KSS2Checks.nb

•Substitution rules

Just to avoid overwriting saved results or saving nonsense.

We will work in the limit =:

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

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

Translating from masses of isostates to particle states (no pi-eta mixing):

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

The Gell-Mann-Okubo mass formula (will be applied only on 4th order expressions):

$toEtaRules = {ParticleMass[PseudoScalar[2], r___]^2 - 4 ParticleMass[PseudoScalar[6], r___]^2 :> -3 * ParticleMass[PseudoScalar[11], r]^2, -ParticleMass[PseudoScalar[2], r___]^2 + 4 ParticleMass[PseudoScalar[6], r___]^2 :> 3 * ParticleMass[PseudoScalar[11], r]^2} ;$
$gellmannOkubo = {ParticleMass[EtaMeson, r___]^2 -> (4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3, ParticleMass[EtaMeson, r___]^n_ -> ((4 ParticleMass[Kaon, r]^2 - ParticleMass[Pion, r]^2)/3)^(n/2)} ;$

Rules for eliminating scalar products:

${p _ 1 · p _ 2 -> 1/2 (-p _ 1^2 - p _ 2^2 + p _ 3^2), p _ 2 · p _ 3 -> 1/2 (p _ 1^2 - p _ 2^2 - p _ 3^2), p _ 1 · p _ 3 -> 1/2 (-p _ 1^2 + p _ 2^2 - p _ 3^2)}$

$onshellrules = {Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2}$

${p _ 1^2 -> (m _ K^(ó ))^2}$

Translation :

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

${N _ 5^( ) -> Ε _ 10 - Ε _ 11, N _ 6^( ) -> Ε _ 11 + 2 Ε _ 12, N _ 7^( ) -> Ε _ 11/2 + Ε _ 13, N _ 8^( ) -> Ε _ 11, N _ 9^( ) -> Ε _ 15, N _ 10^( ) -> Ε _ 1 - Ε _ 5, N _ 11^( ) -> Ε _ 2, N _ 12^( ) -> Ε _ 5 - Ε _ 3, N _ 13^( ) -> -Ε _ 4, N _ 36^( ) -> Ε _ 5}$

$BijnensToKambor = Solve[KamborToBijnens /. Rule -> Equal, {Ε _ 1, Ε _ 2, Ε _ 3, Ε _ 4, Ε _ 5, Ε _ 10, Ε _ 11, Ε _ 12, Ε _ 13, Ε _ 15}] // Flatten$

${Ε _ 1 -> N _ 10^( ) + N _ 36^( ), Ε _ 2 -> N _ 11^( ), Ε _ 3 -> N _ 36^( ) - N _ 12^( ), Ε _ 4 -> -N _ 13^( ), Ε _ 5 -> N _ 36^( ), Ε _ 10 -> N _ 5^( ) + N _ 8^( ), Ε _ 11 -> N _ 8^( ), Ε _ 12 -> 1/2 (N _ 6^( ) - N _ 8^( )), Ε _ 13 -> 1/2 (2 N _ 7^( ) - N _ 8^( )), Ε _ 15 -> N _ 9^( )}$

Beta functions of the :

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

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

$JBarToMr := (LeutwylerJBar[m : Pair[Momentum[p2_], Momentum[p2_]], r__, o___Rule])/(n : Pair[Momentum[p2_], Momentum[p2_]])^2 :> 3/({r}[[1]] - {r}[[-1]])^2 * (Mr[m, r] - 1/(12 * m) (m - 2 ({r}[[1]] + {r}[[-1]])) LeutwylerJBar[m, r, o] - 1/(288 * Pi^2) + k[r]/6 + 1/(96 * Pi^2 * m) (({r}[[1]] + {r}[[-1]]) + deltainv[{r}[[1]], {r}[[-1]]])) ;$
$JBarToKL := {LeutwylerJBar[t : Pair[Momentum[p2_], Momentum[p2_]], r__, o___Rule]/(t : Pair[Momentum[p2_], Momentum[p2_]]) :> 2 * K[t, r]/({r}[[1]] - {r}[[-1]])} ;$

The function expanded to first order in s and various limits:

$Normal[(Series[LeutwylerJBar[s, m12, m22, LeutwylerJBarEvaluation -> "subthreshold"], {s, 0, 1}] /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y} // Simplify) /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y} // Simplify] // StandardForm$

