Hypertriton Production in Relativistic Heavy Ion Collisions ∗ Zhen Zhang , Che Ming Ko

Physics Letters B 780 (2018) 191–195

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Physics Letters B

www.elsevier.com/locate/physletb

Hypertriton production in relativistic heavy ion collisions ∗ Zhen Zhang , Che Ming Ko

Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA a r t i c l e i n f o a b s t r a c t

Article history: Based on the phase-space distributions of freeze-out nucleons and hyperons from a blast-wave model, Received 17 February 2018 we study hypertriton production in the coalescence model. Including both the coalescence of with Received in revised form 28 February 2018 proton and neutron as well as with deuteron, which is itself formed from the coalescence of proton and Accepted 1 March 2018 neutron, we study how the production of hypertriton is affected if nucleons and deuterons are allowed to Available online 6 March 2018 √ stream freely after freeze-out. Using central Pb+Pb collisions at s = 2.76 as an example, we ﬁnd that Editor: W. Haxton NN this only reduces slightly the hypertriton yield, which has a value consistent with the experimental data, even if the volume of the system has expanded to a size similar to the freeze-out volume for a hyertriton if its dissociation cross section by pions in the system is given by its geometric size. Our results thus suggest that the hypertriton yield in relativistic heavy ion collisions is essentially determined at the time when nucleons and deuterons freeze out, although it still undergoes reactions with pions. © 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction been studied using the statistical model [3], in which their abun- dances are determined by assuming that they are in chemical equi- Besides allowing the opportunity to study the properties of librium with other hadrons and nuclei at a chemical freeze-out strongly interacting matter at extreme temperature and density, temperature that is close to that for the quark–gluon plasma to high energy heavy-ion collisions also provide the possibility to hadronic matter phase transition. This is in stark contrast to the produce hypernuclei that contain strange baryons [1–3]. Because coalescence model, which assumes that hypernuclei are formed of the abundant anti-strange quarks produced in ultrarelativis- from the coalescence of protons, neutrons and hyperons at the tic heavy ion collisions, anti-hypernuclei can also be produced in kinetic freeze-out of heavy ion collisions [8,11]. Both models are, these collisions. Indeed, both hypertriton, which is a bound state of however, quite successful in describing the experimental data, al- proton, neutron and hyperon, and its anti-nucleus, i.e., the anti- though slightly earlier freeze out of hyperons than nucleons is hypertriton, have been detected at the Relativistic Heavy Ion Col- introduced in the coalescence model study of Ref. [11]. The rea- lider (RHIC) by the STAR Collaboration [4] and at the Large Hadron son for this may be due to the fact that their numbers remain Collider (LHC) by the ALICE Collaboration [5]through their weak − + unchanged during the hadronic evolution from the chemical to the decays 3 H → 3He + π and 3 H¯ → 3He¯ + π , respectively, with ¯ kinetic freeze out, like the deuteron [12] and other particles, which the same branch ratio of about 25% [6]. The study of hypernuclei has recently been shown to be associated with the conservation of production in high-energy heavy-ion collisions is also of interest entropy per particle [13]. because it can provide information on local baryon and strangeness Given its very small binding energy of about 130 keV [14] and correlations in the collisions [7], if their production is through the large root-mean-square radius of about 4.9 fm [15], the hypertri- coalescence of protons, neutrons and hyperons at the ﬁnal stage ton is, however, expected to be formed later than the kinetic freeze of the collisions [8]. For example, the ratio S = 3 H/(3He × ) [9] 3 p out time for nucleons and hyperons due to its larger dissociation has been suggested as a possible probe of the onset of deconﬁne- cross section by pions and thus shorter mean-free path than these ment in high-energy heavy-ion collisions [10]. hadrons. To illustrate the effect of the large hypertriton size on its In addition to the coalescence model mentioned in the above, production in relativistic heavy ion collisions, we use the coales- hypernuclei production in relativistic heavy ion collisions has also cence model based on the phase-space distributions of freeze-out nucleons and hyperons from a blast-wave model. In particularly, we use the blast-wave models FOAu-N and FOAu- of Refs. [11,16] Corresponding author. √ * + = E-mail addresses: zhenzhang @comp .tamu .edu (Z. Zhang), ko @comp .tamu .edu for central (0%–10% centrality) Pb Pb collisions at sNN 2.76 (C.M. Ko). TeV [5]. Besides the coalescence of proton, neutron and hyperon, https://doi.org/10.1016/j.physletb.2018.03.003 0370-2693/© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 192 Z. Zhang, C.M. Ko / Physics Letters B 780 (2018) 191–195

