Measurements of $\pi ^\pm$, $K^\pm$, p and $\bar{p}$ spectra in $^{40}\hbox {Ar+}^{45}\hbox {Sc}$ collisions at 13A to 150A $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$

Abstract

The NA61/SHINE experiment at the CERN Super Proton Synchrotron studies the onset of deconfinement in strongly interacting matter through a beam energy scan of particle production in collisions of nuclei of varied sizes. This paper presents results on inclusive double-differential spectra, transverse momentum and rapidity distributions and mean multiplicities of $\pi ^\pm$, $K^\pm$, p and $\bar{p}$ produced in $^{40}\hbox {Ar+}^{45}\hbox {Sc}$ collisions at beam momenta of 13A, 19A, 30A, 40A, 75A and 150A $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The analysis uses the 10% most central collisions, where the observed forward energy defines centrality. The energy dependence of the $K^\pm$/$\pi ^\pm$ ratios as well as of inverse slope parameters of the $K^\pm$ transverse mass distributions are placed in between those found in inelastic $p+p$ and central Pb + Pb collisions. The results obtained here establish a system-size dependence of hadron production properties that so far cannot be explained either within statistical or dynamical models.

1 Introduction

This paper presents experimental results on inclusive spectra and mean multiplicities of $\pi ^\pm , K^\pm , p$ and $\bar{p}$ produced in the 10% most central $^{40}\hbox {Ar+}^{45}\hbox {Sc}$ collisions at beam momenta of 13A, 19A, 30A, 40A, 75A, and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ ($\sqrt{s_{NN}}$ = 5.12, 6.12, 7.62, 8.77, 11.9 and 16.8 GeV). These studies form a part of the strong interactions program of NA61/SHINE  [1] at the CERN SPS investigating the properties of the onset of deconfinement and searching for the possible existence of a critical point. The program is mainly motivated by the observed rapid changes in hadron production properties in central Pb+Pb collisions at about 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ by the NA49 experiment [2, 3]. These findings were interpreted as the onset of deconfinement; they were confirmed by the RHIC beam energy program [4] and their interpretation is supported by the LHC results (see Ref. [5] and references therein).

The goals of the NA61/SHINE strong interaction program are pursued experimentally by a two-dimensional scan in collision energy and size of colliding nuclei. This allows us to systematically explore the phase diagram of strongly interacting matter [1]. In particular, the analysis of the existing data within the framework of statistical models suggests that by increasing collision energy one increases the temperature and decreases the baryon chemical potential of the fireball of strongly interacting matter at kinetic freeze-out [6], whereas by increasing the nuclear mass of the colliding nuclei the temperature decreases [6,7,8,9].

Within this program NA61/SHINE recorded data on p+p, Be+Be, Ar+Sc, Xe+La, and Pb+Pb collisions during 2009–2018 running. Further high-statistics measurements of Pb+Pb collisions with an upgraded detector started in 2022 [10, 11]. Comprehensive results on particle spectra and multiplicities have already been published for p+p interactions [12,13,14] and Be+Be collisions [15, 16] at 19A–150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (20–158 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ for p+p). For Ar+Sc collisions, only results on $\pi ^-$ production were published up to now [17].

The Ar+Sc collisions became crucial for the NA61/SHINE scan program. As the results obtained for the Be+Be system closely resemble inelastic p+p interactions, the collisions of Ar+Sc are the lightest of the studied systems for which a significant increase in the $K^+/\pi ^+$ ratio was observed. The properties of measured spectra and multiplicities indicate that the Ar+Sc system is on a boundary between light (p+p, Be+Be) and heavy (Pb+Pb) systems.

The paper is organized as follows. After this introduction, the experiment is briefly presented in Sect. 2. The analysis procedure, as well as statistical and systematic uncertainties, are discussed in Sect. 3. Section 4 presents experimental results and compares them with measurements of NA61/SHINE in inelastic p+p interactions [12,13,14] and central Be+Be [15, 16] collisions, as well as NA49’s results on Pb+Pb, C+C and Si+Si reactions [2, 3]. Section 5 discusses model predictions. A summary in Sect. 6 closes the paper. Additionally, Appendix A, containing plots presenting details of the analysis is included.

The following variables and definitions are used in this paper. The particle rapidity y is calculated in the collision center of mass system (cms), $y=0.5 \cdot \ln {[(E+p_{L}c)/(E}-p_{L}c)]$, assuming proton mass, where E and $p_{L}$ are the particle energy and longitudinal momentum, respectively. The transverse component of the momentum is denoted as $p_{T}$ and the transverse mass $m_{T}$ is defined as $m_{T} = \sqrt{m^2 + (cp_{T})^2}$ where m is the particle mass in $\text{ Ge }\hspace{-1.00006pt}\text{ V }$. The momentum in the laboratory frame is denoted $p$ and the collision energy per nucleon pair in the center of mass by $\sqrt{s_{NN}}$.

The Ar+Sc collisions are selected by requiring a low value of the forward energy – the energy emitted into the region populated by projectile spectators. These collisions are referred to as central collisions and a selection of collisions based on the forward energy is called a centrality selection. The term central is written in italics throughout this paper to denote the specific event selection procedure based on measurements of the forward energy.

2 Experimental setup

The NA61/SHINE experiment is a multi-purpose facility designed to measure particle production in nucleus-nucleus, hadron-nucleus and p+p interactions [18]. The detector is situated at the CERN Super Proton Synchrotron (SPS) in the H2 beamline of the North experimental area. A schematic diagram of the setup is shown in Fig. 1. The main components of the particle detection system used in the 2015 Ar+Sc data-taking campaign are four large-volume Time Projection Chambers (TPC). Two of them, called Vertex TPCs (VTPC), are located downstream of the target inside superconducting magnets with a maximum combined bending power of 9 Tm. The magnetic field was scaled down in proportion to the beam momentum in order to obtain similar $y-p_{T}$ acceptance at all beam momenta. The main TPCs (MTPC) and two walls of pixel Time-of-Flight (ToF-L/R) detectors are placed symmetrically on either side of the beamline downstream of the magnets. The TPCs are filled with Ar:CO$_{2}$ gas mixtures in proportions 90:10 for the VTPCs and 95:5 for the MTPCs. The Projectile Spectator Detector (PSD) is positioned 20.5 m (16.7 m) downstream of the MTPCs at beam momenta of 75A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (13A, 19A, 30A, 40$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$), centered in the transverse plane on the deflected position of the beam. A degrader in the form of a 5 cm diameter brass cylinder was placed in front of the center of the PSD in order to reduce electronic saturation effects and shower leakage from the downstream side. The length of the cylinder was 10 cm except the 19$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ measurements, when the length was 5 cm. No degrader was used at 13$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$.

Fig. 1

The schematic layout of the NA61/SHINE experiment at the CERN SPS [18] showing the components used for the Ar+Sc energy scan. The beam instrumentation is sketched in the inset. The alignment of the chosen coordinate system as shown in the figure. The nominal beam direction is along the z-axis. The magnetic field bends charged particle trajectories in the x–z (horizontal) plane. The drift direction in the TPCs is along the y (vertical) axis

Primary beams of fully ionized ${}^{40}$Ar nuclei were extracted from the SPS accelerator at beam momenta of 13A, 19A, 30A, 40A, 75A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Two scintillation counters, S1 and S2, provide beam trigger definition, together with a veto counter V1 with a 1 cm diameter hole, which defines the beam before the target. The S1 counter also provides the timing reference (start time for all counters). Beam particles are selected by the trigger system requiring the coincidence $\text {S1} \wedge \text {S2} \wedge \overline{\text {V1}}$. Individual beam particles are precisely measured by the three Beam Position Detectors (BPDs) placed upstream of the target [18]. Collimators in the beam line were adjusted to obtain beam rates of the order of $10^4$/s during the 10.4 s spill within a 32.4 s accelerator super cycle.

