Question: Is it possible to dimensionally regularize an amplitude which contains the totally antisymmetric Levi-Civita tensor $\epsilon^{\mu\nu\alpha\beta}$? 

I don't know if it's possible to define 
$\epsilon^{\mu\nu\alpha\beta}$ in e.g. $$d-\eta$$-dimensions where $\eta$ is considered small and which we set to zero in the end. 

So what are your thoughts?

Answer: 
This problem was already mentioned in the original ['t Hooft-Veltman](http://www.staff.science.uu.nl/%7Ehooft101/gthpub/regularization_renormalization.pdf) article and solved by [Breitenlohner and Maison](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-52/issue-1/Dimensional-renormalization-and-the-action-principle/cmp/1103900439.full). This solution is known by the name "HVBM scheme" (after 't Hooft, Veltman, Breitenlohner and Maison).

A clear description of this regularization procedure is given for example in the following [dissertation](https://epub.uni-regensburg.de/10206/1/main.pdf) by Barbara Jäger. It consists basically of splitting the metric into a $4$-dimnsional and $(d-4)-$ parts, assuming the Levi-Civita tensor to have non-vanishing components only in the 4-dimensional subspace. In addition $\gamma\_5$ is assumed to anti-commute with the $\gamma$ matrices of the 4-dimnsional subspace and commutes with the others. This procedure leads to consistent Ward identities.

As mentioned in Jäger 's thesis, this procedure leads to a higher complexity in the computation of the Feynman diagrams, but there exist computer algebra programs implementing this scheme.
this fails since the tensor $ \epsilon ^{a,b,c,d} $ is diemnsion dependent

however in the case of zeta regularization of integrals $ \int\_{a}^{\infty}dx x^{m-s} $ with 's' a regulator we can overcome this problem