KSPiAmplitude.nb

•Calculation of the amplitude

Calculation of the amplitude:

Isospin reduction:

We could use DoSumOver, but prefer a faster customized implementation:

$amplFC2 = ChekcF[(Print["Reducing..."] ; summ = SUNReduce[SUNReduce[#]] & /@ (Print["Expanding..."] ; Expand[#]) ; suminds = (#[[1]]) & /@ Union[Cases[#, _SumOver, Infinity]] ; sums = If[suminds === {}, {I1, 1}, Sequence @@ ((({#, If[FreeQ[summ, #], 1, 8]} & /@ suminds)))] ; Print["Length of expression: ", Length[summ]] ; tmpi = 0 ; res = If[Head[summ] === Plus, (++ tmpi ; WriteString["stdout", tmpi, " "] ; SUNReduce[SUNReduce[Sum[#, Evaluate[sums]], Explicit -> True, HoldSums -> False]]) & /@ summ, SUNReduce[SUNReduce[Sum[summ, Evaluate[sums]], Explicit -> True, HoldSums -> False]]]) & /@ Take[amplFC, {1, -1}], "KSPiamplFC2leading"] ;$

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Specialization to the (6,3) isospin channel, further isospin reduction and change to Mandelstam variables:

$amp[K1_, K2_] := amplFC2 /. {i1 -> K1, i3 -> K2} ;$

The leading order amplitude:

${(8 c _ 5^( ) (!, _ 0^( ))^2)/(p _ 1^2 - (m _ K^(ó ))^2), (8 c _ 5^( ) (!, _ 0^( ))^2)/(p _ 3^2 - (m _ π^(ó ))^2), -(4 (c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 c _ 5^( ) (m _ K^(ó ))^2) (!, _ 0^( ))^2)/((p _ 3^2 - (m _ π^(ó ))^2) (p _ 1^2 - (m _ K^(ó ))^2))}$

On-mass-shell limit, propagators and couplings to sources are divided off and set to 0:

$Cancel[((-I) (Pair[Momentum[p3], Momentum[p3]] - ParticleMass[Pion]^2) (-I) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2))/(2 I DecayConstant[PhiMeson] QuarkCondensate[])^2 * treeamp /. _RenormalizationState -> Sequence[]] /. {Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion]^2, Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2, Pair[Momentum[p2], Momentum[p2]] -> 0} // Simplify$

Converted by Mathematica (July 10, 2003)