arXiv:1812.05175v5 [nucl-th] 20 May 2020 hnafco f2i + olsosat collisions p+p in more by 2 suppressed of is factor d/p ratio a 37– the [7, than particular, collisions In the decreas- in with 39]. multiplicity monotonically decrease charged and to ing measured, found been have TeV are energies 7 they center-of-mass to GeV at 900 from collisions ranging Pb+Pb and p+Pb tteLC h il aisdpand d/p ratios yield the LHC, the at [29–36]. hadronic low the of collisions are density heavy-ion and they of temperature the freeze-out hand, lambda when and kinetic other the the of at coalescence On for the by model described 28]. QGP statistical are the [27, the in are of production as they hadronization particle collisions hand, at these one in produced ener- the created On be binding small to [18–26]. their assumed sizes finite of and because nuclei are gies debate light collisions heavy-ion under these relativistic when still during and mat- produced how are However, interacting strongly [10–13] [14–17]. of point ter diagram critical phase possible further the the for have in search collisions to used heavy-ion been pro- relativistic nuclei discovery light in the recently, More and duction 9]. [4–7] [8, anti-nuclei anti-hypernuclei of of production led the also have to collisions heavy-ion relativistic [1–3], (QGP) ‡ † ∗ [email protected] [email protected] mi:[email protected] email: nrcn esrmnsb h LC Collaboration ALICE the by measurements recent In plasma quark- the of production the Besides upeso flgtnce rdcini olsoso sma of collisions in production nuclei light of Suppression h rtni essprse u oissalrsz n h h 25.75.Dw 25.75.-q, the and numbers: PACS size size. smaller larger much its its to of due result pro suppressed eve the a less in study is helium-3 to to triton used Compared the also LHC. is the at model ove collisions same the heavy-ion The reduces compar which nucleons. helium-3 systems, of small and those of deuteron collisions of in source sizes the of non-negligible freeze-out the kinetic the expl to at be nucleons can p+P of (LHC) in coalescence Collider those the to Large compared the collisions at p+p Collaboration in to helium-3 eso httercnl bevdsprsino h il r yield the of suppression observed recently the that show We .INTRODUCTION I. ntttfu enhsk oanWlgn Goethe-Univers Wolfgang Johann f¨ur Kernphysik, Institut ylto nttt n eateto hsc n Astronom and Physics of Department and Institute Cyclotron ea & nvriy olg tto,Txs783 USA 77843, Texas Station, College University, A&M Texas a-o-aeSr ,648Fakut Germany Frankfurt, 60438 1, Max-von-Laue-Str. 3 epfo p+p, from He/p a-i Sun Kai-Jia √ s NN arnCollider Hadron Dtd a 2 2020) 22, May (Dated: ejmnD¨onigus Benjamin 7 = ∗ n h igKo Ming Che and tsc iheege 4] o ntne h measured the 5 instance, about For is ratio [45]. collisions to energies in proton high zero as nearly such temperature, is deter- at freeze-out potential and are chemical chemical thermal ratios the the in by yield are only their energies mined and LHC equilibrium, the chemical at collisions heavy- in ion produced all ensemble, canonical grand respec- fm, and 10 p+p and fm in 48]. 