KSPiPiAmplitude.nb

•Calculation of the amplitude

Calculation of the amplitude:

$ttmp = I * Collect[Cancel[(amplFC[[2]] /. {i1 -> 7, i3 -> 3, i4 -> 3, I1 -> 7} // WriteOutUMatrices /. subpar /. udrules // MomentumExpand // ExpandScalarProduct) /. D -> Sequence[] //. MandelstamRules /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. Pair[Momentum[p1], Momentum[p1]]^2 -> ParticleMass[Kaon, RenormalizationState[1]]^4], {_DecayConstant, _ParticleMass}] /. _RenormalizationState -> Sequence[] // SUNReduce // Simplify ;$

Checks:

$ttmp /. subpar /. udrules /. {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> ParticleMass[Kaon]^2, MandelstamU -> Pair[Momentum[p2], Momentum[p2]]} // FullSimplify$

$Limit[ttmp /. subpar /. udrules /. {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> ParticleMass[Kaon]^2, MandelstamU -> Pair[Momentum[p2], Momentum[p2]]} /. {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> ParticleMass[Kaon]^2, MandelstamU -> Pair[Momentum[p2], Momentum[p2]]} //. MandelstamRules /. _RenormalizationState -> Sequence[], ParticleMass[Pion] -> 0] // Simplify$

Isospin reduction:

$amplFC2 = CheckF[(summ = (* SUNReduce[SUNReduce[#]] & /@ *) (Print["Expanding..."] ; tmp = Expand[#] ; Print["Reducing..."] ; tmp) ; suminds = (#[[1]]) & /@ Union[Cases[#, _SumOver, Infinity]] ; sums = If[suminds === {}, {I1, 1}, Sequence @@ ((({#, If[FreeQ[summ, #], 1, 8]} & /@ suminds)))] ; Print["Summing over: ", sums, "\n"] ; Print["Length of expression: ", Length[summ]] ; tmpii = 0 ; res = (If[IntegerQ[tmpii/100], WriteString["stdout", tmpii, " "]] ; ++ tmpii ; SUNReduce[SUNReduce[Sum[WriteOutUMatrices[#], Evaluate[sums]], Explicit -> True, HoldSums -> False]]) & /@ summ) & /@ Take[amplFC, {1, -1}], "KSPiPiCTs.m"] ;$
$amp[K1_, K2_, K3_] := amplFC2 /. {i1 -> K1, i3 -> K2, i4 -> K3} ;$

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.

