•Renormalization

FeynCalc counts all fields as incoming, but we want p1 and p2 to be the same, so we substitute p2->-p1:

amptree1

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

amptree3

-1/(9 C^( ) f _ π^(ó  )) (2 i p _ 2^μ _ 1 (9 (2 k _ 3^( ) - k _ 4^( ) + k _ 9^( )) (f _ π^(ó  ))^4 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2) - ((10 (k _ 1^( ) + k _ 2^( )) + 9 k _ 9^( )) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 C^( ) l _ 4^( ) (m _ π^0^(ó  ))^2) δ _ (I _ 1 I _ 2)^(2)))

ampinfinities

-1/(96 π^2 f _ π^(ó  )) (i (1/(C^( ) p _ 3^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)) (3 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) ((2 p _ 1^μ _ 1 - p _ 3^μ _ 1) p _ 3^2 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 - (32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2) - 16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 3^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) (2 p _ 1^μ _ 1 p _ 3^2 - p _ 3^μ _ 1 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + p _ 3^2))) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2)) (f _ π^(ó  ))^4) + 8 p _ 1^μ _ 1 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (δ _ (1 I _ 1)^(2) δ _ (1 I _ 2)^(2) + δ _ (2 I _ 1)^(2) δ _ (2 I _ 2)^(2) - 2 δ _ (I _ 1 I _ 2)^(2)) (m _ π^+^(ó  ))^2 + (32 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2 (δ _ (3 I _ 1)^(2) δ _ (3 I _ 2)^(2) - δ _ (I _ 1 I _ 2)^(2)))))

The first order tree amplitude is wave function renormalized:

zpionZero = CheckF[dum, "ChPTVirtualPhotons2P40o2.Fac"]

(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)

zpionPlus = CheckF[dum, "ChPTVirtualPhotons2P30o2.Fac"]

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)

zpion = (1 - SU2Delta[3, I1]) zpionPlus + SU2Delta[3, I1] zpionZero ;

amp1Zero = amptree1 * (1 + (2 - zpion))/2 /. Momentum[p2] -> -Momentum[p1] /. {I1 -> 3, I2 -> 3, i1 -> 3, i2 -> 3} // ChargeEliminate // SUNReduce // Simplify

-1/(72 π^2 C^( ) f _ π^(ó  )) (i p _ 1^μ _ 1 (3 C^( ) (48 π^2 (f _ π^(ó  ))^2 + (log((m _ π^+^(ó  ))^2/μ^2) - 64 π^2 λ) (m _ π^+^(ó  ))^2 + 96 π^2 λ (m _ π^0^(ó  ))^2) - 8 π^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) + 9 (k _ 4^(r ) - 2 k _ 3^(r ))) (f _ π^(ó  ))^4 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2)))

amp3Zero = amptree3 /. Momentum[p2] -> -Momentum[p1] /. {I1 -> 3, I2 -> 3, i1 -> 3, i2 -> 3} // SUNReduce // Simplify

-1/(9 C^( ) f _ π^(ó  )) (2 i p _ 1^μ _ 1 (10 k _ 1^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 10 k _ 2^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 (k _ 4^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 2 k _ 3^( ) ((m _ π^0^(ó  ))^2 - (m _ π^+^(ó  ))^2) (f _ π^(ó  ))^4 + C^( ) l _ 4^( ) (m _ π^0^(ó  ))^2)))

amploopZero = ampinfinities /. Momentum[p2] -> -Momentum[p1] /. {I1 -> 3, I2 -> 3, i1 -> 3, i2 -> 3} // SUNReduce // Simplify

(i (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) p _ 1^μ _ 1 (m _ π^+^(ó  ))^2)/(6 π^2 f _ π^(ó  ))

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

ffZero = amp1Zero + amp3Zero + amploopZero // Simplify

-1/(72 π^2 C^( ) f _ π^(ó  )) (i p _ 1^μ _ 1 (8 π^2 (20 k _ 1^( ) - 10 k _ 1^(r ) + 20 k _ 2^( ) - 10 k _ 2^(r ) - 36 k _ 3^( ) + 18 k _ 3^(r ) + 18 k _ 4^( ) - 9 k _ 4^(r )) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 C^( ) (16 π^2 (f _ π^(ó  ))^2 - (64 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + 16 π^2 (l _ 4^( ) + 2 λ) (m _ π^0^(ó  ))^2)))

