•The seventh diagram

melsimplified3 = CheckF[dum, "4MesonsSVertexR.m"] ;

melsimplified3 // LeafCount

126126

melsimplified4 = Collect[melsimplified3, {_Pair, _ParticleMass, _DecayConstant, _CouplingConstant, _SU3Delta}] ;

melsimplified4 // LeafCount

101535

LeafCount /@ List @@ melsimplified4

{2330, 2330, 2330, 2330, 2330, 2330, 32966, 34292, 20296}

mel = Table[WriteString["stdout", Length[tmp = (melsimplified4 /. {I1 -> 3, I2 -> 6, I3 -> ii, I4 -> ii} // Expand)], " "] ; ((WriteString["stdout", "."] ; SUNReduce[#, Explicit -> True, HoldSums -> False]) & /@ tmp) // Simplify, {ii, 8}]

911 ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................911 ...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................362 ..........................................................................................................................................................................................................................................................................................................................................................................1205 .....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................1205 .....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................340 ....................................................................................................................................................................................................................................................................................................................................................792 ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................939 ...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

{-(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) (4 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4)))/(3 (f _ ϕ^(ó  ))^4), -(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) (4 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4)))/(3 (f _ ϕ^(ó  ))^4), -(i (3 c _ 5^( ) (m _ K^0^(ó  ))^2 + 2 c _ 2^( ) (p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 + p _ 2 · p _ 3 + p _ 2 · p _ 4 - p _ 3 · p _ 4)))/(3 (f _ ϕ^(ó  ))^4), -(i c _ 2^( ) (6 p _ 1 · p _ 2 + p _ 1 · p _ 3 + p _ 1 · p _ 4 - 7 p _ 2 · p _ 3 - 7 p _ 2 · p _ 4 + 6 p _ 3 · p _ 4))/(6 (f _ ϕ^(ó  ))^4), -(i c _ 2^( ) (6 p _ 1 · p _ 2 + p _ 1 · p _ 3 + p _ 1 · p _ 4 - 7 p _ 2 · p _ 3 - 7 p _ 2 · p _ 4 + 6 p _ 3 · p _ 4))/(6 (f _ ϕ^(ó  ))^4), -(2 i c _ 5^( ) (m _ K^0^(ó  ))^2)/(f _ ϕ^(ó  ))^4, -(2 i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) (4 p _ 1 · p _ 2 - 2 (p _ 1 · p _ 3 + p _ 1 · p _ 4))))/(3 (f _ ϕ^(ó  ))^4), -(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + 2 c _ 2^( ) (3 p _ 1 · p _ 2 - p _ 1 · p _ 3 - p _ 1 · p _ 4 - p _ 2 · p _ 3 - p _ 2 · p _ 4 + p _ 3 · p _ 4)))/(3 (f _ ϕ^(ó  ))^4)}

mel1 = SUNReduce[#, FullReduce -> True] & /@ mel /. {p4 -> -p3, p2 -> -p1} // Simplify

{(i (4 c _ 2^( ) p _ 1^2 - c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (i (4 c _ 2^( ) p _ 1^2 - c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (i (2 c _ 2^( ) (p _ 1^2 - p _ 3^2) - 3 c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (i c _ 2^( ) (p _ 1^2 + p _ 3^2))/(f _ ϕ^(ó  ))^4, (i c _ 2^( ) (p _ 1^2 + p _ 3^2))/(f _ ϕ^(ó  ))^4, -(2 i c _ 5^( ) (m _ K^0^(ó  ))^2)/(f _ ϕ^(ó  ))^4, (2 i (4 c _ 2^( ) p _ 1^2 - c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (i (2 c _ 2^( ) (3 p _ 1^2 + p _ 3^2) - c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4)}

mel2 = OneLoop[p3, #] & /@ Join[FeynAmpDenominator[PropagatorDenominator[Momentum[p3], ParticleMass[Pion]]] * mel1[[{1, 2, 3}]], FeynAmpDenominator[PropagatorDenominator[Momentum[p3], ParticleMass[Kaon]]] * mel1[[{4, 5, 6, 7}]], FeynAmpDenominator[PropagatorDenominator[Momentum[p3], ParticleMass[EtaMeson]]] * mel1[[{8}]]]

