•Loop amplitude

Construction of topologies:

tops = CreateTopologies[1, 1 -> 2, Adjacencies -> {3}, ExcludeTopologies -> {WFCorrections, SelfEnergies}, CountertermOrder -> 0] ;

Paint[tops, AutoEdit -> False, ColumnsXRows -> {2, 1}] ;

[Graphics:../HTMLFiles/index_115.gif]

Fields insertion:

inserttops = InsertFields[tops, {Photon[0]} -> {Electron[0], -Electron[0]}, Model -> "Automatic", GenericModel -> "Automatic", InsertionLevel -> Classes] ;

Graphical representation of the process:

Paint[inserttops, PaintLevel -> {Classes}, AutoEdit -> False, SheetHeader -> False, Numbering -> False, ColumnsXRows -> {1, 1}] ;

[Graphics:../HTMLFiles/index_118.gif]

Calculation of the amplitude:

amplFC = CreateFCAmp[inserttops, DiracTraceEvaluate -> False][[1]]

(i ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . (e^(0 ) γ^μ _ 2) . (γ · q _ 1 + m _ ψ^(ó  0 )) . (e^(0 ) γ^μ _ 1) . (γ · p _ 3 + γ · p _ 4 + γ · q _ 1 + m _ ψ^(ó  0 )) . (e^(0 ) γ^μ _ 3) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) µ _ μ _ 1(p _ 1) g^(μ _ 2 μ _ 3))/(16 π^4 (q _ 1^2 - (m _ ψ^(ó  0 ))^2) . ((p _ 3 + q _ 1)^2 - (m _ A^(ó  0 ))^2) . ((p _ 3 + p _ 4 + q _ 1)^2 - (m _ ψ^(ó  0 ))^2))

The polarization vector we divide off (Weinberg (11.3.1));

af = amplFC/Pair[LorentzIndex[μ1, D], Momentum[Polarization[p1, I], D]] /. q1 -> q1 - p3 // MomentumExpand // MomentumCombine // DiracGammaCombine // Simplify

(i ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . (e^(0 ) γ^μ _ 2) . (γ · ( q _ 1 - p _ 3 ) + m _ ψ^(ó  0 )) . (e^(0 ) γ^μ _ 1) . (γ · ( p _ 4 + q _ 1 ) + m _ ψ^(ó  0 )) . (e^(0 ) γ^μ _ 3) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) g^(μ _ 2 μ _ 3))/(16 π^4 ((q _ 1 - p _ 3)^2 - (m _ ψ^(ó  0 ))^2) . (q _ 1^2 - (m _ A^(ó  0 ))^2) . ((p _ 4 + q _ 1)^2 - (m _ ψ^(ó  0 ))^2))

The one-loop integrals are simplified and put on-mass-shell:

aff = OneLoopSimplify[af, q1, Dimension -> D] ;

The loop integrals are expressed in terms of Passarino-Veltman symbols:

SetOptions[B0, B0Real -> True] ;

ampreduced = OneLoop[q1, af, ReduceGamma -> True, Dimension -> D, ReduceToScalars -> True (* , WriteOutPaVe -> "tmp/" *)] // Simplify ;

Remembering that FeynCalc has all particles ingoing, we set q^2= (p _ 3 + p _ 4)^2, and make the replacement p _ 3 · p _ 4->q^2/2-(m _ ψ^(0 ))^2:

ampred = FullSimplify /@ (ampreduced /. {Pair[Momentum[p3, ___], Momentum[p3, ___]] -> ParticleMass[Electron, RenormalizationState[0]]^2, Pair[Momentum[p4, ___], Momentum[p4, ___]] -> ParticleMass[Electron, RenormalizationState[0]]^2} /. Pair[Momentum[p3, ___], Momentum[p4, ___]] -> q^2/2 - ParticleMass[Electron, RenormalizationState[0]]^2 (* /. Pair[Momentum[p3, d___], Momentum[p4, ___]] -> (q^2 - Pair[Momentum[p3, d], Momentum[p3, d]] - Pair[Momentum[p4, d], Momentum[p4, d]])/2 *) // DiracSimplify // Collect[#, {_StandardMatrixElement, _A0, _B0, _C0}] &) /. Times[f : (A0[__] | B0[__] | C0[__]), r__] :> f * Simplify[Times[r]]

