Introduction

Quarkonium production in hadronic interactions is an excellent case of study for understanding hadronization in quantum chromodynamics (QCD), the theory of strong interactions [1]. In particular, the production of the \(\text {J}/\psi \) meson, a bound state of a charm and an anti-charm quark and the lightest vector charmonium state, is the subject of many theoretical calculations. The cornerstone of all the theoretical approaches is the factorization theorem, according to which the \(\text {J}/\psi \) production cross section can be factorized into a short distance part describing the \(\text {c}\overline{\text {c}}\) production and a long distance part describing the subsequent formation of the bound state. In this way, the \(\text {c}\overline{\text {c}}\) pair production cross section can be computed perturbatively. The widely used Non-Relativistic QCD (NRQCD) approach [2] describes the transition probabilities of the pre-resonant \(\text {c}\overline{\text {c}}\) pairs to bound states with a set of long-distance matrix elements (LDME) fitted to experimental data, assumed to be universal. Next-to-leading order (NLO) calculations involving collinear parton densities are able to describe production yields for transverse momentum (\(p_{\mathrm{T}}\)) larger than the mass of the bound state [3, 4], but have difficulties describing the measured polarization [5, 6]. Calculations employing the \(k_{\mathrm{T}}\)-factorization approach [7] can reach lower \(p_{\mathrm{T}}\) but have similar difficulties when compared to data [8]. The low-\(p_{\mathrm{T}}\) range of quarkonium production is modelled also within the Color Glass Condensate effective theory coupled to leading order NRQCD calculations [9], which involves a saturation of the small Bjorken-x gluon densities that dampens the heavy-quark pair production yields. An alternative to the universal LDME approach to hadronization used in the NRQCD framework is provided by the Color Evaporation Model (CEM) [10, 11] and its more recent implementation using the \(k_{\mathrm{T}}\)-factorization approach, the Improved CEM (ICEM) [12]. In the ICEM, the transition probability to a given bound state is proportional to the \(\text {c}\overline{\text {c}}\) pair production cross section integrated over an invariant-mass range spanning between the mass of the bound state and twice the mass of the lightest charmed meson. Finally, in the Color Singlet Model (CSM) [13,14,15], the pre-resonant \(\text {c}\overline{\text {c}}\) pair is produced directly in the color-singlet state with the same quantum numbers as the bound state. Calculations within this model at NLO precision are known to strongly underpredict the measured production cross sections [16]. In this context, a \(p_{\mathrm{T}}\)-differential measurement of \(\text {J}/\psi \) production cross section covering a wide \(p_{\mathrm{T}}\) range, starting from \(p_{\mathrm{T}}\) \(=\) 0 and up to high-\(p_{\mathrm{T}}\), can discriminate between the different models of quarkonium production.

In this paper, we present the integrated, and the \(p_{\mathrm{T}}\) and rapidity (y) differential production cross sections of inclusive \(\text {J}/\psi \) production at midrapidity (\(|y|<0.9\)) in proton–proton (pp) collisions at the center-of-mass energy \(\sqrt{s}~=~13\) TeV. The inclusive \(\text {J}/\psi \) yields include contributions from directly produced \(\text {J}/\psi \), feed-down from prompt decays of higher-mass charmonium states, and non-prompt \(\text {J}/\psi \) from the decays of beauty hadrons. The \(p_{\mathrm{T}}\)-differential production cross section of inclusive \(\text {J}/\psi \) is measured in the \(0<p_{\mathrm{T}} <15\) GeV/\(c\) interval using a minimum-bias triggered data sample and in the \(15<p_{\mathrm{T}} <40\) GeV/\(c\) interval using an Electromagnetic Calorimeter triggered data sample. These results complement existing measurements at midrapidity at \(\sqrt{s}~=~13\) TeV performed by the CMS Collaboration [17], which report the prompt \(\text {J}/\psi \) production cross section for \(p_{\mathrm{T}}\) \(>20\) GeV/\(c\). Previous measurements of the \(\text {J}/\psi \) production cross section in pp collisions performed by the ALICE Collaboration at midrapidity at lower energies were published in Refs. [18,19,20]. The inclusive \(\text {J}/\psi \) production cross section in pp collisions at \(\sqrt{s}~=~13\) TeV was published by the ALICE Collaboration at forward rapidity in Ref. [21] and the prompt and non-prompt production cross sections were reported by the LHCb Collaboration in Ref. [22].

