KSPiPiAmplitude.nb

•Preliminaries

We will work in the limit =:

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

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

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

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

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 (the pions on-mass-shell):

$MandelstamRules0 = {Pair[Momentum[p1], Momentum[p2]] -> (MandelstamS - Pair[Momentum[p1], Momentum[p1]] - Pair[Momentum[p2], Momentum[p2]])/2, Pair[Momentum[p3], Momentum[p4]] -> (MandelstamS - ParticleMass[Pion, RenormalizationState[1]]^2 - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p1], Momentum[p4]] -> (MandelstamT - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p2], Momentum[p3]] -> (MandelstamT - Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p1], Momentum[p3]] -> (MandelstamU - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p2], Momentum[p4]] -> (MandelstamU - Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion, RenormalizationState[1]]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion, RenormalizationState[1]]^2, MandelstamS + MandelstamT + MandelstamU -> 2 ParticleMass[Pion, RenormalizationState[1]]^2 + Pair[Momentum[p1], Momentum[p1]] + Pair[Momentum[p2], Momentum[p2]], -MandelstamS - MandelstamT - MandelstamU -> -(2 ParticleMass[Pion, RenormalizationState[1]]^2 + Pair[Momentum[p1], Momentum[p1]] + Pair[Momentum[p2], Momentum[p2]])}$

${p _ 1 · p _ 2 -> 1/2 (s - p _ 1^2 - p _ 2^2), p _ 3 · p _ 4 -> 1/2 (s - 2 (m _ π^(ó r ))^2), p _ 1 · p _ 4 -> 1/2 (-(m _ π^(ó r ))^2 + t - p _ 1^2), p _ 2 · p _ 3 -> 1/2 (-(m _ π^(ó r ))^2 + t - p _ 2^2), p _ 1 · p _ 3 -> 1/2 (-(m _ π^(ó r ))^2 + u - p _ 1^2), p _ 2 · p _ 4 -> 1/2 (-(m _ π^(ó r ))^2 + u - p _ 2^2), p _ 3^2 -> (m _ π^(ó r ))^2, p _ 4^2 -> (m _ π^(ó r ))^2, s + t + u -> 2 (m _ π^(ó r ))^2 + p _ 1^2 + p _ 2^2, -s - t - u -> -2 (m _ π^(ó r ))^2 - p _ 1^2 - p _ 2^2}$

Rules for translating from momenta to Mandelstam variables (all particles and the source off-mass-shell):

${p _ 1 · p _ 2 -> 1/2 (s - p _ 1^2 - p _ 2^2), p _ 3 · p _ 4 -> 1/2 (s - p _ 3^2 - p _ 4^2), p _ 1 · p _ 4 -> 1/2 (t - p _ 1^2 - p _ 4^2), p _ 2 · p _ 3 -> 1/2 (t - p _ 2^2 - p _ 3^2), p _ 1 · p _ 3 -> 1/2 (u - p _ 1^2 - p _ 3^2), p _ 2 · p _ 4 -> 1/2 (u - p _ 2^2 - p _ 4^2)}$

Rules for translating from momenta to Mandelstam variables (the pions on-mass-shell, the scalar source on-mass-shell with zero mass):

$MandelstamRules2 = {Pair[Momentum[p1], Momentum[p2]] -> (MandelstamS - Pair[Momentum[p1], Momentum[p1]])/2, Pair[Momentum[p3], Momentum[p4]] -> MandelstamS/2 - ParticleMass[Pion, RenormalizationState[1]]^2, Pair[Momentum[p1], Momentum[p4]] -> (MandelstamT - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p2], Momentum[p3]] -> (MandelstamT - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p1], Momentum[p3]] -> (MandelstamU - Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Pion, RenormalizationState[1]]^2)/2, Pair[Momentum[p2], Momentum[p4]] -> (MandelstamU - ParticleMass[Pion, RenormalizationState[1]]^2)/2 (* , Pair[Momentum[p2], Momentum[p2]] -> 0 *), Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion, RenormalizationState[1]]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion, RenormalizationState[1]]^2}$

${p _ 1 · p _ 2 -> 1/2 (s - p _ 1^2), p _ 3 · p _ 4 -> s/2 - (m _ π^(ó r ))^2, p _ 1 · p _ 4 -> 1/2 (-(m _ π^(ó r ))^2 + t - p _ 1^2), p _ 2 · p _ 3 -> 1/2 (t - (m _ π^(ó r ))^2), p _ 1 · p _ 3 -> 1/2 (-(m _ π^(ó r ))^2 + u - p _ 1^2), p _ 2 · p _ 4 -> 1/2 (u - (m _ π^(ó r ))^2), p _ 3^2 -> (m _ π^(ó r ))^2, p _ 4^2 -> (m _ π^(ó r ))^2}$

Rules for cancelling one Mandelstam variable:

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

Converted by Mathematica (July 10, 2003)