${-1/(f _ ϕ^(ó ))^4 (i c _ 2^( ) µ _ μ _ 1(p _ 1) (-8 N _ 21^( ) p _ 2^μ _ 1 (m _ π^(ó ))^2 - 2 N _ 22^( ) p _ 2^μ _ 1 (m _ π^(ó ))^2 - 4 N _ 23^( ) p _ 2^μ _ 1 (m _ π^(ó ))^2 + 8 N _ 21^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 + 2 N _ 22^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 + 4 N _ 23^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 + 8 N _ 24^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 - 4 N _ 21^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 - 2 N _ 22^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 - 4 N _ 23^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 - 8 N _ 24^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 + N _ 26^( ) p _ 2^μ _ 1 p _ 1^2 - N _ 26^( ) p _ 1^μ _ 1 p _ 1 · p _ 2 + 2 N _ 19^( ) p _ 3^μ _ 1 p _ 1 · p _ 2 + N _ 25^( ) (p _ 1^μ _ 1 p _ 1 · p _ 2 - p _ 2^μ _ 1 p _ 1^2) - 4 N _ 19^( ) p _ 2^μ _ 1 p _ 1 · p _ 3 + 2 N _ 19^( ) p _ 3^μ _ 1 p _ 2^2 + 2 N _ 19^( ) p _ 1^μ _ 1 p _ 2 · p _ 3 - 2 N _ 19^( ) p _ 2^μ _ 1 p _ 2 · p _ 3 + 4 N _ 19^( ) p _ 3^μ _ 1 p _ 2 · p _ 3 + N _ 20^( ) (-2 p _ 3^μ _ 1 (p _ 1 · p _ 2 + p _ 2^2 + 2 p _ 2 · p _ 3) + p _ 2^μ _ 1 (p _ 1^2 - 2 (p _ 2^2 + p _ 2 · p _ 3)) - p _ 1^μ _ 1 (p _ 1 · p _ 2 + 2 (p _ 2^2 + p _ 2 · p _ 3))) - 4 N _ 19^( ) p _ 2^μ _ 1 p _ 3^2)), -1/(3 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ ϕ^(7 ))^2)) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (24 N _ 21^( ) p _ 1 · p _ 2 (m _ π^(ó ))^2 + 12 N _ 22^( ) p _ 1 · p _ 2 (m _ π^(ó ))^2 + 24 N _ 23^( ) p _ 1 · p _ 2 (m _ π^(ó ))^2 + 4 N _ 21^( ) p _ 2^2 (m _ π^(ó ))^2 + 6 N _ 22^( ) p _ 2^2 (m _ π^(ó ))^2 + 12 N _ 23^( ) p _ 2^2 (m _ π^(ó ))^2 - 24 N _ 21^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 12 N _ 22^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 24 N _ 23^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 36 N _ 24^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 4 N _ 21^( ) p _ 2^2 (m _ K^+^(ó ))^2 - 6 N _ 22^( ) p _ 2^2 (m _ K^+^(ó ))^2 - 12 N _ 23^( ) p _ 2^2 (m _ K^+^(ó ))^2 - 12 N _ 24^( ) p _ 2^2 (m _ K^+^(ó ))^2 + 18 N _ 21^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 + 9 N _ 22^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 + 18 N _ 23^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 + 36 N _ 24^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 + 6 N _ 21^( ) p _ 2^2 (m _ K^0^(ó ))^2 + 3 N _ 22^( ) p _ 2^2 (m _ K^0^(ó ))^2 + 6 N _ 23^( ) p _ 2^2 (m _ K^0^(ó ))^2 + 12 N _ 24^( ) p _ 2^2 (m _ K^0^(ó ))^2 + 6 N _ 19^( ) (-p _ 1^2 p _ 2 · p _ 3 - p _ 1 · p _ 3 (p _ 2^2 + 2 p _ 2 · p _ 3) + p _ 1 · p _ 2 (p _ 1 · p _ 3 + p _ 2 · p _ 3 + 2 p _ 3^2)) - 6 N _ 20^( ) (p _ 1 · p _ 3 p _ 2^2 + 2 p _ 2 · p _ 3 p _ 2^2 + 2 p _ 3^2 p _ 2^2 - 2 p _ 1 · p _ 3 p _ 2 · p _ 3 - p _ 1^2 (p _ 2^2 + p _ 2 · p _ 3) + p _ 1 · p _ 2 (p _ 1 · p _ 3 - p _ 2^2 + p _ 2 · p _ 3 + 2 p _ 3^2)))), 1/(27 (f _ ϕ^(ó ))^4 (p _ 2^2 - (m _ ϕ^(7 ))^2)) (8 i c _ 5^( ) µ _ μ _ 1(p _ 1) ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2) (18 L _ 4^( ) p _ 1^μ _ 1 (m _ π^(ó ))^2 + 126 L _ 4^( ) p _ 2^μ _ 1 (m _ π^(ó ))^2 - 14 L _ 5^( ) p _ 2^μ _ 1 (m _ π^(ó ))^2 + 18 L _ 4^( ) p _ 1^μ _ 1 (m _ K^+^(ó ))^2 + 126 L _ 