•Calculation and reduction of the amplitude

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert] /. I1 -> j /. _SumOver -> 1 // SUNReduce // Simplify

{(p _ 1^μ _ 1 µ _ μ _ 1^*(p _ 3) δ _ (i _ 1 i _ 2)^(2))/(12 π^4 f _ π^(ó  ) (q _ 1^2 - (m _ π^(ó  ))^2))}

The polarization vector is divided off:

afg = amplFC/Pair[LorentzIndex[μ1, D], Momentum[Polarization[p3, -I], D]]

{(p _ 1^μ _ 1 δ _ (i _ 1 i _ 2)^(2))/(12 π^4 f _ π^(ó  ) (q _ 1^2 - (m _ π^(ó  ))^2))}

The loop integral is expressed in terms of Passarino-Veltman symbols:

ampreduced = OneLoop[q1, #] & /@ afg

{(i A _ 0 ( (m _ π^(ó  ))^2 ) p _ 1^μ _ 1 δ _ (i _ 1 i _ 2)^(2))/(12 π^2 f _ π^(ó  ))}

The divergence is singled out:

ampinfinities = VeltmanExpand[ampreduced[[1]], LeutwylerJBarEvaluation -> "subthreshold", ExplicitLeutwylerJ0 -> True] // Simplify

-(i (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) p _ 1^μ _ 1 (m _ π^(ó  ))^2 δ _ (i _ 1 i _ 2)^(2))/(12 π^2 f _ π^(ó  ))

Converted by Mathematica  (July 10, 2003)