•Calculation and reduction of the amplitude

$ConstantIsoIndices = Union[$ConstantIsoIndices, {i1, i2, I1}]

{i _ 1, I _ 1, i _ 2, I _ 2, I _ 3, I _ 4, I _ 5, I _ 6}

amplFC = CreateFCAmp[mesontreeinsert, EqualMasses -> False, Sum -> True] /. $Gauge -> 1 // DoSumOver // SUNReduce // Simplify

{1/(96 π^4 (f _ π^(ó  ))^4) (i (4 C^( ) ((δ _ (i _ 1 i _ 2)^(2) - δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2))/(q _ 1^2 - (m _ π^(3 ))^2) + (4 δ _ (1 i _ 1)^(2) δ _ (1 i _ 2)^(2) - δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) + 2 δ _ (i _ 1 i _ 2)^(2))/(q _ 1^2 - (m _ π^(1 ))^2) + (4 δ _ (2 i _ 1)^(2) δ _ (2 i _ 2)^(2) - δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) + 2 δ _ (i _ 1 i _ 2)^(2))/(q _ 1^2 - (m _ π^(2 ))^2)) (e^( ))^2 + (f _ π^(ó  ))^2 (((2 δ _ (1 i _ 1)^(2) δ _ (1 i _ 2)^(2) + δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2 + 2 q _ 1^2 (δ _ (1 i _ 1)^(2) δ _ (1 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) + p _ 1 · p _ 3 (2 δ _ (i _ 1 i _ 2)^(2) - 2 δ _ (1 i _ 1)^(2) δ _ (1 i _ 2)^(2)))/(q _ 1^2 - (m _ π^(1 ))^2) + ((2 δ _ (2 i _ 1)^(2) δ _ (2 i _ 2)^(2) + δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2 + 2 q _ 1^2 (δ _ (2 i _ 1)^(2) δ _ (2 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) + p _ 1 · p _ 3 (2 δ _ (i _ 1 i _ 2)^(2) - 2 δ _ (2 i _ 1)^(2) δ _ (2 i _ 2)^(2)))/(q _ 1^2 - (m _ π^(2 ))^2) + ((2 δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) + δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2 + 2 q _ 1^2 (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) + p _ 1 · p _ 3 (2 δ _ (i _ 1 i _ 2)^(2) - 2 δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2)))/(q _ 1^2 - (m _ π^(3 ))^2)))), (i (e^( ))^2 g^(μ _ 1 μ _ 2)^2 (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))/(16 π^4 (q _ 1^2 - (m _ γ^(ó  ))^2)), 0, -(i (e^( ))^2 g^(μ _ 1 μ _ 2) (p _ 1^μ _ 1 + q _ 1^μ _ 1) (p _ 3^μ _ 2 - q _ 1^μ _ 2) ((δ _ (1 i _ 1)^(2) δ _ (1 i _ 2)^(2))/(q _ 1^2 - (m _ π^(2 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2) + (δ _ (2 i _ 1)^(2) δ _ (2 i _ 2)^(2))/(q _ 1^2 - (m _ π^(1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2)))/(16 π^4)}

aff = amplFC /. i2 -> i1 // SUNReduce[#, FullReduce -> True] & // Simplify

{1/(96 π^4 (f _ π^(ó  ))^4) (i (4 C^( ) ((4 δ _ (1 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2)/(q _ 1^2 - (m _ π^(1 ))^2) + (4 δ _ (2 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2)/(q _ 1^2 - (m _ π^(2 ))^2) - (δ _ (3 i _ 1)^(2) - 1)/(q _ 1^2 - (m _ π^(3 ))^2)) (e^( ))^2 + (f _ π^(ó  ))^2 (((2 δ _ (1 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 - 2 p _ 1 · p _ 3 (δ _ (1 i _ 1)^(2) - 1) + 2 q _ 1^2 (δ _ (1 i _ 1)^(2) - 1))/(q _ 1^2 - (m _ π^(1 ))^2) + ((2 δ _ (2 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 - 2 p _ 1 · p _ 3 (δ _ (2 i _ 1)^(2) - 1) + 2 q _ 1^2 (δ _ (2 i _ 1)^(2) - 1))/(q _ 1^2 - (m _ π^(2 ))^2) + ((2 δ _ (3 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 - 2 p _ 1 · p _ 3 (δ _ (3 i _ 1)^(2) - 1) + 2 q _ 1^2 (δ _ (3 i _ 1)^(2) - 1))/(q _ 1^2 - (m _ π^(3 ))^2)))), (i (e^( ))^2 g^(μ _ 1 μ _ 2)^2 (δ _ (3 i _ 1)^(2) - 1))/(16 π^4 (q _ 1^2 - (m _ γ^(ó  ))^2)), 0, -(i (e^( ))^2 g^(μ _ 1 μ _ 2) (p _ 1^μ _ 1 + q _ 1^μ _ 1) (p _ 3^μ _ 2 - q _ 1^μ _ 2) (δ _ (1 i _ 1)^(2)/(q _ 1^2 - (m _ π^(2 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2) + δ _ (2 i _ 1)^(2)/(q _ 1^2 - (m _ π^(1 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ γ^(ó  ))^2)))/(16 π^4)}

