Strange hadron production in a quark combination model in Au+Au collisions at energies available at the BNL Relativistic Heavy Ion Collider

Hadron production from final state partons in high energy collisions is a complex Quantum Chromodynamics (QCD) process. Due to the difficulty of non-perturbative QCD, phenomenological mechanisms and models have to be applied to describe the production of hadrons at hadronization Andersson et al. (1983); Webber (1984); Das and Hwa (1977); Anisovich and Shekhter (1973); Bjorken and Farrar (1974); Xie and Liu (1988); Becattini (1996); Braun-Munzinger et al. (1996). In relativistic heavy-ion collisions at high RHIC and LHC energies, quark-gluon plasma (QGP) is created in early stage of collisions and the hadronization of QGP can be microscopically described by the quark (re-)combination/coalescence mechanism Greco et al. (2003); Fries et al. (2003); Molnar and Voloshin (2003); Lin and Molnar (2003); Shao et al. (2005); Chen and Ko (2006). The enhanced ratio of baryon to meson and number of constituent quark scaling for elliptic flows of hadrons at the intermediate transverse momentum ($p_{T}$) are typical experimental signals for quark combination mechanism at hadronization and have been widely observed in relativistic heavy-ion collisions Adare et al. (2007); Adamczyk et al. (2016a); Abelev et al. (2015); Acharya et al. (2018); Adamczyk et al. (2013); Adcox et al. (2002a); Abelev et al. (2006, 2013).

In our recent studies in high-multiplicity events in $pp$ and $p$-Pb collisions at LHC energies where the mini-QGP is possibly created and re-scattering of hadrons is weak, we found an interesting quark number scaling property for $p_{T}$ spectra of hadrons at mid-rapidity Song et al. (2017); Zhang et al. (2020). This scaling property is a direct consequence of equal velocity combination (EVC) of constituent quarks and antiquarks at hadronization. Our studies showed that the EVC of up/down, strange and charm quarks can provide a good and systematic description on $p_{T}$ spectra of light, strange and charm hadrons in ground-state in $pp$ and $p$-Pb collisions at LHC energies Song et al. (2017); Gou et al. (2017); Zhang et al. (2020); Song et al. (2018); Li et al. (2018). In latest work Song et al. (2020), we further found that the experimental data for $p_{T}$ spectra of $\Omega$ and $\phi$ at mid-rapidity in heavy-ion collisions in a broad collision energy range ($\sqrt{s_{NN}}=11.5-2760$ GeV) also satisfy the quark number scaling property. This is a clear signal of EVC at hadronization even in heavy-ion collisions. Therefore, it is interesting to systematically test this mechanism by production of more hadron species in relativistic heavy-ion collisions.

Recently, STAR collaboration reported their precise experimental data for the production of strange hadrons in Au+Au collisions at $\sqrt{s_{NN}}=$7.7-39 GeV Adam et al. (2020). This provides us a good opportunity to systematically study the EVC mechanism of hadron production in these collisions. In this paper, we apply a quark combination model with EVC to carry out a systematic study on yield densities and $p_{T}$ spectra of strange hadrons in Au+Au collisions at $\sqrt{s_{NN}}=$ 7.7, 11.5, 19.6, 27, 39 and 200 GeV. We put particular emphasis on the self-consistency of the model in explaining the experimental data for different kinds of hadrons and on the regularity in multi-hadron production correlations which is sensitive to hadronization mechanism. Taking advantage of precise data for strange hadrons, we also study the strangeness neutralization in the midrapidity region in these collisions. We extract quark $p_{T}$ distributions at hadronization from data of hadrons and study properties of the relative abundance for strange quarks as the function of collision energy. Furthermore, we discuss the key physics in current quark combination model which are responsible for explaining successfully experimental data of hadronic $p_{T}$ spectra and yields, and discuss the creation of QGP or the deconfinement at low RHIC energies.

