•Reduction of the loop amplitude

res4 = Pair[LorentzIndex[μ1], Momentum[Polarization[p1, i]]] * res4 ;

res5 = ((WriteString["stdout", "."] ; (Simplify[# /. massshellrules /. p4 -> -p1 - p3]) // MomentumExpand // ExpandScalarProduct // MomentumCombine // Simplify) & /@ res4) ;

..................

res5[[7]]

(i (5 A _ 0 ( (m _ π^(ó  ))^2 ) + 12 A _ 0 ( (m _ K^(ó  ))^2 ) + 9 A _ 0 ( (m _ η^(ó  ))^2 )) c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2))/(480 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)

ampinfinities = ((WriteString["stdout", "."] ; VeltmanExpand[#, ExplicitLeutwylerJ0 -> True]) & /@ res5) /. Pair[Momentum[p1], Momentum[p1]] -> MandelstamS /. gellmannOkubo ;

..................

ampinfonshell = (WriteString["stdout", "."] ; Cancel[# * (MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2 /. gellmannOkubo) & /@ ampinfinities ;

..................

If[# === 0, Null, Length[Cases[#, _LeutwylerJBar, Infinity]]] & /@ ampinfinities

{0, 0, 4, 19, 19, 0, 0, 20, 20, 0, 15, 0, 11, 11, 4, 15, 15, 10}

looplambdas = Simplify[ScalarProductExpand[Coefficient[#, LeutwylerLambda[]]]] & /@ ampinfinities

{-(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 (m _ π^(ó  ))^2 + 25 (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), 1/(135 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 c _ 5^( ) (43 (m _ π^(ó  ))^4 + 28 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 71 (m _ K^(ó  ))^4) + 5 c _ 2^( ) (-58 (m _ π^(ó  ))^4 + 2 (9 s - 56 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 35 (m _ K^(ó  ))^4 + 117 s (m _ K^(ó  ))^2))), -(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-16 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 9 s))/(6 (f _ ϕ^(ó  ))^4), -(2 i µ _ μ _ 1(p _ 1) (c _ 5^( ) (2 p _ 1^μ _ 1 + 3 p _ 3^μ _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (p _ 1^μ _ 1 (2 (m _ π^(ó  ))^2 - 11 (m _ K^(ó  ))^2) + 6 p _ 3^μ _ 1 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))))/(9 (f _ ϕ^(ó  ))^4), (2 i µ _ μ _ 1(p _ 1) (c _ 5^( ) (p _ 1^μ _ 1 + 3 p _ 3^μ _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (6 p _ 3^μ _ 1 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 1^μ _ 1 (4 (m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2))))/(9 (f _ ϕ^(ó  ))^4), -(2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^2))/(15 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), -1/(27 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-8 (m _ π^(ó  ))^4 + (6 s - 62 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 25 (m _ K^(ó  ))^4 + 48 s (m _ K^(ó  ))^2) - 3 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (4 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 + s))), -1/(27 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-8 (m _ π^(ó  ))^4 + (6 s - 62 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 25 (m _ K^(ó  ))^4 + 48 s (m _ K^(ó  ))^2) - 3 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (4 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 + s))), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 + s) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), 1/(9 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (6 c _ 5^( ) (-4 (m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 7 s) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (56 (m _ π^(ó  ))^4 - 4 (27 s - 4 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 3 (-2 (m _ K^(ó  ))^4 + 7 s (m _ K^(ó  ))^2 + 9 s^2)))), (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (75 (m _ π^(ó  ))^4 - ((m _ K^(ó  ))^2 + 27 s) (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - 3 s)))/(135 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), -(2 i c _ 5^( ) (20 p _ 1^μ _ 1 + 33 p _ 3^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (2 i c _ 5^( ) (13 p _ 1^μ _ 1 + 33 p _ 3^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (8 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 - 21 s))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (8 (m _ π^(ó  ))^4 + 38 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 47 (m _ K^(ó  ))^4 - 21 s (m _ K^(ó  ))^2))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (8 (m _ π^(ó  ))^4 + 38 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 47 (m _ K^(ó  ))^4 - 21 s (m _ K^(ó  ))^2))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (8 (m _ π^(ó  ))^4 - 8 (3 (m _ K^(ó  ))^2 + 2 s) (m _ π^(ó  ))^2 - 6 (m _ K^(ó  ))^4 + 21 s^2 + 9 s (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s))}

looplambdasOnshell = Simplify[Coefficient[#, LeutwylerLambda[]]] & /@ ampinfonshell