4^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 - 20 L _ 5^( ) p _ 2^μ _ 1 (m _ K^+^(ó ))^2 + 18 L _ 4^( ) p _ 1^μ _ 1 (m _ K^0^(ó ))^2 - 9 L _ 5^( ) p _ 1^μ _ 1 (m _ K^0^(ó ))^2 + 126 L _ 4^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 + 115 L _ 5^( ) p _ 2^μ _ 1 (m _ K^0^(ó ))^2 - 432 L _ 1^( ) p _ 2^μ _ 1 p _ 1 · p _ 3 - 432 L _ 1^( ) p _ 2^μ _ 1 p _ 2 · p _ 3 - 216 L _ 2^( ) ((p _ 1^μ _ 1 + p _ 2^μ _ 1) p _ 2 · p _ 3 + p _ 3^μ _ 1 (p _ 1 · p _ 2 + p _ 2^2 + 2 p _ 2 · p _ 3)) - 432 L _ 1^( ) p _ 2^μ _ 1 p _ 3^2 - 54 L _ 3^( ) (p _ 1^μ _ 1 p _ 2 · p _ 3 + p _ 3^μ _ 1 (p _ 1 · p _ 2 + p _ 2^2 + 2 p _ 2 · p _ 3) + p _ 2^μ _ 1 (2 p _ 1 · p _ 3 + 3 p _ 2 · p _ 3 + 2 p _ 3^2)))), -(8 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2) (4 L _ 6^( ) (m _ π^(ó ))^4 + 4 L _ 8^( ) (m _ π^(ó ))^4 - 8 L _ 8^( ) (m _ K^+^(ó ))^2 (m _ π^(ó ))^2 + 60 L _ 6^( ) (m _ K^0^(ó ))^2 (m _ π^(ó ))^2 + 32 L _ 8^( ) (m _ K^0^(ó ))^2 (m _ π^(ó ))^2 + L _ 4^( ) p _ 1^2 (m _ π^(ó ))^2 - L _ 5^( ) p _ 1^2 (m _ π^(ó ))^2 + 28 L _ 4^( ) p _ 1 · p _ 2 (m _ π^(ó ))^2 + 4 L _ 5^( ) p _ 1 · p _ 2 (m _ π^(ó ))^2 - 2 L _ 4^( ) p _ 1 · p _ 3 (m _ π^(ó ))^2 - 2 L _ 5^( ) p _ 1 · p _ 3 (m _ π^(ó ))^2 + L _ 4^( ) p _ 2^2 (m _ π^(ó ))^2 - L _ 5^( ) p _ 2^2 (m _ π^(ó ))^2 - 2 L _ 4^( ) p _ 2 · p _ 3 (m _ π^(ó ))^2 - 2 L _ 5^( ) p _ 2 · p _ 3 (m _ π^(ó ))^2 - 2 L _ 4^( ) p _ 3^2 (m _ π^(ó ))^2 - 2 L _ 5^( ) p _ 3^2 (m _ π^(ó ))^2 - 4 L _ 6^( ) (m _ K^+^(ó ))^4 + 4 L _ 8^( ) (m _ K^+^(ó ))^4 + 8 L _ 6^( ) (m _ K^0^(ó ))^4 + 32 L _ 8^( ) (m _ K^0^(ó ))^4 + L _ 4^( ) p _ 1^2 (m _ K^+^(ó ))^2 + L _ 5^( ) p _ 1^2 (m _ K^+^(ó ))^2 + 4 L _ 4^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 4 L _ 5^( ) p _ 1 · p _ 2 (m _ K^+^(ó ))^2 - 2 L _ 4^( ) p _ 1 · p _ 3 (m _ K^+^(ó ))^2 + 2 L _ 5^( ) p _ 1 · p _ 3 (m _ K^+^(ó ))^2 + L _ 4^( ) p _ 2^2 (m _ K^+^(ó ))^2 + L _ 5^( ) p _ 2^2 (m _ K^+^(ó ))^2 - 2 L _ 4^( ) p _ 2 · p _ 3 (m _ K^+^(ó ))^2 + 2 L _ 5^( ) p _ 2 · p _ 3 (m _ K^+^(ó ))^2 - 2 L _ 4^( ) p _ 3^2 (m _ K^+^(ó ))^2 + 2 L _ 5^( ) p _ 3^2 (m _ K^+^(ó ))^2 + 4 L _ 6^( ) (m _ K^+^(ó ))^2 (m _ K^0^(ó ))^2 - 32 L _ 8^( ) (m _ K^+^(ó ))^2 (m _ K^0^(ó ))^2 + L _ 4^( ) p _ 1^2 (m _ K^0^(ó ))^2 - 2 L _ 5^( ) p _ 1^2 (m _ K^0^(ó ))^2 + 4 L _ 4^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 + 4 L _ 5^( ) p _ 1 · p _ 2 (m _ K^0^(ó ))^2 - 26 L _ 4^( ) p _ 1 · p _ 3 (m _ K^0^(ó ))^2 - 8 L _ 5^( ) p _ 1 · p _ 3 (m _ K^0^(ó ))^2 + L _ 4^( ) p _ 2^2 (m _ K^0^(ó ))^2 - 2 L _ 5^( ) p _ 2^2 (m _ K^0^(ó ))^2 - 26 L _ 4^( ) p _ 2 · p _ 3 (m _ K^0^(ó ))^2 - 8 L _ 5^( ) p _ 2 · p _ 3 (m _ K^0^(ó ))^2 - 26 L _ 4^( ) p _ 3^2 (m _ K^0^(ó ))^2 - 8 L _ 5^( ) p _ 3^2 (m _ K^0^(ó ))^2 - 18 L _ 3^( ) p _ 1 · p _ 2 p _ 1 · p _ 3 - 6 L _ 3^( ) p _ 1 · p _ 3 p _ 2^2 - 6 L _ 3^( ) p _ 1^2 p _ 2 · p _ 3 - 18 L _ 3^( ) p _ 1 · p _ 2 p _ 2 · p _ 3 - 12 L _ 3^( ) p _ 1 · p _ 3 p _ 2 · p _ 3 - 24 L _ 2^( ) (p _ 1^2 p _ 2 · p _ 3 + p _ 1 · p _ 2 (p _ 1 · p _ 3 + p _ 2 · p _ 3) + p _ 1 · p _ 3 (p _ 2^2 + 2 p _ 2 · p _ 3)) - 12 L _ 3^( ) p _ 1 · p _ 2 p _ 3^2 - 48 L _ 1^( ) p _ 1 · p _ 2 (p _ 1 · p _ 3 + p _ 2 · p _ 3 + p _ 3^2)))/(3 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ ϕ^(7 ))^2) (p _ 2^2 - (m _ ϕ^(7 ))^2))}$

