1 Introduction

Light (anti)nuclei are abundantly produced in ultrarelativistic heavy-ion collisions [1,2,3] at the Large Hadron Collider (LHC), but their measurement in pp collisions is challenging due to their lower production yields. As a consequence, until few years ago there were only few measurements of the production rates of (anti)nuclei in small collision systems [1, 4,5,6]. This has recently changed thanks to the large pp data samples collected by ALICE at the LHC, which allow us to perform more precise and differential measurements of the production of light (anti)nuclei. In this paper, we present the detailed study of the multiplicity and transverse momentum dependence of (anti)proton, (anti)deuteron and (anti)\(^3\hbox {He}\) production in pp collisions at \(\sqrt{s} = 5.02\) TeV. The results shown in the following are the most accurate obtained so far in small systems and represent the full compilation of data available for pp collisions at different energies at the end of the LHC Run 2.

The production mechanism of light (anti)nuclei in high-energy hadronic collisions is not fully understood. The classes of models used for comparison with the experimental results are the Statistical Hadronisation Models (SHM) and the coalescence models. SHMs assume that particles originated from an excited region evenly occupy all the available states in phase space [7]. Pb–Pb collisions, characterised by a large extension of the particle-emitting source and hence considered as large systems, are described according to a grand canonical ensemble [8]. On the contrary, pp and p–Pb collisions, which are characterised by a small size and are considered as small systems, must be described based on a canonical ensemble, requiring the local conservation of the appropriate quantum numbers [9]. The expression Canonical Statistical Model (CSM) is used to underline the canonical description.

An important observable that provides information on the production mechanism is the ratio between the \(p_{\mathrm {T}}\)-integrated yields of nuclei and protons. The measured d/p and \(^3\)He/p ratios show a rather constant behaviour as a function of centrality in Pb–Pb collisions. In contrast to that, they increase in pp and p–Pb collisions with increasing multiplicity, finally reaching the values measured in Pb–Pb collisions [1, 10, 11]. The constant nuclei-to-proton ratios in large collision systems is predicted by the SHMs [12], while the experimentally determined difference between small and large systems can be qualitatively explained as an effect of the canonical suppression of the nuclei yields for small system sizes. The prediction of the CSM saturates towards the grand canonical value at larger system size [13] .

In coalescence models, (anti)nuclei are formed by nucleons close in phase space [14]. In this approach, the coalescence parameter \(B_{\mathrm {A}}\) relates the production of (anti)protons to the one of \(\text {(anti)nuclei}\). \(B_{\mathrm {A}}\) is defined as

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

where \(p_{\mathrm {T}}\) is the transverse momentum, y the rapidity and N the number of particles. The labels p and A are used to denote properties related to protons and nuclei with mass number A, respectively. The production spectra of the \(\text {(anti)protons}\) are evaluated at the transverse momentum of the nucleus divided by the mass number, so that \(p_{\mathrm {T}}^{\mathrm {p}} = p_{\mathrm {T}}^{\mathrm {A}} /A\). Neutron spectra are assumed to be equal to proton spectra, due to the isospin symmetry restoration in hadron collisions at the LHC. Since the coalescence process is expected to occur at the late stages of the collision, the \(B_{\mathrm {A}}\) parameter is related to the emission volume. In a simple coalescence approach, which describes the uncorrelated particle emission from a point-like source, \(B_\mathrm {A}\) is expected to be independent of \(p_{\mathrm {T}}\)  and multiplicity. In this context, the measurements of the nuclei-to-proton ratios and of the \(B_\mathrm {A}\) parameters in pp collisions at \(\sqrt{s} = 5.02\) TeV reported in this paper are important to complete the present picture of the production of light nuclei in small systems. In addition, the increased statistics exploited in the present analysis will allow us to better constrain the models, thus to provide important inputs to both the theoretical and experimental communities.

