KSS2Amplitude.nb

•Mass and wave function renormalization, etc.

(2 (64 π^2 (3 L _ 4^(r ) + 3 L _ 5^(r ) + λ) - log((m _ π^(ó ))^2/μ^2)) (m _ π^(ó ))^2 + (768 π^2 L _ 4^(r ) + 64 π^2 λ - log((m _ K^(ó ))^2/μ^2)) (m _ K^(ó ))^2)/(48 π^2 (f _ ϕ^(ó ))^2) + 1

1 - 1/(64 π^2 (f _ ϕ^(ó ))^2) ((32 π^2 λ + log((m _ π^(ó ))^2/μ^2)) (m _ π^(ó ))^2 + 2 (-256 π^2 L _ 5^(r ) - 64 π^2 λ + log((m _ K^(ó ))^2/μ^2)) (m _ K^(ó ))^2 + (32 π^2 λ + log((m _ η^(ó ))^2/μ^2)) (m _ η^(ó ))^2 - 64 π^2 (8 L _ 4^(r ) + λ) ((m _ π^(ó ))^2 + 2 (m _ K^(ó ))^2))

((-2304 π^2 (L _ 4^(r ) + L _ 5^(r ) - 2 (L _ 6^(r ) + L _ 8^(r ))) + 9 log((m _ π^(ó ))^2/μ^2) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^4 - 4 (1152 π^2 (L _ 4^(r ) - 2 L _ 6^(r )) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 (m _ K^(ó ))^2)/(288 π^2 (f _ ϕ^(ó ))^2)

(4 (log((m _ η^(ó ))^2/μ^2) - 288 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 2 (2 L _ 6^(r ) + L _ 8^(r )))) (m _ K^(ó ))^4 - (1152 π^2 (L _ 4^(r ) - 2 L _ 6^(r )) + log((m _ η^(ó ))^2/μ^2)) (m _ π^(ó ))^2 (m _ K^(ó ))^2)/(144 π^2 (f _ ϕ^(ó ))^2)

$massrenormalization = {ParticleMass[Pion, RenormalizationState[0]]^2 -> (ParticleMass[Pion, RenormalizationState[1]]^2 - mpionfac), ParticleMass[Kaon, RenormalizationState[0]]^2 -> (ParticleMass[Kaon, RenormalizationState[1]]^2 - mkaonfac), ParticleMass[Kaon, RenormalizationState[0]]^(-2) -> (1 + mkaonfac/ParticleMass[Kaon, RenormalizationState[1]]^2) * ParticleMass[Kaon, RenormalizationState[1]]^(-2), ParticleMass[EtaMeson, RenormalizationState[0]]^2 -> (ParticleMass[EtaMeson, RenormalizationState[1]]^2 - metafac), ParticleMass[Pion, RenormalizationState[0]]^4 -> (ParticleMass[Pion, RenormalizationState[1]]^2 - mpionfac)^2, ParticleMass[Kaon, RenormalizationState[0]]^4 -> (ParticleMass[Kaon, RenormalizationState[1]]^2 - mkaonfac)^2, ParticleMass[EtaMeson, RenormalizationState[0]]^4 -> (ParticleMass[EtaMeson, RenormalizationState[1]]^2 - metafac)^2} ;$

The leading order amplitude with the above corrections multiplied on:

Change to Mandelstam variables:

Check that the leading order terms are still ok:

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

${-(4 c _ 5^( ) !, _ 0^( ) (δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)))/f _ ϕ^(ó ), (4 c _ 5^( ) ((m _ K^(ó ))^2 - (m _ π^(ó ))^2) !, _ 0^( ) (6 δ _ (0 i)^(3) - 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)))/(3 f _ ϕ^(ó ) (m _ K^(ó ))^2)}$

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

$end2all = CheckF[{amp2[[1]], (DiscardOrders[(Numerator[amp2[[2]]] /. Log[x_] :> Log[x /. ParticleMass -> pm] /. massrenormalization) /. ParticleMass[p_, RenormalizationState[0]] -> ParticleMass[p, RenormalizationState[1]] /. pm -> ParticleMass, PerturbationOrder -> 4]/Denominator[amp2[[2]]])} /. gellmannOkubo /. toEtaRules // Together // Simplify, "KSS2end2all"]$

