•Calculation of the amplitude

Calculation of the amplitude:

amplFC = CreateFCAmp[mesontreeinsert, EqualMasses -> False, Sum -> True] ;

$ConstantIsoIndices = {I1, I2} ;

amplFC2 = CheckF[(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 ; SUNReduce[SUNReduce[Sum[WriteOutUMatrices[#], Evaluate[sums]], Explicit -> True, HoldSums -> False]]) & /@ summ) & /@ Take[amplFC, {1, -1}], "KSPiCTs.1.m"] ;

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

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.

tmp = (((amp[6, 3] // SUNReduce /. D -> Sequence[])) // PropagatorDenominatorExplicit) // Simplify ;

tmp

{(8 c _ 2^( ) (2 N _ 23^( ) (p _ 2 + p _ 3) . (p _ 2) + N _ 22^( ) ((p _ 2 + p _ 3) . (p _ 2) - p _ 2 · p _ 3)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (-(m _ ϕ^(6 ))^2 + p _ 2^2 + 2 p _ 2 · p _ 3 + p _ 3^2)), -(8 c _ 2^( ) (2 N _ 23^( ) p _ 2 · p _ 3 + N _ 22^( ) (p _ 1 · p _ 2 + p _ 2 · p _ 3)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ ϕ^(3 ))^2)), (8 c _ 2^( ) (N _ 22^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ π^(ó  ))^2 + 2 N _ 23^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ π^(ó  ))^2 - N _ 22^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^+^(ó  ))^2 - 2 N _ 23^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^+^(ó  ))^2 - 4 N _ 24^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^+^(ó  ))^2 - N _ 22^( ) p _ 2 · p _ 3 (m _ K^0^(ó  ))^2 - 2 N _ 23^( ) p _ 2 · p _ 3 (m _ K^0^(ó  ))^2 + N _ 22^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^0^(ó  ))^2 + 2 N _ 23^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^0^(ó  ))^2 + 4 N _ 24^( ) (p _ 2 + p _ 3) . (p _ 2) (m _ K^0^(ó  ))^2 + 2 N _ 20^( ) (p _ 2 · p _ 3 - (p _ 2 + p _ 3) . (p _ 2)) (p _ 2 + p _ 3) . (p _ 3) + 2 N _ 21^( ) (p _ 2 + p _ 3) . (p _ 2) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ ϕ^(3 ))^2) (-(m _ ϕ^(6 ))^2 + p _ 2^2 + 2 p _ 2 · p _ 3 + p _ 3^2))}

tmp /. p3 -> -p1 - p2 /. p2 -> 0 // Simplify

{0, 0, 0}

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

res1 = CheckF[((Simplify[ExpandScalarProduct[SUNReduce[SUNReduce[#]]] /. D -> Sequence[] /. subpar /. udrules] // PropagatorDenominatorExplicit) /. MandelstamRules // Simplify // MomentumCombine) & /@ amp[6, 3] /. {p1 + p2 -> -p3, -p1 - p2 -> p3, p2 + p3 -> -p1, -p2 - p3 -> p1, p1 + p3 -> -p2, -p1 - p3 -> p2} // Simplify, "KSPires1"]

{(8 c _ 2^( ) (N _ 22^( ) p _ 2^2 + N _ 23^( ) (p _ 1^2 + p _ 2^2 - p _ 3^2)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), (8 c _ 2^( ) (N _ 22^( ) p _ 2^2 + N _ 23^( ) (-p _ 1^2 + p _ 2^2 + p _ 3^2)) (!, _ 0^( ))^2)/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), -1/((f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (4 c _ 2^( ) (2 N _ 20^( ) p _ 2^2 (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 N _ 21^( ) (p _ 1^2 + p _ 2^2 - p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - (N _ 22^( ) + 2 N _ 23^( )) (p _ 1^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 3^2 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) + p _ 2^2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)}

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

res1 /. {Pair[Momentum[p3], Momentum[p3]] -> Pair[Momentum[p1], Momentum[p1]], p2 -> 0} /. RenormalizationState[1] -> RenormalizationState[0] // Simplify

{0, 0, 0}

Converted by Mathematica  (July 10, 2003)