In the next sections, the ALICE detector and the data sample are described in Sect. 2, and the data analysis and the determination of the systematic uncertainties are described in Sects. 3 and 4, respectively. The results are presented and discussed together with recent model calculations in Sect. 5 and conclusions are drawn in Sect. 6.

The ALICE detector, data set and event selection

A detailed description of the ALICE detector and its performance is provided in Refs. [23, 24]. Here we mention only the detector systems used for the reconstruction of the \(\text {J}/\psi \) mesons decaying in the \(\text {e}^{+}\text {e}^{-}\) channel at midrapidity. Unless otherwise specified, the term electrons will be used throughout the text to refer to both electrons and positrons.

The reconstruction of charged-particle tracks is performed using the Inner Tracking System (ITS) [25] and the Time Projection Chamber (TPC) [26], which are placed inside a solenoidal magnet providing a uniform magnetic field of \(B = 0.5~\text {T}\) oriented along the beam direction. The ITS is a silicon detector consisting of six cylindrical layers surrounding the beam pipe at radii between 3.9 and \(43.0~\text {cm}\). The two innermost layers consist of silicon pixel detectors (SPD), followed by two layers of silicon drift (SDD) and two layers of silicon strip (SSD) detectors. The TPC is a cylindrical gas drift chamber which extends radially between 85 and \(250~\text {cm}\) and longitudinally over \(250~\text {cm}\) on each side of the nominal interaction point. Both TPC and ITS have full coverage in azimuth and provide tracking in the pseudorapidity range \(|\eta | < 0.9\). Additionally, the measurement of the specific energy loss (\(\text {d}E/\text {d}x\)) in the TPC active gas volume is used for electron identification. The Electromagnetic Calorimeter (EMCal) and the Di-jet Calorimeter (DCal) [27,28,29] are employed for triggering and electron identification. The EMCal/DCal is a Shashlik-type lead-scintillator sampling calorimeter located at a radius of 4.5 m from the beam vacuum tube. The EMCal detector covers a pseudorapidity range of \(|\eta |<0.7\) over an azimuthal angle of \( 80^\circ< \varphi < 187^\circ \), and the DCal covers \(0.22<|\eta |< 0.7\) for \(260^\circ< \varphi < 320^\circ \) and \(|\eta |<0.7\) for \(320^\circ< \varphi < 327^\circ \). The EMCal and DCal have identical granularity and intrinsic energy resolution, and they form a two-arm electromagnetic calorimeter, which in this paper will be referred to jointly as EMCal.

In addition to these central barrel detectors, the V0 detectors, composed of two scintillator arrays [30] placed along the beam line on either side of the interaction point and covering the pseudorapidity intervals \(-3.7< \eta < -1.7\) and \(2.8< \eta < 5.1\), respectively, are used for event triggering. Together with the SPD detector, the V0 is also used to reject background from beam-gas collisions and pileup events.

The measurement of the \(p_{\mathrm{T}}\)-integrated and \(p_{\mathrm{T}}\)-differential production cross sections up to \(p_{\mathrm{T}} = 15\) GeV/\(c\), the upper limit being determined by the available integrated luminosity, utilizes the minimum-bias (MB) trigger, defined as the coincidence of signals in both V0 scintillator arrays. For the \(p_{\mathrm{T}}\) interval from 15 up to 40 GeV/\(c\), the EMCal trigger is employed to select events with high-\(p_{\mathrm{T}}\) electrons. The lower \(p_{\mathrm{T}}\) limit is chosen such that the trigger efficiency does not vary with \(p_{\mathrm{T}}\) above this value, thus avoiding systematic uncertainties related to trigger threshold effects. The EMCal trigger is an online trigger which includes a Level 0 (L0) and a Level 1 (L1) component [27]. The calorimeter is segmented into towers and Trigger Region Units (TRUs), the latter being composed of 384 towers each [27]. The L0 trigger is based on the analog charge sum of \(4\times 4\) groups of adjacent towers evaluated within each TRU, in coincidence with the MB trigger. The L1 trigger decision requires the L0 trigger and, in addition, scans for \(4\times 4\) groups of adjacent towers across the entire EMCal surface. The EMCal-triggered analysis presented in this paper uses the L1 trigger, and requires that at least one of the charge sums of the \(4\times 4\) adjacent towers is above 9 GeV.

