AMPT Simulations

To investigate the effects of different hadronization schemes on baryon number transport dynamics, we used both the default and string melting (SM) versions of the AMPT model to simulate Au+Au collisions. The initial phase space is provided by the HIJING model [33-36], which is a Monte Carlo event generator for parton and particle production in high-energy hadronic and nuclear collisions. In the default version of AMPT, minijets and their parent nucleons form excited strings after partonic interactions with minijets, and these strings fragment into hadrons through Lund string fragmentation [37]. In the SM version, strings are converted through the "string melting" mechanism into partons, which subsequently interact during evolution, and the coalescence mechanism is used to combine partons into hadrons. The parton interactions are described by the Boltzmann equation and solved using the ZPC model [38], which includes only two-body elastic scatterings. In the quark coalescence process, the two or three nearest partons (quarks and anti-quarks) in phase space recombine to form a meson or baryon. This study uses the AMPT version that strictly conserves the net electric charge, strangeness, and baryon number. We generated more than 50 million minimum bias events of Au+Au collisions at both 7.7 and 14.6 GeV.

Ω Production and Conservation of Strangeness and Baryon Number

Ω^- production is governed by both strangeness conservation (SC) and baryon number conservation. In addition, the presence of net Ω hyperons at midrapidities indicates the involvement of baryon number transport (BNT) dynamics. Figure 2 illustrates the two basic scenarios based on the quark coalescence picture for the Ω^- production. In Scenario 1, or the general associate production (left side of Fig. 2), three $\overline{s}$ quarks are pair-produced with the three s quarks in Ω^- and may combine with u or d quarks from the incident nuclei to form three kaons. In this case, the Ω^- hyperon does not have an accompanying anti-baryon and carries a fraction of the total net baryon number transported from the colliding nuclei. In exotic dynamics involving gluon junction interactions, the production of three s-$\overline{s}$ pairs may be combined with a gluon junction to yield Ω^-. Consequently, general associate production is inclusive of a possible exotic BNT via gluon junction to the Ω hyperon, whereas valence u/d quarks form the kaons [27]. In the default version, the associated production of Ω and kaons can yield features that are qualitatively similar to the coalescence picture. Therefore, general associate production encompasses contributions from both ordinary physical processes and gluon junction dynamics. The AMPT model in our simulation does not explicitly include gluon junction dynamics. We use Scenario 1, or general associate production, to denote a general process in which a net baryon number is transported to the Ω hyperon regardless of the underlying dynamics. In Scenario 2, or general pair production (right side of Fig. 2), the three pair-produced $\overline{s}$ quarks combine with u($\overline{u}$) or d($\overline{d}$) quarks to form an anti-hyperon and kaon. For example, ${\overline{\text{Ξ}}}^{+}\left({\overline{\text{Ξ}}}^0\right)$ and K⁰(K⁺) can emerge alongside the Ω^- hyperon. In this scenario, the baryon number in Ω^- is balanced by other types of anti-hyperon, and no net baryon number from the incident nuclei is present.

To characterize the numbers of kaons and anti-baryons associated with Ω production, we introduce $\text{Δ}{N}_{\text{A}}\equiv {\langle A\rangle }_{\text{w}\mathrm{.}{\text{Ω}}^{-}}-{\langle A\rangle }_{\text{w}\mathrm{.}\text{o}\mathrm{.}{\text{Ω}}^{-}}\mathrm{,}$ (1) where ${\langle A\rangle }_{\text{w}\mathrm{.}{\text{Ω}}^{-}}$ and ${\langle A\rangle }_{\text{w}\mathrm{.}\text{o}\mathrm{.}{\text{Ω}}^{-}}$ denote the average numbers of particles A in events with one Ω^- and without any Ω^-, respectively. Here, we assume that all other aspects of the two event classes are the same. Table 1 lists the Δ NK and $\text{Δ}{N}_{\overline{B}}$ values expected by the two scenarios, where K represents both K⁺ and K⁰, and $\overline{B}$ refers to anti-baryons ($\overline{\text{Λ}}$, $\overline{�}$, $\overline{\text{Ξ}}$, and so on) associated with the Ω^- production. In our study, we chose the beam energies of 7.7 and 14.6 GeV to limit the fraction of Au+Au events that produce multiple Ω ($\overline{\text{Ω}}$) hyperons. Compared with general pair production, general associate production exhibits a stronger Ω^--K correlation and a weaker Ω^--$\overline{B}$ correlation.

