ ϕ^(ó  ))^2) - (A _ 0 ( (m _ η^(ó  ))^2 ) !, _ 0^( ))/(48 π^2 (f _ ϕ^(ó  ))^2), -(B _ 0 (p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2, (m _ η^(ó  ))^2, (m _ η^(ó  ))^2) !, _ 0^( ) (m _ π^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^2) - (B _ 0 (p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2, (m _ K^(ó  ))^2, (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 + p _ 3^2 + 3 p _ 3 · p _ 4 + p _ 4^2) !, _ 0^( ))/(24 π^2 (f _ ϕ^(ó  ))^2) - (B _ 0 (p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 + 4 p _ 3^2 + 12 p _ 3 · p _ 4 + 4 p _ 4^2) !, _ 0^( ))/(48 π^2 (f _ ϕ^(ó  ))^2) + (A _ 0 ( (m _ π^(ó  ))^2 ) !, _ 0^( ))/(12 π^2 (f _ ϕ^(ó  ))^2) + (A _ 0 ( (m _ K^(ó  ))^2 ) !, _ 0^( ))/(24 π^2 (f _ ϕ^(ó  ))^2)}

ampinfinitiesfull = VeltmanExpand[#, ExplicitLeutwylerJ0 -> True, ExplicitLeutwylerSigma -> True, B0Evaluation -> "jbar"] & /@ ampsimple // Simplify ;

Converted by Mathematica  (July 10, 2003)