The target was a stack of six Sc plates of 1 mm thickness and 2 $\times$ 2 cm$^2$ area placed 75 cm upstream of VTPC-1. Mass concentrations of impurities were measured at 0.3% resulting in an estimated increase of the produced pion multiplicity by less than 0.2% due to the small admixture of heavier elements [19]. No correction was applied for this negligible contamination. Data were taken with target inserted (93%) and target removed (7%).

Interactions in the target are selected with the trigger system by requiring an incoming Ar ion and a signal below that of beam ions from S5, a small 2 cm diameter scintillation counter placed on the beam trajectory behind the MTPCs. This minimum bias trigger selects inelastic collisions of the beam ion with the target and with matter between the target and S5. In addition, central collisions were selected by requiring an energy signal below a threshold set on the summed signal from the 16 central modules of the PSD, which measure mainly the energy of projectile spectators. The cut was set to retain only the event triggers with roughly 30% smallest energies in the PSD, which was studied quantitatively in offline analysis. The central event trigger condition thus was $\text {S1} \wedge \text {S2} \wedge \overline{\text {V1}}\wedge \overline{\text {S5}}\wedge \overline{\text {PSD}}$. The statistics of recorded events are summarized in Table 1.

Table 1 Basic beam properties and the number of events recorded and used in the analysis of 10% most central Ar+Sc interactions. Event selection criteria will be discussed in Sect. 3

3 Analysis procedure

This section starts with a brief overview of the data analysis procedure and the corrections applied to the experimental results. It also defines to which species of particles the final results correspond. A description of the detector calibration and the track and vertex reconstruction procedures can be found in Ref. [12].

The analysis procedure consists of the following steps:

• (i) application of event and track selection criteria,

• (ii) determination of raw spectra of identified charged hadrons using the selected events and tracks,

• (iii) evaluation of corrections to the raw spectra based on experimental data and simulations,

• (iv) calculation of the corrected spectra and mean multiplicities,

• (v) calculation of statistical and systematic uncertainties.

Corrections for the following biases were evaluated:

• (a) contribution from off-target interactions,

• (b) losses of in-target interactions due to the event selection criteria,

• (c) geometric acceptance,

• (d) reconstruction and detector inefficiencies,

• (e) losses of tracks due to track selection criteria,

• (f) contribution of particles other than primary (see below) charged particles produced in Ar+Sc collisions,

• (g) losses of primary charged particles due to their decays and secondary interactions.

Correction (a) was found to be negligible ($\mathcal {O}(10^{-4})$) and was therefore not applied.

Corrections (b)–(g) were estimated by data and simulations. MC events were generated with the Epos1.99 model (version CRMC 1.5.3) [20], passed through detector simulation employing the Geant 3.21 package [21] and then reconstructed using standard procedures, exactly matching the ones used in the processing of experimental data. The selection of central events in the simulation was based on the number of projectile spectator nucleons available in the Epos model.

The final results refer to particles produced in central Ar+Sc collisions by strong and electromagnetic processes. Such hadrons are referred to as primary hadrons. The definition of central collisions is given in Sect. 3.1.

3.1 Central collisions

A short description of the procedure for defining central collisions is given below. For more details, see Refs. [17, 22, 23].

The final results presented in this paper refer to the 10% of Ar+Sc collisions with the lowest value of the forward energy $E_\text {F}$ (central collisions). The quantity $E_\text {F}$ is defined as the sum of energies (measured in the laboratory reference frame) of all particles produced in Ar+Sc collisions via strong and electromagnetic processes in the forward momentum region defined by the acceptance map in Ref. [24]. The forward region defined by the acceptance map roughly corresponds to polar angles (angle between beam momentum and secondary particle momentum vectors in LAB frame of reference) $\theta$ smaller than $\theta _\text {max} = 0.1-0.2$ (charged particles) and $\theta _\text {max} = 0.03-0.05$ (neutral particles). The final results on central collisions, derived using this procedure, allow a precise comparison with model predictions without additional information about the NA61/SHINE setup and used magnetic field. Using this definition, the mean number of wounded nucleons $\langle W \rangle$ was calculated in the Wounded Nucleon Model (WNM) [25] implemented in Epos  [26].

Fig. 2

PSD modules used in the online and offline event selection. The online central trigger is derived from the energy in the central 16 modules, while the set of modules used to determine the offline PSD energy $E_\text {PSD}$ changes with beam momentum

For data analysis, the event selection was based on the 10% of collisions with the lowest value of the energy $E_\text {PSD}$ measured by a subset of PSD modules (see Fig. 2) optimized for the sensitivity to projectile spectators. The acceptance in the definition of the forward energy $E_\text {F}$ corresponds closely to the acceptance of this subset of PSD modules at all energies [15, 17].

Online event selection by the central hardware trigger used a threshold on the sum of electronic signals from the 16 central modules of the PSD set to accept approx. 30% of the inelastic interactions. Measured distributions of $E_\text {PSD}$ for minimum-bias and central trigger selected events, calculated in the offline analysis, are shown in Fig. 3 at beam momenta of 19A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The accepted region corresponding to the 10% of most central collisions is indicated by shading. The minimum-bias distribution was obtained using the data from the beam trigger with an offline selection of events by requiring an event vertex in the target region. A properly normalized spectrum for target-removed events was subtracted.

Fig. 3

Distributions of the energy $E_\text {PSD}$ measured by the PSD calorimeter for 19A (left plot) and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (right plot) beam momentum, for minimum-bias selected (blue histograms) and central trigger selected (red histograms) events. Histograms are normalized to agree in the overlap region (from the beginning of the distribution to the black dashed line). The on-line central trigger was set to accept approximately 30% of most central inelastic events. The shaded area indicates 10% collisions with the smallest $E_\text {PSD}$

The forward energy $E_\text {F}$ cannot be measured directly. However, both $E_\text {F}$ and $E_\text {PSD}$ can be obtained from simulations using the Epos1.99 (version CRMC 1.5.3) [20, 26, 27] model. A global factor $c_{\text {cent}}$ (listed in Table 2) was calculated as the ratio of mean negatively charged pion multiplicities obtained with the two selection procedures in the 10% most central events. A possible dependence of the scaling factor on rapidity and transverse momentum was neglected. The resulting factors $c_{\text {cent}}$ range between 1.002 and 1.005, corresponding to a correction at least an order of magnitude smaller compared to the systematic uncertainties of the measured particle multiplicities (see Sect. 3.5.2). The correction was therefore not applied and its possible impact was neglected in the final uncertainty calculation.

Finally, the Epos WNM [26] simulation was used to estimate the average number of wounded nucleons $\langle W \rangle$ for the 10% of events with the smallest number of spectator nucleons and with the smallest value of $E_\text {F}$. The average impact parameter $\langle b \rangle$ was also obtained for the latter selection. Results are listed in Table 2. Example distributions of $E_F-W$ for 19A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ beam momenta are shown in Fig. 4. These distributions are quite broad and emphasize the importance of proper simulation of the centrality selection when comparing model calculations with the experimental results.

