•The renormalized second order tree amplitude

•Generation of topologies and insertion of fields

•Calculation of the amplitude

•Mass and wave function renormalization, etc.

Various previously calculated and saved factors to multiply on the leading order amplitude:

zpion = CheckF[dum, ToFileName[{$FeynCalcDirectory, "Phi", "Factors"}, "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 - 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))

fpionfac = CheckF[dum, "ChPT3A00P20o2.Fac"]

(128 π^2 L _ 5^(r ) (m _ π^(ó  ))^2 - 2 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 - log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 128 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))/(32 π^2 (f _ ϕ^(ó  ))^2) + 1

fkaonfac = CheckF[dum, "ChPT3A00P60o2.Fac"]

(-3 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + log((m _ η^(ó  ))^2/μ^2) (m _ π^(ó  ))^2 + 512 π^2 L _ 5^(r ) (m _ K^(ó  ))^2 - 6 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 4 log((m _ η^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 + 512 π^2 L _ 4^(r ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))/(128 π^2 (f _ ϕ^(ó  ))^2) + 1

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"]

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))

scalarrenfac = CheckF[dum, "ChPTW3P70S10o2.Fac"] /. toEtaRules /. p1 -> p2

1/(576 π^2 c _ 5^( ) (f _ ϕ^(ó  ))^2 ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^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)) - 2 c _ 5^( ) ((m _ π^(ó  ))^2 - (m _ K^(ó  ))^2) (-288 π^2 (f _ ϕ^(ó  ))^2 + (256 π^2 λ + 9 log((m _ π^(ó  ))^2/μ^2) - log((m _ η^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (352 π^2 λ + 9 log((m _ K^(ó  ))^2/μ^2) + 2 log((m _ η^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))

Changing from unrenormalized to renormalized masses:

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, (Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Kaon, RenormalizationState[0]]^2)^(-1) -> (Pair[Momentum[p2], Momentum[p2]] - ParticleMass[Kaon, RenormalizationState[1]]^2)^(-1), (ParticleMass[Kaon, RenormalizationState[0]]^2 - Pair[Momentum[p2], Momentum[p2]])^(-1) -> (ParticleMass[Kaon, RenormalizationState[1]]^2 - Pair[Momentum[p2], Momentum[p2]])^(-1)} ;

The leading order amplitude with the above corrections multiplied on:

ampl2mult = {(2 * ((3 - zpion)/2) + fkaonfac + 2 * fpionfac - 4) restree[[1]], (2 * ((3 - zpion)/2) + ((3 - zkaon)/2) + fkaonfac + 2 * fpionfac - 5) restree[[2]], (2 * ((3 - zpion)/2) + 2 * ((3 - zkaon)/2) + fkaonfac + 2 * fpionfac + scalarrenfac - 1 - 6) restree[[3]], (2 * ((3 - zpion)/2) + 3 * ((3 - zkaon)/2) + fkaonfac + 2 * fpionfac + scalarrenfac - 1 - 7) restree[[4]]} // Cancel // Simplify ;

Converted by Mathematica  (July 10, 2003)