KSS2Amplitude.nb

•Calculation of the amplitude

Calculation of the amplitude:

$amplFC2 = (summ = SUNReduce[SUNReduce[#]] & /@ (Print["Expanding..."] ; tmp = Expand[#] ; Print["Reducing..."] ; tmp) ; suminds = (#[[1]]) & /@ Union[Cases[#, _SumOver, Infinity]] ; sums = If[suminds === {}, {I1, 1}, Sequence @@ ((({#, If[FreeQ[summ, #], 1, 8]} & /@ suminds)))] ; Print["Length of expression: ", Length[summ]] ; tmpii = 0 ; res = (If[IntegerQ[tmpii/100], Print[tmpii]] ; ++ tmpii ; Sum[SUNReduce[SUNReduce[Sum[WriteOutUMatrices[#], Evaluate[sums]], Explicit -> True, HoldSums -> False]]]) & /@ summ) & /@ Take[amplFC, {1, -1}] ;$

${(4 c _ 2^( ) (2 N _ 21^( ) p _ 1 · p _ 2 - N _ 22^( ) p _ 2 · p _ 3) !, _ 0^( ) (σ _ (2 2)^i - σ _ (3 3)^i))/(f _ ϕ^(ó ))^3}$

The new (as compared to the K->2π amplitude) contributions with a weak counterterm vertex are proportional to the 'scalar' momentum and vanish when it's set to zero.

Further isospin reduction and change to Mandelstam variables:

${-(2 c _ 2^( ) (N _ 22^( ) (p _ 1^2 - p _ 2^2 - p _ 3^2) + 2 N _ 21^( ) (p _ 1^2 + p _ 2^2 - p _ 3^2)) !, _ 0^( ) (σ _ (2 2)^i - σ _ (3 3)^i))/(f _ ϕ^(ó ))^3}$

The new (as compared to the K->π amplitude) contributions with a weak counterterm vertex are proportional to the 'scalar' momentum and vanish when it's set to zero. The contributions with a leading order counterterm vertex are not proportional to the 'scalar' momentum and don't vanish when it's set to zero

Converted by Mathematica (July 10, 2003)