The paper is organized as follows. In Sec. II, we introduce a quark combination model with equal velocity combination approximation. In Sec. III, we study the strangeness neutralization in mid-rapidity region. In Sec. IV, we show results of $p_{T}$ spectra for hadrons and compare them with experimental data. In Sec. V, we study the split in yield between hadrons and their antiparticles. In Sec. VI, we study properties for the obtained numbers and $p_{T}$ spectra of quarks at hadronization at different collision energies. The summary and discussion are given at last in Sec. VII.

The quark combination is one of phenomenological mechanisms for hadron production at hadronization. The basic idea of quark combination was firstly proposed in 1970s Anisovich and Shekhter (1973); Bjorken and Farrar (1974); Das and Hwa (1977) and has many successful applications in high energy reactions Xie and Liu (1988); Buschbeck et al. (1980); Liang and Xie (1991); Braaten et al. (2002). In relativistic heavy-ion collisions, quark combination mechanism is often used to describe the hadron production at QGP hadronization Zimányi et al. (2000); Hwa and Yang (2003); Greco et al. (2003); Fries et al. (2003); Molnar and Voloshin (2003); Shao et al. (2005); Chen and Ko (2006). Recently, we found that quark combination can also well explain experimental data of hadron production in high-multiplicity $pp$ and $p$-Pb collisions at LHC energies Song et al. (2017); Gou et al. (2017); Shao et al. (2017); Zhang et al. (2020).

In this paper, inspired by the quark number scaling property of hadronic $p_{T}$ spectra Song et al. (2017); Zhang et al. (2020); Song et al. (2020), we adopt a specific version of quark combination model Song et al. (2017); Gou et al. (2017). This model assumes the combination of constituent quarks and antiquarks with equal velocity to form baryons and mesons at hadronization. The unknown non-perturbative dynamics at hadronization are parameterized and their values are assumed to be stable in high energy reactions and are fixed by experimental data. This model is essentially a statistical model based on the constituent quark degrees of freedom at hadronization and the constituent quark structure of hadrons. It is different from the popular re-combination/coalescence models which adopt the Wigner wave function method under instantaneous hadronization approximation Fries et al. (2003); Greco et al. (2003). The brief description of the model is as follows.

We start from the general formula for the production of the baryon $B_{j}$ composed of $q_{1}q_{2}q_{3}$ and the production of the meson $M_{j}$ composed of $q_{1}\bar{q}_{2}$ in quark combination mechanism

Here $f_{q_{1}q_{2}q_{3}}(p_{1},p_{2},p_{3})$ is the joint momentum distribution for $q_{1}$, $q_{2}$ and $q_{3}$. The combination kernel function ${\cal R}_{B_{j}}(p_{1},p_{2},p_{3};p_{B})$ denotes the probability density for a given $q_{1}q_{2}q_{3}$ with momenta $p_{1},p_{2}$ and $p_{3}$ to combine into a baryon $B_{j}$ with momentum $p_{B}$. It is similar for mesons. We emphasize that Eqs. (1) and (2) are generally suitable for hadron production in momentum space of any dimension. In this paper, we study the one-dimensional transverse momentum ($p_{T}$) distribution of hadrons at midrapidity $y=0$. In this case, $p_{i}$ simply denotes $p_{T,i}$ and the distribution function $f_{h}$$\left(p\right)$ denotes the $dN_{h}/dp_{T}$ at midrapidity.

The combination functions ${\cal R}_{B_{j}}(p_{1},p_{2},p_{3};p_{B})$ and ${\cal R}_{M_{j}}(p_{1},p_{2};p_{M})$ contain the key information of combination dynamics which is not clear at present due to the non-perturbative difficulty of hadronization. In our recent works Song et al. (2017); Zhang et al. (2020); Song et al. (2020), we observed an interesting quark number scaling property for the $p_{T}$ spectra of hadrons in high-multiplicity $pp$ and $p$-Pb collisions as well as in heavy-ion collisions. This scaling property supports the combination of constituent quarks and antiquarks with equal velocity. This suggests an effective form for the combination kernel functions, i.e.,

