Superfields and the Inconsistency of regularization by dimensional reduction

Question

Question:

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

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Background and some references:

Regularization by dimensional reduction (DRed) was introduced by Siegel in 1979 and was shortly after seen to be inconsistent Siegel in 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 $$ \varepsilon^{\mu_1\mu_2\mu_3\mu_4} \varepsilon^{\nu_1\nu_2\nu_3\nu_4} \propto \det\big((g^{\mu_i\nu_j})\big) $$ and 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.

This post imported from StackExchange Physics at 2014-04-16 01:58 (UCT), posted by SE-user Simon

Comments

@Jose: That does not address the question being asked.

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user Simon

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@Simon : A little bit off-topic and very speculative, but I wonder if it would be possible, around $d=4$, to use $2$ different dimensions, $d_B=4-\epsilon_B$ for bosons, and $d_F=4-\epsilon_F$ for fermions, related by $(d_{B} - 2) = 2^{\large (\frac{d_{F}}{2} -1)}$ (this gives : $\ln(1-\frac{\large \epsilon_B}{2}) = - \frac{\large\epsilon_F}{2}\ln 2$). Very naively, it could "preserve" supersymmetry.

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user Trimok

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@Trimok: I think that if the number of fields don't match then you can't have supersymmetry. However the $2^{d/2}$ structure was considered by people like Delbourgo and others back in the 70s and 80s. I can't remember the details...

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user Simon

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@Undo - thanks for the bounty!

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user Simon

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This question actually got something like 8 wasted bounties on PSE, I hope there is someone here who can answer it.  

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comment: I don't think that it is possible in any way to show the inconsistency of regularization without reducing to components.

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user user28569

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@Humphrey: I thought that maybe swapping the order of some D-algebra (4D) and momentum integration ($(4-\epsilon)$-D) would yield an inconsistency. For example, the former could give you a 4D Kronecker delta and the latter would give you a ($(4-\epsilon)$-D) delta... And more importantly, if it's not possible to get inconsistent results in the superfield formulation, then why not just declare the results of such calculations to be what we mean by DRed and be done with it?

This post imported from StackExchange Physics at 2014-04-16 01:59 (UCT), posted by SE-user Simon