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Yield ratio of hypertriton to light nuclei in heavy-ion collisions from sNN = 4.9 GeV to 2.76 TeV

  • We argue that the difference in the yield ratio S3=N3ΛH/NΛN3He/Np measured in Au+Au collisions at sNN = 200 GeV and in Pb-Pb collisions at sNN = 2.76 TeV is mainly owing to the different treatment of the weak decay contribution to the proton yield in the Au+Au collisions at sNN = 200 GeV. We then use the coalescence model to extract from measured S3 the information about the Λ and nucleon density fluctuations at the kinetic freeze-out of heavy-ion collisions. We also show, using available experimental data, that the yield ratio S2=N3ΛHNΛNd is a more promising observable than S3 for probing the local baryon-strangeness correlation in the produced medium.
  • The correlation coefficient CBS=3BSBSS2S2 between the baryon number B and the strangeness number S in a strongly interacting matter was first proposed in Refs. [1-3] for probing the properties of the matter produced in relativistic heavy-ion collisions. A later study suggested, however, that the strangeness population factor S3=N3ΛH/NΛN3He/Np measured in these collisions could serve as a better probe of the baryon number and strangeness correlation in the produced matter, owing to its different behaviors in QGP and the hadronic matter [4, 5]. Experimentally, S3 increases from heavy-ion collisions at AGS [6] to RHIC energy [7], and then decreases to a small value in collisions at the LHC energy [8]. Compared with the predictions of the statistical model [5, 9, 10], the values of S3 extracted from the RHIC data, which has large statistical uncertainties, are larger, which led to questioning the data and its interpretation. As to the different values of S3 measured at RHIC and LHC, a possible explanation was provided in Ref. [11] by assuming an early freeze-out of Λ compared with nucleons from hadronic matter, and a longer freeze-out time difference at RHIC than at LHC. This idea was further explored in Ref. [12] to study the production of light nuclei in relativistic heavy-ion collisions by considering their finite sizes compared with the size of the produced hadronic matter at the kinetic freeze out.

    Since the first theoretical estimate (in the 1970s) of the abundance of hypernuclei that could be produced in heavy-ion collisions [13], many studies addressed this very interesting problem, providing increasingly better estimations [6-8, 14-17] (more details can be found in recent topical reviews [18-20]). For the lightest hypernucleus 3ΛH, the separation energy of its Λ was very small in early measurements, with a typical value of 130±50 keV [21], but a larger value of 410±120±110 keV has been suggested based on some recent measurements and using a more precise method [22]. Owing to this significantly smaller Λ separation energy than the nucleon separation energy in normal nuclei with a similar mass number [23], 3ΛH can be considered as a loosely bound dΛ 2-body system. In relativistic heavy-ion collisions, the hypertriton is expected to be in chemical equilibrium with Λ and deuteron as well as with proton and neutron; then, its yield is the same whether it is calculated from the coalescence of Λ with deuteron or with proton and neutron. As shown in Ref. [17] based on the coalescence model, the yields of hypertritons from these two processes are approximately the same, and the difference is owing to the simplification of taking deuteron as a point particle in this calculation. Therefore, the same 3ΛH yield appears in both S2 and S3, and the ratio S2=N3ΛHNΛNd can thus also be used as an observable for probing the correlation between baryon and strangeness in relativistic heavy-ion collisions.

    In this paper, we study the yield ratios S3=N3ΛH/NΛN3He/Np and S2=N3ΛHNΛNd in the framework of the coalescence model. We first revisit the study of S3 and find that the discrepancy between the ALICE and STAR measurements may be partially owing to the difference in the primordial proton yield used in the two analyses. We then show that the ratio S2, particularly its ratio S2/B2 with respect to B2, which is the coalescence parameter for the production of a deuteron from a proton and neutron pair, is a cleaner probe of the baryon-strangeness correlation in the produced hadronic matter from relativistic heavy-ion collisions.

    In this Section, we first review the experimental data on S3 from relativistic heavy-ion collisions and then use the coalescence model to extract from these data the correlation between Λ and nucleon density fluctuations in the produced matter.

    The value of S3was measured to be 0.36±0.26 in central 11.5A GeV/c Au + Pt collisions [6] and increased to 1.08±0.22±0.16 for Au+Au collisions at sNN = 200 GeV with a mixed event sample of a central trigger and a minimal bias trigger [7]. The measured value of S3 decreases, however, to 0.60±0.13±0.21 in central Pb-Pb collisions at sNN = 2.76 TeV [8]. Preliminary data with an improved precision from Au+Au collisions at sNN = 200 GeV revealed a 20% reduction in the value of S3 [24], which makes the results from STAR and ALICE comparable within their experimental uncertainties. The proton yield used in the analysis of the S3 ratio by the STAR Collaboration is based on the subtraction of protons from the decay of hyperons in its measurements. Replacing the proton yield in the STAR analysis with that from the PHENIX data, which is obtained from a theoretical model for the same collision system and energy [25], can also reduce the value of S3 at sNN = 200 GeV to a similar value as measured by ALICE [7, 8, 24]. The difference between the results from STAR and ALICE is thus partially owing to the different treatments in the subtraction of the weak decay contribution to the primordial proton yield. Table 1 summarizes the published S3 results, together with the value using the proton yield from the PHENIX data in Au+Au collisions at sNN = 200 GeV. It is seen that the values of S3 from STAR and ALICE are now comparable within their large uncertainties.

    Table 1

    Table 1.  Values of S3 from AGS, STAR and ALICE, with PH indicating that the proton yield is taken from PHENIX [25]. See text for details.
    experimentS3
    AGS0.36±0.26
    STAR1.08±0.22±0.16
    STAR + PH0.90±0.22±0.15
    ALICE0.60±0.13±0.21
    DownLoad: CSV
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    Experimentally, there also exists a puzzle related to the 3ΛH/3He ratio [10, 11]. Its value is 0.82±0.16±0.12 in the 0-80% centrality of Au+Au collisions at sNN = 200 GeV at RHIC [7], which is considerably larger than the value of 0.47±0.10±0.13 in Pb-Pb collisions at sNN = 2.76 TeV and the 0-10% centrality at the LHC [8]. Although a preliminary measurement with improved precision by STAR in Au+Au collisions at sNN = 200 GeV revealed a reduction in the 3ΛH/3He ratio to a value comparable with that assessed by the ALICE study, its large uncertainty [26] indicates that more precise measurements of 3ΛH in high energy heavy-ion collisions are needed.

