Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
The chiral anomaly falls into this category, and this is usually cited as an example where dim reg fails (because of the epsilon problem), but allegedly there is a workaround (but paywalled, so I haven't read it) – twistor59 Apr 12 '13 at 6:44
Thanks for the link...I'll check it out. – Faraday Apr 12 '13 at 11:23
up vote 3 down vote accepted

This problem was already mentioned in the original 't Hooft-Veltman article and solved by Breitenlohner and Maison. 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 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.

share|cite|improve this answer
Awesome...that's exactly what I needed. Thanks a bunch. – Faraday Apr 15 '13 at 10:54

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.