2 The ALICE apparatus

A detailed description of the ALICE detectors can be found in [15, 16] and references therein. In the following more information is given on the sub-detectors used to perform the analysis presented in this work, namely the V0, the Inner Tracking System (ITS), the Time Projection Chamber (TPC) and the Time-of-Flight (TOF). All of them are located inside a solenoidal magnet creating a magnetic field parallel to the beam line, with an intensity of 0.5 T for the data sample here considered.

The V0 detector [17] is formed by two arrays of scintillation counters placed around the beam pipe on either side of the interaction point. They cover the pseudorapidity ranges \(2.8 \le \eta \le 5.1\) (V0A) and \(-3.7 \le \eta \le -1.7\) (V0C). The collision multiplicity is estimated using the signal amplitude in the V0 detector, which is also used as a trigger detector. More details will be given in Sect. 3.

The ITS [18] provides high resolution track points in the proximity of the interaction region and consists of three subsystems. Going from the innermost to the outermost subsystem, we find: two layers of Silicon Pixel Detectors (SPD), two layers of Silicon Drift Detectors (SDD) and two layers equipped with double-sided Silicon Strip Detectors (SSD). The ITS extends radially from 3.9 to 43 cm, it is hermetic in azimuth and it covers the pseudorapidity range \(|\eta |<0.9\).

The same pseudorapidity range is covered by the TPC [19], 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/\(\hbox {CO}_2\)/\(\hbox {N}_2\) gas mixture at atmospheric pressure, has an inner radius of about 85 cm and an outer radius of about 250 cm. 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 reconstructed by combining the hits in the ITS and the measured clusters in the TPC. The TPC is also used for particle identification (PID) by measuring the specific energy loss (\(\mathrm {d} E/\mathrm {d} x\)) in the TPC gas. In pp collisions, the \(\mathrm {d} E/\mathrm {d} x\) in the TPC is measured with a resolution of \(\approx 5.2\%\) [15].

The TOF [20] covers the full azimuth for the pseudorapidity interval \(|\eta |<0.9\). The detector is based on the Multigap Resistive Plate Chambers (MRPC) technology and is located, with a cylindrical symmetry, at an average distance of 380 cm from the beam axis. The particle identification is based on the difference between the measured time of flight and its expected value, computed for each mass hypothesis from track momentum and length. A precise starting signal for the measurement of the time of flight by the TOF is provided by the T0 detector, consisting of two arrays of Cherenkov counters, T0A and T0C, which cover the pseudorapidity regions \(4.61 \le \eta \le 4.92\) and \(3.28 \le \eta \le 2.97\), respectively [21]. The overall resolution on the particles time of flight, including the start time, is \(\approx 80\) ps.

3 Data sample

This analysis is based on approximately 900 million pp collisions (events) at \(\sqrt{s}=5.02\) TeV collected in 2017 by ALICE at the LHC. Events are selected by a minimum-bias (MB) trigger, requiring at least one hit in each of the two V0 detectors. An additional offline rejection is performed to remove events with more than one reconstructed primary vertex (pile-up events) and events triggered by interactions of the beam with the residual gas in the LHC beam pipe [17]. In total, 1.8% of the collected events are rejected due to these selections.

The production of (anti)nuclei is measured around midrapidity, within a rapidity range of \(|y|<0.5\), and within the pseudorapidity interval \(|\eta |<0.8\) to maximise the detector performance. The selected tracks are required to have at least 70 reconstructed points in the TPC and two points in the ITS in order to guarantee good track momentum and \(\mathrm {d} E/\mathrm {d} x\) resolution in the relevant \(p_{\mathrm {T}}\) ranges. In addition, at least one hit in the SPD is required to ensure a resolution of the distance of closest approach to the primary vertex better than 300 \(\upmu \)m, both along the beam axis (\(\hbox {DCA}_\mathrm {z}\)) and in the transverse plane (\(\hbox {DCA}_\mathrm {xy}\)) [