    To demonstrate the physics that can be extracted from the ratio S3, we adopted the coalescence model for the present study. According to the coalescence formula denoted as COAL-SH in Ref. [27], the yield of a certain nucleus consisting of Ni constituent species i (proton, neutron, and Λ) of mass mi from the kinetically frozen-out hadronic matter of local temperature TK and volume VK in a heavy-ion collision can be written as

    NA=grelgsizegA(Aimi)3/2[Ai=1Nim3/2i]×A1i=1(4π/ω)3/2VKx(1+x2)(x21+x2)liG(li,x).

    (1)

    In the above, gA=(2S+1)/(Ai=1(2si+1)) is the statistical factor for A nucleons and/or Λ of spin si=1/2 to form a nucleus of spin S; grel is the relativistic correction to the effective volume in the momentum space and is set to 1 in the present study, owing to the much larger nucleon mass than the effective temperature of the hadronic matter at kinetic freeze-out; and gsize is the correction owing to the finite size of the produced nucleus and is also taken to be 1 in our study because of the much larger size of the hadronic matter than the sizes of produced light nuclei. The symbols li and ω denote, respectively, the orbital angular momentum of the nucleon or Λ in the nucleus and the oscillator constant used in its wave function. The value of x=(2TK/ω)1/2 is significantly larger than one because of the much larger size of the nucleus than the thermal wavelength of its constituents in the hadronic matter. Since the light nuclei considered in the present study all involve only the l=0 s-wave, the suppression factor G(li,x) owing to the orbital angular momentum in the above equation is simply one.

    For the production of 3ΛH and 3He, their yields according to Eq. (1), after taking into account the approximations mentioned above, are given by

    N3ΛH=g3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K,N3He=g3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K.

    (2)

    According to Refs. [28, 29], including possible nucleon and Λ density fluctuations in heavy-ion collisions at lower energies owing to the spinodal instability during the QGP to hadronic matter phase transition [30-32], modifies the above equations to

    N3ΛHg3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K(1+αΛp+αΛn+αnp),N3Heg3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K(1+Δp+2αnp).

    (3)

    In the above, the proton relative density fluctuation is denoted by Δp=(δp)2/p2, where

    p=1VKnp(r)dr,(δp)2=1VK[np(r)np]2dr,

    (4)

    with np(r) being the proton density distribution. The quantities αΛp, αΛn, and αnp are, respectively, the Λ-proton, Λ-neutron, and proton-neutron density fluctuation correlation coefficients αn1n2=δn1δn2/(n1n2) with n1 and n2 denoting Λ or a nucleon.

    Taking the same masses for proton and neutron, i.e., mp=mn=m, the yield ratio S3=N3ΛH/NΛN3He/Np is then

    S3=g1+αΛp+αΛn+αnp1+Δp+2αnp,

    (5)

    with g=(mΛ+2m3mΛ)3/20.845. From the above equation, the sum of the correlation coefficients αΛp+αΛn between the Λ density and the proton or neutron density fluctuations can be determined from S3, Δp, and αnp according to

    αΛp+αΛn=S3g×(1+Δp+2αnp)αnp1.

    (6)

    For the value of αnp, we follow the method in Ref. [29] by using the deuteron yield after including in the coalescence formula COAL-SH the proton and neutron density fluctuations, i.e.,

    Nd=23/2gd(2πmTK)3/2NpNnVK(1+αnp).

    (7)

    In terms of gdp=123/2gd(2π)3=21/23(2π)30.0019, Odp=Nd/N2p, Rnp=Np/Nn, and Vph=(2πmTK)3/2VK, the value of αnp can then be calculated from

    αnp=gdpRnpVphOdp1.

    (8)

    For the proton density fluctuation Δp, we consider the ratio

    N3HeN3p×NpNn=33/2g3He(2πmTK)31V2K(1+Δp+2αnp),

    (9)

    and determine it from the relation

    Δp=g3HepV2phRnpO3Hep2αnp1,

    (10)

    where g3Hep=133/2g3He(2π)6=433/2(2π)61.25×105 and O3Hep=N3He/N3p, and the factors Vph, Rnp and αnp are the same as above.

    The effective phase-space volume Vph occupied by nucleons in the hadronic matter at kinetic freeze-out can be evaluated from its value at chemical freeze-out, using the relation T3/2KVK=λT3/2chVch, where Tch and Vch are, respectively, the temperature and volume of the system at the chemical freeze-out, and λ is a parameter. For collisions at RHIC energies, we take the value of Tch from the grand canonical ensemble fits to the particle yields in Ref. [33] and that of Vch to be Vch=4πR3/3 as in Ref. [33] for collisions at various centralities, except for collisions at the 0-80% centrality, where it is taken to be proportional to the charged particle multiplicity obtained from Ref. [34]. Using the results from Ref. [33] based on the strangeness suppressed canonical ensemble fits gives almost the same results. The different values of Vch extracted from collisions at sNN = 11.5 and 19.6 GeV are owing to the neglect of experimental uncertainties in our analysis [33]. The values of Tch and Vch used in the present study for collisions at the AGS energy are taken from Ref. [35], and for collisions at the LHC energy, they are taken from the COAL-SH model used in Ref. [27]. In our calculations, all hadrons are taken as point particles. Including an exclusive volume for each hadron increases Vch [35]. The value of λ is determined by assuming that the entropy associated with a single nucleon is the same at the chemical and the kinetic freeze-outs. Explicitly, we use the well-known expression for the entropy associated with a single nucleon in a system in thermal equilibrium, i.e., S/N=5/2+ln(Vph/N), where Vph=(2πmT)3/2V is the phase-space volume of N nucleons of mass m in the system. The constancy of S/N then requires T3/2chVch/Nch=T3/2KVK/NK , with Nch and NK being the numbers of nucleons at the chemical and kinetic freeze-outs, respectively, which then leads to λ=NK/Nch. The values given in Table 2 for λ are obtained from the value NK measured in experiments and the value Nch given by the statistical hadronization model that includes all the resonances in PDG [36]. Our approach differs from the naive approach of a hadronic matter of constant number of nucleons expanding with a constant total nucleon entropy after a chemical freeze-out, which would give λ the value of unity, and also that in Ref. [29] based on a multiphase transport model that only includes a small number of resonances and thus gives a smaller value of λ1.6. We note that the below results of our study are not qualitatively affected by these variations in λ. As to the value of Rnp=Np/Nn, it can be determined from the measured ratio of charged pions according to the relation Np/Nn=(Nπ+/Nπ)1/2 from the statistical model. In Table 2, we summarize the values of the above parameters for central heavy-ion collisions. For reference, we also provide in Table 3 their values for collisions at the centralities of 0-80% or 0-60%.

