On anomaly-free 4d $$ \mathcal{N} $$ = 4 and 6d (2,0) conformal supergravities and UV finiteness of Poincaré supergravities

A bstract We review the structure of superconformal anomalies in 4d $$ \mathcal{N} $$ N = 4 conformal supergravity (CSG) coupled to a number N v of $$ \mathcal{N} $$ N = 4 vector multiplets and 6d (2,0) CSG coupled to N T of (2,0) tensor multiplets. Anomalies cancel if N v = 4 and N T = 26 respectively. If the CSG part of the action is dropped and N v = 6 + n v , the first theory is classically equivalent to the 4d $$ \mathcal{N} $$ N = 4 Poincaré supergravity (PSG) coupled to n v vector multiplets, while the second one with N T = 5 + n T is classically equivalent to the 6d (2,0) PSG coupled to n T tensor multiplets. We suggest that these facts imply that divergences in the 4d PSG with n v vectors should be proportional to n v + 2 and similarly in the 6d PSG with n T tensors to n T – 21. This conjecture appears to be consistent with most of the known results of explicit scattering amplitude computations in these 4d and 6d PSG theories, apart from one coefficient in the 3-loop mixed vector scattering amplitude computed in arXiv:1305.4876.