I am following some notes on the computation of the vector two point function in QCD and I would like somebody to make some intermediate steps more explicit. Let's consider
Πμν=iμ2ϵ∫ddxeiqx⟨Ω|T{jμ(x)jν(0)}|Ω⟩=(qμqν−ημνq2)Π,
where μ is a mass scale, ϵ is the regulator defined in d=4−2ϵ where d is the dimensionality of space-time and jμ(x)=ˉq(x)γμq(x).
The quantity I want to compute is Π. To do that we first multiply the equation above with ημν on both sides to obtain
Π=−iμ2ϵ(d−1)q2∫ddxeiqx⟨Ω|T{jμ(x)jμ(0)}|Ω⟩=…
My notes claim that this leads to
…=−iNcμ2ϵ(d−1)q2∫ddxeiqxTr[S(x)γμS(−x)γμ],
where Nc is the number of colors and S(x) is the free quark propagator.
I want somebody to make the steps between the last two equations explicit, particularly I am interested on where do the traces come from.