1 Introduction

Quarkonia are bound states of either a charm and anti-charm quark pair (charmonia, e.g. \({\mathrm{J}/\psi }\), \(\chi _c\) and \({\psi (\mathrm{2S})}\)) or a bottom and anti-bottom quark pair (bottomonia, e.g. \(\Upsilon \)(1S), \(\Upsilon \)(2S), \(\chi _b\) and \(\Upsilon \)(3S)). While the production of the heavy quark pairs in \(\mathrm{pp}\) collisions is relatively well understood in the context of perturbative QCD calculations [13], their binding into quarkonium states is inherently a non-perturbative process and the understanding of their production in hadronic collisions remains unsatisfactory despite the availability of large amounts of data and the considerable theoretical progress made in recent years [4]. For instance none of the models are able to describe simultaneously different aspects of quarkonium production such as polarization, transverse momentum and energy dependence of the cross sections.

There are mainly three approaches used to describe the hadronic production of quarkonium: the Color-Singlet Model (CSM), the Color Evaporation Model (CEM) and the Non-Relativistic QCD (NRQCD) framework.

In the CSM [57], perturbative QCD is used to model the production of on-shell heavy quark pairs, with the same quantum numbers as the quarkonium into which they hadronize. This implies that only color-singlet quark pairs are considered. Historically, CSM calculations performed at leading order (LO) in \(\alpha _s\), the strong interaction coupling constant, have been unable to reproduce the magnitude and the \({p_\mathrm{T}}\) dependence of the \({\mathrm{J}/\psi }\) production cross section measured by CDF at the Tevatron [8]. Several improvements to the model have been worked out since then: the addition of all next-to-leading order (NLO) diagrams [9] as well as some of the next-to-next-to-leading order (NNLO) [10, 11]; the inclusion of other processes to the production of high \({p_\mathrm{T}}\) quarkonia such as gluon fragmentation [12] or the production of a quarkonium in association with a heavy quark pair [13] and the relaxation of the requirement that the heavy quark pair is produced on-shell before hadronizing into the quarkonium [14]. All these improvements contribute to a better agreement between theory and data but lead to considerably larger theoretical uncertainties and/or to the introduction of extra parameters that are fitted to the data.

In the CEM [1517], the production cross section of a given quarkonium state is considered proportional to the cross section of its constituting heavy quark pair, integrated from the sum of the masses of the two heavy quarks to the sum of the masses of the lightest corresponding mesons (D or B). The proportionality factor for a given quarkonium state is assumed to be universal and independent of its transverse momentum \({p_\mathrm{T}}\) and rapidity \(y\). It follows that the ratio between the yields of two quarkonium states formed out of the same heavy quarks is independent of the collision energy as well as of \({p_\mathrm{T}}\) and \(y\). This model is mentioned here for completeness but is not confronted to the data presented in this paper.

Finally, in the framework of NRQCD [18], contributions to the quarkonium cross section from the heavy-quark pairs produced in a color-octet state are also taken into account, in addition to the color-singlet contributions described above. The neutralization of the color-octet state into a color-singlet is treated as a non-perturbative process. It is expanded in powers of the relative velocity between the two heavy quarks and parametrized using universal long-range matrix elements which are considered as free parameters of the model and fitted to the data. This approach has recently been extended to NLO [1921] and is able to describe consistently the production cross section of quarkonia in \({\mathrm{p}\overline{\mathrm{p}}}\) and \(\mathrm{pp}\) collisions at Tevatron, RHIC and, more recently, at the LHC. However, NRQCD predicts a sizable transverse component to the polarization of the \({\mathrm{J}/\psi }\) meson, which is in contradiction with the data measured for instance at Tevatron [22] and at the LHC [2326].

Most of the observations and discrepancies described above apply primarily to charmonium production. For bottomonium production, theoretical calculations are more robust due to the higher mass of the bottom quark and the disagreement between data and theory is less pronounced than in the case of charmonium [27, 28]. Still, the question of a complete and consistent description of the production of all quarkonium states remains open and the addition of new measurements in this domain will help constraining the various models at hand.

