1 Introduction

High-energy collisions at the large hadron collider (LHC) create a suitable environment for the production of light (anti-)nuclei. In ultra-relativistic heavy-ion collisions light (anti-)nuclei are abundantly produced [1,2,3], but in elementary pp collisions their production is lower [1, 4,5,6]. As a consequence, there are only few detailed measurements of (anti-)nuclei production rate in pp collisions. However, with the recently collected large data sample it is now possible to perform more differential measurements of light (anti-)nuclei production as a function of multiplicity and transverse momentum. In this paper, we present the detailed study of the multiplicity dependence of (anti-)deuteron production in pp collisions at \(\sqrt{s} =\) 13 TeV, the highest collision energy so far delivered at the LHC.

The production mechanism of light (anti-)nuclei in high-energy hadronic collisions is not completely understood. However, two groups of models have turned out to be particularly useful, namely statistical hadronisation models (SHM) and coalescence models. The SHMs, which assume particle production according to the thermal equilibrium expectation, have been very successful in explaining the yields of light (anti-)nuclei along with other hadrons in Pb–Pb collisions [7], suggesting a common chemical freeze-out temperature for light (anti-)nuclei and other hadron species. The ratio between the \(p_{\mathrm {T}}\)-integrated yields of deuterons and protons (d/p ratio) in Pb–Pb collisions remains constant as a function of centrality, but rises in pp and p–Pb collisions with increasing multiplicity, finally reaching the value observed in Pb–Pb [1, 8, 9]. The constant d/p ratio in Pb–Pb collisions as a function of centrality is consistent with thermal production, suggesting that the chemical freeze-out temperature in Pb–Pb collisions does not vary with centrality [10]. Assuming thermal production in pp collisions as well, the lower d/p ratio would indicate a lower freeze-out temperature [10]. On the other hand, the ratio between the \(p_{\mathrm {T}}\)-integrated yields of protons and pions (p/\(\pi \) ratio) does not show a significant difference between pp and Pb–Pb collisions [11, 12]. Also, for p–Pb collisions the freeze-out temperature obtained with SHMs using only light-flavoured particles is constant with multiplicity and its value is similar to that obtained in Pb–Pb collisions [13]. Thus, the increase of the d/p ratio with multiplicity for smaller systems cannot be explained within the scope of the grand-canonical SHM as is done in case of Pb–Pb. It is also not consistent with a simple SHM that the d/p and p/\(\pi \) ratios behave differently as a function of multiplicity even though numerator and denominator differ in both cases by one unit of baryon number. Nonetheless, a process similar to the canonical suppression of strange particles might be worth considering also for baryons. A recent calculation within the SHM approach with exact conservation of baryon number, electric charge, and strangeness focuses on this aspect [14].

In coalescence models (anti-)nuclei are formed by nucleons close in phase-space [15]. In this approach, the coalescence parameter \(B_2\) quantitatively describes the production of \(\text {(anti-)deuterons}\). \(B_2\) is defined as

$$\begin{aligned} B_{2}\left( p_{\mathrm {T}}^p\right)= & {} E_{d}\frac{\mathrm {d}^3N_{d}}{\mathrm {d}p_{d}^3}\bigg /\left( E_{p}\frac{\mathrm {d}^3N_{p}}{\mathrm {d}p_{p}^3}\right) ^2\nonumber \\= & {} \frac{1}{2\pi p_{\mathrm {T}}^d}\frac{\mathrm {d}^2N_{d}}{\mathrm {d}y\mathrm {d}p^{d}_{\mathrm {T}}} \; \bigg / \left( \frac{1}{2\pi p_{\mathrm {T}}^p}\frac{\mathrm {d}^2N_{p}}{\mathrm {d}y\mathrm {d}p_{\mathrm {T}} ^{p}}\right) ^2 , \end{aligned}$$

