•Mass and wave function renormalization

zpion = CheckF[dum, "ChPT3P20o2.Fac"] // Renormalize // Simplify

(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

zkaon = CheckF[dum, "ChPT3P60o2.Fac"] // Renormalize // Simplify

1 - ((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))/(64 π^2 (f _ ϕ^(ó  ))^2)

mpionfac = CheckF[dum, "ChPT3P20o2.Mass"] /. toEtaRules

((-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)

mkaonfac = CheckF[dum, "ChPT3P60o2.Mass"] /. toEtaRules

(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)

metafac = CheckF[dum, "ChPT3P110o2.Mass"] /. toEtaRules

1/(864 π^2 (f _ ϕ^(ó  ))^2) (36864 π^2 L _ 7^(r ) (m _ π^(ó  ))^4 + 13824 π^2 L _ 8^(r ) (m _ π^(ó  ))^4 - 27 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - 7 log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 - 73728 π^2 L _ 7^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 36864 π^2 L _ 8^(r ) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 44 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 36864 π^2 L _ 7^(r ) (m _ K^(ó  ))^4 + 36864 π^2 L _ 8^(r ) (m _ K^(ó  ))^4 + 72 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 64 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 6912 π^2 L _ 5^(r ) (m _ η^(ó  ))^4 + 2304 π^2 L _ 4^(r ) ((m _ π^(ó  ))^4 - 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4) - 4608 π^2 L _ 6^(r ) ((m _ π^(ó  ))^4 - 2 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 8 (m _ K^(ó  ))^4))

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:

ampl2mult = {(2 - zkaon) restree[[1]], (2 - zpion) restree[[2]], (2 - zpion + 2 - zkaon - 1) restree[[3]]} // Cancel // Simplify ;

Converted by Mathematica  (July 10, 2003)