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

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 these is suppression the contribution double twist operators. Firstly, we apply Wilson-Fisher model in $d=4-\epsilon$ dimensions. re-derive many known results order $\epsilon^4$ and make new predictions. No assumption analyticity down spin $0$ was made. Secondly, study holographic CFTs. dispersive obtain tree-level one-loop anomalous Finally, briefly discuss heavy operators UV complete theories.