•Renormalization

The beta functions:

ampct = ampf4 /. p2 -> -p1 // Simplify

-20/9 (e^( ))^2 (l _ 5^( ) - 2 h _ 2^( ) + k _ 15^( )) p _ 1^2

ff2 = ampf2 /. p2 -> -p1 // Simplify

p _ 1^2

amploop = (Plus @@ ampsimple /. p3 -> -p1) //. Times[f__, SumOver[i_]] :> Sum[Times[f], {i, 1, 3}] // IsoToChargedMasses // SUNReduce[#, FullReduce -> True] & // VeltmanExpand[#, ExplicitLeutwylerJ0 -> True] & // Expand

-4/3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2 (e^( ))^2 + 1/3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) p _ 1^2 (e^( ))^2 - 2/3 λ p _ 1^2 (e^( ))^2 - (log((m _ π^+^(ó  ))^2/μ^2) p _ 1^2 (e^( ))^2)/(48 π^2) - (p _ 1^2 (e^( ))^2)/(144 π^2)

Limit[amploop, Pair[Momentum[p1], Momentum[p1]] -> 0] // Simplify

0

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

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

-1/(144 π^2) (192 π^2 (e^( ))^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2 + ((e^( ))^2 (16 π^2 (20 l _ 5^( ) - 40 h _ 2^( ) + 20 k _ 15^( ) - 3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) + 6 λ) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 1) - 144 π^2) p _ 1^2)

This is the photon self-energy contribution to the charge renormalization (e _ 0^2A in Mandl & Shaw p. 185).

SetOptions[LeutwylerJBar, LeutwylerJBarEvaluation -> "subthreshold", ExplicitLeutwylerSigma -> True] ;

Limit[(amploop + ampct // Renormalize)/Pair[Momentum[p1], Momentum[p1]], Pair[Momentum[p1], Momentum[p1]] -> 0] // Simplify

-((e^( ))^2 (320 π^2 l _ 5^(r ) - 640 π^2 h _ 2^(r ) + 320 π^2 k _ 15^(r ) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 3))/(144 π^2)

SetOptions[LeutwylerJBar, LeutwylerJBarEvaluation -> "none", ExplicitLeutwylerSigma -> False] ;

The factor Z can be directly read off the above expression.

z = ff4/ff2

-1/(144 π^2 p _ 1^2) (192 π^2 (e^( ))^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2 + ((e^( ))^2 (16 π^2 (20 l _ 5^( ) - 40 h _ 2^( ) + 20 k _ 15^( ) - 3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) + 6 λ) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 1) - 144 π^2) p _ 1^2)

Mass shell limit:

SetOptions[LeutwylerJBar, LeutwylerJBarEvaluation -> "subthreshold", ExplicitLeutwylerSigma -> True] ;

Limit[z, Pair[Momentum[p1], Momentum[p1]] -> 0] /. CouplingConstant[c_[4], n_, r___] :> If[RenormalizationCoefficients[c[4]][[n]] =!= 0, RenormalizationCoefficients[c[4]][[n]]/(32 Pi^2) (CouplingConstant[c[4], n, r] + Log[ParticleMass[PionPlus, RenormalizationState[0]]^2/ScaleMu^2]), 1/(16 Pi^2) (CouplingConstant[c[4], n, r])] // Simplify

((5 l _ 5^( ) + 5 h _ 2^( ) - k _ 15^( ) - 288 π^2 λ - 9) (e^( ))^2 + 432 π^2)/(432 π^2)

$VeryVerbose = 2 ;

CheckF[z, "ChPTVirtualPhotons2V10o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2V10o2.Fac

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)