This analysis includes all the data recorded by the ALICE Collaboration during the LHC Run 2 data-taking campaigns of 2016, 2017 and 2018 for pp collisions at \(\sqrt{s}~=~13\) TeV. The maximum interaction rate for the dataset was \(260~\text {kHz}\), with a maximum pileup probability in the same bunch crossing of 0.5 \(\times \) 10\(^{-3}\). The events selected for analysis were required to have a reconstructed vertex within the interval \(|z_{\text {vtx}}|<10~\text {cm}\) to ensure a uniform detector acceptance. Beam-gas events and pileup collisions occurring within the readout time of the SPD were rejected offline using timing selections based on the V0 detector information. Pileup collisions occurring within the same LHC bunch crossing were rejected using offline algorithms which identify multiple vertices [24]. The remaining fraction of pileup events surviving the selections is negligible for both the MB and EMCal data samples.

The analyzed MB sample, satisfying all the quality selections, consists of about \(2\times \)10\(^9\) events, corresponding to an integrated luminosity \(L_{\text {int}} = 32.2~\text {nb}^{-1} \pm 1.6\%~(\text {syst})\), and the EMCal-triggered sample consists of approximately \(9\times \)10\(^7\) events which corresponds to an integrated luminosity \(L_{\text {int}} = 8.3~\mathrm {pb}^{-1} \pm 2.0\%~(\text {syst})\). The integrated luminosities are obtained based on the MB trigger cross section (\(\sigma _{\mathrm {MB}}\)), measured in a van der Meer scan [31], separately for each year, as described in Ref. [32]. For each of the used triggers, MB and EMCal, the integrated luminosity is obtained as

$$\begin{aligned} L_{\mathrm {int}} = \frac{N_{\mathrm {MB}}}{\sigma _{\mathrm {MB}}} \times \frac{ds_{\mathrm {trig}}}{ds_{\mathrm {MB}}} \times \frac{LT_{\mathrm {trig}}}{LT_{\mathrm {MB}}} \end{aligned}$$
(1)

where \(N_{\mathrm {MB}}\) is the number of MB-triggered events in the triggered sample, \(ds_{\mathrm {trig}}\) is the downscaling factor applied to the considered trigger by the ALICE trigger processor and \(LT_{\mathrm {trig}}\) is the trigger live time, i.e. the fraction of time where the detector clusterFootnote 1 assigned to the trigger was available for readout.

\(\text {J}/\psi \) reconstruction

In this work we study the integrated, and the rapidity- and \(p_{\mathrm{T}}\)-differential inclusive \(\text {J}/\psi \) production at midrapidity (\(|y|<0.9\)) reconstructing the \(\text {J}/\psi \) from the \(\text {e}^{+}\text {e}^{-}\) decay channel. The MB sample analysis follows closely the one performed in pp collisions at \(\sqrt{s}~=~5.02\) TeV [20].

Track selection

Electron-track candidates are reconstructed employing the ITS and TPC. They are required to be within the acceptance of the central barrel (\(|\eta | < 0.9\)), and to have a minimum transverse momentum of 1 GeV/\(c\), which suppresses the background with only a moderate \(\text {J}/\psi \) efficiency loss. The tracks are selected to have at least 2 hits in the ITS, one of which having to be in one of the SPD layers, and share at most one hit with other tracks. A minimum of 70 out of a maximum of 159 clusters are required in the TPC.

In order to reject tracks originating from weak decays and interactions with the detector material, a selection based on the distance-of-closest approach (DCA) to the primary vertex is applied to the tracks. For the MB analysis the tracks are required to have a minimum DCA lower than 0.2 cm in the transverse direction and 0.4 cm along the beam axis. Such tight selection criterion is used in order to improve the signal-to-background ratio and the signal significance. It was checked with Monte Carlo (MC) simulations that these requirements do not lead to efficiency loss for the non-prompt \(\text {J}/\psi \) relative to the prompt \(\text {J}/\psi \). For the EMCal-triggered event analysis, a looser selection on the DCA to the primary vertex at 1 and 3 cm is applied to avoid rejecting non-prompt \(\text {J}/\psi \) from highly boosted beauty hadron decays.

