•Reduction of the amplitude

The one-loop integrals are simplified:

af = OneLoopSimplify[amplFC, q1]

-(i c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) δ _ (7 i _ 3)^(3))/(6 π^4 (f _ ϕ^(ó  ))^3 (q _ 1^2 - (m _ K^0^(ó  ))^2)) - (i (2 c _ 5^( ) (m _ π^(ó  ))^2 - 3 c _ 2^( ) (m _ π^+^(ó  ))^2 - 2 c _ 5^( ) (m _ K^+^(ó  ))^2) δ _ (7 i _ 3)^(3))/(24 π^4 (f _ ϕ^(ó  ))^3 (q _ 1^2 - (m _ π^+^(ó  ))^2)) - (i (2 c _ 5^( ) (m _ π^(ó  ))^2 - 3 c _ 2^( ) (m _ π^0^(ó  ))^2 - 2 c _ 5^( ) (m _ K^+^(ó  ))^2) δ _ (7 i _ 3)^(3))/(48 π^4 (f _ ϕ^(ó  ))^3 (q _ 1^2 - (m _ π^0^(ó  ))^2)) - (i (2 c _ 5^( ) (m _ π^(ó  ))^2 + 3 c _ 2^( ) (m _ K^+^(ó  ))^2 - 2 c _ 5^( ) (m _ K^+^(ó  ))^2) δ _ (7 i _ 3)^(3))/(24 π^4 (f _ ϕ^(ó  ))^3 (q _ 1^2 - (m _ K^+^(ó  ))^2)) - (i (2 c _ 5^( ) (m _ π^(ó  ))^2 - 2 c _ 5^( ) (m _ K^+^(ó  ))^2 + 3 c _ 2^( ) (m _ η^(ó  ))^2) δ _ (7 i _ 3)^(3))/(48 π^4 (f _ ϕ^(ó  ))^3 (q _ 1^2 - (m _ η^(ó  ))^2))

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

ampreduced = Collect[OneLoop[q1, af], {Pi, _DecayConstant, _A0, _B0, _ParticleMass, _Pair}] /. aa_ a_Plus /; ! FreeQ[aa, Pair, Infinity, Heads -> True] -> aa tog[a] /. tog -> Together /. massrules // Simplify

1/(48 π^2 (f _ ϕ^(ó  ))^3) ((4 A _ 0 ( (m _ K^+^(ó  ))^2 ) c _ 5^( ) (m _ π^(ó  ))^2 + 8 A _ 0 ( (m _ K^0^(ó  ))^2 ) c _ 5^( ) (m _ π^(ó  ))^2 + 6 A _ 0 ( (m _ K^+^(ó  ))^2 ) c _ 2^( ) (m _ K^+^(ó  ))^2 - 4 A _ 0 ( (m _ K^+^(ó  ))^2 ) c _ 5^( ) (m _ K^+^(ó  ))^2 - 8 A _ 0 ( (m _ K^0^(ó  ))^2 ) c _ 5^( ) (m _ K^+^(ó  ))^2 + A _ 0 ( (m _ π^(ó  ))^2 ) (6 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) - 9 c _ 2^( ) (m _ π^(ó  ))^2) + A _ 0 ( 1/3 (2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) - (m _ π^(ó  ))^2) ) (2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) + c _ 2^( ) (2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) - (m _ π^(ó  ))^2))) δ _ (7 i _ 3)^(3))

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

1/(144 π^2 (f _ ϕ^(ó  ))^3) ((c _ 2^( ) ((832 π^2 λ + 27 log((m _ π^(ó  ))^2/μ^2) - log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ π^(ó  ))^4 + 4 (32 π^2 λ + log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) (m _ π^(ó  ))^2 - 2 ((352 π^2 λ + 9 log((m _ K^+^(ó  ))^2/μ^2) + 2 log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ K^+^(ó  ))^4 + 4 (32 π^2 λ + log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ K^0^(ó  ))^2 (m _ K^+^(ó  ))^2 + 2 (32 π^2 λ + log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ K^0^(ó  ))^4)) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) ((256 π^2 λ + 9 log((m _ π^(ó  ))^2/μ^2) - log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ π^(ó  ))^2 + 2 ((128 π^2 λ + 3 log((m _ K^+^(ó  ))^2/μ^2) + log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ K^+^(ó  ))^2 + (224 π^2 λ + 6 log((m _ K^0^(ó  ))^2/μ^2) + log(-((m _ π^(ó  ))^2 - 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2))/(3 μ^2))) (m _ K^0^(ó  ))^2))) δ _ (7 i _ 3)^(3))

Coefficient[ampinfinities, LeutwylerLambda[]] // FullSimplify

1/(9 (f _ ϕ^(ó  ))^3) (4 (c _ 2^( ) (13 (m _ π^(ó  ))^4 + 2 ((m _ K^+^(ó  ))^2 + (m _ K^0^(ó  ))^2) (m _ π^(ó  ))^2 - 11 (m _ K^+^(ó  ))^4 - 2 (m _ K^0^(ó  ))^4 - 4 (m _ K^+^(ó  ))^2 (m _ K^0^(ó  ))^2) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^+^(ó  ))^2) (7 (m _ K^0^(ó  ))^2 + 4 ((m _ π^(ó  ))^2 + (m _ K^+^(ó  ))^2))) δ _ (7 i _ 3)^(3))

Converted by Mathematica  (July 10, 2003)