•Calculation of the amplitude

Calculation of the amplitude:

amplFC = CreateFCAmp[mesonins, AmplitudeLevel -> Classes, MomentumConservation -> True, Sum -> True, EqualMasses -> False] ;

Isospin reduction:

amplFC2 = CheckF[(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 = (++ tmpi ; WriteString["stdout", tmpi, " "] ; SUNReduce[SUNReduce[Sum[#, Evaluate[sums]], Explicit -> True, HoldSums -> False]]) & /@ summ) & /@ Take[amplFC, {1, -1}], "KSPiPiamplFC2leadingS"] ;

Specialization to the (7,3,3) isospin channel, further isospin reduction and change to Mandelstam variables:

amp[K1_, K2_, K3_] := amplFC2 /. {i1 -> K1, i3 -> K2, i4 -> K3} ;

restree = (Simplify[(((SUNReduce[SUNReduce[#]])) /. D -> Sequence[] /. subpar /. udrules)] & /@ amp[7, 3, 3] /. {Momentum[p2] + Momentum[p3] + Momentum[p4] -> -Momentum[p1], -Momentum[p2] - Momentum[p3] - Momentum[p4] -> Momentum[p1]} // PropagatorDenominatorExplicit) /. MandelstamRules // Simplify

{-(i c _ 2^( ) (p _ 3^μ _ 1 + p _ 4^μ _ 1) µ _ μ _ 1(p _ 1))/(f _ ϕ^(ó  ))^2, -(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-6 (m _ π^(ó  r ))^2 + 6 s - 3 t - 3 u + 6 p _ 1^2) + 8 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/(6 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), (2 i c _ 5^( ) (2 p _ 2^μ _ 1 - p _ 3^μ _ 1 - p _ 4^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^2 (p _ 2^2 - (m _ K^(ó  ))^2)), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 2 s - t - u))/(3 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2))}

The leading order amplitude:

restree1 = (Simplify[(((SUNReduce[SUNReduce[#]])) /. D -> Sequence[] /. subpar /. udrules)] & /@ amp[7, 3, 3] /. {Momentum[p2] + Momentum[p3] + Momentum[p4] -> -Momentum[p1], -Momentum[p2] - Momentum[p3] - Momentum[p4] -> Momentum[p1]} // PropagatorDenominatorExplicit) /. MandelstamRules /. _RenormalizationState -> Sequence[] // Simplify

{-(i c _ 2^( ) (p _ 3^μ _ 1 + p _ 4^μ _ 1) µ _ μ _ 1(p _ 1))/(f _ ϕ^(ó  ))^2, -(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-6 (m _ π^(ó  ))^2 + 6 s - 3 t - 3 u + 6 p _ 1^2) + 8 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/(6 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), (2 i c _ 5^( ) (2 p _ 2^μ _ 1 - p _ 3^μ _ 1 - p _ 4^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^2 (p _ 2^2 - (m _ K^(ó  ))^2)), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 2 s - t - u))/(3 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - p _ 2^2))}

The leading order amplitude with both pions and the kaon on-mass-shell (and p _ 2^2 = 0):

(Cancel[-I * (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)/(-DecayConstant[PhiMeson]) * Plus @@ restree1] /. cancelU1 /. _RenormalizationState -> Sequence[] /. {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2, Pair[Momentum[p2], Momentum[p2]] -> 0, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion]^2}) /. Pair[_LorentzIndex, ___] -> 1 // Simplify

(c _ 2^( ) (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s) (m _ K^(ó  ))^2 + 2 c _ 5^( ) (s - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(2 (f _ ϕ^(ó  ))^3 (m _ K^(ó  ))^2)

The propagator and the coupling to the source are divided off and p _ 2^2 set to 0:

Plus @@ Cancel[-I * (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)/(-DecayConstant[PhiMeson]) * restree1] /. cancelU1 /. {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2, Pair[Momentum[p2], Momentum[p2]] -> 0, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion]^2} /. MandelstamS -> ParticleMass[Pion]^2 /. Pair[_LorentzIndex, ___] -> 1 // Simplify

(((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (c _ 2^( ) (m _ K^(ó  ))^2 + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/(2 (f _ ϕ^(ó  ))^3 (m _ K^(ó  ))^2)

The limit p _ 2 -> 0:

Plus @@ Cancel[-I * (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2)/(-DecayConstant[PhiMeson]) * restree1] /. cancelU1 /. {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2, Pair[Momentum[p2], Momentum[p2]] -> 0, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion]^2} /. MandelstamS -> ParticleMass[Kaon]^2 /. Pair[_LorentzIndex, ___] -> 1 // Simplify

(2 c _ 2^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(f _ ϕ^(ó  ))^3

Converted by Mathematica  (July 10, 2003)