The electrons are identified using the specific energy loss \(\text {d}E/\text {d}x\) in the TPC gas. Their \(\text {d}E/\text {d}x\) is required to be within a band of \([-2, 3]~\sigma \) relative to the expectation for electrons at the given track momentum, with \(\sigma \) being the \(\text {d}E/\text {d}x\) resolution. The contamination from protons which occurs in the momentum range \(p<1.5\) GeV/\(c\) and from pions for momenta above 2 GeV/\(c\) is mitigated by rejecting tracks compatible with the proton or pion hypothesis within \(3\sigma \).

For the analysis of EMCal-triggered events, both the TPC and the EMCal are used for electron identification. At least one of the \(\text {J}/\psi \) decay-electron tracks, initially identified by the TPC, is required to be matched to an EMCal cluster (a group of adjacent towers belonging to the same electromagnetic shower). In order to ensure a constant trigger efficiency on the selected events, the matched clusters are selected to have a minimal energy of 14 GeV, a value that is significantly higher than the applied online threshold of 9 GeV. Electrons are identified by applying a selection on the energy-to-momentum ratio of the EMCal matched track of \(0.8<E/p<1.3\) and on the \(\text {d}E/\text {d}x\) in the TPC of \([-2.25, 3]~\sigma \). Due to the additional use of the EMCal for electron identification with respect to the MB based analysis, no explicit hadron rejection was used for the EMCal sample.

Secondary electrons from photon conversions, the main background source for both analyses, are rejected using the requirement of a hit in the SPD detector. This requirement rejects most of the electrons from photon conversions occurring beyond the SPD layers. An additional selection based on track pairing, as described in detail in Ref. [20], is applied to further reject conversion electrons, especially those from photons converting in the beam pipe or in the SPD.

Signal extraction

The number of reconstructed \(\text {J}/\psi \) mesons is extracted from the invariant mass (\(m_{\mathrm{ee}} \)) distribution of all possible opposite-sign (OS) pairs constructed combining the selected electron tracks within the same event (SE). Besides the \(\text {J}/\psi \) signal, i.e. pairs of electrons originating from the decay of a common \(\text {J}/\psi \) mother, the invariant mass distribution contains a background with contributions from combinatorial and correlated sources.

In the MB analysis, the combinatorial background, i.e. pairs of electrons originating from uncorrelated processes, is estimated using the event-mixing technique (ME), in which pairs are built from opposite-sign electrons belonging to different events. The mixing is done considering events from the same run (a collection of events taken during a period of time of up to a few hours) with a similar vertex position. The normalized combinatorial background distribution \(B_{\mathrm{comb}}(m_{\mathrm{ee}})\) is obtained as

$$\begin{aligned} B_{\mathrm{comb}}(m_{\mathrm{ee}}) = N_{\mathrm{OS}}^{\mathrm{ME}}(m_{\mathrm{ee}}) \times \frac{\sum _{m_{\mathrm{i}}} N_{\mathrm{LS}}^{\mathrm{SE}}(m_{\mathrm{i}})}{\sum _{m_\mathrm{i}} N_{\mathrm{LS}}^{\mathrm{ME}}(m_{\mathrm{i}})} \end{aligned}$$
(2)

where \(N_{\mathrm{LS}}^{\mathrm{SE}}\), \(N_{\mathrm{OS}}^{\mathrm{ME}}\) and \(N_\mathrm{LS}^{\mathrm{ME}}\) are the number of same-event like-sign (LS), mixed-event OS and mixed-event LS pairs, respectively. Here, the mixed-event OS distribution is normalized using the ratio of SE to ME like-sign pairs since these are not expected to contain any significant correlated source. The summation extends over all the mass bins \(m_{\mathrm{i}}\) between 0 and 5 GeV/\(c^2\) to minimize the statistical uncertainty on the background matching. The correlated background in the mass region relevant for this analysis originates mainly from semi-leptonic decays of heavy-flavor hadrons [33]. In order to extract the number of reconstructed \(\text {J}/\psi \), \(N_{\text {J}/\psi }\), the combinatorial background-subtracted invariant mass distribution is fitted with a two-component function: one empirical function to describe the correlated background shape, which is a second order polynomial at low pair \(p_{\mathrm{T}}\) (\(p_{\mathrm{T}}^{\mathrm{ee}} <1\) GeV/\(c\)) and an exponential at high-\(p_{\mathrm{T}}\) (\(p_{\mathrm{T}}^{\mathrm{ee}} >1\) GeV/\(c\)), plus a template shape obtained from MC simulations, described in Sect. 3.3, for the \(\text {J}/\psi \) signal.

