•Renormalization

The beta functions:

Cases[amp4, CouplingConstant[__], Infinity] // Union // Renormalize

{e^( ), l _ 3^(r ) - λ/2, k _ 1^(r ) + (-C^( )/(5 (f _ π^(ó  ))^4) - 27/20) λ, k _ 2^(r ) + (2 C^( ) λ)/(f _ π^(ó  ))^4, k _ 3^(r ) - (3 λ)/4, k _ 4^(r ) + (2 C^( ) λ)/(f _ π^(ó  ))^4, k _ 5^(r ) + (-C^( )/(5 (f _ π^(ó  ))^4) - 1/4) λ, k _ 6^(r ) + ((2 C^( ))/(f _ π^(ó  ))^4 + 1/4) λ, k _ 7^(r ), k _ 8^(r ) + (1/8 - C^( )/(f _ π^(ó  ))^4) λ, k _ 13^(r ) + (-(12 (C^( ))^2)/(5 (f _ π^(ó  ))^8) - (3 C^( ))/(5 (f _ π^(ó  ))^4) - 3) λ, k _ 14^(r ) + ((12 (C^( ))^2)/(f _ π^(ó  ))^8 + (3 C^( ))/(f _ π^(ó  ))^4 + 3/2) λ}

The contributions:

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

(2 C^( ) (δ _ (3 i _ 1)^(2) - 1) (e^( ))^2 + (f _ π^(ó  ))^2 (p _ 1^2 - (m _ π^0^(ó  ))^2))/(f _ π^(ó  ))^2

amploop = ((Plus @@ ampinfinities /. Momentum[p3] -> -Momentum[p1]) // Simplify) /. delrules // Simplify

-1/(96 π^2) (1/((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) (6 (-32 π^2 λ (m _ π^+^(ó  ))^4 - log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^4 + 32 π^2 λ (m _ γ^(ó  ))^4 + log((m _ γ^(ó  ))^2/μ^2) (m _ γ^(ó  ))^4 + 16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) (2 (m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + 2 p _ 1^2) + p _ 1^2 (2 (32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2 - 2 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)) ((δ _ (1 i _ 1)^(2))^2 + (δ _ (2 i _ 1)^(2))^2) (e^( ))^2) - 12 (64 π^2 λ + 2 log((m _ γ^(ó  ))^2/μ^2) + 1) (m _ γ^(ó  ))^2 (δ _ (3 i _ 1)^(2) - 1) (e^( ))^2 - 1/(f _ π^(ó  ))^4 (2 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) ((f _ π^(ó  ))^2 (-(δ _ (3 i _ 1)^(2) + 1) (m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2 (δ _ (3 i _ 1)^(2) - 2) - p _ 1^2 (δ _ (3 i _ 1)^(2) + 1)) - 4 C^( ) (e^( ))^2 (3 δ _ (3 i _ 1)^(2) - 4)) (m _ π^+^(ó  ))^2 + (32 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2 ((f _ π^(ó  ))^2 ((4 δ _ (3 i _ 1)^(2) - 1) (m _ π^0^(ó  ))^2 + 2 p _ 1^2 (δ _ (3 i _ 1)^(2) - 1)) - 4 C^( ) (e^( ))^2 (δ _ (3 i _ 1)^(2) - 1))))

ampwf4 = amp4 /. Momentum[p2] -> -Momentum[p1] // Renormalize // Simplify

1/(9 (f _ π^(ó  ))^6) (2 (5 k _ 13^(r ) (f _ π^(ó  ))^8 + 10 k _ 14^(r ) (f _ π^(ó  ))^8 + 27 C^( ) ((f _ π^(ó  ))^4 + 4 C^( )) λ) (δ _ (3 i _ 1)^(2) - 1) (e^( ))^4 + (f _ π^(ó  ))^2 (-20 k _ 5^(r ) (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^4 - 92 k _ 6^(r ) (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^4 - 4 k _ 7^(r ) (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^4 - 72 k _ 8^(r ) (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^4 - 27 λ (m _ π^0^(ó  ))^2 (f _ π^(ó  ))^4 + 20 k _ 1^(r ) p _ 1^2 (f _ π^(ó  ))^4 + 20 k _ 2^(r ) p _ 1^2 (f _ π^(ó  ))^4 - 27 λ p _ 1^2 (f _ π^(ó  ))^4 + 72 k _ 6^(r ) (m _ π^0^(ó  ))^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 + 72 k _ 8^(r ) (m _ π^0^(ó  ))^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 + 27 λ (m _ π^0^(ó  ))^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 - 36 k _ 3^(r ) p _ 1^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 + 18 k _ 4^(r ) p _ 1^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 + 27 λ p _ 1^2 δ _ (3 i _ 1)^(2) (f _ π^(ó  ))^4 - 108 C^( ) λ (m _ π^0^(ó  ))^2 + 36 C^( ) λ p _ 1^2 + 72 C^( ) λ (m _ π^0^(ó  ))^2 δ _ (3 i _ 1)^(2) + 36 C^( ) λ p _ 1^2 δ _ (3 i _ 1)^(2)) (e^( ))^2 + 9 (f _ π^(ó  ))^4 (λ - 2 l _ 3^(r )) (m _ π^0^(ó  ))^4)

