Inclusive particle production

The data for all light-flavor particles in this paper represent the sum of particles and anti-particles, whereas the data for heavy-flavor particles are the sum of particles and anti-particles divided by 2, following experimental analysis conventions. For inclusive hadron production, events were selected according to the minbias trigger used in the ALICE experiment, which requires signals accepted by either side of the V0 detector. The transverse momentum spectra for charged pions, kaons, and protons with |y|<0.5 are shown in Fig. 1 produced in inelastic pp collisions at $\sqrt{s}=13\text{ TeV}$. We observe that the ALICE data of the pion and kaon spectra can be roughly described by the allFSI model, with a slight overestimation of the pion production at p_T of approximately 1 GeV/c. In contrast, the proton spectra were overestimated by the allFSI model with p_T above 1.5 GeV/c. The overestimation of protons occurred because the current coalescence model in AMPT can affect hadrons even in the high-p_T region. Coalescence overpredicted the baryon spectra at high p_T because it assumed that quarks combined directly by adding their momenta, neglecting the dominance of fragmentation at high pT, where baryon production became less efficient. In addition, hadronic rescattering, which induced radial flow-like effects to the proton spectra, enhanced proton production at higher p_T. Similar comparisons are presented in Sect. 3.3. The CR model results reasonably described the kaon p_T distribution, whereas the pion spectra were overpredicted, even in the higher p_T region. The proton yield was significantly overestimated in the entire p_T regime owing to the enhanced junction formation in the BLC CR model. This enhanced junction formation can result in significantly increased baryon production with different flavor components and must be regulated with the diquark production parameter in the string fragmentation model, as discussed in Ref. [45].

Fig. 1Full-screen View

(Color online) Transverse momentum spectra of charged ${\pi }^{\pm }$ (black), $K^{\pm }$ (red), and $p+\overline{p}$ (blue) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The lines represent model calculations from AMPT with all final-state interactions (a) and PYTHIA8 model with color reconnection effects (b), whereas the open markers represent ALICE data [63]

We compare the transverse momentum dependence of the neutral strange hadrons $K_{\text{S}}^0$ and Λ production in Fig. 2. The strange meson $K_{\text{S}}^0$ spectra were well described in both the allFSI and color reconnection models, with a slight overestimation at a p_T of approximately 0.5 to 1 GeV/c at a similar level. The Λ p_T distribution was well described by the color reconnection model, whereas allFSI overestimated the yield at a higher p_T. The overestimation in allFSI was largely related to the application of the coalescence mechanism over the entire transverse momentum region. The overall yield of Λ in allFSI roughly agreed with the experimental data because of the sensible choice of $r_{\text{BM}}^{\text{s}}$ in strange baryon production.

Fig. 2Full-screen View

(Color online) Transverse momentum spectra of $K_S^0$ (black) and $\text{Λ}+\overline{\text{Λ}}$ (red) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The lines represent model calculations from AMPT with all final-state interactions (a) and PYTHIA8 model with color reconnection effects (b), whereas the open markers represent ALICE data [63]

The transverse momentum spectra for D⁰, $D_{\text{s}}^{\pm }$, and ${\text{Λ}}_{\text{c}}^{\pm }$ are shown in Fig. 3. We converted the ALICE data from ${\text{d}}^2\sigma /\text{d}{p}_{\text{T}}\text{d}y$ to ${\text{d}}^{2}N/\text{d}{p}_{\text{T}}\text{d}y$ using the inelastic cross section ${\sigma }_{\text{inel}}=77.6\text{ mb}$ estimated in [64] to make a consistent comparison with our model calculations. The p_T distributions for all three charmed hadron species were well described in the allFSI model with the refined r_BM parameter for charm baryon formation. The calculations from both the allFSI model and color reconnection models agreed reasonably well with the experimental data.

