•Reduction of counterterm amplitude

test1 = Cancel[# (* * (MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) *) ] & /@ rest1 (* /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2 *) // Simplify

{-(2 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-2 N _ 9^( ) (m _ π^(ó  ))^2 + N _ 22^( ) (m _ π^(ó  ))^2 + 2 N _ 23^( ) (m _ π^(ó  ))^2 - 2 N _ 5^( ) (m _ K^(ó  ))^2 + N _ 20^( ) (s - 2 (m _ π^(ó  ))^2) - N _ 8^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(f _ ϕ^(ó  ))^4, -1/(3 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-3 N _ 8^( ) (m _ π^(ó  ))^4 + 6 N _ 10^( ) (m _ π^(ó  ))^4 + 14 N _ 11^( ) (m _ π^(ó  ))^4 + 6 N _ 12^( ) (m _ π^(ó  ))^4 - 6 N _ 8^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 6 N _ 9^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2 N _ 10^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 10 N _ 11^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 6 N _ 12^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 s N _ 8^( ) (m _ π^(ó  ))^2 + 3 s N _ 9^( ) (m _ π^(ó  ))^2 - 4 N _ 10^( ) (m _ K^(ó  ))^4 - 4 N _ 11^( ) (m _ K^(ó  ))^4 + 6 s N _ 8^( ) (m _ K^(ó  ))^2 + 3 s N _ 9^( ) (m _ K^(ó  ))^2 + 6 N _ 7^( ) (s - 2 (m _ π^(ó  ))^2) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 3 N _ 5^( ) (-2 (m _ π^(ó  ))^4 + s (m _ π^(ó  ))^2 + s (m _ K^(ó  ))^2))), (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (2 N _ 10^( ) (m _ K^(ó  ))^2 + N _ 11^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (8 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (2 L _ 4^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) - L _ 5^( ) (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 + s) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (2 N _ 10^( ) (m _ K^(ó  ))^2 + N _ 11^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), 1/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (16 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-L _ 5^( ) ((m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - s (m _ K^(ó  ))^2) + L _ 4^( ) (-(m _ π^(ó  ))^4 + (s - 14 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 8 s (m _ K^(ó  ))^2) + 2 (L _ 8^( ) ((m _ π^(ó  ))^4 + 6 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4) + L _ 6^( ) ((m _ π^(ó  ))^4 + 15 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4))))}

ctlambdas = Coefficient[Renormalize[# /. KamborToBijnens /. RenormalizeBijnens /. CouplingConstant[ChPTW3[4], ii_] -> CouplingConstant[ChPTW3[4], ii, RenormalizationState[0]]], LeutwylerLambda[]] & /@ test1

{-(2 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((3 c _ 5^( ) (m _ π^(ó  ))^2)/(2 c _ 2^( )) - 5/3 (m _ π^(ó  ))^2 - (3 c _ 5^( ) (m _ K^(ó  ))^2)/(2 c _ 2^( )) - 2 (m _ K^(ó  ))^2 + (3 s)/4))/(f _ ϕ^(ó  ))^4, -1/(3 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((55 c _ 5^( ) (m _ π^(ó  ))^4)/(18 c _ 2^( )) - 47/18 (m _ π^(ó  ))^4 + (19 c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(18 c _ 2^( )) - 119/18 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 3/2 s (m _ π^(ó  ))^2 + (3 s c _ 5^( ) (m _ π^(ó  ))^2)/c _ 2^( ) - (37 c _ 5^( ) (m _ K^(ó  ))^4)/(9 c _ 2^( )) + 2/9 (m _ K^(ó  ))^4 + 21/2 s (m _ K^(ó  ))^2 - (3 s c _ 5^( ) (m _ K^(ó  ))^2)/c _ 2^( ))), (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) ((11 c _ 5^( ) (m _ π^(ó  ))^2)/(18 c _ 2^( )) - 13/18 (m _ π^(ó  ))^2 + (37 c _ 5^( ) (m _ K^(ó  ))^2)/(18 c _ 2^( )) - 1/9 (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (8 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (1/4 (m _ π^(ó  ))^2 + 1/8 (m _ K^(ó  ))^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 + s) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) ((11 c _ 5^( ) (m _ π^(ó  ))^2)/(18 c _ 2^( )) - 13/18 (m _ π^(ó  ))^2 + (37 c _ 5^( ) (m _ K^(ó  ))^2)/(18 c _ 2^( )) - 1/9 (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), (16 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-5/36 (m _ π^(ó  ))^4 + 2/3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1/8 s (m _ π^(ó  ))^2 + 37/72 (m _ K^(ó  ))^4 + 11/8 s (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s))}

ctlambdasOnshell = Cancel[ctlambdas * (MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2

