Higher-order condensate corrections to $\Upsilon$ masses, leptonic decay rates and sum rules

With the recent completion of NNNLO results, the perturbative description of the $\Upsilon$ system has reached a very high level of sophistication. We consider the non-perturbative corrections as an expansion in terms of local condensates, following the approach pioneered by Voloshin and Leutwyler. The leading order corrections up to dimension eight and the potential NLO corrections at dimension four are computed and given in analytical form. We then study the convergence of the expansion for the masses, the leptonic decay rates and the non-relativistic moments of the $\Upsilon$ system. We demonstrate that the condensate corrections to the $\Upsilon(1S)$ mass exhibit a region with good convergence, which allows us to extract $\overline{m}_b(\overline{m}_b) = 4214\pm37\,(\text{pert.})\,_{-22}^{+20}\,(\text{non-pert.})\text{ MeV}$, and show that non-perturbative contributions to the moments with $n\approx10$ are negligible.