Fig. 3Full-screen View

(Color online) Transverse momentum spectra of $D^0+{\overline{D}}^0$ (black), $D_{\text{s}}^{\pm }$ (red), and ${\text{Λ}}_{\text{c}}^{\pm }$ (blue) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The experimental data and model data in the figure are both the sum of particles and anti-particles divided by 2. The lines represent model calculations from AMPT with all final-state interactions (a) and PYTHIA8 model with color reconnection effects (b), whereas the open markers represent ALICE data [41]

The particle ratios p/π, K/π and $\text{Λ}/K_{\text{S}}^0$ varying with the transverse momentum are shown in Fig. 4. The baryon-to-meson ratios in allFSI for both p/π and $\text{Λ}/K_{\text{S}}^0$ were consistent with the experimental data within a p_T range of less than 2 GeV/c but were significantly overestimated at higher p_T. This observation was an outcome of our implementation of the quark coalescence hadronization mechanism over the entire pT range. The strange-to-non-strange meson ratio K/π was slightly underestimated at p_T greater than 1 GeV/c. Figure. 4(b) shows that the color reconnection model provided a satisfactory description of the baryon-to-meson ratios for p/π and $\text{Λ}/K_{\text{S}}^0$ over the examined p_T region, emphasizing the importance of BLC junction formation to the understanding of baryon production in string fragmentation models. However, the K/π ratio was significantly lower than the data for p_T values greater than 0.5 GeV/c with the CR effect.

Fig. 4Full-screen View

(Color online) p_T -differential p/π ratio (black), K/π ratio (red), and $\text{Λ}/K_{\text{S}}^0$ ratio (blue) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. Here, $\text{Λ}/K_{\text{S}}^0$ is actually $(\text{Λ}+\overline{\text{Λ}})/2K_{\text{S}}^0$. The lines represent model calculations from AMPT with all final-state interactions (a) and PYTHIA8 model with color reconnection effects (b), whereas the open markers represent ALICE data [63]

Figure 5 shows the p_T dependence of the heavy-flavored particle ratios, namely $D_{\text{s}}^{+}/D^0$ and ${\text{Λ}}_{\text{c}}^{+}/D^0$. For the $D_{\text{s}}^{+}/D^0$ ratio, both the allFSI and color reconnection model results agreed with the ALICE data within the uncertainties. No sizable p_T dependence of the $D_{\text{s}}^{+}/D^0$ ratio was found in either model within the examined p_T range. For the charmed baryon-to-meson ratio ${\text{Λ}}_{\text{c}}^{+}/D^0$, the allFSI model described the experimentally observed magnitude within the uncertainties, whereas the color-reconnection model predicted a slightly higher ratio. A characteristic peak appeared at a p_T of approximately 1.5 GeV/c in the ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio from both the AMPT model with all final-state interactions and the string fragmentation model with color reconnection effects, indicating the qualitative similarities of these two approaches.

Fig. 5Full-screen View

(Color online) p_T -differential $D_{\text{s}}^{+}/D^0$ ratio (black) and ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio (red) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The lines represent model calculations from AMPT with all final-state interactions (a) and PYTHIA8 model with color reconnection effects (b), whereas the open markers represent ALICE data [41]

