•Renormalization

FeynCalc counts all fields as incoming, but we want p _ 1 and p _ 2 to be the same, so we substitute p _ 2->-p _ 1:

ff2 = amp2 /. Momentum[p2] -> -Momentum[p1]

p _ 1^2 - (m _ π^(ó  ))^2

amploop = ampinfinities /. Momentum[p3] -> -Momentum[p1] // Simplify

((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2 ((m _ π^(ó  ))^2 - 4 p _ 1^2))/(96 π^2 (f _ π^(ó  ))^2)

ampwf4 = amp4 /. Momentum[p2] -> -Momentum[p1] /. SUNDelta[__] -> 1 // Simplify

(8 (2 L _ 4^( ) + L _ 5^( )) p _ 1^2 (m _ π^(ó  ))^2 - 16 (2 L _ 6^( ) + L _ 8^( )) (m _ π^(ó  ))^4)/(f _ π^(ó  ))^2

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

ff4 = ff2 + amploop + ampwf4 // Expand // FullSimplify

1/(96 π^2 (f _ π^(ó  ))^2) ((32 π^2 (λ - 48 (2 L _ 6^( ) + L _ 8^( ))) + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^4 + 4 (32 π^2 (12 L _ 4^( ) + 6 L _ 5^( ) - λ) - log((m _ π^(ó  ))^2/μ^2)) p _ 1^2 (m _ π^(ó  ))^2 + 96 π^2 (f _ π^(ó  ))^2 (p _ 1^2 - (m _ π^(ó  ))^2))

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

zpion = Collect[Coefficient[ff4, Pair[Momentum[p1], Momentum[p1]]], {_DecayConstant, _ParticleMass}] // Simplify

((384 π^2 L _ 4^( ) + 192 π^2 L _ 5^( ) - 32 π^2 λ - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2) + 1

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 working to O(p^4), we only need Cm to first order in m _ π^2.

Cm = -ff4 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[PseudoScalar[2], RenormalizationState[0]]^2 // Renormalize // FullSimplify

((log((m _ π^(ó  ))^2/μ^2) - 256 π^2 (2 L _ 4^(r ) + L _ 5^(r ) - 2 (2 L _ 6^(r ) + L _ 8^(r )))) (m _ π^(ó  ))^4)/(32 π^2 (f _ π^(ó  ))^2)

Save results for later use:

$VeryVerbose = 2 ;

CheckF[zpion, "ChPT2P20o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT2P20o2.Fac

File does not exist, evaluating

Saving

CheckF[Cm, "ChPT2P20o2.Mass"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT2P20o2.Mass

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

Numerical value of the mass renormalization constant:

Cm/ParticleMass[Pion, RenormalizationState[0]]^2 /. {ParticleMass[Pion, RenormalizationState[0]] -> 140.97, DecayConstant[Pion, RenormalizationState[0]] -> 87.7, CouplingConstant[ChPT2[4], 4, RenormalizationState[1]] -> 0, CouplingConstant[ChPT2[4], 5, RenormalizationState[1]] -> 2.3 * 10^(-3), CouplingConstant[ChPT2[4], 6, RenormalizationState[1]] -> 0, CouplingConstant[ChPT2[4], 8, RenormalizationState[1]] -> 1.2 * 10^(-3), ScaleMu -> 548.8}

-0.02017190208012836`

Converted by Mathematica  (July 10, 2003)