$KLToJBar := {K[t_ ? ((# =!= 0) &), r__] :> ({r}[[1]] - {r}[[-1]])/(2 * t) * LeutwylerJBar[t, r], K[0, r_, rr_] :> (r - rr)/2 * (r^2 - rr^2 + 2 r rr Log[rr/r])/(32 (r - rr)^3 π^2) /; (r =!= rr && r =!= 0 && rr =!= 0), K[0, 0, rr_] :> (-rr)/2 * 1/(32 π^2 rr), K[0, r_, 0] :> (r)/2 * 1/(32 π^2 r), K[0, r_, r_] :> 1/(96 r π^2)} ;$

$fixzeros = {LeutwylerJBar[p_, 0, pp_, o___Rule] :> LeutwylerJBar[p, ParticleMass[Pion]^2, pp, o] /; p =!= pp, LeutwylerJBar[p_, 0, p_, o___Rule] :> 1/(16 π^2), k[0, p_] :> k[ParticleMass[Pion]^2, p], K[a___, 0, b___] :> K[a, ParticleMass[Pion]^2, b]} ;$

We load this lagrangian just to have the coupling constants displayed nicely

Some factors that we'll need:

(128 π^2 L _ 5^(r ) (m _ π^(ó ))^2 - 2 log((m _ π^(ó ))^2/μ^2) (m _ π^(ó ))^2 - log((m _ K^(ó ))^2/μ^2) (m _ K^(ó ))^2 + 128 π^2 L _ 4^(r ) ((m _ π^(ó ))^2 + 2 (m _ K^(ó ))^2))/(32 π^2 (f _ ϕ^(ó ))^2) + 1

1/(128 π^2 (f _ ϕ^(ó ))^2) (-3 log((m _ π^(ó ))^2/μ^2) (m _ π^(ó ))^2 + log((m _ η^(ó ))^2/μ^2) (m _ π^(ó ))^2 + 512 π^2 L _ 5^(r ) (m _ K^(ó ))^2 - 6 log((m _ K^(ó ))^2/μ^2) (m _ K^(ó ))^2 - 4 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó ))^2 + 512 π^2 L _ 4^(r ) ((m _ π^(ó ))^2 + 2 (m _ K^(ó ))^2)) + 1

1/2 (1/(144 π^2 (f _ ϕ^(ó ))^2) (144 π^2 (f _ ϕ^(ó ))^2 + (-1152 π^2 (L _ 4^(r ) + L _ 5^(r ) - 4 (L _ 6^(r ) + L _ 8^(r ))) - 9 log((m _ π^(ó ))^2/μ^2) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 - (2304 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) + 9 log((m _ K^(ó ))^2/μ^2) + 4 log((m _ η^(ó ))^2/μ^2)) (m _ K^(ó ))^2) + 1)

1/2 (3 - 1/(144 π^2 (f _ ϕ^(ó ))^2) (144 π^2 (f _ ϕ^(ó ))^2 + (-1152 π^2 (L _ 4^(r ) + L _ 5^(r ) - 4 (L _ 6^(r ) + L _ 8^(r ))) - 9 log((m _ π^(ó ))^2/μ^2) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 - (2304 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) + 9 log((m _ K^(ó ))^2/μ^2) + 4 log((m _ η^(ó ))^2/μ^2)) (m _ K^(ó ))^2))

1/2 (1/(576 π^2 (f _ ϕ^(ó ))^2) (576 π^2 (f _ ϕ^(ó ))^2 + (-4608 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) - 27 log((m _ π^(ó ))^2/μ^2) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 - 2 (2304 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 8 L _ 6^(r ) - 4 L _ 8^(r )) + 27 log((m _ K^(ó ))^2/μ^2) + 2 log((m _ η^(ó ))^2/μ^2)) (m _ K^(ó ))^2) + 1)

1/2 (3 - 1/(576 π^2 (f _ ϕ^(ó ))^2) (576 π^2 (f _ ϕ^(ó ))^2 + (-4608 π^2 (L _ 4^(r ) - 4 L _ 6^(r )) - 27 log((m _ π^(ó ))^2/μ^2) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 - 2 (2304 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 8 L _ 6^(r ) - 4 L _ 8^(r )) + 27 log((m _ K^(ó ))^2/μ^2) + 2 log((m _ η^(ó ))^2/μ^2)) (m _ K^(ó ))^2))

Converted by Mathematica (July 10, 2003)