•The eta

ff2 = amp2Eta /. Momentum[p2] -> -Momentum[p1]

p _ 1^2 - (m _ η^(ó  ))^2

ampwf4 = (amp4Eta /. Momentum[p2] -> -Momentum[p1] // Simplify)

1/(3 (f _ ϕ^(ó  ))^2) (4 (-9 (2 L _ 6^( ) + 3 L _ 8^( )) (m _ η^(ó  ))^4 + 6 (-3 L _ 6^( ) (m _ π^(ó  ))^2 + L _ 5^( ) p _ 1^2 + L _ 8^( ) ((m _ π^(ó  ))^2 + 4 (m _ K^(ó  ))^2)) (m _ η^(ó  ))^2 - 18 L _ 7^( ) ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^2)^2 - L _ 8^( ) (7 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^4) + 9 L _ 4^( ) p _ 1^2 ((m _ π^(ó  ))^2 + (m _ η^(ó  ))^2)))

amploop = ampinfinitiesEta // Simplify

1/(384 π^2 (f _ ϕ^(ó  ))^2) (3 (96 π^2 λ + 4 log((m _ π^(ó  ))^2/μ^2) - log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^4 - 2 ((-128 π^2 λ + log((m _ K^(ó  ))^2/μ^2) - 5 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ K^(ó  ))^2 + 2 (128 π^2 λ + log((m _ η^(ó  ))^2/μ^2) + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ η^(ó  ))^2) (m _ π^(ó  ))^2 - 512 π^2 λ (m _ K^(ó  ))^4 - 8 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 - 8 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^4 + 224 π^2 λ (m _ η^(ó  ))^4 + 16 log((m _ η^(ó  ))^2/μ^2) (m _ η^(ó  ))^4 - 9 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ η^(ó  ))^4 + 768 π^2 λ (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + 6 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 + 18 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2 - 6 p _ 1^2 ((32 π^2 λ + log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 - 2 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^2 + 96 π^2 λ (m _ η^(ó  ))^2 + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ η^(ó  ))^2))

ff4 = ff2 + amploop + ampwf4 // Expand // 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])])

-((9216 π^2 L _ 7^( ) + 3584 π^2 L _ 8^( ) - 288 π^2 λ - 12 log((m _ π^(ó  ))^2/μ^2) + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^2) + (((32 L _ 8^( ))/3 + (2 λ)/3 - log((m _ K^(ó  ))^2/μ^2)/(192 π^2) + (5 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)))/(192 π^2)) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2 + ((-24 L _ 6^( ) + 48 L _ 7^( ) + 8 L _ 8^( ) - (4 λ)/3 - log((m _ η^(ó  ))^2/μ^2)/(96 π^2) - log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))/(32 π^2)) (m _ η^(ó  ))^2 (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2 - ((8192 π^2 L _ 8^( ) + 512 π^2 λ + 8 log((m _ K^(ó  ))^2/μ^2) + 8 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ K^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^2) - ((9216 π^2 L _ 6^( ) + 9216 π^2 L _ 7^( ) + 13824 π^2 L _ 8^( ) - 224 π^2 λ - 16 log((m _ η^(ó  ))^2/μ^2) + 9 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ η^(ó  ))^4)/(384 π^2 (f _ ϕ^(ó  ))^2) + ((32 L _ 8^( ) + 2 λ + log((m _ K^(ó  ))^2/μ^2)/(64 π^2) + (3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)))/(64 π^2)) (m _ K^(ó  ))^2 (m _ η^(ó  ))^2)/(f _ ϕ^(ó  ))^2 - (m _ η^(ó  ))^2 + p _ 1^2 (1/(f _ ϕ^(ó  ))^2 (-((-4608 π^2 L _ 4^( ) + 192 π^2 λ + 6 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^2)/(384 π^2) - ((12 log((m _ K^(ó  ))^2/μ^2) - 12 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ K^(ó  ))^2)/(384 π^2) - ((-4608 π^2 L _ 4^( ) - 3072 π^2 L _ 5^( ) + 576 π^2 λ + 18 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ η^(ó  ))^2)/(384 π^2)) + 1)

ff4 - tt // Simplify

0

z = Collect[Coefficient[ff4, Pair[Momentum[p1], Momentum[p1]]], {_DecayConstant, _ParticleMass}] /. log -> Log /. gellmannOkubo // FullSimplify

((32 π^2 (24 L _ 4^( ) + 16 L _ 5^( ) - 3 λ) - 3 log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2 + 16 π^2 (3 (f _ ϕ^(ó  ))^2 + 8 (3 L _ 4^( ) - L _ 5^( )) (m _ π^(ó  ))^2))/(48 π^2 (f _ ϕ^(ó  ))^2)

CheckF[z, "ChPT3P110o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P110o2.Fac

File does not exist, evaluating

Saving

CMEta = -ff4 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[EtaMeson, RenormalizationState[0]]^2 /. udrules // FullSimplify

1/(384 π^2 (f _ ϕ^(ó  ))^2) (-288 π^2 λ (m _ π^(ó  ))^4 - 12 log((m _ π^(ó  ))^2/μ^2) (m _ π^(ó  ))^4 + 3 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ π^(ó  ))^4 - 256 π^2 λ (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 2 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 - 10 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 512 π^2 λ (m _ K^(ó  ))^4 + 8 log((m _ K^(ó  ))^2/μ^2) (m _ K^(ó  ))^4 + 8 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2)) (m _ K^(ó  ))^4 + (32 π^2 (11 λ - 48 (3 L _ 4^( ) + 2 L _ 5^( ) - 6 L _ 6^( ))) - 16 log((m _ η^(ó  ))^2/μ^2) + 27 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ η^(ó  ))^4 + 2 ((32 π^2 (-72 L _ 4^( ) + 144 L _ 6^( ) + 11 λ) + 2 log((m _ η^(ó  ))^2/μ^2) + 9 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ π^(ó  ))^2 + 3 (-128 π^2 λ + log((m _ K^(ó  ))^2/μ^2) - 5 log(((m _ π^(ó  ))^2 - 2 (m _ K^(ó  ))^2 + 3 (m _ η^(ó  ))^2)/(2 μ^2))) (m _ K^(ó  ))^2) (m _ η^(ó  ))^2 + 9216 π^2 L _ 7^( ) ((m _ π^(ó  ))^2 - (m _ η^(ó  ))^2)^2 + 512 π^2 L _ 8^( ) (7 (m _ π^(ó  ))^4 - 8 (m _ K^(ó  ))^2 (m _ π^(ó  ))^2 + 16 (m _ K^(ó  ))^4 + 27 (m _ η^(ó  ))^4 - 6 ((m _ π^(ó  ))^2 + 4 (m _ K^(ó  ))^2) (m _ η^(ó  ))^2))

CMrenEta = ((Renormalize[CMEta] // Simplify) /. gellmannOkubo // Simplify) /. log -> Log /. toEtaRules // FullSimplify

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))

CheckF[CMrenEta, "ChPT3P110o2.Mass", ForceSave -> True] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P110o2.Mass

File does not exist, evaluating

Saving

Coefficient[CMrenEta, LeutwylerLambda[]] // Simplify

0

Converted by Mathematica  (July 10, 2003)