Abstract
We compute the energy dependence of the $$P_T$$ -integrated cross section of directly produced quarkonia in pp collisions at next-to-leading order (NLO), namely up to $$\alpha _S^3$$ , within nonrelativistic QCD (NRQCD), treating the $$P_T$$ -integrated and the $$P_T$$ -differential cross sections as two different observables. The colour-octet NRQCD parameters needed to predict the $$P_T$$ -integrated yield can thus be extracted from the fits of the $$P_T$$ -differential cross sections at mid and large $$P_T$$ . For the first time, the total cross section is evaluated in NRQCD at full NLO accuracy using the recent NLO fits of the $$P_T$$ -differential yields. Both the normalisation and the energy dependence of the $$J/\psi $$ , $$\psi '$$ and $$\Upsilon (1S)$$ we obtained disagree with the data except when using the fit results of Butenschoen and Kniehl. If one disregards the colour-octet contribution, the existing data in the TeV range are well described by the $$\alpha _S^3$$ contribution in the colour-singlet model – which, at $$\alpha _S^4$$ , however, shows an unphysical energy dependence. A similar observation is made for $$\eta _{c,b}$$ . All this underlines the necessity for a resummation of initial-state radiations in both channels, which is, however, beyond the scope of this article. In any case, past claims that colour-octet transitions are dominantly responsible for low- $$P_T$$ quarkonium production are not supported by our results.
Highlights
Whereas, based on an analysis of the sole early RHIC data, Cooper et al argued [14] that the universality of nonrelativistic QCD (NRQCD) was safe and that colour-singlet contributions to the PT integrated J/ψ yields were negligible, the global analysis of Maltoni et al at next-to-leading order (NLO) showed [15] that the colour-octet LongDistance Matrix Elements (LDMEs) required to describe the total prompt J/ψ yield from fixed-target energies to RHIC were one tenth of that expected from the – leading-order – fit of the PT -differential cross sections at Tevatron energies
As a point of comparison, we had a look at ColourEvaporation-Model-like (CEM) predictions derived from NRQCD following the work of [26] and we found that it cannot reproduce PT -integrated yields using the LDMEs obtained following the relations of [26] after identifying the minimal singlet transition to that of the CSM
Our results would indicate a severe breakdown of NRQCD universality – in line with the previous analysis of Maltoni et al [15] – unless one keeps the LDMEs close to the fit of Butenschoen and Kniehl, which, disagrees with the J/ψ polarisation measurements and the ηc cross sections
Summary
Whereas, based on an analysis of the sole early RHIC data, Cooper et al argued [14] that the universality of NRQCD was safe and that colour-singlet contributions to the PT integrated J/ψ yields were negligible, the global analysis of Maltoni et al at NLO showed [15] that the colour-octet LongDistance Matrix Elements (LDMEs) required to describe the total prompt J/ψ yield from fixed-target energies to RHIC were one tenth of that expected from the – leading-order – fit of the PT -differential cross sections at Tevatron energies Such fits of the PT -differential J/ψ cross sections have recently been extended to NLO – i.e. one-loop – accuracy on the prompt J/ψ yields – some of them focusing on the larger PT data and explicitly including the feed-down contributions [17,18], some enlarging the analysis beyond hadroproduction and including rather low-PT data [19] – and on the Υ(nS) yields [20,21].
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