where E is the energy, p is the momentum, \(p_{\mathrm {T}}\)  is the transverse momentum and y is the rapidity. The labels p and d are used to denote properties related to protons and deuterons, respectively. The invariant spectra of the \(\text {(anti-)protons}\) are evaluated at half of the transverse momentum of the deuterons, so that \(p_{\mathrm {T}}^p = p_{\mathrm {T}}^d / 2\). Neutron spectra are assumed to be equivalent to proton spectra, since neutrons and protons belong to the same isospin doublet. Since the coalescence process is expected to occur at the late stage of the collision, the parameter \(B_2\) is related to the emission volume. In a simple coalescence approach, which describes the uncorrelated particle emission from a point-like source, \(B_2\) is expected to be independent of \(p_{\mathrm {T}}\) and multiplicity. However, it has been observed that \(B_2\) at a given transverse momentum decreases as a function of multiplicity, suggesting that the nuclear emission volume increases with multiplicity [2, 9, 16]. In Pb–Pb collisions the \(B_2\) parameter as a function of \(p_T\) shows an increasing trend, which is usually attributed to the position-momentum correlations caused by radial flow or hard scatterings [17, 18]. Such an increase of \(B_2\) as a function of \(p_{\mathrm {T}}\) has in fact also been observed in pp collisions at \(\sqrt{s}=7\) TeV  [6]. However, if pp collisions are studied in separate intervals of multiplicity, \(B_2\) is found to be almost constant as a function of \(p_{\mathrm {T}}\)  [8]. Similarly, \(B_2\) does not depend on \(p_{\mathrm {T}}\) in multiplicity selected p–Pb collisions [9]. Moreover, the highest multiplicities reached in pp collisions are comparable with those obtained in p–Pb collisions and not too far from peripheral Pb–Pb collisions. Therefore, the measure of \(B_2\) as a function of \(p_{\mathrm {T}}\) for finer multiplicity intervals in pp collisions at \(\sqrt{s} =\) 13 TeV gives the opportunity to compare different collision systems and to evaluate the dependence on the system size.

The paper is organized as follows. Section 2 discusses the details of the ALICE detector. Section 3 describes the data sample used for the analysis and the corresponding event and track selection criteria. Section 4 presents the data analysis steps in detail, such as raw yield extraction and various corrections, as well as the systematic uncertainty estimation. In Sect. 5, the results are presented and discussed. Finally, conclusions are given in Sect.  6.

2 The ALICE detector

A detailed description of the ALICE detectors can be found in [19] and references therein. For the present analysis the main sub-detectors used are the V0, the inner tracking system (ITS), the time projection chamber (TPC) and the time-of-flight (TOF), which are all located inside a 0.5 T solenoidal magnetic field.

The V0 detector [20] is formed by two arrays of scintillation counters placed around the beampipe on either side of the interaction point: one covering the pseudorapidity range \(2.8< \eta < 5.1\) (V0A) and the other one covering \(-3.7< \eta < -1.7\) (V0C). The collision multiplicity is estimated using the counts in the V0 detector, which is also used as trigger detector. More details will be given in Sect. 3.

The ITS [21], designed to provide high resolution track points in the proximity of the interaction region, is composed of three subsystems of silicon detectors placed around the interaction region with a cylindrical symmetry. The silicon pixel detector (SPD) is the subsystem closest to the beampipe and is made of two layers of pixel detectors. The third and the fourth layers consist of silicon drift detectors (SDD), while the outermost two layers are equipped with double-sided silicon strip detectors (SSD). The inner radius of the SPD, 3.9 cm, is essentially given by the radius of the beam pipe, while the inner field cage of the TPC limits the radial span of the entire ITS to be 43 cm. The ITS covers the pseudorapidity range \(|\eta |<0.9\) and it is hermetic in azimuth.

The same pseudorapidity range is covered by the TPC [22], which is the main tracking detector, consisting of a hollow cylinder whose axis coincides with the nominal beam axis. The active volume, filled with a Ne/CO\(_2\)/N\(_2\) gas mixture (Ar/CO\(_2\)/N\(_2\) in 2016), at atmospheric pressure, has an inner radius of about 85 cm, an outer radius of about 250 cm, and an overall length along the beam direction of 500 cm. The gas is ionised by charged particles traversing the detector and the ionisation electrons drift, under the influence of a constant electric field of \(\sim \) 400 V/cm, towards the endplates, where their position and arrival time are measured. The trajectory of a charged particle is estimated using up to 159 combined measurements (clusters) of drift times and radial positions of the ionisation electrons. The charged-particle tracks are then formed by combining the hits in the ITS and the reconstructed clusters in the TPC. The TPC is used for particle identification by measuring the specific energy loss (\(\mathrm {d} E/\mathrm {d} x\)) in the TPC gas.

Table 1 Summary of the relevant information about the multiplicity classes and the fits to the measured transverse momentum spectra of anti-deuterons. \(\langle \mathrm {d}N_{ch}/\mathrm {d}\eta \rangle \) is the mean pseudorapidity density of the primary charged particles [25]. n and C are the parameters of the Lévy–Tsallis fit function [27]. \(\mathrm {d}N/\mathrm {d}y\) is the integrated yield, with statistical uncertainties, multiplicity-un