The full propagator (to fourth order) can be written (1 - Z)/(p^2 + Σ(p^2) - (1 - c) m^2)~~(1 - Z)/(p^2 - m^2) - (Σ(p^2) + c m^2)/(m^2 - p^2)^2. Thus the amplitude ff4 can be written (p^2 - m^2) (1 - Z) - Σ(p^2) - c m^2.

ff4 = ff2 + amploop + ampwf4 // Simplify ;

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.

Cm = (Collect[(-ff4 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[Pion, SUNIndex[i1], RenormalizationState[0]]^2 /. pimassrule), LeutwylerLambda[]] // SUNReduce[#, FullReduce -> True] & // Simplify) //. delrules // Simplify ;

The mass shift is finite:

Coefficient[Cm, LeutwylerLambda[]] /. $ChargeEliminate /. delrules // Simplify

-(3 (f _ π^(ó  ))^2 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (m _ γ^(ó  ))^2 (δ _ (3 i _ 1)^(2) - 1))/C^( )

Here follow then the mass renormalization of the charged and neutral pions in terms of scale independent coupling constants (they agree with Knecht & Urech):

CmPlus = (ILimit[Cm /. i1 -> 1, ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify) // Simplify // ChargedNeutralMassesCancel

1/(288 π^2 (f _ π^(ó  ))^6) (6912 π^2 (C^( ))^2 λ (e^( ))^4 + 12 C^( ) (f _ π^(ó  ))^2 (48 π^2 (3 λ (e^( ))^2 + 1) (f _ π^(ó  ))^2 - 8 (44 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + (256 π^2 λ - log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) (e^( ))^2 + (f _ π^(ó  ))^4 (320 π^2 (e^( ))^4 (k _ 13^(r ) + 2 k _ 14^(r )) (f _ π^(ó  ))^4 + 2 (e^( ))^2 (16 π^2 (20 k _ 5^(r ) + 92 k _ 6^(r ) + 4 k _ 7^(r ) + 72 k _ 8^(r ) + 27 λ) (m _ π^0^(ó  ))^2 - (320 π^2 k _ 1^(r ) + 320 π^2 k _ 2^(r ) + 432 π^2 λ + 27 log((m _ π^+^(ó  ))^2/μ^2) - 36) (m _ π^+^(ó  ))^2) (f _ π^(ó  ))^2 + 3 (4 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^4 + 2 (-32 π^2 λ - 2 log((m _ π^+^(ó  ))^2/μ^2) + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2 (m _ π^+^(ó  ))^2 + (192 π^2 l _ 3^(r ) - 64 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^4 - 192 π^2 C^( ) (e^( ))^2)))

CmPlus /. $ChargeEliminate // Simplify

1/(288 π^2 (C^( ))^2 (f _ π^(ó  ))^2) (80 π^2 (k _ 13^(r ) + 2 k _ 14^(r )) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2)^2 (f _ π^(ó  ))^8 - C^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) ((320 π^2 k _ 1^(r ) + 320 π^2 k _ 2^(r ) + 27 log((m _ π^+^(ó  ))^2/μ^2) - 36) (m _ π^+^(ó  ))^2 - 64 π^2 (5 k _ 5^(r ) + 23 k _ 6^(r ) + k _ 7^(r ) + 18 k _ 8^(r )) (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 (C^( ))^2 (-4 log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^4 + 4 log((m _ π^+^(ó  ))^2/μ^2) (m _ π^0^(ó  ))^2 (m _ π^+^(ó  ))^2 + (64 π^2 l _ 3^(r ) + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^4))

