Universal correction to higher spin entanglement entropy

We consider conformal field theories in $1+1$ dimensions with $\mathcal{W}$-algebra symmetries, deformed by a chemical potential $\mu$ for the spin-three current. We show that the order ${\mu }^2$ correction to the Rényi and entanglement entropies of a single interval in the deformed theory, on the infinite spatial line and at finite temperature, is universal. The correction is completely determined by the operator product expansion of two spin-three currents, and by the expectation values of the stress tensor, its descendants and its composites, evaluated on the $n$-sheeted Riemann surface branched along the interval. This explains the recently found agreement of the order ${\mu }^2$ correction across distinct free field CFTs and higher spin black hole solutions holographically dual to CFTs with $\mathcal{W}$ symmetry.