KSPiPiAmplitude.nb

•Reduction of the K->2π counterterm amplitude

We divide off the kaon propagator and put the kaon on-mass-shell:

$end4old1 = (Cancel[(res1old /. cancelU) * (Pair[Momentum[p1], Momentum[p1]] - (ParticleMass[Kaon, RenormalizationState[0]])^2)] /. {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2} /. CouplingConstant[c_[4], n_] -> CouplingConstant[c[4], n, RenormalizationState[0]] /. RenormalizationState[1] -> RenormalizationState[0]) // Simplify$

${0, -1/(3 (f _ ϕ^(ó ))^4) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-12 N _ 8^( ) (m _ π^(ó ))^4 + 24 N _ 10^( ) (m _ π^(ó ))^4 + 56 N _ 11^( ) (m _ π^(ó ))^4 + 24 N _ 12^( ) (m _ π^(ó ))^4 - 21 N _ 8^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 18 N _ 9^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 8 N _ 10^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 40 N _ 11^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 24 N _ 12^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 9 s N _ 8^( ) (m _ π^(ó ))^2 + 6 s N _ 9^( ) (m _ π^(ó ))^2 - 3 N _ 8^( ) p _ 2^2 (m _ π^(ó ))^2 - 6 N _ 9^( ) p _ 2^2 (m _ π^(ó ))^2 + 6 N _ 8^( ) (m _ K^(ó ))^4 - 16 N _ 10^( ) (m _ K^(ó ))^4 - 16 N _ 11^( ) (m _ K^(ó ))^4 + 18 s N _ 8^( ) (m _ K^(ó ))^2 + 12 s N _ 9^( ) (m _ K^(ó ))^2 - 6 N _ 8^( ) p _ 2^2 (m _ K^(ó ))^2 + 24 N _ 7^( ) (s - 2 (m _ π^(ó ))^2) ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) + 6 N _ 5^( ) (-4 (m _ π^(ó ))^4 + 2 s (m _ π^(ó ))^2 + (m _ K^(ó ))^2 ((m _ K^(ó ))^2 + s - p _ 2^2)))), 0, 1/(3 (f _ ϕ^(ó ))^4 (p _ 2^2 - (m _ K^(ó ))^2)) (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) (6 L _ 3^( ) (m _ π^(ó ))^4 - 4 L _ 4^( ) (m _ π^(ó ))^4 - 4 L _ 5^( ) (m _ π^(ó ))^4 + 8 L _ 6^( ) (m _ π^(ó ))^4 + 8 L _ 8^( ) (m _ π^(ó ))^4 + 18 L _ 3^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 82 L _ 4^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 18 L _ 5^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 120 L _ 6^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 48 L _ 8^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 - 18 s L _ 3^( ) (m _ π^(ó ))^2 - 12 t L _ 3^( ) (m _ π^(ó ))^2 + 30 s L _ 4^( ) (m _ π^(ó ))^2 + 6 s L _ 5^( ) (m _ π^(ó ))^2 + 18 L _ 3^( ) p _ 2^2 (m _ π^(ó ))^2 - 26 L _ 4^( ) p _ 2^2 (m _ π^(ó ))^2 - 6 L _ 5^( ) p _ 2^2 (m _ π^(ó ))^2 - 4 L _ 4^( ) (m _ K^(ó ))^4 - 2 L _ 5^( ) (m _ K^(ó ))^4 + 16 L _ 6^( ) (m _ K^(ó ))^4 + 8 L _ 8^( ) (m _ K^(ó ))^4 - 9 s L _ 3^( ) (m _ K^(ó ))^2 - 6 t L _ 3^( ) (m _ K^(ó ))^2 + 36 s L _ 4^( ) (m _ K^(ó ))^2 + 6 s L _ 5^( ) (m _ K^(ó ))^2 + 6 L _ 3^( ) p _ 2^2 (m _ K^(ó ))^2 - 4 L _ 4^( ) p _ 2^2 (m _ K^(ó ))^2 - 2 L _ 5^( ) p _ 2^2 (m _ K^(ó ))^2 + 9 s^2 L _ 3^( ) + 6 t^2 L _ 3^( ) + 6 s t L _ 3^( ) - 9 s L _ 3^( ) p _ 2^2 - 6 t L _ 3^( ) p _ 2^2 + 24 L _ 1^( ) (s - 2 (m _ π^(ó ))^2) (-(m _ K^(ó ))^2 + s - p _ 2^2) + 12 L _ 2^( ) (2 (m _ π^(ó ))^4 - 2 (-(m _ K^(ó ))^2 + s + 2 t) (m _ π^(ó ))^2 + s^2 + 2 t^2 - s (m _ K^(ó ))^2 - 2 t (m _ K^(ó ))^2 + 2 s t - p _ 2^2 (-2 (m _ π^(ó ))^2 - 2 (m _ K^(ó ))^2 + s + 2 t))))}$

We need only the first two amplitudes (well, actually only the first, since the kaon will be put on-mass-shell), since the last two have already been included above.

Converted by Mathematica (July 10, 2003)