Conformal Bootstrap in Mellin Space

TL;DR

This paper develops a Mellin-space version of the conformal bootstrap using Polyakov-type crossing-symmetric Witten exchange diagrams as the fundamental building blocks. Spurious pole cancellation in the Mellin amplitude yields infinite sets of constraints on operator dimensions and OPE coefficients, which the authors exploit to analyze the Wilson-Fisher fixed point in $d=4-\epsilon$. They reproduce known results for $\Delta_{\phi}$ and $\Delta_{\phi^2}$ to high order in $\epsilon$, and derive new OPE data, including higher-spin anomalous dimensions, OPE coefficients, and the central charge $c_T$ to $O(\epsilon^3)$, with improved agreement for the 3d Ising model at $\epsilon=1$. The work demonstrates analytic leverage of Mellin-space techniques and hints at deeper connections to AdS/CFT and potential dual descriptions, providing a complementary and potentially simpler route to CFT data than traditional diagrammatic methods.

Abstract

We propose a new approach towards analytically solving for the dynamical content of Conformal Field Theories (CFTs) using the bootstrap philosophy. This combines the original bootstrap idea of Polyakov with the modern technology of the Mellin representation of CFT amplitudes. We employ exchange Witten diagrams with built in crossing symmetry as our basic building blocks rather than the conventional conformal blocks in a particular channel. Demanding consistency with the operator product expansion (OPE) implies an infinite set of constraints on operator dimensions and OPE coefficients. We illustrate the power of this method in the epsilon expansion of the Wilson-Fisher fixed point by reproducing anomalous dimensions and, strikingly, obtaining OPE coefficients to higher orders in epsilon than currently available using other analytic techniques (including Feynman diagram calculations). Our results enable us to get a somewhat better agreement of certain observables in the 3d Ising model, with the precise numerical values that have been recently obtained.

Figures (2)

• Figure 1: Plot of $\beta_\ell$ as a function of $t$ for different values of $\epsilon$.
• Figure 2: Crossed channel sum over spins at $O(\epsilon^4)$ is absolutely convergent. Here $r_\ell=c_{\Delta,\ell} q_{\Delta,\ell}^{(2,t)}$ and $\beta_\ell=|\frac{r_{\ell+2}}{r_\ell}|$. In the $\epsilon$-expansion only the higher spin currents $J^{(\ell)}$ contribute at this order. Assuming this continues to hold for $\epsilon\sim O(1)$ and using our $O(\epsilon^3)$ results, we have obtained the above plot.