In this paper we present measurements of the inclusive production cross section of several quarkonium states (namely \({\mathrm{J}/\psi }\), \({\psi (\mathrm{2S})}\), \(\Upsilon \)(1S) and \(\Upsilon \)(2S)) using the ALICE detector at forward rapidity (\(2.5<y<4\)) in \(\mathrm{pp}\) collisions at \(\sqrt{s}=7\) TeV. Inclusive measurements contain, in addition to the quarkonium direct production, contributions from the decay of higher mass excited states: predominantly \({\psi (\mathrm{2S})}\) and \({\chi _c}\) for the \({\mathrm{J}/\psi }\); \(\Upsilon \)(2S), \({\chi _b}\) and \(\Upsilon \)(3S) for the \(\Upsilon \)(1S), and \(\Upsilon \)(3S) and \({\chi _b}\) for the \(\Upsilon \)(2S). For \({\mathrm{J}/\psi }\) and \({\psi (\mathrm{2S})}\), they contain as well contributions from non-prompt production, mainly from the decay of \(b\)-mesons. For the \({\mathrm{J}/\psi }\) meson, these measurements represent an increase by a factor of about 80 in terms of luminosity with respect to published ALICE results [29, 30]. For the \({\psi (\mathrm{2S})}\) and the \(\Upsilon \), we present here the first ALICE measurements in \(\mathrm{pp}\) collisions.

This paper is organized as follows: a brief description of the ALICE detectors used for this analysis and of the data sample is provided in Sect. 2; the analysis procedure is described in Sect. 3; in Sect. 4 the results are presented and compared to those obtained by other LHC experiments; finally, in Sect. 5 the results are compared to several theoretical calculations.

2 Experimental apparatus and data sample

2.1 Experimental apparatus

The ALICE detector is extensively described in [31]. The analysis presented in this paper is based on muons detected at forward pseudo-rapidity (\(-4<\eta <-2.5\)) in the muon spectrometer [29]Footnote 1. In addition to the muon spectrometer, the Silicon Pixel Detector (SPD) [32] and the V0 scintillator hodoscopes [33] are used to provide primary vertex reconstruction and a Minimum Bias (MB) trigger, respectively. The T0 Čerenkov detectors [34] are also used for triggering purposes and to evaluate some of the systematic uncertainties on the integrated luminosity determination. The main features of these detectors are listed in the following paragraphs.

The muon spectrometer consists of a front absorber followed by a 3 Tm dipole magnet, coupled to tracking and triggering devices. The front absorber, made of carbon, concrete and steel is placed between 0.9 and 5 m from the Interaction Point (IP). It filters muons from hadrons, thus decreasing the occupancy in the first stations of the tracking system. Muon tracking is performed by means of five stations, positioned between 5.2 and 14.4 m from the IP, each one consisting of two planes of Cathode Pad Chambers. The total number of electronic channels is close to 1.1\(\times 10^{6}\) and the intrinsic spatial resolution is about 70 \(\upmu \)m in the bending direction. The first and the second stations are located upstream of the dipole magnet, the third station is embedded inside its gap and the fourth and the fifth stations are placed downstream of the dipole, just before a 1.2 m thick iron wall (7.2 interaction lengths) which absorbs hadrons escaping the front absorber and low momentum muons (having a total momentum \(p<1.5\) GeV/c at the exit of the front absorber). The muon trigger system is located downstream of the iron wall and consists of two stations positioned at 16.1 and 17.1 m from the IP, each equipped with two planes of Resistive Plate Chambers (RPC). The spatial resolution achieved by the trigger chambers is better than 1 cm, the time resolution is about 2 ns and the efficiency is higher than 95 % [35]. The muon trigger system is able to deliver single and dimuon triggers above a programmable \({p_\mathrm{T}}\) threshold, via an algorithm based on the RPC spatial information [36]. For a given trigger configuration, the threshold is defined as the \({p_\mathrm{T}}\) value for which the single muon trigger efficiency reaches 50 % [35]. Throughout its entire length, a conical absorber (\(\theta <2^{\circ }\)) made of tungsten, lead and steel, shields the muon spectrometer against secondary particles produced by the interaction of large-\(\eta \) primary particles in the beam pipe.

Primary vertex reconstruction is performed using the SPD, the two innermost layers of the Inner Tracking System (ITS) [32]. It covers the pseudo-rapidity ranges \(|\eta |<2\) and \(|\eta |<1.4\), for the inner and outer layers respectively. The SPD has in total about \(10^7\) sensitive pixels on 240 silicon ladders, aligned using \(\mathrm{pp}\) collision data as well as cosmic rays to a precision of 8 \(\upmu \)m.

