•Renormalization

The coefficients of the infinities:

c1 = Coefficient[Plus @@ ampinfinitiesfull, LeutwylerLambda[]] /. p1 -> -p3 - p4 /. gellmannOkubo // ExpandScalarProduct // FullSimplify

-(16 (-3 (m _ π^(ó  ))^2 + 7 (m _ K^(ó  ))^2 + 8 p _ 3^2 + 9 p _ 3 · p _ 4 + 8 p _ 4^2) (!, _ 0^( ))^3)/(3 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

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

-(64 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) (!, _ 0^( ))^3)/(3 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

c3 = Coefficient[Plus @@ Renormalize[amp4], LeutwylerLambda[]] // Simplify

(16 (5 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2 + 8 p _ 3^2 + 9 p _ 3 · p _ 4 + 8 p _ 4^2) (!, _ 0^( ))^3)/(3 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2))

c1 + c2 + c3 // Simplify

0

This is then the full renormalized amplitude:

ampfinal = (Collect[Plus @@ ampinfinitiesfull + Plus @@ ampl2mult + Plus @@ amp4 /. p1 -> -p3 - p4 /. gellmannOkubo // ExpandScalarProduct // SUNReduce // Renormalize, LeutwylerLambda[]] // Expand) /. toEtaRules // FullSimplify

1/(12 π^2 (p _ 3^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - p _ 4^2)) ((96 π^2 (f _ ϕ^(ó  ))^2 - 1536 π^2 L _ 4^(r ) (m _ π^(ó  ))^2 - 1536 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 + 3072 π^2 L _ 8^(r ) (m _ π^(ó  ))^2 - 144 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) (m _ π^(ó  ))^2 - 96 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) (m _ π^(ó  ))^2 - 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) (m _ π^(ó  ))^2 + 6 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 6 log((m _ K^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 2 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 16 (m _ π^(ó  ))^2 - 2 (1536 π^2 (L _ 4^(r ) - 3 L _ 6^(r )) + 3 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + (16 π^2 (288 L _ 6^(r ) + 96 L _ 8^(r ) + 9 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) + 6 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) + Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2)) - 9 log((m _ π^(ó  ))^2/μ^2) - 6 log((m _ K^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2) - 16) p _ 3^2 + 2304 π^2 L _ 4^(r ) p _ 3 · p _ 4 + 768 π^2 L _ 5^(r ) p _ 3 · p _ 4 + 192 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) p _ 3 · p _ 4 + 96 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) p _ 3 · p _ 4 - 12 log((m _ π^(ó  ))^2/μ^2) p _ 3 · p _ 4 - 6 log((m _ K^(ó  ))^2/μ^2) p _ 3 · p _ 4 - 18 p _ 3 · p _ 4 + 4608 π^2 L _ 6^(r ) p _ 4^2 + 1536 π^2 L _ 8^(r ) p _ 4^2 + 144 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) p _ 4^2 + 96 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) p _ 4^2 + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2 + 2 p _ 3 · p _ 4 + p _ 4^2) p _ 4^2 - 9 log((m _ π^(ó  ))^2/μ^2) p _ 4^2 - 6 log((m _ K^(ó  ))^2/μ^2) p _ 4^2 - log((m _ η^(ó  ))^2/μ^2) p _ 4^2 - 16 p _ 4^2) (!, _ 0^( ))^3)

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 /. massshellrules] // FullSimplify

1/(12 π^2) ((96 π^2 (f _ ϕ^(ó  ))^2 + 2 (8 π^2 (Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 1^2) - 3 (16 (5 L _ 4^(r ) + 3 L _ 5^(r ) - 12 L _ 6^(r ) - 8 L _ 8^(r )) + Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2))) + 1) (m _ π^(ó  ))^2 - 2 (1536 π^2 (L _ 4^(r ) - 3 L _ 6^(r )) + 3 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 3 (16 π^2 (24 L _ 4^(r ) + 8 L _ 5^(r ) + 2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) + Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 1^2)) - 2 log((m _ π^(ó  ))^2/μ^2) - log((m _ K^(ó  ))^2/μ^2) - 3) p _ 1^2) (!, _ 0^( ))^3)

And produce some nice L A T E X output:

PhiToLaTeX[ampfin /. Pair[Momentum[p1], Momentum[p1]] -> MandelstamS]

(B_0^3 (96 f^2 \pi^2 - 2 m_{\rm K}^2 (1536 (L_{4} - 3 L_{6}) \pi^2 + 2 \log(m_{\rm \eta}^2/\mu^2)
+ 3 \log(m_{\rm K}^2/\mu^2)) + 3 s (-3 - \log(m_{\rm K}^2/\mu^2) - 2 \log(m_{\rm \pi}^2/\mu^2) +
16 \pi^2 (24 L_{4} + 8 L_{5} + \overline{J}(s, m_{\rm K}^2) + 2 \overline{J}(s, m_{\rm \pi}^2))) +
2 m_{\rm \pi}^2 (1 + 8 \pi^2 (\overline{J}(s, m_{\rm \eta}^2) - 3 (16 (5 L_{4} + 3 L_{5} - 12
L_{6} - 8 L_{8}) + \overline{J}(s, m_{\rm \pi}^2))))))/(12 \pi^2)

Converted by Mathematica  (July 10, 2003)