•Reduction of the mass and wave function renormalized tree amplitude

test2 = Cancel /@ ((* (# * (MandelstamS - (ParticleMass[Kaon, RenormalizationState[0]])^2)) & /@ *) ampl2mult) (* /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[1]]^2 *) /. _Log -> 0 // Simplify

{(i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 (f _ ϕ^(ó  ))^2 + 4 (-2 λ (m _ π^(ó  ))^2 + (3 L _ 5^(r ) - λ) (m _ K^(ó  ))^2 + 3 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))))/(3 (f _ ϕ^(ó  ))^4), 1/(18 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (s - (m _ π^(ó  r ))^2) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (-12 (f _ ϕ^(ó  ))^2 + 35 λ (m _ π^(ó  ))^2 + 48 L _ 5^(r ) (m _ K^(ó  ))^2 + 40 λ (m _ K^(ó  ))^2 - 3 λ (m _ η^(ó  ))^2 + 48 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))), (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-6 (f _ ϕ^(ó  ))^2 + 19 λ (m _ π^(ó  ))^2 + 24 L _ 5^(r ) (m _ K^(ó  ))^2 + 32 λ (m _ K^(ó  ))^2 - 3 λ (m _ η^(ó  ))^2 + 24 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), 1/(18 (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) ((m _ π^(ó  ))^2 - (m _ π^(ó  r ))^2 + (m _ K^(ó  ))^2 + s) (-12 (f _ ϕ^(ó  ))^2 + 41 λ (m _ π^(ó  ))^2 + 144 L _ 5^(r ) (m _ K^(ó  ))^2 + 88 λ (m _ K^(ó  ))^2 - 9 λ (m _ η^(ó  ))^2 + 144 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))}

end2 = DiscardOrders[(* (test2[[2]] /. massrenormalization) + (test2[[4]] /. massrenormalization /.) *) (Plus @@ test2) /. propren /. massrenormalization /. _Log -> 0, PerturbationOrder -> 4] /. gellmannOkubo // Simplify

1/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  r ))^2 ((m _ K^(ó  r ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (3 (f _ ϕ^(ó  ))^2 (-2 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 + s) - 2 (-9 λ (m _ π^(ó  r ))^4 + (5 λ (s - (m _ K^(ó  r ))^2) - 12 L _ 5^(r ) (m _ K^(ó  r ))^2) (m _ π^(ó  r ))^2 + 6 L _ 4^(r ) (-2 (m _ π^(ó  r ))^2 - (m _ K^(ó  r ))^2 + 3 s) ((m _ π^(ó  r ))^2 + 2 (m _ K^(ó  r ))^2) + (m _ K^(ó  r ))^2 (6 L _ 5^(r ) (3 s - (m _ K^(ó  r ))^2) + λ (2 (m _ K^(ó  r ))^2 + 7 s)))) (m _ K^(ó  r ))^2 + 2 c _ 5^( ) ((m _ π^(ó  r ))^2 - (m _ K^(ó  r ))^2) (48 L _ 6^(r ) (m _ π^(ó  ))^4 + 48 L _ 8^(r ) (m _ π^(ó  ))^4 + 144 L _ 6^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 96 L _ 6^(r ) (m _ K^(ó  ))^4 + 48 L _ 8^(r ) (m _ K^(ó  ))^4 + 15 λ (m _ K^(ó  r ))^4 + s λ (m _ π^(ó  r ))^2 + 3 λ (m _ π^(ó  r ))^2 (m _ K^(ó  r ))^2 + 5 s λ (m _ K^(ó  r ))^2 - 24 L _ 5^(r ) ((m _ π^(ó  ))^4 + (m _ K^(ó  ))^4 - (m _ K^(ó  r ))^2 ((m _ K^(ó  r ))^2 + s)) - 24 L _ 4^(r ) ((m _ π^(ó  ))^4 + 3 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - (m _ π^(ó  r ))^2 ((m _ K^(ó  r ))^2 + s) - 2 (-(m _ K^(ó  ))^4 + (m _ K^(ó  r ))^4 + s (m _ K^(ó  r ))^2)))))

Cancel[((MandelstamS - ParticleMass[Kaon, RenormalizationState[1]]^2) end2)] /. _RenormalizationState -> Sequence[] /. MandelstamS -> ParticleMass[Kaon]^2 // FullSimplify

1/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2) (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (24 (L _ 4^( ) + L _ 5^( ) - 2 (L _ 6^( ) + L _ 8^( ))) c _ 5^( ) (m _ π^(ó  ))^4 - (36 (2 L _ 4^( ) + L _ 5^( )) c _ 2^( ) + 24 (2 L _ 4^( ) + L _ 5^( ) + 4 L _ 6^( ) + 2 L _ 8^( )) c _ 5^( ) + (27 c _ 2^( ) + 20 c _ 5^( )) λ) (m _ K^(ó  ))^4 + (4 c _ 5^( ) (6 L _ 4^( ) - 36 L _ 6^( ) - λ) (m _ π^(ó  ))^2 + 9 c _ 2^( ) ((f _ ϕ^(ó  ))^2 - (4 L _ 4^( ) + 3 λ) (m _ π^(ó  ))^2)) (m _ K^(ó  ))^2))

multinfs = Coefficient[Renormalize[DiscardOrders[#, PerturbationOrder -> 4] & /@ (test2 /. propren /. massrenormalization /. _Log -> 0) /. gellmannOkubo /. KamborToBijnens /. RenormalizeBijnens /. CouplingConstant[c_[4], n_] -> CouplingConstant[c[4], n, RenormalizationState[0]]] /. D -> Sequence[], LeutwylerLambda[]] /. RenormalizationState[1] -> RenormalizationState[0] /. gellmannOkubo // FullSimplify

