Production of (anti-)$^3$He and (anti-)$^3$H in p-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV

The transverse momentum ($p_{\rm{T}}$) differential yields of (anti-)$^3$He and (anti-)$^3$H measured in p-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV with ALICE at the LHC are presented. The ratios of the $p_{\rm{T}}$-integrated yields of (anti-)$^3$He and (anti-)$^3$H to the proton yields are reported, as well as the $p_{\rm{T}}$ dependence of the coalescence parameters $B_3$ for (anti-)$^3$He and (anti-)$^3$H. For (anti-)$^3$He, the results obtained in four classes of the mean charged-particle multiplicity density are also discussed. These results are compared to predictions from a canonical statistical hadronization model and coalescence approaches. An upper limit on the total yield of $^4\bar{\mathrm{He}}$ is determined.


Introduction
In ultra-relativistic nuclear collisions, midrapidity production yields of ordinary hadrons, i.e. mesons and baryons, can be described within the Statistical Hadronization Model (SHM), for which the temperature and the baryo-chemical potential are the parameters regulating hadron production [1,2]. In this model, hadrons are produced from an expanding medium in local thermodynamic equilibrium. Their abundances are fixed when the rate of inelastic collisions becomes negligible. This chemical freeze-out is associated with a characteristic temperature which is found to be T chem ≈ 156 MeV in Pb-Pb collisions at the LHC [1]. The yields of hadrons in central Pb-Pb collisions are reproduced by this approach [2] within uncertainties. Elastic and quasi-elastic scattering might still occur among hadrons during the further evolution of the system. The transverse momentum distributions can be modified until also the elastic interactions cease at the kinetic freeze-out. At LHC energies, baryon number transport from the initial nuclei at beam rapidity to midrapidity is completely negligible. This implies that particles and their corresponding anti-particles are produced in approximately equal amounts which is accounted for by a vanishing baryo-chemical potential µ B .
Light (anti-)nuclei are composite objects of (anti-)baryons with radii that are substantially larger than those of ordinary hadrons and their sizes reach a significant fraction of the volume of the expanding medium. Their production yields can also be described within the SHM. This may be surprising as the separation energy of nucleons is much smaller than the system temperature, thus raising the question of how nuclei can survive during the hadronic phase. Alternative approaches were developed that are able to describe production yields of light nuclei via the coalescence of protons and neutrons which are close by in phase space at kinetic freeze-out [3,4]. In this simplified approach, the invariant yield of nuclei with mass number A, E A (d 3 N A /dp 3 A ), is related to that of nucleons via where E p (d 3 N p /dp 3 p ) is the invariant yield of protons, which is expected to be identical to that of neutrons at midrapidity and LHC energies [5]. Here, the coalescence probability is given by the parameter B A . Both the SHM and the coalescence approach result in similar predictions, as demonstrated for the production of deuterons [6][7][8]. A review can be found in [9].
However, recent studies [5] have shown a sizeable difference for the B A parameter as a function of the size of the particle emitting source between predictions by the SHM with kinetic freeze-out conditions from a simple hydrodynamical model and the coalescence model. Here, information from Hanbury Brown-Twiss (HBT) correlations is used to determine the source size. This effect is more pronounced for (hyper-)nuclei with larger radii. Thus, the ideal benchmark would be to study the production of hypertriton ( 3 Λ H) as a function of the mean charged-particle multiplicity density, which is not yet possible due to the size of the data sets available. Thus, the difference between the production of 3 He and 3 H was studied, which is expected to offer similar insight on the comparison of the SHM and the coalescence approaches, especially for smaller collision systems [5,10]. The production yields for (anti-) 3 He in pp and Pb-Pb [11,12] collisions measured by ALICE do not cover completely the evolution from small to large source sizes. To bridge this gap, measurements in p-Pb collisions are needed which cover the intermediate source sizes.
In a broader context, the measurement of the production of nuclei in pp and p-Pb collisions contributes significantly and decisively to indirect searches for segregated primordial antimatter and dark matter via satellite-borne instruments, such as AMS-02 [13]. These experiments search for an excess in the measured production of anti-nuclei above the background stemming from pp and p-A collisions in the interstellar medium. This background is predicted by calculations [14] that use measurements of the production of anti-nuclei in accelerator experiments as a key ingredient.
Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration T0 detector [21]. The latter consists of two arrays of Cherenkov counters (T0C and T0A) positioned around the beam pipe, on both sides of the nominal interaction point. A weighted average is performed when both detectors have measured the start time [22]. The total time resolution for the analysed data sample is ∼ 80 ps.
The last detector used for this analysis is the V0, which consists of two scintillator hodoscopes (V0C and V0A) [23], covering the pseudorapidity regions −3.7 < η lab < −1.7 and 2.8 < η lab < 5.1. It is used to define the minimum-bias trigger, which requires a coincident signal in V0A and V0C to reduce the contamination from single-diffractive and asymmetric electromagnetic interactions. In addition, the V0A signal is proportional to the mean charged-particle multiplicity density in the direction of the Pb beam. The minimum-bias data sample is divided into four multiplicity classes defined as percentiles of the V0A signal. These are summarized in Table 1, where the corresponding mean charged-particle multiplicity densities at midrapidity dN ch /dη lab |η lab |<0.5 are also listed. These values and their uncertainties are taken from [24].