${0, 0, (8 i c _ 5^( ) (((-p _ 1 - p _ 3)^μ _ 1) + p _ 3^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2) (2 L _ 4^( ) ((m _ π^(ó ))^2 + (m _ K^+^(ó ))^2 + (m _ K^0^(ó ))^2) - L _ 5^( ) (m _ K^0^(ó ))^2))/(3 (f _ ϕ^(ó ))^4 (m _ ϕ^(7 ))^2), -(8 i c _ 5^( ) ((-p _ 1)^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2) (L _ 5^( ) (2 ( -p _ 1 - p _ 3 ) · p _ 3 ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2 + 4 (m _ K^0^(ó ))^2) - (((-p _ 1 - p _ 3) + p _ 3) . (-p _ 1 - p _ 3) + ((-p _ 1 - p _ 3) + p _ 3) . (p _ 3)) ((m _ π^(ó ))^2 - (m _ K^+^(ó ))^2 + 2 (m _ K^0^(ó ))^2)) + L _ 4^( ) ((((-p _ 1 - p _ 3) + p _ 3) . (-p _ 1 - p _ 3) + ((-p _ 1 - p _ 3) + p _ 3) . (p _ 3)) ((m _ π^(ó ))^2 + (m _ K^+^(ó ))^2 + (m _ K^0^(ó ))^2) + 2 ( -p _ 1 - p _ 3 ) · p _ 3 ((m _ π^(ó ))^2 + (m _ K^+^(ó ))^2 + 13 (m _ K^0^(ó ))^2)) + 4 (L _ 6^( ) ((m _ π^(ó ))^4 + 15 (m _ K^0^(ó ))^2 (m _ π^(ó ))^2 - (m _ K^+^(ó ))^4 + 2 (m _ K^0^(ó ))^4 + (m _ K^+^(ó ))^2 (m _ K^0^(ó ))^2) + L _ 8^( ) ((m _ π^(ó ))^4 - 2 ((m _ K^+^(ó ))^2 - 4 (m _ K^0^(ó ))^2) (m _ π^(ó ))^2 + (m _ K^+^(ó ))^4 + 8 (m _ K^0^(ó ))^4 - 8 (m _ K^+^(ó ))^2 (m _ K^0^(ó ))^2))))/(3 (f _ ϕ^(ó ))^4 (m _ ϕ^(7 ))^2 (-(m _ ϕ^(7 ))^2 + ((-p _ 1 - p _ 3)^2) + 2 ( -p _ 1 - p _ 3 ) · p _ 3 + p _ 3^2))}$

