Correlations, multiplicity distributions, and the ridge in pp and p-Pb collisions

Measurements made by the ALICE Collaboration of single- and two-particle distributions in high-energy pp and p-Pb collisions are used to characterize the interactions in small collision systems, tune models of particle production in QCD, and serve as a baseline for heavy-ion observables. The measurements of charged-particle multiplicity density, $\langle dN_{ch}/d\eta\rangle$, and multiplicity distributions are shown in pp and p-Pb collisions, including data from the top center-of-mass energy achieved at the Large Hadron Collider (LHC), $\sqrt{s}$ = 13 TeV. Two-particle angular correlations in p-Pb collisions are studied in detail to investigate long-range correlations in pseudorapidity which are reminiscent of structures previously thought unique to heavy-ion collisions.


Introduction
In high-energy hadronic collisions, studies of inclusive single-particle distributions are used to investigate particle production in QCD. The charged-particle multiplicity density in pp and p-Pb collisions is measured by ALICE over a range of centre-of-mass energies, including the top LHC energy of √ s = 13 TeV. The multiplicity distributions are also shown, and all the experimental data is compared with Monte Carlo models. Since the produced multiplicity is dominated by soft (low-momentum) particle production, which is in the non-perturbative regime of QCD, these measurements can be used to further constrain and tune models.
Beyond single-particle inclusive measurements, two-particle correlation studies have yielded surprising results in small collision systems, showing the presence of correlations between particles over large ranges in pseudorapidity in high-multiplicity pp and p-Pb collisions. These correlations are reminiscent of features observed in heavy-ion collisions where they are commonly attributed to anisotropic flow (v n ). The transverse momentum (p T ), pseudorapidity (η), and particle species dependence of v 2 in p-Pb collisions has been measured in ALICE. In particular, in the analysis of correlations between forward muons and mid-rapidity charged hadrons it is possible to measure the v 2 for large values of pseudorapidity in both the proton-going and Pb-going directions. These observations will be used to deepen our understanding of possible collective effects in small collision systems and their implications for heavy-ion physics.

Multiplicity density
In ALICE, the charged-particle multiplicity density, dN ch /dη has been measured across a wide range in center-of-mass energy, at √ s = 0.9, 2.36, 2.76, 7, 8, and 13 TeV. The multiplicity is measured in different classes of events, including inelastic events ('INEL'), inelastic events with at least one charged particle produced within |η| < 1 ('INEL>0'), and non-single-diffractive events ('NSD'). Figure 1 shows the results for the INEL and INEL>0 classes, which demonstrate power-law scaling with √ s. Results from p-Pb collisions are also shown in Fig. 1 [2]. Additionally, the charged particle multiplicity density has been measured as a function of pseudorapidity in INEL and NSD events at √ s = 0.9 and 2.36 TeV, and in INEL and INEL>0 events at 13 TeV, as shown in Fig. 2. The distributions are also compared to multiple Monte Carlo (MC) models Figure 1. The mid-rapidity charged-particle multiplicity density, dN ch /dη , is shown as a function of center-ofmass energy for (left) pp and (right) pp, p-Pb, and Pb-Pb collisions at the LHC [2,7]. on the detector acceptance and efficiency due to the limited hit statistics and the current alignment precision of the detector is estimated by this method to be 1.5 %. The uncertainty in background corrections was estimated according to the description in Section 3. The total systematic uncertainty on the pseudorapidity density measurement at 0.9 TeV is smaller than 2.5 % for INEL collisions and is about 3.3 % for NSD collisions. At 2.36 TeV, the corresponding uncertainties are below 6.7 % and 3.7 % for INEL and NSD collisions, respectively. For all cases, they are dominated by uncertainties in the cross sections of diffractive processes and their kinematics.
To evaluate the systematic error on the multiplicity distribution, a new response matrix was generated for each change listed above and used to unfold the measured spectrum. The difference between these unfolded spectra and the unfolded spectrum produced with the unaltered response matrix determines the systematic uncertainty.
Additional systematic uncertainties originate from the unfolding method itself, consisting of two contributions. The first one arises from statistical fluctuations due to the finite number of events used to produce the response matrix as well as the limited number of events in the measurement. The unfolding procedure was repeated 100 times while randomizing the input measurement and the response matrix according to their respective statistical uncertainties. The resulting uncertainty due to the response matrix fluctuations is negligible. The uncertainty on the measured multiplicity distribution due to the event statistics reproduces the uncertainty obtained with the minimization procedure, as expected.
A second contribution arises from the influence of the regularization on the distribution. The bias introduced by the regularization was estimated using the prescription described in [39] and is significantly lower than the statistical error inferred from the χ 2 minimization, except in the lowmultiplicity region. In that region, the bias is about 2 %, but the statistical uncertainty is negligible. Therefore, we added the estimated value of the bias to the statistical uncertainty in this region. The correction procedure is insensitive to the shape of the multiplicity distribution of the events, which produce the response matrix. Table 2 summarizes the systematic uncertainties for the multiplicity distribution measurements. Note that the uncertainty is a function of the multiplicity which is reflected by the ranges of values. Further details about the analysis, corrections, and the evaluation of the systematic uncertainties are in [38].
Both the pseudorapidity density and multiplicity distribution measurements have been cross-checked by a second analysis employing the Time-Projection Chamber (TPC) [1]. It uses tracks and vertices reconstructed in the TPC in the pseudorapidity region |η| < 0.8. The pseudorapidity density is corrected using a method similar to that used for the SPD analysis. The results of the two independent analyses are consistent.

