•Final expression

The results below are in agreement with  Bijnens and Cornet 1987.

This is the full renormalized isospin 1 amplitude:

ampfinalren = Renormalize[ainf + amult + a4, InfinityFactor -> LeutwylerLambda[]] // ExpandAll // Simplify // MomentumCombine // FullSimplify

((e^( ))^2 (96 π^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)) (f _ π^(ó  ))^2 + 3 µ ( p _ 2 ) · µ^* ( p _ 4 ) (-2 (m _ π^(ó  ))^2 + s + t) (s - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - 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 - 64 π^2 (L _ 9^(r ) + L _ 10^(r )) + 1)))/(48 π^2 (f _ π^(ó  ))^2 ((m _ π^(ó  ))^4 - (s + t) (m _ π^(ó  ))^2 + s t))

We may split it in the leading order and next to leading order contributions and use u instead of s:

(ampfinalren // DiscardOrders[#, PerturbationOrder -> 4] & // MandelstamReduce[#, OnMassShell -> True, Cancel -> MandelstamS, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] &) /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I] // Factor // FullSimplify

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

((ampfinalren - DiscardOrders[ampfinalren, PerturbationOrder -> 4]) // MandelstamReduce[#, OnMassShell -> True, Cancel -> MandelstamS, Masses -> {ParticleMass[Pion, RenormalizationState[0]], 0, ParticleMass[Pion, RenormalizationState[0]], 0}] &) /. Polarization[-p1 - p2 - p3, -I] -> Polarization[p4, -I] // FullSimplify

-(u (e^( ))^2 µ ( p _ 2 ) · µ^* ( p _ 4 ) (-2 C _ 0(0, 0, u, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2, (m _ π^(ó  ))^2) (m _ π^(ó  ))^2 + 64 π^2 (L _ 9^(r ) + L _ 10^(r )) - 1))/(16 π^2 (f _ π^(ó  ))^2)

Converted by Mathematica  (July 10, 2003)