In the framework of the nonrelativistic quantum chromodynamics factorization formalism, we study the processes of ψ(nS) and Υ(nS) decay into a lepton pair or a charm pair associated with two jets up to the next-to-leading order in velocity expansion. We present the analytic expressions for the differential decay rate to the invariant mass of the lepton pair or charm pair. We find that the ratio of the next-to-leading order short-distance coefficient to the leading order one is in the range from −5.5 to −12.4. The relativistic corrections are so large that they modify the leading order prediction significantly. Utilizing the analytic expressions, we also investigate the relativistic corrections in different kinematic regions and their dependence on the masses of the initial-state quarkonium and the final-state fermion. In addition, we study the momentum distribution of D * + in the process Υ(1S) → ccgg → D * + X.

In the framework of the NRQCD factorization formalism, we calculate the decay rate for the process Υ(1S) → ccgg to the next-to-leading order (NLO) in the relative velocity v of the b quark in the bottomonium rest frame. We also study the momentum distributions of the charm quark and the charmed-hadron in the decay. The momentum distribution of the charmed-hadron is obtained by convolving the charm quark momentum distribution with a fragmentation function of the charm quark into the hadron. In addition, we fit the nonperturbative NRQCD matrix element v 2 Υ through comparing the theoretical prediction with the measurement from the BaBar collaboration for the decay rate of Υ(1S) → D * + X. In return, taking this matrix element as an input parameter, we predict the decay rates as well as the momentum distributions for a collection of charmed-hadrons in the process Υ(1S) → ccgg → hX.

In this work, we determine the short-distance coefficients for Υ inclusive decay into a charm pair through relative order v 4 within the framework of NRQCD factorization formula. The shortdistance coefficient of the order-v 4 color-singlet NRQCD matrix element is obtained through matching the decay rate of bb( 3 S[1] 1 ) → ccgg 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-leadingorder renormalization of the operator O( 3 S [8] 1 ) and the next-to-leading-order renormalization of the operators O( 3 P [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 Υ → cc + 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. 12.38.Bx, 12.39.St, 13.20.Gd where m indicates the mass of the bottom quark b, O( 2S+1 L [c] J ) indicates the NRQCD operator with the spectroscopic state 2S+1 L [c] J with the spin S, orbital angular momentum L, total angular momentum J, and color quantum number c, and O( 2S+1 L [c] J ) H ≡ H|O( 2S+1 L [c] J )|H indicates the NRQCD matrix element averaged over the spin states of H. The color quantum number c = 1 and 8 in the NRQCD operators denote the color-singlet and the color-octet respectively. The NRQCD operators in the factorization formula (1) are1 When we were doing this work, a work about order-v 4 corrections to gluon fragmentation appeared at arXiv recently [12]. In that work, the authors applied the similar techniques to cancel IR divergences. (46)We can utilize (25) to calculate the decay rates in full QCD. First, we need to obtain H( 3 P [8] J ) defined in (26). To understand the IR structure and show the IR cancelation, here we separate H( 3 P [8] J ) into two parts: H( 3 P [8] J ) = H d ( 3 P [8] J ) + H s ( 3 P [8] J ). H d ( 3 P [8] J ) include the terms proportional to 1/(1−z), which contribute the whole IR divergences to the decay rates from the region with the real gluon being soft. H s ( 3 P [8] J ) take in charge of the remainder, which is absent of any singularity. Correspondingly, we use the subscripts d, s in both dΓ( 3 P [8] J ) and dF ( 3 P [8]J ) to denote the contributions from the two parts. The advantage of this classification will be recognized when we determine the short-distance coefficient