•Reduction of the amplitude - charged pions

amp1 = (WriteString["stdout", "."] ; Simplify[MomentumCombine[#]]) & /@ (amplFC /. {i1 -> 1, i2 -> 1}) ;

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

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

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

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

ampsimple = (WriteString["stdout", "."] ; Simplify[#]) & /@ amploop ;

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

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

ampres = Simplify /@ (MandelstamReduce[ampCMS, 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}) ;

ampfinalpi1 = FullSimplify /@ ampres

{1/(48 π^2 (f _ π^(ó  ))^2) ((e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) ((2 - 4 B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2)) (m _ π^(ó  ))^2 - 3 (s + t) - 3 (-2 (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)))), -(3 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(8 π^2 (f _ π^(ó  ))^2), (5 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 1 · µ ( p _ 2 ) p _ 3 · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(12 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - s)), (5 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - t)), ((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) (-2 (m _ π^(ó  ))^2 + s + t)))/(48 π^2 (f _ π^(ó  ))^2), (5 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - t)), ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(6 π^2 (f _ π^(ó  ))^2), (5 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 1 · µ ( p _ 2 ) p _ 3 · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(12 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - s)), ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(6 π^2 (f _ π^(ó  ))^2), ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 1 · µ ( p _ 2 ) p _ 3 · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(4 π^2 (f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2)), -((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (m _ π^(ó  ))^2)/(8 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - t)), ((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 3 · µ ( p _ 2 ) (p _ 1 · µ^* ( p _ 4 ) - p _ 3 · µ^* ( p _ 4 )) (m _ π^(ó  ))^2)/(8 π^2 (f _ π^(ó  ))^2 (t - (m _ π^(ó  ))^2)), -((B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) (e^( ))^2 p _ 1 · µ ( p _ 2 ) p _ 3 · µ^* ( p _ 4 ) (m _ π^(ó  ))^2)/(4 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - s))}

ampp[1, 1] := Plus @@ ampfinalpi1 ; ampp[2, 2] := Plus @@ ampfinalpi1 ; ampp[0, 0] := ampfinalpi0 ; ampp[_, _] := 0 ;

AmplitudeProjection[amppp, Channel -> {{PionPlus} -> {PionPlus}}, OnMassShell -> True] /. amppp -> ampp /. 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/(48 π^2 (f _ π^(ó  ))^2) ((e^( ))^2 ((16 (B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 1) p _ 1 · µ^* ( p _ 4 ) (m _ π^(ó  ))^2 (p _ 1 · µ ( p _ 2 ) (t - (m _ π^(ó  ))^2) + p _ 3 · µ ( p _ 2 ) ((m _ π^(ó  ))^2 - s)))/((m _ π^(ó  ))^4 - (s + t) (m _ π^(ó  ))^2 + s t) - µ ( p _ 2 ) · µ^* ( p _ 4 ) (2 (2 B _ 0 (0, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) + 3 C _ 0 ( 0 , 0 , 2 (m _ π^(ó  ))^2 - s - t , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 , (m _ π^(ó  ))^2 ) (-2 (m _ π^(ó  ))^2 + s + t) - 1) (m _ π^(ó  ))^2 + 3 (s + t))))

The Passarino-Veltman integrals are expanded in 1/(D-4):

ampinfinitiesfull = FullSimplify /@ VeltmanExpand[ampfinalpi1, ExplicitLeutwylerJ0 -> True, ExplicitLeutwylerSigma -> True, B0Evaluation -> "jbar", C0Evaluation -> None] ;

Converted by Mathematica  (July 10, 2003)