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

Renormalize[ffZero] // Simplify

-1/(72 π^2 C^( ) f _ π^(ó  )) (i p _ 1^μ _ 1 (8 π^2 (10 k _ 1^(r ) + 10 k _ 2^(r ) + 9 (k _ 4^(r ) - 2 k _ 3^(r ))) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 C^( ) (16 π^2 (f _ π^(ó  ))^2 - log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^2 + 16 π^2 l _ 4^(r ) (m _ π^0^(ó  ))^2)))

Coefficient[Renormalize[ffZero], LeutwylerLambda[]] // Simplify

0

amp1Charged = amptree1 * (1 + (2 - zpion))/2 /. Momentum[p2] -> -Momentum[p1] /. {I1 -> 1, I2 -> 1, i1 -> 1, i2 -> 1} // ChargeEliminate // SUNReduce // Simplify

-1/(144 π^2 C^( ) f _ π^(ó  )) (i p _ 1^μ _ 1 (3 C^( ) (96 π^2 (f _ π^(ó  ))^2 + (log((m _ π^+^(ó  ))^2/μ^2) - 64 π^2 λ) (m _ π^+^(ó  ))^2 + (128 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) - (f _ π^(ó  ))^4 (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)) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2)))

amp3Charged = amptree3 /. Momentum[p2] -> -Momentum[p1] /. {I1 -> 1, I2 -> 1, i1 -> 1, i2 -> 1} // SUNReduce // Simplify

-(2 i p _ 1^μ _ 1 (10 k _ 1^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 10 k _ 2^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 (k _ 9^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + C^( ) l _ 4^( ) (m _ π^0^(ó  ))^2)))/(9 C^( ) f _ π^(ó  ))

amploopCharged = ampinfinities /. Momentum[p3] -> -Momentum[p1] /. {I1 -> 1, I2 -> 1, i1 -> 1, i2 -> 1} // SUNReduce // Simplify

(i p _ 1^μ _ 1 (3 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + 3 p _ 1^2) + 3 p _ 1^2 ((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2 - (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)) (f _ π^(ó  ))^4 + 8 C^( ) p _ 1^2 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + (32 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)))/(96 π^2 C^( ) f _ π^(ó  ) p _ 1^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2))

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

ffCharged = amp1Charged + amp3Charged + amploopCharged // Simplify

-1/(288 C^( ) f _ π^(ó  )) (i p _ 1^μ _ 1 (1/π^2 (2 (3 C^( ) (96 π^2 (f _ π^(ó  ))^2 + (log((m _ π^+^(ó  ))^2/μ^2) - 64 π^2 λ) (m _ π^+^(ó  ))^2 + (128 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) - (f _ π^(ó  ))^4 (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)) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2))) + 64 (10 k _ 1^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 10 k _ 2^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + 9 (k _ 9^( ) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^4 + C^( ) l _ 4^( ) (m _ π^0^(ó  ))^2)) - 1/(π^2 p _ 1^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)) (3 (3 ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + 3 p _ 1^2) + 3 p _ 1^2 ((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2 - (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)) (f _ π^(ó  ))^4 + 8 C^( ) p _ 1^2 ((32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + (32 π^2 λ + log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)))))

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

Renormalize[ffCharged] // Expand // FullSimplify

-(i p _ 1^μ _ 1 (((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (p _ 1^2 ((16 π^2 (20 k _ 1^(r ) + 20 k _ 2^(r ) + 36 k _ 9^(r ) - 27 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2)) + 9 (log((m _ π^+^(ó  ))^2/μ^2) - 2 log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) + 2)) (m _ π^+^(ó  ))^2 - (16 π^2 (20 k _ 1^(r ) + 20 k _ 2^(r ) + 36 k _ 9^(r ) - 27 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2)) - 9 (2 log((m _ π^+^(ó  ))^2/μ^2) - 3 log((m _ γ^(ó  ))^2/μ^2) + 2 log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) - 2)) (m _ γ^(ó  ))^2) - 144 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)^2) (f _ π^(ó  ))^4 + 18 C^( ) p _ 1^2 (32 π^2 (f _ π^(ó  ))^2 - log((m _ π^+^(ó  ))^2/μ^2) (m _ π^+^(ó  ))^2 + (32 π^2 l _ 4^(r ) - log((m _ π^0^(ó  ))^2/μ^2)) (m _ π^0^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)))/(288 π^2 C^( ) f _ π^(ó  ) p _ 1^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2))