Fig. 4

Distributions of number of wounded nucleons as a function of $E_F$ for all inelastic Ar+Sc collisions at 19A and 150A GeV/c beam momentum calculated from Epos  [26]

Table 2 The average number of wounded nucleons $\langle W \rangle$ and the average impact parameter $\langle b\rangle$ in the 10% most central Ar+Sc collisions estimated from simulations using Epos  [20, 26, 27]. Results from Epos  WNM are for centrality selection using the smallest number of spectators, whereas the Epos  $E_\text {F}$ results are obtained using the forward energy $E_\text {F}$ within the acceptance map in Ref. [24]. The last line presents numerical values of the $c_\text {cent}$ factor for all the beam momenta (see text and Ref. [17] for more details)

3.2 Event and track selection

3.2.1 Event selection

For further analysis, Ar+Sc events were selected using the following criteria:

• (i) No off-time beam particle detected within a time window of $\pm 4~\upmu \hbox {s}$ around the trigger particle and no other event trigger detected within a time window of $\pm 25~\upmu \hbox {s}$ around the trigger particle, reducing pile-up events in the data sample.

• (ii) Beam particle detected in at least three planes out of four of BPD-1 and BPD-2 and in both planes of BPD-3, providing good reconstruction of the beam trajectory.

• (iii) A well-reconstructed interaction vertex with z-coordinate (fitted using the beam trajectory and TPC tracks) not farther away than 10 cm from the center of the Sc target.

• (iv) An upper limit on the measured energy $E_{PSD}$ selecting 10% of all inelastic collisions.

The event statistics after applying the selection criteria is summarized in Table 1.

3.2.2 Track selection

To select tracks of primary charged hadrons and to reduce the contamination by particles from secondary interactions and weak decays, the following track selection criteria were applied:

• (i) Fitted x component of particle rigidity $q \cdot p$ should be positive. This selection minimizes the angle between the track trajectory and the TPC pad direction for the chosen magnetic field direction, reducing uncertainties of the reconstructed cluster position, energy deposition and track parameters.

• (ii) Total number of reconstructed points on the track should be greater than 30, ensuring good resolution of $\text {d}E/\text {d}x$ measurement.

• (iii) Sum of the number of reconstructed points inside the vertex magnets (VTPC-1 and VTPC-2) should be greater than 15, which ensures good accuracy of track momentum fit.

• (iv) The distance between the track extrapolated to the interaction plane and the reconstructed vertex (track impact parameter) should be smaller than 4 cm in the horizontal (bending) plane and 2 cm in the vertical (drift) plane.

In the case of $\text {d}E/\text {d}x$ analysis, an additional criterion was used:

• (i) track azimuthal angle $\phi$ should be within 30$^\circ$ with respect to the horizontal plane (x–z).

Similarly, specifically for $tof -\text {d} E/\text {d} x$ analysis, the following supplementary cuts were implemented:

• (i) the extrapolated trajectory (as measured in the TPCs) reaches one of the ToF walls,

• (ii) the last measured point on the track is located at least 70 cm upstream of the back wall of MTPCs, and its distance from the fitted track is within 4 cm,

• (iii) measured flight time and charge for the pixel are of good quality (as in Refs. [28, 29]).

3.3 Identification techniques

Charged particle identification in NA61/SHINE is based on the ionization energy loss, $\text {d}E/\text {d}x$ , in the gas of the TPCs and the time of flight, $tof$, obtained from the ToF-L and ToF-R walls. In the region of the relativistic rise of the ionization at large momenta, the measurement of $\text {d}E/\text {d}x$ alone allows identification. At lower momenta, the $\text {d}E/\text {d}x$ bands for different particle species overlap, and an additional measurement of $tof$ is used for unambiguous particle identification. These two methods allow covering most of the relevant space in rapidity and transverse momentum, in particular the mid-rapidity region of $K^+$ and $K^-$ spectra, which is an important part of the strong interaction program of NA61/SHINE. The acceptance of the two methods is shown in Figs. 5 and 6 for the 10% most central Ar+Sc interactions at 40A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$, respectively. The figures also display the $h^-$ analysis method [17], which provides large-acceptance measurements of $\pi ^-$ yields. At low beam energies, the $tof -\text {d} E/\text {d} x$ method extends the identification acceptance, while at the top SPS energy it overlaps with the $\text {d}E/\text {d}x$ method.

Fig. 5

Acceptance of the $tof -\text {d} E/\text {d} x$ and $\text {d}E/\text {d}x$ methods for identification of pions, kaons and protons in the 10% most central Ar+Sc interactions at 40$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Negatively charged pion yield is also calculated using a so-called $h^-$ method (see Ref. [17])

Fig. 6

Acceptance of the $tof -\text {d} E/\text {d} x$ and $\text {d}E/\text {d}x$ methods for identification of pions, kaons and protons in the 10% most central Ar+Sc interactions at 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Negatively charged pion yield is also calculated using a so-called $h^-$ method [17]

3.3.1 Identification based on energy loss measurement $\text {d}E/\text {d}x$

Time projection chambers provide measurements of energy loss $\text {d}E/\text {d}x$ of charged particles in the chamber gas along their trajectories. Simultaneous measurements of $\text {d}E/\text {d}x$ and $p$   allow extraction of information on particle mass. The mass assignment follows the procedure that was developed for the analysis of p+p reactions as described in Ref. [13]. Values of $\text {d}E/\text {d}x$ are calculated as the truncated mean (smallest 50%) of ionization energy loss measurements along the track trajectory. As an example, $\text {d}E/\text {d}x$ measured in Ar+Sc interactions at 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$  is presented in Fig. 7, for positively and negatively charged particles, as a function of momentum.

Fig. 7

Distribution of charged particles in the $\text {d}E/\text {d}x$ –$p$  plane. The energy loss (in units of the minimum ionizing particle) in the TPCs is shown for different charged particles produced in Ar+Sc collisions at 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Expectations for the dependence of the mean $\text {d}E/\text {d}x$ on  $p$ for the considered particle types are shown by the curves calculated based on the Bethe-Bloch function

In the $\text {d}E/\text {d}x$ method the contributions of $e^+, e^-, \pi ^+, \pi ^-$, $K^+, K^-, p,\,\bar{p} \text {~and~} d$ are obtained by fitting the $\text {d}E/\text {d}x$ distributions in bins of laboratory momentum $p$ and transverse momentum $p_{T}$. The data are divided into 13 logarithmic bins in $p$ in the range 5–100 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ and into linear bins in $p_{T}$. Thin binning in $p_{T}$ is used up to $p_{T} =0.6$ $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (bin width 0.05 GeV/c) and wider bins are used above this value (0.1 GeV/c). Due to the crossing of Bethe-Bloch curves at low momenta, the applicability of particle identification based solely on $\text {d}E/\text {d}x$ measurement is limited to tracks with $p >5$ $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Only bins with a total number of selected tracks greater than 100 were used in the further analysis.