    Table 2

    Table 2.  Values of parameters used for 0-10% central collisions.
    sNN/GeVTch/GeVVch/fm3Rnpαnpλ
    4.90.1326400.9250.781±0.0262.23
    7.70.1448060.9660.744±0.0242.60
    11.50.1518750.9770.763±0.0192.76
    19.60.1588430.9870.830±0.0142.92
    270.1608460.9880.848±0.0122.97
    390.1609510.9900.834±0.0133.00
    62.40.16412150.9920.792±0.0373.16
    2000.16813340.9920.726±0.0383.30
    27600.15643201.000.717±0.0232.94
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    Table 3

    Table 3.  Same as Table 2 for values of parameters used in the calculations for 0-80% centrality in collisions at RHIC energies and for 0-60% centrality at the LHC energy.
    sNN/GeVTch/GeVVch/fm3Rnpαnpλ
    7.70.1442680.9660.775±0.0202.60
    11.50.1512920.9770.793±0.0182.76
    19.60.1582810.9870.851±0.0142.92
    270.1602820.9880.867±0.0122.97
    390.1603170.9900.859±0.0133.00
    2000.1684450.9920.747±0.0363.30
    27600.15618001.000.710±0.0242.94
    DownLoad: CSV
    Show Table

    We note that if we have assumed instead that the total entropy is the same at the chemical and kinetic freeze-outs, i.e., T3KVK=T3chVch [37], a somewhat smaller VK would have been obtained. Since it has been shown in Ref. [38] that the total entropy increases from the chemical to the kinetic freeze-out, and the entropy associated with a baryon is essentially constant, we adopt in the present study the condition T3/2KVK=λT3/2chVch of constant entropy associated with a nucleon. Although the values of TK are not used in our calculations, it is useful to show them as references. According to Ref. [33] based on a blast-wave model fit to measured proton, pion, and kaon transverse momentum spectra, the values of TK are 126, 117, 119, 114, 116, 118, 88, 88 MeV for central heavy-ion collisions at sNN = 4.9, 7.7, 11.5, 19.6, 27, 39, 200, and 2760 GeV, respectively.

    As shown in Tables 2 and 3, the extracted values for the neutron and proton density fluctuation correlation αnp are negative with appreciable magnitude, indicating that neutrons and protons are anti-correlated in the matter produced in relativistic heavy-ion collisions, similar to the findings in Ref. [29]. For the proton density fluctuation Δp, the extracted values are 0.656 ± 0.049, 0.536 ± 0.083, and 0.727 ± 0.085 for collisions at sNN = 4.9, 200, and 2760 GeV, respectively [6, 7, 39]. It shows a non-monotonic behavior as a function of the collision energy, from sNN = 7.7 GeV to 200 GeV, using the preliminary data of 3He yield from Ref. [24], similar to that of the neutron density fluctuation extracted from the yield ratio NtNpN2d [40]. From the measured values of S3 and extracted values for αnp and  Δp, one can then determine the values of αΛn+αΛp from heavy-ion collisions at various energies, according to Eq. (6). These results will be shown in the next Section, to compare with the correlation coefficient αΛd of the Λ and deuteron density fluctuations that is extracted from measured S2=N3ΛHNΛNd ratio.

    In this Section, we first consider the S2=N3ΛHNΛNd ratio in the framework of the coalescence model. Its ratio S2/B2 with respect to the coalescence parameter B2 for the production of a deuteron from the coalescence of a proton and a neutron is then studied. Based on the experimental data from the RHIC and LHC, we further extract the correlation coefficient αΛd between the Λ and deuteron density fluctuations.

    Approximating 3ΛH as a bound system of Λand a deuteron, the yield of 3ΛH in heavy-ion collisions can be calculated from the coalescence of Λ and a deuteron using Eq. (1). Including the effect of the deuteron and Λ density fluctuations, it is given by

    N3ΛH=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/2NΛNdVK(1+αΛd),

    (11)

    with αΛd=δnΛδnd/(nΛnd) being the correlation coefficient between the deuteron and Λ density fluctuations. The S2 ratio is then

    S2=N3ΛHNΛNd=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/21VK(1+αΛd),

    (12)

    with gS2=(13(mΛ+md)3/2m3/2Λm3/2d(2π)3/2)10.12, from which we can express the density fluctuation correlation coefficient αΛd in terms of S2 as

    αΛd=gS2S2T3/2KVK1.

    (13)

    In the left plot of Fig. 1, we show by solid symbols the extracted ratio S2=N3ΛHNΛNd using the experimental data from AGS [6, 41, 42], RHIC [7, 43, 44], and LHC [8, 39, 45]. Open symbols represent the results for the 3ΛH yield in the collision at the 0-10% centrality, where the RHIC data are obtained from multiplying the measured 3ΛH data at the 0-80% centrality by a factor of 3. We have checked, using data available from the RHIC BES program, that the deuteron to Λ yield ratio is larger by a factor of 3 for collisions at the 0-10% centrality than at the 0-80% centrality, independent of the collision energy [43, 46]. Solid and dashed lines in the left window of Fig. 1 are results from calculations based on the statistical model of Refs. [47-49] as in Ref. [50] using the parameters in Tables 2 and 3 and after taking into account the feed-down correction to the Λ yield. Clearly, although there are missing data points in the large collision energy range, the ratio S2 in collisions at the 0-10% centrality seems to be independent of the collision energy sNN considering the large uncertainty of the AGS data. In addition, the model calculation describes reasonably well the data at the LHC energy, but over-predicts the results for collisions at the AGS and RHIC energies.

    Figure 1

    Figure 1.  (color online) Collision energy dependence of the S2=N3ΛHNΛNd ratio (left) and the density fluctuation correlation coefficients αΛp+αΛn and αΛd (right) extracted from the experimental data (symbols) and calculated from the statistical model (horizontal bars) using parameters in Tables 2 and 3. See text for details.

