Transverse momentum spectra as a function of charged-particle multiplicity
The \(p_{\mathrm{T}}\) distributions of charged particles, measured in \(|\eta |<0.8\) for pp collisions at \(\sqrt{s}=5.02\) and 13 TeV, are shown in Fig. 2 for the different multiplicity classes selected using the estimator based on \(N_{\mathrm{SPD\,tracklets}}\). The bottom panels depict the ratios to the \(p_{\mathrm{T}}\) distribution of the INEL\(\,>0\) event class. The features of the spectra, i.e. the change of the spectral shape going from low- to high-multiplicity values, are qualitatively the same for both energies. The only significant difference is the multiplicity reach which is higher at 13 TeV than that at 5.02 TeV. In the following we discuss the results for pp collisions at the highest energy. As shown in Fig. 2, the \(p_{\mathrm{T}}\) spectra become harder as the multiplicity increases, which contributes to the increase of the average transverse momentum with multiplicity. The ratios to the INEL\(>0\) \(p_{\mathrm{T}}\) distribution exhibit two distinct behavior. While at low \(p_{\mathrm{T}}\) (\(<0.5\) GeV/c) the ratios exhibit a modest \(p_{\mathrm{T}}\) dependence, for \(p_{\mathrm{T}} >0.5\) GeV/c they strongly depend on multiplicity and \(p_{\mathrm{T}}\).
Figure 3 shows the multiplicity dependent \(p_{\mathrm{T}}\) spectra using a multiplicity selection based on the V0M amplitude. Results for pp collisions at \(\sqrt{s}=5.02\) and 13 TeV are shown. The average multiplicity values are significantly smaller than those reached with the mid-pseudorapidity estimator (based on \(N_{\mathrm{SPD\,tracklets}}\)). For example, in pp collisions at \(\sqrt{s}=13\) TeV, while the average charged-particle multiplicity density amounts to 56.55 for the highest \(N_{\mathrm{SPD\,tracklets}}\) class, it only reaches 27.61 for the highest V0M multiplicity class. We note that for similar average particle densities, e.g. the multiplicity classes II (V0M) and VII’ (SPD tracklets) in pp collisions at \(\sqrt{s}=13\) TeV, the ratios measured using the V0M amplitude and the \(N_{\mathrm{SPD\,tracklets}}\) are similar. The comparison of the \(p_{\mathrm{T}}\) spectra for these multiplicity classes is shown in Fig. 4. We observe that for transverse momentum below 0.5 GeV/c, the spectra exhibit the same shape. For transverse momenta within 0.5–3 GeV/c the spectra for the multiplicity class II is harder than that for the VII” class. At higher \(p_{\mathrm{T}}\), the spectral shapes are the same, but the yield of the class II is \(\sim \)15% higher than that for the VII’ class. Similar results are obtained if we compare the multiplicity classes I and VI’ for pp collisions at 5.02 TeV.
Commonly, the particle production is characterized by quantities like integrated yields, or any fit parameter of the curve extracted from fits to the data, for example, the so-called inverse slope parameter reported by ALICE in Ref. [39]. This facilitates the visualization of the evolution of the particle production as a function of multiplicity and the comparison among different colliding systems. Several publications have adopted this strategy for soft (\(p_{\mathrm{T}} <2\) GeV/c) [2, 6, 21] physics and others to describe the particle production for intermediate and high \(p_{\mathrm{T}}\) (\(2 \le p_{\mathrm{T}} <20\) GeV/c) [40]. It is interesting and important to define a common quantity to compare the shape of the high-\(p_{\mathrm{T}}\) part of the spectra of different particle species and collision systems. The natural choice is fitting a power-law function (\(\alpha \times p_{\mathrm{T}} ^{-n}\)) to the invariant yield and studying the multiplicity dependence of the exponent (n) extracted from the fit. Figure 5 illustrates the results considering particles with transverse momentum within 6–20 GeV/c for pp at \(\sqrt{s}=13\) TeV. It is worth mentioning that within uncertainties the power-law function describes rather well the data in that \(p_{\mathrm{T}}\) interval. Similarly, the \(p_{\mathrm{T}}\) spectra simulated with the different generators are well described (within 2%) by the power-law function.