CmZero = (ILimit[Cm /. i1 -> 3, ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify) // collectk

1/(288 π^2 (f _ π^(ó  ))^4) (-9 (f _ π^(ó  ))^2 (64 π^2 (λ - l _ 3^(r )) + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^4 + 2 (32 π^2 ((-10 k _ 1^(r ) - 10 k _ 2^(r ) + 18 k _ 3^(r ) - 9 k _ 4^(r ) + 2 (5 (k _ 5^(r ) + k _ 6^(r )) + k _ 7^(r ))) (f _ π^(ó  ))^4 - 18 C^( ) λ) (e^( ))^2 + 3 (f _ π^(ó  ))^2 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2) (m _ π^0^(ó  ))^2 + 12 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 ((f _ π^(ó  ))^2 (m _ π^+^(ó  ))^2 - 2 C^( ) (e^( ))^2))

Collect[CmZero /. $ChargeEliminate, _LeutwylerLambda] // Simplify

1/(288 π^2 (f _ π^(ó  ))^2) ((m _ π^0^(ó  ))^2 ((32 π^2 (-10 k _ 1^(r ) - 10 k _ 2^(r ) + 18 k _ 3^(r ) - 9 k _ 4^(r ) + 2 (5 (k _ 5^(r ) + k _ 6^(r )) + k _ 7^(r ))) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4)/C^( ) + 18 log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^2 + 576 π^2 l _ 3^(r ) (m _ π^0^(ó  ))^2 - 9 log((m _ π^0^(ó  ))^2/μ^2) (m _ π^0^(ó  ))^2))

EM counter-terms for easy comparison with Knecht & Urech

(CmPlus - (CmPlus /. CouplingConstant[ChPTVirtualPhotons2[4], _, ___] -> 0) // FullSimplify) /. $ChargedMassesEliminate // FullSimplify

1/(9 (f _ π^(ó  ))^2) (2 (5 ((k _ 13^(r ) + 2 k _ 14^(r )) (f _ π^(ó  ))^4 - 4 C^( ) (k _ 1^(r ) + k _ 2^(r ))) (e^( ))^4 - 2 (5 k _ 1^(r ) + 5 k _ 2^(r ) - 5 k _ 5^(r ) - 23 k _ 6^(r ) - k _ 7^(r ) - 18 k _ 8^(r )) (f _ π^(ó  ))^2 (m _ π^0^(ó  ))^2 (e^( ))^2 + 9 l _ 3^(r ) (m _ π^0^(ó  ))^4))

CmZero - (CmZero /. CouplingConstant[ChPTVirtualPhotons2[4], _, ___] -> 0) /. $ChargedMassesEliminate // FullSimplify

(2 l _ 3^(r ) (m _ π^0^(ó  ))^4)/(f _ π^(ó  ))^2 - 2/9 (e^( ))^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 18 k _ 3^(r ) + 9 k _ 4^(r ) - 2 (5 (k _ 5^(r ) + k _ 6^(r )) + k _ 7^(r ))) (m _ π^0^(ó  ))^2

Adding the mass shift to the full amputated two-point function, (p _ 1^2 - (m _ π^(ó  ))^2)^2 times the two-point function (compare Urech's thesis formula 1.55), gives (1-Z)(p^2 - m^2). Thus Z (plus the linear coefficient Overscript[Σ, _] of Σ(p^2)) is found by dividing off (p^2 - m^2) and taking the limit p^2 -> m^2.

zz = (ff4 + Cm // SUNReduce[#, FullReduce -> True] &) /. delrules ;

zzz = zz/ff2 // Simplify ;

z = ILimit[zzz, ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify ;

The neutral residue:

zZero = z /. i1 -> 3 /. Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[PionZero, RenormalizationState[0]] // Simplify

(8 π^2 (2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 18 k _ 3^(r ) + 9 k _ 4^(r )) (e^( ))^2 + 9) (f _ π^(ó  ))^4 - 3 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 (f _ π^(ó  ))^2 + 576 π^2 C^( ) (e^( ))^2 λ)/(72 π^2 (f _ π^(ó  ))^4)

The charged residue:

zPlus1 = z /. i1 -> 1 // Simplify ;

zPlus = ILimit[ILimit[(zPlus1 // ChargeEliminate) /. LeutwylerJBar[a__, ___Rule] :> LeutwylerJBar[a, ExplicitLeutwylerSigma -> True, LeutwylerJBarEvaluation -> "subthreshold"], Pair[Momentum[p1], Momentum[p1]] -> ParticleMass[PionPlus, RenormalizationState[0]]^2], ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify // ChargedNeutralMassesCancel // Simplify

