•Renormalization

The coefficients of the infinities:

c1 = Coefficient[Plus @@ ampinfinitiesfull, LeutwylerLambda[]] // Simplify

-(8 (2 (m _ π^(ó  ))^2 + 9 p _ 3^2 + 12 p _ 3 · p _ 4 + 9 p _ 4^2) (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(3 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

c2 = Coefficient[Plus @@ Renormalize[ampl2mult], LeutwylerLambda[]] // Simplify

(128 (m _ π^(ó  ))^2 (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(3 ((m _ π^(ó  ))^2 - p _ 3^2) ((m _ π^(ó  ))^2 - p _ 4^2))

c3 = Coefficient[Plus @@ Renormalize[ampl4], LeutwylerLambda[]] // Simplify

(8 (6 (m _ π^(ó  ))^2 + 3 p _ 3^2 + 4 p _ 3 · p _ 4 + 3 p _ 4^2) (!, _ 0^( ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/((p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

c1 + c2 + c3 // SUNReduce // Simplify

0

This is then the full renormalized amplitude:

ampfinal = Collect[Plus @@ ampinfinitiesfull + Plus @@ ampl2mult + Plus @@ ampl4 // SUNReduce // Renormalize, LeutwylerLambda[]] // MandelstamReduce[#, OnMassShell -> False] & // FullSimplify

((!, _ 0^( ))^3 (512 π^2 (2 L _ 7^(r ) + L _ 8^(r )) ((p _ 4^2 - (m _ π^(ó  ))^2) δ _ (0 i _ 3)^(2) δ _ (i _ 1 i _ 2)^(2) + (p _ 3^2 - (m _ π^(ó  ))^2) δ _ (0 i _ 2)^(2) δ _ (i _ 1 i _ 3)^(2)) + (32 π^2 (f _ π^(ó  ))^2 + (16 π^2 (64 (2 L _ 6^(r ) + L _ 8^(r )) - 3 Overscript[J, _] _ (m _ π^(ó  ))^2(s)) + 2 log((m _ π^(ó  ))^2/μ^2) + 3) (m _ π^(ó  ))^2 - 2 s + 32 s π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) - 2 s log((m _ π^(ó  ))^2/μ^2) + 1024 π^2 L _ 6^(r ) p _ 3^2 + 512 π^2 L _ 8^(r ) p _ 3^2 + 16 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) p _ 3^2 - log((m _ π^(ó  ))^2/μ^2) p _ 3^2 - p _ 3^2 + 1024 π^2 L _ 6^(r ) p _ 4^2 + 512 π^2 L _ 8^(r ) p _ 4^2 + 16 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(s) p _ 4^2 - log((m _ π^(ó  ))^2/μ^2) p _ 4^2 - p _ 4^2 + 256 π^2 L _ 4^(r ) (-4 (m _ π^(ó  ))^2 + s - p _ 3^2 - p _ 4^2) + 128 π^2 L _ 5^(r ) (-4 (m _ π^(ó  ))^2 + s - p _ 3^2 - p _ 4^2)) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2)))/(4 π^2 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

We may amputate and put it on the mass shell:

ampfin = (Cancel[((Pair[Momentum[p3], Momentum[p3]] - ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2) (-Pair[Momentum[p4], Momentum[p4]] + ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2)) ampfinal] // MandelstamReduce[#, OnMassShell -> True] &) /. RenormalizationState[0] -> RenormalizationState[1] // FullSimplify

1/(4 π^2) (((1 - 16 π^2 (96 L _ 4^(r ) + 48 L _ 5^(r ) - 128 (2 L _ 6^(r ) + L _ 8^(r )) + Overscript[J, _] _ (m _ π^(ó  r ))^2(s))) (m _ π^(ó  r ))^2 - 2 s + 32 π^2 ((f _ π^(ó  r ))^2 + s (8 L _ 4^(r ) + 4 L _ 5^(r ) + Overscript[J, _] _ (m _ π^(ó  r ))^2(s))) - 2 s log((m _ π^(ó  r ))^2/μ^2)) (!, _ 0^(r ))^3 δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))

And make some L A T E X output:

PhiToLaTeX[ampfin /. {i2 -> a, i3 -> b, i1 -> c}]

(B_0^3 \delta_{0c} \delta_{ab} (-2 s - 2 s \log(m_{\rm \pi}^2/\mu^2) + 32 \pi^2 (f^2 +
s (8 L_{4} + 4 L_{5} + \overline{J}(s, m_{\rm \pi}^2))) + m_{\rm \pi}^2 (1 - 16 \pi^2
(96 L_{4} + 48 L_{5} - 128 (2 L_{6} + L_{8}) + \overline{J}(s, m_{\rm \pi}^2)))))/(4 \\pi^2)

Converted by Mathematica  (July 10, 2003)