•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:

amptree1

-i f _ π^(ó  ) p _ 2^μ _ 1 δ _ (I _ 1 I _ 2)

amptree3

-(8 i (2 L _ 4^( ) + L _ 5^( )) p _ 2^μ _ 1 (m _ π^(ó  ))^2 δ _ (I _ 1 I _ 2))/f _ π^(ó  )

ampinfinities

-(i (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) p _ 1^μ _ 1 (m _ π^(ó  ))^2 δ _ (i _ 1 i _ 2)^(2))/(12 π^2 f _ π^(ó  ))

The first order tree amplitude is wave function renormalized:

zpion = CheckF[dum, "ChPT2P20o2.Fac"]

((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

amp1 = amptree1 * (1 + (2 - zpion))/2 /. Momentum[p2] -> -Momentum[p1] /. SUNDelta[__] -> 1 // Simplify

1/2 i f _ π^(ó  ) p _ 1^μ _ 1 (2 - ((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))

amp3 = amptree3 /. Momentum[p2] -> -Momentum[p1] /. SUNDelta[__] -> 1

(8 i (2 L _ 4^( ) + L _ 5^( )) p _ 1^μ _ 1 (m _ π^(ó  ))^2)/f _ π^(ó  )

amploop = ampinfinities /. SU2Delta[__] -> 1 // Simplify

-(i (32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) p _ 1^μ _ 1 (m _ π^(ó  ))^2)/(12 π^2 f _ π^(ó  ))

The full unrenormalized amplitude (to third order in the energy):

ff0 = amp1 + amp3 + amploop

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

After renormalization of the coupling constants of the counterterm lagrangian, the infinite λ-terms drop out:

Renormalize[amp1 + amp3 + amploop] // Simplify

(i p _ 1^μ _ 1 (16 π^2 (f _ π^(ó  ))^2 + (128 π^2 L _ 4^(r ) + 64 π^2 L _ 5^(r ) - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2))/(16 π^2 f _ π^(ó  ))

The coefficient c of f _ π is then the renormalization factor relating the unrenormalized f _ π^0 to the renormalized f _ π = c f _ π^0:

c = Collect[Coefficient[ff0/I/DecayConstant[PseudoScalar[2], RenormalizationState[0]], Pair[LorentzIndex[μ1], Momentum[p1]]], _DecayConstant] // FullSimplify

((32 π^2 (4 L _ 4^( ) + 2 L _ 5^( ) - λ) - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(16 π^2 (f _ π^(ó  ))^2) + 1

$VeryVerbose = 2 ;

CheckF[c, "ChPT2A00P20o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPT2A00P20o2.Fac

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

The numerical value of f _ π:

cren = c // Renormalize // Simplify

(16 π^2 (f _ π^(ó  ))^2 + (128 π^2 L _ 4^(r ) + 64 π^2 L _ 5^(r ) - log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(16 π^2 (f _ π^(ó  ))^2)

cren /. {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), ScaleMu -> 548.8}

1.0682485433808555`

Converted by Mathematica  (July 10, 2003)