•Divergencies

The remaining divergencies after renormalization are all proportional to c _ 5- we still need to calculate the c _ 5renormalization beta functions.

llc = LOOPlambdaCoeff /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2 (* /. C5 -> 0 *) // Simplify

1/(27 (f _ ϕ^(ó  ))^3 ((m _ K^(ó  ))^2 - p _ 2^2)) (!, _ 0^( ) (3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (-16 (15 δ _ (0 i)^(3) - δ _ (3 i)^(3) + 4 3^(1/2) δ _ (8 i)^(3)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (240 δ _ (0 i)^(3) + 89 δ _ (3 i)^(3) - 41 3^(1/2) δ _ (8 i)^(3)) + 15 (p _ 2^2 - 3 p _ 3^2) (δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3))) + 2 c _ 5^( ) (60 δ _ (0 i)^(3) (m _ π^(ó  ))^4 - 30 δ _ (3 i)^(3) (m _ π^(ó  ))^4 - 10 3^(1/2) δ _ (8 i)^(3) (m _ π^(ó  ))^4 - 300 (m _ K^(ó  ))^2 δ _ (0 i)^(3) (m _ π^(ó  ))^2 + 117 (m _ K^(ó  ))^2 δ _ (3 i)^(3) (m _ π^(ó  ))^2 - 25 3^(1/2) (m _ K^(ó  ))^2 δ _ (8 i)^(3) (m _ π^(ó  ))^2 + 240 (m _ K^(ó  ))^4 δ _ (0 i)^(3) + 93 (m _ K^(ó  ))^4 δ _ (3 i)^(3) - 145 3^(1/2) (m _ K^(ó  ))^4 δ _ (8 i)^(3) + 27 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (12 δ _ (0 i)^(3) - 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + 3 p _ 2^2 ((84 δ _ (0 i)^(3) - 19 δ _ (3 i)^(3) + 15 3^(1/2) δ _ (8 i)^(3)) (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 (-84 δ _ (0 i)^(3) - 41 δ _ (3 i)^(3) + 45 3^(1/2) δ _ (8 i)^(3))))))

clc = CTlambdaCoeff /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon]^2 (* /. C5 -> 0 *) // Simplify

-1/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) (!, _ 0^( ) (2 c _ 5^( ) (24 δ _ (0 i)^(3) (m _ π^(ó  ))^4 - 12 δ _ (3 i)^(3) (m _ π^(ó  ))^4 - 4 3^(1/2) δ _ (8 i)^(3) (m _ π^(ó  ))^4 + 12 (m _ K^(ó  ))^2 δ _ (0 i)^(3) (m _ π^(ó  ))^2 - 39 (m _ K^(ó  ))^2 δ _ (3 i)^(3) (m _ π^(ó  ))^2 - 21 3^(1/2) (m _ K^(ó  ))^2 δ _ (8 i)^(3) (m _ π^(ó  ))^2 - 36 (m _ K^(ó  ))^4 δ _ (0 i)^(3) + 87 (m _ K^(ó  ))^4 δ _ (3 i)^(3) - 11 3^(1/2) (m _ K^(ó  ))^4 δ _ (8 i)^(3) - 9 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (12 δ _ (0 i)^(3) - 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) + 3 p _ 2^2 ((m _ π^(ó  ))^2 (36 δ _ (0 i)^(3) - 11 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)) - (m _ K^(ó  ))^2 (36 δ _ (0 i)^(3) + δ _ (3 i)^(3) - 13 3^(1/2) δ _ (8 i)^(3)))) + c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (4 (60 δ _ (0 i)^(3) - 15 δ _ (3 i)^(3) - 5 3^(1/2) δ _ (8 i)^(3) - 11 σ _ (2 2)^i - 37 σ _ (3 3)^i) (m _ π^(ó  ))^2 + 15 (p _ 2^2 - 3 p _ 3^2) (σ _ (2 2)^i - σ _ (3 3)^i) + (m _ K^(ó  ))^2 (-240 δ _ (0 i)^(3) + 60 δ _ (3 i)^(3) + 20 3^(1/2) δ _ (8 i)^(3) + 149 σ _ (2 2)^i + 43 σ _ (3 3)^i))))

llc /. i -> 0 // Simplify

(8 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (c _ 5^( ) (5 (m _ π^(ó  ))^2 - 20 (m _ K^(ó  ))^2 + 21 p _ 2^2 + 27 p _ 3^2) - 30 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2)) !, _ 0^( ))/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

clc /. UGenerator[_][__] -> 0 /. i -> 0 // Simplify

(8 ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (10 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) + c _ 5^( ) (2 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2 + 9 p _ 2^2 - 9 p _ 3^2)) !, _ 0^( ))/(3 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

% + %% // Simplify

(8 c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (48 p _ 2^2 + 11 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) !, _ 0^( ))/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

llc /. i -> 3 // Expand // Simplify

1/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((2 c _ 5^( ) (10 (m _ π^(ó  ))^4 - 39 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 31 (m _ K^(ó  ))^4 + 27 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 (19 (m _ π^(ó  ))^2 + 41 (m _ K^(ó  ))^2)) - c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (16 (m _ π^(ó  ))^2 + 89 (m _ K^(ó  ))^2 + 15 p _ 2^2 - 45 p _ 3^2)) !, _ 0^( ))

clc /. i -> 3 // Simplify

1/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (16 (m _ π^(ó  ))^2 + 89 (m _ K^(ó  ))^2 + 15 p _ 2^2 - 45 p _ 3^2) + 6 c _ 5^( ) (4 (m _ π^(ó  ))^4 + 13 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 29 (m _ K^(ó  ))^4 - 9 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 2^2 (11 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))) !, _ 0^( ))

% + %% // Simplify

(4 c _ 5^( ) (11 (m _ π^(ó  ))^4 - 59 (m _ K^(ó  ))^4 + p _ 2^2 (26 (m _ π^(ó  ))^2 + 22 (m _ K^(ó  ))^2)) !, _ 0^( ))/(9 (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

llc /. i -> 8 // Expand // Simplify

1/(9 3^(1/2) (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((3 c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (64 (m _ π^(ó  ))^2 + 41 (m _ K^(ó  ))^2 + 15 p _ 2^2 - 45 p _ 3^2) + 2 c _ 5^( ) (27 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 45 p _ 2^2 ((m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^2) + 5 (2 (m _ π^(ó  ))^4 + 5 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 29 (m _ K^(ó  ))^4))) !, _ 0^( ))

clc /. i -> 8 // Simplify

-1/(3 3^(1/2) (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2)) ((c _ 2^( ) (p _ 2^2 - (m _ K^(ó  ))^2) (64 (m _ π^(ó  ))^2 + 41 (m _ K^(ó  ))^2 + 15 p _ 2^2 - 45 p _ 3^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^4 + 21 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^4 + 3 p _ 2^2 ((m _ π^(ó  ))^2 - 13 (m _ K^(ó  ))^2) - 9 p _ 3^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))) !, _ 0^( ))

% + %% // Simplify

(4 c _ 5^( ) (11 (m _ π^(ó  ))^4 + 44 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 89 (m _ K^(ó  ))^4 - 18 p _ 2^2 ((m _ π^(ó  ))^2 + 7 (m _ K^(ó  ))^2)) !, _ 0^( ))/(9 3^(1/2) (f _ ϕ^(ó  ))^3 (p _ 2^2 - (m _ K^(ó  ))^2))

Converted by Mathematica  (July 10, 2003)