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Applications of dispersive sum rules: $ε$-expansion and holography

by Dean Carmi, Joao Penedones, Joao A. Silva, Alexander Zhiboedov

This is not the current version.

Submission summary

As Contributors: Joao A. Silva
Arxiv Link: (pdf)
Date submitted: 2021-02-05 13:03
Submitted by: A. Silva, Joao
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in $d=4-\epsilon$ dimensions. We re-derive many of the known results to order $\epsilon^4$ and we make new predictions. No assumption of analyticity down to spin $0$ was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission scipost_202104_00018v1 on 15 April 2021

Reports on this Submission

Anonymous Report 2 on 2021-3-27 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2009.13506v2, delivered 2021-03-27, doi: 10.21468/SciPost.Report.2734


1. Compact and clear.
2. Useful new results.


1. Unnecessary paragraph breaks after equations, punctuation might help.
2. The appendices are a bit disjoint.


This paper provides interesting applications of Mellin space dispersion relations. It fits nicely into recent work developing this toolkit.

I would recommend this for publication in Sci Post.

Requested changes

It's fine as is. If the authors are going to make the changes recommended by the other reviewer they may want to take into account the comments above about the appendices, equation punctuations, and paragraph breaks, as well as standardize the spelling of Fisher's name.

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: good
  • grammar: perfect

Anonymous Report 1 on 2021-3-14 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2009.13506v2, delivered 2021-03-13, doi: 10.21468/SciPost.Report.2692


1) The paper is well-written and goes through many concrete applications for dispersive sum rules.

2) There are new results for the Wilson-Fisher in the $\epsilon$ expansion, including OPE coefficients at order $\epsilon^4$.

3) The dispersive sum rules are used to derive bounds on AdS EFTs and to determine tree and loop-level anomalous dimensions.


1) Its less clear which results in the holographic section are new and which ones are a rederivation of previous results.

From my understanding, the results for the one-loop bubble diagram and for the contribution of heavy operators to the sum rule are new and the results for scalar exchange diagrams and the bound on $(\partial \phi)^4$ couplings are rederivations of previous results. It may be useful to clarify this in the text.


The paper is well-written and presents new results on CFTs using dispersive sum rules. I would recommend this paper be published in scipost, with some minor changes.

Requested changes

1) It would useful to define the term "collinear family" and "collinear Mack polynomials". The authors are referring to the contribution of the leading SL(2,R) primaries, although this is not defined in the text.

2) At one-loop and in integer dimensions the bubble diagram has UV divergences. If possible, it would be interesting if the authors could explain how these UV divergences appear using their method.

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

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