•Reduction of the amplitude

aff = OneLoopSimplify[#, q1, Dimension -> D] & /@ amplFC ;

ampreduced = OneLoop[q1, #, Dimension -> D] & /@ aff ;

ampsimple = Simplify /@ ampreduced

{(5 i A _ 0 ( (m _ π^(ó  ))^2 ) e^( ) (p _ 3 · µ ( p _ 1 ) - p _ 4 · µ ( p _ 1 )) f _ (3 i _ 1 i _ 2)^(2))/(48 π^2 (f _ π^(ó  ))^2), -1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) (p _ 3 · µ ( p _ 1 ) - p _ 4 · µ ( p _ 1 )) (12 (B _ 0 (p _ 1^2, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (m _ π^(ó  ))^2 + 6 A _ 0 ( (m _ π^(ó  ))^2 ) - (3 B _ 0 (p _ 1^2, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 2) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))}

symmrules1 = {Pair[Momentum[p3], Momentum[Polarization[p1, I]]] -> -Pair[Momentum[p4 - p3], Momentum[Polarization[p1, I]]]/2, Pair[Momentum[p4], Momentum[Polarization[p1, I]]] -> Pair[Momentum[p4 - p3], Momentum[Polarization[p1, I]]]/2}

{p _ 3 · µ ( p _ 1 ) -> -1/2 ( p _ 4 - p _ 3 ) · µ ( p _ 1 ), p _ 4 · µ ( p _ 1 ) -> 1/2 ( p _ 4 - p _ 3 ) · µ ( p _ 1 )}

ampinfinitiesfull = VeltmanExpand[#, ExplicitLeutwylerJ0 -> True, ExplicitLeutwylerSigma -> True, B0Evaluation -> "jbar"] & /@ ampsimple /. symmrules1 // Simplify

{(5 i e^( ) (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (m _ π^(ó  ))^2 f _ (3 i _ 1 i _ 2)^(2))/(48 π^2 (f _ π^(ó  ))^2), -1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (6 (-32 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + 96 π^2 λ + 3 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) - 96 π^2 λ - 3 log((m _ π^(ó  ))^2/μ^2) - 1) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))}

amploopfull = ampinfinitiesfull[[1]] + ampinfinitiesfull[[2]] // Simplify

1/(288 π^2 (f _ π^(ó  ))^2) (i e^( ) ( p _ 4 - p _ 3 ) · µ ( p _ 1 ) (12 (16 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + 32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (-48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + 96 π^2 λ + 3 log((m _ π^(ó  ))^2/μ^2) + 1) p _ 1^2) f _ (3 i _ 1 i _ 2)^(2))

Converted by Mathematica  (July 10, 2003)