Data analysis
In this section the analysis technique is described. In particular, the criteria used for the event and track selection, the signal extraction techniques used for 3 H and 3 He, the corrections based on Monte Carlo (MC) simulations and the evaluation of the systematic uncertainties are illustrated and discussed.
The reconstruction efficiencies of (anti-) 3 H and (anti-) 3 He, the estimate of the contribution of secondary nuclei produced by spallation in the detector material, and the subtraction of the feed-down from the weak decay of hypertriton are obtained using Monte Carlo simulations. Nuclei and anti-nuclei were generated with a flat distribution in transverse momentum and rapidity within 0 ≤ p T ≤ 8 GeV/c and −1 ≤ y cms ≤ 1. Ten deuterons, 3 H, 3 He, and 4 He as well as their anti-nuclei were injected into each p-Pb collision simulated with the EPOS-LHC event generator [25]. In addition, twenty hypertritons and antihypertritons were injected per event. For particle propagation and simulation of the detector response, GEANT 3 is used [26].

Event and track selection
In order to keep the conditions of the detectors as uniform as possible, to avoid edge effects, and reject residual background collisions, the coordinate of the primary vertex along the beam axis is required to be within ±10 cm from the nominal interaction point. The primary vertices are identified either using tracks reconstructed in the full central barrel or with the SPD. The contamination from pile-up events is reduced to a negligible level by rejecting events with multiple vertices. Pile-up vertices identified with the SPD are required to be reconstructed using a minimum number of contributors dependent on the total number of SPD track segments (tracklets) in the event and have to be compatible with the expected collision region. A tracklet is defined as a straight line connecting two SPD hits which points back to the primary vertex. For the vertices identified using tracks reconstructed in the full central barrel, a minimum number of contributing tracks and a maximum χ 2 per contributor for the vertex fit are required to reject Production of (anti-) 3 He and (anti-) 3  Due to the different magnetic rigidity and the 2-in-1 magnet design of the LHC, the momenta of the particle beams are different for asymmetric collision systems such as p-Pb. As a consequence, the centerof-mass system (CMS) is shifted in the laboratory frame by a rapidity offset ∆y = 0.465 in the direction of the proton beam. Primary track candidates with transverse momentum p T > 1.5 GeV/c, pseudorapidity |η lab | ≤ 0.9 and −1 ≤ y cms < 0 are selected from those reconstructed both in the ITS and TPC by applying quality criteria that were optimized to ensure a good track momentum and dE/dx resolution.
Tracks are required to have a minimum number of reconstructed space points in the TPC (N TPC cls ) of 70 for 3 He and 120 for 3 H out of a maximum of 159 clusters, respectively. For 3 H candidates, a stronger selection is used in order to reduce the contamination from other particle species. In addition, at least two hits in the ITS (N ITS cls ≥ 2), with at least one in the SPD, are requested. The latter requirement significantly suppresses the contribution of secondary tracks. During the data taking, the SDD was only read out for about half of the events recorded in order to maximize the data acquisition speed. To maximize the size of the data set and to unify the reconstruction of the events, the information from the SDD was not used for the current analyses, which reduces the maximum number of hits in the ITS to 4.
The quality of the track fit is quantified by the value of χ 2 /N TPC cls , which is required to be less than 4. In addition, the ratio of the number of reconstructed TPC clusters to the number of findable TPC clusters is required to be larger than 80%. The number of findable clusters is the maximum number of geometrically possible clusters which can be assigned to a track.
The contribution from secondary tracks that are produced, e.g. by spallation in the detector material, is further suppressed by restricting the DCA to the primary vertex. The absolute value of the DCA in the transverse plane (DCA xy ) and in the beam direction (DCA z ) are required to be smaller than 0.1 cm and 1 cm, respectively.

Particle identification
The identification of tracks as 3 He and 3 H is based on the specific energy loss dE/dx measured by the TPC. For 3 He, this provides excellent separation from other particle species due to the quadratic dependence of dE/dx on the particle charge. The only relevant contamination is caused by secondary 3 H due to the similar specific energy loss in the kinetic region of p T < 3 GeV/c. As shown in the left panel of Figure 1, the fraction of contamination is estimated from data by fitting the slope on the left side of the 3 He peak in the dE/dx distribution with a Gaussian function. This contamination is found to be below 0.5% for 3 He, while the signal extraction of 3 He is not affected. For 3 H, the PID signal in the TPC contains a large background from other, more abundant particle species because 3 H has only one elementary charge. This background is largely suppressed by applying a pre-selection based on the measured time-of-flight, which is required to be within 3σ TOF from the value expected for 3 H, where σ TOF is the resolution of the time-of-flight measurement. At p T > 2.0 GeV/c, the TOF pre-selection does not efficiently suppress the contamination by other particles, like electrons and pions, anymore which leads to an increasingly large contamination for higher p T . The contamination of the signal is estimated following the same approach as for the signal extraction of 3 He. For 3 H ( 3 H) in the transverse momentum regions p T = 2 − 2.5 GeV/c and p T = 2.5 − 3 GeV/c, the contamination is found to be ∼ 7(9)% and ∼ 34(21)%, respectively. The 3 He ( 3 H) candidates are selected using the difference between the measured dE/dx and the expected value for 3 He ( 3 H), in units of the energy loss resolution of the TPC, n TPC σ . The signal is extracted by subtracting the contamination and counting the number of candidates inside the interval [−3σ , 3σ ].

