•Preliminaries

SetOptions[MandelstamReduce, Masses -> {ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]]}] ;

These rules can be applied to fourth order expressions:

dcren = {DecayConstant[p_, RenormalizationState[0]] -> DecayConstant[p, RenormalizationState[1]], ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]]} ;

Simplify logs

logRule = Log[ParticleMass[p__]^2] :> Log[ParticleMass[p]^2/ScaleMu^2] - Log[1/ScaleMu^2] ;

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

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

{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)}

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]]] ;

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} ;

NOTICE:  The default convention of Phi is at variance with the default convention of FeynCalc w.r.t. Mandelstam variables (t<->u).  Below u is correspondingly written before t, setting the FeynCalc Mandelstam variables in agreement with the Phi default convention. Uncommenting should speed things up a great deal.

manrules = {-MandelstamS - MandelstamT :> MandelstamU - 4 ParticleMass[Kaon, RenormalizationState[1]]^2, MandelstamS + MandelstamT :> -MandelstamU + 4 ParticleMass[Kaon, RenormalizationState[1]]^2, a_ MandelstamS + a_ MandelstamT :> a (-MandelstamU + 4 ParticleMass[Kaon, RenormalizationState[1]]^2)}

{-s - t :> u - 4 (m _ Kaon^(ó  r ))^2, s + t :> 4 (m _ Kaon^(ó  r ))^2 - u, s a_ + t a_ :> a (4 (m _ Kaon^(ó  r ))^2 - u)}

(* SetMandelstam[MandelstamS, MandelstamU, MandelstamT, p1, p2, p3, p4, ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]], ParticleMass[Kaon, RenormalizationState[1]]] ; *)

Converted by Mathematica  (July 10, 2003)