KSPiAmplitude.nb

•Preliminaries

We will work in the limit =:

$subpar = Table[(ParticleMass[PseudoScalar[1], SUNIndex[i], r___] -> ParticleMass[Select[$IsoSpinProjectionRules, (! FreeQ[#, {i}] &)][[1]][[1]], r]), {i, 8}]$

${ParticleMass(ϕ, 1, r___) -> ParticleMass(π^+, r), ParticleMass(ϕ, 2, r___) -> ParticleMass(π^+, r), ParticleMass(ϕ, 3, r___) -> ParticleMass(π^0, r), ParticleMass(ϕ, 4, r___) -> ParticleMass(K^+, r), ParticleMass(ϕ, 5, r___) -> ParticleMass(K^+, r), ParticleMass(ϕ, 6, r___) -> ParticleMass(K^0, r), ParticleMass(ϕ, 7, r___) -> ParticleMass(K^0, r), ParticleMass(ϕ, 8, r___) -> ParticleMass(η, r)}$

Translating from masses of isostates to particle states (no pi-eta mixing):

${π^+ -> π, π^0 -> π, K^+ -> K, K^0 -> K}$

The Gell-Mann-Okubo mass formula (will be applied only on 4th order expressions):

$gellmannOkubo = {ParticleMass[EtaMeson, r___]^2 -> (4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3, ParticleMass[EtaMeson, r___]^n_ -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^(n/2)} ;$
$toEtaRules = {ParticleMass[PseudoScalar[2], r___]^2 - 4 ParticleMass[PseudoScalar[6], r___]^2 :> -3 * ParticleMass[PseudoScalar[11], r]^2, -ParticleMass[PseudoScalar[2], r___]^2 + 4 ParticleMass[PseudoScalar[6], r___]^2 :> 3 * ParticleMass[PseudoScalar[11], r]^2} ;$

Rules for translating from momenta to Mandelstam variables:

${p _ 1 · p _ 2 -> 1/2 (-p _ 1^2 - p _ 2^2 + p _ 3^2), p _ 2 · p _ 3 -> 1/2 (p _ 1^2 - p _ 2^2 - p _ 3^2), p _ 1 · p _ 3 -> 1/2 (-p _ 1^2 + p _ 2^2 - p _ 3^2)}$

We load this lagrangian just to have the coupling constants displayed nicely

Converted by Mathematica (July 10, 2003)