•Loop amplitude

Construction of topologies:

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

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

[Graphics:../HTMLFiles/index_16.gif]

Fields insertion:

inserttops = InsertFields[tops, {Photon[0]} -> {Photon[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_19.gif]

Calculation of the amplitude, the two polarization vectors are divided off:

afc = CreateFCAmp[inserttops, AmplitudeLevel -> Classes][[1]]/(Pair[LorentzIndex[μ1, D], Momentum[Polarization[p1, i], D]] Pair[LorentzIndex[μ2, D], Momentum[Polarization[p3, -i], D]] ) // Simplify

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

amplFC = afc /. DiracTrace -> Tr // Simplify

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

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

ampreduced = OneLoop[q1, amplFC, Dimension -> D, ScalarProductCancel -> False] // Simplify

((e^(0 ))^2 (p _ 3^μ _ 1 p _ 3^μ _ 2 - g^(μ _ 1 μ _ 2) p _ 3^2) (-6 (B _ 0 (p _ 3^2, (m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2) + 1) (m _ ψ^(ó  0 ))^2 + 6 A _ 0 ( (m _ ψ^(ó  0 ))^2 ) + (1 - 3 B _ 0 (p _ 3^2, (m _ ψ^(ó  0 ))^2, (m _ ψ^(ó  0 ))^2)) p _ 3^2))/(36 π^2 p _ 3^2)

Current conservation:

Pair[LorentzIndex[μ1], Momentum[p3]] ampreduced // Contract

0

The Passarino-Veltman integrals are evaluated:

ampinfinitiesfull = VeltmanExpand[ampreduced, B0Evaluation -> "direct1", ExpandGammas -> False, DimensionExpand -> False, SmallEpsilon -> 0, TaylorOrder -> 2] /. p3 -> -p1 // FullSimplify

1/(36 π^4 p _ 1^2 (m _ ψ^(ó  0 ))^2) (μ^(-D) (e^(0 ))^2 (p _ 1^μ _ 1 p _ 1^μ _ 2 - g^(μ _ 1 μ _ 2) p _ 1^2) (6 π^(D/2) μ^4 Γ(2 - D/2) ((m _ ψ^(ó  0 ))^2)^(D/2) + (m _ ψ^(ó  0 ))^2 (π^2 μ^D p _ 1^2 - 3 π^(D/2) μ^4 Γ(2 - D/2) (Underoverscript[∫, 0, arg3] ((m _ ψ^(ó  0 ))^2 + (x - 1) x p _ 1^2)^(D - 4)/2 d x) (2 (m _ ψ^(ó  0 ))^2 + p _ 1^2))))

The value of the scalar integral at p _ 1^2= 0:

tensor = (-Pair[LorentzIndex[μ1], Momentum[p1]] Pair[LorentzIndex[μ2], Momentum[p1]] + Pair[LorentzIndex[μ1], LorentzIndex[μ2]] Pair[Momentum[p1], Momentum[p1]])

g^(μ _ 1 μ _ 2) p _ 1^2 - p _ 1^μ _ 1 p _ 1^μ _ 2

pi0full = (Limit[(Simplify[Cancel[ampinfinitiesfull/tensor]] // DimensionExpand[#, TaylorOrder -> 1, Dimension -> D] & // ExpandGammas[#, TaylorOrder -> 2] &) /. Pair[Momentum[p1], Momentum[p1]] -> p2 /. IntegrateHeld -> Integrate, p2 -> 0] // Simplify) /. Sqrt[x_^2] -> x // Simplify

1/(144 (D - 4) π^2) ((e^(0 ))^2 ((D - 4) log(μ) - 1) (-÷J log(π) D^3 + 18 ÷J log(π) D^2 - 2 log(π) D^2 - 2 ÷J D^2 - 96 ÷J log(π) D + 28 log(π) D + 28 ÷J D + 4 (D - 4)^2 log(μ) + 3 (D - 4) (÷J (D - 4) + 2) (log(π) D - 4 log(π) + 2) log((m _ ψ^(ó  0 ))^2) + 160 ÷J log(π) - 80 log(π) - 80 ÷J + 24))

Converted by Mathematica  (July 10, 2003)