2 [47, radii about tively source are that Gaussian collisions the Pb+Pb interferome- gives (HBT) which measurements Twiss try, This correlation Hanbury-Brown two-pion the . the through produced produced by of of confirmed number difference size total is only the the the is or systems and matter colliding 46], p+Pb these same [45, p+p, between the collisions in almost Pb+Pb occurs and Therefore, to temperature transformed freeze-out [44]. is chemical QGP matter produced hadronic initially the the which at ottesm eprtr of al- reaches temperature equal it same and nearly the , of most and consists particles of matter number produced the energies, o eta bP olsosat collisions Pb+Pb central for e 3]cmae ota ncnrlP+bcollisions Pb+Pb central in that to at compared [38] TeV ebe o h il aiso / and en- d/p canonical of grand ratios yield the the on statisti- For based the semble. with prediction consistent model is cal which collisions, Pb+Pb olsoso uhasalsse,tesaitclmodel statistical the system, in small nuclei [38]. a light LHC such of the production of at much suppressed collisions collisions however, the p+p explain are, from To values those the than These with agreement larger nice in data. also experimental are model statistical the itdrsetv auso bu 3.6 about of values respective dicted ntesaitclhdoiainapoc ae nthe on based approach hadronization statistical the In √ ecliin.Ti upeso sattributed is suppression This collisions. se s NN lpo hi nenlwv ucin with functions wave internal their of rlap ‡ t flwcagdpril multiplicity, particle charged low of nts dt h ieo h ulo emission the of size the to ed prrtni vnmr upesdas suppressed more even is ypertriton ie flgtnce r rdcdfrom produced are nuclei light if ained rP+bcliin yteALICE the by collisions Pb+Pb or b 2 = uto ftio n yetio in hypertriton and triton of duction too etrnt rtnadof and proton to deuteron of atio . † 6TV[] eas ftehg collision high the of Because [7]. TeV 76 ta Frankfurt,it¨at lssesa h Large the at systems ll y, × √ 10 T c s − NN × ≈ 2 10 npp +band p+Pb p+p, in 5 e [40–43] MeV 154 2 = − 3 3 ep h pre- the He/p, n 1 and . 6TVfrom TeV 76 . 0 × 10 − 5 2 has been modified to use the canonical ensemble to take For deuteron production in heavy-ion collisions, its num- into account the conservation of baryon number, electric ber from the coalescence model is given by charge and the strangeness [49]. The resulting ratios of 3 3 3 3 light nuclei to proton in these collisions are, however, too Nd = gd d x1 d k1 d x2 d k2fn(x1, k1) small compared with the experimental data unless the Z Z Z Z x k x x k k canonical correlation volume for exact charge conserva- fp( 2, 2)Wd( 1 2, ( 1 2)/2), (1) − − tions is taken to span three units of rapidity, instead of where g = 3/4 is the statistical factor for forming a the usual one unit of rapidity for collisions with large d one deuteron from spin half proton and [35, particle multiplicity, or using a higher chemical freeze- 55], f (x, k) are the neutron and proton phase-space out temperature of 170 MeV than the usual value of 155 p,n distributions, and W (x, k) is the Wigner function of the MeV for collisions of large systems. d deuteron. In the coalescence model, the formation probability of Since the nucleon coalescence is a local process, one can a light nucleus in a heavy-ion collision depends not only neglect the effect of collective flow on nucleons and take on the thermal properties and volume of the nucleon and their phase-space distributions in a thermalized expand- emission source but also on the internal wave ing spherical fireball of kinetic freeze-out temperature T function of the light nucleus. The small size of the emis- K and radius R to be sion source in p+p collisions is expected to significantly 2 2 k x