Within uncertainties, going from low to high multiplicity n decreases taking values from 6 to 5, respectively. A similar behavior has been reported for heavy-ion collisions [41]. Moreover, the results using the two multiplicity estimators are consistent within the overlapping multiplicity interval. This result is consistent with that shown in Fig. 4. PYTHIA 6 and 8 simulations describe the trends very well, but a strong deviation between EPOS LHC and data is observed. In PYTHIA 8, it has been shown that the number of high-\(p_{\mathrm{T}}\) jets increases with event multiplicity. Moreover, for a given event multiplicity and fixed jet \(p_{\mathrm{T}}\), the high-\(p_{\mathrm{T}}\) tails of the charged-particle spectra are very similar in low- and high-multiplicity events [16]. Therefore, based on PYTHIA 8 studies, the reduction of the power-law exponent with increasing multiplicity can be attributed to an increasing number of high-\(p_{\mathrm{T}}\) jets.
As pointed out above, the ratios to the INEL\(\,>0\) \(p_{\mathrm{T}}\) distributions for \(\langle \mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \rangle \lesssim 25\) exhibit a weak \(p_{\mathrm{T}}\)-dependence for \(p_{\mathrm{T}} >4\) GeV/c. This applies to both energies and to all multiplicity estimators. To illustrate better the behaviour of the yields at high momenta, we adopted a representation previously used for heavy-flavour hadrons [42] to point out to the similarities between the two results. The trend at high-\(p_{\mathrm{T}}\) is highlighted in Fig. 6, which shows the integrated yields for three transverse momentum intervals (\(2<p_{\mathrm{T}} <10\) GeV/c, \(4<p_{\mathrm{T}} <10\) GeV/c, and \(6<p_{\mathrm{T}} <10\) GeV/c) as a function of the average mid-pseudorapidity multiplicity. Both the charged-particle yields and the average multiplicity are self-normalized, i.e. they are divided by their average value for the INEL\(\,>0\) sample. The high-\(p_{\mathrm{T}}\) (\(>4\) GeV/c) yields of charged particles increase faster than the charged-particle multiplicity, while the increase is smaller when we consider lower-\(p_{\mathrm{T}}\) particles. The trend of the data is qualitatively well reproduced by PYTHIA 8, but for \(p_{\mathrm{T}} >6\) GeV/c the model significantly overestimates the ratio by a factor larger than 1.5. Although the shapes of the spectra (characterized by n) are not well reproduced by EPOS LHC, the model gives the best description of the self-normalized yields. Despite the large uncertainties, it is clear the data show a non-linear increase.
Double-differential study of the average transverse momentum
The spherocity-integrated average \(p_{\mathrm{T}}\) as a function of \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \) for pp collisions at \(\sqrt{s}=13\) TeV is shown in Fig. 7. In accordance with measurements at lower energies [21], the \(\langle p_{\mathrm{T}} \rangle \) increases with \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \). In PYTHIA 8 the effect is enhanced by color reconnection, which allows the interaction among partons originating from multiple semi-hard scatterings via color strings. The minimum-bias data are compared with analogous measurements for the most jet-like structure (0 – 10%) and isotropic (90 – 100%) event classes. Studying observables as a function of spherocity reveals interesting features. On one hand, for isotropic events the average \(p_{\mathrm{T}}\) stays systematically below the spherocity-integrated \(\langle p_{\mathrm{T}} \rangle \) over the full multiplicity range; on the other hand, for jet-like events the \(\langle p_{\mathrm{T}} \rangle \) is higher than that for spherocity-integrated events. Moreover, within uncertainties the overall shape of the correlation, i.e. a steep linear rise below \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta =10\) followed by a less steep but still linear rise above, is not spherocity-dependent.