Due to the characteristic long tails in the distribution of charge deposited in a single ionization cluster, the mean $\text {d}E/\text {d}x$ for a given track is calculated using 50% of the lowest charge deposits. Such a truncation may introduce an asymmetry of the final $\text {d}E/\text {d}x$ distribution, shift of the peak and it also affects its width. Therefore, the $\text {d}E/\text {d}x$ spectrum in each $p, p_T$ bin is fitted by the sum of asymmetric Gaussians with widths $\sigma _{i, l}$ depending on the particle type i and the number of points l measured in the TPCs (the method is based on previous work described in Refs. [30, 31]):

$$\begin{aligned} \rho (x)= & {} \sum _{i=\pi ^\pm ,~K^\pm ,p,\bar{p},d,e^\pm } N_i \frac{1}{\sum _l n_l}\sum _l \frac{n_l}{\sqrt{2\pi }\sigma _{i,l}}\nonumber \\{} & {} \times \exp \left[ -\frac{1}{2}\left( \frac{x-x_i+\frac{2}{\sqrt{2\pi }} \delta _{l} \sigma _{i,l}}{(1 \pm \delta _{l})~\sigma _{i,l}} \right) ^2\right] , \end{aligned}$$

(1)

where truncated mean energy loss $\text {d}E/\text {d}x$ is denoted with x, the amplitude of the contribution of particles of type i is expressed as $N_i$ and variable $n_l$ is the number of tracks with the number of points l in the sample. The peak position of the $\text {d}E/\text {d}x$ distribution for particle type i is expressed as $x_i$ and the expression $2 \delta _{l} \sigma /\sqrt{2\pi }$ accounts for the drift of the peak related to the asymmetry of the distribution introduced with the parameter $\delta _{l}=\delta _0/l$, which is taken with a negative sign if $x<x_i-\frac{2}{\sqrt{2\pi }} \delta _{l} \sigma _{i,l}$ and with a positive sign otherwise. The width, $\sigma _{i,l}$ depends on the particle species and the track length in the following way:

$$\begin{aligned} \sigma _{i,l} = \frac{\sigma _0}{\sqrt{l}} \left( \frac{x_i}{x_{\pi }} \right) ^{\alpha }, \end{aligned}$$

(2)

where $\sigma _0$ is common for all particle types and $\alpha$ is a universal constant. The details about the fitting procedure can be found in Ref. [32]. Examples of final fits are shown in Fig. 8.

Fig. 8

The $\text {d}E/\text {d}x$ distributions for negatively (left) and positively (right) charged particles in a selected $p-p_{T}$ bin produced in central Ar+Sc collisions at 75$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The fits by a sum of contributions from different particle types are shown by solid lines. The corresponding residuals (the difference between the data and fit divided by the statistical uncertainty of the data) are shown in the bottom plots

3.3.2 Identification based on time of flight and energy loss measurements ($tof -\text {d} E/\text {d} x$)

Identification of $\pi ^{+}$, $\pi ^{-}$, $K^{+}$, $K^{-}$, p and $\bar{p}$ at low momenta (0.5–10 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$) is possible when measurement of $\text {d}E/\text {d}x$ is combined with time-of-flight information $tof$. Timing signals from the constant-fraction discriminators and signal amplitude information are recorded for each tile of the ToF-L/R walls. The coordinates of the track intersection with the front face are used to match the track to tiles with valid $tof$ hits. The position of the extrapolation point on the scintillator tile is used to correct the measured value of $tof$ for the propagation time of the light signal inside the tile. The distribution of the difference between the corrected $tof$ measurement and the value calculated from the extrapolated track trajectory length with the assumed mass hypothesis can be described well by a Gaussian with a standard deviation of 80 ps for ToF-R and 100 ps for ToF-L. These values represent the $tof$ resolution including all detector effects.

Momentum phase space is subdivided into bins of 1 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ in $p$  and 0.1 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$  in $p_{T}$. Only bins with more than 1000 entries were used for extracting yields with the $tof -\text {d} E/\text {d} x$ method.

Fig. 9

Distribution of particles in the plane laboratory momentum and mass squared derived using time-of-flight measured by ToF-R (right) and ToF-L (left) produced in central Ar+Sc collisions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The lines show the expected mass squared values for different hadron species

The square of the particle mass $m^2$ is obtained from $tof$, from the momentum $p$ and from the fitted trajectory length L:

$$\begin{aligned} m^2=(cp)^2 \left( \frac{c^2~tof ^2}{L^2}-1 \right) . \end{aligned}$$

(3)

For illustration distributions of $m^2$ versus $p$  are plotted in Fig. 9 for positively (left) and negatively (right) charged hadrons produced in 10% central Ar+Sc interactions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Bands that correspond to different particle types are visible.

Example distributions of particles in the $m^2$–$\text {d}E/\text {d}x$ plane for the selected Ar+Sc interactions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ are presented in Fig. 10. Simultaneous $\text {d}E/\text {d}x$ and $tof$ measurements lead to improved separation between different hadron types. In this case, a simple Gaussian parametrization of the $\text {d}E/\text {d}x$ distribution for a given hadron type can be used.

Fig. 10

Example distributions of particles in $m^2$ and $\text {d}E/\text {d}x$ plane in a single bin ($2~\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c<p <3~\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$, $0.5~\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c<p_{T} <0.6~\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$) for positively (left) and negatively (right) charged particles measured in central Ar+Sc collisions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$

The $tof -\text {d} E/\text {d} x$ identification method proceeds by fitting the two-dimensional distribution of particles in the $\text {d}E/\text {d}x$ –$m^2$ plane. Fits were performed in the momentum range from 2 to 10 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ and transverse momentum range 0–1.5 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Particles with total momentum less than 2 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ are identified based on $m^2$ measurement alone, as different species of particles are separated enough. Contamination of electrons to pions identified in such a way is removed with a dedicated $\text {d}E/\text {d}x$ cut (see Fig. 10 and Ref. [29]). For positively charged particles the fit function included contributions of p, $K^+$, $\pi ^+$, and $e^+$, and for negatively charged particles the corresponding anti-particles were considered. The deuterons are not accounted for in the fits, as they are removed with a cut on measured $m^2$. The fit function for a given particle type was assumed to be the product of a Gaussian function in $\text {d}E/\text {d}x$ and a sum of two Gaussian functions in $m^2$ (in order to properly describe the tails of the $m^2$ distributions). In order to simplify the notation in the fit formulae, the peak positions of the $\text {d}E/\text {d}x$ and $m^2$ Gaussians for particle type j are denoted as $x_{j}$ and $y_{j}$, respectively. The fitted function reads:

$$\begin{aligned} \rho (x,y)&= \sum _{j = p,\pi ,K,e}\rho _j(x, y)= \sum _{j}\ \frac{N_j}{2\pi \ \sigma _{x}}\exp \left[ {-\frac{(x-x_j)^2}{2 \sigma ^2 _{x}}} \right] \nonumber \\&\quad \times \Bigg ( \frac{f}{\sigma _{y_1}}\exp \left[ {\frac{(y-y_j )^2}{2 \sigma ^2 _{y_1}}} \right] + \frac{(1-f)}{\sigma _{y_2}} \exp \left[ {\frac{(y-y_j)^2}{2 \sigma ^2 _{y_2}}} \right] \Bigg ) , \end{aligned}$$

(4)

where $N_{j}$ and f are amplitude parameters, $x_j$, $\sigma _{x}$ are means and width of the $\text {d}E/\text {d}x$ Gaussians and $y_j$, $\sigma _{y1}$, $\sigma _{y2}$ are means and widths of the $m^2$ Gaussians, respectively. The total number of parameters in Eq. 4 is 16. Imposing the constraint of normalization to the total number of tracks N in the kinematic bin

$$\begin{aligned} N=\sum _{i} N_i, \end{aligned}$$

(5)

the number of parameters is reduced to 15. Two additional assumptions were adopted:

• (i) the fitted amplitudes were required to be greater than or equal to 0,

• (ii) $\sigma _{y1} < \sigma _{y2}$ and $f > 0.7$, the ”core” distribution dominates the $m^2$ fit.