+ → 3 we also include the coalescence process d H between the particles are thus determined by the parameters T , ρ0, R0, τ0, τ , deuteron d and , which is found to enhance the hypertriton yield ξp , ξn, and ξ in the blast-wave model. by about a factor of two, to take account of the fact that the hyper- We note that by assuming freeze-out at constant longitudinal triton wave function is dominated by a hyperon that is loosely proper, we have neglected the effect that the edge of the fireball bonded to a deuteron [5,15,17]. We find that the yield of hyper- freezes out much earlier than the center. Although this effect can triton obtained from the coalescence model does not change much be included by free-streaming the frozen-out particles from the if it is calculated with nucleons and that are allowed to stream realistic decoupling surface to one of constant longitudinal proper freely after freeze out to a volume that is nineteen times larger, time, which is expected to render these particles no longer per- similar to the hypertriton freeze-out volume that is obtained with fectly thermal, it is not expected to significantly change our results a dissociation cross section given by its geometrical size. as the parameters used in the present study are tuned to repro- This paper is organized as follows. In Secs. 2 and 3, we briefly duce the experimental data on the transverse momentum spectra introduce the blast-wave fireball model and the coalescence model, of nucleons and hyperons. respectively. Our results and related√ discussions on the hypertriton yield in central Pb+Pb collisions at sNN = 2.76 TeV are presented 3. The coalescence model in Sec. 4. Finally, we give the conclusion in Sec. 5. In the coalescence model, the production probability of a clus- 2. The blast wave model ter is determined by the overlap of the Wigner function of its internal wave function with the phase-space distributions of the Following Refs. [11,16], we use a blast-wave model to describe constituent particles at the kinetic freeze-out of heavy ion colli- the phase-space distributions of nucleons and hyperons at the sions. The multiplicity of a cluster containing M particles can then kinetic freeze out of a heavy ion collision.√ With the assumption be written as [16,18–23] = 2 − 2 that the longitudinal proper time τ t z for the freeze-out M d3 p hypersurface μ has a Gaussian distribution, = μ 3 i NM gM dτi J(τi)pi d σiμ f (xi, pi) Ei − 2 i=1 1 (τ τ0) J(τ ) = √ exp − , (1) × f (x , ··· , x ; p , ··· , p ) (6) 2πτ 2(τ )2 M 1 M 1 M ··· ; ··· where f M (x1, , xM p1, , pM ) is the Wigner function of the with a mean value τ0 and a dispersion τ , the single-particle in- = + M + variant momentum spectrum of nucleons or is then given by cluster and gM (2S 1)/ i=1(2Si 1) is the statistical factor with Si and S being the spins of the ith constituent particle d3 N E = d J( ) d3 pμ f (x, p), (2) and the cluster, respectively. The coordinate xi and momentum pi 3 τ τ σμ d p are those of the ith particle in the fireball frame, while xi and pi μ are the corresponding ones after Lorentz transforming to the rest μ frame of the produced cluster and then propagating earlier freeze- where σμ is the covariant normal vector to μ and p is the four- momentum of the emitted particle. The invariant distribution of out particles freely to the time when the last particle in the cluster these emitted particles from the hyper-surface is freezes out [20,22]. The