# Area law and OPE blocks in conformal field theory

### Submission summary

 As Contributors: Jiang Long Preprint link: scipost_202010_00028v4 Date submitted: 2021-05-21 03:50 Submitted by: Long, Jiang Submitted to: SciPost Physics Proceedings Proceedings issue: 4th International Conference on Holography, String Theory and Discrete Approach in Hanoi Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

This is an introduction to the relationship between area law and OPE blocks in conformal field theory.

###### Current status:
Editor-in-charge assigned

1) The referee says: I still fail to understand the purpose of c. I can't find where a canonical normalization for Q is given. So why can't c be removed completely?

Our response: the constant c can be removed in principle. However, sometimes the normalization of the operator O is given, then Q still has the freedom to choose its own normalization. For example, usually, the stress tensor in integral of the modular Hamiltonian is defined unambiguously, then one should choose c=2\pi such that tr_A \rho_A=1.

2) The referee says: This sentence seems to imply that for m>2, D is not related to the normalization of operators. If so, why?

Our response: I rewrote the sentence. D also depends on the normalization of the operators.

3) The referee says: The universal constant $p_q$ in (3.1) depends on your operator normalization. How are you normalizing O?

Our response: I agree the constant $p_q$ depends on the operator normalization. I don't choose a definite normalization when I state general results. The normalization doesn't affect the validity of my results.

4) The referee says: Are you saying that there is no unambiguous way to regularize the divergences for non-integer weights...

Our response: No. I rewrote the sentence. I can't regularize the integral by straightforward computation for non-integer weights, but this doesn't rule out the possibility that the constant $p_q$ is still defined unambiguously by other means.

### List of changes

1-line 25-26: I have changed the sentence to "...is a relatively unexplored topic in conformal field theory, though it has been defined and discussed at the early stages of conformal field theory."

2-line 41-42: I put "area law" in quotation marks and add a footnote to explain it.

3-line 42-43: I changed the sentence as the referee suggested.

4-line 47-48: I changed the sentence to "In all examples we studied, we found q = 0, 1, 2, but in general we do not rule out the possibility of other values."

5-line 85: I changed "nature number" in the original version to "nonnegative number".

6-line 163 : I rewrote the sentence as " we have restored the radius R that was previously set to 1". There are similar modification in line 229-230.

7-line 223-224: I changed the sentence to "For $m\ge 2$, the coefficients $D^{(d)}[\mathcal{O}_1, cdots,\mathcal{O}_m]$ are related to the normalization of the primary operators. For any $m\ge 3$, it also contains dynamical information of the theory."

8-(3.1): I added a footnote 3 to address the point that $p_q$ also depends on the normalization of the operators.

9-(2.47): I added a footnote 2 to explain equation (2.47).

10-line 312: I deleted original argument on cyclic identity, just mention that we could read out a cyclic identity (3.29) from (3.28).

11-line 334-336: I rewrote a comment on the regularization of the integral for non-integer conformal weight.

12-line 337: I changed the sentence to "For m\ge 4, the cyclic identity are ..."

13: Iine 28-30: I changed the sentence as the referee suggested.

14: line 262: I change "far away to" to "far away from".

### Submission & Refereeing History

Resubmission scipost_202010_00028v4 on 21 May 2021
Resubmission scipost_202010_00028v3 on 21 February 2021
Resubmission scipost_202010_00028v2 on 30 December 2020
Submission scipost_202010_00028v1 on 30 October 2020