Secondary nuclei from material
Secondary nuclei are produced as spallation fragments in the interactions between primary particles and nuclei in the detector material or in the beam pipe. The contribution of secondary nuclei can be experimentally separated from that of primary nuclei using the DCA xy to the primary vertex. The DCA xy distribution of primary nuclei is peaked at zero, while the one of secondary nuclei is flat over most of the DCA xy range and has a small peak around DCA xy = 0 cm for low p T , as shown in Figure 2. This structure is artificially created by the tracking algorithm and is due to incorrect cluster association in the first ITS layer. Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration The DCA xy distribution of 3 He in data is obtained by applying stricter PID requirements compared to those described in Subsection 3.2 to ensure a pure 3 He sample. In particular, the difference between the measured dE/dx and the expected average for 3 He is required to be in the range [−2σ , 3σ ] for p T < 2 GeV/c and in the range [−2.5σ , 3σ ] for 2 < p T < 2.5 GeV/c. The remaining contamination is at maximum 0.1% for 3 He and 1.2% for 3 H for p T < 2.5 GeV/c.
The fraction of primary nuclei is obtained by a two-component fit to the measured DCA xy distribution, one for the signal and the other for the secondaries. The distribution of both components is obtained from Monte Carlo simulations. Due to the lack of secondary 3 He in the MC simulation, the distributions of secondary deuterons are used as a proxy. For a given p T , the template of deuterons at p T /2 is used to compensate for the charge difference. The different multiple scattering for deuterons and 3 He has a negligible impact on the DCA xy distribution. This is confirmed by comparing the DCA xy distributions of antideuteron and 3 He candidates in data for the same interval of transverse rigidity (p T /q). For p T > 2.5 GeV/c, the DCA xy distributions of 3 He and 3 H are well reproduced using only the template for primary nuclei, which implies that the fractions of secondary 3 He and 3 H are negligible or below the sensitivity of this measurement. The fraction of primary nuclei is calculated in the range |DCA xy | ≤ 0.1 cm. The resulting values are summarized in Table 2.
The fractions of primary nuclei calculated in different multiplicity intervals are consistent with those calculated for the minimum-bias data sample within uncertainties. Due to the limited number of 3 He candidates, the fit is highly unstable for the lowest multiplicity. Therefore, the primary fraction is calculated using the minimum-bias data sample and used to correct the spectra in all the multiplicity intervals.

Efficiency and acceptance
The product of the acceptance and the efficiency is calculated as the ratio between reconstructed and generated primary nuclei in the MC simulation within −1 ≤ y cms < 0 and 1 ≤ p T < 5 GeV/c or 1 ≤ p T < 3 GeV/c for (anti-) 3 He and (anti-) 3 H, respectively. The same track selection criteria that are used in data are applied to the reconstructed particles in the simulation. The acceptance × efficiency of (anti-) 3 H and (anti-) 3 He are shown in Figure 3 as a function of p T .
The efficiency for (anti-) 3 H is lower compared to that of (anti-) 3 He due to the larger number of TPC clusters required and the additional requirement of a hit in the TOF detector. The latter implies the crossing of the additional material between the TPC and the TOF detector. Nuclear absorption and multiple Coulomb scattering reduce the TPC-TOF matching efficiency, leading to a lower efficiency for 3 H. Furthermore, the efficiency and the acceptance of the TOF detector have to be taken into account. The efficiency for the anti-nuclei is reduced compared to the one for the nuclei due to annihilation processes with the beam pipe and the detector material.

