Loop contribution:
Wavefuntion-renormalized lowest order contribution:
Converted by Mathematica (July 10, 2003)
![Overscript[J, _]](../HTMLFiles/index_95.gif){kind=small}
's, logs and lowest order contributions on-mass-shell's, logs and lowest order contributions on-mass-shell![finloops = Collect[Cancel[(Pair[Momentum[p3], Momentum[p3]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2) * finalloops (* /. _LeutwylerJBar -> 0 *)] /. MomentaRules /. onshellrules /. gellmannOkubo, {_DecayConstant, _LeutwylerJBar, _Log}] /. toEtaRules /. {Log[a_] * b_ :> Log[a] * Simplify[b], LeutwylerJBar[a__] * b_ :> LeutwylerJBar[a] * Simplify[b]}](../HTMLFiles/index_96.gif)
![1/(f _ ϕ^(ó ))^2 ((Overscript[J, _] _ ((m _ π^(ó ))^2 (m _ K^(ó ))^2)(p _ 2^2) (2 c _ 5^( ) (m _ K^(ó ))^2 + c _ 2^( ) (-(m _ π^(ó ))^2 - (m _ K^(ó ))^2 + p _ 2^2)) (5 p _ 2^4 - 2 ((m _ π^(ó ))^2 + (m _ K^(ó ))^2) p _ 2^2 - 3 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2)^2) (!, _ 0^( ))^2)/(2 p _ 2^2) + (log((m _ π^(ó ))^2/μ^2) (m _ π^(ó ))^2 (31 (m _ π^(ó ))^2 - 43 (m _ K^(ó ))^2 + 15 p _ 2^2) (c _ 2^( ) ((m _ π^(ó ))^2 + (m _ K^(ó ))^2 - p _ 2^2) - 2 c _ 5^( ) (m _ K^(ó ))^2) (!, _ 0^( ))^2)/(96 π^2 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2)) + (Overscript[J, _] _ ((m _ K^(ó ))^2 (m _ η^(ó ))^2)(p _ 2^2) (2 c _ 5^( ) (m _ K^(ó ))^2 + c _ 2^( ) (-(m _ K^(ó ))^2 - (m _ η^(ó ))^2 + p _ 2^2)) (9 p _ 2^4 - (7 (m _ π^(ó ))^2 + 2 (m _ K^(ó ))^2 + 3 (m _ η^(ó ))^2) p _ 2^2 - 9 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) ((m _ K^(ó ))^2 - (m _ η^(ó ))^2)) (!, _ 0^( ))^2)/(18 p _ 2^2) + 1/(288 π^2 ((m _ η^(ó ))^2 - (m _ K^(ó ))^2)) (log((m _ η^(ó ))^2/μ^2) (m _ η^(ó ))^2 (2 c _ 5^( ) (7 (m _ π^(ó ))^2 + 29 (m _ K^(ó ))^2 - 24 (m _ η^(ó ))^2 - 9 p _ 2^2) (m _ K^(ó ))^2 + c _ 2^( ) (-69 (m _ K^(ó ))^4 + 47 (m _ η^(ó ))^2 (m _ K^(ó ))^2 + 12 (m _ η^(ó ))^4 - 9 p _ 2^4 + p _ 2^2 (7 (m _ π^(ó ))^2 + 62 (m _ K^(ó ))^2 - 39 (m _ η^(ó ))^2) + (m _ π^(ó ))^2 (37 (m _ η^(ó ))^2 - 51 (m _ K^(ó ))^2))) (!, _ 0^( ))^2) + (log((m _ K^(ó ))^2/μ^2) (m _ K^(ó ))^2 (2 c _ 5^( ) (7 (m _ π^(ó ))^4 + (171 (m _ η^(ó ))^2 - 173 (m _ K^(ó ))^2) (m _ π^(ó ))^2 + 5 (m _ K^(ó ))^2 (26 (m _ K^(ó ))^2 - 27 (m _ η^(ó ))^2) - 9 p _ 2^2 ((m _ π^(ó ))^2 - 6 (m _ K^(ó ))^2 + 5 (m _ η^(ó ))^2)) (m _ K^(ó ))^2 + c _ 2^( ) (-9 ((m _ π^(ó ))^2 - 6 (m _ K^(ó ))^2 + 5 (m _ η^(ó ))^2) p _ 2^4 + (7 (m _ π^(ó ))^4 + (225 (m _ η^(ó ))^2 - 209 (m _ K^(ó ))^2) (m _ π^(ó ))^2 + 76 (m _ K^(ó ))^4 - 99 (m _ K^(ó ))^2 (m _ η^(ó ))^2) p _ 2^2 + 2 (-109 (m _ K^(ó ))^6 + 114 (m _ η^(ó ))^2 (m _ K^(ó ))^4 + (m _ π^(ó ))^4 (86 (m _ K^(ó ))^2 - 93 (m _ η^(ó ))^2) + (m _ π^(ó ))^2 (59 (m _ K^(ó ))^4 - 57 (m _ K^(ó ))^2 (m _ η^(ó ))^2)))) (!, _ 0^( ))^2)/(288 π^2 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) ((m _ K^(ó ))^2 - (m _ η^(ó ))^2)))](../HTMLFiles/index_97.gif)
