Abstract
Using nonrelativistic QCD (NRQCD) factorization, we calculate the yields for , , and hadroproduction at GeV and 115 GeV including the next-to-leading order QCD corrections. Both these center-of-mass energies correspond to those obtained with 7 TeV and 2.76 TeV nucleon beam impinging a fixed target. We study the cross section integrated in as a function of the (center-of-mass) rapidity as well as the differential cross section in the central rapidity region. Using different NLO fit results of the NRQCD long-distance matrix elements, we evaluate a theoretical uncertainty which is certainly much larger than the projected experimental uncertainties with the expected 20 fb−1 to be collected per year with AFTER@LHC for collision at the center of mass energy GeV.
1. Introduction
Nonrelativistic quantum chromodynamics (NRQCD) [1] is the most systematic factorization scheme to describe the decay and production of heavy quarkonia. It allows one to organize the theoretical calculations as double expansions in both the coupling constant and the heavy-quark relative velocity . In the past few years, significant progress has been made in next-to-leading order (NLO) QCD calculations based on NRQCD. Calculations and fits of NRQCD long-distance matrix elements (LDMEs) for both the yield and polarization in hadroproduction have been carried out [2–6] as well as for hadroproduction [7, 8]. Using these LMDEs, one can in principle predict the transverse momentum differential cross section at any energies. In addition, in a recent study [9], we have discussed the implication of these fits on the energy dependence of the cross sections integrated in .
In this paper, we predict these differential cross sections for the kinematics of a fixed-target experiment using the LHC beams (AFTER@LHC) [10]. In practice, 7 TeV protons on targets yield to a c.m.s energy close to 115 GeV and 72 GeV for 2.76 TeV nucleons (as in the case of a Pb beam). This corresponds to a range very seldom explored so far, significantly higher than that at CERN-SPS and not far from BNL-RHIC. With the typical luminosity of the fixed-target mode, which allows for yearly luminosities as large as 20 fb−1 in collision at 115 GeV, AFTER@LHC is expected to be a quarkonium and heavy-flavor observatory [10, 11]. In general, the opportunities of a fixed-target experiment using the LHC beam for spin and heavy-ion physics are discussed in [10, 12–14]. With the calculation at GeV, which is supposed to be a baseline rate where nuclear effects would be added, we confirm that charmonium yields can easily reach per year and for bottomonium at 115 GeV.
2. Next-to-Leading Order Calculation
Following the NRQCD factorization formalism [1], the cross section for quarkonium hadroproduction can be expressed as where is either a proton or an antiproton, is the parton distribution function (PDF) of , the indices run over all possible partonic species, and denotes the color, spin, and angular momentum states of the intermediate pair. For and , namely, the quarkonium sates, their leading CO states of relative order are , , and . Along with the CS transition , we call the total CS + CO contributions as direct production. The short-distance coefficient (SDC) will be calculated perturbatively, while the long-distance matrix elements (LDMEs) are governed by nonperturbative QCD effects.
Now let us take a look at the parton level processes relevant to this work. As it is well known, the CO contributions to hadroproduction appear at [15] and their Born contributions are where denotes the light quarks (antiquarks).
Up to , QCD corrections include real and virtual corrections. One inevitably encounters ultra-violet (UV), infrared (IR), and Coulomb divergences when dealing with the virtual corrections. UV divergences from self-energy and triangle diagrams are canceled upon the renormalization procedure. For the real emission corrections, three kinds of processes should be considered: some of which involve IR singularities in phase space integration and we adopt the two-cutoff phase space slicing method [16] to isolate these singularities by introducing two small cutoffs, and . For technical details, we refer readers to [17, 18].
One has to note that in (3), the production in fusion is not really correction. Strictly speaking, it is only the Born-order contribution for hadroproduction with a jet. In fact, all the real emission processes in (3) will be taken as Born-order contributions of quarkonium-jet production.
As regards to the dependent differential cross section, and the QCD NLO corrections in this case are up to , which involves the real emission processes where , denote light quarks with different flavors and can be either , , , or . One can find the details of such computations at this order in [18, 19] and some examples in [2, 3, 6–8].
All of these calculations are made with the newly updated Feynman Diagram Calculation package [20].
3. Constrains on the LDMEs
The color-singlet (CS) LDMEs are estimated from the wave functions at the origin by , where the wave functions are obtained via potential model calculation [21]. This gives , , and . In the following, we will refer to this contribution as the CSM results when performed separately.
The color-octet (CO) LDMEs can only be extracted from data. As for now, SDC are known up to NLO accuracy and the fits of LDMEs can be thus performed at NLO. However, different results are obtained when different dataset is used. We made a selection of these fits in order to assess the theoretical uncertainty induced by the LDMEs. We briefly discuss below these different fit results.
In the case, seven groups of LDMEs [2, 5, 6, 22–25] are collected in Table 1. They are extracted by fitting the data of hadroproduction yield [2] or combined with polarization [5, 6] on collisions. The first one [22] was based on a wider set of data including and system with GeV. In [5, 6], the data with GeV are excluded in their fit. The fit in [23, 24] took the measurement ( GeV) into consideration. Only one of them is used [24] since their results are almost the same. The last one incorporates the leading-power fragmentation corrections together with the QCD NLO corrections, which results in a different SDC and may result in different LDMEs. In [2], Ma et al. fit the data with GeV by two linear combinations of LDMEs: from which we extract the value of LDMEs by restricting and to be positive to get a loose constraint on the range, from which we choose the center value in order to obtain the three LDMEs (Ma et al. (2011) in Table 1).