Figure 8 shows that within uncertainties, PYTHIA 8 with color reconnection gives an adequate description of the spherocity-integrated event class. It is worth mentioning that color reconnection was originally introduced to explain the rise of \(\langle p_{\mathrm{T}} \rangle \) with multiplicity [43]. However, PYTHIA 6 shows a steeper rise of \(\langle p_{\mathrm{T}} \rangle \) with \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \) than that seen in data. The Perugia 2011 tune relies on Tevatron and SPS minimum-bias data, while the Monash tune was constrained using the early LHC measurements [38]. The comparison of data with EPOS LHC is also shown. Clearly, the quantitative agreement is as good as that achieved by PYTHIA 8. The EPOS LHC model uses a different approach in order to simulate the hadronic interactions. Namely, the model considers a collective hadronization which depends only on the geometry and the density [37].
For the 0 – 10% and 90 – 100% spherocity classes, Fig. 8 also shows comparisons between data and Monte Carlo generators (PYTHIA 6, PYTHIA 8 and EPOS LHC). It is worth mentioning that we also used spherocity percentiles in all the Monte Carlo event generators reported in this paper because their spherocity distributions do not differ much from those measured in data. For further Monte Carlo comparisons the spherocity binning which was used in the analysis is provided as HEP data. In low-multiplicity events (\(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta <10\)), the deviations between data and PYTHIA 8 (without color reconnection) are smaller and larger respectively for the 0 – 10% and 90 – 100% spherocity classes than those seen for the 0 – 100% spherocity class. The effect could be a consequence of the reduction of color reconnection contribution in events containing jets surrounded by a small underlying event activity. For isotropic events the three models quantitatively describe the correlation. Even for PYTHIA 6, the size of the discrepancy which was pointed out for the spherocity-integrated event class is reduced. On the contrary, for jet-like events both PYTHIA 6 and 8 exhibit a larger disagreement with the data. These models produce three distinct multiplicity regions, for \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \lesssim 7\) the models give a steeper rise of \(\langle p_{\mathrm{T}} \rangle \) than data. Within the intermediate multiplicity interval (\(7\lesssim \mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \lesssim 25\)), the slope of \(\langle p_{\mathrm{T}} \rangle \) given by models is more compatible with that seen in data, although the models overestimate the average \(p_{\mathrm{T}}\). While in data the average \(p_{\mathrm{T}}\) increases at a constant rate with multiplicity for \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \gtrsim 7\), PYTHIA 6 and 8 shows a third change of the slope of \(\langle p_{\mathrm{T}} \rangle \), observed for \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \gtrsim 25\). The data to model ratio indicates a discrepancy larger than 10%, which is larger than the systematic uncertainties associated to \(\langle p_{\mathrm{T}} \rangle \) in that multiplicity interval.