The two V0 hodoscopes, with 32 scintillator tiles each, are placed on opposite sides of the IP, covering the pseudo-rapidity ranges \(2.8<\eta < 5.1\) and \(-3.7<\eta <-1.7\). Each hodoscope is segmented into eight sectors and four rings of equal azimuthal and pseudo-rapidity coverage, respectively. A logical AND of the signals from the two hodoscopes constitutes the MB trigger, whereas the timing information of the two is used offline to reject beam-halo and beam-gas events, thanks to the intrinsic time resolution of each hodoscope which is better than 0.5 ns.

The T0 detectors are two arrays of 12 quartz Čerenkov counters, read by photomultiplier tubes and located on opposite sides of the IP, covering the pseudo-rapidity ranges \(4.61<\eta <4.92\) and \(-3.28<\eta <-2.97\), respectively. They measure the time of the collision with a precision of \({\sim }40\) ps in \(\mathrm{pp}\) collisions and this information can also be used for trigger purposes.

2.2 Data sample and integrated luminosity

The data used for the analysis were collected in 2011. About 1,300 proton bunches were circulating in each LHC ring and the number of bunches colliding at the ALICE IP was ranging from 33 to 37. The luminosity was adjusted by means of the beam separation in the transverse (horizontal) direction to a value of \(\sim 2\times 10^{30}\) cm\(^{-2}\) s\(^{-1}\). The average number of interactions per bunch crossing in such conditions is about 0.25, corresponding to a pile-up probability of \(\sim \)12 %. The trigger condition used for data taking is a dimuon-MB trigger formed by the logical AND of the MB trigger and an unlike-sign dimuon trigger with a \({p_\mathrm{T}}\) threshold of 1 GeV/c for each of the two muons.

About 4\(\times \)10\(^{6}\) dimuon-MB-triggered events were analyzed, corresponding to an integrated luminosity \(L_\mathrm{{int}}=1.35\pm 0.07\) pb\(^{-1}\). The integrated luminosity is calculated on a run-by-run basis using the MB trigger counts measured with scalers before any data acquisition veto, divided by the MB trigger cross section and multiplied by the dimuon-MB trigger lifetime (75.6 % on average). The MB trigger counts are corrected for the trigger purity (fraction of events for which the V0 signal arrival times on the two sides lie in the time window corresponding to beam-beam collisions) and for pile-up. The MB trigger cross section is measured with the van der Meer (vdM) scan method [37]. The result of the vdM scan measurement [38] is corrected by a factor \(0.990\pm 0.002\) arising from a small modification of the V0 high voltage settings which occurred between the vdM scan and the period when the data were collected. The resulting trigger cross section is \(\sigma _\mathrm{MB}=53.7\pm 1.9(\mathrm{syst})\) mb.

3 Data analysis

The quarkonium production cross section \(\sigma \) is determined from the number of reconstructed quarkonia \(N\) corrected by the branching ratio in dimuon \(\mathrm{BR}_{\mu ^+\mu ^-}\) and the mean acceptance times efficiency \({\langle {A\epsilon }\rangle }\) to account for detector effects and analysis cuts. The result is normalized to the integrated luminosity \(L_\mathrm{{int}}\):

$$\begin{aligned} \sigma =\frac{1}{L_\mathrm{{int}}} \frac{N}{\mathrm{BR}_{\mu ^+\mu ^-} \times {\langle {A\epsilon }\rangle }}, \end{aligned}$$
(1)

with \(\mathrm{BR}_{\mu ^+\mu ^-}=(5.93\pm 0.06)\) %, \((0.78\pm 0.09)\) %, \((2.48\pm 0.05)\) % and \((1.93\pm 0.17)\) % for \({\mathrm{J}/\psi }\), \({\psi (\mathrm{2S})}\), \(\Upsilon \)(1S) and \(\Upsilon \)(2S), respectively [39]. Pile up events have no impact on the reconstruction of the quarkonium yields and are properly accounted for by the luminosity measurement.

3.1 Signal extraction

Quarkonia are reconstructed in the dimuon decay channel and the signal yields are evaluated using a fit to the \(\mu ^+\mu ^-\) invariant mass distributions, as detailed in [29]. In order to improve the purity of the dimuon sample, the following selection criteria are applied:

  • both muon tracks in the tracking chambers must match a track reconstructed in the trigger system;

  • tracks are selected in the pseudo-rapidity range \(-4\le \eta \le -2.5\);

  • th