# Tripartite information at long distances

### Submission summary

 As Contributors: Cesar Agon · Pablo Bueno Preprint link: scipost_202111_00022v1 Date submitted: 2021-11-14 19:23 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.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2109.09179v2 on 17 March 2022

Submission scipost_202111_00022v1 on 14 November 2021

## Reports on this Submission

### Anonymous Report 2 on 2022-2-26 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202111_00022v1, delivered 2022-02-26, doi: 10.21468/SciPost.Report.4524

### Strengths

1- interesting topic
2- analytic results and corresponding numerical checks

### Weaknesses

1-straightforward application of existing concepts and methodologies

### Report

The authors investigate the tripartite information in higher dimensional CFT for three equal sphere.
In particular, they focus on the asymptotic regime where the distances between the spheres are much larger than their radius.
The analysis is performed by extending to this interesting case the techniques discussed in refs. [6] and [7].
The main analytic result is eq. (34), as clearly stated also by the authors, and I find it insightful also the analysis involving the conformal spinors in sec. 2.3. It is also important the numerical analysis reported in sec. 4.2: it has allowed to check the analytic result corresponding to the massless scalar and to find a numerical prediction for the fermionic model.

I find this paper very interesting and well written; hence I strongly recommend its publication in Scipost.

### Requested changes

I strongly suggest a little effort to improve the comparison of the results presented in this paper with the existing ones in 1+1 dimensions, starting e.g. from the following missing references:

1) the relevant paper 1011.5482, where the expansion of the mutual information in 1+1 CFT has been first studied
and the method employed through the paper has been established, finding also the result (19) in one space dimension.

2) a qualitative comparison can be discussed between the results obtained in this paper with the ones for the tripartite information obtained in 1+1 dimensions e.g. in 1309.2189 and 1501.04311

• validity: good
• significance: high
• originality: good
• clarity: good
• formatting: good
• grammar: excellent

### Anonymous Report 1 on 2022-1-4 (Invited Report)

• Cite as: Anonymous, Report on arXiv:scipost_202111_00022v1, delivered 2022-01-04, doi: 10.21468/SciPost.Report.4139

### Strengths

1- considers an important and timely problem: the tripartite information in CFTs, holographic and otherwise
2- presents detailed calculations and many novel results, including the qualitatively novel fact that the leading contribution to the tripartite information depends on a certain OPE coefficient (and not just the spectrum)
3- includes a thorough analysis of the result, and a check against lattice calculations
4- is very clearly written and organized

### Weaknesses

None that I could detect.

### Report

The manuscript has the strengths listed above, and no weakness that I could detect. It covers an important and timely subject in a comprehensive and clear manner. I recommend publications without changes, except a single typo I detected.

### Requested changes

Equation (22) has a typo in the form of a spurious + sign, which is somewhat confusing.

• validity: top
• significance: top
• originality: top
• clarity: top
• formatting: perfect
• grammar: excellent