{0, (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (47 c _ 2^( ) (m _ π^(ó  ))^4 - 55 c _ 5^( ) (m _ π^(ó  ))^4 + 146 c _ 2^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 73 c _ 5^( ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 193 c _ 2^( ) (m _ K^(ó  ))^4 + 128 c _ 5^( ) (m _ K^(ó  ))^4))/(27 (f _ ϕ^(ó  ))^4), 0, 0, (4 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (13 c _ 2^( ) (m _ π^(ó  ))^2 - 11 c _ 5^( ) (m _ π^(ó  ))^2 + 2 c _ 2^( ) (m _ K^(ó  ))^2 - 37 c _ 5^( ) (m _ K^(ó  ))^2))/(27 (f _ ϕ^(ó  ))^4), -(2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-10 (m _ π^(ó  ))^4 + 57 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 136 (m _ K^(ó  ))^4))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)}

ctlambdas - Cancel[ctlambdasOnshell/(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] // Simplify

{-(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-20 (m _ π^(ó  ))^2 - 24 (m _ K^(ó  ))^2 + 9 s) + 18 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/(6 (f _ ϕ^(ó  ))^4), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) ((m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^2) + 2 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2)))/(f _ ϕ^(ó  ))^4, (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) - c _ 5^( ) (11 (m _ π^(ó  ))^2 + 37 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2) - c _ 5^( ) (11 (m _ π^(ó  ))^2 + 37 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^4 + 10 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 11 (m _ K^(ó  ))^4))/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)}

end1 = Plus @@ test1 // Expand // FullSimplify

1/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (-2 (2 N _ 5^( ) + 4 N _ 7^( ) + N _ 8^( ) - 2 (N _ 10^( ) + 2 N _ 11^( ) + N _ 12^( ))) (m _ π^(ó  ))^4 + s (2 N _ 5^( ) + 4 N _ 7^( ) + N _ 8^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ π^(ó  ))^2 + 2 (N _ 5^( ) + N _ 8^( )) (m _ K^(ó  ))^4 - ((-8 N _ 7^( ) + 3 N _ 8^( ) + 2 N _ 9^( ) + 4 N _ 10^( ) + 8 N _ 11^( ) + 4 N _ 12^( ) + N _ 22^( ) + 2 N _ 23^( )) (m _ π^(ó  ))^2 - 2 s (-2 N _ 7^( ) + N _ 8^( ) + N _ 9^( ))) (m _ K^(ó  ))^2 + N _ 20^( ) (s - 2 (m _ π^(ó  ))^2) (s - (m _ K^(ó  ))^2)) (m _ K^(ó  ))^2 + 4 c _ 5^( ) (m _ π^(ó  ) - m _ K^(ó  )) (m _ π^(ó  ) + m _ K^(ó  )) (2 (L _ 4^( ) + L _ 5^( ) - 2 (L _ 6^( ) + L _ 8^( ))) (m _ π^(ó  ))^4 - (4 L _ 4^( ) - L _ 5^( ) + 8 L _ 6^( ) + 4 L _ 8^( )) (m _ K^(ó  ))^4 - (3 s (4 L _ 4^( ) + L _ 5^( )) - 2 (13 L _ 4^( ) + 3 (L _ 5^( ) - 10 L _ 6^( ) - 4 L _ 8^( ))) (m _ π^(ó  ))^2) (m _ K^(ó  ))^2)))

Cancel[(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2) (end1)] /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2 // Simplify

-1/(3 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2) (4 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (4 c _ 5^( ) (L _ 4^( ) ((m _ π^(ó  ))^4 + 13 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4) + L _ 5^( ) ((m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - (m _ K^(ó  ))^4) - 2 (L _ 8^( ) ((m _ π^(ó  ))^4 + 6 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^4) + L _ 6^( ) ((m _ π^(ó  ))^4 + 15 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^4))) - 3 c _ 2^( ) (m _ K^(ó  ))^2 (N _ 8^( ) (m _ π^(ó  ))^2 - 2 N _ 10^( ) (m _ π^(ó  ))^2 - 4 N _ 11^( ) (m _ π^(ó  ))^2 - 2 N _ 12^( ) (m _ π^(ó  ))^2 + 2 N _ 8^( ) (m _ K^(ó  ))^2 + N _ 9^( ) (m _ K^(ó  ))^2 + N _ 7^( ) (4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2) + N _ 5^( ) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))))

Converted by Mathematica  (July 10, 2003)