![finTrees = (Cancel[(Pair[Momentum[p3], Momentum[p3]] - ParticleMass[Pion]^2) (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[Kaon]^2) * (Plus @@ final2all /. _RenormalizationState -> Sequence[])] /. onshellrules /. gellmannOkubo // Simplify) /. toEtaRules](../HTMLFiles/index_98.gif)
![finpoly = (finloops + finTrees) /. {_Log -> 0, _LeutwylerJBar -> 0, CouplingConstant[_[4], ___] -> 0} // Simplify](../HTMLFiles/index_100.gif){kind=small}
![finJBars = finloops + finTrees - finpoly /. {_Log -> 0, CouplingConstant[_[4], ___] -> 0} // Simplify](../HTMLFiles/index_102.gif){kind=small}
![1/(18 (f _ ϕ^(ó ))^2 p _ 2^2) ((9 Overscript[J, _] _ ((m _ π^(ó ))^2 (m _ K^(ó ))^2)(p _ 2^2) (2 c _ 5^( ) (m _ K^(ó ))^2 + c _ 2^( ) (-(m _ π^(ó ))^2 - (m _ K^(ó ))^2 + p _ 2^2)) (5 p _ 2^4 - 2 ((m _ π^(ó ))^2 + (m _ K^(ó ))^2) p _ 2^2 - 3 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2)^2) + Overscript[J, _] _ ((m _ K^(ó ))^2 (m _ η^(ó ))^2)(p _ 2^2) (2 c _ 5^( ) (m _ K^(ó ))^2 + c _ 2^( ) (-(m _ K^(ó ))^2 - (m _ η^(ó ))^2 + p _ 2^2)) (9 p _ 2^4 - (7 (m _ π^(ó ))^2 + 2 (m _ K^(ó ))^2 + 3 (m _ η^(ó ))^2) p _ 2^2 - 9 ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) ((m _ K^(ó ))^2 - (m _ η^(ó ))^2))) (!, _ 0^( ))^2)](../HTMLFiles/index_103.gif)
![finJBarsKLM = Simplify /@ Collect[Expand[finJBars] /. JBarToKL /. cancelLogs /. gellmannOkubo /. {Log[l_] :> Log[l /. toEtaRules], K[l__] :> K[Sequence @@ ({l} /. toEtaRules)], LeutwylerJBar[l__] :> LeutwylerJBar[Sequence @@ ({l} /. toEtaRules)]}, {_LeutwylerJBar, _K}]](../HTMLFiles/index_104.gif)
![-(K[p _ 2^2, (m _ K^(ó ))^2, (m _ η^(ó ))^2] ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) (6 c _ 5^( ) (m _ K^(ó ))^2 + c _ 2^( ) ((m _ π^(ó ))^2 - 7 (m _ K^(ó ))^2)) (!, _ 0^( ))^2)/(3 (f _ ϕ^(ó ))^2) + (3 K[p _ 2^2, (m _ π^(ó ))^2, (m _ K^(ó ))^2] ((m _ π^(ó ))^2 - (m _ K^(ó ))^2) (c _ 2^( ) ((m _ π^(ó ))^2 + (m _ K^(ó ))^2) - 2 c _ 5^( ) (m _ K^(ó ))^2) (!, _ 0^( ))^2)/(f _ ϕ^(ó ))^2 - (Overscript[J, _] _ ((m _ π^(ó ))^2 (m _ K^(ó ))^2)(p _ 2^2) (2 c _ 5^( ) (2 ((m _ π^(ó ))^2 + (m _ K^(ó ))^2) - 5 p _ 2^2) (m _ K^(ó ))^2 + c _ 2^( ) ((m _ π^(ó ))^4 - 10 (m _ K^(ó ))^2 (m _ π^(ó ))^2 + (m _ K^(ó ))^4 - 5 p _ 2^4 + 7 p _ 2^2 ((m _ π^(ó ))^2 + (m _ K^(ó ))^2))) (!, _ 0^( ))^2)/(2 (f _ ϕ^(ó ))^2) + (Overscript[J, _] _ ((m _ K^(ó ))^2 (m _ η^(ó ))^2)(p _ 2^2) (6 c _ 5^( ) (3 p _ 2^2 - 2 ((m _ π^(ó ))^2 + (m _ K^(ó ))^2)) (m _ K^(ó ))^2 + c _ 2^( ) (-5 (m _ π^(ó ))^4 + 18 (m _ K^(ó ))^2 (m _ π^(ó ))^2 + 11 (m _ K^(ó ))^4 + 9 p _ 2^4 - 3 p _ 2^2 ((m _ π^(ó ))^2 + 9 (m _ K^(ó ))^2))) (!, _ 0^( ))^2)/(18 (f _ ϕ^(ó ))^2)](../HTMLFiles/index_105.gif)
![finLogs = Simplify /@ Collect[Simplify[(finloops + finTrees - finpoly) /. gellmannOkubo /. {_LeutwylerJBar -> 0, CouplingConstant[_[4], ___] -> 0}], {_Log}] /. Log[l_] :> Log[l /. toEtaRules] // Simplify](../HTMLFiles/index_106.gif)
