# Tripartite information at long distances

### Submission summary

 As Contributors: Cesar Agon · Pablo Bueno Arxiv Link: https://arxiv.org/abs/2109.09179v2 (pdf) Date accepted: 2022-04-22 Date submitted: 2022-03-17 05:06 Submitted by: Agon, Cesar Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

We compute the leading term of the tripartite information at long distances for three spheres in a CFT. This falls as $r^{-6\Delta}$, where $r$ is the typical distance between the spheres, and $\Delta$, the lowest primary field dimension. The coefficient turns out to be a combination of terms coming from the two- and three-point functions and depends on the OPE coefficient of the field. We check the result with three-dimensional free scalars in the lattice finding excellent agreement. When the lowest-dimensional field is a scalar, we find that the mutual information can be monogamous only for quite large OPE coefficients, far away from a perturbative regime. When the lowest-dimensional primary is a fermion, we argue that the scaling must always be faster than $r^{-6\Delta_f}$. In particular, lattice calculations suggest a leading scaling $r^{-(6\Delta_f+1)}$. For free fermions in three dimensions, we show that mutual information is also non-monogamous in the long-distance regime.

Published as SciPost Phys. 12, 153 (2022)

### List of changes

From report 1:

We corrected Eq. (20). Changed spurious + sign to a $\times$ sign

From report 2:

We added reference [7] (arXiv:1011.5482) in the second paragraph of page 2 to appropriately cite the coefficient in the long-distance mutual information of disjoint intervals in CFT (spheres in $d=2$). In this regard, we added some extra comments in footnote 2, appearing on page 5.

We added a paragraph on page 9 (the last paragraph of the page), where we explain the prospects of comparing our long-distance tripartite information result Eq (34) with the exact results in $d=2$ presented in references [28] (arXiv:1309.2189) and [29] (arXiv:1501.04311).