Np,n − − 2 reduce the phase-space volume in which a light nucleus x k 2mT 2R fp,n( , )= 3 e K , (2) 3 2 can be formed, leading to a suppression of its production. (2π) (mTK R ) 2 Using a schematic coalescence model based on nucleons from the UrQMD model by allowing a deuteron to be with m being the nucleon mass, and they are nor- formed from a pair of proton and neutron when their malized to their numbers Np,n according to Np,n = 3x 3k x k separation in phase-space is less than certain value, it is d d fp,n( , ). found in Ref. [50] that this model can give a good de- R UsingR the harmonic oscillator wave function for the in- scription of the experimental data on the d/p ratio in ternal wave function of the deuteron, which is usually as- p+p, p+A, and A+A collisions at the LHC. sumed in the coalescence model for deuteron production, In this Letter, we use a more realistic coalescence its Wigner function then has the Gaussian form [31–33], 2 model to study the system size or charged particle multi- x 2 2 x k − 2 −σ k plicity dependence of the d/p and 3He/p ratios by taking Wd( , )=8 e σ e , (3) 3 into account the finite size of deuteron and He through 3 3 3 with the normalization d x d k Wd(x, k) = (2π) . their internal wave functions. Our results on these yield x Transforming the protonR and neutronR coordinates 1 and ratios are found in good agreement with available exper- x k k 3 2 as well as their momenta 1 and 2 to their center- imental data. We also confirm that the He/p ratio has of-mass reference frame, a stronger system size dependence than the d/p ratio as x + x helium-3 has three nucleons and is thus more sensitive X = 1 2 , x = x x , to the spatial distribution of nucleons in the emission 2 1 − 2 source. For the triton 3H, we find that its production is k k K = k + k , k = 1 − 2 , (4) 10%-30% larger than that of helium-3 and thus less sup- 1 2 2 pressed in p+p collisions because of its smaller matter the integrals in Eq. (1) can then be straightforwardly radius. For the hypertriton 3 H, the 3 H/Λ ratio in colli- Λ Λ evaluated, leading to sions with small charged particle multiplicity is found, on 3 2 X 1 1 2 the other hand, much more suppressed than the He/p 8gdNpNn 3 − 2 3 −( 2 + 2 )x X R x σ 4R Nd = 6 2 3 d e d e ratio, and the suppression further depends on whether (2π) (mTK R ) Z Z 3 2 the ΛH is produced from the coalescence of n-p-Λ or d-Λ. K 2 2 1 3 − 4 3 −k (σ + ) d K e mTK d k e mTK Z Z 3NnNp 1 1 II. LIGHT NUCLEI PRODUCTION IN = 2 . (5) 4(mT R2)3/2 3/2 σ 3/2 COALESCENCE MODEL K 1 (1 + 4R2 ) 1+ mT σ2  K  Although the coalescence model has been used in var- The parameter σ in Eq.(3) is related to the root-mean- ious ways for studying light nuclei production in nuclear square matter radius rd =1.96 fm of deuteron [62] by σ = reactions [51–59], we follow in the present study that em- 8/3 r 3.2 fm. For the kinetic freeze-out temperature d ≈ ployed in Refs. [14, 15]. In this approach, the formation TpK of nucleons, it is typically of the order of 100 MeV. 2 probability of a light nucleus in heavy-ion collisions is We therefore have mTK 1/σ , and the yield ratio d/p given by the overlap of the nucleon phase-space distri- is then approximately given≫ by bution functions in the emission source with the Wigner N 3N 1 function of the light nucleus, which is obtained from the d n 2 3/2 3/2 (6) Np ≈ 4(mTKR ) 1.6 fm 2 Wigner transform of its internal wave function [60, 61]. 1 + ( R ) .   3

