•Overscript[J, _]'s, logs, non-CT polynomials and lowest order contributions on-mass-shell

Loop contribution:

finLoops = ((Simplify /@ #) & /@ Collect[su2IsoscalarProj[endloops] /. _LeutwylerLambda -> 0 /. MomentaRules /. onshellrules /. gellmannOkubo /. toEtaRules /. _RenormalizationState -> Sequence[], {_Pair, _LeutwylerJBar, _Log, _CouplingConstant}]) // Simplify

1/(288 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((2 c _ 5^( ) (-32 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ π^(ó  ))^4 - 9 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2 (m _ π^(ó  ))^4 - 432 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 432 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 54 log((m _ π^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 9 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 53 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 96 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ η^(ó  ))^2 (m _ π^(ó  ))^2 + 9 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 (m _ π^(ó  ))^2 + 6 (m _ η^(ó  ))^2 (m _ π^(ó  ))^2 + 432 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^4 + 432 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^4 + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^4 - 27 log((m _ π^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 27 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 55 (m _ K^(ó  ))^4 + 96 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 - 3 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 - 6 (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + 9 (48 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) + 48 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) - 3 log((m _ π^(ó  ))^2/μ^2) - 3 log((m _ K^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2) - 7) p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 ((432 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) + 432 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) - 45 log((m _ π^(ó  ))^2/μ^2) - 27 log((m _ K^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2) - 55) (m _ π^(ó  ))^2 + (-432 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) - 432 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) - 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) + 27 log((m _ π^(ó  ))^2/μ^2) - 9 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2) + 55) (m _ K^(ó  ))^2 - 6 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2)) - 3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (576 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (m _ π^(ó  ))^2 - 72 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 36 (m _ π^(ó  ))^2 - 144 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 - 144 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 + 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ K^(ó  ))^2 + 9 log((m _ π^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 21 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 17 (m _ K^(ó  ))^2 - 64 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (m _ η^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 4 (m _ η^(ó  ))^2 + (144 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) - 48 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) - 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) - 9 log((m _ π^(ó  ))^2/μ^2) + 3 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2) - 5) p _ 2^2 - 3 (144 π^2 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) - 48 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) - 16 π^2 Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) - 9 log((m _ π^(ó  ))^2/μ^2) + 3 log((m _ K^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2) - 5) p _ 3^2)) !, _ 0^( ))

Wavefuntion-renormalized lowest order contribution:

finTrees = (su2IsoscalarProj[Plus @@ end2all] /. Log -> log /. _LeutwylerLambda -> 0 /. onshellrules /. gellmannOkubo // Simplify) /. _RenormalizationState -> Sequence[] /. toEtaRules /. log -> Log

1/(288 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((c _ 5^( ) (9216 π^2 L _ 5^( ) (m _ π^(ó  ))^4 - 18432 π^2 L _ 6^( ) (m _ π^(ó  ))^4 - 18432 π^2 L _ 8^( ) (m _ π^(ó  ))^4 - 45 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - 9 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - 13824 π^2 L _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 18432 π^2 L _ 6^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 9 log((m _ π^(ó  ))^2/μ^2) p _ 2^2 (m _ π^(ó  ))^2 - 3 log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ π^(ó  ))^2 + 9216 π^2 L _ 5^( ) (m _ K^(ó  ))^4 + 36864 π^2 L _ 6^( ) (m _ K^(ó  ))^4 + 18432 π^2 L _ 8^( ) (m _ K^(ó  ))^4 - 4608 π^2 L _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 + 18 log((m _ K^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2 + 12 log((m _ η^(ó  ))^2/μ^2) p _ 2^2 (m _ K^(ó  ))^2 + 1152 π^2 (f _ ϕ^(ó  ))^2 ((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + p _ 2^2) - 4608 π^2 L _ 4^( ) ((m _ π^(ó  ))^4 - 4 (m _ K^(ó  ))^4 + p _ 2^2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))) - 2 c _ 2^( ) (-27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - 2304 π^2 N _ 10^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2304 π^2 N _ 10^( ) (m _ K^(ó  ))^4 + 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 1152 π^2 N _ 21^( ) p _ 2^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 1152 π^2 N _ 11^( ) ((m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^4))) !, _ 0^( ))

finPoly = finLoops + finTrees /. {_Log -> 0, _LeutwylerJBar -> 0, CouplingConstant[_[4], ___] -> 0} // Expand // Simplify

