•Reduction of the amplitude - neutral pions

amp1 = (WriteString["stdout", "."] ; Simplify[MomentumCombine[#]]) & /@ (amplFC /. {i1 -> 3, i2 -> 3}) ;

.............

The loop integrals are expressed in terms of Passarino-Veltman symbols:

SetOptions[A0, A0ToB0 -> True] ; SetOptions[B0, BReduce -> False] ;

amploop = (WriteString["stdout", "."] ; PaVeReduce[OneLoop[q1, #, Dimension -> D]]) & /@ amp1 ;

.............

ampsimple = Simplify[#] & /@ amploop ;

ampCMS = ampsimple /. {Pair[Momentum[p2, ___], Momentum[Polarization[p4, -I], ___]] -> 0} ;

ampres = FullSimplify /@ (MandelstamReduce[ampCMS, OnMassShell -> True, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I])

{1/(24 π^2 (f _ π^(ó  ))^2) ((e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (-(4 B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (m _ π^(ó  ))^2 - 3 (s + t) - 3 (-(m _ π^(ó  ))^2 + s + t) (2 C _ 0 ( 0 , 0 , 2 (m _ π^(ó  ))^2 - s - t , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 ) (m _ π^(ó  ))^2 + B _ 0 (2 (m _ π^(ó  ))^2 - s - t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2)))), -((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(12 π^2 (f _ π^(ó  ))^2), 0, 0, ((e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (2 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (m _ π^(ó  ))^2 + 3 B _ 0 (2 (m _ π^(ó  ))^2 - s - t, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) (-(m _ π^(ó  ))^2 + s + t)))/(24 π^2 (f _ π^(ó  ))^2), 0, ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(12 π^2 (f _ π^(ó  ))^2), 0, ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(12 π^2 (f _ π^(ó  ))^2), 0, 0, 0, 0}

ampfinalpi0 = Plus @@ ampres // FullSimplify

-((e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (-(m _ π^(ó  ))^2 + s + t) (2 C _ 0 ( 0 , 0 , 2 (m _ π^(ó  ))^2 - s - t , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 ) (m _ π^(ó  ))^2 + 1))/(8 π^2 (f _ π^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)