Specialization to the (7,3,3) isospin channel, further isospin reduction and change to Mandelstam variables. We only need the 2nd and 4th amplitudes (since we'll put the kaon on-mass-shell):

${-1/(6 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ K^(ó ))^2)) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (48 N _ 21^( ) (m _ π^(ó r ))^2 (m _ π^(ó ))^2 + 24 N _ 22^( ) (m _ π^(ó r ))^2 (m _ π^(ó ))^2 + 48 N _ 23^( ) (m _ π^(ó r ))^2 (m _ π^(ó ))^2 - 24 t N _ 21^( ) (m _ π^(ó ))^2 - 24 u N _ 21^( ) (m _ π^(ó ))^2 - 12 t N _ 22^( ) (m _ π^(ó ))^2 - 12 u N _ 22^( ) (m _ π^(ó ))^2 - 24 t N _ 23^( ) (m _ π^(ó ))^2 - 24 u N _ 23^( ) (m _ π^(ó ))^2 + 8 N _ 21^( ) p _ 2^2 (m _ π^(ó ))^2 + 12 N _ 22^( ) p _ 2^2 (m _ π^(ó ))^2 + 24 N _ 23^( ) p _ 2^2 (m _ π^(ó ))^2 - 12 N _ 21^( ) (m _ π^(ó r ))^2 (m _ K^(ó ))^2 - 6 N _ 22^( ) (m _ π^(ó r ))^2 (m _ K^(ó ))^2 - 12 N _ 23^( ) (m _ π^(ó r ))^2 (m _ K^(ó ))^2 + 6 t N _ 21^( ) (m _ K^(ó ))^2 + 6 u N _ 21^( ) (m _ K^(ó ))^2 + 3 t N _ 22^( ) (m _ K^(ó ))^2 + 3 u N _ 22^( ) (m _ K^(ó ))^2 + 6 t N _ 23^( ) (m _ K^(ó ))^2 + 6 u N _ 23^( ) (m _ K^(ó ))^2 + 4 N _ 21^( ) p _ 2^2 (m _ K^(ó ))^2 - 6 N _ 22^( ) p _ 2^2 (m _ K^(ó ))^2 - 12 N _ 23^( ) p _ 2^2 (m _ K^(ó ))^2 - 3 N _ 20^( ) (2 (m _ π^(ó r ))^4 - 2 p _ 1^2 (m _ π^(ó r ))^2 + 2 p _ 2^4 + s t + s u - 2 t u - 6 p _ 2^2 (s - (m _ π^(ó r ))^2)) + 3 N _ 19^( ) (2 (m _ π^(ó r ))^4 - 2 p _ 2^2 (m _ π^(ó r ))^2 + s t + s u - 2 t u + 2 p _ 1^2 (p _ 2^2 - (m _ π^(ó r ))^2)))), 1/(3 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ K^(ó ))^2) ((m _ K^(ó ))^2 - p _ 2^2)) (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) (8 L _ 6^( ) (m _ π^(ó ))^4 + 8 L _ 8^( ) (m _ π^(ó ))^4 + 48 L _ 4^( ) (m _ π^(ó r ))^2 (m _ π^(ó ))^2 + 8 L _ 5^( ) (m _ π^(ó r ))^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 + 4 s L _ 4^( ) (m _ π^(ó ))^2 - 26 t L _ 4^( ) (m _ π^(ó ))^2 - 26 u L _ 4^( ) (m _ π^(ó ))^2 - 6 t L _ 5^( ) (m _ π^(ó ))^2 - 6 u L _ 5^( ) (m _ π^(ó ))^2 - 30 L _ 3^( ) (m _ π^(ó r ))^4 + 16 L _ 6^( ) (m _ K^(ó ))^4 + 8 L _ 8^( ) (m _ K^(ó ))^4 - 6 L _ 3^( ) p _ 2^4 + 18 s L _ 3^( ) (m _ π^(ó r ))^2 + 18 t L _ 3^( ) (m _ π^(ó r ))^2 + 18 u L _ 3^( ) (m _ π^(ó r ))^2 - 12 L _ 3^( ) p _ 2^2 (m _ π^(ó r ))^2 - 48 L _ 4^( ) (m _ π^(ó r ))^2 (m _ K^(ó ))^2 - 8 L _ 5^( ) (m _ π^(ó r ))^2 (m _ K^(ó ))^2 + 32 s L _ 4^( ) (m _ K^(ó ))^2 - 4 t L _ 4^( ) (m _ K^(ó ))^2 - 4 u L _ 4^( ) (m _ K^(ó ))^2 + 4 s L _ 5^( ) (m _ K^(ó ))^2 - 2 t L _ 5^( ) (m _ K^(ó ))^2 - 2 u L _ 5^( ) (m _ K^(ó ))^2 - 9 s t L _ 3^( ) - 9 s u L _ 3^( ) - 6 t u L _ 3^( ) + 6 s L _ 3^( ) p _ 2^2 + 6 t L _ 3^( ) p _ 2^2 + 6 u L _ 3^( ) p _ 2^2 - 24 L _ 1^( ) (s - 2 (m _ π^(ó r ))^2) (-2 (m _ π^(ó r ))^2 + t + u) - 12 L _ 2^( ) (-2 (m _ π^(ó r ))^4 - 2 p _ 2^2 (m _ π^(ó r ))^2 + s t + s u + 2 t u - 2 p _ 1^2 ((m _ π^(ó r ))^2 + p _ 2^2))))}$