1/(144 π^2 (f _ π^(ó  ))^2) (2 ((160 π^2 k _ 1^(r ) + 160 π^2 k _ 2^(r ) + 9 (8 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2) + log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) - 1)) (e^( ))^2 + 72 π^2) (f _ π^(ó  ))^2 + 3 (64 π^2 λ - log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 - 3 (128 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2)

Check the strong part:

zZero /. $ChargeEliminate /. {PionPlus -> Pion, PionZero -> Pion} // Simplify

1 - ((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2)

zPlus /. $ChargeEliminate /. {PionPlus -> Pion, PionZero -> Pion} // Simplify

1 - ((32 π^2 λ + log((m _ π^(ó  ))^2/μ^2)) (m _ π^(ó  ))^2)/(24 π^2 (f _ π^(ó  ))^2)

Finally, the non-linear part Overscript[Overscript[Σ, _], _](p^2) of Σ(p^2) is found. Overscript[Overscript[Σ, _], _](p^2) vanishes non-linearly for p^2 -> m^2.

sigmaZero = (ILimit[ff4 + CmZero - zZero (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionZero, RenormalizationState[0]]^2) /. i1 -> 3, ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify) // ChargedMassesEliminate // Simplify

0

sigmaPlus = (ILimit[ff4 + CmPlus - zPlus (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionPlus, RenormalizationState[0]]^2) /. i1 -> 1, ParticleMass[Photon, RenormalizationState[0]] -> 0] // Simplify) // ChargeEliminate // Simplify // NeutralMassesEliminate // Simplify

-((e^( ))^2 ((16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) - log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) - 1) (m _ π^+^(ó  ))^2 + (16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) + log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) - 1) p _ 1^2))/(8 π^2)

sigmaPlus /. _Log -> 0 // Simplify

-((e^( ))^2 (16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) - 1) ((m _ π^+^(ó  ))^2 + p _ 1^2))/(8 π^2)

zFullZero = (sigmaZero + zZero (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionZero, RenormalizationState[0]]^2))/(Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionZero, RenormalizationState[0]]^2) // FullSimplify

(8 π^2 ((2 (10 k _ 1^(r ) + 10 k _ 2^(r ) - 18 k _ 3^(r ) + 9 k _ 4^(r )) (e^( ))^2 + 9) (f _ π^(ó  ))^4 + 72 C^( ) (e^( ))^2 λ) - 3 (f _ π^(ó  ))^2 (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)/(72 π^2 (f _ π^(ó  ))^4)

zFullPlus = FullSimplify /@ Collect[(sigmaPlus + zPlus (Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionPlus, RenormalizationState[0]]^2))/(Pair[Momentum[p1], Momentum[p1]] - ParticleMass[PionPlus, RenormalizationState[0]]^2) // Expand, {Pi, _DecayConstant, _CouplingConstant}] // FullSimplify

1/(144 π^2) (2 (9 (log((m _ π^+^(ó  ))^2/μ^2) + (2 p _ 1^2)/(p _ 1^2 - (m _ π^+^(ó  ))^2) - 2) + 8 π^2 (20 k _ 1^(r ) + 20 k _ 2^(r ) + 9 (λ + Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((4 p _ 1^2)/((m _ π^+^(ó  ))^2 - p _ 1^2) + 2)))) (e^( ))^2 + (3 (48 π^2 (f _ π^(ó  ))^2 + (64 π^2 λ - log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 - (128 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2))/(f _ π^(ó  ))^2)

Save for later use.

$VeryVerbose = 2 ;

CheckF[CmPlus, "ChPTVirtualPhotons2P30o2.Mass"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2P30o2.Mass

File exists, force evaluating

Saving

CheckF[CmZero, "ChPTVirtualPhotons2P40o2.Mass"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2P40o2.Mass

File exists, force evaluating

Saving

CheckF[z, "ChPTVirtualPhotons2P20o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2P20o2.Fac

File exists, force evaluating

Saving

CheckF[zPlus, "ChPTVirtualPhotons2P30o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2P30o2.Fac

File exists, force evaluating

Saving

CheckF[zZero, "ChPTVirtualPhotons2P40o2.Fac"] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2P40o2.Fac

File exists, force evaluating

Saving

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)