# Multiplicity and transverse momentum fluctuations in inelastic proton–proton interactions at the CERN Super Proton Synchrotron

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## Abstract

Measurements of multiplicity and transverse momentum fluctuations of charged particles were performed in inelastic p+p interactions at 20, 31, 40, 80, and 158 \(\text{ GeV }\!/\!c\) beam momentum. Results for the scaled variance of the multiplicity distribution and for three strongly intensive measures of multiplicity and transverse momentum fluctuations \(\varDelta [P_{T},N]\), \(\varSigma [P_{T},N]\) and \(\varPhi _{p_T}\) are presented. For the first time the results on fluctuations are fully corrected for experimental biases. The results on multiplicity and transverse momentum fluctuations significantly deviate from expectations for the independent particle production. They also depend on charges of selected hadrons. The string-resonance Monte Carlo models Epos and Urqmd do not describe the data. The scaled variance of multiplicity fluctuations is significantly higher in inelastic p+p interactions than in central Pb+Pb collisions measured by NA49 at the same energy per nucleon. This is in qualitative disagreement with the predictions of the Wounded Nucleon Model. Within the statistical framework the enhanced multiplicity fluctuations in inelastic p+p interactions can be interpreted as due to event-by-event fluctuations of the fireball energy and/or volume.

## Keywords

Transverse Momentum Multiplicity Distribution Beam Momentum Intensive Measure Super Proton Synchrotron## 1 Introduction and motivation

This paper presents experimental results on event-by-event fluctuations of multiplicities and transverse momenta of charged particles produced in inelastic p+p interactions at 20, 31, 40, 80 and 158 \(\text{ GeV }\!/\!c\). The measurements were performed by the multi-purpose NA61/SHINE [1, 2] experiment at the CERN Super Proton Synchrotron (SPS). They are part of the strong interaction program devoted to the study of the properties of the onset of deconfinement and search for the critical point of strongly interacting matter. Within this program a two-dimensional scan in collision energy and size of colliding nuclei is in progress. Data on p+p, Be+Be and Ar+Sc collisions were already recorded and data on p+Pb and Xe+La collisions will be registered within the coming years. The expected signal of a critical point is a non-monotonic dependence of various fluctuation measures in such a scan, for recent review see Ref. [3].

The NA49 experiment [4] published results for central Pb+Pb collisions in the collision energy range 20*A* to 158\(A\,\text{ GeV }\), as well as for p+p, C+C and Si+Si reactions at 158\(A\,\text{ GeV }\). Multiplicity fluctuations were measured in terms of the scaled variance of the multiplicity distribution [5, 6] and fluctuations of the transverse momentum of the particles were studied employing measures \(\varPhi _{p_T}\) [7, 8], recently \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) [9].

Also, at SPS energies results on event-by-event fluctuations in Pb+Au collisions on mean transverse momentum were published by the CERES experiment [10] and in Pb+Pb collisions by the WA98 Collaboration on charged particle multiplicity, transverse energy [11] as well as the ratio of the charged to neutral pion multiplicity [12].

An interpretation of the experimental results on nucleus–nucleus collisions relies to a large extent on a comparison with the corresponding data on p+p and p+A interactions. However, in particular the available data on fluctuations are sparse. Suitable fluctuation measurements for p+p interactions only exist at 158 GeV/c beam momentum [5, 7]. Moreover, fluctuation measurements cannot be corrected in a model independent manner for partial phase space acceptance. Thus all measurements of the scan should be performed in the same phase space region.

In nucleus–nucleus reactions the impact parameter of the collisions cannot be tightly controlled. This problem results in additional unwanted contributions to fluctuations the effect of which needs to be suppressed by employing so-called strongly intensive measures. In addition to \(\varPhi _{p_T}\) two recently proposed strongly intensive quantities \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) are studied in this publication.

The paper is organized as follows. In Sect. 2 strongly intensive fluctuation measures are introduced. The experimental setup is presented in Sect. 3. Data processing and simulation and the analysis procedure are described in Sects. 4 and 5, respectively. Results are presented and discussed in Sect. 6 and compared to model calculations in Sect. 7. A summary and outlook in Sect. 8 closes the paper.

