•The pion

FeynCalc counts all fields as incoming, so we substitute p _ 2->-p _ 1:

ff2 = amp2Pion /. Momentum[p2] -> -Momentum[p1]

p _ 1^2 - (m _ π^(ó  ))^2

ampwf4 = (amp4Pion /. Momentum[p2] -> -Momentum[p1] // Simplify)

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

amploop = ampinfinitiesPion // Simplify

1/(96 π^2 (f _ ϕ^(ó  ))^2) ((m _ π^(ó  ))^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) - 2 p _ 1^2 (2 (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (32 π^2 λ + log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2))

The full amplitude (to fourth order) ff4 differs from the lowest order amplitude ff4 by a factor Z, ff4 = Z^(-1) ff2.  This is equivalent to a redefinition of the pion field, π _ r= Z^(-1/2)π.

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

We may beautify this a bit:

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])])

((-1536 π^2 L _ 6^( ) - 1536 π^2 L _ 8^( ) + 32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4)/(96 π^2 (f _ ϕ^(ó  ))^2) + ((-32 L _ 6^( ) + (2 λ)/3 + log((m _ K^(ó  ))^2/μ^2)/(48 π^2)) (m _ K^(ó  ))^2 (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2 + ((λ/3 + log((m _ η^(ó  ))^2/μ^2)/(96 π^2)) (m _ η^(ó  ))^2 (m _ π^(ó  ))^2)/(f _ ϕ^(ó  ))^2 - (m _ π^(ó  ))^2 + p _ 1^2 ((((768 π^2 L _ 4^( ) + 768 π^2 L _ 5^( ) - 128 π^2 λ - 4 log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(96 π^2) + ((1536 π^2 L _ 4^( ) - 64 π^2 λ - 2 log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)/(96 π^2))/(f _ ϕ^(ó  ))^2 + 1)

ff4 - tt // Simplify

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The factor Z can be directly read off the above expression as the coefficient of p _ 1^2.

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

(2 (32 π^2 (6 (L _ 4^( ) + L _ 5^( )) - λ) - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 + (32 π^2 (24 L _ 4^( ) - λ) - log((m _ K^(ó  ))^2/μ^2)) (m _ K^(ó  ))^2)/(48 π^2 (f _ ϕ^(ó  ))^2) + 1

$VeryVerbose = 1 ;

CheckF[z, "ChPT3P20o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P20o2.Fac

File does not exist, evaluating

Saving

Now we demand that ff4 be zero on the mass shell with  p^2=m _ (π, r)^2, where  m _ (π, r)^2= m _ π^2+Cm is the renormalized mass. Since we are only working to O(p^4), we only need Cm to first order in m _ π^2.

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

1/(288 π^2 (f _ ϕ^(ó  ))^2) ((64 π^2 (5 λ - 36 (L _ 4^( ) + L _ 5^( ) - 2 (L _ 6^( ) + L _ 8^( )))) + 9 log((m _ π^(ó  ))^2/μ^2) + log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ π^(ó  ))^4 - 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)

CMrenPion = Renormalize[CMPion] // FullSimplify

1/(288 π^2 (f _ ϕ^(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 _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ π^(ó  ))^4 - 4 (1152 π^2 (L _ 4^(r ) - 2 L _ 6^(r )) + log(-((m _ π^(ó  ))^2 - 4 (m _ K^(ó  ))^2)/(3 μ^2))) (m _ π^(ó  ))^2 (m _ K^(ó  ))^2)

CheckF[CMrenPion, "ChPT3P20o2.Mass"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT3P20o2.Mass

File does not exist, evaluating

Saving

Coefficient[CMrenPion, LeutwylerLambda[]] // Simplify

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Converted by Mathematica  (July 10, 2003)