1/(16 π^2 (q^2 - 4 (m _ ψ^(ó  0 ))^2)) ((e^(0 ))^3 (B _ 0 (q^2, (m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) (3 q^2 - 12 (m _ ψ^(ó  0 ))^2 + 2 (m _ A^(ó  0 ))^2) + 2 (q^2 - 4 (m _ ψ^(ó  0 ))^2 + B _ 0 ((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2) (-2 q^2 + 8 (m _ ψ^(ó  0 ))^2 - (m _ A^(ó  0 ))^2) + C _ 0 ( (m _ ψ^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 , q^2 , (m _ ψ^(ó  0 ))^2 , (m _ A^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 ) (8 (m _ ψ^(ó  0 ))^4 - 2 (3 q^2 + 4 (m _ A^(ó  0 ))^2) (m _ ψ^(ó  0 ))^2 + (q^2 + (m _ A^(ó  0 ))^2)^2))) {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . γ^μ _ 1 . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |}) - 1/(8 π^2 m _ ψ^(ó  0 ) (q^2 - 4 (m _ ψ^(ó  0 ))^2)^2) ((e^(0 ))^3 (p _ 3^μ _ 1 - p _ 4^μ _ 1) (6 C _ 0 ( (m _ ψ^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 , q^2 , (m _ ψ^(ó  0 ))^2 , (m _ A^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 ) (m _ ψ^(ó  0 ))^2 (m _ A^(ó  0 ))^4 - (2 (C _ 0 ( (m _ ψ^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 , q^2 , (m _ ψ^(ó  0 ))^2 , (m _ A^(ó  0 ))^2 , (m _ ψ^(ó  0 ))^2 ) (8 (m _ ψ^(ó  0 ))^2 - 2 q^2) - 3 B _ 0 (q^2, (m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2)) (m _ ψ^(ó  0 ))^2 + B _ 0 (0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2) (q^2 - 4 (m _ ψ^(ó  0 ))^2) + B _ 0 ((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2) (10 (m _ ψ^(ó  0 ))^2 - q^2)) (m _ A^(ó  0 ))^2 + (B _ 0 (0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2) + B _ 0 (q^2, (m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) - 2 B _ 0 ((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2) + 1) (m _ ψ^(ó  0 ))^2 (q^2 - 4 (m _ ψ^(ó  0 ))^2)) {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |})

The Passarino-Veltman integrals are evaluated:

ampinfinitiesfull = VeltmanExpand[ampred, B0Evaluation -> "direct1", C0Evaluation -> None, ExpandGammas -> True, DimensionExpand -> True, SmallEpsilon -> 0] // Simplify

1/(16 π^2 (q^2 - 4 (m _ ψ^(ó  0 ))^2)^2) ((e^(0 ))^3 ((q^2 - 4 (m _ ψ^(ó  0 ))^2) {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . γ^μ _ 1 . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |} (1/(4 (D - 4)) ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log((x - 1) x q^2 + (m _ ψ^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) (3 q^2 - 12 (m _ ψ^(ó  0 ))^2 + 2 (m _ A^(ó  0 ))^2)) + 2 (q^2 - 4 (m _ ψ^(ó  0 ))^2 - 1/(4 (D - 4)) ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) (2 q^2 - 8 (m _ ψ^(ó  0 ))^2 + (m _ A^(ó  0 ))^2)) + (8 (m _ ψ^(ó  0 ))^4 - 2 (3 q^2 + 4 (m _ A^(ó  0 ))^2) (m _ ψ^(ó  0 ))^2 + (q^2 + (m _ A^(ó  0 ))^2)^2) C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, q^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2))) - 1/m _ ψ^(ó  0 ) (2 (p _ 3^μ _ 1 - p _ 4^μ _ 1) {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |} (6 (m _ ψ^(ó  0 ))^2 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, q^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) (m _ A^(ó  0 ))^4 - 1/4 (8 (-(3 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log((x - 1) x q^2 + (m _ ψ^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(4 (D - 4)) - 2 (q^2 - 4 (m _ ψ^(ó  0 ))^2) C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, q^2, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2)) (m _ ψ^(ó  0 ))^2 - 1/(D - 4) ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) (q^2 - 10 (m _ ψ^(ó  0 ))^2)) + 1/(D - 4) ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) (q^2 - 4 (m _ ψ^(ó  0 ))^2))) (m _ A^(ó  0 ))^2 + 1/4 (((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log((x - 1) x q^2 + (m _ ψ^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) - (2 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + 4) (m _ ψ^(ó  0 ))^2 (q^2 - 4 (m _ ψ^(ó  0 ))^2)))))

f0 = Coefficient[ampinfinitiesfull, StandardMatrixElement[Spinor[-Momentum[p3], ParticleMass[Electron, RenormalizationState[0]], 1] . DiracGamma[LorentzIndex[μ1]] . Spinor[Momentum[p4], ParticleMass[Electron, RenormalizationState[0]], 1]]] /. q -> 0 // Simplify

-1/(64 π^2 (m _ ψ^(ó  0 ))^2) ((e^(0 ))^3 (((÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) ((D - 4) log((m _ ψ^(ó  0 ))^2) + 2) ((m _ A^(ó  0 ))^2 - 6 (m _ ψ^(ó  0 ))^2))/(2 (D - 4)) + 2 (-4 (m _ ψ^(ó  0 ))^2 - 1/(4 (D - 4)) ((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) ((m _ A^(ó  0 ))^2 - 8 (m _ ψ^(ó  0 ))^2)) + (8 (m _ ψ^(ó  0 ))^4 - 8 (m _ A^(ó  0 ))^2 (m _ ψ^(ó  0 ))^2 + (m _ A^(ó  0 ))^4) C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2))))

c3 = Coefficient[ampinfinitiesfull, Pair[LorentzIndex[μ1], Momentum[p3]]] /. q -> 0 // Simplify