    The right plot in Fig. 1 shows the value of the correlation coefficients αΛd as a function of the collision energy for collisions at the 0-10% centrality (solid symbols) and at the 0-80% centrality (open symbols). The large uncertainty at sNN = 2.76 TeV is owing to the propagation of the standard error from the large volume Vch used in collisions at the LHC energy. Within current experimental uncertainties, the value of αΛd, which is negative and thus indicates an anti-correlation between the Λ and deuteron density fluctuations, becomes slightly less negative as the collision energy increases and approaches zero at sNN = 2.76 TeV. The negative αΛd and the negative αnp, shown in Tables 2 and 3, respectively, could be owing to the underestimation of the value of the λ parameter or the kinetic freeze-out volume used in our study. Full understanding of these results requires detailed studies based on the microscopic models of light cluster production in high-energy heavy-ion collisions [51, 52], which is, however, beyond the scope of the present study. Compared with the correlation coefficient αΛp+αΛn extracted from S3, which seems to vary very little over a broad range of collision energies, the value of αΛd shows a more visible sNN dependence. Also, the deviation of αΛp+αΛn from zero is larger than that of αΛp+αΛn. This may suggest that αΛd is a cleaner observable than αΛp+αΛn for studying the sNN dependence of baryon density fluctuations and their correlations, as seen from the comparison of Eq. (12) to Eq. (5). Future experimental measurements in a broad range of collision energies from AGS to RHIC will be very useful for shedding light on the underlying physics.

    In the coalescence model, the yield ratio S2=N3ΛH/(NΛNd) is the coalescence parameter for the production of 3ΛH if it is considered as a bound system of Λ and a deuteron. Because of the strangeness carried by Λ, the S2 may be different from the coalescence parameter B2 for the deuteron production following the coalescence of a proton and a neutron [19, 53-57].

    From Eq. (12) for S2 and a similar equation for B2, given by

    B2=NdNpNn=gd1m3/2p(2πTK)3/21VK(1+αnp),

    (14)

    taking Nn=Np then leads to

    S2B2=N3ΛHNΛNd/NdNpNn=g1+αΛd1+αnp,

    (15)

    where g=g3ΛHgdm3/2p(mΛ+md)3/2m3/2Λm3/2d0.23. The ratio S2/B2 thus carries information about the difference between αnp and αΛd and thus about the difference between the baryon-baryon correlation and the baryon-strangeness correlation. We note that the B2 coalescence parameter here refers to the ratio of integrated yields, whereas in the literature it is determined differentially in momentum [55, 58-60].

    Similarly, we can introduce the coalescence parameter B3=N3HeNpNpNn for the production of 3He from the three-body coalescence of two protons and a neutron, and the coalescence parameter Bs3=N3ΛHNΛNpNn for the production of 3ΛH from the three-body coalescence of Λ, a proton, and a neutron. Their ratio is exactly the value of S3 , as discussed in Section 2. Since the ratio S2/B2 does not involve the proton density fluctuation Δp and other mixed density fluctuation correlations, it seems a more sensitive observable than S3 for studying the Λ density fluctuation.

    The left plot in Fig. 2 shows the results for the yield ratio S2/B2 = N3ΛH/(NΛNd)/(Nd/N2p) from experimental data (solid triangles) and those predicted by the statistical model (solid horizontal bars) using the proton, deuteron, and 3ΛH yields from the full pT range, and including the feed-down correction for the proton yield. It is seen that the measured yield ratio increases slightly with increasing collision energy, as predicted by the statistical model. We note that the value of B2 has also been determined in experiments from the proton and deuteron momentum spectra in a small pT window [43]. The S2/B2 ratio obtained from collisions at the LHC energy for momentum per constituent pT/A=1.4GeV/c [8, 39, 45] is 0.899±0.171. Within their uncertainties, this value is similar to that obtained using yields from the full pT range. However, the S2/B2 ratios measured for different pT bins are unavailable from experiments in the energy range available at AGS and RHIC, and this is owing to the lack of 3ΛHpT spectra at these energies. Future measurements of 3ΛH spectra over a broad range of energies are needed for extracting the Λ and deuteron density correlation coefficient αΛd discussed below.

    Figure 2

    Figure 2.  (color online) Left: The S2/B2 ratio extracted from experimental data (solid triangles) and predicted by the statistical model (solid horizontal bars) for collision at the 0-10% centrality [39, 43]. Right: Values of αΛd extracted from experimental results according to Eqs. (16) and (13).

    The Λ and deuteron correlation coefficient αΛd can also be extracted from the yield ratio S2/B2 given in Eq. (15), that is

    αΛd=(N3ΛHNΛNd(Nd/N2p)/g)×(1+αnp)1,

    (16)

    by taking advantage of the empirical fact that the neutron and proton density fluctuation correlation αnp is less affected by TK and VK. Shown in the right plot of Fig. 2 by triangles are the values of αΛd extracted from the experimental results using Eq. (16). They are seen to have similar values to those obtained from Eq. (13) using S2=N3ΛHNΛNd, which are shown by closed circles and also in Fig. 1 where it is compared with the density fluctuation correlation coefficient αΛp+αΛn extracted from S3=N3ΛH/NΛN3He/Np.

    In summary, we have argued that both the ratio S2 and the ratio S2/B2, where S2 and B2 are, respectively, the coalescence parameter for the production of hypertriton from Λ and a deuteron, and of a deuteron from a proton and a neutron, are more sensitive observables than the previously proposed ratio S3 = N3ΛH/NΛN3He/Np for studying the local baryon-strangeness correlation in the matter produced in relativistic heavy-ion collisions. We have substantiated this argument in the framework of baryon coalescence by demonstrating that the correlation coefficient αΛd between Λ and deuteron density fluctuations extracted from measured S2/B2 shows a stronger dependence on the energy of heavy-ion collisions than the correlation coefficients αΛp+αΛn between Λ and nucleon density fluctuations extracted from the measured S3. Although the results in the present study are obtained without including the feed-down contribution to nucleons from Δ resonances, they will not be qualitatively affected because of the low kinetic freeze-out temperature of ~100 MeV, which only contribues ~20% to the nucleon yield. Experimental measurements of the ratio S2/B2 are expected to provide a promising way to study the strangeness and baryon correlation in the matter produced from heavy-ion collisions as the collision energy or the baryon chemical potential of produced matter is varied, which in turn can shed light on the properties of the QGP to hadronic matter phase transition during collisions.