Results
In this section, pseudorapidity density and multiplicity distribution results are presented for two centre-of-mass energies and compared to results of other experiments and to models. For the model comparisons we have used QGSM [6], three different tunes of PYTHIA, tune D6T [9], tune ATLAS-CSC [10] and tune Perugia-0 [11], and PHO-JET [12]. The PYTHIA tunes have been developed by three independent groups extensively comparing Monte Carlo distributions to underlying-event and minimum-bias  high multiplicities and for the 0.9 TeV sample, the PHO-JET model agrees well with the data. The PYTHIA tunes D6T and Perugia-0 underestimate the data at high multiplicities and the ATLAS-CSC tune is above the data in this region. At 2.36 TeV, ATLAS-CSC tune of PYTHIA and, to some extent, PHOJET are close to the data. The ratios of data over Monte Carlo calculations are very similar in all three pseudorapidity ranges and suggests that the stronger rise with energy seen in the charged-particle density is, at least partly, due to a larger fraction of highmultiplicity events.
From these multiplicity distributions we have calculated the mean multiplicity and first reduced moments 17 Table 4. Mean multiplicity and Cq-moments (5) of the multiplicity distributions measured by UA5 [19] in proton-antiproton collisions at √ s = 0.9 TeV, and by ALICE at √ s = 0.9 TeV and 2.36 TeV, for NSD events in three different pseudorapidity intervals. The first error is statistical and the second systematic.
UA5 pp ALICE pp √ s = 0.9 TeV √ s = 2.36 TeV |η| < 0.5 ⟨Nch⟩ 3.61 ± 0.04 ± 0.12 3.60 ± 0.02 ± 0.11 4.47 ± 0.03 ± 0.10 C2 1   Fig. 11. Energy dependence of the Cq-moments (5) of the multiplicity distributions measured by UA5 [19] and ALICE at both energies for NSD events in two different pseudorapidity intervals. The error bars represent the combined statistical and systematic uncertainties. The data at 0.9 TeV are displaced horizontally for visibility. summarized in Table 4. For |η| < 0.5 and |η| < 1.0 our results are compared to the UA5 measurement for pp collisions at √ s = 0.9 TeV [19]. Note that the mean multiplicities quoted in this table are those calculated from the multiplicity distributions and are therefore slightly different from the values given in Table 3. The value of the pseudorapidity density obtained when averaging the multiplicity distribution for |η| < 0.5 is consistent with the value obtained in the pseudorapidity-density analysis. This is an important consistency check, since the correction methods in the pseudorapidity-density and multiplicity-distribution analyses are different. Our data are consistent with UA5 proton-antiproton measurements at 900 GeV ( Fig. 8a and Table 4). The en- KNO scaling is also shown (right). [8] including PYTHIA 6 [3], PYTHIA 8 [4], PHOJET [5], and EPOS LHC [6]. It can be observed in Fig. 2 that the model in best agreement with the √ s = 13 TeV data is PYTHIA 6. These results will be used for further tuning of the Monte Carlo generators.

Multiplicity distributions
The charged-particle multiplicity distributions, P(N ch ), were measured at √ s = 0.9 and 2.36 TeV, as shown in Fig. 3. The experimental data were compared to results from PHOJET and three PYTHIA 6 tunes (Perugia-0, ATLAS-CSC, and D6T). The best agreement with the data is achieved by PHOJET at √ s = 0.9 TeV and the ATLAS-CSC tune of PYTHIA 6 at √ s = 2.36 TeV. Furthermore, the multiplicity distributions were scaled by the mean multiplicity to obtain the distribution of z = N ch / N ch , also shown in Fig. 3. The hypothesis that the distributions of N ch P(z) are independent of center-of-mass energy is known as KNO scaling [9], and these experimental results indicate that KNO scaling holds up to approximately z = 4.

