Combined fixed-order and effective-theory approach to $b \bar{b}$ sum rules

Understand

We combine the fixed-order evaluation of the $b\bar{b}$ sum rules with a non-relativistic effective-theory approach.

• The combined result for the $n$-th moment includes all terms suppressed with respect to the leading-order result by ${\cal O}(\alpha_s^3)$ and ${\cal O}((\alpha_s \sqrt{n})^l \alpha_s^2)$, counting $\alpha_s \sqrt{n} \sim 1$.
• When compared to experimental data, the moments thus obtained show a remarkable consistency and allow for an analysis in the whole range $1\le n\lesssim 16$.