ampreduced = OneLoop[q1, #] & /@ aff ;

ampsimple = ampreduced // SUNReduce[#, FullReduce -> True] & // Simplify

{-1/(96 π^2 (f _ π^(ó  ))^4) (A _ 0 ( (m _ π^(1 ))^2 ) (4 C^( ) (4 δ _ (1 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2) (e^( ))^2 + (f _ π^(ó  ))^2 ((2 δ _ (1 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(1 ))^2 (δ _ (1 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (1 i _ 1)^(2) - 1))) + A _ 0 ( (m _ π^(2 ))^2 ) (4 C^( ) (4 δ _ (2 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2) (e^( ))^2 + (f _ π^(ó  ))^2 ((2 δ _ (2 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(2 ))^2 (δ _ (2 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (2 i _ 1)^(2) - 1))) + A _ 0 ( (m _ π^(3 ))^2 ) ((f _ π^(ó  ))^2 ((2 δ _ (3 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(3 ))^2 (δ _ (3 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (3 i _ 1)^(2) - 1)) - 4 C^( ) (e^( ))^2 (δ _ (3 i _ 1)^(2) - 1))), -((e^( ))^2 (2 A _ 0 ( (m _ γ^(ó  ))^2 ) - (m _ γ^(ó  ))^2) (δ _ (3 i _ 1)^(2) - 1))/(8 π^2), 0, 1/(32 π^2 p _ 3^2) ((e^( ))^2 (-B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) δ _ (1 i _ 1)^(2) p _ 3^4 - B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) δ _ (2 i _ 1)^(2) p _ 3^4 + 3 B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) (m _ γ^(ó  ))^2 δ _ (1 i _ 1)^(2) p _ 3^2 - 2 B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) (m _ γ^(ó  ))^2 δ _ (1 i _ 1)^(2) p _ 3^2 - 3 B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) (m _ π^(2 ))^2 δ _ (1 i _ 1)^(2) p _ 3^2 + A _ 0 ( (m _ π^(2 ))^2 ) δ _ (1 i _ 1)^(2) p _ 3^2 + 3 B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) p _ 1 · p _ 3 δ _ (1 i _ 1)^(2) p _ 3^2 + B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) (m _ γ^(ó  ))^2 δ _ (2 i _ 1)^(2) p _ 3^2 - 3 B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) (m _ π^(1 ))^2 δ _ (2 i _ 1)^(2) p _ 3^2 + A _ 0 ( (m _ π^(1 ))^2 ) δ _ (2 i _ 1)^(2) p _ 3^2 + 3 B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) p _ 1 · p _ 3 δ _ (2 i _ 1)^(2) p _ 3^2 - 3 A _ 0 ( (m _ γ^(ó  ))^2 ) (δ _ (1 i _ 1)^(2) + δ _ (2 i _ 1)^(2)) p _ 3^2 - B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2 δ _ (1 i _ 1)^(2) + B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2 δ _ (1 i _ 1)^(2) + B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) p _ 1 · p _ 3 (m _ π^(2 ))^2 δ _ (1 i _ 1)^(2) - B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(2 ))^2) p _ 1 · p _ 3 (m _ π^(2 ))^2 δ _ (1 i _ 1)^(2) - B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2 δ _ (2 i _ 1)^(2) + B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) p _ 1 · p _ 3 (m _ γ^(ó  ))^2 δ _ (2 i _ 1)^(2) + B _ 0 (0, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) p _ 1 · p _ 3 (m _ π^(1 ))^2 δ _ (2 i _ 1)^(2) - B _ 0 (p _ 3^2, (m _ γ^(ó  ))^2, (m _ π^(1 ))^2) p _ 1 · p _ 3 (m _ π^(1 ))^2 δ _ (2 i _ 1)^(2)))}

ampinfinities = VeltmanExpand[ampsimple, ExplicitLeutwylerJ0 -> True] // Simplify

