Production of deuterons, tritons, $^{3}$He nuclei and their anti-nuclei in pp collisions at $\mathbf{\sqrt{{\textit s}}}$ = 0.9, 2.76 and 7 TeV

Invariant differential yields of deuterons and anti-deuterons in pp collisions at $\sqrt{s}$ = 0.9, 2.76 and 7 TeV and the yields of tritons, $^{3}$He nuclei and their anti-nuclei at $\sqrt{s}$ = 7 TeV have been measured with the ALICE detector at the CERN Large Hadron Collider. The measurements cover a wide transverse momentum ($p_{\text{T}}$) range in the rapidity interval $|y|<0.5$, extending both the energy and the $p_{\text{T}}$ reach of previous measurements up to 3 GeV/$c$ for $A=2$ and 6 GeV/$c$ for $A=3$. The coalescence parameters of (anti-)deuterons and $^{3}\overline{\text{He}}$ nuclei exhibit an increasing trend with $p_{\text{T}}$ and are found to be compatible with measurements in pA collisions at low $p_{\text{T}}$ and lower energies. The integrated yields decrease by a factor of about 1000 for each increase of the mass number with one (anti-)nucleon. Furthermore, the deuteron-to-proton ratio is reported as a function of the average charged particle multiplicity at different center-of-mass energies.

The high luminosity provided by the LHC allows these measurements to be extended to higher energies and transverse momenta (p T ) than in previous experiments, and provides in addition the possibility to detect for the first time heavier anti-nuclei than anti-deuterons in pp collisions. Many of these measurements have been explained as the result of the coalescence of protons and neutrons that are nearby in space and have similar velocities [23,24], but this has not been experimentally tested in high p T regimes in small systems. On the other hand, statistical hadronization models [10,25] have been successful in describing particle yields over a wide range of energies in AA collisions, with the chemical freeze-out temperature and baryochemical potential being constrained by measurements of particle ratios. In this sense, the deuteron-to-proton ratio could serve as a test for possible thermal-statistical behavior in pp collisions at LHC energies.
On a broader level, this subject may also have an impact on cosmology. Big-Bang Nucleosynthesis is the dominant natural source of deuterons [26] and in the absence of baryogenesis, one could assume that the same holds for anti-deuterons. These anti-nuclei and even heavier anti-nuclei can also be produced in pp and pA collisions in interstellar space, representing a background source in the searches for segregated primordial anti-matter and dark matter [27][28][29]. As it turns out, the low momentum characteristic yield of anti-nuclei at central rapidities (compared to forward) lie in an energy region which is best suited for identification by most satellite-borne (low magnetic-field) instruments, such as AMS-02 [27].
While the differential yields of deuterons in pp collisions at √ s = 7 TeV have been reported in [10], this paper complements the previously published results by providing the corresponding measurements of anti-deuterons at the same collision energy. In addition, results for (anti-)deuterons at √ s = 0.9 TeV and 2.76 TeV as well as for (anti-)tritons and 3 He (anti-)nuclei at √ s = 7 TeV are given. The paper is organized as follows: Section 2 gives a description of the experimental apparatus. Section 3 describes the analysis procedure of the experimental data along with the estimation of the systematic uncertainties. In Section 4, the distributions of (anti-)deuterons, (anti-)tritons and 3 He (anti-)nuclei are presented. The integrated yields, the deuteron-to-proton ratios and the coalescence parameters, which relate the production of nuclei with those of the nucleons, are obtained in Section 5 and the summary and conclusions are presented in Section 6. from secondary nuclei via the determination of the distance of closest approach of the track to the primary vertex. The TPC [34], the main tracking component of ALICE, is a large drift detector with a low material budget to reduce multiple scattering and secondary particle production. In combination with the ITS, it is used to measure particle momenta. The TPC is also used to identify particles via their specific ionization energy loss with a resolution of 5% in pp collisions [35]. The TOF [36] detector is a large-area array of Multigap Resistive Plate Chambers covering the full azimuth 0 ≤ φ < 2π and |η| < 0.9, except the region 260 • < φ < 320 • and |η| < 0.12 to avoid covering the Photon Spectrometer with more material. In pp collisions, it measures the time of flight of particles with an overall resolution of about 120 ps, allowing the identification of light nuclei and anti-nuclei with transverse momenta above 3 GeV/c, depending on the available data. The start time for the time of flight is provided by the T0 detector, with a time resolution of ∼ 40 ps. The T0 consists of two arrays of Cherenkov counters, T0A and T0C, placed on opposite sides of the interaction point at z = 375.0 cm and z = −72.7 cm, respectively. If there is no T0 signal, the TOF detector is used to measure the start time when at least three particles reach the TOF [37].
Between the TPC and TOF there is a Transition Radiation Detector (TRD) [32] to discriminate between electrons and pions above 1 GeV/c. Only 7 modules out of 18 were installed for the pp run of 2010, leaving the major part of space between TPC and TOF free of additional material. The V0 detector [38], two hodoscopes of 32 scintillator cells each which cover the pseudo-rapidity ranges 2.8 < η < 5.1 and −3.7 < η < −1.7, provides in combination with the SPD the trigger for inelastic pp collisions.