Multiplicity dependence of particle ratio

After examining the inclusive hadron production, in this section, we investigate the variations in the hadron chemical compositions by exploring the ratios of different particle species with the final-state charged particle density. Figure 6 shows the p_T integrated particle ratios of K/π, p/π, and $\text{Λ}/K_{\text{S}}^0$ as functions of $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$. To quantify the impact of different transport stages on the multiplicity dependence of hadron production, we turned on the final-state parton and hadron interactions in the AMPT model in a step-by-step manner. Figure 6(a) shows that no multiplicity dependence of the light-flavor particle ratios was generated if all the final-state interactions were turned off in the AMPT model. The $\text{Λ}/K_{\text{S}}^0$ ratio increased rapidly with the event multiplicity at a low $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$ and became almost saturated in the high-$\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$ region when the final-state parton interactions and coalescence hadronization are included, as shown in Fig. 6(b). As indicated in Fig. 6(c), further hadronic rescattering slightly suppressed the increase in $\text{Λ}/K_{\text{S}}^0$ ratio and resulted in a weak p/π multiplicity dependence. The AMPT model, including all final-state interactions, effectively reproduced the multiplicity dependence of the light-flavor particle ratios observed in the experimental data [16, 19], except for the underestimation of the $\text{Λ}/K_{\text{S}}^0$ ratio. This underestimation can be understood as the initial conditions from PYTHIA8 providing too small a strange-baryon yield, as shown in Fig. 6(a). Final-state interactions, together with the coalescence hadronization mechanism, were required to describe the multiplicity dependence of light hadron production in the experiment, particularly for the strange baryon-to-meson ratio within the AMPT framework. In contrast, the string fragmentation model with CR effects predicted a strong multiplicity dependence of the baryon-to-meson ratio for both p/π and $\text{Λ}/K_{\text{S}}^0$ shown in Fig. 6(d). The multiplicity dependence of the unusual baryon-to-meson ratio observed in the experiment was well captured by the CR model while the p/π ratio was overestimated, and the K/π ratio was lower than the experimental data.

Fig. 6Full-screen View

(Color online) Integrated yield ratios of K/π (red), p/π(black), and $\text{Λ}/K_{\text{S}}^0$ (blue) as a function of the charged-particle multiplicity density in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. Here, $\text{Λ}/K_{\text{S}}^0$ represents $(\text{Λ}+\overline{\text{Λ}})/2K_{\text{S}}^0$. The lines represent model calculations from AMPT noFSI(a), AMPT pFSI(b), AMPT allFSI(c), and PYTHIA8 CR(d), whereas the point markers represent ALICE data [16, 19]

In Fig. 7, we compare the charm hadron ratios $D_{\text{s}}^{+}/D^0$ and ${\text{Λ}}_{\text{c}}^{+}/D^0$ as a function of $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$ from both AMPT and CR model, compared with the experimental data. This comparison shows that the $D_{\text{s}}^{+}/D^0$ ratio barely changed with the final-state interactions implemented in the current AMPT, and the measured $D_{\text{s}}^{+}/D^0$ ratio was well described by the model calculations. Nevertheless, the ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio was flat over the entire multiplicity range and was significantly lower than that of the ALICE data if no final-state interactions were included in the AMPT, as shown in Fig. 7(a). The incorporation of parton rescatterings in AMPT resulted in a significant multiplicity dependence of the ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio. The charmed baryon-to-meson ratio increased significantly with $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$, which was seemingly faster than the experimental data. The increase in this fraction indicated that more charm quarks are involved in partonic scatterings in high-multiplicity events and will be hadronized via the coalescence process with a higher baryon formation probability. The string fragmentation model with CR effects also predicts a very rapid increase in ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio, which varied with event multiplicity, and a smooth $D_{\text{s}}^{+}/D^0$ ratio, which was consistent with the experimental data.

Fig. 7Full-screen View

(Color online) Integrated yield ratios of $D_{\text{s}}^{+}$/D⁰ (black) and ${\text{Λ}}_{\text{c}}^{+}/D^0$ (red) as a function of charged-particle multiplicity density in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The lines represent model calculations from AMPT noFSI (a), AMPT pFSI (b), AMPT allFSI (c), and PYTHIA8 CR (d), whereas the point markers represent ALICE data [21]