{0, (4 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 5^( ) (43 (m _ π^(ó  ))^2 + 71 (m _ K^(ó  ))^2) - 5 c _ 2^( ) (29 (m _ π^(ó  ))^2 + 76 (m _ K^(ó  ))^2)))/(135 (f _ ϕ^(ó  ))^4), 0, 0, 0, 0, 0, (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (8 (m _ π^(ó  ))^4 + 56 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 73 (m _ K^(ó  ))^4) + 3 c _ 5^( ) (4 (m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^4)))/(27 (f _ ϕ^(ó  ))^4), (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (8 (m _ π^(ó  ))^4 + 56 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 73 (m _ K^(ó  ))^4) + 3 c _ 5^( ) (4 (m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^4)))/(27 (f _ ϕ^(ó  ))^4), (4 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (28 (m _ π^(ó  ))^4 - 46 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 21 (m _ K^(ó  ))^4) - 3 c _ 5^( ) (4 (m _ π^(ó  ))^4 - 9 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^4)))/(9 (f _ ϕ^(ó  ))^4), (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-75 (m _ π^(ó  ))^4 + 28 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 26 (m _ K^(ó  ))^4))/(135 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), 0, 0, 0, (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-4 (m _ π^(ó  ))^6 - 15 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 6 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^6))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-4 (m _ π^(ó  ))^6 - 15 (m _ K^(ó  ))^2 (m _ π^(ó  ))^4 + 6 (m _ K^(ó  ))^4 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^6))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (8 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) ((m _ π^(ó  ))^4 - 5 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 3 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)}

looplambdas - Cancel[looplambdasOnshell/(MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] // Simplify

{-(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 (m _ π^(ó  ))^2 + 25 (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (2 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), -(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-16 (m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2 + 9 s))/(6 (f _ ϕ^(ó  ))^4), -(2 i µ _ μ _ 1(p _ 1) (c _ 5^( ) (2 p _ 1^μ _ 1 + 3 p _ 3^μ _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (p _ 1^μ _ 1 (2 (m _ π^(ó  ))^2 - 11 (m _ K^(ó  ))^2) + 6 p _ 3^μ _ 1 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))))/(9 (f _ ϕ^(ó  ))^4), (2 i µ _ μ _ 1(p _ 1) (c _ 5^( ) (p _ 1^μ _ 1 + 3 p _ 3^μ _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (6 p _ 3^μ _ 1 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + p _ 1^μ _ 1 (4 (m _ π^(ó  ))^2 + 5 (m _ K^(ó  ))^2))))/(9 (f _ ϕ^(ó  ))^4), -(2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^2))/(15 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), -(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) + 2 c _ 2^( ) ((m _ π^(ó  ))^2 + 8 (m _ K^(ó  ))^2)))/(9 (f _ ϕ^(ó  ))^4), -(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 5^( ) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) + 2 c _ 2^( ) ((m _ π^(ó  ))^2 + 8 (m _ K^(ó  ))^2)))/(9 (f _ ϕ^(ó  ))^4), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (c _ 2^( ) (13 (m _ π^(ó  ))^2 + 17 (m _ K^(ó  ))^2) - 2 c _ 5^( ) (4 (m _ π^(ó  ))^2 + 11 (m _ K^(ó  ))^2)))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (14 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + c _ 2^( ) (-36 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^2 + 9 s)))/(3 (f _ ϕ^(ó  ))^4), (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-9 (m _ π^(ó  ))^4 - 4 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 13 (m _ K^(ó  ))^4))/(45 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), -(2 i c _ 5^( ) (20 p _ 1^μ _ 1 + 33 p _ 3^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (2 i c _ 5^( ) (13 p _ 1^μ _ 1 + 33 p _ 3^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (8 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 - 21 s))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (7 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (7 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))/(9 (f _ ϕ^(ó  ))^4), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (-16 (m _ π^(ó  ))^2 + 30 (m _ K^(ó  ))^2 + 21 s))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)}

looplambdas[[7]]

(2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) ((m _ π^(ó  ))^2 + 12 (m _ K^(ó  ))^2))/(15 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)

looplambdas[[8]]

-(i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (c _ 2^( ) (-8 (m _ π^(ó  ))^4 + (6 s - 62 (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 25 (m _ K^(ó  ))^4 + 48 s (m _ K^(ó  ))^2) - 3 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (4 (m _ π^(ó  ))^2 + 6 (m _ K^(ó  ))^2 + s)))/(27 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2))

Simplify[(looplambdas[[16]] + looplambdas[[17]] + looplambdas[[18]])/(-ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 + ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2)] (-ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 + ParticleMass[PseudoScalar[6], RenormalizationState[0]]^2)

-(i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (40 (m _ π^(ó  ))^4 + (4 (m _ K^(ó  ))^2 - 48 s) (m _ π^(ó  ))^2 + 76 (m _ K^(ó  ))^4 + 63 s^2 - 15 s (m _ K^(ó  ))^2))/(27 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s))

fin1 = Plus @@ looplambdas // 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)