This is the sum of all unrenormalized amplitudes:
The infinities exactly cancel:
This is then the full renormalized amplitude:
Converted by Mathematica (July 10, 2003)
![cc1 = Coefficient[Renormalize[MandelstamReduce[ampfinal]], LeutwylerLambda[]] // Simplify](../HTMLFiles/index_40.gif){kind=small}
![ampfinalren = FullSimplify /@ ((Simplify /@ Collect[Renormalize[ampfinal] // Expand, _LeutwylerLambda]) //. manrules // Expand // Collect[#, {_DecayConstant, _SU2Delta, _Log, _LeutwylerJBar, _ParticleMass, MandelstamS, MandelstamT, MandelstamU}] &)](../HTMLFiles/index_42.gif)
![((t - (m _ π^(ó r ))^2) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + (u - (m _ π^(ó r ))^2) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2) + (s - (m _ π^(ó r ))^2) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2))/(f _ π^(ó r ))^2 + 1/(288 (f _ π^(ó r ))^4) (3 (((1536 π^2 (2 L _ 1^(r ) + 2 L _ 2^(r ) + L _ 3^(r ) - 2 L _ 4^(r ) - L _ 5^(r ) + 2 L _ 6^(r ) + L _ 8^(r )) + 7) (m _ π^(ó r ))^4)/π^2 - (8 t (288 π^2 (4 L _ 1^(r ) + 2 (L _ 2^(r ) + L _ 3^(r ) - L _ 4^(r )) - L _ 5^(r )) + 1) (m _ π^(ó r ))^2)/(3 π^2) + 4/3 s u (1/π^2 - 576 L _ 2^(r )) + 96 t^2 (8 L _ 1^(r ) + 4 L _ 2^(r ) + 4 L _ 3^(r ) - 5/(144 π^2)) - (log((m _ π^(ó r ))^2/μ^2) (-7 (m _ π^(ó r ))^4 + 4 t (m _ π^(ó r ))^2 + 3 t^2 + (s - u)^2))/π^2 - 16 Overscript[J, _] _ (m _ π^(ó r ))^2(u) (-2 (m _ π^(ó r ))^4 + (-3 s + t + 3 u) (m _ π^(ó r ))^2 + (s - u) u) + 48 Overscript[J, _] _ (m _ π^(ó r ))^2(t) (t^2 - (m _ π^(ó r ))^4) + 16 Overscript[J, _] _ (m _ π^(ó r ))^2(s) (2 (m _ π^(ó r ))^4 - (3 s + t - 3 u) (m _ π^(ó r ))^2 + s (s - u)) - s^2/π^2 - u^2/π^2) δ _ (i _ 1 i _ 4)^(2) δ _ (i _ 2 i _ 3)^(2) + 1/π^2 ((3 (16 π^2 (2 (96 L _ 2^(r ) + 48 L _ 3^(r ) - 96 L _ 4^(r ) - 48 L _ 5^(r ) + 96 L _ 6^(r ) + 48 L _ 8^(r ) + Overscript[J, _] _ (m _ π^(ó r ))^2(s) + Overscript[J, _] _ (m _ π^(ó r ))^2(t)) - 3 Overscript[J, _] _ (m _ π^(ó r ))^2(u)) + 7 (log((m _ π^(ó r ))^2/μ^2) + 1)) (m _ π^(ó r ))^4 - 4 (2 (u + 6 π^2 (3 (s - t) (Overscript[J, _] _ (m _ π^(ó r ))^2(s) - Overscript[J, _] _ (m _ π^(ó r ))^2(t)) + u (96 L _ 2^(r ) + 96 L _ 3^(r ) - 96 L _ 4^(r ) - 48 L _ 5^(r ) + Overscript[J, _] _ (m _ π^(ó r ))^2(s) + Overscript[J, _] _ (m _ π^(ó r ))^2(t)))) + 3 u log((m _ π^(ó r ))^2/μ^2)) (m _ π^(ó r ))^2 - 3 s^2 - 3 t^2 - 10 u^2 + 2304 π^2 L _ 1^(r ) (u - 2 (m _ π^(ó r ))^2)^2 + 4 s t + 48 π^2 (3 (8 L _ 3^(r ) + Overscript[J, _] _ (m _ π^(ó r ))^2(u)) u^2 + 24 (u^2 - 2 s t) L _ 2^(r ) + (s - t) (s Overscript[J, _] _ (m _ π^(ó r ))^2(s) - t Overscript[J, _] _ (m _ π^(ó r ))^2(t))) - 3 ((s - t)^2 + 3 u^2) log((m _ π^(ó r ))^2/μ^2)) δ _ (i _ 1 i _ 3)^(2) δ _ (i _ 2 i _ 4)^(2)) + 3 (((1536 π^2 (2 L _ 1^(r ) + 2 L _ 2^(r ) + L _ 3^(r ) - 2 L _ 4^(r ) - L _ 5^(r ) + 2 L _ 6^(r ) + L _ 8^(r )) + 7) (m _ π^(ó r ))^4)/π^2 - (8 s (288 π^2 (4 L _ 1^(r ) + 2 (L _ 2^(r ) + L _ 3^(r ) - L _ 4^(r )) - L _ 5^(r )) + 1) (m _ π^(ó r ))^2)/(3 π^2) + 4/3 t u (1/π^2 - 576 L _ 2^(r )) + 96 s^2 (8 L _ 1^(r ) + 4 L _ 2^(r ) + 4 L _ 3^(r ) - 5/(144 π^2)) - (log((m _ π^(ó r ))^2/μ^2) (-7 (m _ π^(ó r ))^4 + 4 s (m _ π^(ó r ))^2 + 3 s^2 + (t - u)^2))/π^2 - 16 Overscript[J, _] _ (m _ π^(ó r ))^2(u) (-2 (m _ π^(ó r ))^4 + (s - 3 t + 3 u) (m _ π^(ó r ))^2 + (t - u) u) + 48 Overscript[J, _] _ (m _ π^(ó r ))^2(s) (s^2 - (m _ π^(ó r ))^4) + 16 Overscript[J, _] _ (m _ π^(ó r ))^2(t) (2 (m _ π^(ó r ))^4 - (s + 3 t - 3 u) (m _ π^(ó r ))^2 + t (t - u)) - t^2/π^2 - u^2/π^2) δ _ (i _ 1 i _ 2)^(2) δ _ (i _ 3 i _ 4)^(2))](../HTMLFiles/index_43.gif)