Feed-down from hypertriton
The transverse momentum distribution of (anti-) 3 He and (anti-) 3 H contains a contribution from weak decays of (anti-)hypertriton, 3 Λ H → 3 He + π − and 3 Λ H → 3 H + π 0 and charge conjugates. The (anti-) hypertriton represents the only relevant source of feed-down at LHC energies. The goal of this study is the measurement of primary (anti-) 3 He and (anti-) 3 H produced in the collision. For this reason, the con-Production of (anti-) 3 He and (anti-) 3  tribution of secondary (anti-) 3 He and (anti-) 3 H produced in weak decays of (anti-)hypertriton, estimated using the simulations, is subtracted from the inclusive p T distribution.
The fraction of secondary (anti-) 3 He from (anti-)hypertriton decays is given by: where ε feed-down and ε3 He are the reconstruction efficiencies of secondary 3 He from (anti-)hypertriton decays and primary 3 He, respectively. The DCA selection introduced to suppress the secondaries from the interaction with material also reduces the reconstruction efficiency for feed-down 3 He by about 40% compared to the one for primary 3 He. BR denotes the branching ratio of the decay of 3 Λ H into 3 He which is about 25% [27]. The (anti-) 3 Λ H-to-(anti-) 3 He ratio is extrapolated to the analyzed multiplicity class from those measured as a function of dN/dη lab in Pb-Pb collisions at √ s NN = 2.76 TeV [28].
An upper limit for this contribution to 3 H is evaluated as half of the contribution for 3 He since the branching ratio of the two-body decay with neutral daughters is half the one with charged particles [27]. The measured p T spectra of (anti-) 3 He and (anti-) 3 H are corrected for the fraction of secondary (anti-) 3 He and (anti-) 3 H from (anti-)hypertriton decays, which is estimated to be about 3.7% and 1.9%, respectively.

Systematic uncertainties
The main sources of systematic uncertainties on the (anti-) 3 He and (anti-) 3 H yields are summarized in Table 3 and discussed in the following. The procedures used for the evaluation of the systematic uncertainties quantify effects due to residual discrepancies between the data and the MC used to evaluate the reconstruction efficiency. The total systematic uncertainties are calculated as the sum in quadrature of the individual contributions assuming that they are uncorrelated.
The systematic uncertainty related to track reconstruction contains contributions coming from the different matching efficiencies between ITS and TPC for 3 He and 3 H, and between TPC and TOF for 3 H in data and MC and a contribution due to the track selection criteria used in the analysis. The latter is estimated by varying the track selection criteria, both for data and in the MC for the efficiency calculation. For each Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration transverse momentum interval, the systematic uncertainty is given by the root mean square (RMS) of the spread of data points, each corresponding to a given track selection. The corresponding uncertainty was found to be 4 − 5%. The uncertainties due to the different ITS-TPC and TPC-TOF matching efficiencies are both about 1%. The total tracking systematic uncertainty is obtained as the sum in quadrature of each contribution and is found to be about 4 − 5% for both 3 He and 3 H, independent of p T .
Similarly, the uncertainty from the particle identification is estimated by varying the fitting ranges in the TPC and TOF for the signal extraction as well as for the evaluation of the contamination. The latter has only a minor effect on the uncertainty for 3 He due to its clear separation from other charged particles. In contrast, the effect of the contamination on 3 H is much larger because the separation from other charged particles, which are much more abundant, decreases with increasing p T . The systematic uncertainty on the PID and the contamination is found to be at maximum 3% for 3 He and 30% for 3 H.
The systematic uncertainty associated with the fraction of primary nuclei contains three sources: the uncertainty of the template fit, the stability against including more secondaries, and the possible bias of the templates used. For the latter contribution, a Gaussian function is used to describe the DCA xy distribution of secondary nuclei, while the distribution of anti-nuclei is used as a template for the primary nuclei. The parameters of the Gaussian function are obtained by fitting the DCA xy distribution excluding the region |DCA xy | ≤ 0.1 cm. The fraction of primary nuclei is calculated using two methods: in one case, the template for primary nuclei and the Gaussian background are used, in the other case only the Gaussian function is used. In addition, the primary fraction is calculated using MC templates from secondary 3 H scaled in the same way as the deuteron templates used as default. The maximum difference between the fraction of primary nuclei obtained from these methods is divided by √ 12. The stability of the primary fraction correction is tested by varying the DCA selection and, thus, varying the number of secondary nuclei taken into account. The primary fraction should adjust accordingly. This uncertainty is evaluated using an RMS approach. The total uncertainty linked to the primary fraction estimate is given by the sum in quadrature of the three components and is found to be at maximum 9% for 3 He and 6% for 3 H following a decreasing trend with p T .
The material budget of the detector is known with an uncertainty of 4.5% [16] which leads to an uncertainty on the reconstruction efficiency. The impact of this uncertainty on the results is studied by evaluating the relative uncertainty on the reconstruction efficiency using a dedicated MC production with 4.5% higher or lower material budget. The relative uncertainty σ material budget is calculated via Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration where ε max and ε min are the largest and the smallest efficiencies obtained in a given p T interval. ε default denotes the efficiency calculated with the default material budget. The effect is larger for 3 H than for 3 He because of the additional detector material which has to be taken into account when including the TOF detector in the analysis.
To evaluate the reconstruction efficiency GEANT3 was used to propagate the particles through the detectors. In the GEANT3 version used for this analysis, an empirical parametrization of the antideuteron absorption cross section, based on the measurements carried out at the U-70 Serpukhov accelerator [29,30], is used. Elastic scattering processes are not taken into account by this description. In GEANT4 [31], a Glauber model based on the well-measured total and elastic pp cross section is implemented [32]. Thus, the systematic effect due to the incomplete knowledge about the hadronic interaction cross section of nuclei is evaluated using half of the relative difference between the reconstruction efficiency evaluated with GEANT3 and GEANT4. This contribution is found to be smaller than 12% for (anti-) 3 He and (anti-) 3 H.
The last contribution to the systematic uncertainties is the feed-down from weak decays of hypertritons. In Subsection 3.5, this contribution is estimated using an extrapolation of the measured 3 Λ H-to-3 He ratio assuming a linear trend with the charged particle multiplicity. This extrapolation is repeated shifting the measured data points up and down by their uncertainties such that the resulting slope is maximal or minimal. The resulting maximal or minimal 3 Λ H-to-3 He ratios are used to calculate the relative uncertainty on the feed-down contribution given by the difference of the maximum (6.3%) and the minimum (1.1%) feed-down contribution divided by √ 12. The corresponding contribution to the total systematic uncertainty is found to be 1.5% for 3 He and 0.75% 3 H.

