# Conformal partial waves in momentum space

### Submission summary

 As Contributors: Marc Gillioz Arxiv Link: https://arxiv.org/abs/2012.09825v3 (pdf) Date accepted: 2021-03-26 Date submitted: 2021-03-09 13:07 Submitted by: Gillioz, Marc Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

The decomposition of 4-point correlation functions into conformal partial waves is a central tool in the study of conformal field theory. We compute these partial waves for scalar operators in Minkowski momentum space, and find a closed-form result valid in arbitrary space-time dimension $d \geq 3$ (including non-integer $d$). Each conformal partial wave is expressed as a sum over ordinary spin partial waves, and the coefficients of this sum factorize into a product of vertex functions that only depend on the conformal data of the incoming, respectively outgoing operators. As a simple example, we apply this conformal partial wave decomposition to the scalar box integral in $d = 4$ dimensions.

Published as SciPost Phys. 10, 081 (2021)

I would like to thank both referees for their useful comments. I have incorporated all suggested changes in the new version.

As pointed out by one of the referees (Slava Rychkov), the status of the time-ordered functions in CFT is not fully satisfactory. To get a completely unambiguous definition, one should probably begin with considerations about advanced and retarded commutators. This is mentioned at the bottom of page 17 (unchanged from the first version). In fact, this is an issue that I am currently investigating as part of another project, and while it is not possible for me to develop too much on this rich topic in the current manuscript, I can give you a quick summary of what might be the essential points.

In the case where only two operators are time-ordered, as in the vertex functions appearing in section 3, the time-ordered product can be constructed from Wightman functions using the Jost-Lehmann-Dyson representation. This is a representation for the commutator of two operators. The commutator is certainly a tempered distribution as it is the difference between two Wightman functions. From the JLD representation, one can construct retarded and advanced functions, as well as a time-ordered function that corresponds to the difference between the retarded function and one of the Wightman functions. I believe that this construction gives a precise definition of the time-ordered product of 2 operators, and that it establishes rigorously its domain of analyticity.

For time-ordered products involving more than 2 operators, the problem is more difficult. It might still be possible to give a precise definition starting with R-products and using ideas that date back to the work of Steinmann, but this is certainly not easy. From this perspective, the CFT optical theorem (4.2) is not established on rigorous grounds, but its validity has been tested in several examples (including this work).

### List of changes

- Both referees have pointed out that some of the results in section 2 were lacking proofs. I have added a technical appendix with proofs of eqs. (2.18), (2.19), (2.21), (2.23), (2.27), and reference to this appendix in the text before eqs. (2.18), (2.23) and (2.27).

- I have remodeled section 2.4: instead of writing the completeness relation as (identity) = (sum over spin eigenstates), which as pointed out by one of the referees was only true in some vaguely defined subspace, I have given a new derivation of eq. (2.38) starting with (2.31). The new derivation is more rigorous and hopefully easier to follow for the reader.

- On page 12, before eq. (3.3), I have added a footnote that explains why no action on spin indices of O appears in that equation.

- On page 15 and beginning of p. 16, I have extended the discussion of the Wightman 3-point function, in particular about the matching condition between eq. (3.27) and the results of ref. [64]. I did not extend the discussion of the time-ordered function as it was already more detailed than that of the Wightman function.

### Submission & Refereeing History

Resubmission 2012.09825v3 on 9 March 2021
Submission 2012.09825v2 on 29 December 2020

## Reports on this Submission

### Anonymous Report 2 on 2021-3-17 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2012.09825v3, delivered 2021-03-17, doi: 10.21468/SciPost.Report.2710

### Report

The author has addressed all the points raised in my previous report. I especially appreciate the edits to section (2.4), which I now find much easier to understand. I recommend publication in the present form.

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

### Report 1 by Slava Rychkov on 2021-3-16 (Invited Report)

• Cite as: Slava Rychkov, Report on arXiv:2012.09825v3, delivered 2021-03-16, doi: 10.21468/SciPost.Report.2705

### Report

The revision addressed the bulk of my concerns. Some more concerns were answered in the author's reply (although those remarks did not enter the manuscript). I am recommending the paper for publication in SciPost.

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