Two-particle correlations
Two-particle angular correlations, which are distributions in relative azimuthal angle (∆ϕ = ϕ trig − ϕ assoc ) and relative pseudorapidity (∆η = η trig − η assoc ) between trigger and associated particles, are used to study many aspects of the physics of heavy-ion collisions, in particular jet fragmentation and collective effects. In elementary collisions and small collision systems such as pp they show characteristic features attributed to jet production, while in heavy-ion collisions the same jet features are observed in addition to structures around ∆ϕ = 0 (nearside) and ∆ϕ = π (awayside) extended in ∆η. These long-range correlations, known as 'ridges,' are often attributed to hydrodynamic flow behavior in the quark-gluon plasma (QGP) and are typically quantified by the coefficients of a Fourier cosine series, v n .
It was therefore surprising when a nearside ridge was observed in high multiplicity collisions of small systems, pp [10] and p-Pb [11]. Furthermore, it was observed that in p-Pb collisions at √ s NN = 5.02 TeV the nearside peak yields are mostly independent of multiplicity [12], meaning that for the same trigger and associated p T the same jet population is selected regardless of multiplicity. This served as justification to subtract the correlations in low-multiplicity events from the high-multiplicity correlation functions in order to remove correlations due to jet and minijet fragmentation. This subtraction procedure (illustrated in Fig. 4) showed the nearside ridge more clearly and also revealed a symmetric ridge on the awayside [13,14]. This 'double ridge' structure was decomposed into Fourier coefficients in order to extract the parameter v 2 in p-Pb collisions. The analysis was repeated with identified particles and it was observed that the v 2 shows similar mass ordering as was observed in Pb-Pb collisions. Figure 5 shows v 2 in p-Pb collisions as a function of p T for unidentified hadrons, pions, kaons, and protons [15]. Results from CMS show similar behavior for K 0 S mesons and Λ baryons [16]. The v 2 in p-Pb collisions was also measured with the two-and multi-particle cumulant methods [17][18][19]. It is important to note, however, that while the v 2 measured in p-Pb collisions shows qualitatively similar features as v 2 measured in heavy ion collisions, the physical mechanism leading to a non-zero v 2 is still under theoretical debate and the presence of v 2 does not necessarily imply the existence of hydrodynamics or a QGP in small collision systems.

Muon-hadron correlations
In order to gain more information about potential collective effects and constrain theoretical calculations, it is important to measure the strength of the ridge to larger ∆η and to measure the dependence of v 2 on pseudorapidity. Both of these points are addressed in the muon-hadron analysis performed in ALICE [20], in which correlation functions between muons at forward rapidities and charged hadrons  at mid-rapidity are constructed in order to investigate the long-range behavior of the double ridge structure for −5 < ∆η < −1.5.
The correlations between muons detected in the FMS and tracklets reconstructed in the ITS were measured in high-multiplicity (the top 20% of the analyzed event sample) and in low-multiplicity (60-100%) events. As in [13], the low-multiplicity correlations are subtracted from the high-multiplicity correlation functions to remove structures associated with jet fragmentation. After subtraction, the correlation functions were projected onto ∆ϕ, and then fit with a Fourier cosine series to extract v 2 for the muons detected at forward rapidities. The resulting v µ 2 {2PC,sub} values are shown in Fig. 6 for muons heading in the proton-and Pb-going directions. The data are compared with an AMPT [21] simulation in which the muon decay products are scaled to account for the efficiency of the absorber in the ALICE FMS. In Fig. 6 it is seen that while AMPT qualitatively describes the p T -dependence at low p T , there are significant quantitative differences in the p T -dependence and η-dependence between data and the model. At high p T (above p T ∼ 2 GeV/c), where muon production is dominated by heavy flavor decays, AMPT does not describe the data well. This could be because heavy flavor muons have a non-zero v 2 , or the parent particle composition or v 2 values in data and AMPT are different. The ratio of v µ 2 {2PC,sub} in the Pb-going and p-going directions is also shown in Fig. 6 where it is observed to be independent of p T within the statistical and systematic uncertainties. A constant fit to the data points shows that the v 2 is (16 ± 6)% higher in the Pb-going than in the p-going direction. These results are qualitatively in agreement with model predictions. However, current theoretical calculations cannot be directly compared with experimental results, because the effects of the absorber are included in the experimental data (unfolding such effects could not be done in a model-independent way). Future model calculations should use the efficiencies provided in [20] in order to compare directly to the experimental results.

Conclusions
Single-particle inclusive and two-particle correlation measurements are used to characterize the pp and p-Pb collision systems. The charged-particle multiplicity density has been measured across a range of energies including the top LHC energy of √ s = 13 TeV. The pseudorapidity dependence of dN ch /dη has been compared with MC generators in order to further tune the models. The multiplicity distributions were also compared with models and demonstrate KNO scaling up to z ∼ 4. In two-particle measurements, long-range correlations in pseudorapidity are observed up to η ∼ 4 and ∆η ∼ 5. The presence of these correlations is reminiscent of Pb-Pb collisions where the structures are frequently attributed to hydrodynamic flow, with similar mass ordering being observed in both small and large systems. The v 2 of forward muons in the Pb-going direction is observed to be higher EPJ Web of Conferences than in the p-going direction. While these features are similar to correlations observed in heavy-ion collisions, further theoretical and phenomenological investigations are needed before any inferences about collectivity in small systems can be drawn.