Correlation Using the Event Mixing Technique

To divide the combinatorial background, we first analyze the correlations between Ω^- and strange hadrons with traditional normalization using mixed events, which is similar to the scheme used in other analyses (e.g., [39]). The correlation function is expressed as $C\left(k^{*}\right)=\mathcal{N}\frac{A\left(k^{*}\right)}{B\left(k^{*}\right)}\mathrm{,}$ (2) where A(k^*) is the same-event distribution, B(k^*) is the mixed-event distribution, and $k^{*}\equiv \frac{1}{2}\left|k_1-k_2\right|$ is the reduced momentum in the pair rest frame. In addition, 𝒩 is the normalization factor determined by matching the same- and mixed-event correlations in the uncorrelated phase space, for example, by requiring C(k^*>1 GeV/c)=1.

The event mixing normalization technique enables the correlation length between two types of particles in momentum space to be investigated by analyzing their distributions. The underlying assumption is that the distribution of mixed events should be equivalent to that of the same events in the absence of any correlation. By dividing this mixed-event background, we can obtain the correlation between the two particles of interest. To achieve meaningful results, selecting an appropriate k^* range for normalization is crucial. Typically, the k^* range with a high counting density is selected for normalization because it provides a reliable background estimate. This method has also been applied to examine the correlation function for Ω^--K⁺, Ω^--${\overline{\text{Λ}}}^0$, and Ω^--${\overline{\text{Ξ}}}^{+}$ in terms of the relative rapidity and relative transverse momentum. However, with the normalization procedure, this correlation function cannot fully capture the magnitude of multiple kaons correlated with Ω production, nor can it adequately represent the sensitivity of the correlation to the production dynamics.

Correlations Using Combinatorial Background Subtraction

To make the correlation measurement sensitive to the number of associated hadrons, we introduce combinatorial background subtracted (CBS) correlations by taking the difference between the Ω- and $\overline{\text{Ω}}$-hadron correlations, each normalized by the corresponding number of Ω or $\overline{\text{Ω}}$ hyperons. For example, the Ω^--K⁺ correlation is defined as $C_{{\text{Ω}}^{-}{K}^{+}}^{\text{CBS}}\left(k^{*}\right)=\frac{\text{d}{N}_{{\text{Ω}}^{-}{K}^{+}}/\text{d}{k}^{*}}{N_{{\text{Ω}}^{-}}}-\frac{\text{d}{N}_{{\overline{\text{Ω}}}^{+}{K}^{+}}/\text{d}{k}^{*}}{N_{{\overline{\text{Ω}}}^{+}}}\mathrm{.}$ (3) Due to the SC in the Ω^--K⁺ pairs, this background subtraction approach is intended to extract the main component of the correlation. The opposite-sign pair (Ω^--K⁺ or ${\overline{\text{Ω}}}^{+}$-K^-) distribution in the first term contains the signal, whereas the same-sign pair distribution in the second term (${\overline{\text{Ω}}}^{+}$-K⁺ or Ω^--K^-) models the uncorrelated background. This subtraction scheme is sensitive to the difference in the number of kaons between events with Ω^- and those with ${\overline{\text{Ω}}}^{+}$ as well as to the phase-space distribution of the extra kaons. There may be variations in the kinematic phase spaces of these pairs in nuclear collisions, and these effects can be mitigated using this normalization scheme and fine collision centrality bins. Similar approaches have been applied to study the correlations between Ω and other hadrons.

Strangeness Conservation and Strange Hadron Yields

Table 2 lists the AMPT simulations for determining the differences in the numbers of $s\overline{s}$ pairs between events with one Ω^- and those without any Ω^- or ${\overline{\text{Ω}}}^{+}$ in the Au+Au collisions at $\sqrt{s_{\text{NN}}}=$ 7.7 and 14.6 GeV. For events with one Ω, at least three $s\overline{s}$ pairs are produced because of SC. The AMPT results are slightly greater than three, and the slight excess indicates that the other strangeness productions do not significantly affect the number of $s\overline{s}$. Compared with the SM version of AMPT, the default version yields a slightly greater number of $s\overline{s}$ pairs and a smaller Ω formation probability. This correlation is presumably due to differences in the formation dynamics and/or the $s\overline{s}$ phase space. AMPT simulations suggest that numerous strange quarks per event are needed to favorably form an Ω^- hyperon during collisions at lower energies.