{-1/(96 π^2 (f _ π^(ó  ))^4) (-(32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) (4 C^( ) (4 δ _ (1 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2) (e^( ))^2 + (f _ π^(ó  ))^2 ((2 δ _ (1 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(1 ))^2 (δ _ (1 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (1 i _ 1)^(2) - 1))) (m _ π^(1 ))^2 - (32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) (m _ π^(2 ))^2 (4 C^( ) (4 δ _ (2 i _ 1)^(2) - δ _ (3 i _ 1)^(2) + 2) (e^( ))^2 + (f _ π^(ó  ))^2 ((2 δ _ (2 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(2 ))^2 (δ _ (2 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (2 i _ 1)^(2) - 1))) - (32 π^2 λ + log((m _ π^(3 ))^2/μ^2)) (m _ π^(3 ))^2 ((f _ π^(ó  ))^2 ((2 δ _ (3 i _ 1)^(2) + 1) (m _ π^(ó  ))^2 + 2 (m _ π^(3 ))^2 (δ _ (3 i _ 1)^(2) - 1) - 2 p _ 1 · p _ 3 (δ _ (3 i _ 1)^(2) - 1)) - 4 C^( ) (e^( ))^2 (δ _ (3 i _ 1)^(2) - 1))), ((e^( ))^2 (64 π^2 λ + 2 log((m _ γ^(ó  ))^2/μ^2) + 1) (m _ γ^(ó  ))^2 (δ _ (3 i _ 1)^(2) - 1))/(8 π^2), 0, ((e^( ))^2 (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(2 ))^2)(p _ 3^2) ((m _ γ^(ó  ))^2 - (m _ π^(1 ))^2) ((m _ γ^(ó  ))^2 - (m _ π^(2 ))^2) (p _ 1 · p _ 3 ((m _ γ^(ó  ))^2 - (m _ π^(2 ))^2 + 3 p _ 3^2) - p _ 3^2 (2 (m _ γ^(ó  ))^2 + p _ 3^2)) δ _ (1 i _ 1)^(2) + p _ 1 · p _ 3 (3 p _ 3^2 (((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (2 i _ 1)^(2)) (m _ γ^(ó  ))^4 - (((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2)) (m _ π^(1 ))^2 + (m _ π^(2 ))^2 ((32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (2 i _ 1)^(2))) (m _ γ^(ó  ))^2 + (m _ π^(1 ))^2 (m _ π^(2 ))^2 ((32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2))) - 16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) ((m _ γ^(ó  ))^2 - (m _ π^(1 ))^2)^2 ((m _ γ^(ó  ))^2 - (m _ π^(2 ))^2) δ _ (2 i _ 1)^(2)) + p _ 3^2 (p _ 3^2 (-((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (2 i _ 1)^(2)) (m _ γ^(ó  ))^4 + (((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2)) (m _ π^(1 ))^2 + (m _ π^(2 ))^2 ((32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (2 i _ 1)^(2))) (m _ γ^(ó  ))^2 - (m _ π^(1 ))^2 (m _ π^(2 ))^2 ((32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (-16 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2))) - 2 (((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + (8 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (2 i _ 1)^(2)) (m _ γ^(ó  ))^6 - (((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) δ _ (1 i _ 1)^(2) + 32 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) δ _ (2 i _ 1)^(2)) (m _ π^(1 ))^2 + (8 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ π^(2 ))^2 δ _ (2 i _ 1)^(2)) (m _ γ^(ó  ))^4 - ((-24 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2) (m _ π^(1 ))^4 - 32 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) (m _ π^(2 ))^2 δ _ (2 i _ 1)^(2) (m _ π^(1 ))^2 + (32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) (m _ π^(2 ))^4 δ _ (1 i _ 1)^(2)) (m _ γ^(ó  ))^2 + (m _ π^(1 ))^2 (m _ π^(2 ))^2 ((-24 π^2 Overscript[J, _] _ ((m _ γ^(ó  ))^2 (m _ π^(1 ))^2)(p _ 3^2) + 32 π^2 λ + log((m _ π^(1 ))^2/μ^2)) δ _ (2 i _ 1)^(2) (m _ π^(1 ))^2 + (32 π^2 λ + log((m _ π^(2 ))^2/μ^2)) (m _ π^(2 ))^2 δ _ (1 i _ 1)^(2))))))/(32 π^2 p _ 3^2 ((m _ γ^(ó  ))^2 - (m _ π^(1 ))^2) ((m _ π^(2 ))^2 - (m _ γ^(ó  ))^2))}

Converted by Mathematica  (July 10, 2003)