The last factor in the above equation describes the sup- pression of deuteron production due to its finite size rel- ative to that of the nucleon emission source. Its value Pb+Pb @ 2.76 TeV approaches unity as the source radius R becomes much p+p @ 7 TeV

p+p @ 2.76 TeV

larger than the size of deuteron, while it is significantly p+p @ 900 GeV

smaller than unity when R is close to or less than 1.6 fm. Fit (N = 0.0223 * dN /d ) p ch

3Nn p The factor C1 = 2 3/2 in Eq. (6) corresponds to 4(mTK R ) N the d/p ratio in the limit of large nucleon emission source when the suppression effect due to finite deuteron size is negligible, and it is directly related to the entropy per nucleon in a nuclear collision, which remains essentially (a) unchanged after chemical freeze-out [63]. Therefore, the value of C1 is expected to be similar in p+p, p+Pb and Pb+Pb @2.76 TeV

Pb+Pb collisions at the LHC. From the d/p ratio mea- Fit sured in central Pb+Pb collisions at √sNN =2.76 TeV, a value of about 4.0 10−3 is obtained from Eq.(6) for × C1. Using this value, Eq. (6) can be rewritten as (MeV) K

−3 T Nd 4.0 10 × 3/2 , (7) Np ≈ 1.6 fm 2 1 + ( R )   (b) where the value of R can be calculated from

Our prediction

R at k = 0.2-0.3 GeV (Pb+Pb @ 2.76 TeV) 1/3 inv T (3Nn) R at k = 0.2-0.3 GeV (p+Pb @ 5.02 TeV) R = 3/2 1/3 . (8) inv T [4C1(mTK) ]

using the neutron number Nn, which is the same as (fm) R the proton number in collisions at the LHC energies be- cause of the vanishing isospin chemical potential, and the kinetic freeze-out temperature TK extracted from mea- (c) sured charged particle spectra. Shown in panel (a) of Fig. 1 by symbols with error bars is the charged particle multiplicity dependence of dN /d ch the proton number measured by the ALICE Collabo- ration [64, 65]. The dependence is seen to be essen- FIG. 1: Charged particle multiplicity dependence of the pro- tially linear and can be well parametrized by Np = ton number Np (panel (a)), kinetic freeze-out temperature TK 0.0223 dN /dη shown by the solid line. (panel (b)) and the radius of emission source R (panel (c)). × ch Panel (b) of Fig. 1 shows the charged particle multi- [47, 64, 65]. The solid line in panel (a) represents a linear fit to plicity dependence of the kinetic freeze-out temperature. the data. The solid line in panel (b) is the fit to the data using The solid circles with error bars are from the ALICE Col- Eq.(9) with the shadow region surrounding the line denoting the uncertainties. The uncertainties of the data points shown laboration based on a blast wave model fit to the experi- in panel (b) are only statistical [64]. The dashed line in panel mental data [64]. It is seen that TK increases as dNch/dη (c) is the predicted radius of emission source with uncertain- decreases, and it can be fitted by the function ties given by the shaded band. Except the experimental data shown by solid squares with error bars in panel (c), which are 1 − −1 from the ATLAS Collaboration [66], all other experimental dN /dη q T = T + T 1 + (q 1) ch , (9) data are from the ALICE Collaboration [47, 64, 65]. K 0 1  − × M  in terms of the four parameters T0 = 80.6 31.0 MeV, by T = 83.0 46.9 MeV, M = 67.3 76.±3, and q = 1 −1 −1 3.33 3.25± after taking into account± the errors in the 1.27 4.33 10 1.80 10 9.33 ± 4.33 10−1 1.93 × 10−1 6.90 × 10−2 2.71 extracted TK . The corresponding uncertainty of TK at H =   . 1.80 × 10−1 6.90 × 10−2 2.79 × 10−2 1.26 any charged particle multiplicity can be obtained from × × × 1/2  9.33 2.71 1.26 7.44 101  ∂TK −1 ∂TK   ∆TK = 2 (H )ij , where xi is one of × i,j ∂xi ∂xj (10) h i the four parametersP in Eq.(9). The Hessian matrix H in The fitted charged particle multiplicity dependence of TK our chi-square fit to the empirically extracted TK is given is shown in panel (b) of Fig. 1 by the solid line with the 4 shaded band denoting its uncertainties. We note that the function in Eq.(9) has a similar form as the Tsallis distri- bution for the single particle energies in a non-extensive ALICE published system [67]. p+p @ 900 GeV

p+p @ 2.76 TeV

With the above determined values of Nn and TK , we p+p @ 7 TeV ) -3 can evaluate from Eq. (8) the charged particle multiplic- Pb+Pb @ 2.76 TeV ity dependence of the radius R of emission source. The result is depicted in panel (c) of Fig. 1 by the dashed line with the theoretical uncertainties given by the shaded (x10 d/p band, which turns out to be quite small. Also shown COAL. in this panel by solid squares and stars with error bars

(a) are the one-dimensional femtoscopic radius Rinv of the Gaussian emission source extracted, respectively, by the ATLAS Collaboration [66] and by the ALICE Collab- oration [47] from the two-pion interferometry measure- ments [68] for pion pairs of transverse momentum kT = 0.2-0.3 GeV. It is seen that the predicted R is larger than

