•Reduction of the mass and wave function renormalized tree amplitude

We divide off the kaon propagator and put the kaon on-mass-shell:

amp2 = Cancel /@ ((# * (Pair[Momentum[p1], Momentum[p1]] - (ParticleMass[Kaon, RenormalizationState[0]])^2)) & /@ ampl2mult) /. MandelstamRules /. cancelU /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[1]]^2 // Simplify

{1/(384 π^2 (f _ ϕ^(ó  ))^4) (i c _ 2^( ) (p _ 3^μ _ 1 + p _ 4^μ _ 1) µ _ μ _ 1(p _ 1) (384 π^2 (f _ ϕ^(ó  ))^2 - 1024 π^2 λ (m _ π^(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 - 512 π^2 λ (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)) ((m _ K^(ó  ))^2 - (m _ K^(ó  r ))^2)), 1/(2304 π^2 (f _ ϕ^(ó  ))^4) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (8 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 3 c _ 2^( ) (-4 (m _ π^(ó  r ))^2 + (m _ K^(ó  r ))^2 + 3 s - p _ 2^2)) (-384 π^2 (f _ ϕ^(ó  ))^2 + (1120 π^2 λ + 38 log((m _ π^(ó  ))^2/μ^2) - 3 log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 1280 π^2 λ (m _ K^(ó  ))^2 + 28 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 12 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 96 π^2 λ (m _ η^(ó  ))^2 - 3 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2)), 1/(1728 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i (2 p _ 2^μ _ 1 - p _ 3^μ _ 1 - p _ 4^μ _ 1) µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 - (m _ K^(ó  r ))^2) (c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-1152 π^2 (f _ ϕ^(ó  ))^2 + 4672 π^2 λ (m _ π^(ó  ))^2 + 141 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 13 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 4608 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 + 8960 π^2 λ (m _ K^(ó  ))^2 + 138 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 52 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 576 π^2 λ (m _ η^(ó  ))^2 - 18 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)) - 2 c _ 2^( ) (1152 π^2 N _ 11^(r ) (m _ π^(ó  ))^4 + 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1152 π^2 N _ 11^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 960 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^4 - 2304 π^2 N _ 11^(r ) (m _ K^(ó  ))^4 - 960 π^2 λ (m _ K^(ó  ))^4 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^(r ) p _ 2^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 960 π^2 λ p _ 2^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2)))), -1/(3456 π^2 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (-2 (m _ π^(ó  ))^2 + 2 (m _ π^(ó  r ))^2 - 2 (m _ K^(ó  ))^2 + (m _ K^(ó  r ))^2 - 3 s + p _ 2^2) (c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-1152 π^2 (f _ ϕ^(ó  ))^2 + 4960 π^2 λ (m _ π^(ó  ))^2 + 132 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 13 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 9216 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 + 11264 π^2 λ (m _ K^(ó  ))^2 + 120 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 52 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 864 π^2 λ (m _ η^(ó  ))^2 - 27 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 + 9216 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) - 2 c _ 2^( ) (1152 π^2 N _ 11^(r ) (m _ π^(ó  ))^4 + 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 1152 π^2 N _ 11^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 960 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 8 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 2304 π^2 N _ 10^(r ) (m _ K^(ó  ))^4 - 2304 π^2 N _ 11^(r ) (m _ K^(ó  ))^4 - 960 π^2 λ (m _ K^(ó  ))^4 - 18 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 16 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 1152 π^2 N _ 21^(r ) p _ 2^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) - 960 π^2 λ p _ 2^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2))))}

This makes the 1st and 3rd amplitudes vanish:

amp2 /. RenormalizationState[1] -> RenormalizationState[0] /. _Log -> 0 // Simplify

{0, (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) (8 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) + 3 c _ 2^( ) (-4 (m _ π^(ó  ))^2 + (m _ K^(ó  ))^2 + 3 s - p _ 2^2)) (λ (35 (m _ π^(ó  ))^2 + 40 (m _ K^(ó  ))^2 - 3 (m _ η^(ó  ))^2) - 12 (f _ ϕ^(ó  ))^2))/(72 (f _ ϕ^(ó  ))^4), 0, -1/(108 (f _ ϕ^(ó  ))^4 (p _ 2^2 - (m _ K^(ó  ))^2)) (i p _ 1^μ _ 1 µ _ μ _ 1(p _ 1) ((m _ K^(ó  ))^2 + 3 s - p _ 2^2) ((m _ K^(ó  ))^2 - (m _ π^(ó  ))^2) (c _ 5^( ) (-36 (f _ ϕ^(ó  ))^2 + 155 λ (m _ π^(ó  ))^2 + 288 L _ 5^( ) (m _ K^(ó  ))^2 + 352 λ (m _ K^(ó  ))^2 - 27 λ (m _ η^(ó  ))^2 + 288 L _ 4^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) - 12 c _ 2^( ) (6 N _ 11^( ) (m _ π^(ó  ))^2 + 12 N _ 10^( ) (m _ K^(ó  ))^2 + 12 N _ 11^( ) (m _ K^(ó  ))^2 + 5 λ (m _ K^(ó  ))^2 - 6 N _ 21^( ) p _ 2^2 - 5 λ p _ 2^2)))}

The amplitudes are rexpressed in terms of renormalized masses and simplified:

end2all = CheckF[DiscardOrders[(amp2[[2]] /. Log[x_] :> Log[x /. {ParticleMass -> pm, Pair -> pa}] /. massrenormalization) + (amp2[[4]] /. Log[x_] :> Log[x /. {ParticleMass -> pm, Pair -> pa}] /. massrenormalization) /. Log[x_] :> Log[x /. {ParticleMass -> pm, Pair -> pa}] /. ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]], PerturbationOrder -> 6] /. gellmannOkubo /. {pm -> ParticleMass, pa -> Pair} // Simplify, "KSPiPiend2allS"] ;

Converted by Mathematica  (July 10, 2003)