SciPost logo

SciPost Submission Page

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

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

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Joao A. Silva
Submission information
Preprint Link: scipost_202104_00018v1  (pdf)
Date accepted: 2021-06-03
Date submitted: 2021-04-15 18:07
Submitted by: A. Silva, Joao
Submitted to: SciPost Physics
Ontological classification
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.

Author comments upon resubmission

We thank the referees for the suggestions they made. In our resubmission, we address all of the comments made by the referees.

List of changes

- we clarify which results in the holographic section are new and which results are a rederivation of previous results in footnote 8.

- we clarify the use of our word "collinear", see in page 22 the phrase : "The collinear Mack polynomial captures the contribution of the leading SL(2,R) exchanged primaries".

- one referee asked how does the UV divergence of the phi^4 one loop diagram affect our functional. We explain this on page 23 in the paragraph that starts by "The one loop diagram ...".

The UV divergence of the one loop diagram corresponds to a divergent counterterm for the quartic coupling. We know that a purely quartic interaction (no derivatives) only contributes to the anomalous dimensions of spin 0 double twist operators. Therefore, the UV divergence should not appear in our formula for the one loop anomalous dimensions, which is valid only for spin greater or equal to 2.

- We corrected our spelling of Fisher's name. We fixed the spacing and punctuation in some of the equations.

Published as SciPost Phys. 10, 145 (2021)

Login to report or comment