Across this paper the pion rapidity is calculated in the collision center-of-mass system: \(y_{\pi } = {{\mathrm{atanh}}}(\beta _L)\), where \(\beta _L = p_L/E\) is the longitudinal (*z*) component of the velocity, \(p_L\) and *E* are pion longitudinal momentum and energy given in the collision center-of-mass system. The transverse component of the momentum is denoted \(p_T\) and the transverse mass \(m_T\) is defined as \(m_T = \sqrt{m_{\pi }^2 + (cp_T)^2}\), where \(m_{\pi }\) is the charged pion mass in \(\text{ GeV }\). The azimuthal angle \(\phi \) is the angle between transverse momentum vector and the horizontal (*x*) axis. The nucleon mass and collision energy per nucleon pair in the center-of-mass system are denoted \(m_\mathrm {N}\) and \(\sqrt{s_\mathrm {NN}}\), respectively.

## 2 Fluctuation measures

### 2.1 Intensive fluctuation measures

*A*

*N*in the events is widely used to quantify multiplicity fluctuations in high-energy heavy-ion experiments.

The scaled variance assumes the value \(\omega [N] = 0\) for \(N = \mathrm{const}.\) and \(\omega [N] = 1\) for a Poisson multiplicity distribution.

### 2.2 Strongly intensive fluctuation measures

Unfortunately, the volume of the matter produced in heavy-ion collisions cannot be fixed and changes significantly from one event to another. Therefore, it is very important to be able to measure the properties of the created matter independently of its volume fluctuations. The quantities which allow this are called strongly intensive measures. They depend neither on the volume nor on the fluctuations of the volume. Ratios of mean multiplicities are both intensive and strongly intensive measures. The situation is, however, much more difficult for the analysis of fluctuations. For example the scaled variance is an intensive but not strongly intensive measure.

*A*and

*B*:

*i*.

There is an important difference between \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\). Only the first two moments: \(\langle P_{T} \rangle \), \(\langle N \rangle \), and \(\langle P_{T}^2 \rangle \), \(\langle N^2 \rangle \) are required to calculate \(\varDelta [P_{T},N]\), whereas \(\varSigma [P_{T},N]\) includes the correlation term \(\langle P_{T}N \rangle - \langle P_{T} \rangle \langle N \rangle \). Thus \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) can be sensitive to various physics effects in different ways. In Ref. [14] strongly intensive quantities including the correlation term are named the \(\varSigma \) family, and those based only on mean values and variances the \(\varDelta \) family.

Properties of \(\varPhi _{p_T}\), \(\varDelta [P_{T},N]\), and \(\varSigma [P_{T},N]\) in the absence of fluctuations, and in the independent particle model (IPM) and the model of independent sources (MIS) (\(N_S\) denotes the number of sources)

Unit | No fluctuations | IPM | MIS | |
---|---|---|---|---|

\(\varPhi _{p_T}\) | \(\text{ MeV }/c\) | \(\varPhi _{p_T}=- \sqrt{ \overline{p_T} \omega [p_T]}\) | \(\varPhi _{p_T}=0\) | Does not depend on \(N_s\) and its fluctuations |

\(\varDelta [P_{T},N]\) | Dimensionless | \(\varDelta [P_{T},N]=0\) | \(\varDelta [P_{T},N]=1\) | Does not depend on \(N_s\) and its fluctuations |

\(\varSigma [P_{T},N]\) | Dimensionless | \(\varSigma [P_{T},N]=0\) | \(\varSigma [P_{T},N]=1\) | Does not depend on \(N_s\) and its fluctuations |

\(\omega [N]\) | Dimensionless | \(\omega [N]=0\) | \(\omega [N]=1\) | Does not depend on \(N_s\) |

## 3 Experimental facility

### 3.1 The NA61/SHINE detector

A set of scintillation and Cherenkov counters as well as beam position detectors (BPDs) upstream of the spectrometer provide timing reference, identification and position measurements of incoming beam particles. Trajectories of individual beam particles were measured in a telescope of beam position detectors placed along the beam line (BPD-1/2/3 in Fig. 1). These counters are small (\(4.8\times 4.8\) cm\(^2\)) proportional chambers with cathode strip readout, providing a resolution of about 100 \(\upmu \)m in two orthogonal directions. Due to properties of the H2 beam line both the beam width and the divergence at the NA61/SHINE target increase with decreasing beam momentum. The trigger scintillator counter S4 placed downstream of the target is used to select events with collisions in the target area. The liquid hydrogen target as well as the proton beams and triggers are described in Sects. 3.2 and 3.3, respectively.