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. 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

$res1 /. {MandelstamT -> ParticleMass[Pion, RenormalizationState[1]]^2, MandelstamU -> ParticleMass[Pion, RenormalizationState[1]]^2} /. RenormalizationState[1] -> RenormalizationState[0] // Simplify$

${-1/(3 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ K^(ó ))^2)) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 N _ 20^( ) (-p _ 2^4 + 3 (s - (m _ π^(ó ))^2) p _ 2^2 + (p _ 1^2 - s) (m _ π^(ó ))^2) + 3 N _ 19^( ) ((s - p _ 2^2) (m _ π^(ó ))^2 + p _ 1^2 (p _ 2^2 - (m _ π^(ó ))^2)) + p _ 2^2 (3 (N _ 22^( ) + 2 N _ 23^( )) (2 (m _ π^(ó ))^2 - (m _ K^(ó ))^2) + 2 N _ 21^( ) (2 (m _ π^(ó ))^2 + (m _ K^(ó ))^2)))), -1/(3 (f _ ϕ^(ó ))^4 (p _ 1^2 - (m _ K^(ó ))^2) ((m _ K^(ó ))^2 - p _ 2^2)) (8 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) (3 L _ 3^( ) (s - p _ 2^2) p _ 2^2 + 2 (-L _ 5^( ) (m _ π^(ó ))^4 + 2 L _ 6^( ) (m _ π^(ó ))^4 + 2 L _ 8^( ) (m _ π^(ó ))^4 - 3 L _ 5^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 30 L _ 6^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 12 L _ 8^( ) (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 4 L _ 6^( ) (m _ K^(ó ))^4 + 2 L _ 8^( ) (m _ K^(ó ))^4 + s L _ 5^( ) (m _ K^(ó ))^2 + 6 L _ 2^( ) ((p _ 2^2 - s) (m _ π^(ó ))^2 + p _ 1^2 ((m _ π^(ó ))^2 + p _ 2^2)) + L _ 4^( ) (-(m _ π^(ó ))^4 + (s - 14 (m _ K^(ó ))^2) (m _ π^(ó ))^2 + 8 s (m _ K^(ó ))^2))))}$

Converted by Mathematica (July 10, 2003)