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Tianhao Shao, Jinhui Chen, Che Ming Ko, Kai-Jia Sun and Zhangbu Xu. Yield ratio of hypertriton to light nuclei in heavy-ion collisions from sNN = 4.9 GeV to 2.76 TeV[J]. Chinese Physics C. doi: 10.1088/1674-1137/abadf0
Tianhao Shao, Jinhui Chen, Che Ming Ko, Kai-Jia Sun and Zhangbu Xu. Yield ratio of hypertriton to light nuclei in heavy-ion collisions from sNN = 4.9 GeV to 2.76 TeV[J]. Chinese Physics C.  doi: 10.1088/1674-1137/abadf0 shu
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Yield ratio of hypertriton to light nuclei in heavy-ion collisions from sNN = 4.9 GeV to 2.76 TeV

  • 1. Key Laboratory of Nuclear Physics and Ion-beam Application (MOE), Institute of Modern Physics, Fudan University, Shanghai 200433, China
  • 2. Shanghai Institute of Applied Physics, Chinese Academy of Science, Shanghai 201800, China
  • 3. University of Chinese Academy of Science, Beijing 100049, China
  • 4. Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA
  • 5. Brookhaven National Laboratory, Upton, New York 11973, USA

Abstract: We argue that the difference in the yield ratio S3=N3ΛH/NΛN3He/Np measured in Au+Au collisions at sNN = 200 GeV and in Pb-Pb collisions at sNN = 2.76 TeV is mainly owing to the different treatment of the weak decay contribution to the proton yield in the Au+Au collisions at sNN = 200 GeV. We then use the coalescence model to extract from measured S3 the information about the Λ and nucleon density fluctuations at the kinetic freeze-out of heavy-ion collisions. We also show, using available experimental data, that the yield ratio S2=N3ΛHNΛNd is a more promising observable than S3 for probing the local baryon-strangeness correlation in the produced medium.

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    1.   Introduction
    • The correlation coefficient CBS=3BSBSS2S2 between the baryon number B and the strangeness number S in a strongly interacting matter was first proposed in Refs. [1-3] for probing the properties of the matter produced in relativistic heavy-ion collisions. A later study suggested, however, that the strangeness population factor S3=N3ΛH/NΛN3He/Np measured in these collisions could serve as a better probe of the baryon number and strangeness correlation in the produced matter, owing to its different behaviors in QGP and the hadronic matter [4, 5]. Experimentally, S3 increases from heavy-ion collisions at AGS [6] to RHIC energy [7], and then decreases to a small value in collisions at the LHC energy [8]. Compared with the predictions of the statistical model [5, 9, 10], the values of S3 extracted from the RHIC data, which has large statistical uncertainties, are larger, which led to questioning the data and its interpretation. As to the different values of S3 measured at RHIC and LHC, a possible explanation was provided in Ref. [11] by assuming an early freeze-out of Λ compared with nucleons from hadronic matter, and a longer freeze-out time difference at RHIC than at LHC. This idea was further explored in Ref. [12] to study the production of light nuclei in relativistic heavy-ion collisions by considering their finite sizes compared with the size of the produced hadronic matter at the kinetic freeze out.

      Since the first theoretical estimate (in the 1970s) of the abundance of hypernuclei that could be produced in heavy-ion collisions [13], many studies addressed this very interesting problem, providing increasingly better estimations [6-8, 14-17] (more details can be found in recent topical reviews [18-20]). For the lightest hypernucleus 3ΛH, the separation energy of its Λ was very small in early measurements, with a typical value of 130±50 keV [21], but a larger value of 410±120±110 keV has been suggested based on some recent measurements and using a more precise method [22]. Owing to this significantly smaller Λ separation energy than the nucleon separation energy in normal nuclei with a similar mass number [23], 3ΛH can be considered as a loosely bound dΛ 2-body system. In relativistic heavy-ion collisions, the hypertriton is expected to be in chemical equilibrium with Λ and deuteron as well as with proton and neutron; then, its yield is the same whether it is calculated from the coalescence of Λ with deuteron or with proton and neutron. As shown in Ref. [17] based on the coalescence model, the yields of hypertritons from these two processes are approximately the same, and the difference is owing to the simplification of taking deuteron as a point particle in this calculation. Therefore, the same 3ΛH yield appears in both S2 and S3, and the ratio S2=N3ΛHNΛNd can thus also be used as an observable for probing the correlation between baryon and strangeness in relativistic heavy-ion collisions.

      In this paper, we study the yield ratios S3=N3ΛH/NΛN3He/Np and S2=N3ΛHNΛNd in the framework of the coalescence model. We first revisit the study of S3 and find that the discrepancy between the ALICE and STAR measurements may be partially owing to the difference in the primordial proton yield used in the two analyses. We then show that the ratio S2, particularly its ratio S2/B2 with respect to B2, which is the coalescence parameter for the production of a deuteron from a proton and neutron pair, is a cleaner probe of the baryon-strangeness correlation in the produced hadronic matter from relativistic heavy-ion collisions.

    2.   The S3=N3ΛH/NΛN3He/Np ratio in relativistic heavy-ion collisions
    • In this Section, we first review the experimental data on S3 from relativistic heavy-ion collisions and then use the coalescence model to extract from these data the correlation between Λ and nucleon density fluctuations in the produced matter.

    • 2.1.   Experimental results on S3

    • The value of S3was measured to be 0.36±0.26 in central 11.5A GeV/c Au + Pt collisions [6] and increased to 1.08±0.22±0.16 for Au+Au collisions at sNN = 200 GeV with a mixed event sample of a central trigger and a minimal bias trigger [7]. The measured value of S3 decreases, however, to 0.60±0.13±0.21 in central Pb-Pb collisions at sNN = 2.76 TeV [8]. Preliminary data with an improved precision from Au+Au collisions at sNN = 200 GeV revealed a 20% reduction in the value of S3 [24], which makes the results from STAR and ALICE comparable within their experimental uncertainties. The proton yield used in the analysis of the S3 ratio by the STAR Collaboration is based on the subtraction of protons from the decay of hyperons in its measurements. Replacing the proton yield in the STAR analysis with that from the PHENIX data, which is obtained from a theoretical model for the same collision system and energy [25], can also reduce the value of S3 at sNN = 200 GeV to a similar value as measured by ALICE [7, 8, 24]. The difference between the results from STAR and ALICE is thus partially owing to the different treatments in the subtraction of the weak decay contribution to the primordial proton yield. Table 1 summarizes the published S3 results, together with the value using the proton yield from the PHENIX data in Au+Au collisions at sNN = 200 GeV. It is seen that the values of S3 from STAR and ALICE are now comparable within their large uncertainties.

      experimentS3
      AGS0.36±0.26
      STAR1.08±0.22±0.16
      STAR + PH0.90±0.22±0.15
      ALICE0.60±0.13±0.21

      Table 1.  Values of S3 from AGS, STAR and ALICE, with PH indicating that the proton yield is taken from PHENIX [25]. See text for details.

      Experimentally, there also exists a puzzle related to the 3ΛH/3He ratio [10, 11]. Its value is 0.82±0.16±0.12 in the 0-80% centrality of Au+Au collisions at sNN = 200 GeV at RHIC [7], which is considerably larger than the value of 0.47±0.10±0.13 in Pb-Pb collisions at sNN = 2.76 TeV and the 0-10% centrality at the LHC [8]. Although a preliminary measurement with improved precision by STAR in Au+Au collisions at sNN = 200 GeV revealed a reduction in the 3ΛH/3He ratio to a value comparable with that assessed by the ALICE study, its large uncertainty [26] indicates that more precise measurements of 3ΛH in high energy heavy-ion collisions are needed.