The main tracking devices of the spectrometer are four large volume Time Projection Chambers (TPCs). Two of them, the vertex TPCs (VTPC-1 and VTPC-2 in Fig. 1), are located in the magnetic fields of two super-conducting dipole magnets with a maximum combined bending power of 9 Tm, which corresponds to about 1.5 T and 1.1 T fields in the upstream and downstream magnets, respectively. In order to optimize the acceptance of the detector at each collision momentum, the field in both magnets was adjusted proportionally to the beam momentum.

Two large TPCs (MTPC-L and MTPC-R) are positioned downstream of the magnets symmetrically to the beam line. The fifth small TPC (GAP-TPC) is placed between VTPC-1 and VTPC-2 directly on the beam line. It closes the gap along the beam axis between the sensitive volumes of the other TPCs.

The TPCs are filled with Ar:CO\(_2\) gas mixtures in proportions 90:10 for the VTPCs and the GAP-TPC, and 95:5 for the MTPCs.

The particle identification capability of the TPCs based on measurements of the specific energy loss, \(\mathrm{d}E\!/\!\mathrm{d}x\), is augmented by time-of-flight measurements using Time-of-Flight (ToF) detectors. The high resolution forward calorimeter, the Projectile Spectator Detector (PSD), measures energy flow around the beam direction, which in nucleus–nucleus collisions is primarily given by the projectile spectators.

The results presented in this paper were obtained using information from the TPCs the Beam Position Detectors as well as from the beam and trigger counters.

### 3.2 Target

NA61/SHINE uses various solid nuclear targets and a liquid hydrogen target (see Sect. 3.3 for details). For data taking on p+p interactions a liquid hydrogen target of 20.29 cm length (2.8% interaction length) and 3 cm diameter was placed 88.4 cm upstream of VTPC-1. The Liquid Hydrogen Target facility (LHT) filled the target cell with para-hydrogen obtained in a closed-loop liquefaction system which was operated at 75 mbar overpressure with respect to the atmosphere. At the atmospheric pressure of 965 mbar the liquid hydrogen density is \(\rho _\mathrm {LH} = 0.07\) g/cm\(^3\).

Data taking with inserted (I) and removed (R) liquid hydrogen (LH) in the LHT was alternated in order to calculate a data-based correction for interactions with the material surrounding the liquid hydrogen.

### 3.3 Beams and triggers

Secondary beams of positively charged hadrons at 20, 31, 40, 80 and 158 \(\text{ GeV }\!/\!c\) were produced from 400 \(\text{ GeV }\!/\!c\) protons extracted from the SPS onto a beryllium target in a slow extraction mode with a flat-top of 10 s. The secondary beam momentum and intensity was adjusted by proper setting of the H2 beam-line magnet currents and collimators. The beam was transported along the H2 beam-line towards the experiment. The precision of the setting of the beam magnet currents was approximately 0.5%. This was verified by a direct measurement of the beam momentum at 31 \(\text{ GeV }\!/\!c\) by bending the incoming beam particles into the TPCs with the maximum magnetic field [19]. Selected beam properties are given in Table 2.

Basic properties of the beam used in the study of p+p interactions at 20, 31, 40, 80, and 158 \(\text{ GeV }\!/\!c\). The first column gives the beam momentum. The second and third columns list typical numbers of beam particles at NA61/SHINE per spill (about 10 s) and the fraction of protons in the beam, respectively

\(p_\mathrm {beam}\) [\(\text{ GeV }\!/\!c\) ] | Particles per spill | Proton fraction (%) |
---|---|---|

20 | 1000k | 12 |

31 | 1000k | 14 |

40 | 1200k | 14 |

80 | 460k | 28 |

158 | 250k | 58 |

## 4 Data processing and simulation

Detector parameters were optimized by a data-based calibration procedure which also took into account their time dependence, for details see Refs. [20, 21].