Here, $\kappa_{B_{j}}$ and $\kappa_{M_{j}}$ are coefficients independent of momentum. Momentum fraction $x_{i}$ is determined by the masses of constituent quarks because $p_{i}=m_{i}\gamma\beta\propto m_{i}$. Specifically, we have $x_{i}=m_{i}/(m_{1}+m_{2}+m_{3})$ for baryon $B_{j}$ with $x_{1}+x_{2}+x_{3}=1$ and $x_{i}=m_{i}/(m_{1}+m_{2})$ for meson $M_{j}$ with $x_{1}+x_{2}=1$. $m_{i}$ is the constituent mass for quark of flavor $i$. Because the mass of the formed hadron under Eqs. (3) and (4) is the sum of these of constituent quarks, we take contituent masses $m_{u}=m_{d}=0.3$ GeV and $m_{s}=0.5$ GeV in order to properly describe the production of baryons and vector mesons studied in this paper.

The joint momentum distributions $f_{q_{1}q_{2}q_{3}}(p_{1},p_{2},p_{3})$ and $f_{q_{1}\bar{q}_{2}}(p_{1},p_{2})$ generally contain the correlation term caused by, for example, the collective flow formed in system evolution before hadronization in heavy-ion collisions. In order to obtain analytical and simple expressions for $f_{B_{j}}(p_{B})$ and $f_{M_{j}}(p_{M})$, we take the independent distribution approximation

Substituting Eqs. (3)-(6) into Eqs. (1) and (2), we obtain

$\kappa_{B_{j}}$ and $\kappa_{M_{j}}$ carry the information of $p_{T}$-independent combination dynamics. In order to determine their forms, we express momentum distributions of hadrons in another form

$N_{B_{j}}$ and $N_{M_{j}}$ are numbers of $B_{j}$ and $M_{j}$, respectively. $f_{B_{j}}^{\left(n\right)}\left(p_{B}\right)$ and $f_{M_{j}}^{\left(n\right)}\left(p_{M}\right)$ are distribution functions normalized to one when integrating over momentum, which can be obtained by those of quarks and antiquarks,

Here, $f_{q_{i}}^{\left(n\right)}\left(p\right)=f_{q_{i}}\left(p\right)/N_{q_{i}}$ is the normalized distribution function of quark $q_{i}$. Normalization coefficients $A_{B_{j}}$ and $A_{M_{j}}$ are defined as

Substituting Eqs. (11) and (12) into Eqs. (9) and (10) and then comparing them with Eqs. (7) and (8), we obtain

From above two equations, we can read out the physical meaning of $\kappa_{B_{j}}$ and $\kappa_{M_{j}}$. It is obvious that $\kappa_{B_{j}}/A_{B_{j}}$ denotes the momentum-integrated probability of $q_{1}q_{2}q_{3}$ forming a baryon $B_{j}$ and $\kappa_{M_{j}}/A_{M_{j}}$ denotes that of $q_{1}\bar{q}_{2}$ forming a meson $M_{j}$. Two probabilities can be further decomposed as

Taking meson for example, $\overline{N}_{M}/N_{q}N_{\bar{q}}$ is used to denote the averaged probability of a $q\bar{q}$ pair forming a meson. Here, $N_{q}N_{\bar{q}}$ is the all possible number of $q\bar{q}$ pair, where $N_{q}=N_{u}+N_{d}+N_{s}$ is the number of all quarks and $N_{\bar{q}}=N_{\bar{u}}+N_{\bar{d}}+N_{\bar{s}}$ is that of all antiquarks. $\overline{N}_{M}$ is the average number of mesons produced by the hadronization of quark system with given quark number $N_{q}$ and antiquark number $N_{\bar{q}}$. The factor $C_{M_{j}}$ denotes further sophisticated tune for the production probability of $M_{j}$ on the basis of the averaged probability $\overline{N}_{M}/N_{q}N_{\bar{q}}$. The baryon formula is similar to meson except a factor $N_{iter}$. This factor is to assure $\sum_{q_{1}q_{2}q_{3}}N_{iter}N_{q_{1}}N_{q_{2}}N_{q_{3}}=N_{q}^{3}$ and equals to $1,3,6$ for $q_{1}q_{2}q_{3}$ with three identical flavor, two identical flavor, and three different flavors, respectively. Finally, we obtain yield formulas of hadrons

