•Final non-loop expressions

The final counter-term expression:

bpp = end1 + end2 /. KamborToBijnens /. _LeutwylerLambda -> 0 /. RenormalizationState[0] -> RenormalizationState[1] // FullSimplify

1/(3 (f _ ϕ^(ó  r ))^4 (m _ K^(ó  r ))^2 ((m _ K^(ó  r ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) ((-2 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 + s) (f _ ϕ^(ó  r ))^2 + 2 N _ 20^( ) (s - 2 (m _ π^(ó  r ))^2) (s - (m _ K^(ó  r ))^2) + 2 (4 L _ 4^(r ) (m _ π^(ó  r ))^4 + 4 Ε _ 1 (m _ π^(ó  r ))^4 + 8 Ε _ 2 (m _ π^(ó  r ))^4 - 4 Ε _ 3 (m _ π^(ó  r ))^4 - 4 Ε _ 10 (m _ π^(ó  r ))^4 - 2 Ε _ 11 (m _ π^(ó  r ))^4 - 8 Ε _ 13 (m _ π^(ó  r ))^4 + 10 L _ 4^(r ) (m _ K^(ó  r ))^2 (m _ π^(ó  r ))^2 + 4 L _ 5^(r ) (m _ K^(ó  r ))^2 (m _ π^(ó  r ))^2 - 6 s L _ 4^(r ) (m _ π^(ó  r ))^2 + N _ 22^( ) (s - (m _ K^(ó  r ))^2) (m _ π^(ó  r ))^2 + 2 N _ 23^( ) (s - (m _ K^(ó  r ))^2) (m _ π^(ó  r ))^2 - 4 (m _ K^(ó  r ))^2 Ε _ 1 (m _ π^(ó  r ))^2 - 8 (m _ K^(ó  r ))^2 Ε _ 2 (m _ π^(ó  r ))^2 + 4 (m _ K^(ó  r ))^2 Ε _ 3 (m _ π^(ó  r ))^2 + 2 s Ε _ 10 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 Ε _ 11 (m _ π^(ó  r ))^2 + s Ε _ 11 (m _ π^(ó  r ))^2 + 8 (m _ K^(ó  r ))^2 Ε _ 13 (m _ π^(ó  r ))^2 + 4 s Ε _ 13 (m _ π^(ó  r ))^2 + 4 L _ 4^(r ) (m _ K^(ó  r ))^4 + 2 L _ 5^(r ) (m _ K^(ó  r ))^4 - 12 s L _ 4^(r ) (m _ K^(ó  r ))^2 - 6 s L _ 5^(r ) (m _ K^(ó  r ))^2 + 2 (m _ K^(ó  r ))^4 Ε _ 10 - 4 s (m _ K^(ó  r ))^2 Ε _ 13 + 2 (s - (m _ π^(ó  r ))^2) (m _ K^(ó  r ))^2 Ε _ 15)) (m _ K^(ó  r ))^2 + 8 c _ 5^( ) (m _ π^(ó  r ) - m _ K^(ó  r )) (m _ π^(ó  r ) + m _ K^(ó  r )) ((L _ 5^(r ) (6 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 - s) - 24 (2 L _ 6^(r ) + L _ 8^(r )) (m _ π^(ó  r ))^2) (m _ K^(ó  r ))^2 + 2 L _ 4^(r ) ((m _ π^(ó  r ))^2 (11 (m _ K^(ó  r ))^2 + s) - 2 ((m _ K^(ó  r ))^4 + 2 s (m _ K^(ó  r ))^2)))))

This expression can be compared directly with Bijnens, Pallante and Prades:

Cancel[(MandelstamS - ParticleMass[Kaon, RenormalizationState[1]]^2) bpp] /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[1]]^2 /. gellmannOkubo // FullSimplify

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

We work with the N's

CTcontrib = end1 + end2 /. CouplingConstant[c_[4], n_] -> CouplingConstant[c[4], n, RenormalizationState[0]] /. D -> Sequence[] /. ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]] // FullSimplify