{-(π^2 (4 A _ 0 ( (m _ π^(ó  ))^2 ) c _ 2^( ) p _ 1^2 - A _ 0 ( (m _ π^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), -(π^2 (4 A _ 0 ( (m _ π^(ó  ))^2 ) c _ 2^( ) p _ 1^2 - A _ 0 ( (m _ π^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), -(π^2 (-2 A _ 0 ( (m _ π^(ó  ))^2 ) c _ 2^( ) (m _ π^(ó  ))^2 - 3 A _ 0 ( (m _ π^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2 + 2 A _ 0 ( (m _ π^(ó  ))^2 ) c _ 2^( ) p _ 1^2))/(3 (f _ ϕ^(ó  ))^4), -(π^2 c _ 2^( ) (A _ 0 ( (m _ K^(ó  ))^2 ) (m _ K^(ó  ))^2 + A _ 0 ( (m _ K^(ó  ))^2 ) p _ 1^2))/(f _ ϕ^(ó  ))^4, -(π^2 c _ 2^( ) (A _ 0 ( (m _ K^(ó  ))^2 ) (m _ K^(ó  ))^2 + A _ 0 ( (m _ K^(ó  ))^2 ) p _ 1^2))/(f _ ϕ^(ó  ))^4, (2 π^2 A _ 0 ( (m _ K^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2)/(f _ ϕ^(ó  ))^4, -(2 π^2 (4 A _ 0 ( (m _ K^(ó  ))^2 ) c _ 2^( ) p _ 1^2 - A _ 0 ( (m _ K^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), -(π^2 (2 A _ 0 ( (m _ η^(ó  ))^2 ) c _ 2^( ) (m _ η^(ó  ))^2 - A _ 0 ( (m _ η^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2 + 6 A _ 0 ( (m _ η^(ó  ))^2 ) c _ 2^( ) p _ 1^2))/(3 (f _ ϕ^(ó  ))^4)}

((Plus @@ mel2 // VeltmanExpand[#, ExplicitLeutwylerJ0 -> True] &) /. udrules /. gellmannOkubo /. _LeutwylerLambda -> 0 // Simplify) /. toEtaRules /. _RenormalizationState -> Sequence[] // Simplify

1/(27 (f _ ϕ^(ó  ))^4) (π^2 (2 c _ 2^( ) ((log((m _ η^(ó  ))^2/μ^2) - 9 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 - 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (27 log((m _ K^(ó  ))^2/μ^2) + 16 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + 9 p _ 1^2 ((5 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (7 log((m _ K^(ó  ))^2/μ^2) + 4 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)) - 3 c _ 5^( ) (m _ K^(ó  ))^2 ((15 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 4 (6 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)))

To get amplFC, first evaluate the relevant part of the notebook "KSPiAmplitude.nb". Check that the number selected below (9) corresponds to the graph calculated "by hand".

amplFC = CheckF[dum, "KSPiLoops.m"] ;

amplFC // LeafCount

77658

dum = Table[amplFC[[7]] /. {SUNDelta[SUNIndex[i1], SUNIndex[I1]] -> 1, SUNDelta[SUNIndex[I2], SUNIndex[i3]] -> 1} /. {I1 -> 6, I2 -> 3}, {I3, 8}] ;

dum1 = (cou = 0 ; tmp = Expand[#] ; WriteString["stdout", "\n Length: ", Length[tmp], "\n"] ; (++ cou ; If[IntegerQ[cou/100], WriteString["stdout", cou, " "]] ; SUNReduce[SUNReduce[SUNReduce[SUNReduce[#]]]]) & /@ tmp) & /@ dum ;


Length: 866
100 200 300 400 500 600 700 800
Length: 866
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Length: 362
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Length: 1187
100 200 300 400 500 600 700 800 900 1000 1100
Length: 1187
100 200 300 400 500 600 700 800 900 1000 1100
Length: 348
100 200 300
Length: 841
100 200 300 400 500 600 700 800
Length: 903
100 200 300 400 500 600 700 800 900

dum2 = SUNReduce /@ dum1 ;

res1

{(8 c _ 2^( ) (N _ 22^( ) p _ 2^2 + N _ 23^( ) (p _ 1^2 + p _ 2^2 - p _ 3^2)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), (8 c _ 2^( ) (N _ 22^( ) p _ 2^2 + N _ 23^( ) (-p _ 1^2 + p _ 2^2 + p _ 3^2)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), -1/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (4 c _ 2^( ) (2 N _ 20^( ) p _ 2^2 (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 N _ 21^( ) (p _ 1^2 + p _ 2^2 - p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - (N _ 22^( ) + 2 N _ 23^( )) (p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 3^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) + p _ 2^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)}

dum3 = (dum2 /. subpar /. D -> Sequence[] // ExpandScalarProduct) /. p2 -> -p1 - p3 // ExpandScalarProduct // Simplify

{-(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(24 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^+^(ó  ))^2) (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(24 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^+^(ó  ))^2) (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i (3 c _ 5^( ) (m _ K^0^(ó  ))^2 + 2 c _ 2^( ) (p _ 1 · p _ 3 + q _ 1^2)) (!, _ 0^( ))^2)/(24 π^4 (f _ ϕ^(ó  ))^2 (q _ 1^2 - (m _ π^0^(ó  ))^2) (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i c _ 2^( ) (p _ 1 · p _ 3 - q _ 1^2) (!, _ 0^( ))^2)/(8 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (q _ 1^2 - (m _ K^+^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i c _ 2^( ) (p _ 1 · p _ 3 - q _ 1^2) (!, _ 0^( ))^2)/(8 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (q _ 1^2 - (m _ K^+^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i c _ 5^( ) (m _ K^0^(ó  ))^2 (!, _ 0^( ))^2)/(4 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (q _ 1^2 - (m _ K^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(12 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (q _ 1^2 - (m _ K^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), -(i (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) (6 p _ 1 · p _ 3 - 2 q _ 1^2)) (!, _ 0^( ))^2)/(24 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2) (q _ 1^2 - (m _ η^(ó  ))^2))}

