Advances in High Energy Physics

Volume 2015, Article ID 726393, 7 pages

http://dx.doi.org/10.1155/2015/726393

## Next-to-Leading Order Differential Cross Sections for , , and Production in Proton-Proton Collisions at a Fixed-Target Experiment Using the LHC Beams

Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918(4), Beijing 100049, China

Received 17 April 2015; Accepted 4 June 2015

Academic Editor: Cynthia Hadjidakis

Copyright © 2015 Yu Feng and Jian-Xiong Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### 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).