An example of the $tof -\text {d} E/\text {d} x$ fit obtained in a single phase-space bin for positively charged particles in central Ar+Sc collisions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ is shown in Fig. 11.

Fig. 11

Example of the $tof -\text {d} E/\text {d} x$ fit (Eq. 4) obtained in a single bin (2 < $p$ < 3 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ and 0.5 < $p_{T}$ < 0.6 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$) for positively charged particles in central Ar+Sc collisions at 30$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$

Fig. 12

Distribution of probabilities of a track being a pion, kaon, proton for positively (solid lines) and negatively (dashed lines) charged particles from $\text {d}E/\text {d}x$ measurements in central Ar+Sc collisions at 19A (top) and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (bottom)

The $tof -\text {d} E/\text {d} x$ method allows fitting the kaon yield close to mid-rapidity. This is not possible using the $\text {d}E/\text {d}x$ method alone. Moreover, the kinematic domain in which pion and proton yields can be fitted is enlarged by the $tof -\text {d} E/\text {d} x$ analysis. The results from both methods partly overlap at the highest beam momenta. In these regions, the results from both PID methods were combined using standard formulae [33].

3.3.3 Probability method

The 1D ($\text {d}E/\text {d}x$ ) and 2D ($tof -\text {d} E/\text {d} x$) models fitted to experimental distributions provide information on the contribution of individual particle species to total measured yields in bins of $p$ and $p_{T}$. In order to unfold these contributions in the $\text {d}E/\text {d}x$ method, for each particle trajectory with measured charge q, $p$, $p_{T}$ and $\text {d}E/\text {d}x$ a probability $P_i$ of being a given species can be calculated as:

$$\begin{aligned} P_{i}(p,p_{T},\text {d}E/\text {d}x) = \frac{\rho _{i}^{p,p_{T}}(\text {d}E/\text {d}x)}{\sum \nolimits _{i=\pi , K, p, e, d}^{} \rho _{i}^{p,p_{T}}(\text {d}E/\text {d}x)}, \end{aligned}$$

(6)

where $\rho _{i}^{p,p_{T}}$ is the probability density according to the model with parameters fitted in a given ($p$, $p_{T}$) bin calculated for $\text {d}E/\text {d}x$ of the particle.

Similarly, in the $tof -\text {d} E/\text {d} x$ method (see Eq. 4) for $p$ >2 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ the particle type probability is given by

$$\begin{aligned}{} & {} P_{i}(p,p_{T},\text {d}E/\text {d}x,m^2) \nonumber \\{} & {} \qquad = \frac{\rho _{i}^{p,p_{T}}(\text {d}E/\text {d}x, m^2)}{\sum \nolimits _{i=\pi , K, p, e}^{} \rho _{i}^{p,p_{T}}(\text {d}E/\text {d}x, m^2)}, \end{aligned}$$

(7)

In the case of low-momentum particles ($p$ <2 $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$), the assigned probability is either 0 or 1 based on the measured $m^2$. For illustration, particle type probability distributions for positively and negatively charged particles produced in central Ar+Sc collisions at 19A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ are presented in Fig. 12 for the $\text {d}E/\text {d}x$ fits and in Fig. 13 for the $tof -\text {d} E/\text {d} x$ fits. In the case of perfect particle type identification, the probability distributions in Figs. 12 and 13 will show entries at 0 and 1 only. In the case of incomplete particle identification (overlap** $\text {d}E/\text {d}x$ or $tof -\text {d} E/\text {d} x$ distributions) values between these extremes will also be populated.

Fig. 13

Distribution of probabilities of a track being a pion, kaon, proton for positively (solid lines) and negatively (dashed lines) charged particles from $tof -\text {d} E/\text {d} x$ measurements in central Ar+Sc collisions at 19A (top) and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ (bottom)

The probability method allows transforming the fit results performed in ($p$, $p_{T}$) bins to results in ($y$, $p_{T}$) bins. Hence, for the probability method the mean number of identified particles in a given kinematical bin (e.g. ($y$, $p_{T}$)) is given by [34]:

$$\begin{aligned} n[i]^{\text {raw}}_{\text {d}E/\text {d}x}(y,p_{T})=\frac{1}{N_{\text {ev}}}\sum _{j=1}^{N_{\text {trk}}}P_{i}(p,p_{T},\text {d}E/\text {d}x), \end{aligned}$$

(8)

for the $\text {d}E/\text {d}x$ identification method and:

$$\begin{aligned} n[i]^{\text {raw}}_{\text {d}E/\text {d}x,m^2}(y,p_{T})=\frac{1}{N_{\text {ev}}}\sum _{j=1}^{N_{\text {trk}}}P_{i}(p,p_{T},\text {d}E/\text {d}x,m^2), \end{aligned}$$

(9)

for the $tof -\text {d} E/\text {d} x$ procedure, where $P_i$ is the probability of particle type i given by Eqs. 6 and 7, j the summation index running over all entries $N_{\text {trk}}$ in the bin, $N_{\text {ev}}$ is the number of selected events. In the case of the $\text {d}E/\text {d}x$ analysis, the probabilities $P_i$ were linearly interpolated in the $(p,p_{T})$ plane in order to minimize bin-edge effects.

3.4 Corrections and uncertainties

In order to estimate the true number of each type of identified particle produced in Ar+Sc interactions, a set of corrections was applied to the extracted raw results. These were obtained from a simulation of the NA61/SHINE detector followed by event reconstruction using the standard reconstruction chain. Only inelastic Ar+Sc collisions were simulated in the target material. The Epos1.99 model [20] was selected to generate primary inelastic interactions as it best describes the NA61/SHINE measurements [12]. A Geant3-based program chain was used to track particles through the spectrometer, generate decays and secondary interactions, and simulate the detector response (for details see Ref. [12]). Simulated events were then processed using the standard NA61/SHINE reconstruction chain. The reconstructed tracks were matched to the simulated particles based on the cluster positions of the reconstructed simulated tracks. The selection of central events was based on the number of forward spectators. Corrections depend on the particle identification technique (i. e. $\text {d}E/\text {d}x$ or $tof -\text {d} E/\text {d} x$). Hadrons that were not produced in the primary interaction can amount to a significant fraction of the selected tracks, thus a special effort was undertaken to evaluate and subtract this contribution. The correction factors were calculated in the same bins of $y$ and $p_{T}$ as the particle spectra. The magnitude of correction factors reflects the effects of detector acceptance, track selection criteria, and reconstruction efficiency. The generated Epos events are referred to as “MCgen” and the label “MCsel” is given to the events with simulated detector response and reconstructed using standard NA61/SHINE chain with event and track selection criteria matching the ones used in the analysis of the experimental data.

