•Renormalization

The coefficients of the infinities:

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

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

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

(16 (m _ π^(ó  ))^2 !, _ 0^( ) δ _ (0 i _ 1)^(2) δ _ (i _ 2 i _ 3)^(2))/(3 (f _ π^(ó  ))^2)

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

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

c1 + c2 + c3 // SUNReduce // Simplify

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

This is then the full renormalized amplitude:

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

We may amputate and put it on the mass shell:

ampfin = ampfinal // MandelstamReduce[#, OnMassShell -> True] & // FullSimplify

-1/(16 π^2 (f _ π^(ó  ))^2) (((-16 π^2 (32 (2 L _ 4^(r ) + L _ 5^(r ) - 2 (2 L _ 6^(r ) + L _ 8^(r ))) + Overscript[J, _] _ (m _ π^(ó  ))^2(s)) + 2 log((m _ π^(ó  ))^2/μ^2) + 1) (m _ π^(ó  ))^2 - 2 s + 32 π^2 ((f _ π^(ó  ))^2 + s (8 L _ 4^(r ) + 4 L _ 5^(r ) + Overscript[J, _] _ (m _ π^(ó  ))^2(s))) - 2 s log((m _ π^(ó  ))^2/μ^2)) !, _ 0^( ) 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 \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 + 2 \\log(m_{\rm \pi}^2/\mu^2) - 16 \pi^2 (32 (2 L_{4} + L_{5} - 2 (2 L_{6} + L_{8})) + \\overline{J}(s, m_{\rm \pi}^2)))))/ (16 f^2 \pi^2)

Converted by Mathematica  (July 10, 2003)