-1/(256 π^2 (m _ ψ^(ó  0 ))^3) ((e^(0 ))^3 {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |} (12 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) (m _ A^(ó  0 ))^4 - (-(2 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + (5 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + 4 (8 (m _ ψ^(ó  0 ))^2 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) - (3 (÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) ((D - 4) log((m _ ψ^(ó  0 ))^2) + 2))/(4 (D - 4)))) (m _ A^(ó  0 ))^2 - 2 (((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) - (2 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + ((÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log((m _ ψ^(ó  0 ))^2) + 2) ((D - 4) log(μ) - 1))/(D - 4) + 4) (m _ ψ^(ó  0 ))^2))

c4 = Coefficient[ampinfinitiesfull, Pair[LorentzIndex[μ1], Momentum[p4]]] /. q -> 0 // Simplify

1/(256 π^2 (m _ ψ^(ó  0 ))^3) ((e^(0 ))^3 {| ϕ ( -p _ 3 , m _ ψ^(ó  0 ) ) . ϕ ( p _ 4 , m _ ψ^(ó  0 ) ) |} (12 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) (m _ A^(ó  0 ))^4 - (-(2 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + (5 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + 4 (8 (m _ ψ^(ó  0 ))^2 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) - (3 (÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1) ((D - 4) log((m _ ψ^(ó  0 ))^2) + 2))/(4 (D - 4)))) (m _ A^(ó  0 ))^2 - 2 (((÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) - (2 (÷J (D - 4) + 2) ((D - 4) Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log(μ) - 1))/(D - 4) + ((÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) ((D - 4) log((m _ ψ^(ó  0 ))^2) + 2) ((D - 4) log(μ) - 1))/(D - 4) + 4) (m _ ψ^(ó  0 ))^2))

c3 + c4 // Simplify

0

g0 = c3 2 ParticleMass[Electron, RenormalizationState[0]] ;

The definition of e^(0 )as the real unit charge implies (see e.g. Weinberg p. 489) that at q^2= 0 the coefficient f0 of γ^μ _ 1in the full amplitude plus the coefficient g0 of ((p _ 3)^μ _ 1- (p _ 4)^μ _ 1)/(2 m _ ψ^(0 )) in the full amplitude must equal one.  g0 however gets no contribution from counterterms and photon self energy, so we can simply use the loop value of g0 to find f0:

g00 = Limit[g0 /. _StandardMatrixElement :> Sequence[], ParticleMass[Photon, RenormalizationState[0]] -> 0] /. D -> 4 + δ // Simplify

Underscript[lim, m _ A^(ó  0 ) -> 0] -1/(128 π^2 (m _ ψ^(ó  0 ))^2) ((e^(0 ))^3 (12 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) (m _ A^(ó  0 ))^4 - (-(2 (÷J δ + 2) (δ Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) δ + 2) (δ log(μ) - 1))/δ + (5 (÷J δ + 2) (δ Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) δ + 2) (δ log(μ) - 1))/δ + 4 (8 (m _ ψ^(ó  0 ))^2 C _ 0((m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2, 0, (m _ ψ^(ó  0 ))^2, (m _ A^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) - (3 (÷J δ + 2) (log(π) δ + 2) (δ log(μ) - 1) (δ log((m _ ψ^(ó  0 ))^2) + 2))/(4 δ))) (m _ A^(ó  0 ))^2 - 2 (((÷J δ + 2) (δ Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) δ + 2) (δ log(μ) - 1))/δ - (2 (÷J δ + 2) (δ Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2 - (x - 1) (m _ A^(ó  0 ))^2) d x + 2) (log(π) δ + 2) (δ log(μ) - 1))/δ + ((÷J δ + 2) (log(π) δ + 2) (δ log((m _ ψ^(ó  0 ))^2) + 2) (δ log(μ) - 1))/δ + 4) (m _ ψ^(ó  0 ))^2))

This is then the Schwinger correction to the electron magnetic moment (Weinberg (11.3.16):

SchwingerCorrection = Limit[g00, δ -> 0] // FullSimplify

-((e^(0 ))^3 (Underoverscript[∫, 0, arg3] log(x (m _ ψ^(ó  0 ))^2) d x - 2 Underoverscript[∫, 0, arg3] log(x^2 (m _ ψ^(ó  0 ))^2) d x + log((m _ ψ^(ó  0 ))^2) - 1))/(16 π^2)

sc0 = SchwingerCorrection /. IntegrateHeld -> Integrate // Simplify

-(e^(0 ))^3/(8 π^2)

Converted by Mathematica  (July 10, 2003)