Table 3 lists the AMPT calculations for Δ NK and $\text{Δ}{N}_{\overline{B}}$ as defined in Eq. (1). Compared with the expectations listed in Table 1, which correspond to the two Ω production scenarios, these numbers indicate that Ω production is likely to receive contributions from both scenarios. The SM version seems to favor the dominance of general associate production, with Δ NK values close to three and to the lower $\text{Δ}{N}_{\overline{B}}$ values. This may be explained by the fact that in the SM version, for the Ω^- that do not originate from pair production, the production depends on the s-quark threshold. In addition, in the coalescence version, production does not increase, causing an enhancement in the general associate production scenario.

Strangeness and baryon numbers can be represented by various particle types in the final states of nuclear collision. For example, $\overline{s}$ quarks may exist in K⁺ and K⁰. The actual distributions of strangeness and baryon numbers in final-state particles may be sensitive to nuclear dynamics and may also depend on the beam energy. Figures 3 and 4 show the AMPT results of the differences in strange hadron yields between Ω^- and non-Ω^- events of Au+Au collisions at 7.7 and 14.6 GeV, respectively. At 7.7 and 14.6 GeV, the correlations between Ω^- and kaons are stronger in the SM version. By contrast, those between Ω^- and anti-hyperons are stronger in the default version. For Ω^- events, the number of kaons is much greater than the number of anti-hyperons, suggesting that the correlations between Ω^- and kaons are much stronger than those between Ω^- and anti-hyperons. According to Table 1, these numbers seem to support the idea that general associate production contributes more significantly to Ω yields in both versions.

Fig. 3Full-screen View

(Color online) AMPT calculations of the difference in strange hadron yields between Ω^- and non-Ω^- events of Au+Au collisions at 7.7 GeV

Fig. 4Full-screen View

(Color online) AMPT calculations of the difference in strange hadron yields between Ω^- and non-Ω^- events of Au+Au collisions at 14.6 GeV

Correlations between Ω^± and Strange Hadrons

We first show the correlations between Ω and strange hadrons as a function of the reduced momentum k^* in the pair rest frame using event mixing normalization. Figure 5 presents the correlation functions for the (a) Ω^--K⁺, (b) Ω^--$\overline{\text{Λ}}$, and (c) Ω^--${\overline{\text{Ξ}}}^{+}$ pairs for 0–5% centrality Au + Au collisions at 7.7 GeV. The normalization between the same- and mixed-event distributions is determined by the k^* region of 0.6–1.5 GeV/c. A distinct correlation exists between Ω^- and K⁺, indicating that SC plays a major role in the Ω^- and K⁺ yields with a typical correlation length on the order of less than k^* of 0.5 GeV/c. Ω^- may also be correlated with the anti-hyperons of $\overline{\text{Λ}}$ and ${\overline{\text{Ξ}}}^{+}$. However, the current simulated results do not have sufficient statistics to be definitive.

Fig. 5Full-screen View

Correlation functions between Ω^- and strange hadrons using the event mixing normalization in the 0–5% centrality range of Au + Au collisions at 7.7 GeV (|η|<1)

Figure 6 shows (a) Ω^--K⁺, (b) Ω^--$\overline{\text{Λ}}$, and (c) Ω^--${\overline{\text{Ξ}}}^{+}$ correlations for the 0–5% centrality Au + Au collisions at 14.6 GeV. The Ω^--K⁺ correlation at 14.6 GeV seems to be less prominent than that at 7.7 GeV. This can be explained by our normalization scheme, in which the normalization factor is the number of uncorrelated Ω^--K⁺ pairs. More kaons are produced at higher energies, resulting in more uncorrelated kaons pairing with Ω at 14.6 GeV, diluting the correlation between the Ω^--K⁺ due to the combinatorial background.

