•The infinities

CTlambdaCoeff = Coefficient[end4 + end4old /. cancelU /. kaonOnShell /. Pair[___, _LorentzIndex] -> 1 // Renormalize, LeutwylerLambda[]] /. _RenormalizationState -> Sequence[] ;

CTlambdaCoeff // Expand

(142 i c _ 5^( ) (m _ π^(ó  ))^6)/(27 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (80 i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (14 i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^4)/(3 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (4 i s c _ 5^( ) (m _ π^(ó  ))^4)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (12 i t c _ 5^( ) (m _ π^(ó  ))^4)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (148 i c _ 2^( ) (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4) + (292 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4) + (38 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^2)/(9 (f _ ϕ^(ó  ))^4) - (193 i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(3 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (7 i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (6 i s c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (6 i t c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i s^2 c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i t^2 c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i s t c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (4 i t c _ 2^( ) (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^4 - (383 i c _ 2^( ) (m _ K^(ó  ))^4)/(108 (f _ ϕ^(ó  ))^4) + (3 i c _ 2^( ) p _ 2^4)/(4 (f _ ϕ^(ó  ))^4) + (41 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2)/(18 (f _ ϕ^(ó  ))^4) - (39 i s c _ 2^( ) (m _ K^(ó  ))^2)/(4 (f _ ϕ^(ó  ))^4) - (2 i t c _ 2^( ) (m _ K^(ó  ))^2)/(f _ ϕ^(ó  ))^4 - (5 i s c _ 2^( ) p _ 2^2)/(4 (f _ ϕ^(ó  ))^4) - (2 i t c _ 2^( ) p _ 2^2)/(f _ ϕ^(ó  ))^4 - (29 i c _ 5^( ) (m _ K^(ó  ))^6)/(27 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (13 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4)/(3 (f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (3 i s c _ 5^( ) (m _ K^(ó  ))^4)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i t c _ 5^( ) (m _ K^(ó  ))^4)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i s c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) + (6 i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (6 i s^2 c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (6 i t^2 c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (6 i s t c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 ((m _ K^(ó  ))^2 - p _ 2^2)) - (i s^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 + (2 i t^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 + (2 i s t c _ 2^( ))/(f _ ϕ^(ó  ))^4

ctl = CTlambdaCoeff /. CouplingConstant[ChPTW3[2], 2] -> 0 // Expand

(148 i c _ 2^( ) (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4) + (292 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4) + (38 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^2)/(9 (f _ ϕ^(ó  ))^4) - (4 i t c _ 2^( ) (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^4 - (383 i c _ 2^( ) (m _ K^(ó  ))^4)/(108 (f _ ϕ^(ó  ))^4) + (3 i c _ 2^( ) p _ 2^4)/(4 (f _ ϕ^(ó  ))^4) + (41 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2)/(18 (f _ ϕ^(ó  ))^4) - (39 i s c _ 2^( ) (m _ K^(ó  ))^2)/(4 (f _ ϕ^(ó  ))^4) - (2 i t c _ 2^( ) (m _ K^(ó  ))^2)/(f _ ϕ^(ó  ))^4 - (5 i s c _ 2^( ) p _ 2^2)/(4 (f _ ϕ^(ó  ))^4) - (2 i t c _ 2^( ) p _ 2^2)/(f _ ϕ^(ó  ))^4 - (i s^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 + (2 i t^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 + (2 i s t c _ 2^( ))/(f _ ϕ^(ó  ))^4

LOOPlambdaCoeff = Coefficient[end2all + endloops /. cancelU /. kaonOnShell /. Pair[___, _LorentzIndex] -> 1 /. _RenormalizationState -> Sequence[] // Renormalize, LeutwylerLambda[]] ;