Transverse momentum spectra
The production yields of (anti-) 3 He and (anti-) 3 H as a function of p T are obtained by multiplying the observed number of candidate nuclei after the statistical subtraction of the contamination (N obs ) with the fraction of primary nuclei ( f prim ) and correcting for the reconstruction efficiency (ε) in each p T interval. Afterwards the feed-down nuclei from hypertriton decays are subtracted. The corrected number of observed nuclei is divided by the number of selected events (N events ), the width of the transverse momentum bins (∆p T ) and the rapidity interval (∆y). The resulting p T -differential yields of (anti-) 3 He and (anti-) 3 H correspond to the ones in INEL > 0 events because the signal and event loss due to the event selection and the trigger were found to match within less 1% and thus no corrections are applied.
The minimum-bias p T -differential yields of (anti-) 3 He and (anti-) 3 H measured in p-Pb collisions at √ s NN = 5.02 TeV and the corresponding antiparticle-to-particle ratios are shown in Figure 4. The antiparticle-to-particle ratio is consistent with unity within uncertainties. This indicates that matter and antimatter are produced in equal amounts in p-Pb collisions at √ s NN = 5.02 TeV. This is also observed for other light (anti-)nuclei in different collision systems and center-of-mass energies at the LHC [11,12]. For the calculation of the systematic uncertainty of the antiparticle-to-particle ratio, the systematic uncertainties on the spectra were propagated, taking into account that some of them are correlated between antiparticles and particles, i.e. the uncertainty linked to the tracking, the material budget, and the feeddown.
The p T spectra, which are the average of 3 He and 3 He, are summarized in Figure 5 for different multiplicity classes and INEL > 0 events. The p T spectra of 3 He and 3 He, as well as of 3

H and 3 H have
Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration  to be extrapolated to the unmeasured regions in order to obtain the integrated yield (dN/dy). For the extrapolation, the measured p T spectra are fitted with the following functional forms: p T -exponential, m T -exponential, Boltzmann, Bose-Einstein, and Fermi-Dirac function.
The extrapolated yield is calculated by integrating each of these functions outside the measured p T range and taking the average. The result is added to the integral of the measured spectrum to obtain the total p Tintegrated yield. For the calculation of the statistical uncertainty on the yield, the transverse momentum spectrum is modified by shifting the data points for different transverse momentum bins independently by random numbers with Gaussian distributions centered around the measured values with a width given by the statistical uncertainties. In addition, the extrapolated yields at p T below and above the measured range are varied following a Gaussian function centered at the default value with a width given by the uncertainty on the extrapolated yield. The standard deviation of the distribution of measured yields determines the statistical uncertainty for each functional form fitted.
For the systematic uncertainty of the total yield, for each of the functional forms the part correlated in p T , i.e. the material budget, the hadronic cross section, feed-down uncertainty, and the uncertainty linked to the estimation of the primary fraction, is treated separately from the remaining uncertainty. It is evaluated as the average difference between the default value and the yield obtained by shifting the measured points up or down by the correlated part of the systematic uncertainties. The remaining part of the total uncertainty, i.e. the uncertainty linked to the track selection, PID, and contamination, is partially uncorrelated between p T -bins. Therefore, the Gaussian sampling procedure is also used to evaluate the contributions of these sources to the systematic uncertainty of the p T -integrated yield. The contribution for each functional form is given by the sum in quadrature of the uncorrelated and the correlated uncertainty. To obtain the total systematic uncertainty on the integrated yield, the average of the contributions from the different functional forms is calculated and added in quadrature to the uncertainty given by the spread of the values obtained with the different functional forms. The latter is calculated as the difference of the maximum and the minimum yield divided by √ 12. The extrapolated Production of (anti-) 3 He and (anti-) 3  fraction of the integrated yield below and above the measured p T interval is summarized in Table 4.  [12] in the left panel of Figure 6. The p T measured in p-Pb collisions increases with the mean charged-particle multiplicity density, connecting the measured results in pp [11] and Pb-Pb collisions [12] in a smooth way. This indicates a hardening of the p T spectra with increasing mean charged-particle multiplicity density, which might be caused by production in jets [35] or by collective expansion effects [24]. The latter would also result in a shift of the maximum of the p T distribution, which cannot be observed in the present measurements due to the limited statistical precision.
If the system evolves following a hydrodynamic expansion, the mean transverse momentum of different particle species should follow a mass ordering, as a result of the radial flow. In the right panel of Figure 6, the p T as a function of the particle mass is shown for different mean charged-particle multiplicity densities. For similar dN ch /dη lab , a clear mass ordering is observed for the different particle species.
Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration He/ 3 Figure 6: Left: Mean transverse momentum of (anti-) 3 He as a function of the mean charged-particle multiplicity density in p-Pb collisions at √ s NN = 5.02 TeV. Statistical and systematic uncertainties are indicated by vertical bars and boxes, respectively. The published results from pp [11] and Pb-Pb [12] collisions are shown with diamonds and rectangles, respectively. Right: Mean transverse momentum measured in p-Pb collisions at √ s NN = 5.02 TeV as a function of the particle mass is shown for different mean charged-particle multiplicity densities. The linear scaling with the mass found for the results for π, K, p, Λ [24], Ξ, and Ω [33] is indicated by dashed lines. The deuteron p T is taken from [34].
The measurements for the nuclei prefer a scaling different than linear or with a different offset compared to the results for π, K, p, Λ [24], Ξ, and Ω [33].

