•Calculation of the amplitude

Calculation of the amplitude:

ampp4 = CreateFCAmp[mesontreeinsert, AmplitudeLevel -> Classes, Sum -> False, EqualMasses -> True] /. D -> Sequence[] // SUNReduce // Contract // Collect[#, {_DecayConstant, _CouplingConstant, _Pair, _SU2Delta}] & // Simplify

{1/(f _ π^(ó  ))^2 (4 (e^( ))^2 (4 (2 L _ 4^( ) + L _ 5^( )) µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2 + L _ 10^( ) (2 p _ 2 · p _ 4 µ ( p _ 2 ) · µ^* ( p _ 4 ) - 2 p _ 2 · µ^* ( p _ 4 ) p _ 4 · µ ( p _ 2 )) + L _ 9^( ) (p _ 1 · µ ( p _ 2 ) p _ 2 · µ^* ( p _ 4 ) + p _ 3 · µ ( p _ 2 ) p _ 2 · µ^* ( p _ 4 ) + p _ 1 · µ^* ( p _ 4 ) p _ 4 · µ ( p _ 2 ) + p _ 3 · µ^* ( p _ 4 ) p _ 4 · µ ( p _ 2 ) - p _ 1 · p _ 2 µ ( p _ 2 ) · µ^* ( p _ 4 ) - p _ 1 · p _ 4 µ ( p _ 2 ) · µ^* ( p _ 4 ) - p _ 2 · p _ 3 µ ( p _ 2 ) · µ^* ( p _ 4 ) - p _ 3 · p _ 4 µ ( p _ 2 ) · µ^* ( p _ 4 ))) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))), -1/((f _ π^(ó  ))^2 ((-p _ 3 - p _ 4)^2 - (m _ π^(ó  ))^2)) (4 (e^( ))^2 p _ 3 · µ^* ( p _ 4 ) (p _ 1 · µ ( p _ 2 ) - p _ 3 · µ ( p _ 2 ) - p _ 4 · µ ( p _ 2 )) (L _ 9^( ) (p _ 3 · p _ 4 - (p _ 3 + p _ 4) . (p _ 4)) - 4 (2 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^2) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))), -1/((f _ π^(ó  ))^2 ((-p _ 2 - p _ 3)^2 - (m _ π^(ó  ))^2)) (4 (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - p _ 2 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (L _ 9^( ) (p _ 2 · p _ 3 - (p _ 2 + p _ 3) . (p _ 2)) - 4 (2 L _ 4^( ) + L _ 5^( )) (m _ π^(ó  ))^2) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))), 1/((f _ π^(ó  ))^2 ((-p _ 2 - p _ 3)^2 - (m _ π^(ó  ))^2)) (8 (e^( ))^2 p _ 3 · µ ( p _ 2 ) (2 (2 L _ 4^( ) + L _ 5^( )) (p _ 1 · µ^* ( p _ 4 ) - p _ 2 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (m _ π^(ó  ))^2 + L _ 9^( ) (p _ 1 · p _ 4 (p _ 2 · µ^* ( p _ 4 ) + p _ 3 · µ^* ( p _ 4 )) - p _ 1 · µ^* ( p _ 4 ) (p _ 2 + p _ 3) . (p _ 4))) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))), -1/((f _ π^(ó  ))^2 ((-p _ 3 - p _ 4)^2 - (m _ π^(ó  ))^2)) (8 (e^( ))^2 p _ 3 · µ^* ( p _ 4 ) (L _ 9^( ) (p _ 1 · µ ( p _ 2 ) (p _ 3 + p _ 4) . (p _ 2) - p _ 1 · p _ 2 (p _ 3 · µ ( p _ 2 ) + p _ 4 · µ ( p _ 2 ))) - 2 (2 L _ 4^( ) + L _ 5^( )) (p _ 1 · µ ( p _ 2 ) - p _ 3 · µ ( p _ 2 ) - p _ 4 · µ ( p _ 2 )) (m _ π^(ó  ))^2) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))}

ampll4 = (((Plus @@ PropagatorDenominatorExplicit[ampp4]) // Simplify // MandelstamReduce[#, OnMassShell -> True, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] &) /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I] /. {Pair[Momentum[p2, ___], Momentum[Polarization[p4, -I], ___]] -> 0, Pair[Momentum[p4, ___], Momentum[Polarization[p2, I], ___]] -> 0}) // Collect[#, {_Pair, _DecayConstant, _CouplingConstant, (SU2Delta[3, SUNIndex[i1]] SU2Delta[3, SUNIndex[i2]] - SU2Delta[SUNIndex[i1], SUNIndex[i2]]), ((MandelstamS - ParticleMass[Pion, RenormalizationState[0]]^2) * _)^-1}] &