LOOPlambdaCoeff // Simplify // Expand

(142 i c _ 5^( ) (m _ π^(ó  ))^6)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (148 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (19 i c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (148 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (25 i c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^4)/(3 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (25 i s c _ 5^( ) (m _ π^(ó  ))^4)/(9 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (12 i t c _ 5^( ) (m _ π^(ó  ))^4)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (292 i c _ 2^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (329 i c _ 5^( ) (m _ K^(ó  ))^4 (m _ π^(ó  ))^2)/(54 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (38 i c _ 2^( ) p _ 2^4 (m _ π^(ó  ))^2)/(9 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (3 i c _ 5^( ) p _ 2^4 (m _ π^(ó  ))^2)/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (178 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (26 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(9 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (4 i t c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (79 i s c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(18 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i t c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (4 i t c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i s c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^2)/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (6 i t c _ 5^( ) p _ 2^2 (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i s^2 c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i t^2 c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i s t c _ 5^( ) (m _ π^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (383 i c _ 2^( ) (m _ K^(ó  ))^6)/(108 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (83 i c _ 5^( ) (m _ K^(ó  ))^6)/(54 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (629 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^4)/(108 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (49 i c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^4)/(9 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (39 i s c _ 2^( ) (m _ K^(ó  ))^4)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (2 i t c _ 2^( ) (m _ K^(ó  ))^4)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (29 i s c _ 5^( ) (m _ K^(ó  ))^4)/(18 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i t c _ 5^( ) (m _ K^(ó  ))^4)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (3 i c _ 2^( ) p _ 2^6)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (5 i s c _ 2^( ) p _ 2^4)/(4 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (2 i t c _ 2^( ) p _ 2^4)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (55 i c _ 2^( ) p _ 2^4 (m _ K^(ó  ))^2)/(36 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (3 i c _ 5^( ) p _ 2^4 (m _ K^(ó  ))^2)/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (17 i s c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2)/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i s c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2)/(2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (6 i t c _ 5^( ) p _ 2^2 (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (i s^2 c _ 2^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (2 i t^2 c _ 2^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (2 i s t c _ 2^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (6 i s^2 c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (6 i t^2 c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (6 i s t c _ 5^( ) (m _ K^(ó  ))^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) + (i s^2 c _ 2^( ) p _ 2^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (2 i t^2 c _ 2^( ) p _ 2^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) - (2 i s t c _ 2^( ) p _ 2^2)/((f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2))

loopl = LOOPlambdaCoeff /. CouplingConstant[ChPTW3[2], 2] -> 0 // Simplify // Expand

-(148 i c _ 2^( ) (m _ π^(ó  ))^4)/(27 (f _ ϕ^(ó  ))^4) - (292 i c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(27 (f _ ϕ^(ó  ))^4) - (38 i c _ 2^( ) p _ 2^2 (m _ π^(ó  ))^2)/(9 (f _ ϕ^(ó  ))^4) + (4 i t c _ 2^( ) (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^4 + (383 i c _ 2^( ) (m _ K^(ó  ))^4)/(108 (f _ ϕ^(ó  ))^4) - (3 i c _ 2^( ) p _ 2^4)/(4 (f _ ϕ^(ó  ))^4) - (41 i c _ 2^( ) p _ 2^2 (m _ K^(ó  ))^2)/(18 (f _ ϕ^(ó  ))^4) + (39 i s c _ 2^( ) (m _ K^(ó  ))^2)/(4 (f _ ϕ^(ó  ))^4) + (2 i t c _ 2^( ) (m _ K^(ó  ))^2)/(f _ ϕ^(ó  ))^4 + (5 i s c _ 2^( ) p _ 2^2)/(4 (f _ ϕ^(ó  ))^4) + (2 i t c _ 2^( ) p _ 2^2)/(f _ ϕ^(ó  ))^4 + (i s^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 - (2 i t^2 c _ 2^( ))/(f _ ϕ^(ó  ))^4 - (2 i s t c _ 2^( ))/(f _ ϕ^(ó  ))^4

ctl + loopl // Simplify // Expand

0

Converted by Mathematica  (July 10, 2003)