Ratio to protons
The ratio of the integrated yields of (anti-) 3 He to those of (anti-)protons ( 3 He/p) is calculated for the four multiplicity classes used in this analysis, while the yield ratio of (anti-) 3

H to (anti-)protons ( 3 H/p) is calculated for INEL > 0 p-Pb collisions at
√ s NN = 5.02 TeV. The p T -integrated proton yields are taken from [24]. The 3 He/p and the 3 H/p ratios are shown as a function of the mean charged-particle multiplicity density in Figure 7, together with the ones from pp collisions at √ s = 7 TeV [11] and from Pb-Pb collisions at √ s NN = 2.76 TeV [12]. The measured ratio is larger in Pb-Pb collisions with respect to pp collisions. The value measured in central Pb-Pb collisions is consistent with the prediction of the grand canonical version of the SHM [1, 36]. The results obtained in p-Pb collisions show an increasing trend as a function of the mean charged-particle multiplicity density and indicate a smooth transition from pp to Pb-Pb collisions.
In Figure 7, the data are compared to the expectations from the Canonical Statistical hadronization Model (CSM) [8] and two coalescence approaches [37]. The trend observed in the data can be qualitatively reproduced over the full multiplicity range using the CSM approach, which is based on exact conservation of charges across the correlation volume V c [8]. The predictions were calculated using a temperature T = 155 MeV and a correlation volume extending across one unit (V c = dV /dy) and three units (V c = 3 dV /dy) of rapidity. The temperature value is constrained by the ratio measured in Pb-Pb collisions [12]. It is very close to the chemical freeze out temperature which results in the best description by the grand canonical SHM [1] of the ALICE measurements of the integrated yields of particles measured in most-central Pb-Pb collisions. For the mean charged-particle multiplicity density region covered by the results obtained in Pb-Pb collisions, the CSM has reached the grand canonical limit and, thus, matches the version of the SHM using the grand canonical ensemble.
Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration  [11,12] as a function of the mean charged-particle multiplicity density, together with the 3 H/p ratio. Statistical and systematic uncertainties are indicated by vertical bars and boxes, respectively. The expectations for the canonical statistical hadronization model (Thermal-FIST [8]) and two coalescence approaches [37] are shown. For the thermal model two different values of the correlation volume are displayed. The uncertainties of the coalescence calculations, which are due to the theoretical uncertainties on the emission source radius, are denoted as shaded bands. The 3 He/p and 3 H/p ratios measured in p-Pb collisions, which cover the gap in the multiplicity between the existing measurements in pp and Pb-Pb collisions, favour a small correlation volume V c = dV /dy, while the ratios of the deuteron to the proton yield measured in pp collisions are more compatible with a larger correlation volume [8]. The 3 He/p ratio as a function of the mean charged-particle multiplicity density has a similar trend as the d/p ratio. However, the increase between the pp and the Pb-Pb results is about a factor of 3-4 larger for 3 He/p than for d/p [34]. The simplified version of the CSM presented in this paper, which assumes a constant freeze-out temperature as a function of the system size, shows some tensions with data for the p/π and K/π ratios and fails to describe the measured φ /π ratio [38].
With increasing mean charged-particle multiplicity density, the number of protons and neutrons produced in the collision also increases. The more protons and neutrons are available, the more likely nucleons can be close enough in phase space to form a nucleus. Therefore, an increasing trend for the 3 He/p ratio as a function of the mean charged-particle multiplicity density is expected in the coalescence approach. The measured ratio is compared to coalescence predictions [37] which take the radii of the source and the emitted nucleus into account. The case of three-body coalescence, where the nuclei are directly produced from protons and neutrons, as well as the expectation for two-body coalescence, where an intermediate formation of a deuteron is needed, are shown. For both coalescence approaches, the theoretical uncertainties are given by the uncertainty on the emission source radius. Both calculations are in overall agreement with the data at low and intermediate multiplicities while they underestimate the experimental results for higher multiplicities. The measured 3 He/p ratio shows a slight preference for the two-body coalescence approach, even though this is not yet conclusive due to the uncertainties on both the data and the theoretical description.

