Ultraviolet structure of half-maximal supergravity with matter multiplets at two and three loops

Using the duality between color and kinematics, we construct the two- and three-loop amplitudes of half-maximal supergravity with matter multiplets and show that new divergences occur in $D=4$ and $D=5$. In a recent paper, Bossard, Howe and Stelle conjecture the existence of 16-supercharge off-shell harmonic superspaces in order to explain the ultraviolet finiteness of pure half-maximal supergravity with no matter multiplets in $D=4$ at three loops and in $D=5$ at two loops. By assuming the required superspace exists in $D=5$, they argue that no new divergences should occur at two loops even with the addition of Abelian-vector matter multiplets. Up to possible issues with the $SL(2,\mathbb{R})$ global anomaly of the theory, they reach a similar conclusion in $D=4$ for two and three loops. The divergences we find contradict these predictions based on the existence of the desired off-shell superspaces. Furthermore, our $D=4$ results are incompatible with the new divergences being due to the anomaly. We find that the two-loop divergences of half-maximal supergravity are directly controlled by the divergences appearing in ordinary nonsupersymmetric Yang-Mills theory coupled to scalars, explaining why half-maximal supergravity develops new divergences when matter multiplets are added. We also provide a list of one- and two-loop counterterms that should be helpful for constraining any future potential explanations of the observed vanishings of divergences in pure half-maximal supergravity.

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