Abstract
We compute the color-singlet and color-octet nonrelativistic QCD (NRQCD) long-distance matrix elements for inclusive production of P-wave quarkonia in the framework of potential NRQCD. In this way, the color-octet NRQCD long-distance matrix element can be determined without relying on measured cross section data, which has not been possible so far. We obtain inclusive cross sections of χ_{cJ} and χ_{bJ} at the LHC, which are in good agreement with data. In principle, the formalism developed in this Letter can be applied to all inclusive production processes of heavy quarkonia.
Highlights
The mechanism underlying heavy quarkonium production is a key to understanding the dynamics of strongly coupled systems [1,2,3,4,5]
Much of the difficulty stems from our limited knowledge of the nonrelativistic QCD (NRQCD) long-distance matrix elements (LDMEs), which describe the nonperturbative evolution of the heavy quark Q and antiquark Qinto a quarkonium
Phenomenological determinations of the unknown LDMEs based on different choices of observables have led to inconsistent sets of LDMEs, which have resulted in contradicting predictions, in particular, leaving open the long-standing problem of the polarization of quarkonium produced in hadron colliders [16]
Summary
We use spectroscopic notation for the angular momentum state of the QQ , while the superscripts 1 and 8 denote the color state of the QQ : color singlet (CS) and σccooQelQofðfr3iSco1⁄218iceÞtneattsr(eC(SOtDh)e,Crspe)es.prtWeucretbivahteailvvyee.lTyuhsceeadqlcutuhalenatbhitleieeasvsyhσoQqrQtu-ða3dPri1⁄2Jk1stÞasanpncinde symmetry to reduce the χQJ LDMEs into LDMEs of χQ0, which are defined by hOχQ0 ð3P1⁄201Þi. For the state jn; x1; x2i to have a nonvanishing overlap with PQðPÞ, its static component jn; x1; x2ið0Þ must be of the form Trfψ†ðx1Þχðx2ÞgTrfjn; x1; x2ið0Þg=Nc in the limit x1 − x2 → 0, Tr being the trace over color and Nc the number of colors [22] This condition follows from requiring that, in the absence of gluonic excitations carried by 1=m corrections, the quark-antiquark fields at the same point are in a color-singlet configuration, which, in turn, is a necessary requirement for the state jn; x1; x2i to overlap with a state containing a quarkonium. In strongly coupled pNRQCD [14,20,21], color-singlet heavy quark-antiquark pairs in the presence of light degrees of freedom in a state n are described by the field Sn and the Hamiltonian. At leading order in v, it holds φQðn;PÞðx; x2Þ ≈ e−iP·ðx1þx2Þ=2φðQ0ðÞnÞðx; x2Þ; ð8Þ where exp1⁄2−iP · ðx þ x2Þ=2 is a plane wave describing the center of mass motion and φðQ0ðÞnÞðx; x2Þ is an eigenfunction of the leading-order pNRQCD Hamiltonian
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