R for central Pb+Pb collisions. This is likely due to He/p inv 3

the large radial flow in central Pb+Pb collisions, which COAL. (d-p) would lead to a smaller apparent Gaussian radii of an COAL. (p-p-n) emission source. With the information on the radius of nucleon emission (b) source, the d/p ratio can then be calculated from Eq. (7), and its dependence on the charged particle multiplicity is shown in panel (a) of Fig. 2. Compared with the mea-

Two-body COAL.

sured ratio in central Pb+Pb collisions at √sNN =2.76 Three-body COAL. TeV [7, 64] and in p+p collisions at √sNN = 900 GeV, He 2.76 TeV and 7 TeV [38, 65], the theoretical results are 3 H/ in nice agreement with the data for all charged particle 3 multiplicities. Our results are consistent with those from a schematic coalescence model based on kinetic freeze- out nucleons from the UrQMD model [50]. We note that the finite deuteron size suppresses not only the total yield ratio of deuteron to proton as studied here but also their

dN /d ratio as a function of transverse momentum [25]. We ch also note that the size effect in the coalescence model was firstly studied in Refs. [69, 70] at early 1960s and FIG. 2: Charged particle multiplicity dependence of the yield 3 3 3 then comprehensively studied in Refs. [30]. ratios d/p, He/p and H/ He. The lines denote the predic- Similarly, we can calculate the charged particle mul- tions of coalescence model with theoretical uncertainties on tiplicity dependence of the 3He/p ratio by extending the emission source radius given by the shaded band. Ex- the formalism for deuteron production from proton and perimental data from the ALICE Collaboration are shown by symbols with error bars [7, 38, 64, 65]. neutron coalescence to the production of helium-3 from the coalescence of two and one neutron as in Refs. [31–35]. The resulting yield ratio 3He/p is given by We also consider 3He production from the coalescence N3He NnNp 1 of a deuteron and a proton. In this case, the root- , (11) 3 2 3 2 3 mean-square radius of He can be estimated as r3 Np ≈ 4(mTKR ) r3 He He 1/2 2 2 ≈ 1+ 2R2 (3/8) r =1.15 fm with r 2.6 fm being h pdi h pdi ≈   the distancep between proton andp the center of mass of where r3He = 1.76 fm is the matter radius of helium- the deuteron inside the helium-3. Using the statistical 3 [62]. In obtaining the above equation, we have included factor of 1/3 for the coalescence of a spin 1 deuteron and the statistical factor of 1/4 for forming a spin 1/2 helium- a spin 1/2 proton to 3He, the 3He/p ratio is then 3 from three spin 1/2 nucleons and used the condition 2 NnNp −6 mTK 1/r3 . With the factor C2 = 2 3 = N3He 7.1 10 ≫ He 4(mTK R ) × , (13) 2 −6 1.15 fm 3/2 1.6 fm 3/2 4C1 /9=7.1 10 , determined from the value of C1, Np ≈ 1 + ( )2 1 + ( )2 × R R Eq.(11) becomes     −6 where the suppression factor for deuteron production N3He 7.1 10 has been included. As shown in panel (b) of Fig. 2, × 3 . (12) Np ≈ 1.24 fm 2 1 + ( R ) the contribution from the coalescence of deuteron and   5 proton is larger than that from the coalescence of two protons and one neutron in collisions of small charged particle multiplicities, although the two processes give similar contributions to 3He production in collisions of large charged particle multiplicities. Besides, the the- oretical results are found in nice agreement with the

Pb+Pb @ 2.76 T eV

data at dN /dη < 1000, while they are slightly smaller H/ ch COAL. (d- ) 3 than the data in the most central Pb+Pb collisions at COAL. (n-p- ) √sNN =2.76 TeV. The above calculation for helium-3 production can be straightfowardly extended to triton (3H) production. Be- (a) cause of its smaller radius of r3H = 1.59 fm [62] than helium-3, triton production in collisions with low multi- plicities is expected to be less suppressed than helium-3. For instance, the 3H/3He ratio is