Data analysis
The pp events used in this paper were collected by the ALICE Collaboration during 2010 and 2011. The recorded integrated luminosity for each analyzed sample is 0.124 nb −1 , 0.692 nb −1 and 4.20 nb −1 for the center-of-mass energies of √ s = 0.9, 2.76 and 7 TeV, respectively.

Event and track selection
The pp events were triggered by requiring a hit in both sides of the V0, i.e., two charged particles separated by approximately 4.5 units of pseudo-rapidity, which suppresses single diffractive events. The presence of passing bunches was detected by two beam-pickup counters. Contamination from beaminduced background was rejected offline using the timing information of the V0. Additionally, a cut on the correlation between the number of SPD clusters and the number of small track segments (tracklets) in the SPD detector was applied. Furthermore, in order to maintain a uniform acceptance and to reduce beam-induced noise, collision vertices were required to be within 10 cm of the center of the detector in the beam direction and within 1 cm in the transverse direction. Pile-up events were reduced by requiring that more than three tracklets or tracks contribute to the reconstructed vertex. In case of multiple vertices which are separated by more than 0.8 cm, the vertex reconstruction with the SPD allows these events to be tagged as pile-up and hence not considered in the analysis. The events analyzed here consist mostly of non-single diffractive events, which represent a fraction of the total inelastic cross-section equal to 0.763 +0.022 −0.008 , 0.760 +0.052 −0.028 and 0.742 +0.050 −0.020 for √ s = 0.9, 2.76 and 7 TeV [39], respectively. Those fractions were used to extrapolate the measurements to inelastic pp collisions assuming that the production of nuclei in single-diffractive events is not significant with respect to non-single diffractive events based on Monte-Carlo estimates (less than 3%).
For each track at least 2 track points were required in the ITS and 70 out of a maximum of 159 in the TPC. A pseudo-rapidity cut of |η| < 0.8 was also required to avoid edge effects. Tracks with kinks, typically originating from weak decays inside the TPC volume, were treated as two separate tracks and only the track pointing to the primary vertex was kept.
The measurements are reported for the rapidity interval |y| < 0.5 and have been corrected for detector efficiency based on the GEANT3 particle propagation code [40]. Track matching between the TPC and TOF detectors in GEANT3 was further improved by a data driven method based on a study of tracks not crossing the TRD material, resulting in a 6% difference. Since at low p T many nuclei in |y| < 0.5 are outside |η| < 0.8, their number was extrapolated using a Monte Carlo simulation where the rapidity distribution was approximated by a flat distribution.
In order to allow for a consistent comparison of the anti-deuteron-to-deuteron ratio across different center-of-mass energies with an identical GEANT version, a re-analysis of the deuteron differential yield at √ s = 7 TeV is presented here. The results are found to be consistent with the previous measurements shown in [10] within the systematic uncertainties.