    • 2.2.   S3 in the coalescence model and the nucleon and Λ density fluctuations

    • To demonstrate the physics that can be extracted from the ratio S3, we adopted the coalescence model for the present study. According to the coalescence formula denoted as COAL-SH in Ref. [27], the yield of a certain nucleus consisting of Ni constituent species i (proton, neutron, and Λ) of mass mi from the kinetically frozen-out hadronic matter of local temperature TK and volume VK in a heavy-ion collision can be written as

      NA=grelgsizegA(Aimi)3/2[Ai=1Nim3/2i]×A1i=1(4π/ω)3/2VKx(1+x2)(x21+x2)liG(li,x).

      (1)

      In the above, gA=(2S+1)/(Ai=1(2si+1)) is the statistical factor for A nucleons and/or Λ of spin si=1/2 to form a nucleus of spin S; grel is the relativistic correction to the effective volume in the momentum space and is set to 1 in the present study, owing to the much larger nucleon mass than the effective temperature of the hadronic matter at kinetic freeze-out; and gsize is the correction owing to the finite size of the produced nucleus and is also taken to be 1 in our study because of the much larger size of the hadronic matter than the sizes of produced light nuclei. The symbols li and ω denote, respectively, the orbital angular momentum of the nucleon or Λ in the nucleus and the oscillator constant used in its wave function. The value of x=(2TK/ω)1/2 is significantly larger than one because of the much larger size of the nucleus than the thermal wavelength of its constituents in the hadronic matter. Since the light nuclei considered in the present study all involve only the l=0 s-wave, the suppression factor G(li,x) owing to the orbital angular momentum in the above equation is simply one.

      For the production of 3ΛH and 3He, their yields according to Eq. (1), after taking into account the approximations mentioned above, are given by

      N3ΛH=g3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K,N3He=g3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K.

      (2)

      According to Refs. [28, 29], including possible nucleon and Λ density fluctuations in heavy-ion collisions at lower energies owing to the spinodal instability during the QGP to hadronic matter phase transition [30-32], modifies the above equations to

      N3ΛHg3ΛH(mΛ+mp+mn)3/2m3/2Λm3/2pm3/2n(2πTK)3NΛNpNnV2K(1+αΛp+αΛn+αnp),N3Heg3He(2mp+mn)3/2m3pm3/2n(2πTK)3N2pNnV2K(1+Δp+2αnp).

      (3)

      In the above, the proton relative density fluctuation is denoted by Δp=(δp)2/p2, where

      p=1VKnp(r)dr,(δp)2=1VK[np(r)np]2dr,

      (4)

      with np(r) being the proton density distribution. The quantities αΛp, αΛn, and αnp are, respectively, the Λ-proton, Λ-neutron, and proton-neutron density fluctuation correlation coefficients αn1n2=δn1δn2/(n1n2) with n1 and n2 denoting Λ or a nucleon.

      Taking the same masses for proton and neutron, i.e., mp=mn=m, the yield ratio S3=N3ΛH/NΛN3He/Np is then

      S3=g1+αΛp+αΛn+αnp1+Δp+2αnp,

      (5)

      with g=(mΛ+2m3mΛ)3/20.845. From the above equation, the sum of the correlation coefficients αΛp+αΛn between the Λ density and the proton or neutron density fluctuations can be determined from S3, Δp, and αnp according to

      αΛp+αΛn=S3g×(1+Δp+2αnp)αnp1.

      (6)

      For the value of αnp, we follow the method in Ref. [29] by using the deuteron yield after including in the coalescence formula COAL-SH the proton and neutron density fluctuations, i.e.,

      Nd=23/2gd(2πmTK)3/2NpNnVK(1+αnp).

      (7)

      In terms of gdp=123/2gd(2π)3=21/23(2π)30.0019, Odp=Nd/N2p, Rnp=Np/Nn, and Vph=(2πmTK)3/2VK, the value of αnp can then be calculated from

      αnp=gdpRnpVphOdp1.

      (8)

      For the proton density fluctuation Δp, we consider the ratio

      N3HeN3p×NpNn=33/2g3He(2πmTK)31V2K(1+Δp+2αnp),

      (9)

      and determine it from the relation

      Δp=g3HepV2phRnpO3Hep2αnp1,

      (10)

      where g3Hep=133/2g3He(2π)6=433/2(2π)61.25×105 and O3Hep=N3He/N3p, and the factors Vph, Rnp and αnp are the same as above.

      The effective phase-space volume Vph occupied by nucleons in the hadronic matter at kinetic freeze-out can be evaluated from its value at chemical freeze-out, using the relation T3/2KVK=λT3/2chVch, where Tch and Vch are, respectively, the temperature and volume of the system at the chemical freeze-out, and λ is a parameter. For collisions at RHIC energies, we take the value of Tch from the grand canonical ensemble fits to the particle yields in Ref. [33] and that of Vch to be Vch=4πR3/3 as in Ref. [33] for collisions at various centralities, except for collisions at the 0-80% centrality, where it is taken to be proportional to the charged particle multiplicity obtained from Ref. [34]. Using the results from Ref. [33] based on the strangeness suppressed canonical ensemble fits gives almost the same results. The different values of Vch extracted from collisions at sNN = 11.5 and 19.6 GeV are owing to the neglect of experimental uncertainties in our analysis [33]. The values of Tch and Vch used in the present study for collisions at the AGS energy are taken from Ref. [35], and for collisions at the LHC energy, they are taken from the COAL-SH model used in Ref. [27]. In our calculations, all hadrons are taken as point particles. Including an exclusive volume for each hadron increases Vch [35]. The value of λ is determined by assuming that the entropy associated with a single nucleon is the same at the chemical and the kinetic freeze-outs. Explicitly, we use the well-known expression for the entropy associated with a single nucleon in a system in thermal equilibrium, i.e., S/N=5/2+ln(Vph/N), where Vph=(2πmT)3/2V is the phase-space volume of N nucleons of mass m in the system. The constancy of S/N then requires T3/2chVch/Nch=T3/2KVK/NK , with Nch and NK being the numbers of nucleons at the chemical and kinetic freeze-outs, respectively, which then leads to λ=NK/Nch. The values given in Table 2 for λ are obtained from the value NK measured in experiments and the value Nch given by the statistical hadronization model that includes all the resonances in PDG [36]. Our approach differs from the naive approach of a hadronic matter of constant number of nucleons expanding with a constant total nucleon entropy after a chemical freeze-out, which would give λ the value of unity, and also that in Ref. [29] based on a multiphase transport model that only includes a small number of resonances and thus gives a smaller value of λ1.6. We note that the below results of our study are not qualitatively affected by these variations in λ. As to the value of Rnp=Np/Nn, it can be determined from the measured ratio of charged pions according to the relation Np/Nn=(Nπ+/Nπ)1/2 from the statistical model. In Table 2, we summarize the values of the above parameters for central heavy-ion collisions. For reference, we also provide in Table 3 their values for collisions at the centralities of 0-80% or 0-60%.

