•Preliminaries

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

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)}

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

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

WFFactor1[_[__, Scalar2[0, ___]]] := 0 ; WFFactor1[_[__, PseudoScalar0[0, ___]]] := 0 ;

WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {1}]]] := WFFactor[Propagator[p][v, PseudoScalar2[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {2}]]] := WFFactor[Propagator[p][v, PseudoScalar2[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {3}]]] := WFFactor[Propagator[p][v, PseudoScalar2[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {4}]]] := WFFactor[Propagator[p][v, PseudoScalar6[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {5}]]] := WFFactor[Propagator[p][v, PseudoScalar6[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {6}]]] := WFFactor[Propagator[p][v, PseudoScalar6[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {7}]]] := WFFactor[Propagator[p][v, PseudoScalar6[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar1[0, {8}]]] := WFFactor[Propagator[p][v, PseudoScalar11[0]]] ;

gellmannOkubo = {ParticleMass[EtaMeson, r___]^2 -> (4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3, ParticleMass[EtaMeson, r___]^4 -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^2, ParticleMass[EtaMeson, r___]^6 -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^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}

massshellrules = {Pair[Momentum[p3], Momentum[p4]] -> Pair[Momentum[p1], Momentum[p1]]/2 - ParticleMass[Pion, RenormalizationState[0]]^2, Pair[Momentum[p1], Momentum[p3]] -> -Pair[Momentum[p1], Momentum[p1]]/2, Pair[Momentum[p1], Momentum[p4]] -> -Pair[Momentum[p1], Momentum[p1]]/2, Pair[Momentum[p3], Momentum[p3]] -> ParticleMass[Pion, RenormalizationState[0]]^2, Pair[Momentum[p4], Momentum[p4]] -> ParticleMass[Pion, RenormalizationState[0]]^2}

{p _ 3 · p _ 4 -> p _ 1^2/2 - (m _ π^(ó  ))^2, p _ 1 · p _ 3 -> -p _ 1^2/2, p _ 1 · p _ 4 -> -p _ 1^2/2, p _ 3^2 -> (m _ π^(ó  ))^2, p _ 4^2 -> (m _ π^(ó  ))^2}

toEtaRules = {ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 - 4 ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2 :> -3 * ParticleMass[PseudoScalar[11], RenormalizationState[0]]^2, -ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 + 4 ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2 :> 3 * ParticleMass[PseudoScalar[11], RenormalizationState[0]]^2} ;

Cosmetics:

couplingunrenormalization = Flatten[Solve[(Equal @@ #) & /@ Transpose[{CouplingConstant[ChPT3[4], #, RenormalizationState[0]] & /@ Range[10], Renormalize[CouplingConstant[ChPT3[4], #, RenormalizationState[0]] & /@ Range[10]]}], Evaluate[CouplingConstant[ChPT3[4], #, RenormalizationState[1]] & /@ Range[10]]]]

{L _ 1^(r ) -> L _ 1^( ) - (3 λ)/32, L _ 2^(r ) -> L _ 2^( ) - (3 λ)/16, L _ 3^(r ) -> L _ 3^( ), L _ 4^(r ) -> L _ 4^( ) - λ/8, L _ 5^(r ) -> L _ 5^( ) - (3 λ)/8, L _ 6^(r ) -> L _ 6^( ) - (11 λ)/144, L _ 7^(r ) -> L _ 7^( ), L _ 8^(r ) -> L _ 8^( ) - (5 λ)/48, L _ 9^(r ) -> L _ 9^( ) - λ/4, L _ 10^(r ) -> L _ 10^( ) + λ/4}

JBarToB = {LeutwylerJBar[s_, m1_, ___Rule] -> -DecayConstant[PhiMeson, RenormalizationState[0]]^2 B[m1, m1, s] + (Log[m1/ScaleMu^2] + 1)/(16 π^2), LeutwylerJBar[s_, m1_, m1_, ___Rule] -> -DecayConstant[PhiMeson, RenormalizationState[0]]^2 B[m1, m1, s] + (Log[m1/ScaleMu^2] + 1)/(16 π^2), LeutwylerJBar[s_, m1_, m2_, ___Rule] -> -DecayConstant[PhiMeson, RenormalizationState[0]]^2 B[m1, m2, s] + (m1 Log[m1/ScaleMu^2] - m2 Log[m2/ScaleMu^2])/(16 π^2 (m1 - m2))}

{Overscript[J, _] _ (m1_ ___Rule)(s_) -> (log (m1/μ^2) + 1)/(16 π^2) - B (m1, m1, s) (f _ ϕ^(ó  ))^2, Overscript[J, _] _ (m1_ m1_ ___Rule)(s_) -> (log (m1/μ^2) + 1)/(16 π^2) - B (m1, m1, s) (f _ ϕ^(ó  ))^2, Overscript[J, _] _ (m1_ m2_ ___Rule)(s_) -> (m1 log (m1/μ^2) - m2 log (m2/μ^2))/(16 (m1 - m2) π^2) - B (m1, m2, s) (f _ ϕ^(ó  ))^2}

logstomus = Log[ParticleMass[p_, RenormalizationState[0]]^2/ScaleMu^2] -> 32 π^2 DecayConstant[PseudoScalar[1], RenormalizationState[0]]^2 μ _ p/ParticleMass[p, RenormalizationState[0]]^2

log((m _ p_^(ó  ))^2/μ^2) -> (32 π^2 (f _ ϕ^(ó  ))^2 μ _ p)/(m _ p^(ó  ))^2

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

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

Converted by Mathematica  (July 10, 2003)