As regards the , only two NLO analyses have been done (see [2, 6]), both of which excluded the data with GeV in their fit. To extract the LDME values from the fit results of Ma et al., the same method is used as for the . For , we use three groups of LDMEs [8, 26, 27]. Both of them have exactly accounted for the direct production and the feed-down contributions. In the fit of [26], only the data in GeV region are used, while in [8, 27] the region is GeV. They all describe the high yield data at Tevatron and LHC very well. We gather the LDMEs of and in Table 2.
4. Numerical Results
The differential cross sections as a function of (center-of-mass) rapidity and transverse momentum are considered in this section. In both cases, the CTEQ6M parton distribution functions [28] and the corresponding two-loop QCD coupling constants are used. The charm quark mass is set to be GeV, while the bottom quark mass is taken to be GeV. The renormalization and factorization scales are chosen as for rapidity distribution plots, while for the plots of transverse momentum distribution they are , with . The NRQCD scale is chosen as . It is important to note that different choices of these scales may be adopted which can bring some additional uncertainties in our predictions. We show these uncertainties from the scales combined to those from the quark masses for the plot of the rapidity distribution of the yield (and only for the CO contributions). The scale dependence is estimated by varying , , by a factor of 1/2 and 2 with respect to their central values and quark masses varying 0.1 GeV up and down for , as well as 0.25 GeV for . Branching ratios are taken from PDG [29], which give = 0.0596, = 0.0079, and = 0.0248, respectively. The two phase space cutoffs and are chosen and the insensitivity of the results on different choices for these cutoffs has been checked.
4.1. up to
First, we study the -integrated cross section (where the whole region is integrated) as a function of rapidity. The QCD NLO corrections are up to in this case. In Figures 1 and 2, we show the rapidity distribution of direct , , and production cross section at center of mass energy GeV and 115 GeV, respectively. We first discuss the “unnormalized” CO channel contributions shown in Figure 1, where the CO LDMEs are set to unity for all three production channels. For , the CSM is different from only by a factor, we therefore do not perform it separately. Obviously, the CSM results (red lines) for both and are small compared with the CO channels. The dominant CO channel for is transition, while for it is . Besides, these “unnormalized” contributions for display a clear hierarchy, but for , little difference between and contributions shows up.
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Adopting the LDMEs in Tables 1 and 2, we present the rapidity distribution of cross section for various cases in Figure 2. The lines are the central values with different groups of LDMEs, while the colored areas are the uncertainties from scales and quark masses. Only the boundary lines are shown with scales and mass uncertainties. For the , six groups of NRQCD results are shown as a band, the boundaries of which has a distance within factor 10. The values of the cross sections are roughly in the region of pb. The CSM results are systematically below the full NRQCD band, again by a factor 10. Without a surprise, the CSM seems to be negligible for total NRQCD results. However, some words of cautious are in order here. In fact, as we have discussed in [9], the LO CSM contribution explains the data very well from the RHIC to LHC energies, while the CO LDMEs extracted from -differential NLO correction would lead to -integrated cross sections overshooting the data. Only the fit from Butenschoen and Kniehl [22], including rather low data, provides an acceptable description of the pt-integrated cross section. In other words, most of the predictions in Figure 2 may overshoot the data. One should indeed stress that most of the fits which we used are based on large data, while this rapidity distribution bears on the whole region. We suppose the band from the LDMEs of Butenschoen and Kniehl [22]; namely, the lowest band (red dashed line) would probably give the best prediction of the yields, although their LDMEs clearly face difficulty to describe the polarization data. To be complete, let us mention that, following an observations also done in [9], the CSM yield may underestimate the measurements below RHIC energy.
As regards the , two groups of LDMEs lead to a consistent predictions which give a cross section around pb at both GeV and 115 GeV. Accounting for the uncertainties of the scales and quark masses, the cross sections reach pb in the central rapidity region. Nevertheless, these results overestimated the data as discussed in [9].
In the case, two curves are close and a third one is slightly different. Yet, their difference is only in pb units. At RHIC energies and below [9], this reproduces quite well the data. This should thus also be the case at the energies considered here.
4.2. up to
Now let us discuss the differential cross sections in the transverse momentum . In Figure 3, the distributions of direct , , and hadroproduction are presented. For and , the yields are dominated by the CO contributions, which is larger than that of the CSM at NLO by at least one order of magnitude. The various groups of LDMEs predict and differential yields which are much less spread than for the integrated yields; this is expected since the fits are based on a similar distribution but at different energies. Only the one from [25] (the light blue dot-dashed line) seems to depart from the other ones, being from 2 to 10 times larger in case. This may be understood by the fact that the fits in [25] have a different SDC compared with others, which would be the source of the difference.
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For , the red dashed and blue dot-dashed lines are almost parallel with a tiny difference, while the green dot line is obviously lower at low region and crosses the other ones as increases. This explains the behavior of in Figure 2 with a visible difference between the green curve and the other two.
5. Summary
We evaluated the NLO QCD corrections for the direct , , and production at fixed-target LHC energies. We studied the cross section integrated in as a function of the rapidity as well as the differential cross section in the central rapidity region, including QCD corrections up to and contributions, respectively. To perform a reliable prediction, various sets of NRQCD long-distance matrix elements obtained from different fitting methods are considered as well as the uncertainties from the scales and quark masses. With the typical luminosity of the fixed-target mode, which allows for yearly luminosities as large as 20 fb−1 with 7 TeV proton beams, our predictions confirm that charmonium yields can easily reach per year and for bottomonium at the center of mass energy GeV in collision.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are grateful to Jean-Philippe Lansberg for his generous help in this work.