1.24 fm 2 3 3 T wo-body COAL. N H 1 + ( R ) 3 3 (14) T hree-body COAL. N3He ≈  1.12 fm 2 S 1 + ( R )   from the three-body coalescence and

1.15 fm 3/2 (b) 3 2 N H 1 + ( R ) 3/2 (15) N3He ≈  1.039 fm 2 1 + ( R )

dN /d   ch from the two-body coalescence. Shown in panel (c) of 3 3 Fig. 1 is the H/ He yield ratio as a function of charged FIG. 3: Charged particle multiplicity dependence of the yield 3 particle multiplicity. It is seen that this ratio indeed in- ratio ΛH/Λ and the S3 factor. Predictions from the coales- creases with decreasing charged particle multiplicity, par- cence model are shown by solid lines for the three-body coa- ticularly for triton and helium-3 production from three- lescence and dashed lines for the two-body coalescence with body coalescence. For instance, this ratio in p+p colli- theoretical uncertainties given by shaded bands. Experimen- sions at dNch/dη = 5 is predicted to be 1.1 if triton and tal data from the ALICE Collaboration [7, 9] are shown by helium-3 are produced from two-body coalescence but solid stars with error bars. increases to 1.3 if they are produced from three-body coalescence, suggesting a 10%-30% enhancement in the important than that of the d-Λ coalescence for hyper- production of triton than heilium-3 in p+p collisions. Fu- trion production, and the hypertriton yield in relativistic ture measurements of the triton yield in p+p collisions heavy-ion collisions is essentially determined at the time can be used to testify this result. when nucleons and deuterons freeze out, although it still undergoes reactions with . 3 3 Similar to helium-3 production, the yield ratio ΛH/Λ III. ΛH PRODUCTION IN COALESCENCE MODEL is given by −6 N3 H 7.1 10 Λ (16) To study the production of 3 H in collisions of small × 3 Λ NΛ ≈ 1 + ( 3.46 fm )2 systems, we first note that 3 H is the lightest known nu- R Λ   cleus with strangeness, and it has a small binding energy for hypertriton production from the coalescence of pro- of only BΛ= 2.35 MeV and a large root-mean-square ra- ton, neutron and Λ-hyperon, and 3 dius of rΛH 4.9 fm [71]. Besides being a ≈ N3 −6 of proton, neutron and Λ-hyperon, the hypertriton can ΛH 7.1 10 × 3/2 . (17) also be considered as a bound state of a deuteron and a NΛ ≈ 1 + ( 4.2 fm )2)3/2(1 + ( 1.6 fm )2 Λ-hyperon with a binding energy B = 0.13 0.05 MeV R R Λ ±   [72] and a distance of rΛd 10 fm [71] between deuteron for hypertriton production from the coalescence of d and and Λ-hyperon. Because≈ of its large size, the produc- Λ. In obtaining Eq.(16) for the three-body coalescence 3 tion of ΛH in collisions of small systems is expected to be process, we have taken the root-mean-square radius of 3 3 1/2 2 much more suppressed than that of helium-3. We note ΛH as rΛH (3/8) rΛd = 4.2 fm. Also, we have 3 ≈ h i that the study of ΛH production in relativistic heavy-ion neglected the mass differencep of the constituent particles collisions including both the coalescence of p-n-Λ and of in obtaining above expressions since its effect is small. d-Λ has recently been reported in Ref. [73]. According In panel (a) of Fig. 3, we show the charged particle mul- 3 to this study, the process of p-n-Λ coalescence is more tiplicity dependence of the yield ratio ΛH/Λ in Pb+Pb 6 collisions at √sNN = 2.76 TeV. The dashed and solid lisions with its temperature taken from the empirical fit lines represent results from the two-body and the three- to measured particle spectra and its size determined by body coalescence, respectively. No significant difference assuming that the entropy per baryon is independent of is seen between these two processes when dNch/dη > 100, the colliding system. We have found that the yield ratios and both agree very well the experimental data shown by d/p and 3He/p are significantly reduced once the charged the solid star with error bar measured by the ALICE Col- particle multiplicity is below about 100 as a result of 3 laboration for central collisions. For dNch/dη 10, both the non-negligible deuteron and