For the analysis of the EMCal-triggered event sample, due to the relatively large contribution from correlated sources at high \(p_{\mathrm{T}}\), the event mixing technique is not used. Instead, a fit of the invariant mass distribution is performed using the MC template for the signal and a third-order polynomial function to describe both the combinatorial and correlated background contributions.

In both analyses, the contribution from \(\psi (2S)\) decaying in the dielectron channel is not included in the fit as the expected number of such pairs is \(\sim \)1% of the \(\text {J}/\psi \) raw yield and it is statistically not significant in the analyzed data samples. The number of \(\text {J}/\psi \) is obtained by counting the number of \(\text {e}^{+}\text {e}^{-}\) pairs in the mass range \(2.92 \le m_{\text {ee}} \le 3.16\) GeV/\(c^2\) remaining after subtracting the background. The SE-OS dielectron invariant mass distribution for a few of the \(p_{\mathrm{T}}\) intervals is shown in Fig. 1, together with the estimated signal and background components.

Fig. 1
figure 1

Invariant-mass distributions for SE \(\text {e}^{+}\text {e}^{-}\) pairs in two \(p_{\mathrm{T}}^{\mathrm{ee}}\) intervals from the MB event analysis (left panels) and two \(p_{\mathrm{T}}^{\mathrm{ee}}\) intervals from the EMCal-triggered event analysis (right panels). The signal and background components obtained from the fit procedure are shown separately. For the top-right panel, the distributions are scaled for convenience by a factor of 2

Corrections

The double differential \(\text {J}/\psi \) production cross section is calculated as

$$\begin{aligned}&\frac{\text {d}^2\sigma _{\text {J}/\psi }}{\text {d}y\text {d}p_{\mathrm{T}}}\nonumber \\&\quad = \frac{N_{\text {J}/\psi }}{\text {BR}(\text {J}/\psi \rightarrow \text {e}^{+}\text {e}^{-}) \times \langle A \times \epsilon \rangle \times \Delta y \times \Delta p_{\mathrm{T}} \times L_{\text {int.}}} \, , \end{aligned}$$
(3)

where \(N_{\text {J}/\psi }\) is the number of reconstructed \(\text {J}/\psi \) in a given interval of rapidity \(\Delta y\) and transverse momentum \(\Delta \) \(p_{\mathrm{T}}\), \(\text {BR}(\text {J}/\psi \rightarrow \text {e}^{+}\text {e}^{-})\) is the decay branching ratio into the dielectron channel [34], \(\langle A \times \epsilon \rangle \) is the average acceptance and efficiency factor and \(L_{\text {int}}\) is the integrated luminosity of the data sample.

The correction for acceptance and efficiency is the product of the kinematical acceptance factor, the reconstruction efficiency, which includes both tracking and particle-identification (PID) efficiency, and the fraction of signal in the signal counting mass window. For the EMCal-triggered events analysis, the efficiency for the EMCal cluster reconstruction is also considered. The efficiency related to the EMCal trigger is estimated using a parameterized simulation of the L1 trigger which includes decalibration and noise based on measured data, and takes into account the time-dependent detector conditions. With the exception of the PID efficiency, all the corrections are obtained based on a MC simulation of unpolarized \(\text {J}/\psi \) mesons embedded in inelastic pp collisions simulated using PYTHIA 6.4 [35] with the Perugia 2011 tune [36]. The prompt \(\text {J}/\psi \) are generated with a flat rapidity distribution and a \(p_{\mathrm{T}}\) spectrum obtained from a phenomenological interpolation of \(\text {J}/\psi \) measurements at RHIC, CDF and the LHC at lower energies [37]. For the non-prompt \(\text {J}/\psi \), \(\text {b}\overline{\text {b}}\) pairs are generated using the PYTHIA Perugia 2011 tune. The \(\text {J}/\psi \) decays are simulated using PHOTOS [38], which includes the radiative component of the \(\text {J}/\psi \) decay. The generated particles are transported through the ALICE detector setup using the GEANT3 package [39].

