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

TL;DR

This work develops dispersive sum rules for CFTs in Mellin space, leveraging Polyakov conditions to suppress double-twist contributions and extract OPE data. It systematically applies the framework to the Wilson-Fisher fixed point in d=4-ε, reproducing known results to ε^4 and delivering new predictions for twist-4 data and OPE coefficients, all without assuming analyticity down to spin zero. The authors extend the approach to holographic CFTs, deriving tree-level and one-loop anomalous dimensions from dispersive sum rules and exploring the role of heavy operators in UV-complete theories. Overall, the paper provides a versatile, nonperturbative toolkit for obtaining precise CFT data and establishing connections between IR dispersive data and UV holographic completions.

Abstract

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 $d=4-ε$ dimensions. We re-derive many of the known results to order $ε^4$ 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.

Figures (5)

• Figure 1: The Mellin-Mandelstam plane. The axis at $120^\circ$ ensure that every point on the plane satisfies $\gamma_{12}+\gamma_{13}+\gamma_{14}=\Delta$. The accumulation points $\gamma_{12}=-n$ for $n=0, 1, ...$ are shown in blue. The region of convergence for sum rules with a function $F$ with a simple/double pole at one of these points is shown in red/green. Notice that the red region contains the green region.
• Figure 2: A plot of the $\omega_{1,1,1}^{\tau,\ell} \equiv \sum_{m=0}^\infty \omega_{1,1,1}^{\tau,\ell,m}$ functional as a function of the twist $\tau$ of the exchanged operator. We picked $\Delta= \frac{3}{5}$, $\gamma_{13} = \frac{3}{4}$, $\ell=2$ and $d=3$. The plot is qualitatively the same for other values. We sum in $m$ from $0$ to $50$, since this is enough to have an accurate plot of the functional. Notice that the functional does not vanish for twists $\tau= 2\Delta=1$ and $\tau= 2\Delta +2=3$. However, it has double zeros for all $\tau= 2 \Delta + 2n$, for $n \geq 2$.
• Figure 3: On the left, a plot of the $\omega_{1,0,0}$ functional as a function of the twist $\tau$ of the exchanged operator. We picked $\Delta= 1.1$, $\gamma_{13} =\frac{1}{3}$, $\ell=2$ and $d=4$. The plot is qualitatively similar for other values. We sum in $m$ from $0$ to $50$, since this is enough to have an accurate plot of the functional. Notice that the functional has single zeros for twists $\tau= 2\Delta=2$ (though it is not very clear from the left plot) and $\tau= 2\Delta +2=4$. However, it has double zeros for all $\tau= 2 \Delta + 2n$, where $n \geq 2$. On the right, we zoom in to the region around $\tau= 2\Delta$, so that we can observe the single zero at $\tau= 2\Delta$.
• Figure 4: Partial sum $S_ J(\gamma_{13})$ defined in \ref{['eq:partsum']} for $d=3$ and $\Delta=\frac{3}{5}$. On the left, one can see that only for $\Delta <\gamma_{13} <1$ the partial sum converges to zero as expected. On the right, we fix $\gamma_{13}=\frac{4}{5}$ and use a log-log plot to exhibit the large $J$ behavior predicted by \ref{['eq:log-logslope']}. The straight orange line is a fit (to the points $20\le J \le 60$) with slope given by \ref{['eq:log-logslope']}.
• Figure 5: The functional $\Lambda_{\tau,\ell,m}$ given in \ref{['eq:pdkkkA']}, and its contribution to the sum rule \ref{['eq:ghj0A']}. The red dots signify double zeros of the functional, $\Lambda_{\tau,\ell,m}$, and the blue dots signify $single$ zeros of the functional. At subleading order, the sum rule \ref{['eq:ghj0A']} receives two types of contributions. A contribution at $\ell=0$ and $n=0,1,2,\ldots$ coming from contact diagrams in AdS. And a contribution from the leading tracjectory $\tau=2\Delta$ and $\ell=2,4,6, \ldots$ coming from the 1-loop bubble diagrams in AdS.