Wigner function of a cluster is obtained from the Wigner g −1 f (x, p) = exp(−pμu /T )/ξ ± 1 , (3) transform of its internal wave function, which is usually taken to 3 μ (2π) be the product of harmonic oscillator wave functions [16,20–23]. For example, using the ground-state wave function of a harmonic where g is the spin degeneracy factor of the particle, uμ is the oscillator for the deuteron, its Wigner function is given by flow four-velocity, ξ is the fugacity parameter determined by the number of emitted particles, and T is the temperature of the fire- ρ2 ball. Taking the longitudinal flow velocity to be v = z/t, the lon- f (ρ, p ) = 8exp − − p2 2 , (7) L d ρ 2 ρσρ = 1 [ + − ] σρ gitudinal flow rapidity is then ηflow 2 ln (1 v L )/(1 v L ) and = 1 [ + − ] is identical to the space–time rapidity η 2 ln (t z)/(t z) . In where the relative coordinate ρ and momentum pρ between the = 1 [ + − ] proton and neutron are defined, respectively, by terms of η, the transverse flow rapidity ρ 2 ln (1 β)/(1 β) with β being the magnitude of the transverse flow velocity, one = √1 − then has ρ (xp xn), (8) 2 μ √ − p uμ = mT coshρcosh(η − y) − pT sinhρcos φp − φb , (4) mn p p mp pn pρ = 2 , (9) μ 3 mp + mn p d σμ = τmT cosh(η − y)dηrdrdφ. (5) with mp and mn being the masses of proton and neutron, re- = 2 + 2 In the above, pT and mT m pT are the transverse momen- spectively. The width parameter σρ in Eq. (7)is related to the tum and mass with m being the particle mass, φp and φb are the root-mean-squared charge radius of deuteron by azimuthal angles of the particle transverse momentum and the 2 3mp transverse flow velocity with respect to the reaction plane, and r2= 2. (10) d 2 σρ r and φ are the radial and angular coordinates of the particle in (mn + mp) the transverse plane. For central heavy-ion collisions considered in 2= = the present study, the azimuthal angle of the transverse flow ve- Using the deuteron charge radius rd 2.142 fm [24] and mp locity φb is equal to φ and the transverse flow rapidity of the fluid mn = 0.939 GeV, we find the width parameter to have the value element in the fireball can be parametrized ρ = ρ0r/R0, with ρ0 σρ = 2.473 fm. In the present study, we will use the Wigner func- being the maximum transverse flow rapidity and R0 the transverse tion given by Eq. (7)in the coalescence model to study deuteron radius of the fireball. The phase-space distributions of freeze-out production. Z. Zhang, C.M. Ko / Physics Letters B 780 (2018) 191–195 193 2 Table 1 4vm vnλ vm − vn −3/2 × exp − cos 2 p · λ , (13) Gaussian fit coefficients ai and bi (in fm ), and + + λ −2 vm vn vm vn width parameters ui and vi (in fm ) to the hypertriton wave function from Ref. [15]. with pρ similarly defined as in Eq. (9) and pλ defined by − − a1 1.8385E 1 u1 5.1800E 1 a2 1.2015E−2 u2 2.6409E−2 + − + 3 m(p p pn) (mp mn)p a 6.7806E−2 u 1.1221E−1 = 3 3 pλ . (14) 2 mp + mn + m b1 −5.9323E−2 v1 2.3992E−1 b2 2.6936E−1 v2 2.5200E+3 + + → 3 Besides the coalescence process n p H, we also con- b −2.8946E−2 v 4.0743E−2 3 3 sider the process d + → 3 Hfor hypertriton production by treat- b4 −7.3758E−3 v4 4.7214E−3 ing deuteron as a point particle. In this case, the Wigner func- 3 tion of His simply given by the second factor in Eq. (13)with λ and pλ now denoting the relative coordinate and momentum + + between deuteron and after replacing mp mn, pp pn and + + (mp xp mnxn)/(mp mn) in Eqs. (12) and (14)with md, pd and xd, respectively.

