Production of heavy quarks and heavy quarkonia

Uncertainties in the next-to-leading-order calculations of heavyquark (Q) production are investigated. Predictions for total cross sections, single-inclusive distributions of heavy quarks and heavy flavoured hadrons, as well as for QQ̄ correlations, are compared with charm and bottom data. The description of heavy-quarkonium production requires a separation of the short-distance scale of QQ̄ production, which is set by the heavy-quark mass from the longer-distance scales associated with the bound-state formation. Various factorization approaches are compared, in particular with respect to the different constraints imposed on the colour and angular-momentum states of the QQ̄ pair(s) within a specific quarkonium state. Theoretical predictions are confronted with data on heavy-quarkonium production at fixed-target experiments and also at pp̄ colliders, where fragmentation gives the leading-twist cross section in 1/pT and 1/m 2 at high transverse momentum. a Heisenberg Fellow. CERN–TH/95–75 March 1995 1 Production of heavy quarks 1.1 Theoretical status Total cross sections and single-inclusive distributions of open heavy-quark production have been calculated in perturbative QCD (pQCD) to next-to-leadingorder (NLO) accuracy for essentially all high-energy reactions, hadroproduction [1], photoproduction [2], leptoproduction (deep-inelastic lepton–nucleon scattering) [3], γγ collisions [4], and deep-inelastic eγ scattering [5]. In the case of photoand hadroproduction, the NLO calculations have even been implemented in a Monte Carlo programme [6], such that also double-differential distributions can be studied. Attempts are ongoing to develop such an exclusive programme also for deep-inelastic scattering [7]. In pQCD, the differential cross section of heavy-quark (Q) production in the collision of hadrons A and B is given by the factorized expression dσ[AB → QQ̄X](pA, pB) = ∑ i,j ∫ dx1 dx2fi/A(x1, μ 2 F ) fj/B(x2, μ 2 F )dσ̂[ij → QQ̄X ′](x1pA, x2pB, μF , μR) . (1) Here i, j represent the interacting partons (gluons, light quarks and antiquarks) and the functions fi/A(x, μ 2 F ) are their number densities, the parton distribution functions (PDF) evaluated at momentum fraction x and factorization scale μF . The short-distance cross section σ̂ is calculable as a perturbation series in αs(μR) where the strong coupling constant is evaluated at the renormalization scale μR. Leading-order (LO) diagrams for heavy-quark hadroproduction are shown in Fig. 1. The obvious question is, of course, whether pQCD does describe the experimental data on charm and bottom production. To answer this question the uncertainties in the theoretical predictions have to be investigated. These arise from four main sources: 1. The heavy-quark mass mQ. Conservative ranges are 1.2 < mc < 1.8GeV and 4.5 < mb < 5.0GeV. 2. The size of unknown higher-order corrections. An estimate may be obtained by varying the factorization scale μF and the renormalization scale μR around their ‘natural’ value μR = μF = mQ, say between m/2 and 2m. In the case of the charm quark, one has to note that most PDF fi/H(x, μ 2 F ) are not valid for μF <∼ 2GeV and that the running of αs(μR) is no longer purely perturbative for μR <∼ mc/2. Proper error estimates are therefore not as easy as in the case of the b quark (see below). 3. The value of the QCD scale Λ and the shape of the PDF. Since these are strongly correlated (in particular Λ and the gluon density), the choice of different Λ values (in 60 < Λ5 < 300MeV, say) in the partonic cross sections should