Massive Higher-Spin Multiplets and Asymptotic Freedom in Quantum Gravity

We consider massive higher-spin multiplets that contain both standard and purely virtual degrees of freedom coupled to quantum gravity and compute their contributions to the gravitational beta functions at one loop. We extract the necessary number of fields to obtain asymptotic freedom by solving the relevant renormalization-group equations in the ultraviolet limit. We show that in the case of irreducible rank-$s$ bosons with minimal kinetic term only certain number of fields with $0\leq s \leq 3$ are compatible with asymptotic freedom. No solution involving fermionic rank-$s$ fields are found. We also consider the cases of rank-$1$ and symmetric rank-$2$ tensor fields with nonminimal kinetic terms and repeat the analysis, finding that the allowed number of fields is enlarged. Possible implications for metric affine theories of gravity are pointed out.