(Anti-)nuclei identification
The identification of nuclei and anti-nuclei is based on their specific energy loss in the TPC and the estimation of their mass with the TOF detector. Figure 1 shows the energy loss signal recorded by the TPC of different nucleus species versus the rigidity (p TPC /|Z|), where p TPC is the momentum estimated at the inner wall of the TPC.
Deuterons and anti-deuterons can be identified cleanly up to p TPC 1.2 GeV/c, which corresponds to a maximum p T of 1 GeV/c. For p T > 1 GeV/c a coincidence with a TOF signal was required, in addition to a ±3σ cut around their expected energy loss in the TPC, extending the identification up to p T = 3 GeV/c. For this, tracks were propagated to the outer radius of the TOF and if a hit was found close enough to the trajectory, the corresponding time of flight was assigned to the track. Then, the squared mass, m 2 = p 2 (t 2 /l 2 − 1), where p is the reconstructed momentum, t the time of flight and l the track length, was calculated. Figure 2 shows the squared mass distribution for several p T bins in the region of the anti-deuteron squared mass. The anti-deuteron signal is approximately Gaussian, centered at the deuteron squared mass and with an exponential tail on the high mass side. This exponential tail is also present in the signal of other particle species such as π, K and p and extends to the anti-deuteron squared mass, producing an exponential background. The signal was extracted by combining a Gaussian with an exponential tail and an exponential background ( Figure 2).
Tritons and anti-tritons were identified by selecting tracks within ±3σ of their expected energy loss in the TPC and by also requiring a match to a TOF hit. The minimum p T = 1.2 GeV/c was chosen to be the same as for the 3 He nuclei. Due to the small number of tritons, it was not possible to use the signal extraction procedure used for deuterons. In this case, the selected tracks were required to have an associated mass within ±3σ (σ 0.05 GeV/c 2 ) of the triton mass and the maximum p T was limited to 1.8 GeV/c. The result is shown in Figures 1 and 3, with 6 anti-triton candidates in the interval 1.2 < p T < 1.8 GeV/c. Unlike deuterons and tritons, 3 He and 3 He nuclei can be identified throughout the p T range with the TPC, since for nuclei with |Z| = 2 the energy deposition is well separated from particles with |Z| = 1. In total, 17 candidates for 3 He nuclei were observed, based on the specific energy loss in the TPC (Figure 1), out of which 14 candidates were in the interval 1.2 < p T < 6 GeV/c and these were used in the measurements. Their identity was confirmed for those particles that were matched to a TOF hit (10 out of 14) with a mass measurement based on their time of flight as shown in Figure 3. A few 3 He nuclei (6 candidates) were also observed at the center-of-mass energy of 2.76 TeV.

Secondary (anti-)nuclei
Secondary nuclei are copiously produced in spallation reactions induced in the detector material by the impact of primary particles. They are also produced in the decays of Λ-hypernuclei, with the π-mesonic decay of the (anti-)hypertriton being the most important contribution [41].
The distance of closest approach (DCA) of the track to the primary vertex in the transverse plane (DCA xy ) and along the beam direction (DCA z ) were used to distinguish and reduce the number of secondary nuclei. Since they are produced far away from the primary vertex, they have a broader and flatter DCA xy distribution than primary nuclei, which have a narrow DCA xy distribution peaked at zero, similar to antinuclei. Figure 4 illustrates the different DCA xy distributions for deuterons and anti-deuterons at low and high p T . A positive DCA xy was assigned when the primary vertex was inside the radius of curvature of the track and a negative value in the opposite case. The number of secondary nuclei was greatly reduced by requiring |DCA xy | < 0.2 cm and |DCA z | < 3 cm, corresponding to a cut of ±10σ in the measured DCA resolution in the lowest p T bin.
The fraction of secondary nuclei with respect to primary nuclei was estimated with DCA xy templates from Monte Carlo simulations for each p T bin. The templates were fitted to the measured distribution with a maximum likelihood procedure which relies on a Poisson distribution and takes into account both the measured distribution and Monte Carlo statistical uncertainties [42]. This fraction was found to fall exponentially as a function of p T and subtracted from the measurements.
The production of anti-nuclei from interactions of primary particles with the detector material was neglected, since anti-nuclei exhibit a narrow DCA xy distribution peaked at zero (Figure 4). Due to the small production cross section of (anti-)hypernuclei in pp collisions, the feed-down contribution of (anti-)nuclei was not subtracted, but instead included as a systematic uncertainty.  Table 1 summarizes the values of the systematic uncertainties for the lowest and highest p T bins. These uncertainties take into account the identification procedure, the track selection criteria, secondary nuclei originating in the detector material and from feed-down, the (anti-)nucleus-nucleus interactions simulated in GEANT and the material budget.

