•Calculation of the amplitude

Cosmetics:

LoadLagrangian[ChPT2[4]] ;

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert, EqualMasses -> False, Sum -> True, WFRenormalize -> True] /. pirule // Contract // SUNReduce // Simplify

{1/(36 π^2 C^( ) (f _ π^(ó  ))^4) ((m _ π^0^(ó  ))^2 (3 C^( ) (12 π^2 (f _ π^(ó  ))^2 + (log((m _ π^+^(ó  ))^2/μ^2) - 64 π^2 λ) (m _ π^+^(ó  ))^2 - 192 π^2 (2 L _ 4^(r ) + L _ 5^(r )) (m _ π^0^(ó  ))^2) - 8 π^2 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) + 10 k _ 10^(r )) (f _ π^(ó  ))^4 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2))), 0, 0, 0}

(* To use PMRenormalize, some mass factors would have to be saved first *) ampl2mult = Simplify /@ Collect[DiscardOrders[Plus @@ amplFC (* // DCRenormalize[#, PhiModel -> ChPT2] & // PMRenormalize *) // PropagatorDenominatorExplicit, PerturbationOrder -> 4] // MandelstamReduce, _DecayConstant] // Simplify

1/(36 π^2 C^( ) (f _ π^(ó  ))^4) ((m _ π^0^(ó  ))^2 (3 C^( ) (12 π^2 (f _ π^(ó  ))^2 + (log((m _ π^+^(ó  ))^2/μ^2) - 64 π^2 λ) (m _ π^+^(ó  ))^2 - 192 π^2 (2 L _ 4^(r ) + L _ 5^(r )) (m _ π^0^(ó  ))^2) - 8 π^2 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) + 10 k _ 10^(r )) (f _ π^(ó  ))^4 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2)))

zpionZero = CheckF[dum, "ChPTEM2P40o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]]

1/(72 π^2 C^( ) (f _ π^(ó  ))^2) (8 π^2 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) + 10 k _ 10^(r )) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 72 π^2 C^( ) (f _ π^(ó  ))^2 + 3 C^( ) ((64 π^2 λ - log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + 192 π^2 (2 L _ 4^(r ) + L _ 5^(r )) (m _ π^0^(ó  ))^2))

zpionPlus = CheckF[dum, "ChPTEM2P30o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]]

1/(144 π^2 C^( ) (f _ π^(ó  ))^2) ((160 π^2 k _ 2^(r ) + 160 π^2 k _ 10^(r ) + 9 (8 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2) - 1)) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 144 π^2 C^( ) (f _ π^(ó  ))^2 + 3 C^( ) ((64 π^2 λ - log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + (768 π^2 L _ 4^(r ) + 384 π^2 L _ 5^(r ) + 64 π^2 λ - log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2))

zpion = (1 - SU2Delta[3, I1]) zpionPlus + SU2Delta[3, I1] zpionZero ;

zphoton = CheckF[dum, "ChPTEM2V10o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]]

-(192 π^2 (e^( ))^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2 + ((e^( ))^2 (16 π^2 (20 (L _ 10^(r ) + 2 H _ 1^(r ) + k _ 15^(r )) - 3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2)) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 1) - 144 π^2) p _ 1^2)/(144 π^2 p _ 1^2)

ampl2mult = Simplify /@ Collect[((((3 - zpion)/2) /. {I1 -> i1, p1 -> p1}) + (((3 - zpion)/2) /. {I1 -> i2, p1 -> p2}) + (((3 - zpion)/2) /. {I1 -> i3, p1 -> p3}) + (((3 - zpion)/2) /. {I1 -> i4, p1 -> p4}) - 3) (amplFC [[1]] + amplFC [[2]] + amplFC [[3]] + amplFC [[4]]) + (2 * ((3 - (zphoton /. p1 -> -(p1 + p2)))/2) - 2) amplFC [[2]] + (2 * ((3 - (zphoton /. p1 -> -(p1 + p3)))/2) - 2) amplFC [[3]] + (2 * ((3 - (zphoton /. p1 -> -(p1 + p4)))/2) - 2) amplFC [[4]] // PropagatorDenominatorExplicit // MandelstamReduce[#, Cancel -> None, Masses -> ({ParticleMass[Pion, SUNIndex[i1], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i2], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i3], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i4], RenormalizationState[0]]} /. subpar)] &, _DecayConstant] // Simplify

1/36 (m _ π^0^(ó  ))^2 ((8 (10 k _ 2^(r ) - 18 k _ 3^(r ) - 9 k _ 4^(r ) + 10 k _ 10^(r )) ((m _ π^0^(ó  ))^2 - (m _ π^+^(ó  ))^2))/C^( ) - (3 ((64 π^2 λ - log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + 192 π^2 (2 L _ 4^(r ) + L _ 5^(r )) (m _ π^0^(ó  ))^2))/(π^2 (f _ π^(ó  ))^4) + 36/(f _ π^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)