$\overline{N}_{B}$ and $\overline{N}_{M}$ are functions of $N_{q}$ and $N_{\bar{q}}$ Song and Shao (2013),

where $x=N_{q}+N_{\bar{q}}$ and $z=\left(N_{q}-N_{\bar{q}}\right)/x$. Parameter $a=1+\left(\overline{N}_{M}/\overline{N}_{B}\right)_{z=0}/3$ denotes the production competition of baryon to meson and is tuned to be $a\approx 4.86\pm 0.1$ according to our recent work Shao et al. (2017).

In this paper, we only consider the production of the ground state $J^{P}=0^{-},\,1^{-}$ mesons and $J^{P}=(1/2)^{+},\,(3/2)^{+}$ baryons in flavor SU(3) group. In meson formation, we introduce a parameter $R_{V/P}$ to describe the relative weight of a quark-antiquark pair forming the state of spin 1 to that of spin 0. Here, we take $R_{V/P}=0.55\pm 0.05$ in order to reproduce the measured $K^{*}/K$ and $\phi/K$ data in high energy reactions Adam et al. (2016); Acharya et al. (2020). Factor $C_{M_{j}}$ is then parameterized as

In baryon formation, we introduce a parameter $R_{D/O}$ to describe the relative weight of spin $3/2$ state to $1/2$ state for three quark combination. We take $R_{D/O}=0.5\pm 0.04$ by fitting the experimental data of $\Xi^{*}/\Xi$ and $\Sigma^{*}/\Lambda$ in high energy collisions Adamova et al. (2017). Then we have for $C_{B_{j}}$

except that $C_{\Lambda}=C_{\Sigma^{0}}={1}/{(2+R_{D/O})},\leavevmode\nobreak\ C_{\Sigma^{*0}}={R_{D/O}}/{(2+R_{D/O})},\leavevmode\nobreak\ C_{\Delta^{++}}=C_{\Delta^{-}}=C_{\Omega^{-}}=1$.

Taking $f_{q_{i}}\left(p\right)$ as model inputs, we can calculate $f_{h}\left(p\right)$ and $N_{h}$ of hadrons directly produced at hadronization. Finally, we take the decay contribution of short-life resonances into account according to experimental measurements, and obtain results of final-state hadrons

where the decay function $D_{ij}\left(p^{\prime},p\right)$ is determined by the decay kinematics and decay branch ratios Olive et al. (2014).

In relativistic heavy-ion collisions, strange quark and antiquark are always created in pair in collisions and therefore strangeness is globally conserved. However, for a finite kinetic region such as the mid-rapidity region, the strangeness neutralization is not so explicit, in particular, at low collision energies. In this section, using the precise data for yield densities and $p_{T}$ spectra of identified hadrons Adler et al. (2004); Adams et al. (2007); Abelev et al. (2009a, b); Aggarwal et al. (2011); Adamczyk et al. (2017); Adam et al. (2020), we study the local strangeness in the midrapidity region in Au+Au collisions at STAR BES energies.

In this section, we use the quark combination model (QCM) in Sec. II to study $p_{T}$ spectra of light-flavor hadrons in Au+Au collisions at RHIC energies. The inputs of model are $p_{T}$ spectra of quarks and antiquarks at hadronization. Here, we take $f_{\bar{s}}\left(p_{T}\right)=f_{s}\left(p_{T}\right)$ based on the study of strangeness neutralization in Sec. III. We take $f_{\bar{u}}\left(p_{T}\right)=f_{\bar{d}}\left(p_{T}\right)$ for the newborn up and down anti-quarks. Because a part of up and down quarks comes from the colliding nucleons, $p_{T}$ spectrum of up quarks is not exactly the same as that of down quarks. As discussed in [47], the relative difference in number between up and down quarks is only a few percentages. We have checked that $p_{T}$ spectra of hadrons and yield ratios of anti-hadron to hadron studied in this paper are not sensitive to this small asymmetry. Therefore, in this paper, we take the approximation $f_{u}\left(p_{T}\right)\approx f_{d}\left(p_{T}\right)$ in the mid-rapidity region.

