•The kaon

ff2 = amp2Kaon /. Momentum[p2] -> -Momentum[p1]

p _ 1^2 - (m _ K^(ó  ))^2

ampwf4 = (amp4Kaon /. Momentum[p2] -> -Momentum[p1] // Simplify)

(8 ((L _ 5^( ) p _ 1^2 - 2 (L _ 8^( ) (m _ K^(ó  ))^2 + L _ 6^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))) (m _ K^(ó  ))^2 + L _ 4^( ) p _ 1^2 ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2)))/(f _ ϕ^(ó  ))^2

amploop = ampinfinitiesKaon // Simplify

-1/(192 π^2 (f _ ϕ^(ó  ))^2) (((32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2 - 3 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2) (m _ π^(ó  ))^2 + 3 p _ 1^2 ((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + 2 (32 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) - 3 (2 (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4 + (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2 (m _ K^(ó  ))^2 - (32 π^2 λ + log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^4))

ff4 = ff2 + amploop + ampwf4 // ExpandAll // Simplify ;

tt = (ff2 - Coefficient[ff2, Pair[Momentum[p1], Momentum[p1]]] Pair[Momentum[p1], Momentum[p1]]) + Collect[Coefficient[ff4, Pair[Momentum[p1], Momentum[p1]]], {_DecayConstant, _ParticleMass}] Pair[Momentum[p1], Momentum[p1]] + Plus @@ ((Collect[Coefficient[ff4, ParticleMass[#[[1]], RenormalizationState[0]]^2 ParticleMass[#[[2]], RenormalizationState[0]]^2], {_DecayConstant, _ParticleMass}] ParticleMass[#[[1]], RenormalizationState[0]]^2 ParticleMass[#[[2]], RenormalizationState[0]]^2) & /@ Union[Sort /@ (Flatten[Outer[List, {Pion, Kaon, EtaMeson}, {Pion, Kaon, EtaMeson}], 1])])

-((6144 π^2 L _ 6^( ) + 3072 π^2 L _ 8^( ) - 192 π^2 λ - 6 log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^4)/(192 π^2 (f _ ϕ^(ó  ))^2) + ((-16 L _ 6^( ) + λ/2 + log((m _ π^(ó  ))^2/μ^2)/(64 π^2)) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)/(f _ ϕ^(ó  ))^2 + ((λ/2 + log((m _ η^(ó  ))^2/μ^2)/(64 π^2)) (m _ η^(ó  ))^2 (m _ K^(ó  ))^2)/(f _ ϕ^(ó  ))^2 - (m _ K^(ó  ))^2 - ((96 π^2 λ + 3 log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^4)/(192 π^2 (f _ ϕ^(ó  ))^2) + ((-λ/6 - log((m _ η^(ó  ))^2/μ^2)/(192 π^2)) (m _ π^(ó  ))^2 (m _ η^(ó  ))^2)/(f _ ϕ^(ó  ))^2 + p _ 1^2 ((-((-1536 π^2 L _ 4^( ) + 96 π^2 λ + 3 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(192 π^2) - ((-3072 π^2 L _ 4^( ) - 1536 π^2 L _ 5^( ) + 192 π^2 λ + 6 log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)/(192 π^2) - ((96 π^2 λ + 3 log((m _ η^(ó  ))^2/μ^2)) (m _ η^(ó  ))^2)/(192 π^2))/(f _ ϕ^(ó  ))^2 + 1)

ff4 - tt // Simplify

0

z = Collect[Coefficient[ff4, Pair[Momentum[p1], Momentum[p1]]], {_DecayConstant, _ParticleMass}] // FullSimplify

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 (32 π^2 (λ - 8 L _ 5^( )) + 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 - 512 π^2 L _ 4^( ) ((m _ π^(ó  ))^2 + 2 (m _ K^(ó  ))^2))

CheckF[z, "ChPT3P60o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P60o2.Fac

File does not exist, evaluating

Saving

CMKaon = -ff4 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Kaon, RenormalizationState[0]]^2 /. fromEtaRules /. udrules // FullSimplify

1/(144 π^2 (f _ ϕ^(ó  ))^2) (4 (32 π^2 (λ - 9 (2 L _ 4^( ) + L _ 5^( ) - 2 (2 L _ 6^( ) + L _ 8^( )))) + log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ K^(ó  ))^4 - (32 π^2 (36 L _ 4^( ) - 72 L _ 6^( ) + λ) + log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)

CMrenKaon = Renormalize[CMKaon] /. toEtaRules // FullSimplify

(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)

CheckF[CMrenKaon, "ChPT3P60o2.Mass", ForceSave -> True] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P60o2.Mass

File does not exist, evaluating

Saving

Coefficient[CMrenKaon, LeutwylerLambda[]] // Simplify

0

Converted by Mathematica  (July 10, 2003)