KSPiPiChecksA.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}]$

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

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):

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

Rules for cancelling one Mandelstam variable:

Variable changes:

$symmetrize = {MandelstamT -> 1/2 (ParticleMass[Kaon, RenormalizationState[1]]^2 + Pair[Momentum[p2], Momentum[p2]] + 2 ParticleMass[Pion, RenormalizationState[1]]^2 + ν - MandelstamS), MandelstamU -> 1/2 (ParticleMass[Kaon, RenormalizationState[1]]^2 + Pair[Momentum[p2], Momentum[p2]] + 2 ParticleMass[Pion, RenormalizationState[1]]^2 - ν - MandelstamS)}$

${t -> 1/2 (2 (m _ π^(ó r ))^2 + (m _ K^(ó r ))^2 - s + ν + p _ 2^2), u -> 1/2 (2 (m _ π^(ó r ))^2 + (m _ K^(ó r ))^2 - s - ν + p _ 2^2)}$

$symmetrize1 = {t -> 1/2 (ParticleMass[Kaon]^2 + Pair[Momentum[p2], Momentum[p2]] + 2 ParticleMass[Pion]^2 + ν - MandelstamS), u -> 1/2 (ParticleMass[Kaon]^2 + 2 ParticleMass[Pion]^2 - ν - MandelstamS)}$

${t -> 1/2 (2 (m _ π^(ó ))^2 + (m _ K^(ó ))^2 - s + ν + p _ 2^2), u -> 1/2 (2 (m _ π^(ó ))^2 + (m _ K^(ó ))^2 - s - ν)}$

$symmJ = {LeutwylerJBar[t, a__, b__Rule] -> (Js[a] + Ja[a])/2, LeutwylerJBar[u, a__, b__Rule] -> (Js[a] - Ja[a])/2}$

${Overscript[J, _] _ (a__ b__Rule)(t) -> 1/2 (Ja(a) + Js(a)), Overscript[J, _] _ (a__ b__Rule)(u) -> 1/2 (Js(a) - Ja(a))}$

$JBarToKL := {LeutwylerJBar[t, r__, o___Rule]/MandelstamT :> 2 * K[t, r]/({r}[[1]] - {r}[[-1]]), LeutwylerJBar[u, r__, o___Rule]/MandelstamU :> 2 * K[u, r]/({r}[[1]] - {r}[[-1]])} ;$

(4 t - (t (-4 mm + t))^(1/2) Log[(t + (t (-4 mm + t))^(1/2))^2/(t - (t (-4 mm + t))^(1/2))^2])/(32 π^2 t)

$JBarToKL1 := {LeutwylerJBar[t_, r_, rr_, o___Rule]/t_ :> 2 * K[t, r, rr]/(r - rr) /; r =!= rr} ;$

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

$fixzeros = {LeutwylerJBar[p_, 0, pp_] :> LeutwylerJBar[p, ParticleMass[Pion]^2, pp] /; p =!= pp, LeutwylerJBar[p_, 0, p_] :> 1/(16 π^2), k[0, p_] :> k[ParticleMass[Pion]^2, p], K[a___, 0, b___] :> K[a, ParticleMass[Pion]^2, b]} ;$
$Normal[(Series[LeutwylerJBar[s, m12, m22, LeutwylerJBarEvaluation -> "subthreshold"], {s, 0, 1}] /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y} // Simplify) /. {Sqrt[x_^2] -> x, Sqrt[x_^2 * y_^2] -> x * y} // Simplify] // StandardForm$

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

$JBarToMr := (LeutwylerJBar[m : (t | u), r__, o___Rule])/((n : (MandelstamT | MandelstamU))^2) :> 3/({r}[[1]] - {r}[[-1]])^2 * (Mr[m, r] - 1/(12 * mandelstam[m]) (mandelstam[m] - 2 ({r}[[1]] + {r}[[-1]])) LeutwylerJBar[m, r, o] - 1/(288 * Pi^2) + k[r]/6 + 1/(96 * Pi^2 * mandelstam[m]) (({r}[[1]] + {r}[[-1]]) + deltainv[{r}[[1]], {r}[[-1]]])) /; mandelstam[m] === n ;$
$MrToJBar := {Mr[r_, 0, r_] :> (5 - 3 Log[r/ScaleMu^2])/(576 π^2), Mr[t_, 0, r_] :> 1/(12 * mandelstam[t]) (mandelstam[t] - 2 (r + ParticleMass[Pion]^2)) LeutwylerJBar[t, ParticleMass[Pion]^2, r] + (r - ParticleMass[Pion]^2)^2/(3 * mandelstam[t]^2) LeutwylerJBar[t, ParticleMass[Pion]^2, r] + 1/(288 * Pi^2) - k[ParticleMass[Pion]^2, r]/6 - 1/(96 * Pi^2 * mandelstam[t]) ((r + ParticleMass[Pion]^2) + deltainv[r, ParticleMass[Pion]^2]) /; r =!= t, Mr[t_, r__] :> 1/(12 * mandelstam[t]) (mandelstam[t] - 2 ({r}[[1]] + {r}[[-1]])) LeutwylerJBar[t, r] + ({r}[[1]] - {r}[[-1]])^2/(3 * mandelstam[t]^2) LeutwylerJBar[t, r] + 1/(288 * Pi^2) - k[r]/6 - 1/(96 * Pi^2 * mandelstam[t]) (({r}[[1]] + {r}[[-1]]) + deltainv[{r}[[1]], {r}[[-1]]]) /; ((* {r}[[-1]] =!= t && *) {r}[[1]] =!= 0)} ;$

$cancelLogs1 = {Log[a_ * b_] :> Log[a] + Log[b], Log[a_/b_] :> Log[a] - Log[b], Log[a_^n_] :> n Log[a]} ;$
$moms = {(MandelstamS | MandelstamT | MandelstamU | ParticleMass[a___]^2 | Pair[__]) * (MandelstamS | MandelstamT | MandelstamU | ParticleMass[b___]^2 | Pair[__]) * (MandelstamS | MandelstamT | MandelstamU | ParticleMass[c___]^2 | Pair[__]), (MandelstamS^2 | MandelstamT^2 | MandelstamU^2 | ParticleMass[a___]^4 | Pair[__]^4) * (MandelstamS | MandelstamT | MandelstamU | ParticleMass[b___]^2 | Pair[__]), MandelstamS^3 | MandelstamT^3 | MandelstamU^3 | ParticleMass[___]^6 | Pair[__]^3} ;$

$softPionLimit = {MandelstamS -> ParticleMass[Pion]^2, MandelstamT -> ParticleMass[Kaon]^2, MandelstamU -> Pair[Momentum[p2], Momentum[p2]]} ;$

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

Converted by Mathematica (July 10, 2003)