Coalescence parameter (B 3 )
Following Equation 1, the coalescence parameter B 3 is obtained from the invariant yields of 3 He or 3 H and protons and is shown in Figure 8 as a function of the transverse momentum per nucleon.
Production of (anti-) 3 He and (anti-) 3  The ratio of the yields of (anti-) 3 H and (anti-) 3 He ( 3 H/ 3 He), which is shown in the bottom panel of Figure 8, is expected to be consistent with unity according to a naive coalescence approach. In more advanced coalescence calculations, that take into account the size of the emitting source and the nucleus, this ratio is expected to be above unity [5,37]. The difference in the coalescence expectations is mainly due to a different parametrization of the source radius as a function of the mean charged-particle multiplicity density. Another source of differences between the two coalescence approaches is the use of slightly different values for the radius of 3 H.
The ratio is found to be in slightly better agreement with the coalescence expectations than with unity. In the SHM approach the 3 H/ 3 He ratio is expected to be consistent with unity. Thus, this observable is potentially useful not only to discriminate between different implementations of the coalescence approach but also with respect to SHMs. The increase of the 3 H/ 3 He yield ratio with p T /A observed in data is not reflected in the theoretical predictions.
The coalescence parameter B 3 for 3 He calculated for the four multiplicity classes analyzed is shown in Figure 9 as a function of the transverse momentum per nucleon. A rising trend of B 3 with p T /A is observed in all multiplicity classes, contrary to the expectations of the naive coalescence approach, which predicts a constant B 3 . This behaviour can at least be partially understood by the effect coming from the hardening of the proton spectra with increasing mean charged-particle multiplicity density as explained in [39]. According to this, the coalescence parameter obtained in a wider charged-particle multiplicity interval develops an increasing trend with p T /A even though the coalescence parameter is flat in each small sub-interval. Due to the limited statistical precision, the multiplicity intervals used in this analysis are large and thus a residual effect from the change of the proton spectra inside the multiplicity intervals could cause the observed increase for the sub-intervals. Another explanation would be an increased coalescence probability in jets, which could cause a larger coalescence parameter at higher p T . A detailed study of this effect would require to reduce the size of the multiplicity intervals and, thus, a larger data set.
Production of (anti-) 3 He and (anti-) 3  The multiplicity dependence of B 3 is compared to theoretical model calculations for p T /A = 0.73 GeV/c and p T /A = 0.90 GeV/c in Figure 10. The B 3 values for the measurements in pp, p-Pb, and Pb-Pb [11,12] collisions are shown as a function of the mean charged-particle multiplicity density. In addition, the expected values for the coalescence approach taken from [5] are shown for two different parametrizations of the source radius as a function of the mean charged-particle multiplicity density. The two parametrizations can be understood as an indication of the validity band of the model description, which is expected to be more constrained with future measurements. The measurements are compared to the expected values for the grand canonical version of the SHM, the GSI-Heidelberg model [1,36], assuming that the transverse momentum shape is given by a Blast-Wave parametrization obtained by a simultaneous fit to the pion, kaon, and proton spectra measured in Pb-Pb collisions [40]. Since this model Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration uses a grand-canonical description, it is applicable only for high mean charged-particle multiplicity densities. If canonical suppression is taken into account, the expected B 3 deviates from the grand canonical value, as indicated in Figure 10 by exchanging the GSI-Heidelberg model with the CSM, Thermal-FIST [8]. The change to the canonical ensemble description extends the applicability of the model to intermediate mean charged-particle multiplicity densities. In the low mean charged-particle multiplicity density region, the assumption that the p T shape of the nuclei follows the Blast-Wave parametrization breaks down. This is reflected by the larger deviation of the CSM plus Blast-Wave curve from the measured result in pp collisions for p T /A = 0.73 GeV/c compared to p T /A = 0.90 GeV/c.
The best description of the coalescence parameter B 3 is given by the coalescence expectation for low mean charged-particle multiplicity densities and by the SHM for higher mean charged-particle multiplicity densities. The measurement of B 3 presented in this paper indicates a smooth transition between the regimes that are described by the two different approaches.