Fig. 6Full-screen View

Correlation functions between Ω^- and strange hadrons using the event mixing normalization in the 0–5% centrality range of Au + Au collisions at 14.6 GeV (|η|<1)

Table 1 shows that the correlation between Ω and anti-hyperons may also be susceptible to the Ω-pair production scenario. The middle and right panels in Figs. 5 and 6 show the Ω^--${\overline{\text{Λ}}}^0$ and Ω^--${\overline{\text{Ξ}}}^{+}$ correlation functions, respectively, which were obtained using the event mixing technique at 7.7 and 14.6 GeV, respectively. At both energies, the Ω^--${\overline{\text{Λ}}}^0$ results lack sufficient statistics for a definitive conclusion, whereas some level of Ω^--${\overline{\text{Ξ}}}^{+}$ correlation may exist. However, no significant difference appears between Ω^--${\overline{\text{Ξ}}}^{+}$ and ${\overline{\text{Ω}}}^{+}$-Ξ^-, which is similar to the Ω-K results.

For both beam energies, no significant difference exists in the observed correlations between the two AMPT hadronization schemes. It appears that the correlations using the event mixing technique are sensitive only to the kinematic region where SC produces $s\overline{s}$ pairs. However, the magnitude differences, as listed in Table 3, are not quantitatively reflected in the measured correlations due to the normalization scheme. We explore the CBS correlations within the same event framework and background subtraction scheme.

Our goal is to use the ${\overline{\text{Ω}}}^{+}$-K^- correlation (general pair production only) as a reference for comparison with the Ω^--K⁺ correlation. For each CBS correlation, we again select events with one Ω^- or ${\overline{\text{Ω}}}^{+}$ from 0–5% of the most central collisions. As described in Eq. (3), the combinatorial background of the Ω^--K⁺ correlation is modeled by the ${\overline{\text{Ω}}}^{+}$-K⁺ correlation based on events with one ${\overline{\text{Ω}}}^{+}$. ${\overline{\text{Ω}}}^{+}$ and K⁺ both contain $\overline{s}$ quarks, and the narrow centrality bin makes the ${\overline{\text{Ω}}}^{+}$-K⁺ correlation a good candidate for the combinatorial background in the Ω^--K⁺ correlation. Similarly, we can also use the difference between ${\overline{\text{Ω}}}^{+}$-K^- and Ω^--K^- to calculate the ${\overline{\text{Ω}}}^{+}$-K^- correlation, which represents contributions from only the SC. In summary, Ω^--K⁺ includes both BNT and SC dynamics, whereas ${\overline{\text{Ω}}}^{+}$-K^- includes only the SC. Therefore, any difference between Ω^--K⁺ and ${\overline{\text{Ω}}}^{+}$-K^- correlations will be sensitive to the net BNT dynamics, which are absent in general pair production.

Figure 7 shows the CBS correlations for (left) Ω^--K⁺ and (right) ${\overline{\text{Ω}}}^{+}$-K^- at 7.7 and 14.6 GeV, respectively. The correlations are shown as functions of k^*, rapidity difference (Δ y), and transverse momentum difference (Δ p_T) in the three rows, respectively. At 7.7 GeV, the two AMPT versions exhibit clear differences. Compared with the SM version, the default version shows a stronger ${\overline{\text{Ω}}}^{+}$-K^- correlation, presumably indicating a stronger and more localized SC. For the Ω^--K⁺ correlation, which is sensitive to both the SC and BNT dynamics while the total magnitudes (integral) are similar between the two AMPT versions, the correlation shapes differ as a function of both k^* and Δ p_T. In the SM version, the coalescence formation mechanism and BNT dynamics seem to yield a narrower correlation function between Ω^- and K⁺.

Fig. 7Full-screen View

CBS correlations between ${\text{Ω}}^{-}\left({\overline{\text{Ω}}}^{+}\right)$ and kaons in 0–5% of the most central Au + Au collisions at 7.7 and 14.6 GeV (|η|<1). The left and right columns show the results of $C_{{\text{Ω}}^{-}{K}^{+}}^{\text{CBS}}$ and $C_{{\overline{\text{Ω}}}^{+}{K}^{-}}^{\text{CBS}}$ at 7.7 and 14.6 GeV, respectively

At 14.6 GeV (Fig. 7), the difference between the two hadronization schemes is relatively small. The shape difference in the correlation as a function of k^* and Δ pT is much less prominent than at 7.7 GeV. The CBS correlations at 7.7 and 14.6 GeV suggest that the event-level Ω^--K⁺ correlation is stronger than that of the ${\overline{\text{Ω}}}^{+}$-K^-. In general pair production, Ω^- does not carry a net baryon number transported from the colliding nuclei. Because ${\overline{\text{Ω}}}^{+}$ is produced only by pair production, whereas Ω^- can be produced in both scenarios, the stronger Ω^--K⁺ correlation indicates significant contributions of general associate production to Ω^- production at both beam energies. Particularly at 7.7 GeV, general associate production seems to contribute more to Ω^- production in the SM version, as the difference in correlation amplitudes between Ω^--K⁺ and ${\overline{\text{Ω}}}^{+}$-K^- becomes noticeably larger.