1/(288 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (36 (m _ π^(ó  ))^2 - 17 (m _ K^(ó  ))^2 - 4 (m _ η^(ó  ))^2 + 5 p _ 2^2 - 15 p _ 3^2) + 2 c _ 5^( ) (576 π^2 ((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + p _ 2^2) (f _ ϕ^(ó  ))^2 + ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 (m _ π^(ó  ))^2 + 55 (m _ K^(ó  ))^2 + 6 (m _ η^(ó  ))^2 - 55 p _ 2^2 - 63 p _ 3^2))) !, _ 0^( ))

finJBars = Collect[finLoops + finTrees - (finLoops + finTrees /. _LeutwylerJBar -> 0) /. {_Log -> 0, CouplingConstant[_[4], ___] -> 0} // Expand, {_LeutwylerJBar}] // Simplify

1/(18 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (27 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2) + 27 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2) + Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (-2 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 - 6 (m _ η^(ó  ))^2 + p _ 2^2 + 9 p _ 3^2)) - 3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (9 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (4 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2) - 3 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (3 (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2) + Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) ((m _ K^(ó  ))^2 - 4 (m _ η^(ó  ))^2 - p _ 2^2 + 3 p _ 3^2))) !, _ 0^( ))

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}]

-(3 Overscript[J, _] _ (m _ π^(ó  ))^2(p _ 3^2) (c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (4 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2) + 2 c _ 5^( ) (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) !, _ 0^( ))/(2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) + (Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (6 c _ 5^( ) (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (3 (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2)) !, _ 0^( ))/(2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) + (Overscript[J, _] _ (m _ η^(ó  ))^2(p _ 3^2) (2 c _ 5^( ) (-9 (m _ K^(ó  ))^2 + p _ 2^2 + 9 p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (-4 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2 + 3 p _ 2^2 - 9 p _ 3^2)) !, _ 0^( ))/(18 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

finLogs = (Simplify /@ (Collect[finLoops + finTrees - (finLoops + finTrees /. _Log -> 0) /. {_LeutwylerJBar -> 0, CouplingConstant[_[4], ___] -> 0} // Expand, {_Log}] /. gellmannOkubo)) /. Log[l_] :> Log[l /. toEtaRules]

-1/(32 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) (log((m _ K^(ó  ))^2/μ^2) (c _ 2^( ) (p _ 2^4 - 3 (p _ 3^2 - 2 (m _ K^(ó  ))^2) p _ 2^2 + 3 (m _ K^(ó  ))^2 (p _ 3^2 - (m _ K^(ó  ))^2)) + 6 c _ 5^( ) (-(m _ K^(ó  ))^4 + p _ 2^2 (m _ π^(ó  ))^2 + p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))) !, _ 0^( )) + 1/(32 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) (log((m _ π^(ó  ))^2/μ^2) (3 c _ 2^( ) (2 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4 + p _ 2^4 + 3 p _ 3^2 (m _ K^(ó  ))^2 + p _ 2^2 (8 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 3 p _ 3^2)) + c _ 5^( ) (-7 (m _ π^(ó  ))^4 + 12 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 6 (m _ K^(ó  ))^4 - 6 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 (6 (m _ K^(ó  ))^2 - 9 (m _ π^(ó  ))^2))) !, _ 0^( )) - 1/(288 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) (log((m _ η^(ó  ))^2/μ^2) (c _ 5^( ) (11 (m _ π^(ó  ))^4 - 60 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 10 (m _ K^(ó  ))^4 + 18 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) + c _ 2^( ) (2 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^4 + 3 p _ 2^4 + 9 p _ 3^2 (m _ K^(ó  ))^2 + p _ 2^2 (-8 (m _ π^(ó  ))^2 + 26 (m _ K^(ó  ))^2 - 9 p _ 3^2))) !, _ 0^( ))

Converted by Mathematica  (July 10, 2003)