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In this work, we determine the short-distance coefficients for $\Upsilon$inclusive decay into a charm pair through relative order $v^4$ within theframework of NRQCD factorization formula. The short-distance coefficient of theorder-$v^4$ color-singlet NRQCD matrix element is obtained through matching thedecay rate of $b\bar{b}(^3S^{[1]}_1)\to c\bar{c}gg$ in full QCD to that inNRQCD. The double and single IR divergences appearing in the decay rate areexactly canceled through the next-to-next-to-leading-order renormalization ofthe operator ${\mathcal O}(^3S^{[8]}_1)$ and the next-to-leading-orderrenormalization of the operators ${\mathcal O}(^3P^{[8]}_J)$. To investigatethe convergence of the relativistic expansion arising from the color-singletcontributions, we study the ratios of the order-$v^2$ and -$v^4$ color-singletshort-distance coefficients to the leading-order one. Our results indicate thatthough the order-$v^4$ color-singlet short-distance coefficient is quite large,the relativistic expansion for the color-singlet contributions in the process$\Upsilon\to c\bar{c}+X$ is well convergent due to a small value of $v$. Inaddition, we extrapolate the value of the mass ratio of the charm quark to thebottom quark, and find the relativistic corrections rise quickly with increaseof the mass ratio.

Order-$v^4$ Relativistic Corrections to $\Upsilon$ Inclusive Decay into a Charm pair