3.4.1 Corrections of the spectra

The total correction for biasing effects listed in Sect. 3 items (b)–(g) (influence of item (a) on the final result was found to be negligible) was calculated in the following way:

$$\begin{aligned} n[i]^{\text {corrected}} = \left( n[i]^{\text {raw data}}-n[i]^{\text {MCsel decay}}\right) \cdot \frac{n[i]^{\text {MCgen}}}{n[i]^{\text {MCsel primary}}}, \end{aligned}$$

(10)

where, n[i] stands for the per-event yield in the bin i of the $y-p_{T}$ histogram of a given particle type, specifically:

$$\begin{aligned} \begin{array}{ll} n[i]^{\text {corrected}} &{} \hbox {stands for the corrected yield},\\ n[i]^{\text {raw data}} &{} \hbox {stands for the raw data yield},\\ n[i]^{\text {MCsel decay}} &{} \hbox {is the contribution of feed-down from }\\ &{}\hbox {weak decays in MCsel},\\ n[i]^{\text {MCsel primary}} &{} \hbox {is the contribution of primary particles }\\ &{} \hbox {in MCsel},\\ n[i]^{\text {MCgen}} &{} \hbox {is a pure MC simulated yield}.\\ \end{array} \end{aligned}$$

The correction of spectra due to contamination by weak decays $(n[i]^{\text {MCsel decay}})$ is weakly correlated with the primary hadron yields, therefore this contribution is accounted for in an additive way (later referred to as $c_{\text {add}}$). The combined geometrical and efficiency correction is applied as the quotient in the second term of the Eq. 10 of the numbers of reconstructed primary tracks and all simulated tracks in a given momentum space bin (later referred to as $c_{\text {mult}}$).

The corrections for the spectra obtained with the $tof -\text {d} E/\text {d} x$ PID method account additionally for the ToF tile efficiency $\epsilon _{\text {pixel}}(p,p_{T})$. It was calculated from the measured data as the probability of observing a valid reconstructed ToF hit if there exists an extrapolated TPC track that intersects with a given ToF tile. The ToF hit was considered valid if the signal satisfied the quality criteria given in Ref. [28].

The ToF pixel efficiency factor $\epsilon _{\text {pixel}}(p,p_{T})$ was used in the MC simulation by weighting each reconstructed MC track passing all event and track selection cuts by the efficiency factor of the corresponding ToF tile. Then, the number of selected MC tracks $n[i]^{\text {MCsel primary}}$ originating from primary particles becomes the sum of weights of those tiles which contribute to bin i:

$$\begin{aligned} n[i]^{\text {MCsel primary}} = \sum _{j=1}^{N_{\text {trk}}} \epsilon _{\text {pixel}}^{j}(p,p_{T}). \end{aligned}$$

(11)

$n[i]^{\text {MCsel decay}}$ is defined in the same manner for particles originating from weak decays. Only hits in working tiles, with efficiency higher than 50%, were taken into account in the identification and correction procedures.

The uncertainty of the multiplicative part of the correction was calculated assuming that the “MCsel primary” sample is a subset of the “MCgen” sample and thus has a binomial distribution. The uncertainty of the $c_\text {mult}$ ratio is thus given by:

$$\begin{aligned} \sigma _{\text {stat}}\left( c_{\text {mult}}\right)= & {} \frac{n[i]^{\text {MCgen}}}{n[i]^{\text {MCsel primary}}}\nonumber \\{} & {} \times \sqrt{\frac{N[i]^{\text {MCgen}}-N[i]^{\text {MCsel primary}}}{N[i]^{\text {MCgen}} \cdot N[i]^{\text {MCsel primary}}}}, \end{aligned}$$

(12)

where N[i] is the number of tracks in bin i (not normalized with the number of events, unlike n[i]). Absolute values of correction factors in phase space bins weakly depend on the original shapes of y–$p_{T}$ distributions provided by the model, due to the fact that the core part of the correction is calculated as a ratio (Eq. 10) in small y–$p_{T}$ bins.

The statistical uncertainty of the additive weak-decay feed-down correction ($c_{\text {add}}$) is added to the statistical uncertainty as a quadratic average.

Fig. 14

Contribution per event of reconstructed secondary tracks in the fiducial volume of the detector originating from weak decays and erroneously identified as products of primary interaction (based on Epos at 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$)

Fig. 15

An example of the relative contribution of weak decay products in spectra of hadrons identified with $\text {d}E/\text {d}x$ method at 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ plotted in $y-p_{T}$ plane. The corrections are based on the Epos model with data-derived tuning (see text for details). The feed-down for $\pi ^-$ and $\pi ^+$ is typically on the level of 1%-3%, reaching up to 10% in the case of low-$p_T$ $\pi ^-$. Secondary kaons can only originate from sparsely produced $\Omega ^+$ and $\Omega ^-$decays, thus the correction is well below 1%. Spectra of protons and anti-protons are heavily influenced by the decays of $\Lambda$ and $\Sigma ^+$ ($\bar{\Lambda }$ and $\bar{\Sigma ^+}$ for anti-protons) – in these cases, feed-down correction can reach over 50% at low $p_T$. Similar numbers hold true for all collision energies

3.4.2 Tuning of feed-down in MC corrections

Another source of contamination of experimental results are the secondary particles originating from weak decays, that are reconstructed as primary ones. Figure 14 shows the contribution of decay products originating from different decay parents and Fig. 15 shows the relative cumulative contribution of weak-decay feed-down to the measured particle spectra. While the yield of weak-decay products is negligibly small in the case of kaons ($K^+$ and $K^-$) it is significant in the case of pions ($\pi ^+$ and $\pi ^-$) and (anti-)protons which are contaminated by the decay products from $K_S^0$ and (anti-)hyperons. The Epos model used in the MC simulation does not reproduce the properties of strangeness enhancement in nucleus-nucleus collisions, thus the yields of strange mesons and strange baryons are typically underestimated. A procedure for tuning the contribution of weak decays was developed to improve the precision of the calculated corrections. It is based on data-derived quantities: mean multiplicities of particles estimated in measured data are compared with the ones extracted from MC simulation. Thus, using the preliminary results on charged kaon multiplicities from this analysis it is possible to construct an auto-tuning factor for yields of $K^0_S$:

$$\begin{aligned} K^0_{S~\text {(EPOS tuned)}} = K^0_{S~\text {(EPOS)}} \times \frac{\langle K^+ \rangle _{data}+\langle K^- \rangle _{data}}{\langle K^+ \rangle _{\text {(EPOS)}}+\langle K^- \rangle _{\text {(EPOS)}}}. \end{aligned}$$

(13)

In the absence of measurements of strange (anti-)baryons in Ar+Sc collisions, the best effort was made to estimate their yields using existing data. At the SPS collision energies mean multiplicities of $\Lambda$ baryons are well approximated by the following relation:

$$\begin{aligned} \langle \Lambda \rangle = \alpha \cdot ( \langle K^+ \rangle - \langle K^- \rangle ), \end{aligned}$$

(14)

where $\alpha$ is usually close to unity. A relevant parametrization of $\alpha$ was extracted from NA49’s Pb+Pb data [35] and used to get an approximate estimate of $\Lambda$ yield in Ar+Sc at each collision energy. Scaling of the yields of $\Lambda$ and $\Sigma ^\pm$ are calculated as:

$$\begin{aligned} \Lambda _\text {(EPOS~tuned)} = \Lambda _\text {(EPOS)} \times \frac{\alpha \cdot ( \langle K^+ \rangle - \langle K^- \rangle )}{\langle \Lambda \rangle _\text {(EPOS)}}. \end{aligned}$$

(15)

Obtained tuning factors are presented in Table 3, showing also the uncertainties assigned to these corrections. Furthermore, the yields of other strange and multi-strange baryons were tuned with the same factors as $\Lambda$ and $\Sigma ^{\pm }$. The imperfect description of rapidity and transverse momentum dependence in the Epos model is not accounted for in the presented calculation, hence the large values of assigned uncertainties. In Figs. 17 and 18 the total uncertainty introduced by the contribution of secondary particles is denoted with grey lines.

