•Preliminaries

We will work in the limit m _ u=m _ d:

subpar = Table[(ParticleMass[PseudoScalar[1], SUNIndex[i], RenormalizationState[0]] -> ParticleMass[Select[$IsoSpinProjectionRules, (! FreeQ[#, {i}] &)][[1]][[1]], RenormalizationState[0]]), {i, 8}]

{m _ ϕ^(1 ) -> m _ π^+^(ó  ), m _ ϕ^(2 ) -> m _ π^+^(ó  ), m _ ϕ^(3 ) -> m _ π^0^(ó  ), m _ ϕ^(4 ) -> m _ K^+^(ó  ), m _ ϕ^(5 ) -> m _ K^+^(ó  ), m _ ϕ^(6 ) -> m _ K^0^(ó  ), m _ ϕ^(7 ) -> m _ K^0^(ó  ), m _ ϕ^(8 ) -> m _ η^(ó  )}

massrules = {ParticleMass[PionMinus, RenormalizationState[0]] -> ParticleMass[Pion, RenormalizationState[0]], ParticleMass[PionPlus, RenormalizationState[0]] -> ParticleMass[Pion, RenormalizationState[0]], ParticleMass[PionZero, RenormalizationState[0]] -> ParticleMass[Pion, RenormalizationState[0]], ParticleMass[KaonMinus, RenormalizationState[0]]^2 -> ParticleMass[KaonPlus, RenormalizationState[0]]^2, ParticleMass[EtaMeson, RenormalizationState[0]]^2 -> (2 ParticleMass[KaonPlus, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 - ParticleMass[Pion, RenormalizationState[0]]^2)/3, ParticleMass[EtaMeson, RenormalizationState[0]]^4 -> ((2 ParticleMass[KaonPlus, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 - ParticleMass[Pion, RenormalizationState[0]]^2)/3)^2}

{m _ π^-^(ó  ) -> m _ π^(ó  ), m _ π^+^(ó  ) -> m _ π^(ó  ), m _ π^0^(ó  ) -> m _ π^(ó  ), (m _ K^-^(ó  ))^2 -> (m _ K^+^(ó  ))^2, (m _ η^(ó  ))^2 -> 1/3 (-(m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 + 2 (m _ K^0^(ó  ))^2), (m _ η^(ó  ))^4 -> 1/9 (-(m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 + 2 (m _ K^0^(ó  ))^2)^2}

udrules = {PionPlus -> Pion, PionZero -> Pion, KaonPlus -> Kaon, KaonZero -> Kaon}

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

gellmannOkubo = ParticleMass[EtaMeson, RenormalizationState[0]]^2 -> (4 ParticleMass[Kaon, RenormalizationState[0]]^2 - ParticleMass[Pion, RenormalizationState[0]]^2)/3

(m _ η^(ó  ))^2 -> 1/3 (4 (m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)

etalogs = {Log[-1/(3 ScaleMu^2) (ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 - 4 ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2)] -> Log[1/ScaleMu^2 ParticleMass[PseudoScalar[11], RenormalizationState[0]]^2], Log[-1/ScaleMu^2 (ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 - 4 ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2)] -> Log[3] + Log[1/ScaleMu^2 ParticleMass[PseudoScalar[11], RenormalizationState[0]]^2]}

{log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2)) -> log((m _ η^(ó  ))^2/μ^2), log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/μ^2) -> log((m _ η^(ó  ))^2/μ^2) + log (3)}

fromEtaRules = {ParticleMass[EtaMeson, RenormalizationState[0]]^2 -> (-ParticleMass[Pion, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 + 2 ParticleMass[KaonPlus, RenormalizationState[0]]^2)/3, ParticleMass[EtaMeson, RenormalizationState[0]]^4 -> ((-ParticleMass[Pion, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 + 2 ParticleMass[KaonPlus, RenormalizationState[0]]^2)/3)^2}

{(m _ η^(ó  ))^2 -> 1/3 (-(m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 + 2 (m _ K^0^(ó  ))^2), (m _ η^(ó  ))^4 -> 1/9 (-(m _ π^(ó  ))^2 + 2 (m _ K^+^(ó  ))^2 + 2 (m _ K^0^(ó  ))^2)^2}

FromK0Rules = {ParticleMass[KaonZero, RenormalizationState[0]]^2 -> 3/2 ParticleMass[EtaMeson, RenormalizationState[0]]^2 + 1/2 ParticleMass[Pion, RenormalizationState[0]]^2 - ParticleMass[KaonPlus, RenormalizationState[0]]^2, ParticleMass[KaonZero, RenormalizationState[0]]^4 -> (3/2 ParticleMass[EtaMeson, RenormalizationState[0]]^2 + 1/2 ParticleMass[Pion, RenormalizationState[0]]^2 - ParticleMass[KaonPlus, RenormalizationState[0]]^2)^2}

{(m _ K^0^(ó  ))^2 -> 1/2 (m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + 3/2 (m _ η^(ó  ))^2, (m _ K^0^(ó  ))^4 -> (1/2 (m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2 + 3/2 (m _ η^(ó  ))^2)^2}

Converted by Mathematica  (July 10, 2003)