A new calculation obtains the long-distance expansion of vacuum mutual information between two disjoint balls in conformal field theories (CFTs) in any dimension d, to all orders. The formula avoids analytic continuation in the replica number, the delicate step of traditional methods, and reproduces known results in the two-point sector. In the two-dimensional Ising CFT, the team computes the first ten radial orders, which involve correlators with up to twenty spin insertions.

Mutual information quantifies how much information two separated regions share and is a central tool for studying entanglement in conformal theories. Previous calculations relied on the replica trick, which requires analytic continuation from an integer number of copies of the system, a technically fragile step that is hard to extend to high orders. The new approach reformulates the problem in terms of modular flow, the symmetry associated with the vacuum density matrix of a ball.

The derivation combines a radial sewing expansion of the joint vacuum density matrix with a modular response formula for relative entropy, using the geometric modular flow of a single ball. The expansion coefficients are modular integrals of vacuum correlators evolved with the modular flow. The full two-point sector resums into a modular integral of the thermal partition function on S^(d−1), with an effective temperature depending on the ball configuration and modular time.

In the two-dimensional Ising CFT, analytic reductions together with multidimensional numerical integration make it possible to reach the first ten radial orders. The truncated expansion agrees well with lattice data at finite separation and approximately satisfies crossing symmetry. The analytic calculations also reproduce known coefficients for the two-dimensional free compact boson at self-dual radius and its current sector, as well as for free scalars in higher dimensions.