•Reduction of the mass and wave function renormalized tree amplitude

Change to Mandelstam variables:

amp2 = ampl2mult /. MandelstamRules // Simplify

{1/(8 π^2 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)) (c _ 5^( ) (64 π^2 (f _ ϕ^(ó  ))^2 - 32 π^2 λ (m _ π^(ó  ))^2 + log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 512 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 256 π^2 λ (m _ K^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 32 π^2 λ (m _ η^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 - 512 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2), (c _ 5^( ) (48 π^2 (f _ ϕ^(ó  ))^2 - 384 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 - 128 π^2 λ (m _ π^(ó  ))^2 + 2 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 64 π^2 λ (m _ K^(ó  ))^2 + log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 384 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2)/(6 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), ((c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) - 2 c _ 5^( ) (m _ K^(ó  ))^2) (-192 π^2 (f _ ϕ^(ó  ))^2 + 1536 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 + 608 π^2 λ (m _ π^(ó  ))^2 - 11 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 1536 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 + 1024 π^2 λ (m _ K^(ó  ))^2 - 10 log((m _ K^(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 + 3072 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2)/(48 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))}

Check that the leading order terms are still ok:

amp2 /. {_LeutwylerLambda -> 0, CouplingConstant[_[4], ___] -> 0, _Log -> 0, RenormalizationState[1] -> RenormalizationState[0], p3 -> -p1, Pair[Momentum[p2], Momentum[p2]] -> 0} // Together // Simplify

{(8 c _ 5^( ) (!, _ 0^( ))^2)/(p _ 1^2 - (m _ K^(ó  ))^2), (8 c _ 5^( ) (!, _ 0^( ))^2)/(p _ 1^2 - (m _ π^(ó  ))^2), -(8 (c _ 2^( ) p _ 1^2 - c _ 5^( ) (m _ K^(ó  ))^2) (!, _ 0^( ))^2)/((p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))}

Plus @@ % // Together // Simplify

-(8 (c _ 2^( ) p _ 1^2 + c _ 5^( ) ((m _ π^(ó  ))^2 - 2 p _ 1^2)) (!, _ 0^( ))^2)/((p _ 1^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))

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

end2all = CheckF[{amp2[[1]], amp2[[2]], (DiscardOrders[(Numerator[amp2[[3]]] /. Log[x_] :> Log[x /. ParticleMass -> pm] /. massrenormalization) /. ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]] /. pm -> ParticleMass, PerturbationOrder -> 4]/Denominator[amp2[[3]]])} /. gellmannOkubo /. toEtaRules // Together // Simplify, "KSPiend2all"]

{1/(8 π^2 (f _ ϕ^(ó  ))^2 (p _ 1^2 - (m _ K^(ó  ))^2)) (c _ 5^( ) (64 π^2 (f _ ϕ^(ó  ))^2 - 32 π^2 λ (m _ π^(ó  ))^2 + log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 512 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 256 π^2 λ (m _ K^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 32 π^2 λ (m _ η^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^2 - 512 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2), (c _ 5^( ) (48 π^2 (f _ ϕ^(ó  ))^2 - 384 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 - 128 π^2 λ (m _ π^(ó  ))^2 + 2 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - 64 π^2 λ (m _ K^(ó  ))^2 + log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 384 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) (!, _ 0^( ))^2)/(6 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2)), -((2 c _ 5^( ) (-576 π^2 (f _ ϕ^(ó  ))^2 + 4608 π^2 L _ 5^(r ) (m _ π^(ó  r ))^2 + 9216 π^2 L _ 6^(r ) (m _ π^(ó  r ))^2 + 1824 π^2 λ (m _ π^(ó  r ))^2 - 33 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  r ))^2 - 4 log((m _ η^(ó  r ))^2/μ^2) (m _ π^(ó  r ))^2 + 18432 π^2 L _ 6^(r ) (m _ K^(ó  r ))^2 + 9216 π^2 L _ 8^(r ) (m _ K^(ó  r ))^2 + 3072 π^2 λ (m _ K^(ó  r ))^2 - 30 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  r ))^2 + 16 log((m _ η^(ó  r ))^2/μ^2) (m _ K^(ó  r ))^2 - 288 π^2 λ (m _ η^(ó  r ))^2 - 9 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  r ))^2 + 4608 π^2 L _ 4^(r ) ((m _ π^(ó  r ))^2 + 2 (m _ K^(ó  r ))^2)) (m _ K^(ó  r ))^2 + 3 c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) (192 π^2 (f _ ϕ^(ó  ))^2 - 1536 π^2 L _ 5^(r ) (m _ π^(ó  r ))^2 - 608 π^2 λ (m _ π^(ó  r ))^2 + 11 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  r ))^2 - 1536 π^2 L _ 5^(r ) (m _ K^(ó  r ))^2 - 1024 π^2 λ (m _ K^(ó  r ))^2 + 10 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  r ))^2 + 96 π^2 λ (m _ η^(ó  r ))^2 + 3 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  r ))^2 - 3072 π^2 L _ 4^(r ) ((m _ π^(ó  r ))^2 + 2 (m _ K^(ó  r ))^2))) (!, _ 0^( ))^2)/(144 π^2 (f _ ϕ^(ó  ))^2 (p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2))}

end2 = Plus @@ end2all /. _Log -> 0 // Simplify

1/(3 (f _ ϕ^(ó  ))^2) (2 (-(4 c _ 5^( ) (4 (6 L _ 5^(r ) (m _ π^(ó  ))^2 + 2 λ (m _ π^(ó  ))^2 + λ (m _ K^(ó  ))^2 + 6 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)) - 3 (f _ ϕ^(ó  ))^2))/(p _ 3^2 - (m _ π^(ó  ))^2) - (6 c _ 5^( ) (-2 (f _ ϕ^(ó  ))^2 + λ (m _ π^(ó  ))^2 + 16 L _ 5^(r ) (m _ K^(ó  ))^2 + 8 λ (m _ K^(ó  ))^2 - λ (m _ η^(ó  ))^2 + 16 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(p _ 1^2 - (m _ K^(ó  ))^2) - 1/((p _ 3^2 - (m _ π^(ó  ))^2) (p _ 1^2 - (m _ K^(ó  ))^2)) (2 c _ 5^( ) (-6 (f _ ϕ^(ó  ))^2 + 48 L _ 5^(r ) (m _ π^(ó  r ))^2 + 96 L _ 6^(r ) (m _ π^(ó  r ))^2 + 19 λ (m _ π^(ó  r ))^2 + 192 L _ 6^(r ) (m _ K^(ó  r ))^2 + 96 L _ 8^(r ) (m _ K^(ó  r ))^2 + 32 λ (m _ K^(ó  r ))^2 - 3 λ (m _ η^(ó  r ))^2 + 48 L _ 4^(r ) ((m _ π^(ó  r ))^2 + 2 (m _ K^(ó  r ))^2)) (m _ K^(ó  r ))^2 + c _ 2^( ) (p _ 1^2 - p _ 2^2 + p _ 3^2) (6 (f _ ϕ^(ó  ))^2 - 48 L _ 5^(r ) (m _ π^(ó  r ))^2 - 19 λ (m _ π^(ó  r ))^2 - 48 L _ 5^(r ) (m _ K^(ó  r ))^2 - 32 λ (m _ K^(ó  r ))^2 + 3 λ (m _ η^(ó  r ))^2 - 96 L _ 4^(r ) ((m _ π^(ó  r ))^2 + 2 (m _ K^(ó  r ))^2)))) (!, _ 0^( ))^2)

Converted by Mathematica  (July 10, 2003)