      sNN/GeVTch/GeVVch/fm3Rnpαnpλ
      4.90.1326400.9250.781±0.0262.23
      7.70.1448060.9660.744±0.0242.60
      11.50.1518750.9770.763±0.0192.76
      19.60.1588430.9870.830±0.0142.92
      270.1608460.9880.848±0.0122.97
      390.1609510.9900.834±0.0133.00
      62.40.16412150.9920.792±0.0373.16
      2000.16813340.9920.726±0.0383.30
      27600.15643201.000.717±0.0232.94

      Table 2.  Values of parameters used for 0-10% central collisions.

      sNN/GeVTch/GeVVch/fm3Rnpαnpλ
      7.70.1442680.9660.775±0.0202.60
      11.50.1512920.9770.793±0.0182.76
      19.60.1582810.9870.851±0.0142.92
      270.1602820.9880.867±0.0122.97
      390.1603170.9900.859±0.0133.00
      2000.1684450.9920.747±0.0363.30
      27600.15618001.000.710±0.0242.94

      Table 3.  Same as Table 2 for values of parameters used in the calculations for 0-80% centrality in collisions at RHIC energies and for 0-60% centrality at the LHC energy.

      We note that if we have assumed instead that the total entropy is the same at the chemical and kinetic freeze-outs, i.e., T3KVK=T3chVch [37], a somewhat smaller VK would have been obtained. Since it has been shown in Ref. [38] that the total entropy increases from the chemical to the kinetic freeze-out, and the entropy associated with a baryon is essentially constant, we adopt in the present study the condition T3/2KVK=λT3/2chVch of constant entropy associated with a nucleon. Although the values of TK are not used in our calculations, it is useful to show them as references. According to Ref. [33] based on a blast-wave model fit to measured proton, pion, and kaon transverse momentum spectra, the values of TK are 126, 117, 119, 114, 116, 118, 88, 88 MeV for central heavy-ion collisions at sNN = 4.9, 7.7, 11.5, 19.6, 27, 39, 200, and 2760 GeV, respectively.

      As shown in Tables 2 and 3, the extracted values for the neutron and proton density fluctuation correlation αnp are negative with appreciable magnitude, indicating that neutrons and protons are anti-correlated in the matter produced in relativistic heavy-ion collisions, similar to the findings in Ref. [29]. For the proton density fluctuation Δp, the extracted values are 0.656 ± 0.049, 0.536 ± 0.083, and 0.727 ± 0.085 for collisions at sNN = 4.9, 200, and 2760 GeV, respectively [6, 7, 39]. It shows a non-monotonic behavior as a function of the collision energy, from sNN = 7.7 GeV to 200 GeV, using the preliminary data of 3He yield from Ref. [24], similar to that of the neutron density fluctuation extracted from the yield ratio NtNpN2d [40]. From the measured values of S3 and extracted values for αnp and  Δp, one can then determine the values of αΛn+αΛp from heavy-ion collisions at various energies, according to Eq. (6). These results will be shown in the next Section, to compare with the correlation coefficient αΛd of the Λ and deuteron density fluctuations that is extracted from measured S2=N3ΛHNΛNd ratio.

    3.   The S2=N3ΛHNΛNd ratio
    • In this Section, we first consider the S2=N3ΛHNΛNd ratio in the framework of the coalescence model. Its ratio S2/B2 with respect to the coalescence parameter B2 for the production of a deuteron from the coalescence of a proton and a neutron is then studied. Based on the experimental data from the RHIC and LHC, we further extract the correlation coefficient αΛd between the Λ and deuteron density fluctuations.

    • 3.1.   The S2=N3ΛHNΛNd ratio in the coalescence model

    • Approximating 3ΛH as a bound system of Λand a deuteron, the yield of 3ΛH in heavy-ion collisions can be calculated from the coalescence of Λ and a deuteron using Eq. (1). Including the effect of the deuteron and Λ density fluctuations, it is given by

      N3ΛH=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/2NΛNdVK(1+αΛd),

      (11)

      with αΛd=δnΛδnd/(nΛnd) being the correlation coefficient between the deuteron and Λ density fluctuations. The S2 ratio is then

      S2=N3ΛHNΛNd=g3ΛH(mΛ+md)3/2m3/2Λm3/2d(2πTK)3/21VK(1+αΛd),

      (12)

      with gS2=(13(mΛ+md)3/2m3/2Λm3/2d(2π)3/2)10.12, from which we can express the density fluctuation correlation coefficient αΛd in terms of S2 as

      αΛd=gS2S2T3/2KVK1.

      (13)

      In the left plot of Fig. 1, we show by solid symbols the extracted ratio S2=N3ΛHNΛNd using the experimental data from AGS [6, 41, 42], RHIC [7, 43, 44], and LHC [8, 39, 45]. Open symbols represent the results for the 3ΛH yield in the collision at the 0-10% centrality, where the RHIC data are obtained from multiplying the measured 3ΛH data at the 0-80% centrality by a factor of 3. We have checked, using data available from the RHIC BES program, that the deuteron to Λ yield ratio is larger by a factor of 3 for collisions at the 0-10% centrality than at the 0-80% centrality, independent of the collision energy [43, 46]. Solid and dashed lines in the left window of Fig. 1 are results from calculations based on the statistical model of Refs. [47-49] as in Ref. [50] using the parameters in Tables 2 and 3 and after taking into account the feed-down correction to the Λ yield. Clearly, although there are missing data points in the large collision energy range, the ratio S2 in collisions at the 0-10% centrality seems to be independent of the collision energy sNN considering the large uncertainty of the AGS data. In addition, the model calculation describes reasonably well the data at the LHC energy, but over-predicts the results for collisions at the AGS and RHIC energies.

      Figure 1.  (color online) Collision energy dependence of the S2=N3ΛHNΛNd ratio (left) and the density fluctuation correlation coefficients αΛp+αΛn and αΛd (right) extracted from the experimental data (symbols) and calculated from the statistical model (horizontal bars) using parameters in Tables 2 and 3. See text for details.

