•Reduction of the loop amplitude

Changing from B0's to Overscript[J, _]'s and λ's:

ampinfinities = ((WriteString["stdout", "."] ; VeltmanExpand[Collect[# /. D -> Sequence[], _B0], ExplicitLeutwylerJ0 -> True]) & /@ res4) ;

..........

LeafCount /@ ampinfinities

{237, 237, 237, 237, 582, 359, 626, 582, 359, 5738}

Plus @@ ampinfinities // LeafCount

8720

endloops = CheckF[Collect[Plus @@ ampinfinities, {_QuarkCondensate, _DecayConstant, (Pair[__] - ParticleMass[__]^2)^(-2)}] /. Times[a___, (Pair[b__] - ParticleMass[c__]^2)^p_, d___] :> Times[Collect[Times[a, d], {_Pair, _ParticleMass, _CouplingConstant, _Log, _LeutwylerLambda}], (Pair[b] - ParticleMass[c]^2)^p], "KSPiLoopsOnShellfin.m"] ;

endloops // LeafCount

10557

The coefficient on the λ's:

looplambdas = (Simplify[Coefficient[#, LeutwylerLambda[]] /. gellmannOkubo /. MandelstamRules // MomentumExpand // ExpandScalarProduct] & /@ ampinfinities) /. _RenormalizationState -> Sequence[]

{-(16 c _ 5^( ) (7 (m _ π^(ó  ))^2 + 8 (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), -(16 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), -(16 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), -(16 c _ 5^( ) (7 (m _ π^(ó  ))^2 + 8 (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), (8 (7 (m _ π^(ó  ))^2 + 8 (m _ K^(ó  ))^2) (c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 c _ 5^( ) (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)), -(8 (2 c _ 5^( ) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - 2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)/(3 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)), 1/(27 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (4 (c _ 2^( ) (-16 (m _ π^(ó  ))^4 - 16 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36 p _ 3^2 (m _ π^(ó  ))^2 + 86 (m _ K^(ó  ))^4 + 99 p _ 3^2 (m _ K^(ó  ))^2 + 9 p _ 1^2 (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2) - 9 p _ 2^2 (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)) - 42 c _ 5^( ) (m _ K^(ó  ))^2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2), (8 (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2) (c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 c _ 5^( ) (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/(9 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)), -(8 (2 c _ 5^( ) (m _ K^(ó  ))^2 + c _ 2^( ) (p _ 2^2 - 2 ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)/(3 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), 1/(27 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (2 (6 c _ 5^( ) (10 (m _ π^(ó  ))^2 + 14 (m _ K^(ó  ))^2 + 9 p _ 1^2 - 27 p _ 2^2 + 9 p _ 3^2) (m _ K^(ó  ))^2 + c _ 2^( ) (-52 (m _ π^(ó  ))^4 - 28 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 p _ 3^2 (m _ π^(ó  ))^2 - 100 (m _ K^(ó  ))^4 - 81 p _ 2^4 - 111 p _ 3^2 (m _ K^(ó  ))^2 + 3 p _ 1^2 (-29 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^2 + 9 p _ 2^2) + 9 p _ 2^2 (15 (m _ π^(ó  ))^2 + 23 (m _ K^(ó  ))^2 + 3 p _ 3^2))) (!, _ 0^( ))^2)}

lootLambdaCoeff = Plus @@ looplambdas /. gellmannOkubo // MomentumExpand // ExpandScalarProduct // Simplify

-1/(9 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (2 (c _ 2^( ) (52 (m _ π^(ó  ))^4 + 68 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 93 p _ 3^2 (m _ π^(ó  ))^2 + 27 p _ 2^4 - 129 p _ 3^2 (m _ K^(ó  ))^2 + 3 p _ 1^2 (-21 (m _ π^(ó  ))^2 - 53 (m _ K^(ó  ))^2 + p _ 2^2) + p _ 2^2 (11 (m _ π^(ó  ))^2 + 61 (m _ K^(ó  ))^2 + 3 p _ 3^2)) + 2 c _ 5^( ) (-32 (m _ π^(ó  ))^4 - 108 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 14 (m _ K^(ó  ))^4 + 27 p _ 2^2 (m _ K^(ó  ))^2 + p _ 1^2 (56 (m _ π^(ó  ))^2 + 67 (m _ K^(ó  ))^2) + p _ 3^2 (32 (m _ π^(ó  ))^2 + 91 (m _ K^(ó  ))^2))) (!, _ 0^( ))^2)

Converted by Mathematica  (July 10, 2003)