He sizes compared to 3 ∼ production processes give a yield ratio ΛH/Λ that is two- that of the nucleon emission source. Our results thus order of magnitude less than in central Pb+Pb collisions. provide a natural explanation for the observed suppres- We further investigate the strangeness population sion of deuteron and 3He production in p+p collisions by factor S3, which is a double ratio defined by the ALICE Collaboration at the LHC. They also demon- S =3 H/(3He Λ/p) [74]. As suggested in Ref. [74], the strate the importance of the internal structure of light 3 Λ × value of S3 should be about one in the coalescence model nuclei on their production in collisions of small systems. for particle production. It was also argued in Ref. [75] We have further found that the production of triton is that this factor might be a good signal for studying the 10%-30% larger than that of helium-3 in p+p collisions local correlation between baryon number and strangeness because of its smaller matter radius. This enhancement in a quark-gluon plasma [76], providing thus a valuable of 3H/3He ratio can be tested in future measurements. probe of the onset of deconfinement in relativistic heavy- We have also used this model to study the charged ion collisions. The system size dependence of S3 can be particle multiplicity dependence of hypertriton produc- 3 3 calculated from Eqs. (12) and (16), for He and ΛH pro- tion in Pb+Pb collisions at the LHC by considering duction from three-body coalescence and from Eqs. (13) both the three-body process of p-n-Λ coalescence and and (17) for their production from two-body coalescence. the two-body process of d-Λ coalescence. Because of 3 Results for Pb+Pb collisions at √sNN = 2.76 TeV are the much larger ΛH radius than those of deuteron and 3 3 shown in panel (b) of Fig. 3 for both the two-body He, the yield ratio ΛH/Λ is found to be much more (dashed line) and the three-body (solid line) coalescence. suppressed in collisions with low charged-particle mul- One can see that the S3 factor in central collisions is tiplicity, particularly for the three-body coalescence pro- close to unity in both cases, similar to the experimental cess. We have further studied the charged particle mul- value shown by the solid star with error bar measured tiplicity dependence of the strangeness population fac- 3 3 by the ALICE Collabortion [77]. Also, there is no sig- tor S3 =ΛH/( He Λ/p), and its value in collisions with nificant charged particle multiplicity dependence in the small charged particle× multipilicity is found to be sig- S3 factor given by the two coalescence processes when nificantly less than one expected in collisions with large dNch/dη > 100. However, they start to deviate when charged particle multiplicity. Future experimental mea- 3 dNch/dη becomes smaller, with the three-body coales- surements of the yield ratio ΛH/Λ and the strangeness cence giving a much smaller value than the two-body co- population factor S3 in collisions of low charged particle alescence as a result of the suppressed production of hy- multiplicity will be of great interest because it not only pertriton from three-body coalescence in small systems. can check the prediction of the present study but also provide the possibility to improve our knowledge on the 3 internal structure of ΛH. IV. CONCLUSIONS

In summary, based on the coalescence model in full Acknowledgments phase space, we have studied the dependence of deuteron, heilium-3, and triton production in nuclear collisions at This work was supported in part by the US Depart- energies available from the LHC on the charged parti- ment of Energy under Contract No. de-sc0015266 and cle multiplicity of the collisions. For the nucleon dis- the Welch Foundation under Grant No. A-1358 as well tributions, they are assumed to come from a thermalized as by BMBF through the FSP202 (F¨orderkennzeichen hadronic matter at the kinetic freeze-out of heavy-ion col- 05P15RFCA1).

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