•Calculation of the amplitude

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert, AmplitudeLevel -> Classes, Sum -> False, EqualMasses -> True] // Contract // SUNReduce // Simplify

{2 (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)), (2 (e^( ))^2 p _ 3 · µ^* ( p _ 4 ) (p _ 1 · µ ( p _ 2 ) - (p _ 3 + p _ 4) . (µ ( p _ 2 ))) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))/((-p _ 3 - p _ 4)^2 - (m _ π^(ó  ))^2), (2 (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - (p _ 2 + p _ 3) . (µ^* ( p _ 4 ))) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))/((-p _ 2 - p _ 3)^2 - (m _ π^(ó  ))^2)}

ampleading = ((Plus @@ PropagatorDenominatorExplicit[amplFC] /. {i1 -> 1, i2 -> 1} /. p3 -> -p1 - p2 - p4 // MomentumExpand // ScalarProductExpand) /. {Pair[Momentum[p2, ___], Momentum[Polarization[p4, -I], ___]] -> 0, Pair[Momentum[p4, ___], Momentum[Polarization[p2, I], ___]] -> 0} // MandelstamReduce[#, OnMassShell -> True, Cancel -> MandelstamU, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] &) /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I] // Factor // FullSimplify

-(2 (e^( ))^2 (2 p _ 1 · µ ( p _ 2 ) p _ 1 · µ^* ( p _ 4 ) (-2 (m _ π^(ó  ))^2 + s + t) + µ ( p _ 2 ) · µ^* ( p _ 4 ) (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t)))/((s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))

The tree amplitudes are multiplied with renormalization factors:

zpion = CheckF[dum, "ChPT2P20o2.Fac"] /. ChPT2 -> ChPTPhoton2

((384 π^2 L _ 4^( ) + 192 π^2 L _ 5^( ) - 32 π^2 λ - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2) + 1

ampl2mult = ((2 * ((3 - zpion)/2) - 1) * amplFC[[1]] + (4 * ((3 - zpion)/2) - 3) * amplFC[[2]] + (4 * ((3 - zpion)/2) - 3) * amplFC[[3]] /. D -> Sequence[] // PropagatorDenominatorExplicit // MandelstamReduce[#, OnMassShell -> True, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] &) /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I] /. Pair[Momentum[p3], Momentum[Polarization[p4, -i]]] -> -Pair[Momentum[p1], Momentum[Polarization[p4, -i]]] - Pair[Momentum[p2], Momentum[Polarization[p4, -i]]] /. {Pair[Momentum[p2, ___], Momentum[Polarization[p4, -I], ___]] -> 0} // FullSimplify

1/(12 π^2 (f _ π^(ó  ))^2) ((e^( ))^2 (µ ( p _ 2 ) · µ^* ( p _ 4 ) (24 π^2 (f _ π^(ó  ))^2 + (32 π^2 (λ - 6 (2 L _ 4^( ) + L _ 5^( ))) + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2) + 1/((s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t)) (4 p _ 1 · µ^* ( p _ 4 ) (12 π^2 (f _ π^(ó  ))^2 + (32 π^2 (λ - 6 (2 L _ 4^( ) + L _ 5^( ))) + log(-(i m _ π^(ó  ))/μ) + log((i m _ π^(ó  ))/μ)) (m _ π^(ó  ))^2) (p _ 1 · µ ( p _ 2 ) (t - (m _ π^(ó  ))^2) + p _ 3 · µ ( p _ 2 ) ((m _ π^(ó  ))^2 - s)))) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))

Converted by Mathematica  (July 10, 2003)