•Checks

Simplification and restriction to isospin channel i _ 1=1, i _ 2=1:

amult = (ampl2mult /. {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} // Simplify ;

a4 = (ampll4 /. {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} // Simplify ;

ainf = (Plus @@ ampinfinitiesfull /. p3 -> -p1 - p2 - p4 // MomentumExpand // ScalarProductExpand) /. {Pair[Momentum[p2], Momentum[Polarization[p4, -I]]] -> 0, Pair[Momentum[p4], Momentum[Polarization[p2, I]]] -> 0} // Simplify ;

Coefficients of L _ 4 and L _ 5 cancel:

cc1 = {Coefficient[amult, CouplingConstant[ChPTPhoton2[4], 4, RenormalizationState[0]]], Coefficient[amult, CouplingConstant[ChPTPhoton2[4], 5, RenormalizationState[0]]]} // Simplify

{(32 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/((f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t)), (16 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/((f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))}

cc2 = {Coefficient[a4, CouplingConstant[ChPTPhoton2[4], 4, RenormalizationState[0]]], Coefficient[a4, CouplingConstant[ChPTPhoton2[4], 5, RenormalizationState[0]]]} // Simplify

{-(32 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/((f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t)), -(16 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/((f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))}

cc1 + cc2 // ExpandScalarProduct // Simplify

{0, 0}

Coefficients of the infinities cancel:

c1 = Coefficient[ainf, LeutwylerLambda[]] // Simplify

(8 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/(3 (f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))

c2 = Coefficient[Renormalize[ amult // MomentumExpand // ScalarProductExpand], LeutwylerLambda[]] /. {Pair[Momentum[p2, ___], Momentum[Polarization[p4, -I], ___]] -> 0, Pair[Momentum[p4, ___], Momentum[Polarization[p2, I], ___]] -> 0} // Simplify

(16 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/(3 (f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))

c3 = Coefficient[Renormalize[ a4 // MomentumExpand // ScalarProductExpand], LeutwylerLambda[]] /. {Pair[Momentum[p2], Momentum[Polarization[p4, -I]]] -> 0, Pair[Momentum[p4], Momentum[Polarization[p2, I]]] -> 0} // Simplify

-(8 (e^( ))^2 (m _ π^(ó  ))^2 (4 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)))/((f _ π^(ó  ))^2 (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - t))

c1 + c2 + c3 // Simplify

0

Converted by Mathematica  (July 10, 2003)