- (i)
cluster finding in the TPC raw data, calculation of the cluster center of gravity and total charge,

- (ii)
reconstruction of local track segments in each TPC separately,

- (iii)
matching of track segments into global tracks,

- (iv)
track fitting through the magnetic field and determination of track parameters at the first measured TPC cluster,

- (v)
determination of the interaction vertex using the beam trajectory (

*x*and*y*coordinates) fitted in the BPDs and the trajectories of tracks reconstructed in the TPCs (*z*coordinate), - (vi)
refitting the particle trajectory using the interaction vertex as an additional point and determining the particle momentum at the interaction vertex,

- (vii)
matching of ToF hits with the TPC tracks.

- (i)
generation of inelastic p+p interactions using the EPOS model,

- (ii)
propagation of outgoing particles through the detector material using the GEANT 3.21 package which takes into account the magnetic field as well as relevant physics processes, such as particle interactions and decays,

- (iii)
simulation of the detector response using dedicated NA61/SHINE packages which simulates charge clusters in the TPCs and introduces distortions corresponding to all corrections applied to the real data,

- (iv)
simulation of the interaction trigger selection by checking whether a charged particle hits the S4 counter, see Sect. 3.3,

- (v)
storage of the simulated events in a file which has the same format as the raw data,

- (vi)
reconstruction of the simulated events with the same reconstruction chain as used for the real data,

- (vii)
matching of the reconstructed to the simulated tracks based on the cluster positions.

## 5 Analysis procedure

- (i)
applying event and track selection criteria,

- (ii)
evaluation of the moments of distributions of quantities needed to calculate fluctuations (Eqs. 1, 2, 3, 4),

- (iii)
evaluation of corrections to the moments based on experimental data and simulations,

- (iv)
calculation of the corrected fluctuations.

- (i)
contribution of off-target interactions,

- (ii)
losses of inelastic p+p interactions due to the trigger and the event and track selection criteria,

- (iii)
contribution of particles other than primary charged hadrons,

- (iv)
losses of primary charged hadrons due to the track selection criteria.

Data sets together with the statistics of events recorded and selected for the analysis in target inserted and target removed configurations

\(p_\mathrm{beam}\) | \(\sqrt{s_\mathrm{NN}}\) | \(y^\mathrm{CM}_\mathrm{beam}\) | Target inserted | Target removed | ||
---|---|---|---|---|---|---|

[\(\text{ GeV }\!/\!c\) ] | [\(\text{ GeV }\) ] | All | Selected | All | Selected | |

20 | 6.27 | 1.90 | 1 324 k | 255 k | 122 k | 8 k |

31 | 7.62 | 2.10 | 3 140 k | 1058 k | 332 k | 35 k |

40 | 8.73 | 2.23 | 5 226 k | 2008 k | 528 k | 88 k |

80 | 12.32 | 2.57 | 4 444 k | 1791 k | 458 k | 88 k |

158 | 17.27 | 2.91 | 3 538 k | 1819 k | 426 k | 74 k |

### 5.1 Event selection criteria

- (i)
no off-time beam particle was detected within \(\pm 1.5\ \upmu \)s around the trigger particle,

- (ii)
the beam particle trajectory was measured in BPD-3 and at least one of BPD-1 or BPD-2 detectors,

- (iii)
there was at least one track reconstructed in the TPCs and fitted to the interaction vertex,

- (iv)
events with a single, well-measured positively charged track with absolute momentum close to the beam momentum (\(p > p_\mathrm{beam} -1~\) \(\text{ GeV }\!/\!c\)) were rejected.

