Exact results for corner contributions to the entanglement entropy and Renyi entropies of free bosons and fermions in 3d

In the presence of a sharp corner in the boundary of the entanglement region, the entanglement entropy (EE) and Renyi entropies for 3d CFTs have a logarithmic term whose coefficient, the corner function, is scheme-independent. In the limit where the corner becomes smooth, the corner function vanishes quadratically with coefficient $\sigma$ for the EE and $\sigma_n$ for the Renyi entropies. For a free real scalar and a free Dirac fermion, we evaluate analytically the integral expressions of Casini, Huerta, and Leitao to derive exact results for $\sigma$ and $\sigma_n$ for all $n=2,3,\dots$. The results for $\sigma$ agree with a recent universality conjecture of Bueno, Myers, and Witczak-Krempa that $\sigma/C_T = \pi^2/24$ in all 3d CFTs, where $C_T$ is the central charge. For the Renyi entropies, the ratios $\sigma_n/C_T$ do not indicate similar universality. However, in the limit $n \to \infty$, the asymptotic values satisfy a simple relationship and equal $1/(4\pi^2)$ times the asymptotic values of the free energy of free scalars/fermions on the $n$-covered 3-sphere.