Abstract
We test the nonrelativistic QCD factorization conjecture for inclusive quarkonium production at two loops by carrying out a covariant calculation of the nonrelativistic quantum chromodynamics (NRQCD) long-distance matrix element (LDME) for a heavy-quark pair in an $S$-wave, color-octet state to fragment into a heavy-quark pair in a color-singlet state of arbitrary orbital angular momentum. The NRQCD factorization conjecture for the universality of the LDME requires that infrared divergences that it contains be independent of the direction of the Wilson line that appears in its definition. We find this to be the case in our calculation. The results of our calculation differ in some respects from those of a previous calculation that was carried out by Nayak, Qiu, and Sterman using light-cone methods. We have identified the sources of some of these differences. The results of both calculations are consistent with the NRQCD factorization conjecture. However, the general principle that underlies this confirmation of NRQCD factorization at two-loop order has yet to be revealed.
Highlights
The nonrelativistic quantum chromodynamics (NRQCD) factorization conjecture for the universality of the long-distance matrix element (LDME) requires that infrared divergences that it contains be independent of the direction of the Wilson line that appears in its definition
The nonrelativistic quantum chromodynamics (NRQCD) factorization conjecture for inclusive quarkonium production in hard-scattering QCD processes [1] postulates that the production rates for those processes at large momentum transfer can be written as a sum of products of short-distance coefficients (SDCs) and NRQCD long-distance matrix elements (LDMEs)
The central issue in establishing the NRQCD factorization conjecture for inclusive quarkonium production is the question of the universality of the NRQCD LDMEs
Summary
The nonrelativistic quantum chromodynamics (NRQCD) factorization conjecture for inclusive quarkonium production in hard-scattering QCD processes [1] postulates that the production rates for those processes at large momentum transfer can be written as a sum of products of short-distance coefficients (SDCs) and NRQCD long-distance matrix elements (LDMEs). The central problem in proving NRQCD factorization is to demonstrate that the soft divergences in the LDMEs are independent of the direction(s) of the gauge-completion Wilson line(s) Without such a demonstration, the LDMEs would depend on the directions of the order-mQ gluons, and so universality of the LDMEs would be lost, even for a single type of quarkonium production process. [9], the oneand two-loop contributions were given to all orders in v, explicit expressions were given only for the non-Abelian diagrams, which are the most complicated to evaluate The calculations in these papers confirmed that the soft divergences in the LDMEs are independent of the Wilson-line direction through two loops.
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