  In this work, we determine the short-distance coefficients for $\Upsilon$
inclusive decay into a charm pair through relative order $v^4$ within the
framework of NRQCD factorization formula. The short-distance coefficient of the
order-$v^4$ color-singlet NRQCD matrix element is obtained through matching the
decay rate of $b\bar{b}(^3S^{[1]}_1)\to c\bar{c}gg$ in full QCD to that in
NRQCD. The double and single IR divergences appearing in the decay rate are
exactly canceled through the next-to-next-to-leading-order renormalization of
the operator ${\mathcal O}(^3S^{[8]}_1)$ and the next-to-leading-order
renormalization of the operators ${\mathcal O}(^3P^{[8]}_J)$. To investigate
the convergence of the relativistic expansion arising from the color-singlet
contributions, we study the ratios of the order-$v^2$ and -$v^4$ color-singlet
short-distance coefficients to the leading-order one. Our results indicate that
though 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 $v$. In
addition, we extrapolate the value of the mass ratio of the charm quark to the
bottom quark, and find the relativistic corrections rise quickly with increase
of the mass ratio.