Systematic uncertainties
The identification procedure was affected by an uncertainty coming from the background and signal shapes at high p T , where the signal-to-background ratio was small. It was evaluated by changing the squared mass interval and extracting the signal with two different methods: one by using the procedure described in Section 3.2 and the other by counting the number of entries in the 1 < p T < 1.4 GeV/c interval where the identification is unambiguous. For anti-tritons and 3 He nuclei the identification was clean and the particle identification uncertainty was assumed to be negligible. Systematic uncertainties due to the track selection criteria were estimated to be less than 4% for nuclei and anti-nuclei by changing the conditions for selecting tracks.
The approximations made in the DCA xy templates introduced an uncertainty on the removal of secondary nuclei originating in the detector material. A value of 4% was estimated for deuterons by replacing the simulated DCA xy templates of primary deuterons with the measured DCA xy distribution of antideuterons, which are not affected by contamination from secondary tracks. An uncertainty of ∼20% was estimated following a similar procedure for tritons and 3 He nuclei.
The dominant feed-down contribution for (anti-)nuclei is the π-mesonic decay of (anti-)hypertritons [41]: In pp collisions, the 3 Λ H cross section was estimated to be of the same order of magnitude as the 3 He nucleus cross section [43]. However, the production cross section of deuterons is about 10 4 times greater than that of 3 He nuclei, hence the contamination for (anti-)deuterons can be considered negligible. Additionally, the fraction of hypertritons which passes the track selection in the 3 He (anti-)nucleus channel was estimated with a Monte Carlo simulation and is at most 35%. Assuming a similar value for the (anti-)triton channel and branching ratios of 27% and 13% [41], then less than ∼10% and ∼6% contamination is expected for 3 He (anti-)nuclei and (anti-)tritons, respectively.
The (anti-)nucleus-nucleus elastic and inelastic scattering uncertainty was evaluated by comparing the GEANT3 simulations with the data for two different experimental configurations: one using the detector portion in which the TRD modules were installed between the TPC and the TOF detector and another in which the TRD was not installed. The ratio between the number of (anti-)deuterons counted with the two different detector configurations is related to the number of (anti-)deuterons lost due to hadronic interactions. These ratios were compared with a GEANT3 simulation and a 6% uncertainty was estimated for the amount of nuclei lost in such processes. This comparison, however, was not feasible for (anti-)tritons due to the limited data and a 12% uncertainty was assumed. Unlike deuterons and tritons, the measurements of 3 He (anti-)nuclei presented here only rely on TPC information, hence they are not affected by the TRD material in front of the TOF detector.
Another source of systematic uncertainty is the accuracy in the knowledge of the material budget. This uncertainty was estimated to be +3.4% and −6.2% by comparing the material thickness estimated by analyzing photon conversions in the inner detectors with the material description implemented in the Monte Carlo simulations [44]. To propagate these uncertainties to the results, a Monte Carlo simulation was done in which the material density was varied by ±10% and linearly interpolating to match the uncertainties in the material budget. The result was below 3% at low p T and negligible at high p T for the different (anti-)nuclei.
The extrapolation of the measurements to inelastic pp collisions adds additional systematic uncertainties of +2.

Deuterons and anti-deuterons
The invariant differential yields of deuterons and anti-deuterons have been measured in the p T range 0.8 < p T < 3 GeV/c ( Figure 5) and extrapolated to inelastic pp collisions with the cross sections of reference [39]. At LHC energies, both nucleus species are produced with similar abundance since the anti-deuteron-to-deuteron ratio approaches one as the center-of-mass energy increases ( Figure 6). The ratios are consistent with the (p/p) 2 ratios extracted from references [48,49] and hence in agreement with the expectation from simple coalescence and thermal-statistical models.

Heavier nuclei and anti-nuclei
A recorded luminosity of 4.2 nb −1 allowed anti-tritons and 3 He nuclei to be detected in pp collisions. Since the total number of observed candidates is small, the uncertainties were estimated as a central confidence interval (two-sided), using a coverage probability of 68.27% for a Poisson distribution. The resulting invariant yields for both anti-nucleus species are compatible in the p T range where measurements were possible (Figure 7). Some 3 He nuclei were also observed in the highest p T bin, but since the production rate is very small, it was not possible to evaluate the contamination due to secondary 3 He nuclei, and the bin was then excluded from this measurement. In contrast, 3 He nuclei are not affected by this source of contamination and the three measurements are sufficient to determine the parameters of the Tsallis distribution to extrapolate the yields (see Section 5.2).