Table 3 Multiplicative factors used for tuning the feed-down contribution in the Epos MC simulation of central Ar+Sc collisions

Fig. 16

Two-dimensional distributions ($y$ vs. $p_{T}$) of double differential yields (Eq. 16) of $\pi ^{-}$, $\pi ^{+}$, $K^{-}$, $K^{+}$, p and $\bar{p}$ produced in the 10% most central Ar+Sc interactions at 13A, 19A, 30A, 40A, 75A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$

3.5 Corrected spectra

The final spectra of different types of hadrons produced in Ar+Sc collisions are defined as:

$$\begin{aligned} \frac{\text {d}^{2}n}{\text {d}y~\text {d}p_T} = \frac{1}{\Delta y \cdot \Delta p_{T}} n[i]^{\text {corrected}}, \end{aligned}$$

(16)

where $\Delta y$ and $\Delta p_{T}$ are the bin sizes and $n[i]^{\text {corrected}}$ represents the mean multiplicity of given particle type in the i-th bin in $y$ and $p_{T}$ obtained with either $\text {d}E/\text {d}x$ or $tof -\text {d} E/\text {d} x$ identification method, as introduced in Eq. 10.

The resulting two-dimensional distributions $\frac{\text {d}^{2}n}{\text {d}y~\text {d}p_T}$ of $\pi ^{-}, \pi ^{+}, K^{-}, K^{+}, p$ and $\bar{p}$ produced in the 10% most central Ar+Sc collisions at different SPS energies are presented in Fig. 16.

3.5.1 Statistical uncertainties

Statistical uncertainties of multiplicities calculated in the $tof -\text {d} E/\text {d} x$ method were derived under the assumption of Poissonian statistics in a single bin and no correlation between the bins. The resulting uncertainty in ($p$,$p_{T}$) bin is given by:

$$\begin{aligned} \sigma ^{2}_{stat}(n[i]^{\text {raw}}) = \frac{1}{N_{\text {ev}}} \sum _{j=1}^{N_{\text {trk}}} {P_{i}^j}(p,p_{T},\text {d}E/\text {d}x,m^2)^2. \end{aligned}$$

(17)

An alternative method, bootstrap**, was used to calculate statistical uncertainties in the case of the $\text {d}E/\text {d}x$ identification technique. One hundred bootstrap samples were generated through random sampling with replacement, performed on the level of events. Each bootstrap sample is injected into the procedure of particle identification and calculation of y-$p_{T}$ spectra. The errors are then estimated as standard deviations of yields at all bootstrap samples. It was verified that the number of bootstrap samples was large enough and that the distribution of yields resembles the normal distribution, allowing to assign the standard deviation as the statistical error. It was found that bootstrap** and weighted variances (Eq. 17) methods yield similar values of uncertainty.

The contribution to statistical uncertainties from the MC correction factors (discussed in detail in Sect. 3.4) is propagated into final uncertainties using the standard procedure.

Fig. 17

Systematic uncertainty relative to the measured yield of double-differential distributions obtained with $\text {d}E/\text {d}x$ PID method, integrated in $p_{T}$, shown for each identified species in dependence on rapidity y at $p_{\text {beam}}=150$ $A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Different contributions to the total uncertainty are plotted, along with statistical error (shaded area). Large fluctuations seen at low-rapidity uncertainties of pion spectra are due to narrow acceptance in $p_{T}$

Fig. 18

Systematic uncertainty relative to the measured yield of double-differential distributions in y and $p_{T}$ for charged kaons, obtained with tof- $\text {d}E/\text {d}x$ PID method, shown in dependence on transverse momentum $p_{T}$ at mid-rapidity for central Ar+Sc collisions at $p_{\text {beam}}=13A$, 40A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. Different contributions to the total uncertainty are plotted, along with statistical error (shaded area)

3.5.2 Systematic uncertainties

The following sources of systematic uncertainties were considered in this study:

• (I) Particle identification methods utilized in this analysis provide measurements of particle yields through the fits of multi-parameter models. In order to increase stability, some of the parameters need to be fixed. Moreover, it may happen that the fitted variable reaches the imposed limit. Such cases may lead to biases in the estimation of particle yields and therefore were carefully studied.

  • (a) $\text {d}E/\text {d}x$ method

    In the dE/dx method the fits of peak positions of kaons and protons were found to have the largest influence on particle yields and their ratios, while also having a relatively high variance, in particular in sparsely populated bins. The strategy used in this study (described in Sec. 3.3.1) involved fixing these parameters at pre-fitted values and assuming their independence of transverse momentum. The differences between prefits and results of bin-by-bin fits were studied and the spread within a single momentum bin was found at approx. 0.2%. Therefore in order to determine a potential bias introduced by fixing relative peak positions, they are varied by ±0.1%. Contribution to the biases from other fit parameters was found negligible.

  • (b) tof- $\text {d}E/\text {d}x$

    Systematic uncertainties were estimated by shifting the mean ($x_j$ and $y_j$) of the two-dimensional Gaussians (Eq. 4) fitted to the $m^2$-$\text {d}E/\text {d}x$ distributions by $\pm 1\%$, which corresponds to typical uncertainty of the fitted parameters. Additional systematic uncertainty arises for the $tof -\text {d} E/\text {d} x$ method from the quality requirements on the signals registered in the ToF pixels. In order to estimate this uncertainty the nominal signal selection thresholds were varied by $\pm 10\%$.

• (II) Event selection criteria based on any measurements downstream of the target may also introduce bias in the results. Uncertainties due to this were estimated through an independent variation of criteria listed below:

  • (a) Removal of events with off-time particles – the time window in which no off-time beam particle is allowed was varied by ±2 $\mu$s with respect to the default value of 4 $\upmu$s.

  • (b) Fitted main vertex position – the range of allowed main vertex z-coordinate was varied by ±5 cm at both ends.

• (III) Track selection: The contribution to systematic uncertainty from track selection criteria was estimated by varying the following parameters:

  • (a) The required minimum of the total number of clusters was varied by $+$5 and −5 points.

  • (b) Similarly, the minimum number of clusters in VTPCs was varied by ±5 points. Note that both of the cuts on the number of points affect the acceptance of the $\text {d}E/\text {d}x$ PID method as well, which was also taken into account.

  • (c) The influence of the selection of azimuthal angle was investigated in the case of $\text {d}E/\text {d}x$ -only PID by comparing the results obtained for $|\phi |<30^\circ$ (default value), $|\phi |<20^\circ$ and $|\phi |<40^\circ$.

• (IV) Feed-down correction: Uncertainties of weak decays feed-down correction were accounted for as described in Sect. 3.4.2.

The maximum difference of the particle yields in each bin of y and $p_{T}$ obtained under each varied criterium was assigned as the partial contribution to the systematic uncertainty. The final systematic uncertainty was taken as the square root of squares of its partial contributions. The relative contribution of each of the listed sources to the systematic uncertainties of the final spectra of identified particles is shown in Figs. 17 ($\text {d}E/\text {d}x$ method) and 18 (tof- $\text {d}E/\text {d}x$ method). The total uncertainty is typically 3–10% for charged pions, charged kaons, and protons, while it exceeds 10% in the case of anti-protons. The relative total uncertainties tend to increase at lower collision energies.

