The transverse momentum spectra of deuterons and anti-deuterons in different multiplicity classes as well as INEL>0 pp collisions are reported in Fig. 2. The spectra normalised to inelastic pp collisions (INEL) are included in the data provided with this paper. The mean charged-particle multiplicity \(\left\langle {\mathrm {d} N_{ch}/\mathrm {d} \eta } \right\rangle \) for each class is reported in Table 1. The spectra exhibit a slight hardening with increasing multiplicity: the slope of the spectra becomes less steep and the mean transverse momentum \(\left\langle p_{{\mathrm {T}}}\right\rangle \) moves towards higher values. This effect is similar to that observed in Pb–Pb collisions, where it is explained with the presence of increasing radial flow with centrality [1, 28]. However, in pp collisions the intensity of the hardening is not as dramatic. The ratio between the spectra of anti-deuterons and deuterons for all the multiplicity classes under study is reported in Fig. 2. The ratio is compatible within uncertainties with unity in all multiplicity classes.
To calculate the integrated yield (\(\mathrm {d} N/\mathrm {d} y\)) and the mean \(p_{\mathrm {T}}\) the spectra have been fitted with the Lévy–Tsallis function [27, 29, 30]:
$$\begin{aligned} \frac{\mathrm {d}^2N}{\mathrm {d}y \, \mathrm {d}p_{\mathrm {T}}} = \frac{\mathrm {d}N}{\mathrm {d}y}\frac{p_{\mathrm {T}}\left( n-1\right) \left( n-2\right) }{nC [nC + m\left( n-2\right) ]}\left( 1+\frac{m_{\mathrm {T}}-m}{nC}\right) ^{-n}, \end{aligned}$$
(2)
where m is the particle rest mass (i.e. the mass of the deuteron), \(m_{\mathrm {T}}=\sqrt{m^2+p_{\mathrm {T}}^2}\) is the transverse mass, while n, \(\mathrm {d} N/\mathrm {d} y\) and C are free fit parameters. The Lévy–Tsallis function is used to extrapolate the spectra in the unmeasured regions of \(p_{\mathrm {T}}\). One contribution to the systematic uncertainty is obtained by shifting the data points to the upper border of their systematic uncertainty and to the corresponding lower border. The difference between these values and the reference one is taken as an uncertainty which amounts to \(\sim \) 11%. Another contribution to the systematic uncertainty is estimated by using alternative fit functions such as simple exponentials depending on \(p_{\mathrm {T}}\) and \(m_{\mathrm {T}}\), as well as a Boltzmann function, and is found to be \(\sim \) 3%. The two contributions are summed in quadrature. The extrapolation amounts to 25% of the total yield in the highest multiplicity class, where the widest \(p_{\mathrm {T}}\) range is measured, and increases up to 35% in the lowest multiplicity class.
The statistical uncertainty on the integrated yield is obtained by moving the data points randomly within their statistical uncertainties, using a Gaussian probability distribution centered at the measured data point, with a standard deviation corresponding to the statistical uncertainty. In the unmeasured regions at low and high \(p_{\mathrm {T}}\), the value of the fit function at a given \(p_{\mathrm {T}}\) is considered. In this case the statistical uncertainty is estimated using a Monte Carlo method to propagate the uncertainties on the fit parameters. Following the same procedure, the \(\langle p_{\mathrm {T}}\rangle \) and its statistical and systematic uncertainties are computed. The resulting mean \(p_{\mathrm {T}}\) and \(\mathrm {d} N/\mathrm {d} y\), as well as the parameters of the individual Lévy–Tsallis fits, are listed in Table 1.
The coalescence parameter as a function of the transverse momentum is shown in Fig. 3. The transverse momentum spectra needed for the \(B_{2}\) computation are taken from Ref. [31]. The \(B_2\) values for INEL>0 collisions show a significant deviation from a transverse momentum independent coalescence parameter as expected by the simplest implementation of the coalescence model. However, it has been shown [8] that the the multiplicity-integrated coalescence parameter is distorted because deuterons are biased more towards higher multiplicity than protons, and consequently have harder \(p_{\mathrm {T}}\) spectra than expected from inclusive protons. The coalescence parameter evaluated in fine multiplicity classes is consistent with a flat behaviour, in agreement with the expectation of the simple coalescence model.
The evolution of the coalescence parameter as a function of the charged particle multiplicity is sensitive to the production mechanism of deuterons. Recent formulations of the coalescence model [32, 33] implement an interplay between the size of the collision system and the size of the light nuclei produced via coalescence.
Figure 4 shows how the \(B_2\), for a fixed transverse momentum interval, evolves in different systems as a function of the charged particle multiplicity. \(B_2\) is shown at \(p_{\mathrm {T}} ~=~0.75\) GeV/c, which was measured in all the analyses. However, the trend is the same for other \(p_{\mathrm {T}}\) values. The measurements are compared with the model descriptions detailed in [33]. The two descriptions use different parameterisations for the size of the source. Parameterisation A uses the ALICE measurements of system radii R from HBT studies as a function of multiplicity[34]. These values are fitted with the function:
$$\begin{aligned} R = a \; \langle \mathrm {d}N/\mathrm {d}\eta \rangle ^{1/3} + b, \end{aligned}$$
(3)
where a and b are free parameters. In parameterisation B the free parameters a and b in Eq. 3 are fixed to reproduce the \(B_2\) of deuterons in Pb–Pb collisions at \(\sqrt{s_{\mathrm {NN}}} =2.76\) TeV in the centrality class 0–10%. The first parameterisation (dashed red line) describes well the measured \(B_2\) in pp and p–Pb collisions, while it overestimates the measurements in Pb–Pb collisions. However, as outlined by the authors in [33], a more refined parameterisation of the HBT radius evolution through different systems might reduce the observed discrepancy. The parameterisation of the source size fixed to the \(B_2\) measurement in central \(\text {Pb--Pb}\) collisions already departs from the measurements in peripheral Pb–Pb collisions and it underestimates the coalescence parameter for small colliding systems.
Figure 5 shows the ratio of the \(p_{\mathrm {T}}\)-integrated yields of deuterons and protons for different multiplicities in different collisions systems and at different energies. The ratio increases monotonically with multiplicity for pp and p–Pb collisions and eventually saturates for Pb–Pb collisions. The experimental data are compared with a SHM prediction. In this implementation of the model, called the canonical statistical model (CSM), exact conservation of baryon number (B), charge (Q), and strangeness (S) is enforced using the recently developed THERMAL-FIST package [14]. The calculations with the CSM are performed using 155 MeV for the chemical freeze-out temperature, \(B = Q = S = 0\) and two different values of the correlation volume, which is expressed in terms of rapidity units \(\mathrm {d} V/\mathrm {d} y\), corresponding to one and three units of rapidity, respectively. The model qualitatively reproduces the trend observed in data. This might suggest that for small collision systems the light (anti-) nuclei production could be canonically suppressed and that a canonical correlation volume might exist. The correlation volume required to describe the measurements is larger than one unit of rapidity. However, such a canonical suppression should also affect the p/\(\pi \) ratio in a similar way and this is not observed in the experimental measurements [11, 35].
A full coalescence calculation, taking into account the interplay between the system size and the width of the wave function of the produced \(\text {(anti-)deuterons}\), is also able to describe the measured trend of the d/p ratio [36] and it describes the data consistently better than CSM for all system sizes.