•Reduction of the amplitude

The one-loop integrals are simplified:

aff = SUNReduce[amplFC] // Simplify

-1/(144 π^4 f _ ϕ^(ó  )) (i !, _ 0^( ) ((6 (d _ (1 i _ 1 k1)^(3))^2 + 6 (d _ (2 i _ 1 k1)^(3))^2 + 6 (d _ (3 i _ 1 k1)^(3))^2 + 3 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (1 i _ 1)^(3) + 4 δ _ (2 i _ 1)^(3) + 4 δ _ (3 i _ 1)^(3) + 6)/(q _ 1^2 - (m _ π^(ó  ))^2) + (6 (d _ (4 i _ 1 k1)^(3))^2 + 6 (d _ (5 i _ 1 k1)^(3))^2 + 6 (d _ (6 i _ 1 k1)^(3))^2 + 6 (d _ (7 i _ 1 k1)^(3))^2 - 2 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (4 i _ 1)^(3) + 4 δ _ (5 i _ 1)^(3) + 4 δ _ (6 i _ 1)^(3) + 4 δ _ (7 i _ 1)^(3) + 8)/(q _ 1^2 - (m _ K^(ó  ))^2) + (6 (d _ (8 i _ 1 k1)^(3))^2 - 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (8 i _ 1)^(3) + 2)/(q _ 1^2 - (m _ η^(ó  ))^2)))

af = OneLoopSimplify[aff, q1]

-(i !, _ 0^( ) (6 (d _ (1 i _ 1 k1)^(3))^2 + 6 (d _ (2 i _ 1 k1)^(3))^2 + 6 (d _ (3 i _ 1 k1)^(3))^2 + 3 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (1 i _ 1)^(3) + 4 δ _ (2 i _ 1)^(3) + 4 δ _ (3 i _ 1)^(3) + 6))/(144 π^4 f _ ϕ^(ó  ) (q _ 1^2 - (m _ π^(ó  ))^2)) - (i !, _ 0^( ) (3 (d _ (4 i _ 1 k1)^(3))^2 + 3 (d _ (5 i _ 1 k1)^(3))^2 + 3 (d _ (6 i _ 1 k1)^(3))^2 + 3 (d _ (7 i _ 1 k1)^(3))^2 - 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 2 δ _ (4 i _ 1)^(3) + 2 δ _ (5 i _ 1)^(3) + 2 δ _ (6 i _ 1)^(3) + 2 δ _ (7 i _ 1)^(3) + 4))/(72 π^4 f _ ϕ^(ó  ) (q _ 1^2 - (m _ K^(ó  ))^2)) - (i !, _ 0^( ) (6 (d _ (8 i _ 1 k1)^(3))^2 - 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (8 i _ 1)^(3) + 2))/(144 π^4 f _ ϕ^(ó  ) (q _ 1^2 - (m _ η^(ó  ))^2))

The loop integrals are expressed in terms of Passarino-Veltman symbols.

ampreduced = OneLoop[q1, af] // Simplify

1/(144 π^2 f _ ϕ^(ó  )) (!, _ 0^( ) (A _ 0 ( (m _ π^(ó  ))^2 ) (6 (d _ (1 i _ 1 k1)^(3))^2 + 6 (d _ (2 i _ 1 k1)^(3))^2 + 6 (d _ (3 i _ 1 k1)^(3))^2 + 3 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (1 i _ 1)^(3) + 4 δ _ (2 i _ 1)^(3) + 4 δ _ (3 i _ 1)^(3) + 6) + A _ 0 ( (m _ K^(ó  ))^2 ) (6 (d _ (4 i _ 1 k1)^(3))^2 + 6 (d _ (5 i _ 1 k1)^(3))^2 + 6 (d _ (6 i _ 1 k1)^(3))^2 + 6 (d _ (7 i _ 1 k1)^(3))^2 - 2 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (4 i _ 1)^(3) + 4 δ _ (5 i _ 1)^(3) + 4 δ _ (6 i _ 1)^(3) + 4 δ _ (7 i _ 1)^(3) + 8) + A _ 0 ( (m _ η^(ó  ))^2 ) (6 (d _ (8 i _ 1 k1)^(3))^2 - 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (8 i _ 1)^(3) + 2)))

The divergences are singled out:

ampinfinities = VeltmanExpand[ampreduced, ExplicitLeutwylerJ0 -> True] // Simplify

1/(144 π^2 f _ ϕ^(ó  )) (!, _ 0^( ) (-(32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (6 (d _ (1 i _ 1 k1)^(3))^2 + 6 (d _ (2 i _ 1 k1)^(3))^2 + 6 (d _ (3 i _ 1 k1)^(3))^2 + 3 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 4 δ _ (1 i _ 1)^(3) + 4 δ _ (2 i _ 1)^(3) + 4 δ _ (3 i _ 1)^(3) + 6) (m _ π^(ó  ))^2 - 2 (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 (3 (d _ (4 i _ 1 k1)^(3))^2 + 3 (d _ (5 i _ 1 k1)^(3))^2 + 3 (d _ (6 i _ 1 k1)^(3))^2 + 3 (d _ (7 i _ 1 k1)^(3))^2 - 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) + 2 δ _ (4 i _ 1)^(3) + 2 δ _ (5 i _ 1)^(3) + 2 δ _ (6 i _ 1)^(3) + 2 δ _ (7 i _ 1)^(3) + 4) + (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2 (-6 (d _ (8 i _ 1 k1)^(3))^2 + 3^(1/2) d _ (8 i _ 1 i _ 1)^(3) - 4 δ _ (8 i _ 1)^(3) - 2)))

Converted by Mathematica  (July 10, 2003)