Order-$v^4$ Relativistic Corrections to ${\rm Y}$ Inclusive Decay into a Charm pair

Abstract

In this work, we determine the short-distance coefficients for ${\rm Y}$ 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 $b\bar{b}{(}^3{S}_1^{[1]})\to c\bar{c}gg$ 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 $\mathcal{O}{(}^3{S}_1^{[8]})$ and the next-to-leading-order renormalization of the operators $\mathcal{O}{(}^3{P}_J^{[8]})$. 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 ${\rm Y}\to c\bar{c}+X$ 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.

24 pages, 4 figures, 1 table; Typos corrected

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Cited by in corpus (1)

• QCD and strongly coupled gauge theories: challenges and perspectives