A note on the UV behaviour of maximally supersymmetric Yang-Mills theories

TL;DR

This paper investigates the ultraviolet properties of maximally supersymmetric Yang-Mills theories, confirming protection of certain BPS invariants and identifying unprotected non-BPS invariants through algebraic renormalisation.

Contribution

It provides a theoretical analysis confirming the protection of one-half BPS invariants and the non-protection of non-BPS invariants in specific dimensions and loop orders.

Findings

One-half BPS invariants ($F^4$) are protected.

Double-trace one-quarter BPS invariant ($d^2F^4$) is unprotected at two loops in D=7.

Non-BPS invariants are unprotected.

Abstract

The question of whether BPS invariants are protected in maximally supersymmetric Yang-Mills theories is investigated from the point of view of algebraic renormalisation theory. The protected invariants are those whose cohomology type differs from that of the action. It is confirmed that one-half BPS invariants ($F^4$) are indeed protected while the double-trace one-quarter BPS invariant ($d^2F^4$) is not protected at two loops in D=7, but is protected at three loops in D=6 in agreement with recent calculations. Non-BPS invariants, i.e. full superspace integrals, are also shown to be unprotected.