Coefficient[Renormalize[ffCharged], LeutwylerLambda[]] // Simplify

0

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:

cCharged = Collect[Coefficient[ffCharged/(-2 I DecayConstant[PseudoScalar[2], RenormalizationState[0]]), Pair[LorentzIndex[μ1], Momentum[p1]]], _DecayConstant] // FullSimplify

1/576 (1/C^( ) (((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (640 k _ 1^( ) + 640 k _ 2^( ) + 576 k _ 9^( ) - (2 (8 π^2 (20 (k _ 1^(r ) + k _ 2^(r )) + 9 λ) + 9 (log((m _ π^+^(ó  ))^2/μ^2) + log((m _ γ^(ó  ))^2/(m _ π^+^(ó  ))^2) - 1)))/π^2 - 1/(π^2 p _ 1^2 ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2)) (9 (16 π^2 Overscript[J, _] _ ((m _ π^+^(ó  ))^2 (m _ γ^(ó  ))^2)(p _ 1^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2) ((m _ π^+^(ó  ))^2 - (m _ γ^(ó  ))^2 + 3 p _ 1^2) + 3 p _ 1^2 ((32 π^2 λ + log((m _ γ^(ó  ))^2/μ^2)) (m _ γ^(ó  ))^2 - (32 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)))) (f _ π^(ó  ))^2) + 576 - (18 ((64 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2 + (log((m _ π^0^(ó  ))^2/μ^2) - 32 π^2 l _ 4^( )) (m _ π^0^(ó  ))^2))/(π^2 (f _ π^(ó  ))^2))

cZero = Collect[Coefficient[ffZero/(-2 I DecayConstant[PseudoScalar[2], RenormalizationState[0]]), Pair[LorentzIndex[μ1], Momentum[p1]]], _DecayConstant] // FullSimplify

((20 k _ 1^( ) - 10 k _ 1^(r ) + 20 k _ 2^( ) - 10 k _ 2^(r ) - 9 (4 k _ 3^( ) - 2 (k _ 3^(r ) + k _ 4^( )) + k _ 4^(r ))) ((m _ π^+^(ó  ))^2 - (m _ π^0^(ó  ))^2) (f _ π^(ó  ))^2)/(18 C^( )) + 1 + (16 π^2 (l _ 4^( ) + 2 λ) (m _ π^0^(ó  ))^2 - (64 π^2 λ + log((m _ π^+^(ó  ))^2/μ^2)) (m _ π^+^(ó  ))^2)/(16 π^2 (f _ π^(ó  ))^2)

cStrong = (cZero // ChargedMassesEliminate) /. CouplingConstant[QED[_], ___] -> 0 // Renormalize // Simplify

-(log((m _ π^+^(ó  ))^2/μ^2) (m _ π^0^(ó  ))^2)/(16 π^2 (f _ π^(ó  ))^2) + (l _ 4^(r ) (m _ π^0^(ó  ))^2)/(f _ π^(ó  ))^2 + 1

$VeryVerbose = 2 ;

CheckF[cCharged, "ChPTVirtualPhotons2A00P30o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2A00P30o2.Fac

File does not exist, evaluating

Saving

CheckF[cZero, "ChPTVirtualPhotons2A00P40o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2A00P40o2.Fac

File does not exist, evaluating

Saving

CheckF[cStrong, "ChPTVirtualPhotons2A00P20o2.Fac", Directory -> ToFileName[{$FeynCalcDirectory, "Phi"}, "Factors"]] ;

Using file name D:\\Program Files\\Wolfram Research\\Mathematica\\4.1\\AddOns\\Applications\\HighEnergyPhysics\\Phi\\Factors\\ChPTVirtualPhotons2A00P20o2.Fac

File does not exist, evaluating

Saving

$VeryVerbose = 0 ;

Converted by Mathematica  (July 10, 2003)