•Renormalization

The coefficients of the infinities:

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

(4 (7 (m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2 + 9 p _ 3^2 + 27 p _ 3 · p _ 4 + 9 p _ 4^2) !, _ 0^( ))/(9 (f _ ϕ^(ó  ))^2)

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

(8 (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^2)

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

-(4 (37 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2 + 27 p _ 3 · p _ 4) !, _ 0^( ))/(9 (f _ ϕ^(ó  ))^2)

c1 + c2 + c3 // Simplify

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

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 // Simplify ;

We may put it on the mass-shell:

ampfin = ampfinal /. massshellrules // FullSimplify

-1/(144 π^2 (f _ ϕ^(ó  ))^2) ((288 π^2 (f _ ϕ^(ó  ))^2 - 2 (24 π^2 (96 (2 L _ 4^(r ) + L _ 5^(r ) - 4 L _ 6^(r ) - 2 L _ 8^(r )) + 3 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 1^2) - Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 1^2)) - 9 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2) - 3) (m _ π^(ó  ))^2 - 4 (1152 π^2 (L _ 4^(r ) - 2 L _ 6^(r )) + log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 9 (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^( ))

And produce some nice L A T E X output:

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

-(B_0 (288 f^2 \pi^2 - 4 m_{\rm K}^2 (1152 (L_{4} - 2 L_{6}) \pi^2 + \log(m_{\rm \eta}^2/\mu^2)) +
9 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 (-3 + \\log(m_{\rm \eta}^2/\mu^2) - 9 \log(m_{\rm \pi}^2/\mu^2) + 24 \pi^2 (96 (2 L_{4} + L_{5} - 4 L_{6}
- 2 L_{8}) - \overline{J}(s, m_{\rm \eta}^2) + 3 \overline{J}(s, m_{\rm \pi}^2)))))/(144 f^2 \\pi^2)

Converted by Mathematica  (July 10, 2003)