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Conformal partial waves in momentum space

by Marc Gillioz

This is not the current version.

Submission summary

As Contributors: Marc Gillioz
Arxiv Link: (pdf)
Date submitted: 2020-12-29 17:08
Submitted by: Gillioz, Marc
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2021-2-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2012.09825v2, delivered 2021-02-01, doi: 10.21468/SciPost.Report.2486


The paper computes relatively explicit expressions for scalar conformal blocks in momentum space in Lorentzian signature. While momentum space is often not very natural in a conformal setting, it does appear useful for some applications. For example, the results of this paper provide explicit expressions for scalar conformal blocks in momentum space as analytic functions of the number of space-time dimensions. In position space such expressions are not known, which makes this result especially interesting.

The paper is well organized and the exposition is, for the most part, very clear. I recommend to publish the paper after a minor revision as detailed below.

I only have a couple requests for clarification of some key points. First of all, it would be good to clarify more explicitly for the reader what determines the exact expression in eq (2.18) (e.g. tracelessness, orthogonality to p, ...). The wording "one can take" right above (2.18) suggests some arbitrariness.

Secondly, is it possible to clarify the meaning of (2.32)? Both equations (2.31) and (2.32) are a bit misleading (as clarified below (2.32)), but in the case of (2.31) one can at least directly interpret it as the restriction of the identity operator to some subspace. It does not seem to be possible to interpret (2.32) in this way, at least not naively: morally, (2.32) is supposed to act on polarizations with vector q' but return polarizations with vector q. Ideally, I would suggest expanding a bit more both on the technical meaning and the derivation of (2.32).

Finally, for the reader's benefit, it would be useful to add a short comment about why no action on spin indices of O appears in (3.3). Or, in a sense, why no "action on O" is present. Right now the reader is referred to [65,66], while the argument is rather short: if I understand correctly, it follows because the << ... >> expectation value can be rewritten as <f(p1) f(p2) O(x=0)>, and K acts trivially on O(x=0).

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

Report 1 by Slava Rychkov on 2021-1-30 (Invited Report)

  • Cite as: Slava Rychkov, Report on arXiv:2012.09825v2, delivered 2021-01-30, doi: 10.21468/SciPost.Report.2481


This is a nice paper with many nontrivial computations. The main result is a representation of a scalar conformal four point function in momentum space as a sum of explicitly given blocks (which can be thought of as conformal blocks in momentum space). I believe this explicit representation is new.

The paper is a bit terse at times. The author states many results without derivation. It’s not clear if the author has a proof of these formulas or if he just checked them in a few cases and conjectures that they should be generally true. E.g. does he have a general proof of 2.27? It would be responsible to provide such information for all nontrivial combinatorial identities (following the tradition of Dolan and Osborn).

I would also prefer to see more details about the exact matching conditions mentioned but not stated on p.15 which fix the form of the Wightman and time-ordered functions.

I should say that I’m confused about the status of time-ordered three-point functions.
Wightman three point functions are conformally invariant tempered distributions, hence their F.T.’s satisfy conformal Ward identities. Time ordered three point functions involve multiply Wightman three point functions by theta-functions and combining the pieces. Of course when we multiply a distribution by a theta-function it’s no longer obvious that the result is a distribution. Has it been proven that this operation can be done in a way that time ordered three point functions are conformally invariant tempered distributions? The author seems to tacitly assume that this is the case. A more detailed discussion and references concerning this subtlety would be welcome. Ref. [85] mentioned in the conclusions, and our future work to appear soon, only treat the case of Wightman functions, so it cannot be used for justification of the time-ordered case.

A similar worry concerns the CFT optical theorem, Eq. (4.2). Is it affected by the unclear status of time-ordered functions as distributions?

A revision along these lines would improve the paper, and I’d be happy to see it.

In preparing this report I benefitted from the discussions with Jiaxin Qiao.

Requested changes

I don't make any hard requirements but I'd be happy if the author tries to improve the paper along the lines stated in the report.

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

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