Renormalization of conical entropy at higher orders

Determine whether the standard renormalization procedure for conical-singularity calculations of entanglement entropy remains well-defined at higher orders, particularly when second derivatives of the free energy introduce terms proportional to squared delta-function contributions localized on the horizon.

Background

The paper explains that the conical method produces ultraviolet divergences that can be absorbed into renormalized gravitational couplings at leading order and, for suitable theories, at first order in the conical deficit. However, computing higher thermodynamic derivatives involves products of distributions supported on the horizon.

Specifically, the heat capacity requires second derivatives of the free energy, for which the conical expansion contains a contribution behaving as an integral of the square of a delta function. The authors explicitly leave unresolved whether the usual renormalization framework can consistently handle these higher-order terms.

References

It is also important to note that the conical method is not suitable for computing second derivatives of the free energy, such as those needed for the heat capacity, because at second order the free energy contains a contribution that behaves like $\int dV\delta2(\Sigma)$. Consequently, it is unclear whether the standard renormalization procedure remains well-defined at higher orders.

— Entanglement entropy of black holes from the first law of thermodynamics  (2610.11777 - Diakonov et al., 8 Oct 2026) in Section 1, Introduction