SciPost logo

SciPost Submission Page

Bulk locality from the celestial amplitude

by Chi-Ming Chang, Yu-tin Huang, Zi-Xun Huang, Wei Li

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

As Contributors: Chi-Ming Chang
Preprint link: scipost_202109_00019v2
Date submitted: 2022-01-28 04:16
Submitted by: Chang, Chi-Ming
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


In this paper, we study the implications of bulk locality on the celestial amplitude. In the context of the four-point amplitude, the fact that the bulk S-matrix factorizes locally in poles of Mandelstam variables is reflected in the imaginary part of the celestial amplitude. In particular, on the positive real axes in the complex plane of the boost weight, the imaginary part of the celestial amplitude can be given as a positive expansion on the Poincar\'e partial waves, which are nothing but the projection of flat-space spinning polynomials onto the celestial sphere. Furthermore, we derive the celestial dispersion relation, which relates the imaginary part to the residue of the celestial amplitude for negative even integer boost weight. The latter is precisely the projection of low energy EFT coefficients onto the celestial sphere. We demonstrate these properties explicitly on the open and closed string celestial amplitudes. Finally, we give an explicit expansion of the Poincar\'e partial waves in terms of 2D conformal partial waves.

Current status:
Has been resubmitted

List of changes

1. The “[]” above (2.6) is removed.
2. A typo in (5.1) is corrected.
3. Sentences added around (5.5) to address referee's comment.

Reports on this Submission

Anonymous Report 2 on 2022-3-8 (Invited Report)


The authors have addressed all the comments in my previous report. It is ready for publication in my opinion.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2022-2-27 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202109_00019v2, delivered 2022-02-27, doi: 10.21468/SciPost.Report.4534


The celestial holography program aims to construct a 2D holographic theory, a Celestial Conformal Field Theory (CCFT), that computes scattering amplitudes for 4D quantum gravity in asymptotically flat space as 2D conformal correlators (also known as celestial amplitudes) on the celestial sphere at null infinity. So far a successful example of CCFT has not been constructed yet, and it is an active area of research to understand the properties of such a theory. Other than matching symmetries on the bulk and boundary sides, there is more information about the bulk physics encoded in the celestial amplitudes. Inspired by the parallel development of the S-matrix bootstrap program, one is led to ask: how are bulk unitarity and locality reflected on the analytic structures of celestial amplitudes? This work takes the initiative to explore this direction, with the focus on 4-point massless celestial amplitudes. Here I would like to highlight a few results that I believe will be useful for this research program:

1. The formula (3.5) for the imaginary part of the celestial amplitudes of 4-point massless particles.

2. Positivity of the imaginary part of the celestial amplitude expanded in Poincare partial waves.

3. The discussion of the analytic continuation of the celestial amplitudes in the cross ratio $z$.

4. Dispersion relation (6.4) of celestial amplitudes.

These results are novel, and are checked analytically and numerically with massless scalars and string theory examples. This work will help build a bootstrap approach to celestial amplitudes. The present manuscript is reasonably well-written, and the technical arguments are, for the most part, presented clearly and easy to follow. I would like to make a few comments and recommend some clarifications (see below). Once these are addressed, I would be happy to recommend the present manuscript for publication in SciPost.

Requested changes

1-Before (2.18) it says ``celestial scalar amplitudes", so it is confusing that equations (2.18), (2.19), (2.23) have helicity ($\ell_i$) dependence.

2-After (2.25), in $s$-kinematics we should have $z\geq 1$.

3-I believe the result (2.33) was first obtained in reference [22] and therefore they should be mentioned.

4-It should be mentioned that (3.6) is for scalars.

5-Even though it is somewhat clear from the context, $m,m_i,J,J_i$ are not defined in section 3.2. Also, it seems that there should be a sum over $J_i$ in (3.7) and (3.10). The left hand side of these expressions are certainly not functions of $J$'s.

6-It seems that (3.24) only captures the contributions from the factorization poles (labeled by $a$). It should be clarified. Also, there should be a sum over $J_a$.

7-Related to the last point, I wonder if the authors have any comments on the positivity for the part associated with the branch cut in (3.5).

8-Footnote 6 suggests that the authors have worked out the case for odd spins. I recommend that the more general expressions for all spins should be included.

9-Is (5.5) only true for 4 external scalars, or is it true for any massless particles of any spins?

10-It would be great if the authors can comment on the motivations/justifications for the two assumptions after (6.7).

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

Author:  Chi-Ming Chang  on 2022-03-12  [id 2285]

(in reply to Report 1 on 2022-02-27)

We thank the referee for the comments. We will resubmit the paper with the changes listed below to address referee's points.

1.The “scalar” has been removed. 2. Typo fixed. 3. Reference added. 4. A sentence added above (3.6). 5-7. Sentences addressing these two points are added around (3.8). 8-9. Original footnote 6 removed. Sentences added below (5.5). 10. Explanations added below the analyticity assumption. The boundedness condition is moved to Section 3.1 with some justifications added below it.

Login to report or comment