4. Results

In the present study, we consider, as an example,√ the production of hypertriton in central Pb+Pb collision at sNN = 2.76 TeV. For the parameters in the blast-wave model, we use the FOPb- N and FOPb- parameter sets reported in Refs. [11,16], i.e. T = 121.1MeV, ρ0 = 1.215, R0 = 19.7fm, τ0 = 15.5fm/c, τ = 1.0fm/c, ξp = ξn = 3.72 and ξ = 9.54, which are obtained by fitting the measured spectra of p, d, 3He and from the ALICE Collaboration. Fig. 1. Density distributions of proton or neutron (N) and in hypertriton as func- Using the hypertriton Wigner function in Eq. (13), we find that tions of the distance r from the center-of-mass of proton and neutron. Black dashed + + 3 dN = × lines are theoretical results from Ref. [15], and red solid lines show results obtained the coalescence of n p gives the Hyield 6.7 dy y=0 from the wave functions used in this work. − 10 5, which is almost a factor of two smaller than the measured − yield of 1.54 ± 0.41 × 10 4 [5]. Compared to the hypertriton yield − Because of the long tail of the wave function in the hypertri- of 5.72 ×10 5 in Ref. [11], which is calculated from the hypertriton ton [15], we use instead the following product of sums of Gaussian Wigner function by assuming that the proton, neutron and are 3 functions for the hypertriton wave function: all in the ground state of an isotropic harmonic oscillator, our H number is only 17% larger, indicating that details of the hypertriton 3 4 − 2 − 2 wave function do not significantly affect the hypertriton yield in = ui ρ vi λ (ρ, λ) aie bie . (11) the coalescence model. i i + → 3 Including also the d H coalescence by treating deuteron as a point particle as introduced in Sec. 3 and with the deuteron In the above, ρ is similarly defined as in Eq. (8) and λ is defined phase-space distribution obtained via the coalescence of proton by − and neutron, we obtain a 3 Hyield of 7.9 × 10 5 from the d + + coalescence, which is close to that from the coalescence of neu- 2 mp xp mnxn 3 λ = − x . (12) tron, proton and . The total Hyield from the two contributions + − 3 mp mn is 1.46 ×10 4 and is consistent with the experimental value within its uncertainty. The coefficients a and b as well as the width parameters u and i i i By producing hypertritons from the coalescence of freeze-out vi can be determined by fitting the hypertriton rms radius and nucleons, hyperons, and deuterons as in the above calculations, the density distributions of proton or neutron and in hypertriton obtained from the stochastic variational calculations [15]. We it is assumed that hypertritons also freeze out at similar times. ∼ list their values in Table 1 and show the density distributions of Since the hypertriton is weakly bounded ( 130 keV) and has a 3 size comparable to the volume of the kinetically freeze-out fire- nucleons and in Has functions of the distance from the center- of-mass of proton and neutron in Fig. 1. ball, particularly for the distance between the and the two 3 nucleons, it can be easily dissociated by collisions with the abun- The Wigner function or phase-space density of Hcan be straightforwardly evaluated and is given by dant pions in heavy ion collisions at the LHC energies [25] and ⎡ is thus unlikely to freeze out at the same time as nucleons and 3 3/2 p2 hyperons. To determine the freeze-out time for the hypertriton = ⎣ π − ρ f 3 H 8 aia jexp relative to that of nucleons, one needs its scattering cross sec- ui + u j ui + u j i, j tion with pion compared to that of nucleon. Although the isospin averaged pion–nucleon scatter cross section is known to have a 4u u ρ2 u − u × − i j i j · value of about 100 mb [26], there is no empirical information on exp + cos 2 + pρ ρ ui u j ui u j the hypertriton-pion scattering cross section. Since the typical momentum of a pion at the kinetic freeze-out temperature of about 4 3/2 2 π pλ 120 MeV is about 300 MeV, corresponding to a wave-length of × 8 bmbnexp − v + v v + v m,n m n m n less than 1 fm, which is much smaller than the size of hypertriton, 194 Z. Zhang, C.M. Ko / Physics Letters B 780 (2018) 191–195

