A7PiPiAmplitude.nb

•Preliminaries

We will work in the limit =:

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

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

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

$gellmannOkubo = {ParticleMass[EtaMeson, r___]^2 -> (4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3, ParticleMass[EtaMeson, r___]^4 -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^2, ParticleMass[EtaMeson, r___]^6 -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^3} ;$
$fromEtaRules = {ParticleMass[EtaMeson, RenormalizationState[0]]^2 -> (-ParticleMass[Pion, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 + 2 ParticleMass[KaonPlus, RenormalizationState[0]]^2)/3, ParticleMass[EtaMeson, RenormalizationState[0]]^4 -> ((-ParticleMass[Pion, RenormalizationState[0]]^2 + 2 ParticleMass[KaonZero, RenormalizationState[0]]^2 + 2 ParticleMass[KaonPlus, RenormalizationState[0]]^2)/3)^2}$

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

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

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

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

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

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

Cosmetics:

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

$RenormalizeBijnens = (Ε _ #[[1]] -> Ε _ #[[1]]^r + LeutwylerLambda[] * 1/CouplingConstant[ChPTW3[2], 1] * (CouplingConstant[ChPTW3[2], 1] * #[[2]] + CouplingConstant[ChPTW3[2], 2] * #[[3]])) & /@ Transpose[{{1, 2, 3, 4, 5, 10, 11, 12, 13, 15}, {1/4, -13/18, 0, 0, -5/12, 1, -1/2, 1/8, -7/8, 3/4}, {5/6, 11/18, 0, 0, 5/12, 3/4, 0, 0, 1/2, -3/4}}]$

${Ε _ 1 -> Ε _ 1^r + ((c _ 2^( )/4 + (5 c _ 5^( ))/6) λ)/c _ 2^( ), Ε _ 2 -> Ε _ 2^r + (((11 c _ 5^( ))/18 - (13 c _ 2^( ))/18) λ)/c _ 2^( ), Ε _ 3 -> Ε _ 3^r, Ε _ 4 -> Ε _ 4^r, Ε _ 5 -> Ε _ 5^r + (((5 c _ 5^( ))/12 - (5 c _ 2^( ))/12) λ)/c _ 2^( ), Ε _ 10 -> Ε _ 10^r + ((c _ 2^( ) + (3 c _ 5^( ))/4) λ)/c _ 2^( ), Ε _ 11 -> Ε _ 11^r - λ/2, Ε _ 12 -> Ε _ 12^r + λ/8, Ε _ 13 -> Ε _ 13^r + ((c _ 5^( )/2 - (7 c _ 2^( ))/8) λ)/c _ 2^( ), Ε _ 15 -> Ε _ 15^r + (((3 c _ 2^( ))/4 - (3 c _ 5^( ))/4) λ)/c _ 2^( )}$

$KamborToBijnens = {CouplingConstant[ChPTW3[4], 5] -> Ε _ 10 - Ε _ 11, CouplingConstant[ChPTW3[4], 6] -> Ε _ 11 + 2 Ε _ 12, CouplingConstant[ChPTW3[4], 7] -> 1/2 Ε _ 11 + Ε _ 13, CouplingConstant[ChPTW3[4], 8] -> Ε _ 11, CouplingConstant[ChPTW3[4], 9] -> Ε _ 15, CouplingConstant[ChPTW3[4], 10] -> Ε _ 1 - Ε _ 5, CouplingConstant[ChPTW3[4], 11] -> Ε _ 2, CouplingConstant[ChPTW3[4], 12] -> -Ε _ 3 + Ε _ 5, CouplingConstant[ChPTW3[4], 13] -> -Ε _ 4 (* , CouplingConstant[ChPTW3[4], 36] -> Ε _ 5 *)}$

${N _ 5^( ) -> Ε _ 10 - Ε _ 11, N _ 6^( ) -> Ε _ 11 + 2 Ε _ 12, N _ 7^( ) -> Ε _ 11/2 + Ε _ 13, N _ 8^( ) -> Ε _ 11, N _ 9^( ) -> Ε _ 15, N _ 10^( ) -> Ε _ 1 - Ε _ 5, N _ 11^( ) -> Ε _ 2, N _ 12^( ) -> Ε _ 5 - Ε _ 3, N _ 13^( ) -> -Ε _ 4}$

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

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

Converted by Mathematica (July 10, 2003)