The ultra-violet question in maximally supersymmetric field theories

TL;DR

This work examines the ultraviolet behavior of maximally supersymmetric field theories, focusing on which supersymmetric invariants can serve as counterterms. It employs diverse formalisms—conventional, harmonic, light-cone superspace, off-shell harmonic formulations, and the algebraic renormalisation framework—to classify BPS and non-BPS invariants and to determine the first possible UV divergences. The central finding is that half-BPS counterterms (e.g., $F^4$ in YM and $R^4$ in SG) are generally protected by supersymmetry Ward identities, while divergences are expected to arise from one-quarter BPS or non-BPS invariants; this aligns with explicit multi-loop computations in maximal SYM and informs expectations for maximal supergravity. Collectively, the results highlight a coherent framework across multiple methodologies for predicting UV behavior in maximally supersymmetric theories, while leaving open questions about higher-loop finiteness in SG and potential string-theoretic cancellations.

Abstract

We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang--Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form $F^4$ and $R^4$ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. $d^2 F^4$ and $d^4 R^4$, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D=5 and at five loops in D=4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.

Figures (1)

• Figure 1: 3-loop Feynman diagram types leading to unanticipated 'superfiniteness' of maximal supergravity at this loop order. Diagrams (a)-(g) can be analysed using iterated 2-particle cuts, leading to an expectation of ultraviolet divergence cancellation. Diagrams (h) and (i) cannot be treated this way, but the result of summing all diagrams (a-i) is a deeper cancellation of the leading UV behaviour than anticipated.