In order to study the details of the changes of the functional form of \(\langle p_{\mathrm{T}} \rangle (N_{\mathrm{ch}})\) due to the spherocity selection, Fig. 9 shows the average \(p_{\mathrm{T}}\) of jet-like and isotropic events normalized to that for the spherocity-integrated event class. For jet-like events, the data exhibit a hint of a modest peak at \(\mathrm{d}N_{\mathrm{ch}}/d\eta \sim 7\), which is not significant if we consider the size of the systematic uncertainties. Moreover, within uncertainties the ratio remains constant for \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \gtrsim 25\). EPOS LHC describes rather well the high-multiplicity behavior, however, it overestimates the peak. PYTHIA 6 and 8 show the worst agreement with the data. In this representation, the three distinct regions, which were described before are highlighted. In PYTHIA 8, the peak (at \(\mathrm{d}N_{\mathrm{ch}}/d\eta \sim 7\)) in jet-like events is caused by particles with transverse momentum above 2 GeV/c. The size of the peak is determined by particles with \(p_{\mathrm{T}} >5-6\) GeV/c. In contrast, data do not show a significant peak structure for any specific transverse momentum interval. We also varied the upper \(p_{\mathrm{T}}\) (\(0.15< p_{\mathrm{T}} < p_{\mathrm{T}} ^{\mathrm{max}}\)) limit (\(p_{\mathrm{T}} ^{\mathrm{max}} = 10\) GeV/c is the default) and studied the effect on the extracted \(\langle p_{\mathrm{T}} \rangle \). The \(\langle p_{\mathrm{T}}\rangle \) remains constant within uncertainties for \(4< p_{\mathrm{T}} ^{\mathrm{max}} < 10\) GeV/c in data and for \(6< p_{\mathrm{T}} ^{\mathrm{max}} < 10\) GeV/c in PYTHIA 8. For \(p_{\mathrm{T}} ^{\mathrm{max}}=2\) GeV/c the \(\langle p_{\mathrm{T}} \rangle \) decreases by 23% (29%) in data (PYTHIA 8) compared to \( p_{\mathrm{T}} ^{\mathrm{max}}=10\) GeV/c. The relative difference of \(\langle p_{\mathrm{T}} \rangle \) between data and PYTHIA 8 amounts to 9% (4%) for \(p_{\mathrm{T}} ^{\mathrm{max}}=2\) GeV/c (\(p_{\mathrm{T}} ^{\mathrm{max}}=10\) GeV/c). The results suggest that the power-law tail produces a smaller impact on data than in PYTHIA 8. A similar ratio for isotropic events shows a smaller structure at \(\mathrm{d}N_{\mathrm{ch}}/d\eta \sim 7\). This effect is well reproduced by all models.
Finally, we also examined the evolution of \(\langle p_{\mathrm{T}} \rangle (N_{\mathrm{ch}})\) going from the most jet-like to the most isotropic event classes. Figure 10 shows the spherocity-dependent \(\langle p_{\mathrm{T}} \rangle (N_{\mathrm{ch}})\) in data and models, the data to model ratios are displayed in Fig. 11. The difference between the 0 – 10% and 10 – 20% spherocity classes is smaller for data and EPOS LHC than for PYTHIA 6 and 8. Moreover, within uncertainties PYTHIA 8 describes rather well the data for the 10 – 20% spherocity class. This contrasts with the disagreement between the model and data for the 0 – 10% spherocity class. Other features in PYTHIA 6 and 8 are the reduction of the bump at \(\mathrm{d}N_{\mathrm{ch}}/d\eta \sim 7\) and the disappearance of a third rise of the \(\langle p_{\mathrm{T}} \rangle \) for \(\mathrm{d}N_{\mathrm{ch}}/d\eta \gtrsim 25\) when one goes from the 0 – 10% to the 10 – 20% spherocity classes. The agreement among models and data for the 20 – 100% spherocity classes is similar to that observed for the 10 – 20% spherocity class. Within uncertainties, PYTHIA 8 and EPOS LHC qualitatively describe the data for \(\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta \gtrsim 10\), while PYTHIA 6 overestimates the average \(p_{\mathrm{T}}\).
From previous LHC studies we know that the production cross section of jets in high-multiplicity pp collisions is smaller in data than predicted from the Monte Carlo generators [32, 44, 45]. Therefore, a possible interpretation is that the low-momentum partons, color connected with higher momentum ones (jets), would produce an overall increase of the hadron transverse momentum. This would affect more the low-\(p_{\mathrm{T}}\) part of the spectrum associated with jet-enriched samples, which are achieved by requiring low-spherocity values. The incorporation of these new observables into the PYTHIA 8 tuning could be a challenge because, on one hand, the color reconnection has to be reduced to describe the low-\(S_{0}\) data; on the other hand, the variation should not be too large because the good description of the spherocity-integrated and isotropic classes could be affected.