An interesting observation was that the ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio in the AMPT continued to increase with $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$ even in very high-multiplicity events, unlike the case observed in Fig. 6 for strange baryon-to-meson ratio calculations. In the string-melting AMPT framework, hadron production was not only dependent on the coalescence hadronization mechanism but was also related to the fraction of unmelted initial strings in each event. In high-multiplicity events, the initial strings were more likely to be destroyed by the surrounding parton matter, and the coalescence hadronization effect became more dominant. The saturation of the strange baryon-to-meson ratio at an intermediate to high multiplicity indicates that the initial strings containing the strange quark were fully destroyed, and the hadron production ratios were determined mostly by the coalescence parameter value in this scenario. Charm initial string objects were generated independently of the bulk medium and may have not participated fully in its evolution. A significant fraction of the undestroyed charm strings remained, although this gradually decreased with increasing event multiplicity. This discrepancy suggested a different degree of thermalization for the charm and strange quarks in small systems. Researchers have extensively discussed that canonical suppression in the statistical hadronization model may play a similar role in the determination of multiplicity dependence for both light and heavy flavor hadron production [24, 47, 65], suggesting a suppressed baryon-to-meson ratio at low multiplicity and a saturated value at high-multiplicity events, even for charm and bottom hadrons. Testing the multiplicity dependence of heavy-flavor hadron production ratios using high-precision experimental data is important for understanding the details of these flavor hierarchy structures.

To further identify the multiplicity-dependent shape of the particle ratios with different quark flavor components, we present the self-normalized particle ratios for p/π, K/π, $\text{Λ}/K_{\text{S}}^0$, $D_{\text{s}}^{+}/D^0$ and ${\text{Λ}}_{\text{c}}^{+}/D^0$ in each multiplicity event class divided by the minbias–particle ratio as a function of the charged particle density $\langle \text{d}{N}_{\text{ch}}/\text{d}\eta \rangle$ in Fig. 8. Thus, the difference in the magnitude of the particle ratios with different flavors is cancelled, and the shape of each particle ratio can be quantitatively compared on the same ground, indicating how rapidly a certain particle ratio changes with event multiplicity. Figure 8(a) shows that the self-normalized ratios for all particle species were consistent with unity and independent of the charged particle density if all final-state parton and hadron interactions were switched off in the AMPT model. The multiplicity dependence of the hadron ratios that appeared after the partonic-level evolution stage was included, as shown in Fig. 8(b). Within the coalescence procedure of the AMPT model, a clear flavor hierarchy structure was produced, indicating an increased slope from p/π, $\text{Λ}/K_{\text{S}}^0$ to ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio, which depended on the final-state multiplicity. The strange-to-non-strange meson ratio was flat across the entire multiplicity range. The follow-up hadronic rescattering shown in Fig. 8(c) resulted in a slight modification of the multiplicity dependence of the self-normalized ratio for p/π and $\text{Λ}/K_{\text{S}}^0$, whereas the flavor hierarchy structure remained unchanged. The experimental data were converted from the measured particle ratios in each multiplicity bin divided by the corresponding ratio obtained for the minimum bias events, incorporating all events with different multiplicity bins [16, 19, 21]. The converted experimental data show that the slopes for all particle ratios were quite small, in contrast to the model calculations, although a tantalizing flavor hierarchy structure existed in the three baryon-to-meson ratios. We also examined the same self-normalized particle ratio results from the color reconnection model calculations shown in Fig. 8(d). A much larger slope was also predicted for the baryon-to-meson ratios with different quark flavors by the color-reconnection model compared with the experimental data. However, the baryon-to-meson ratios $\text{Λ}/K_{\text{S}}^0$ and p/π were observed to be quite similar in this framework. The ordering between the non-strange and strange baryon-to-meson ratios obtained with the color reconnection model was not as clear as that observed with the coalescence hadronization process.

Fig. 8Full-screen View

(Color online) Integrated yield ratios normalized to the minbias particle ratio of p/π (red), K/π (black), $\text{Λ}/K_{\text{S}}^0$ (green), $D_{\text{s}}^{+}/D^0$ (blue) and ${\text{Λ}}_{\text{c}}^{+}/D^0$ (magenta) as a function of the charged particle density in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. Here, $\text{Λ}/K_{\text{S}}^0$ represents $(\text{Λ}+\overline{\text{Λ}})/2K_{\text{S}}^0$. The lines represent model calculations from AMPT noFSI (a), AMPT pFSI (b), AMPT allFSI (c), and PYTHIA8 CR (d), whereas the markers represent ALICE data [16, 19, 21]

