•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 (f _ ϕ^(ó  ))^3) (((6 c _ 5^( ) (16 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) - log((m _ K^(ó  ))^2/μ^2) - 1) (-(m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 c _ 5^( ) (5 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 8 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) + (4 c _ 5^( ) ((m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2 + 3 p _ 2^2 - 9 p _ 3^2) (-log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 16 π^2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 3^2) ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^2)) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(π^2 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^2)) + (3 (16 π^2 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) - log((m _ K^(ó  ))^2/μ^2) - 1) (8 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) + 3 c _ 2^( ) (3 (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2)))/π^2 - (12 c _ 2^( ) ((m _ K^(ó  ))^2 - p _ 2^2 + p _ 3^2) (log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2))/(π^2 p _ 3^2) + (12 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2))/(π^2 (p _ 2^2 - (m _ K^(ó  ))^2)) - (12 c _ 2^( ) (log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 3 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2))/π^2 + (12 c _ 5^( ) (3 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 6 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2))/π^2 + 1/p _ 3^2 (192 c _ 2^( ) (1/2 (-(m _ K^(ó  ))^2 + p _ 2^2 - p _ 3^2) (2 (m _ π^(ó  ))^2 - 2 (m _ η^(ó  ))^2 + p _ 3^2) + p _ 3^2 ((m _ π^(ó  ))^2 + (m _ η^(ó  ))^2 - p _ 3^2)) (Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 3^2) + (log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 - log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2)/(16 π^2 ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^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) ((1/(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 - 9 log((m _ π^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 33 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 4608 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 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 12 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 (f _ ϕ^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 4608 π^2 L _ 4^( ) ((m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^4)) + 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))) - 9 c _ 5^( ) (128 π^2 (f _ ϕ^(ó  ))^2 + log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 512 π^2 L _ 5^( ) (m _ K^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 - 512 π^2 L _ 4^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))) !, _ 0^( ))

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

-1/(32 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (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) + 2 c _ 5^( ) (64 π^2 ((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + p _ 2^2) (f _ ϕ^(ó  ))^2 + (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^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 _ 3^2 (p _ 2^2 - (m _ K^(ó  ))^2)) ((2 c _ 5^( ) p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 3^2) ((m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2 + 3 p _ 2^2 - 9 p _ 3^2) - 9 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) (-(m _ K^(ó  ))^2 + p _ 2^2 + p _ 3^2)) + 3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (3 Overscript[J, _] _ (m _ K^(ó  ))^2(p _ 3^2) p _ 3^2 (3 (m _ K^(ó  ))^2 + p _ 2^2 - 3 p _ 3^2) + 2 Overscript[J, _] _ ((m _ π^(ó  ))^2 (m _ η^(ó  ))^2)(p _ 3^2) (-3 p _ 3^4 - ((m _ K^(ó  ))^2 - 4 (m _ η^(ó  ))^2) p _ 3^2 + p _ 2^2 (2 (m _ π^(ó  ))^2 - 2 (m _ η^(ó  ))^2 + p _ 3^2) + 2 (m _ K^(ó  ))^2 ((m _ η^(ó  ))^2 - (m _ π^(ó  ))^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}]

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

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

(log((m _ π^(ó  ))^2/μ^2) (c _ 2^( ) (12 (m _ π^(ó  ))^4 - 16 (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 (4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 - 3 p _ 3^2)) - 2 c _ 5^( ) (7 (m _ π^(ó  ))^2 - 9 (m _ K^(ó  ))^2 + 2 p _ 2^2 + 3 p _ 3^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) !, _ 0^( ) (m _ π^(ó  ))^2)/(64 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)) - 1/(32 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) (log((m _ K^(ó  ))^2/μ^2) (c _ 2^( ) (-11 (m _ K^(ó  ))^4 + 3 p _ 3^2 (m _ K^(ó  ))^2 + p _ 2^4 - 3 p _ 2^2 (p _ 3^2 - 2 (m _ K^(ó  ))^2)) - 2 c _ 5^( ) (-5 (m _ K^(ó  ))^4 + 2 (m _ π^(ó  ))^2 (m _ K^(ó  ))^2 + p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))) !, _ 0^( )) + (log((m _ η^(ó  ))^2/μ^2) ((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2) (2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-11 (m _ π^(ó  ))^2 + 7 (m _ K^(ó  ))^2 + 4 p _ 2^2 - 9 p _ 3^2) + c _ 2^( ) (-4 (m _ π^(ó  ))^4 + 24 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^4 + 3 p _ 2^4 + 9 p _ 3^2 (m _ K^(ó  ))^2 + p _ 2^2 (-4 (m _ π^(ó  ))^2 + 10 (m _ K^(ó  ))^2 - 9 p _ 3^2))) !, _ 0^( ))/(576 π^2 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))

Converted by Mathematica  (July 10, 2003)