Upper limit on the 4 He production
An upper limit on the 4 He production in p-Pb collisions at √ s NN = 5.02 TeV is estimated. The limit is based on the non-observation of 4 He candidates using the same track selection criteria as for 3 He, except for the maximum distance-of-closest approach to the primary vertex. The DCA xy is required to be smaller than 2.4 cm, while the DCA z smaller than 3.2 cm.
The identification of 4 He is based on the time-of-flight, measured by the TOF detector, and the specific energy loss dE/dx in the TPC. These measurements are required to be within ±5σ TOF and ±3σ TPC from the expected values. The analysis is performed in the transverse momentum interval 2 ≤ p T < 10 GeV/c. The left panel of Figure 11 shows the distribution of the specific energy loss compared to the expected one for 4 He (n TPC σ ) after the pre-selection using the TOF. The distribution at n TPC σ < −3, corresponding to 3 He candidates, is fitted with a Gaussian function and extrapolated to the signal region, defined by the range [−3, 3]. The expected background in the signal region is 1 × 10 −5 . The expected background Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration and the non-observation of candidates in the signal region are used to calculate the upper limit at 90% confidence level using the Feldmann-Cousins approach [41]. The resulting number is corrected for the product of the acceptance and the reconstruction efficiency, the rapidity range, and the number of events selected. The product of the acceptance and the reconstruction efficiency was obtained as the average of the values found in smaller p T intervals weighted with the expected shape of the spectrum of 4 He. For the latter, a m T -exponential parametrization of the 4 He p T -spectrum, with parameters identical to those of 3 He except for the mass, which is set equal to the 4 He mass, was used. The obtained value for the upper limit is extrapolated to the full p T range using the m T -exponential parametrization of the 4 He p T -spectrum. A systematic uncertainty of 20%, similar to that of the measurement in Pb-Pb collisions at √ s NN = 2.76 TeV [42], is taken into account following the procedure described in [43,44].
The upper limit on the 4 He total yield (dN/dy) in p-Pb collisions at √ s NN = 5.02 TeV is found to be 2.3 × 10 −8 at 90% confidence level. The upper limit is shown in the right panel of Figure 11 together with the measured dN/dy corrected for the spin degeneracy factor, 2J + 1, of (anti-)protons [24], (anti-)deuterons [34], and (anti-) 3 He. The upper limit is compatible with a penalty factor, i.e. the suppression of the yield for each additional nucleon, of 668 ± 45 obtained by fitting the measurements of the proton, deuteron, and 3 He yields with an exponential function. The value of the penalty factor is consistent with the one obtained in previous measurements [34]. Taking into account this penalty factor, the expected yield of (anti-) 4 He is about 8 × 10 −10 .

Conclusions
The p T -differential yields of 3 H and 3 He nuclei and their anti-nuclei were measured in p-Pb collisions at √ s NN = 5.02 TeV. For (anti-) 3 He, the production was studied in different classes of mean charged-particle multiplicity density.
For the first time, a consistent comparison between experimental results and the canonical statistical hadronization model as well as the coalescence calculations was done for the same observable. The 3 He/p ratio measured in p-Pb collisions at √ s NN = 5.02 TeV bridges the gap between existing measurements in pp and Pb-Pb collisions and is overall in good agreement with the theoretical descriptions. Despite the agreement of the measurement and the CSM model, there is some tension due to the bad matching of the predicted and measured p/π and K/π ratios and the failure to describe the measured φ /π ratio. The coalescence approach has some difficulties to describe the measurements at high mean charged-particle multiplicity densities.
The coalescence parameter is measured as a function of the transverse momentum per nucleon. The result presented in this paper cannot give a clear answer to the question about the origin of the increasing trend with p T /A observed for the different multiplicity classes. The coalescence parameter B 3 is also measured as a function of the mean charged-particle multiplicity density and compared to expectations from the grand canonical and the canonical versions of the SHM. The use of the Blast-Wave parametrizations to define the p T shape breaks down for low multiplicities, which leads to larger discrepancies between the CSM and the measurements for the two p T /A intervals shown. In addition, the measurements are compared to the coalescence expectations for two different parametrizations of the source radius as a function of the mean charged-particle multiplicity density. The presented measurements are in agreement with the coalescence description as well as the SHM description within theoretical and experimental uncertainties. The data indicate a smooth transition between the regimes described best by the coalescence approach and the Statistical Hadronization Model.
These measurements provide the possibility to test the dependence of the production rate on the nuclear radius, for the first time by direct comparison of isospin partner nuclei in the same data set at the LHC. The 3 H/ 3 He ratio is sensitive to the production mechanism within the coalescence approach. The measurement presented in this paper deviates from unity and, therefore, slightly favours the coalescence Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration description including the dependence on the radii of the nucleus and the emitting source but it is not conclusive yet due to the large uncertainties.
An upper limit on the total production yield of 4 He in p-Pb collisions at √ s NN = 5.02 TeV was found which is about two orders of magnitude above the expected result obtained from the exponential fit of the proton, deuteron, and 3 He yields.
Production of (anti-) 3 He and (anti-) 3 H in p-Pb collisions at √ s NN = 5.02 TeV ALICE Collaboration