Transverse momentum dependence of the double ratio

We further investigated the transverse momentum-dependent modifications to the hadron production ratios induced by the event multiplicities and present the double ratios constructed with the particle ratios from central collisions (0%–5% centrality) divided by those from peripheral collisions (64.5%–100% centrality) in Fig. 9. Because the strange-to-non-strange meson ratios K/π and $D_{\text{s}}^{+}/D^0$ were observed to be approximately unity, independent of the event multiplicity and p_T in our model, we only show the baryon-to-meson ratio results in this comparison. This was consistent with the expectation that all particle ratios will be close to unity when the final-state parton and hadron interactions were off. The final-state parton evolution and corresponding coalescence hadronization process resulted in non-trivial p_T dependence for three baryon-to-meson ratios with different flavors and a clear ordering structure at low p_T shown, as shown in Fig. 9(b). Within the investigated p_T range, the $\text{Λ}/K_{\text{S}}^0$ and p/π double ratios increased with p_T whereas ${\text{Λ}}_{\text{c}}^{+}/D^0$ was almost unrelated to p_T. A significant ordering from ${\text{Λ}}_{\text{c}}^{+}/D^0$ to p/π was observed, particularly in the lower p_T region. The follow-up hadronic scattering suppresses the $\text{Λ}/K_{\text{S}}^0$ double ratio significantly at p_T greater than 1 GeV/c whereas the low-p_T region preserved the flavor hierarchy structure shown in Fig. 9(c). Additionally, we observed that the color reconnection model exhibited an increasing behavior that was dependent on p_T for all baryon-to-meson ratios. The charm baryon-to-meson ratio was enhanced much more strongly than that in the light-flavor sectors, but no significant difference was observed between the $\text{Λ}/K_{\text{S}}^0$ and p/π ratios. The charm baryon-to-meson ratio was more sensitive to the color reconnection effect in high-multiplicity events compared with the light-flavor sectors, which was largely related to the fact that heavy flavor baryons can only be produced with baryon junction formations, which were controlled by the reconnection probability of the string objects beyond the leading color level [61]. Additionally, some radial flow-like features were observed in the ${\text{Λ}}_{\text{c}}^{+}/D^0$ ratio in high-multiplicity pp collisions from color reconnection calculations, particularly in the low-p_T region, similar to the observations in the experimental data [21]. This behavior can be attributed to the junction formation process beyond the leading color level. Because charmed hadrons cannot be created in the fragmentation process, enhanced ${\text{Λ}}_{\text{c}}^{+}$ production in color reconnection is always related to the junctions connecting two dipole strings. In low-multiplicity events, the dipole strings are frequently parallel to the beam direction; therefore, the CR-induced ${\text{Λ}}_{\text{c}}^{+}$ favors production in the low-pT region [45]. In contrast, higher p_T jets are produced in high-multiplicity events, resulting in reconnected strings shifting to the higher p_T region. The reconnected junction may capture this p_T increasement and make the CR related ${\text{Λ}}_{\text{c}}^{+}$ shifted to a larger value p_T. The difference between the coalescence hadronization model and color reconnection models on the flavor associated multiplicity dependence of baryon-to-meson ratios should be further tested with more precise experimental data in the future.

Fig. 9Full-screen View

(Color online) p_T dependence of the double ratios, defined as the particle ratios in high-multiplicity events (0%–5%) divided by those in low-multiplicity events (64.5%–100%), for p/π (black), $\text{Λ}/K_{\text{S}}^0$ (red) and ${\text{Λ}}_{\text{c}}^{+}/D^0$ (blue) in $\sqrt{s}=13\text{ TeV}$ pp collisions at mid-rapidity |y|<0.5. The lines represent model calculations from AMPT noFSI(a), AMPT pFSI(b), AMPT allFSI(c), and PYTHIA8 CR(d)