•Renormalization

The beta functions:

ampf4

20/9 (e^( ))^2 (L _ 10^( ) + 2 H _ 1^( ) + k _ 15^( )) (p _ 1 · p _ 2 µ ( p _ 1 ) · µ^* ( p _ 2 ) - p _ 1 · µ^* ( p _ 2 ) p _ 2 · µ ( p _ 1 ))

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

-2/9 (e^( ))^2 (10 L _ 10^(r ) + 20 H _ 1^(r ) + 10 k _ 15^(r ) - 3 λ) p _ 1^2

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

p _ 1^2

amploop = ((Plus @@ ampsimple /. p3 -> -p1) // DoSumOver) /. pirule /. subpar // SUNReduce[#, FullReduce -> True] & // VeltmanExpand[#, ExplicitLeutwylerJ0 -> True] & // Simplify

((e^( ))^2 ((48 π^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) - 96 π^2 λ - 3 log((m _ π^+^(ó  ))^2/μ^2) - 1) p _ 1^2 - 192 π^2 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2) (m _ π^+^(ó  ))^2))/(144 π^2)

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

0

Coefficient[ampct, LeutwylerLambda[]] // Simplify

2/3 (e^( ))^2 p _ 1^2

Coefficient[amploop, LeutwylerLambda[]] // Simplify

-2/3 (e^( ))^2 p _ 1^2

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 _ 10^(r ) + 2 H _ 1^(r ) + k _ 15^(r )) - 3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2)) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 1) - 144 π^2) p _ 1^2)

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

z = ff4/ff2 // Simplify

-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 _ 10^(r ) + 2 H _ 1^(r ) + k _ 15^(r )) - 3 Overscript[J, _] _ (m _ π^+^(ó  ))^2(p _ 1^2)) + 3 log((m _ π^+^(ó  ))^2/μ^2) + 1) - 144 π^2) p _ 1^2)

In the limit p _ 1^2->(m _ π^+^(ó  ))^2 there is agreement with Meissner and Kubis 1999:

Limit[z /. CouplingConstant[ChPTEM2[4], 15, RenormalizationState[1]] -> 2/3/(32 Pi^2) (CouplingConstant[ChPTEM2[4], 15, RenormalizationState[1]] + Log[ParticleMass[PionPlus, RenormalizationState[0]]^2/ScaleMu^2]) /. LeutwylerJBar[a__, b___Rule] :> LeutwylerJBar[a, LeutwylerJBarEvaluation -> "subthreshold", ExplicitLeutwylerSigma -> True, b], Pair[Momentum[p1], Momentum[p1]] -> 0] // Simplify

(432 π^2 - (e^( ))^2 (960 π^2 L _ 10^(r ) + 1920 π^2 H _ 1^(r ) + 20 k _ 15^(r ) + 29 log((m _ π^+^(ó  ))^2/μ^2) + 9))/(432 π^2)

$VeryVerbose = 2 ;

CheckF[z, "ChPTEM2V10o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTEM2V10o2.Fac

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)