In Table 2, we list all inputs and parameters of the model which are needed to calculate $p_{T}$ spectra of hadrons. As introduced in model description in Sec. II, two parameters $R_{V/P}$ and $R_{D/O}$ are taken as 0.55 and 0.5, respectively, at all the studied collision energies. For three inputs, $f_{s}\left(p_{T}\right)$ is fixed by using our model to fit experimental data of $\phi$, and $f_{u}\left(p_{T}\right)$ ($f_{\bar{u}}\left(p_{T}\right)$) is fixed by experimental data of (anti-)baryons containing $u(\bar{u})$ such as (anti-)proton Abelev et al. (2009b); Adamczyk et al. (2017, 2016b), respectively. The extracted results for quark $p_{T}$ spectra in Au+Au collisions for 0-5% centrality at six RHIC energies are shown in Fig. 2.

In Fig. 3, we show the calculated results for $p_{T}$ spectra of hadrons in central Au+Au collisions at $\sqrt{s_{NN}}=$ 200 GeV and compare them with experimental data Abelev et al. (2006); Adams et al. (2007); Abelev et al. (2009a). We see that the agreement between our model results and experimental data is satisfactory. Although there exists many successful explanations on these experimental data in the framework of quark combination mechanism in previous works Fries et al. (2003); Greco et al. (2003); Hwa and Yang (2003); Chen and Ko (2006); Shao et al. (2005, 2009), here we would like to emphasize that the current EVC model can systematically explain these data in a simple way. Furthermore, the good agreement at top RHIC energy provides important basis for the application of our model to lower RHIC energies.

In Figs. 4, 5, 6, 7 and 8, we show results for $p_{T}$ spectra of hadrons in central Au+Au collisions at $\sqrt{s_{NN}}=$ 39, 27, 19.6, 11.5 and 7.7 GeV and compare with experimental data Adamczyk et al. (2016b, 2017); Adam et al. (2020). At $\sqrt{s_{NN}}=$ 39, 27, 19.6, 11.5 GeV, we see a good agreement between our model results and experimental data Adamczyk et al. (2016b, 2017); Adam et al. (2020). In particular, we see that experimental data for baryons ($p$, $\Lambda$, $\Xi$, $\Omega$) and meson $\phi$ can be explained by the model very well. At $\sqrt{s_{NN}}=$ 7.7 GeV, we also see that model results are in good agreement with available experimental data. However, compared with data in Figs. 3-7, the available data at $\sqrt{s_{NN}}=$ 7.7 GeV cover smaller $p_{T}$ range and $\Omega$ data in the most central collisions are absent. Therefore the comparison at $\sqrt{s_{NN}}=$ 7.7 GeV is not as conclusive as those at other five higher energies. We should study this energy point further in the future when more precise data are available.

Using these systematic calculations and comparisons, we would like to emphasize the equal-velocity combination of quarks and antiquarks as the effective description at hadronization. This is manifested by the following two points. First, from panel (d) in Figs. 3-7, we see that experimental data of $\Omega$ and $\phi$ can be perfectly explained by the same $f_{s}\left(p_{T}\right)$ in a very simple way,

We call this property as the quark number scaling for hadronic $p_{T}$ spectra. We have shown that this property not only exists in relativistic heavy-ion collisions but also exists in high-multiplicity $pp$ and $p$-Pb collisions at LHC energies Song et al. (2017); Zhang et al. (2020); Song et al. (2020). Second, we see that data of $\Lambda$ and $\Xi^{-}$ can be simultaneously explained by $f_{s}\left(p_{T}\right)$ from $\phi$ and $f_{u}\left(p_{T}\right)$ from proton,