Coalescence parameter
Many measurements of light nuclei have been successfully explained as the result of the coalescence of protons and neutrons that are nearby in phase-space [23,24]. In this model, the production of a nucleus with mass number A = N + Z is related to the production of nucleons at equal momentum per nucleon by where B A is called the coalescence parameter. This parameter has been found to be constant at low transverse momentum in light-particle collisions [14,50]. In contrast, in AA collisions it has been reported that B A decreases with increasing centrality of the collision and for each centrality increases with p T [7-10].
Using equation (1) and taking the proton and anti-proton distributions from references [45-47], the coalescence parameter (B 2 ) was computed and it is shown in Figure 8. The resulting values for deuterons and anti-deuterons are compatible and do not show any significant dependence on the center-of-mass energy within uncertainties. These measurements extend the p T reach up to three times beyond previous measurements in pp collisions extracted from the CERN ISR [11,12,51] (Figure 9).  To extract the B 2 from the CERN ISR, the anti-proton distribution was taken from [51] and the total cross section of 42.3 ± 0.4 mb from [52]. The distribution was also scaled by a factor of 0.69, estimated with an EPOS (LHC) simulation [43,53], to take into account the feed-down contribution. Figure 9 also includes the B 2 parameter of anti-deuterons from γp collisions and deep inelastic scattering of electrons at HERA [14,50] and B 2 from p-Cu and p-Pb collisions at Bevalac [1]. Our measurement reveals a p T dependence in B 2 not seen in previous experiments, which is significant given that the systematic uncertainties are correlated bin by bin.
This p T dependence can be reproduced with QCD-inspired event generators, such as PYTHIA 8.2 (Monash tune) [54] and EPOS (LHC), when adding a coalescence-based afterburner [43] that takes into account the momentum correlations between nucleons ( Figure 10). The afterburner looks for clusters of nucleons among the final particles produced by the event generators and boosts them to their centerof-mass frame. If the momentum of each individual nucleon is less than a certain value a nucleus is generated. With the afterburner, a constant B 2 is recovered when selecting protons from one event and neutrons from the next event (event mixing), in agreement with the expectation of an uncorrelated distribution of nucleons ( Figure 10). The p T dependence in B 2 is still present in the results from an alternate PYTHIA 8.2 (Monash tune) simulation with color reconnection turned off ( Figure 10). Furthermore, a radial flow effect in B 2 at these low average charged multiplicities is also discarded by the EPOS (LHC) simulation with the afterburner, since this contribution only arises in high multiplicity events, starting from dN ch /dη > 15 [53]. Thus, this p T dependence can be explained as a purely hard scattering effect, in contrast to AA collisions, where it is usually attributed to collective flow.   As in the case of anti-deuterons, the coalescence parameter (B 3 ) of 3 He nuclei also exhibits a p T dependence (Figure 11), and can be reproduced with QCD-inspired event generators with a coalescence-based afterburner [43]. Moreover, low p T values of B 3 are compatible with those obtained in p-C, p-Cu and p-Pb collisions at Bevalac [1].