1/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  r ))^2 ((m _ K^(ó  r ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (m _ K^(ó  r ))^2 (6 (4 L _ 4^(r ) - 4 N _ 5^( ) - 8 N _ 7^( ) - 2 N _ 8^( ) + 4 (N _ 10^( ) + 2 N _ 11^( ) + N _ 12^( )) + 3 λ) (m _ π^(ó  r ))^4 - 2 s (18 L _ 4^(r ) - 3 (2 N _ 5^( ) + 4 N _ 7^( ) + N _ 8^( ) + N _ 22^( ) + 2 N _ 23^( )) + 5 λ) (m _ π^(ó  r ))^2 + 4 (3 (2 L _ 4^(r ) + L _ 5^(r ) + N _ 5^( ) + N _ 8^( )) - λ) (m _ K^(ó  r ))^4 - 2 (s (36 L _ 4^(r ) + 18 L _ 5^(r ) + 12 N _ 7^( ) - 6 (N _ 8^( ) + N _ 9^( )) + 7 λ) - (30 L _ 4^(r ) + 12 L _ 5^(r ) + 24 N _ 7^( ) - 3 (3 N _ 8^( ) + 2 N _ 9^( ) + 4 N _ 10^( ) + 8 N _ 11^( ) + 4 N _ 12^( ) + N _ 22^( ) + 2 N _ 23^( )) + 5 λ) (m _ π^(ó  r ))^2) (m _ K^(ó  r ))^2 + 6 N _ 20^( ) (s - 2 (m _ π^(ó  r ))^2) (s - (m _ K^(ó  r ))^2) + 3 (f _ ϕ^(ó  ))^2 (-2 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 + s)) - 2 c _ 5^( ) (m _ π^(ó  r ) - m _ K^(ó  r )) (m _ π^(ó  r ) + m _ K^(ó  r )) (24 (-L _ 4^( ) + L _ 4^(r ) - L _ 5^( ) + L _ 5^(r ) + 2 (L _ 6^( ) - L _ 6^(r ) + L _ 8^( ) - L _ 8^(r ))) (m _ π^(ó  r ))^4 - s (24 L _ 4^(r ) + λ) (m _ π^(ó  r ))^2 + 3 (16 L _ 4^( ) - 4 (L _ 5^( ) - 8 L _ 6^( ) + 8 L _ 6^(r ) - 4 L _ 8^( ) + 4 L _ 8^(r )) - 5 λ) (m _ K^(ó  r ))^4 + (s (144 L _ 4^( ) - 48 L _ 4^(r ) + 36 L _ 5^( ) - 24 L _ 5^(r ) - 5 λ) - 3 (104 L _ 4^( ) - 16 L _ 4^(r ) + 24 (L _ 5^( ) - 10 L _ 6^( ) + 2 L _ 6^(r ) - 4 L _ 8^( )) + λ) (m _ π^(ó  r ))^2) (m _ K^(ó  r ))^2)))

The coefficient of the dimensional pole:

CTlambdaCoeff = Coefficient[Renormalize[end1 + end2 /. KamborToBijnens /. RenormalizeBijnens /. CouplingConstant[c_[4], n_] -> CouplingConstant[c[4], n, RenormalizationState[0]]] /. D -> Sequence[], LeutwylerLambda[]] /. RenormalizationState[1] -> RenormalizationState[0] /. gellmannOkubo // FullSimplify

1/(54 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (80 (m _ π^(ó  ))^4 - 4 (17 (m _ K^(ó  ))^2 + 39 s) (m _ π^(ó  ))^2 + 48 (m _ K^(ó  ))^4 + 27 s^2 + 69 s (m _ K^(ó  ))^2) (m _ K^(ó  ))^2 + 4 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (10 (m _ π^(ó  ))^4 + 3 (s - 5 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 - 37 (m _ K^(ó  ))^4 + 15 s (m _ K^(ó  ))^2)))

Converted by Mathematica  (July 10, 2003)