Efficient Conformal Block Evaluation with GoBlocks

Abstract

Conformal blocks in odd spacetime dimensions are not known in closed analytic form. To facilitate efficient computations in the conformal bootstrap, we introduce $\texttt{GoBlocks}$: a novel conformal-block generator implemented in the Go programming language, designed for rapid, on-the-fly, parallel evaluation using recursive relations. The package supports both multi-point and derivative-based bootstrap approaches and allows flexible control over accuracy and performance. We benchmark $\texttt{GoBlocks}$ against the $\texttt{scalar_blocks}$ package, finding significant speed improvements in applications where computational speed and moderate accuracy are critical, but ultra-high precision is not essential. As an illustration, we apply $\texttt{GoBlocks}$ to the mixed-correlator bootstrap of the three-dimensional Ising model, formulated as a non-convex optimisation problem in a suitable truncation scheme. We simultaneously optimise over external scaling dimensions and OPE CFT data. In addition, we discuss how the approach scales as we increase the number of mixed correlators in more general $O(N)$ vector models.

Figures (8)

• Figure 1: Instability of the recursive block algorithm for large cross-ratios and external operator dimensions when calculating $g^{\Delta_{ij}, \Delta_{kl}}_{\Delta, \ell}(u,v)$. The recursive algorithm generally fails to converge if $\Delta_{ij}$ and $\Delta_{kl}$ are large and the real part of $z$ is large.
• Figure 2: Runtime of scalar_blocks as order varies, with and without processing to construct block derivatives and convert to derivatives of $F_\pm$. Without post-processing, scalar_blocks runs in under a second for most order values. With post-processing, the runtime can increase to two seconds.
• Figure 3: Runtime of GoBlocks for different parameter settings. The runtime increases with ell_max and decreases with tol, but stays below around $0.15$ over a large range of parameters.
• Figure 4: Accuracy of GoBlocks as ell_max varies, evaluated over an aggregated average for $0\leq\ell\leq6$. The accuracy varies from 2.6% to 0.015% as ell_max increases from $6$ to $12$.
• Figure 5: Comparison of average runtimes between scalar_blocks and GoBlocks for given accuracies. Overall, GoBlocks is around five times faster than scalar_blocks between the 0.1% and 10% accuracy levels.