4 Results

Figure 16 displays two dimensional distributions $\text {d}^{2}n / (\text {d}y\,\text {d}p_\text {T})$ of $\pi ^{-}$, $\pi ^{+}$, $K^{-}$, $K^{+}$, p and $\bar{p}$ produced in 10% most central Ar+Sc collisions at beam momenta of 13A, 19A, 30A, 40A, 75A and 150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The spectra obtained using $\text {d}E/\text {d}x$ and $tof -\text {d} E/\text {d} x$ PID methods were combined to ensure a maximal momentum space coverage. Bins were removed from the final spectrum in the case of insufficient bin entries for the identification methods used in the analysis or if the yield uncertainty, either statistical or systematic, exceeded 80%. The gaps in the acceptance grow with decreasing collision energies, however, reliable measurement of key properties of charged hadron production is still possible even at the lowest beam momentum. In y–$p_\text {T}$ bins where both $tof -\text {d} E/\text {d} x$ and $\text {d}E/\text {d}x$ measurements exist, a weighted average is calculated using standard formulae [33]. For rapidity bins where there is an overlap, a comparison of $tof -\text {d} E/\text {d} x$ and $\text {d}E/\text {d}x$ results is provided in Appendix A.

The transverse momentum spectra of identified hadrons are extrapolated to account for the missing acceptance. Extrapolation of $p_\text {T}$ spectra allows for an accurate calculation of rapidity distribution, which in turn is also extrapolated into regions of missing measurements to calculate mean multiplicities. Only the experimental results up to $p_\text {T}<$1.5 GeV/c are considered since the contribution of misidentified particles becomes large at higher values of $p_{T}$, which results in a higher systematic uncertainty. The contribution of the extrapolation towards high $p_\text {T}$ (>1.5 GeV/c) is typically of the order of 1%. At small $p_\text {T}$ ($\approx ~0-0.1~\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$) the extrapolation or interpolation (when a gap between $tof -\text {d} E/\text {d} x$ and $\text {d}E/\text {d}x$ data exists) ranges from 0 to 40%. The extrapolation methods and their applicability differ for each of the studied particle species and thus are described separately in Sects. 4.1, 4.2 and 4.3. A complete set of plots depicting transverse momentum spectra of all particles in rapidity slices, together with fitted functions is available in Appendix A.

Fig. 19

Example fits to charged pion, $\pi ^+$ (left) and $\pi ^-$ (right), transverse momentum spectra ($p_{\text {beam}}=30$ $A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ and 75$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ at $0.6<y<0.8$, top and bottom panels respectively). Exponential fits (Eq. 18) are performed in two regions separately: $p_T \in$ [0.0, 0.6] and $p_T \in$ [0.6, 1.5] $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$. The fitted functions are used to extrapolate the yields beyond $p_T=1.5$ $\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ and interpolate the yields in case a gap in acceptance appears due to different coverage of PID methods. A full set of transverse momentum spectra with corresponding fits is presented in Figs. 53 and 54 in Appendix A. In rare cases, the yield is extrapolated also in the region of $p_{T}$ $<0.1$ GeV/c. The vertical bars represent statistical uncertainties and the shaded bands stand for the systematic uncertainties

Subsequently, Sect. 5 reviews presented measurements in terms of collision energy and system size dependence, including also a comparison with relevant models. Presented results are then discussed in the context of the onset of deconfinement and an emerging phenomenon of the onset of QGP fireball.

4.1 Charged pions

4.1.1 Transverse momentum spectra

The measured double differential charged pion spectra in rapidity and transverse momentum at 13A–150$A\,\text{ Ge }\hspace{-1.00006pt}\text{ V }\!/\!c$ beam momenta are presented in Fig. 16.

In order to account for the regions outside $\text {d}E/\text {d}x$ and $tof -\text {d} E/\text {d} x$ PID acceptance, the $p_T$ distributions in each bin of rapidity were fitted independently in two separate $p_T$ intervals: [0.0,0.6] and [0.6,1.5] GeV/c. Such a procedure was employed due to the influence of radial flow and a large contribution from resonance decays, which is difficult to model reliably. Dividing the $p_T$ range into two intervals allows for an accurate interpolation as well as the extrapolation of the $p_{T}$ spectra. A fit to combined data points from both PID methods is performed using the following formula:

$$\begin{aligned} f(p_T) = \frac{A\cdot p_T}{T (m_\pi +T)} \exp \left( \frac{m_\pi -m_T}{T}\right) , \end{aligned}$$

(18)

where T is the inverse slope parameter, $m_\pi$ and $m_T$ denote pion’s rest and transverse masses respectively and A is a normalization factor, T and A are fit parameters. Additionally, it is required that the fitted function is continuous between the intervals. Example fit results are shown in Fig. 19. The inverse slope parameter T is decreasing from mid-rapidity towards higher values of rapidity.

Fig. 20

Rapidity spectra of charged pions ($\pi ^+$ and $\pi ^-$) measured with $\text {d}E/\text {d}x$ and tof-$\text {d}E/\text {d}x$ methods in 10% most central Ar+Sc collisions. The line is a sum of two Gaussians, equidistant from mid-rapidity with differing amplitudes (Eq. 19). The statistical uncertainties are shown with error bars (if not visible, they do not exceed the size of the markers) and the systematic uncertainties are shown as shaded bands

4.1.2 Rapidity spectra

The $\text {d}n/\text {d}y$ yields are obtained by integration of the $\text {d}^2n/\text {d}y\,\text {d}p_T$ spectra and the addition of the integral of the fitted functions in the regions of missing acceptance. An additional contribution to the systematic uncertainty of 25% of the extrapolated yield is added to account for a possible bias due to model selection. Figure 20 displays the resulting $\text {d}n/\text {d}y$ distributions for all collision energies. The one-dimensional rapidity spectra are fitted with double-Gaussians with means equidistant from mid-rapidity:

$$\begin{aligned} f(y)&= \frac{A_0 ~ A_\text {rel}}{\sigma _0\sqrt{2\pi }}~ \exp \left( -\frac{(y-y_0)^2}{2\sigma _0^2}\right) \nonumber \\&\quad + \frac{A_0}{\sigma _0\sqrt{2\pi }}~ \exp \left( -\frac{(y+y_0)^2}{2\sigma _0^2}\right) , \end{aligned}$$

(19)

where $A_0$ is the amplitude, $A_\text {rel}$ is a parameter reflecting the asymmetry between forward and backward rapidity hemispheres, $\sigma _0$ is the width of individual peaks and $y_0$ stands for the displacement of contributing distributions from mid-rapidity. Within such parametrization, RMS width of the obtained rapidity distributions, $y_\text {RMS}$, can be calculated as follows:

$$\begin{aligned} y_\text {RMS}=\sqrt{\sigma _0^2+\frac{4A_\text {rel}}{(A_\text {rel}+1)^2}\cdot y_0^2}, \end{aligned}$$

(20)

which in the case of symmetrical rapidity distribution ($A_\text {rel}=1$) reduces to $y_\text {RMS}=\sqrt{\sigma _0^2+ y_0^2}$. The measured data covers only the region of positive rapidity, thus the parameter $A_\text {rel}$ is not fitted, but instead taken from the published results of complementary analyses, with the $h^-$ method, described in detail in Ref. [71, 72]