Fig. 2. Root-mean-square radius r and longitudinal length z of the expanding rms rms 3 + + + fireball as functions of the free streaming time τ . Fig. 3. (Color online.) Hyield from d and n p coalescense together with the total yield as functions of the free streaming time τ . one can approximate the hypertriton dissociation cross section by its quasi-elastic scattering cross section of about three times the thus are most likely to move together, which is especially the case for the process d + → 3 Hin which the deuteron is treated as a pion–nucleon scattering cross section, if one takes the scattering point particle and has a long tail in its relative wave function with cross section of pion and to have the same value as that for respect to the . The weak dependence of the hypertriton yield on pion and nucleon. According to Ref. [27]based on the comparison its freeze-out time suggests that although the hypertriton should of the particle scattering rate with the expansion rate of a fireball be produced at a much later time than the freeze-out times of nu- that undergoes the boost invariant longitudinal expansion with a cleon, hyperons, and deuterons, the hypertriton abundance is constant transverse expansion velocity, the volume of the fireball essentially determined when nucleons and hyperons freeze out at which a particle freezes out is related to its scattering cross sec- from the fireball, and the coalescence calculations based on parti- tion σ by cles at initial freeze-out hypersurface can still reasonably describe 3/2 the production of hypertriton in relativistic heavy ion collisions. 1 Nπ σ V f = √ , (15) 2π 3 5. Conclusions with Nπ being the number of pions in the fireball. The freeze-out volume for a hypertriton in a fireball with a fixed number of pions Using the coalescence model based on the phase-space distri- is thus about five times larger than that for a nucleon. butions of kinetically freeze-out nucleons and hyperons from If the hypertriton-pion scattering is coherent, which would be a blast-wave model, we have studied hypertriton production in a the case for very low momentum pions, its dissociation cross sec- coalescence model, which includes not only the coalescence pro- + + →3 + →3 tion would be nine times larger than that for the pion–nucleon cess p n H but also the coalescence process d H scattering, resulting in a hypertriton freeze-out volume about 27 as the hypertriton can be considered as a loosely bound state of 3 times larger than that for nucleons. On the other hand, using the deuteron and , to study the dependence of the Hyield on its geometric cross section based on the hypertriton rms radius of freeze-out time by letting nucleons, hyperons, and deuterons to R = 4.9fm, the hypertriton dissociation cross section would be stream freely after they have frozen out from the initial fireball, about 750 mb, and the resulting hypertriton freeze-out volume and then carrying out the coalescence calculations for different 3 would be about eighteen times larger than that of protons. free streaming times. We have found that the Hyield, which + To study the dependence of the hypertirton yield on its freeze- reproduces√ the experimental data from central Pb Pb collisions out time, we consider the free expansion of the fireball by letting at sNN = 2.76 TeV at the LHC with the two coalescence pro- nucleons, and deuterons to stream freely with constant veloci- cesses giving similar contributions, decreases slowly with the free ties from their initial positions given by the blast-wave model and expansion of the fireball, especially for those produced from the the coalescence model. The size of the fireball can then be deter- d + coalescence. Our result thus indicates that the hypertriton = 2 + 2 mined from the rms transverse radius rrms(τ ) x y and yield in relativistic heavy ion collisions is essentially determined 2 longitudinal length zrms(τ ) = z after the proper free stream- when nucleons and hyperons freeze out kinetically, although it ing time τ . In Fig. 2, we show the ratios rrms(τ )/rrms(0) and still undergoes scattering with the freeze-out pions. The present zrms(τ )/zrms(0) as functions of τ . It is seen that after letting conclusion is based on freeze-out nucleons and hyperons from all particles stream freely by τ = 35 fm/c, the rrms increases by a a simple blast-wave model. It will be of great interest to carry factor of 2.77 while the zrms increases by a factor 2.47, correspond- out similar studies based on more realistic models, such as the ∝ 2 hybrid hydrodynamical+Boltzmann approach [28] and transport ing to an effective volume ( rrmszrms) increase of about nineteen times larger than its initial value. models [29,30], which treat properly the freeze-out of nucleons Fig. 3 shows the dependence of the hypertriton yield on the and hyperons, to obtain a more accurate description of hypertri- free streaming time τ . It is seen that the number of hypertritons ton production in relativistic heavy ion collisions. from the d + coalescence almost remains unchanged during the free expansion of the fireball, and that from the n + p + coales- Acknowledgements cence also decreases slowly. As a result, the total hypertriton number only decreases by about 20% at τ = 35 fm/c, corresponding We thank Sungtae Cho, Su Houng Lee, Kaijia Sun, and Yifeng to an increase of the fireball volume by a factor of 19. This is due Sun for helpful discussions. This work was supported in part by to the fact that the constituent nucleons and hyperons that are the US Department of Energy under Contract No. DE-SC0015266 likely to form hypertritons have small relative velocities, and they and the Welch Foundation under Grant No. A-1358. Z. Zhang, C.M. Ko / Physics Letters B 780 (2018) 191–195 195

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