Plus @@ dum3 /. udrules // Simplify

-(i (c _ 5^( ) (5/(q _ 1^2 - (m _ π^(ó  ))^2) + 8/(q _ 1^2 - (m _ K^(ó  ))^2) + 1/(q _ 1^2 - (m _ η^(ó  ))^2)) (m _ K^(ó  ))^2 + 2 c _ 2^( ) ((7 p _ 1 · p _ 3 - 3 q _ 1^2)/(q _ 1^2 - (m _ K^(ó  ))^2) + (3 p _ 1 · p _ 3 - q _ 1^2)/(q _ 1^2 - (m _ η^(ó  ))^2) + (5 p _ 1 · p _ 3 + q _ 1^2)/(q _ 1^2 - (m _ π^(ó  ))^2))) (!, _ 0^( ))^2)/(24 π^4 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))

dum4 = OneLoop[q1, #] & /@ dum3 // Simplify

{(A _ 0 ( (m _ π^+^(ó  ))^2 ) (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(24 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ π^+^(ó  ))^2 ) (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(24 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ π^0^(ó  ))^2 ) (3 c _ 5^( ) (m _ K^0^(ó  ))^2 + 2 c _ 2^( ) ((m _ π^0^(ó  ))^2 + p _ 1 · p _ 3)) (!, _ 0^( ))^2)/(24 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ K^+^(ó  ))^2 ) c _ 2^( ) (p _ 1 · p _ 3 - (m _ K^+^(ó  ))^2) (!, _ 0^( ))^2)/(8 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ K^+^(ó  ))^2 ) c _ 2^( ) (p _ 1 · p _ 3 - (m _ K^+^(ó  ))^2) (!, _ 0^( ))^2)/(8 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ K^0^(ó  ))^2 ) c _ 5^( ) (m _ K^0^(ó  ))^2 (!, _ 0^( ))^2)/(4 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ K^0^(ó  ))^2 ) (c _ 5^( ) (m _ K^0^(ó  ))^2 + 4 c _ 2^( ) p _ 1 · p _ 3) (!, _ 0^( ))^2)/(12 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2)), (A _ 0 ( (m _ η^(ó  ))^2 ) (c _ 5^( ) (m _ K^0^(ó  ))^2 + c _ 2^( ) (6 p _ 1 · p _ 3 - 2 (m _ η^(ó  ))^2)) (!, _ 0^( ))^2)/(24 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 - (m _ K^0^(ó  ))^2))}

dum5 = ((Plus @@ dum4 // VeltmanExpand[#, ExplicitLeutwylerJ0 -> True] &) /. p3 -> -p1 /. MomentaRules /. udrules /. gellmannOkubo /. _LeutwylerLambda -> 0 // Simplify) /. toEtaRules /. _RenormalizationState -> Sequence[] // Simplify

((2 c _ 2^( ) ((log((m _ η^(ó  ))^2/μ^2) - 9 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 - 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (27 log((m _ K^(ó  ))^2/μ^2) + 16 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + 9 p _ 1^2 ((5 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (7 log((m _ K^(ó  ))^2/μ^2) + 4 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)) - 3 c _ 5^( ) (m _ K^(ó  ))^2 ((15 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 4 (6 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)) (!, _ 0^( ))^2)/(216 π^2 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))

(ampinfinities[[7]] /. _LeutwylerLambda -> 0 /. p3 -> -p1 /. MomentaRules /. gellmannOkubo // Simplify) /. toEtaRules

((3 c _ 5^( ) (m _ K^(ó  ))^2 ((15 log((m _ π^(ó  ))^2/μ^2) - log((3 (m _ η^(ó  ))^2)/μ^2) + log(3)) (m _ π^(ó  ))^2 + 4 (6 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2) - 2 c _ 2^( ) ((log((m _ η^(ó  ))^2/μ^2) - 9 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 - 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (27 log((m _ K^(ó  ))^2/μ^2) + 16 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + 9 p _ 1^2 ((5 log((m _ π^(ó  ))^2/μ^2) - log((3 (m _ η^(ó  ))^2)/μ^2) + log(3)) (m _ π^(ó  ))^2 + (7 log((m _ K^(ó  ))^2/μ^2) + 4 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)/(216 π^2 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 1^2))

Converted by Mathematica  (July 10, 2003)