1/(f _ π^(ó  ))^2 (µ ( p _ 2 ) · µ^* ( p _ 4 ) (32 L _ 4^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2 + 16 L _ 5^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2 + 8 L _ 10^( ) ((m _ π^(ó  ))^2 - s/2 - t/2) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) + 4 L _ 9^( ) (2 (m _ π^(ó  ))^2 - s - t) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))) (e^( ))^2) + 1/(f _ π^(ó  ))^2 (p _ 3 · µ ( p _ 2 ) p _ 3 · µ^* ( p _ 4 ) (-(32 L _ 4^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(t/2 - 1/2 (m _ π^(ó  ))^2) - (16 L _ 5^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(t/2 - 1/2 (m _ π^(ó  ))^2) + 4 L _ 9^( ) ((1/2 (m _ π^(ó  ))^2 - s/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t))/(t/2 - 1/2 (m _ π^(ó  ))^2) - 1) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))) (e^( ))^2) + 1/(f _ π^(ó  ))^2 (p _ 1 · µ^* ( p _ 4 ) p _ 3 · µ ( p _ 2 ) ((32 L _ 4^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(t/2 - 1/2 (m _ π^(ó  ))^2) + (16 L _ 5^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(t/2 - 1/2 (m _ π^(ó  ))^2) + 4 L _ 9^( ) (-(1/2 (m _ π^(ó  ))^2 - s/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t) - 2 (t/2 - 1/2 (m _ π^(ó  ))^2))/(t/2 - 1/2 (m _ π^(ó  ))^2) - 1) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2))) (e^( ))^2) + p _ 1 · µ ( p _ 2 ) (1/(f _ π^(ó  ))^2 ((e^( ))^2 p _ 3 · µ^* ( p _ 4 ) ((128 L _ 4^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(2 (m _ π^(ó  ))^2 + 2 (s/2 - 1/2 (m _ π^(ó  ))^2) + 2 (t/2 - 1/2 (m _ π^(ó  ))^2) + 2 (1/2 (m _ π^(ó  ))^2 - t/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t)) + 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)) + (64 L _ 5^( ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)) (m _ π^(ó  ))^2)/(2 (m _ π^(ó  ))^2 + 2 (s/2 - 1/2 (m _ π^(ó  ))^2) + 2 (t/2 - 1/2 (m _ π^(ó  ))^2) + 2 (1/2 (m _ π^(ó  ))^2 - t/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t)) + 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)) + 4 L _ 9^( ) (-(2 (s/2 - 1/2 (m _ π^(ó  ))^2))/(2 (m _ π^(ó  ))^2 + 2 (s/2 - 1/2 (m _ π^(ó  ))^2) + 2 (t/2 - 1/2 (m _ π^(ó  ))^2) + 2 (1/2 (m _ π^(ó  ))^2 - t/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t)) + 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)) - (2 (1/2 (m _ π^(ó  ))^2 - s/2))/(2 (m _ π^(ó  ))^2 + 2 (s/2 - 1/2 (m _ π^(ó  ))^2) + 2 (t/2 - 1/2 (m _ π^(ó  ))^2) + 2 (1/2 (m _ π^(ó  ))^2 - t/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t)) + 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)) - (2 (-2 (m _ π^(ó  ))^2 - 2 (s/2 - 1/2 (m _ π^(ó  ))^2) - 2 (t/2 - 1/2 (m _ π^(ó  ))^2) - 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)))/(2 (m _ π^(ó  ))^2 + 2 (s/2 - 1/2 (m _ π^(ó  ))^2) + 2 (t/2 - 1/2 (m _ π^(ó  ))^2) + 2 (1/2 (m _ π^(ó  ))^2 - t/2 + 1/2 (-2 (m _ π^(ó  ))^2 + s + t)) + 2 (1/2 (2 (m _ π^(ó  ))^2 - s - t) - (m _ π^(ó  ))^2)) - 1) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))) - (4 (e^( ))^2 L _ 9^( ) p _ 1 · µ^* ( p _ 4 ) (δ _ (3 i _ 1)^(2) δ _ (3 i _ 2)^(2) - δ _ (i _ 1 i _ 2)^(2)))/(f _ π^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)