{-(4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) (3 c _ 2^( ) (s - (m _ π^(ó  ))^2) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))/((f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  ))^2)), (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (5 (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 + s) (11 (m _ π^(ó  ))^4 + 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 19 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s))}

multinfsOnshell = Cancel[multinfs * (MandelstamS - ParticleMass[Kaon, RenormalizationState[0]]^2)] /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[0]]^2

{0, -(2 i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2) (3 c _ 2^( ) (m _ π^(ó  ))^2 - 2 c _ 5^( ) (m _ π^(ó  ))^2 - 3 c _ 2^( ) (m _ K^(ó  ))^2 + 2 c _ 5^( ) (m _ K^(ó  ))^2))/(f _ ϕ^(ó  ))^4, 0, (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-11 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 19 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4)}

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

{-(4 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))/(3 (f _ ϕ^(ó  ))^4), (6 i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 + (m _ K^(ó  ))^2))/(f _ ϕ^(ó  ))^4, (4 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (5 (m _ π^(ó  ))^4 + 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 7 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2), (2 i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-11 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 19 (m _ K^(ó  ))^4))/(9 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2)}

mult2 = Cancel /@ ((* (# * (MandelstamS - (ParticleMass[Kaon, RenormalizationState[0]])^2)) & /@ *) ampl2mult /. propren) (* /. MandelstamS -> ParticleMass[Kaon, RenormalizationState[1]]^2 *) /. _LeutwylerLambda -> 0 // Simplify

{1/(384 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (384 π^2 (f _ ϕ^(ó  ))^2 - 41 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 3 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 1536 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 34 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 12 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 1536 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))), 1/(576 π^2 (f _ ϕ^(ó  ))^4 (s - (m _ K^(ó  r ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (3 c _ 2^( ) (s - (m _ π^(ó  r ))^2) + 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)) (-384 π^2 (f _ ϕ^(ó  ))^2 + 29 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 1536 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 + 10 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 3 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 1536 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))), 1/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  r ))^2) (i c _ 5^( ) p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-384 π^2 (f _ ϕ^(ó  ))^2 + 35 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 3 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 1536 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 + 22 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 12 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 6 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 1536 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))), -1/(576 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  r ))^2 (s - (m _ K^(ó  r ))^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 - (m _ π^(ó  r ))^2 + (m _ K^(ó  ))^2 + s) (-384 π^2 (f _ ϕ^(ó  ))^2 + 23 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 4608 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 9 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 4608 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))}

multlogs = (DiscardOrders[(* (mult2[[2]] /. a_Log :> (a /. RenormalizationState[0] -> RenormalizationState[1]) /. massrenormalization) + (mult2[[4]] /. a_Log :> (a /. RenormalizationState[0] -> RenormalizationState[1]) /. massrenormalization) *) ((Plus @@ mult2) /. a_Log :> (a /. RenormalizationState[0] -> RenormalizationState[1]) /. propren /. massrenormalization) /. {_LeutwylerLambda -> 0, CouplingConstant[_[4, ___], ___] -> 0}, PerturbationOrder -> 4 (* 6 *)] // Simplify) /. toEtaRules /. RenormalizationState[1] -> RenormalizationState[0] // Simplify

1/(3456 π^2 (f _ ϕ^(ó  ))^4 (m _ K^(ó  ))^2 ((m _ K^(ó  ))^2 - s)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (9 c _ 2^( ) (m _ K^(ó  ))^2 (58 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - ((41 log((m _ π^(ó  ))^2/μ^2) - 20 log((m _ K^(ó  ))^2/μ^2) - 3 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 6 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 17 s log((m _ π^(ó  ))^2/μ^2) + 3 s log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 384 π^2 (f _ ϕ^(ó  ))^2 (-2 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + s) + 2 (-(17 log((m _ K^(ó  ))^2/μ^2) + 6 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + s (7 log((m _ K^(ó  ))^2/μ^2) + 6 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 3 s log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2)) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-4 (9 log((m _ π^(ó  ))^2/μ^2) + log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 + (33 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 36 s log((m _ π^(ó  ))^2/μ^2) - 9 s log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 - 68 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 9 s log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 9 (m _ K^(ó  ))^2 (3 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 8 s log((m _ K^(ó  ))^2/μ^2) + 4 s log((m _ η^(ó  ))^2/μ^2)))))

Converted by Mathematica  (July 10, 2003)