•Scale independence

scaleRule = CouplingConstant[ChPTW3[4], i_, ___] :> CouplingConstant[ChPTW3[4], i] + RenormalizationCoefficients[ChPTW3[4]][[i]]/(32 Pi^2) Log[ParticleMass[Kaon]^2/ScaleMu^2] ;

logRule = Log[a_] :> 0 /; FreeQ[{a}, ScaleMu] ;

logRule1 = Log[a_ * b_] -> Log[a] + Log[b] ;

logcheck1 = strongcts + oldcts + newcts1 /. scaleRule /. CouplingConstant[ChPTW3[4], i_, ___] -> 0 /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. logRule1 /. logRule /. gellmannOkubo /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[] // Simplify

1/(1152 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) log(1/μ^2) (4 (13 s - 45 (m _ K^(ó  ))^2) (m _ π^(ó  ))^4 - 4 (m _ K^(ó  ))^2 (101 (m _ K^(ó  ))^2 + 11 s - 36 t) (m _ π^(ó  ))^2 + 27 p _ 2^6 + p _ 2^4 (132 (m _ π^(ó  ))^2 + 75 (m _ K^(ó  ))^2 - 45 s - 72 t) + (m _ K^(ó  ))^2 (125 (m _ K^(ó  ))^4 + (343 s + 72 t) (m _ K^(ó  ))^2 + 36 (s^2 - 2 t s - 2 t^2)) + p _ 2^2 (180 (m _ π^(ó  ))^4 + 4 (68 (m _ K^(ó  ))^2 + 15 s - 36 t) (m _ π^(ó  ))^2 - 227 (m _ K^(ó  ))^4 - 36 s^2 + 72 t^2 - 366 s (m _ K^(ó  ))^2 + 72 s t)))

logcheck2 = finalLogs /. k -> kk /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. logRule1 /. logRule /. gellmannOkubo /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[] // Simplify

-1/(1152 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i c _ 2^( ) log(1/μ^2) ((52 s - 189 (m _ K^(ó  ))^2) (m _ π^(ó  ))^4 + (m _ K^(ó  ))^2 (-395 (m _ K^(ó  ))^2 - 44 s + 144 t) (m _ π^(ó  ))^2 + 27 p _ 2^6 + p _ 2^4 (123 (m _ π^(ó  ))^2 + 84 (m _ K^(ó  ))^2 - 45 s - 72 t) + (m _ K^(ó  ))^2 (125 (m _ K^(ó  ))^4 + (343 s + 72 t) (m _ K^(ó  ))^2 + 36 (s^2 - 2 t s - 2 t^2)) + p _ 2^2 (189 (m _ π^(ó  ))^4 + 4 (68 (m _ K^(ó  ))^2 + 15 s - 36 t) (m _ π^(ó  ))^2 - 236 (m _ K^(ó  ))^4 - 36 s^2 + 72 t^2 - 366 s (m _ K^(ó  ))^2 + 72 s t)))

logcheck3 = (finaljbars2KLMs /. MrToJBar /. KLToJBar /. k -> kk /. logRule1 /. logRule /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. _LeutwylerJBar -> 0) - (finaljbars2KLMs /. MrToJBar /. KLToJBar /. k -> kk /. logRule1 /. logRule /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. _LeutwylerJBar -> 0 /. _Log -> 0) /. gellmannOkubo /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[] // Simplify

-(i c _ 2^( ) log(1/μ^2) (p _ 2^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(128 π^2 (f _ ϕ^(ó  ))^4)

logcheck1 + logcheck2 + logcheck3 // Simplify

0

newlogchecks1 = end4 + end4old /. scaleRule /. CouplingConstant[ChPTW3[4], i_, ___] -> 0 /. cancelU /. kaonOnShell /. Pair[___, _LorentzIndex] -> 1 /. logRule1 /. logRule /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. _RenormalizationState -> Sequence[] // Simplify

-1/(3456 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) log(1/μ^2) (-592 (m _ π^(ó  ))^4 + 16 (27 t - 73 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 383 (m _ K^(ó  ))^4 + 108 s^2 - 216 t^2 - 81 p _ 2^4 + 1053 s (m _ K^(ó  ))^2 + 216 t (m _ K^(ó  ))^2 - 216 s t + 3 p _ 2^2 (-152 (m _ π^(ó  ))^2 - 82 (m _ K^(ó  ))^2 + 45 s + 72 t)))

newlogchecks2 = Log[1/ScaleMu^2] Coefficient[end2all + endloops /. scaleRule //. logRule1 /. logRule /. CouplingConstant[ChPTW3[2], 2, ___] -> 0 /. _RenormalizationState -> Sequence[], Log[1/ScaleMu^2]] /. cancelU /. kaonOnShell /. _RenormalizationState -> Sequence[] /. Pair[___, _LorentzIndex] -> 1 // Simplify

1/(3456 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) log(1/μ^2) (-592 (m _ π^(ó  ))^4 + 16 (27 t - 73 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 383 (m _ K^(ó  ))^4 + 108 s^2 - 216 t^2 - 81 p _ 2^4 + 1053 s (m _ K^(ó  ))^2 + 216 t (m _ K^(ó  ))^2 - 216 s t + 3 p _ 2^2 (-152 (m _ π^(ó  ))^2 - 82 (m _ K^(ó  ))^2 + 45 s + 72 t)))

newlogchecks1 + newlogchecks2 // Simplify

0

gilbout = ((end4 + end4old /. cancelU /. kaonOnShell /. Pair[___, _LorentzIndex] -> 1 // Renormalize) + (end2all + endloops /. cancelU /. kaonOnShell /. Pair[___, _LorentzIndex] -> 1 /. _RenormalizationState -> Sequence[] // Renormalize)) /. _RenormalizationState -> Sequence[] ;

Coefficient[gilbout /. C5 -> 0, LeutwylerLambda[]] // Simplify // Expand

0

Converted by Mathematica  (July 10, 2003)