# Question:

How can you show the inconsistency of regularization by dimensional reduction in the $\mathcal{N}=1$ superfield approach (without reducing to components)?

# Background and some references:

Regularization by dimensional reduction (DRed) was introduced by Siegel in 1979 and was shortly after seen to be inconsistent (Siegel 1980). Despite this, it is commonly used in supersymmetric calculations since it has most of the advantages of (normal) dimensional regularization (DReg) and (naively) preserves supersymmetry.

The demonstration of the inconsistency of DRed is based on the combination of 4-dimensional identities, such as the product of epsilon-tensors $$\varepsilon^{\mu_1\mu_2\mu_3\mu_4} \varepsilon^{\nu_1\nu_2\nu_3\nu_4} \propto \det\big((g^{\mu_i\nu_j})\big)$$ with the d-dimensional projections of 4-dimensional objects. Details can be found in the references above and below, although the argument is especially clear in Avdeev and Vladimirov 1983.

Various proposals have been made on how to consistently use DRed and most involve restrictions on the use of 4-dimensional identities using epsilon-tensors and $\gamma_5$ matrices. (Note that the treatment of $\gamma_5$ in DReg is also a little tricky...). This means we also have to forgo the use of Fierz identities in the gamma-matrix algebra (which is also a strictly 4-dimensional thing - or whatever integer dimension you're working in). This means we lose most of the advantages that made DRed attractive in the first place - maintaining only the fact that it's better than DReg in SUSY theories. The latest such attempt is Stockinger 2005, but it's also worth looking at the earlier discussions of Delbourgo and Jarvis 1980, Bonneau 1980 and (especially) Avdeev and Vladimirov 1983 & Avdeev and Kamenshchik 1983. The pragmatic discussion in Jack and Jones 1997 is also worth reading - it also contains a fairly complete set of references.

Anyway, all of the "fixes" are hard to do when using superfields, since the $D$-algebra has all of the "bad" 4-dimensional algebra built in.

My question is: What is the easiest way of showing the inconsistency of DRed in the superfield approach? (I want an answer that does not rely on reducing to components!). I'm guessing that it should somehow follow from the $D$-algebra acting on dimensionally reduced superfields.

• @Carl: It's not looking good... To anyone who wants to try to answer this question, the second paper of Siegel linked to above mentions how $\varepsilon_{\mu\nu\kappa\lambda}$ can be created from supergraphs/$D$-algebra. Also, the paper by Avdeev: Dimensional Regularization Of Supergraphs has some good hints. – Simon Jun 18 '11 at 4:13
• Avdeev et. al., Dimensional Regularization Of Supergraphs, Dubna 1982 preprint, available for free here: iaea.org/inis/collection/NCLCollectionStore/_Public/14/784/… – Qmechanic Jun 18 '11 at 15:14
• @Simon Maybe the people at enwp.org/WP:RDS could solve it? I'd ask it there myself, but I don't know the topic behind the question. – Manishearth Feb 8 '12 at 9:58
• @igael - yes, see the references. Although, giving an example using superfields seems to have stumped a few people... – Simon Mar 17 '16 at 10:57
• Congratulations on making the top 5 most bountied questions network-wide. – Emilio Pisanty Jun 30 '16 at 14:39

## 1 Answer

I would go about pointing you to this paper on arxiv, since you seem like an educated expert and can therefore understand the equations.

On page 17 they conclude that

This implies that DReD does not manifestly preserve BRS invariance. As is well known, DReD is algebraically inconsistent because different contractions of three or more ǫ µνρ factors yield different results in d < 3 dimensions