Integrated yields and deuteron-to-proton ratio
Unlike coalescence models, statistical hadronization models only provide predictions for integrated yields. In this case, the integrated yields of light nuclei and the deuteron-to-proton ratio can add additional constraints to these models and could therefore serve as a test for thermal-statistical behavior in small systems at LHC energies.
To find the integrated yields, the measurements were extrapolated to the unmeasured p T region with a statistical distribution that provides an exponential behavior at low p T and a power law behavior at high p T (Figures 5 and 7): where m T = p 2 T + m 2 is the transverse mass, and gV , T and q are free parameters. This distribution can be derived from the Tsallis entropy [55, 56] and gives good description of the data in pp collisions [56]. It was preferred over the Levy-Tsallis used in previous work [10] as it provides a more stable description of the measurements with a limited data set, as in the case of anti-deuterons for the center-of-mass energy of 0.9 TeV or the 3 He nuclei.
The systematic uncertainties of the integrated yields (dN/dy) and mean transverse momenta ( p T ) were evaluated by shifting the data points up and then down by their uncertainties (i.e. assuming full correlation between p T bins). Additionally, the data points were shifted coherently, in a p T -dependent way, within their uncertainties to create maximally hard and maximally soft p T distributions. The values of dN/dy and p T were reevaluated and the largest difference was taken as the systematic uncertainty. Table 2 summarizes the resulting values for the different center-of-mass energies along with the extrapolation fraction due to the unmeasured p T regions. The uncertainty on the extrapolation was estimated by using additional distributions including the Levy-Tsallis [57, 58] and Boltzmann distributions. The change of the default fit function with respect to [10] leads to slightly different values for the obtained dN/dy and p T which are consistent within the respective systematic uncertainties. Figure 12 shows an exponential decrease of the dN/dy as a function of the mass number. The reduction of the yield for each additional nucleon is of about 1000.   Table 2: Integrated yields (dN/dy) and mean transverse momenta ( p T ) for deuterons, anti-deuterons and 3 He nuclei along with the extrapolated fraction (Extr.) due to the unmeasured p T regions. The first uncertainty is the statistical uncertainty and the second one the systematic uncertainty.
The integrated d/p and d/p ratios were calculated from the integrated yields in Table 2 and references [46,47] and are shown in Figure 13 as a function of the average charged particle multiplicity at midrapidity [59, 60]. The dN/dy values for pp collisions at the CERN ISR were computed following the same procedure described above and using the inclusive p distribution from [51] and the d distribution from references [11,12]. The resulting d/p ratio was divided by 0.69 to account for the contributions of feed-down anti-protons, based on an EPOS (LHC) simulation [43]. Figure 13 suggests an increasing trend in this ratio with the average charged particle multiplicity in pp collisions, which is also supported by an EPOS (LHC) simulation with the afterburner, although at ISR energies the d/p ratio is strongly influenced by the baryon number transport at mid-rapidity leading to a higher value than at LHC energies according to the model expectations.
When describing particle ratios such as the d/p ratio, the only free parameter of grand-canonical statistical hadronization models at LHC energies is the chemical freeze-out temperature. In the past, several attempts were made to extend their successful description of AA collisions to smaller collision systems such as pp. In particular, the canonical formulation describes the production of light flavor hadrons, including those with strangeness content [25]. While the p/π ratio is found to be comparable in pp, p-Pb, and Pb-Pb collisions [47,61], indicating a comparable chemical freeze-out temperature among different systems, the d/p ratio in pp collisions at LHC energies is found to be two times lower than the average value in AA collisions. Since the strangeness-canonical formulation only influences the abundance of strange particles with respect to non-strange particles, it can not explain the observed results presented here.

Summary and conclusions
The invariant differential yields of deuterons and anti-deuterons in pp collisions at √ s = 0.9, 2.76 and 7 TeV and the yields of tritons, 3 He nuclei and their anti-nuclei at √ s = 7 TeV have been measured in the rapidity range |y| < 0.5. The measurements cover the p T range 0.8 < p T < 3 GeV/c for (anti-)deuterons, 1.2 < p T < 1.8 GeV/c for (anti-)tritons and 1.2 < p T < 6 GeV/c for 3 He (anti-)nuclei. This extends previous measurements by one order of magnitude in incident energies, a factor of three in p T reach and includes the first ever measurements of anti-tritons and 3 He nuclei in pp collisions.
The present measurements show no significant dependence of the coalescence parameter B 2 on the center-of-mass energy from CERN ISR energies (53 GeV) to the highest LHC energy reported in this paper (7 TeV). Moreover, the values of both B 2 and B 3 are found to be compatible at low p T with those obtained in pA collisions at Bevalac energies.
A previously unobserved p T dependence in pp collisions of the coalescence parameters B 2 and B 3 is also reported. The data are well described by QCD-inspired event generators when a coalescence-based afterburner is added to take into account the momentum correlations between nucleons. According to PYTHIA 8.2 (Monash tune) and EPOS (LHC) with the afterburner, this dependence can be explained purely as a hard scattering effect.
In combination with CERN ISR measurements, our results suggest an increasing trend in the d/p ratio with charged particle multiplicity. While the values reported in central AA collisions are in agreement with a thermal model description of particle yields, the highest d/p ratio reported in this paper is about half the thermal model value, therefore, a thermal-statistical description is disfavored in pp collisions at these low average charged particle multiplicities. Our measurements are expected to contribute to the understanding of the background from pp collisions for the observation of anti-deuterons and 3 He nuclei in cosmic ray experiments and to the estimation of the production rates of the next stable anti-nuclei in pp collisions.     [54] T. Sjöstrand, S. Mrenna, and P. Skands, "A brief introduction to PYTHIA 8.1," Comput. Phys. Commun. 178 (2008) 852-867.