      The right plot in Fig. 1 shows the value of the correlation coefficients αΛd as a function of the collision energy for collisions at the 0-10% centrality (solid symbols) and at the 0-80% centrality (open symbols). The large uncertainty at sNN = 2.76 TeV is owing to the propagation of the standard error from the large volume Vch used in collisions at the LHC energy. Within current experimental uncertainties, the value of αΛd, which is negative and thus indicates an anti-correlation between the Λ and deuteron density fluctuations, becomes slightly less negative as the collision energy increases and approaches zero at sNN = 2.76 TeV. The negative αΛd and the negative αnp, shown in Tables 2 and 3, respectively, could be owing to the underestimation of the value of the λ parameter or the kinetic freeze-out volume used in our study. Full understanding of these results requires detailed studies based on the microscopic models of light cluster production in high-energy heavy-ion collisions [51, 52], which is, however, beyond the scope of the present study. Compared with the correlation coefficient αΛp+αΛn extracted from S3, which seems to vary very little over a broad range of collision energies, the value of αΛd shows a more visible sNN dependence. Also, the deviation of αΛp+αΛn from zero is larger than that of αΛp+αΛn. This may suggest that αΛd is a cleaner observable than αΛp+αΛn for studying the sNN dependence of baryon density fluctuations and their correlations, as seen from the comparison of Eq. (12) to Eq. (5). Future experimental measurements in a broad range of collision energies from AGS to RHIC will be very useful for shedding light on the underlying physics.

    • 3.2.   The S2/B2 ratio

    • In the coalescence model, the yield ratio S2=N3ΛH/(NΛNd) is the coalescence parameter for the production of 3ΛH if it is considered as a bound system of Λ and a deuteron. Because of the strangeness carried by Λ, the S2 may be different from the coalescence parameter B2 for the deuteron production following the coalescence of a proton and a neutron [19, 53-57].

      From Eq. (12) for S2 and a similar equation for B2, given by

      B2=NdNpNn=gd1m3/2p(2πTK)3/21VK(1+αnp),

      (14)

      taking Nn=Np then leads to

      S2B2=N3ΛHNΛNd/NdNpNn=g1+αΛd1+αnp,

      (15)

      where g=g3ΛHgdm3/2p(mΛ+md)3/2m3/2Λm3/2d0.23. The ratio S2/B2 thus carries information about the difference between αnp and αΛd and thus about the difference between the baryon-baryon correlation and the baryon-strangeness correlation. We note that the B2 coalescence parameter here refers to the ratio of integrated yields, whereas in the literature it is determined differentially in momentum [55, 58-60].

      Similarly, we can introduce the coalescence parameter B3=N3HeNpNpNn for the production of 3He from the three-body coalescence of two protons and a neutron, and the coalescence parameter Bs3=N3ΛHNΛNpNn for the production of 3ΛH from the three-body coalescence of Λ, a proton, and a neutron. Their ratio is exactly the value of S3 , as discussed in Section 2. Since the ratio S2/B2 does not involve the proton density fluctuation Δp and other mixed density fluctuation correlations, it seems a more sensitive observable than S3 for studying the Λ density fluctuation.

      The left plot in Fig. 2 shows the results for the yield ratio S2/B2 = N3ΛH/(NΛNd)/(Nd/N2p) from experimental data (solid triangles) and those predicted by the statistical model (solid horizontal bars) using the proton, deuteron, and 3ΛH yields from the full pT range, and including the feed-down correction for the proton yield. It is seen that the measured yield ratio increases slightly with increasing collision energy, as predicted by the statistical model. We note that the value of B2 has also been determined in experiments from the proton and deuteron momentum spectra in a small pT window [43]. The S2/B2 ratio obtained from collisions at the LHC energy for momentum per constituent pT/A=1.4GeV/c [8, 39, 45] is 0.899±0.171. Within their uncertainties, this value is similar to that obtained using yields from the full pT range. However, the S2/B2 ratios measured for different pT bins are unavailable from experiments in the energy range available at AGS and RHIC, and this is owing to the lack of 3ΛHpT spectra at these energies. Future measurements of 3ΛH spectra over a broad range of energies are needed for extracting the Λ and deuteron density correlation coefficient αΛd discussed below.

      Figure 2.  (color online) Left: The S2/B2 ratio extracted from experimental data (solid triangles) and predicted by the statistical model (solid horizontal bars) for collision at the 0-10% centrality [39, 43]. Right: Values of αΛd extracted from experimental results according to Eqs. (16) and (13).

      The Λ and deuteron correlation coefficient αΛd can also be extracted from the yield ratio S2/B2 given in Eq. (15), that is

      αΛd=(N3ΛHNΛNd(Nd/N2p)/g)×(1+αnp)1,

      (16)

      by taking advantage of the empirical fact that the neutron and proton density fluctuation correlation αnp is less affected by TK and VK. Shown in the right plot of Fig. 2 by triangles are the values of αΛd extracted from the experimental results using Eq. (16). They are seen to have similar values to those obtained from Eq. (13) using S2=N3ΛHNΛNd, which are shown by closed circles and also in Fig. 1 where it is compared with the density fluctuation correlation coefficient αΛp+αΛn extracted from S3=N3ΛH/NΛN3He/Np.

    4.   Conclusion
    • In summary, we have argued that both the ratio S2 and the ratio S2/B2, where S2 and B2 are, respectively, the coalescence parameter for the production of hypertriton from Λ and a deuteron, and of a deuteron from a proton and a neutron, are more sensitive observables than the previously proposed ratio S3 = N3ΛH/NΛN3He/Np for studying the local baryon-strangeness correlation in the matter produced in relativistic heavy-ion collisions. We have substantiated this argument in the framework of baryon coalescence by demonstrating that the correlation coefficient αΛd between Λ and deuteron density fluctuations extracted from measured S2/B2 shows a stronger dependence on the energy of heavy-ion collisions than the correlation coefficients αΛp+αΛn between Λ and nucleon density fluctuations extracted from the measured S3. Although the results in the present study are obtained without including the feed-down contribution to nucleons from Δ resonances, they will not be qualitatively affected because of the low kinetic freeze-out temperature of ~100 MeV, which only contribues ~20% to the nucleon yield. Experimental measurements of the ratio S2/B2 are expected to provide a promising way to study the strangeness and baryon correlation in the matter produced from heavy-ion collisions as the collision energy or the baryon chemical potential of produced matter is varied, which in turn can shed light on the properties of the QGP to hadronic matter phase transition during collisions.

Reference (60)

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