The PID efficiency is determined with a data-driven method by using a clean sample of electrons from tagged photon conversion processes, passing the same quality criteria as the electrons selected for the \(\text {J}/\psi \) reconstruction. The PID selection efficiency for single electrons is propagated to the \(\text {J}/\psi \) level using a simulation of the \(\text {J}/\psi \) decay. The acceptance times efficiency correction factor for the MB sample analysis varies with \(p_{\mathrm{T}}\) between \(7.6\%\) and \(16\%\) while in the case of the EMCal-triggered sample analysis it increases with \(p_{\mathrm{T}}\) from 2 to 8%.

Due to the finite size of the \(p_{\mathrm{T}}\) intervals, there is a mild dependence of the correction factors on the shape of the inclusive \(\text {J}/\psi \) \(p_{\mathrm{T}}\) distribution used in the simulation. This is mitigated iteratively by using the corrected \(\text {J}/\psi \) \(p_{\mathrm{T}}\)-differential production cross section to reweight the acceptance times efficiency correction factor and obtain an updated corrected cross section. The procedure is stopped when the difference between the input and output corrected \(p_{\mathrm{T}}\)-differential production cross section drops below 1%, which typically occurs within 1 to 2 iterations, depending on the \(p_{\mathrm{T}}\) interval. Additionally, to check if the default MC used in the analysis could introduce a bias on the EMCal trigger efficiency, due to the enhancement of \(\text {J}/\psi \), another MC simulation, based on a di-jet production generated by PYTHIA8 [40], at \(\sqrt{s}~=~13\) TeV, is used as a cross-check. As a result, the default MC and the di-jet MC lead to a compatible EMCal trigger efficiency.

Systematic uncertainties

There are several sources of systematic uncertainties affecting this analysis, namely the ITS-TPC tracking, the electron identification, the signal extraction procedure, the \(\text {J}/\psi \) input kinematic distributions used in MC simulations, the determination of the integrated luminosity and the branching ratio of the dielectron decay channel. A summary of these is given in Table 1.

Table 1 Summary of contributions to systematic uncertainties of the measured \(\text {J}/\psi \) production cross section (in percentage)

The uncertainty of the ITS-TPC tracking efficiency is one of the dominant sources of systematic uncertainty and has two contributions: one due to the TPC-ITS matching and one related to the track-quality requirements. The former is obtained from the residual difference observed for the ITS-TPC single-track matching between data and MC simulations [41], which is further propagated to \(\text {J}/\psi \) dielectron pairs. It varies between 2.8% and 5.4%, depending on \(p_{\mathrm{T}}\). The uncertainty related to the track-quality requirements is estimated by repeating the analysis with variations of the selection criteria and taking the root mean square (RMS) of the distribution of the results as systematic uncertainty. This uncertainty also depends on \(p_{\mathrm{T}}\) and is equal to 3.7% for \(p_{\mathrm{T}} <5\) GeV/\(c\) and approximately 2% for \(p_{\mathrm{T}} >5\) GeV/\(c\). In Table 1, both contributions are added in quadrature and provided as ranges for the \(p_{\mathrm{T}}\)- and y-differential results.

As described in Sect. 3.3, the particle identification efficiency is determined via a data-driven procedure using a sample of identified electrons from tagged photon conversions. For the MB data sample, the uncertainty of this procedure is estimated by repeating the analysis with a looser and a tighter hadron (pion and proton combined) rejection criteria and taking the largest deviation from the results obtained with the standard PID selection divided by \(\sqrt{12}\). In addition, the statistical uncertainty of the pure electron sample used for the determination of the efficiency, which becomes non-negligible at high \(p_{\mathrm{T}}\), is propagated to the total uncertainty for the