•Calculation and reduction of the amplitude

amplFC = CreateFCAmp[mesontreeinsert, AmplitudeLevel -> Classes, Sum -> Explicit, EqualMasses -> False] /. {I1 -> j, i1 -> I1, i2 -> I2} // Simplify

{(p _ 1^μ _ 1 µ _ μ _ 1^*(p _ 3) ((δ _ (1 I _ 1) δ _ (1 I _ 2) - δ _ (1 1) δ _ (I _ 1 I _ 2))/(q _ 1^2 - (m _ π^(1 ))^2) + (δ _ (2 I _ 1) δ _ (2 I _ 2) - δ _ (2 2) δ _ (I _ 1 I _ 2))/(q _ 1^2 - (m _ π^(2 ))^2) + (δ _ (3 I _ 1) δ _ (3 I _ 2) - δ _ (3 3) δ _ (I _ 1 I _ 2))/(q _ 1^2 - (m _ π^(3 ))^2)))/(12 π^4 f _ π^(ó  )), 0, -((e^( ))^2 f _ π^(ó  ) g^(μ _ 1 μ _ 3) µ _ μ _ 1^*(p _ 3) g^(μ _ 2 μ _ 3) (p _ 1^μ _ 2 + q _ 1^μ _ 2) ((f _ (1 3 I _ 1)^(2) f _ (3 1 I _ 2))/(q _ 1^2 - (m _ π^(1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2) + (f _ (2 3 I _ 1)^(2) f _ (3 2 I _ 2))/(q _ 1^2 - (m _ π^(2 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2)))/(8 π^4)}

afg = amplFC/Pair[LorentzIndex[μ1, D], Momentum[Polarization[p3, -i], D]] // IsoToChargedMasses // ChargeEliminate // Contract // SUNReduce

{((p _ 1^μ _ 1 δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2))/(12 π^4 f _ π^(ó  )) + (p _ 1^μ _ 1 δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2))/(12 π^4 f _ π^(ó  )) - (p _ 1^μ _ 1 δ _ (I _ 1 I _ 2)^(2))/(6 π^4 f _ π^(ó  )))/(q _ 1^2 - (m _ π^+^(ó  ))^2) + ((p _ 1^μ _ 1 δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2))/(12 π^4 f _ π^(ó  )) - (p _ 1^μ _ 1 δ _ (I _ 1 I _ 2)^(2))/(12 π^4 f _ π^(ó  )))/(q _ 1^2 - (m _ π^0^(ó  ))^2), 0, (((p _ 1^μ _ 1 (m _ π^+^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) + (q _ 1^μ _ 1 (m _ π^+^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) - (p _ 1^μ _ 1 (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) - (q _ 1^μ _ 1 (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( ))) δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + ((p _ 1^μ _ 1 (m _ π^+^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) + (q _ 1^μ _ 1 (m _ π^+^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) - (p _ 1^μ _ 1 (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( )) - (q _ 1^μ _ 1 (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^3)/(16 π^4 C^( ))) δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2))/(q _ 1^2 - (m _ π^+^(ó  ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2)}

ampreduced = Simplify[OneLoop[q1, #, Dimension -> D]] & /@ afg

{(i p _ 1^μ _ 1 (A _ 0 ( (m _ π^+^(ó  ))^2 ) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2) - 2 δ _ (I _ 1 I _ 2)^(2)) + A _ 0 ( (m _ π^0^(ó  ))^2 ) (δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2) - δ _ (I _ 1 I _ 2)^(2))))/(12 π^2 f _ π^(ó  )), 0, 1/(32 π^2 C^( ) p _ 3^2) (i (f _ π^(ó  ))^3 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (B _ 0 (0, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) p _ 3^μ _ 1 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) + B _ 0 (p _ 3^2, (m _ π^+^(ó  ))^2, (m _ γ^(ó  ))^2) (2 p _ 1^μ _ 1 p _ 3^2 - p _ 3^μ _ 1 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + p _ 3^2))) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2)))}

ampinfinities = Plus @@ ((VeltmanExpand[#, ExplicitLeutwylerJ0 -> True] // Simplify) & /@ ampreduced) // Simplify

-1/(96 π^2 f _ π^(ó  )) (i (1/(C^( ) p _ 3^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)) (3 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) ((2 p _ 1^μ _ 1 - p _ 3^μ _ 1) p _ 3^2 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 - (32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2) - 16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 3^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) (2 p _ 1^μ _ 1 p _ 3^2 - p _ 3^μ _ 1 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + p _ 3^2))) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2)) (f _ π^(ó  ))^4) + 8 p _ 1^μ _ 1 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2) - 2 δ _ (I _ 1 I _ 2)^(2)) (m _ π^+^(ó  ))^2 + (32 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2 (δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2) - δ _ (I _ 1 I _ 2)^(2)))))

Converted by Mathematica  (July 10, 2003)