Figure 8 shows the CBS correlations between Ω and $\overline{\text{Λ}}$ and their anti-particle pairs as a function of three kinematic variables in 0–5% of the most central Au + Au collisions at 7.7 and 14.6 GeV, respectively. In addition to strangeness and baryon number conservation that govern the correlations between Ω^- and $\overline{\text{Λ}}$, the ${\overline{\text{Ω}}}^{+}$-Λ correlations are also sensitive to BNT dynamics. At 7.7 GeV, the default version of AMPT displays stronger correlations than the SM version.

Fig. 8Full-screen View

CBS correlations between ${\text{Ω}}^{-}\left({\overline{\text{Ω}}}^{+}\right)$ and $\overline{\text{Λ}}(\text{Λ})$ in 0–5% of the most central Au + Au collisions at 7.7 and 14.6 GeV (|η|<1). The left columns show the results of $40\times C_{{\text{Ω}}^{-}\overline{\text{Λ}}}^{\text{CBS}}$ and $C_{{\overline{\text{Ω}}}^{+}\text{Λ}}^{\text{CBS}}$ at 7.7 GeV. The right shows the results of $10\times C_{{\text{Ω}}^{-}\overline{\text{Λ}}}^{\text{CBS}}$ and $C_{{\overline{\text{Ω}}}^{+}\text{Λ}}^{\text{CBS}}$ at 14.6 GeV

For both AMPT versions, the Ω^--$\overline{\text{Λ}}$ correlations are much weaker than those of ${\overline{\text{Ω}}}^{+}$-Λ, which is consistent with the fact that ${\overline{\text{Ω}}}^{+}$ is produced only through general pair production. The default version also indicates a greater discrepancy between ${\overline{\text{Ω}}}^{+}$-Λ and Ω^--$\overline{\text{Λ}}$, which suggests a more significant contribution from general pair production, complementary to the difference between the Ω^--K⁺ and ${\overline{\text{Ω}}}^{+}$-K^- results. At 14.6 GeV, the ${\overline{\text{Ω}}}^{+}$-Λ correlations (right column in Fig. 9) are also higher than those of Ω^--$\overline{\text{Λ}}$ (left column in Fig. 9) by approximately one order of magnitude. However, no significant difference appears between the two AMPT versions at this energy level.

Fig. 9Full-screen View

CBS correlations between ${\text{Ω}}^{-}\left({\overline{\text{Ω}}}^{+}\right)$ and ${\overline{\text{Ξ}}}^{+}\left({\text{Ξ}}^{-}\right)$ in 0–5% of the most central Au + Au collisions at 7.7 and 14.6 GeV (|η|<1). The left and right columns show the results of $5\times C_{{\text{Ω}}^{-}{\overline{\text{Ξ}}}^{+}}^{\text{CBS}}$ and $C_{{\overline{\text{Ω}}}^{+}{\text{Ξ}}^{-}}^{\text{CBS}}$ and of those at 14.6 GeV, respectively

Figure 9 shows the CBS correlations between Ω and $\overline{\text{Ξ}}$ as functions of the three kinematic variables in 0–5% of the most central Au + Au collisions at 7.7 and 14.6 GeV, respectively. At both beam energies, the stronger ${\overline{\text{Ω}}}^{+}$-Ξ^- correlations relative to Ω^--${\overline{\text{Ξ}}}^{+}$ are consistent with the fact that general pair production provides a larger contribution to the ${\overline{\text{Ω}}}^{+}$ production than to that of Ω^-. At 7.7 GeV, the difference in the Ω^--${\overline{\text{Ξ}}}^{+}$ correlations between the two AMPT versions suggests that the default version generates a larger contribution from general pair production to Ω^- production. At 14.6 GeV, the ${\overline{\text{Ω}}}^{+}$-Ξ^- correlations are still stronger than those of Ω^--${\overline{\text{Ξ}}}^{+}$, but no significant difference appears between the two versions.