Dimensional Regularization

[Break1]

Hi guys! I was wondering if there is any difference choosing between d = 4 -e or d = 4 - 2e. If so, what are the impacts ?

 

[Orodruin]

The only difference is that 2e will appear in some places instead of e. There is no actual impact on anything physical.

 

[dextercioby]

Is it e (elementary charge) or $\epsilon$? I think the second.

 

[Orodruin]

Is it e (elementary charge) or $\epsilon$? I think the second.

Yes!

(Expression would not make sense if it was elementary charge.)

 

[vanhees71]

The idea behind "dimensional regularization" is to write down the integrals given by loops in Feynman diagrams in $d$ space-time dimensions and read the results as functions of continuous $d$. Then you do expansions around $d=4$ by setting $d=4-2\epsilon$ and expanding around $\epsilon=0$. The factor $2$ in the expression is just for a bit more convenience but doesn't really matter in any serious way.

The beauty of this regularization technique is that it obeys a lot of symmetries, i.e., Lorentz invariance and many global and local gauge symmetries.

The only difficulty comes into the game when you deal with objects that are specific to 4 space-time dimensions as the Levi-Civita tensor $\epsilon^{\mu \nu \rho \sigma}$ or (closely related with it) $\gamma_5$ in the Dirac-spinor formalism. This difficulties are, e.g., related to the problem of chiral anomalies, where you can choose, which combination of the vector and axial vector current you want to be not conserved due to the anomaly. In QED and QCD you are forced to break the axial-vector current conservation and keep the vector current conserved, because otherwise you break the local gauge symmetry of these theories, and then they become meaningless. The breaking of the $\mathrm{U}_{\mathrm{A}}(1)$ (accidental) symmetry is, however not a bug but a feature, because it resolves the tension about the decay rate for $\pi^0 \rightarrow \gamma \gamma$ and chiral symmetry.

With this application in mind, there's an ad-hoc resolution of the problem with $\gamma_5$ and arbitrary dimensions, invented by 't Hooft and Veltman: make $\gamma_5$ anticommute with $\gamma^0 \ldots \gamma^3$ and commute with all other $\gamma$ matrices ;-).