•Preliminaries

i1 = 3 ; i2 = 3 ; i3 = 3 ; i4 = 3 ;

$Gauge = 1 ;

udrules = {PionZero -> PionPlus}

{π^0 -> π^+}

subpar = Table[(ParticleMass[PseudoScalar[2], i | SUNIndex[i], r___] -> ParticleMass[Select[$IsoSpinProjectionRules, (! FreeQ[#, {i}] &)][[1]][[1]], r]), {i, 3}]

{ParticleMass(π, 1 | 1, r___) -> ParticleMass(π^+, r), ParticleMass(π, 2 | 2, r___) -> ParticleMass(π^+, r), ParticleMass(π, 3 | 3, r___) -> ParticleMass(π^0, r)}

{ParticleMass(π, 1, r___) -> ParticleMass (π^+, r), ParticleMass(π, 2, r___) -> ParticleMass (π^+, r), ParticleMass(π, 3, r___) -> ParticleMass (π^0, r)}

pimassrule = ParticleMass[Pion, i_ ? (! IntegerQ[#] &) | SUNIndex[i_ ? (! IntegerQ[#] &)], r___]^2 -> (SU2Delta[SUNIndex[i], 1] + SU2Delta[SUNIndex[i], 2]) ParticleMass[PionPlus, r]^2 + SU2Delta[SUNIndex[i], 3] ParticleMass[PionZero, r]^2

ParticleMass(π, i_ ? (¬ IntegerQ[#1] &) | i_ ? (¬ IntegerQ[#1] &), r___)^2 -> (δ _ (1 i)^(2) + δ _ (2 i)^(2)) ParticleMass(π^+, r)^2 + ParticleMass(π^0, r)^2 δ _ (3 i)^(2)

pirule = ParticleMass[Pion, RenormalizationState[0]] -> ParticleMass[PionZero, RenormalizationState[0]]

m _ π^(ó  ) -> m _ π^0^(ó  )

delrules = {SU2Delta[1, SUNIndex[i1]] + SU2Delta[2, SUNIndex[i1]] -> (1 - SU2Delta[3, SUNIndex[i1]]), a_ * SU2Delta[1, SUNIndex[i1]] + a_ * SU2Delta[2, SUNIndex[i1]] -> a * (1 - SU2Delta[3, SUNIndex[i1]])}

{δ _ (1 3)^(2) + δ _ (2 3)^(2) -> 1 - δ _ (3 3)^(2), a_ δ _ (1 3)^(2) + a_ δ _ (2 3)^(2) -> a (1 - δ _ (3 3)^(2))}

cancelScales = Log[ParticleMass[PionPlus, RenormalizationState[0]]^2/ScaleMu^2] -> Log[ParticleMass[PionZero, RenormalizationState[0]]^2/ScaleMu^2] + Log[ParticleMass[PionPlus, RenormalizationState[0]]^2/ParticleMass[PionZero, RenormalizationState[0]]^2]

log((m _ π^+^(ó  ))^2/μ^2) -> log((m _ π^+^(ó  ))^2/(m _ π^0^(ó  ))^2) + log((m _ π^0^(ó  ))^2/μ^2)

dmrules = CouplingConstant[QED[1], RenormalizationState[0]]^n_ :> DecayConstant[PseudoScalar[2], RenormalizationState[0]]^n/(2 CouplingConstant[ChPTEM2[2], RenormalizationState[0]])^(n/2) (ParticleMass[PionPlus, RenormalizationState[0]]^2 - ParticleMass[PionZero, RenormalizationState[0]]^2)^(n/2)

(e^( ))^n_ :> ((f _ π^(ó  ))^n ((m _ PionPlus^(ó  ))^2 - (m _ PionZero^(ó  ))^2)^(n/2))/(2 C^( ))^(n/2)

SetOptions[MandelstamReduce, Cancel -> None, Masses -> ({ParticleMass[Pion, SUNIndex[i1], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i2], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i3], RenormalizationState[0]], ParticleMass[Pion, SUNIndex[i4], RenormalizationState[0]]} /. subpar)]

{MomentaSumLeft -> All, OnMassShell -> True, Cancel -> None, MomentumVariablesString -> p, MomentaSumRule -> True, Masses -> {m _ π^0^(ó  ), m _ π^0^(ó  ), m _ π^0^(ó  ), m _ π^0^(ó  )}}

manrul = {MandelstamS + MandelstamU + MandelstamT -> Plus @@ ({ParticleMass[Pion, SUNIndex[i1], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i2], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i3], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i4], RenormalizationState[0]]^2} /. subpar), -MandelstamS - MandelstamU - MandelstamT -> -Plus @@ ({ParticleMass[Pion, SUNIndex[i1], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i2], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i3], RenormalizationState[0]]^2, ParticleMass[Pion, SUNIndex[i4], RenormalizationState[0]]^2} /. subpar)}

{s + t + u -> 4 (m _ π^0^(ó  ))^2, -s - t - u -> -4 (m _ π^0^(ó  ))^2}

dmrulesinv = {(ParticleMass[PionPlus, RenormalizationState[0]]^2 - ParticleMass[PionZero, RenormalizationState[0]]^2) -> 2 CouplingConstant[QED[1], RenormalizationState[0]]^2 CouplingConstant[ChPTEM2[2], RenormalizationState[0]]/DecayConstant[Pion, RenormalizationState[0]]^2, (ParticleMass[PionZero, RenormalizationState[0]]^2 - ParticleMass[PionPlus, RenormalizationState[0]]^2) -> -2 CouplingConstant[QED[1], RenormalizationState[0]]^2 CouplingConstant[ChPTEM2[2], RenormalizationState[0]]/DecayConstant[Pion, RenormalizationState[0]]^2}

{(m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2 -> (2 C^( ) (e^( ))^2)/(f _ π^(ó  ))^2, (m _ π^0^(ó  ))^2 - (m _ π^+^(ó  ))^2 -> -(2 C^( ) (e^( ))^2)/(f _ π^(ó  ))^2}

WFFactor1[Propagator[p_][v__, PseudoScalar2[0, {1}]]] := WFFactor[Propagator[p][v, PseudoScalar3[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar2[0, {2}]]] := WFFactor[Propagator[p][v, PseudoScalar3[0]]] ; WFFactor1[Propagator[p_][v__, PseudoScalar2[0, {3}]]] := WFFactor[Propagator[p][v, PseudoScalar4[0]]] ;

Converted by Mathematica  (July 10, 2003)