${-1/(32 π^2 (f _ ϕ^(ó ))^3) (c _ 5^( ) (128 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^( ) (δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3))), 1/(864 π^2 (f _ ϕ^(ó ))^3 (p _ 2^2 - (m _ K^(ó ))^2)) ((2 c _ 2^( ) (1152 π^2 N _ 11^(r ) (m _ π^(ó r ))^4 + 27 log((m _ π^(ó ))^2/μ^2) (m _ π^(ó r ))^4 - log((m _ η^(ó ))^2/μ^2) (m _ π^(ó r ))^4 + 2304 π^2 N _ 10^(r ) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 1152 π^2 N _ 11^(r ) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 960 π^2 λ (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 8 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 2304 π^2 N _ 10^(r ) (m _ K^(ó r ))^4 - 2304 π^2 N _ 11^(r ) (m _ K^(ó r ))^4 - 960 π^2 λ (m _ K^(ó r ))^4 - 18 log((m _ K^(ó ))^2/μ^2) (m _ K^(ó r ))^4 - 16 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó r ))^4 - 1152 π^2 N _ 21^(r ) p _ 2^2 ((m _ π^(ó r ))^2 - (m _ K^(ó r ))^2) - 960 π^2 λ p _ 2^2 ((m _ π^(ó r ))^2 - (m _ K^(ó r ))^2)) + c _ 5^( ) (9216 π^2 L _ 5^(r ) (m _ π^(ó r ))^4 - 18432 π^2 L _ 6^(r ) (m _ π^(ó r ))^4 - 18432 π^2 L _ 8^(r ) (m _ π^(ó r ))^4 - 1888 π^2 λ (m _ π^(ó r ))^4 - 9 log((m _ π^(ó ))^2/μ^2) (m _ π^(ó r ))^4 - 36 log((m _ π^(ó r ))^2/μ^2) (m _ π^(ó r ))^4 + 4 log((m _ η^(ó ))^2/μ^2) (m _ π^(ó r ))^4 - 4 log((m _ η^(ó r ))^2/μ^2) (m _ π^(ó r ))^4 - 13824 π^2 L _ 5^(r ) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 18432 π^2 L _ 6^(r ) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 7840 π^2 λ (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 9 log((m _ π^(ó ))^2/μ^2) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 18 log((m _ K^(ó ))^2/μ^2) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 20 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 8 log((m _ η^(ó r ))^2/μ^2) (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 + 864 π^2 λ (m _ η^(ó r ))^2 (m _ π^(ó r ))^2 + 27 log((m _ η^(ó ))^2/μ^2) (m _ η^(ó r ))^2 (m _ π^(ó r ))^2 + 4608 π^2 L _ 5^(r ) (m _ K^(ó r ))^4 + 36864 π^2 L _ 6^(r ) (m _ K^(ó r ))^4 + 18432 π^2 L _ 8^(r ) (m _ K^(ó r ))^4 + 9728 π^2 λ (m _ K^(ó r ))^4 + 18 log((m _ K^(ó ))^2/μ^2) (m _ K^(ó r ))^4 + 16 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó r ))^4 + 32 log((m _ η^(ó r ))^2/μ^2) (m _ K^(ó r ))^4 - 864 π^2 λ (m _ K^(ó r ))^2 (m _ η^(ó r ))^2 - 27 log((m _ η^(ó ))^2/μ^2) (m _ K^(ó r ))^2 (m _ η^(ó r ))^2 + 1152 π^2 (f _ ϕ^(ó ))^2 ((m _ π^(ó r ))^2 - (m _ K^(ó r ))^2) - 4608 π^2 L _ 4^(r ) ((m _ π^(ó r ))^4 + (m _ K^(ó r ))^2 (m _ π^(ó r ))^2 - 2 (m _ K^(ó r ))^4))) !, _ 0^( ) (6 δ _ (0 i)^(3) - 3 δ _ (3 i)^(3) - 3^(1/2) δ _ (8 i)^(3)))}$

Converted by Mathematica (July 10, 2003)