- (v)
the vertex

*z*position (fitted using the beam and TPC tracks) was not farther away than 50 cm from the center of the LHT,

Summary of event and track selection criteria used in the analysis

Standard cuts | Loose cuts | Tight cuts | |
---|---|---|---|

T2 trigger | Applied | Applied | Applied |

BPD | Applied | Applied | Applied |

Off-time | \({<}\pm 1.5~\upmu \)s | No cut | \(< \pm 5~\upmu \)s |

Fitted vertex z position | \(\pm 50~\)cm | No cut | \(\pm 10\) cm |

Not elastic scatter | Applied | Applied | Applied |

Total points | \({\ge } 30\) | No cut | \({\ge }30\) |

VTPC (GTPC) points | \({\ge } 15 (5)\) | \({>}10(5)\) | \({\ge }30(6)\) |

\(|b_x|\) | \({\le } 4~\)cm | No cut | \({\le } 2~\)cm |

\(|b_y|\) | \({\le } 2~\)cm | No cut | \({\le } 1~\)cm |

\(p_{T}\) | \({\le }1.5~\text{ GeV }\!/\!c \) | \({\le } 1.5~\text{ GeV }\!/\!c \) | \({\le } 1.5~\text{ GeV }\!/\!c \) |

\(e^{\pm }\) | Applied | Applied | Applied |

### 5.2 Track selection criteria

- (i)
the track momentum fit at the interaction vertex should have converged,

- (ii)
the total number of reconstructed points on the track should be greater than 30,

- (iii)
the sum of the number of reconstructed points in VTPC-1 and VTPC-2 should be greater than 15 or the number of reconstructed points in the GAP-TPC should be greater than 5,

- (iv)
the distance between the track extrapolated to the interaction plane and the interaction point (impact parameter) should be smaller than 4 cm in the horizontal (bending) plane and 2 cm in the vertical (drift) plane,

- (v)
the track should be measured in a high (\({\ge }90\%\)) TPC acceptance and tracking efficiency region (see Sect. 5.3),

- (vi)
tracks with energy loss and total momentum values characteristic for electrons were rejected.

- (vii)
the transverse momentum was required to be less than 1.5 \(\text{ GeV }\!/\!c\).

### 5.3 Determination of the analysis kinematical acceptance

### 5.4 Data-based correction for off-target interactions

*X*) was calculated as

*z*coordinate of the interaction vertex. All vertices far away from the target originate from interactions with the beam line and detector materials. Neglecting the beam attenuation in the target one gets

*z*coordinate of the reconstructed interaction vertex for events recorded with the liquid hydrogen inserted and removed are shown in Fig. 3.

### 5.5 Simulation-based correction for other biases

*N*, \(P_{T}\) and \(P_{T,2} = \sum \nolimits _{i=1}^{N} p^{2}_{T_{i}}\) for positively, negatively and all charged hadrons, separately. The event quantity \(P_{T,2}\) is needed to calculate \(\omega [p_T]\) using only event quantities. \(\omega [p_T]\) appears in the normalization factors \(C_{\varDelta }\) and \(C_{\varSigma }\). Thus there are three three-dimensional tables of correction factors. Then for a given charge selection an event

*i*with

*N*, \(P_{T}\) and \(P_{T,2}\) is weighted with the correction factor \(c_{i}\) from the table of corrections for this charge selection and from the bin which corresponds to

*N*, \(P_{T}\) and \(P_{T,2}\). Thus, in the absence of off-target interactions, the corrected mean value of a quantity

*X*is

### 5.6 The final correction

In order to illustrate the correction procedure and its impact on the results selected distributions of the relevant event quantities, *N*, \(P_T\), \(NP_T\) and \(P_{T,2}\), and results on \(\varDelta [P_{T},N]\), \(\varSigma [P_{T},N]\), \(\varPhi _{p_{T}}\) and \(\omega [N]\) obtained at the subsequent stages of the procedure are presented and discussed.

### 5.7 Statistical uncertainties

The statistical uncertainties of \(\varPhi _{p_{T}}\), \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) were estimated as follows. The whole sample of events was divided into 30 independent sub-samples. The values of \(\varPhi _{p_{T}}\), \(\varDelta [P_{T},N]\), and \(\varSigma [P_{T},N]\) were evaluated for each sub-sample separately (following all the procedures described above, including target removed and Monte Carlo corrections) and the dispersions (\(D_{\varPhi }\), \(D_{\varDelta }\), and \(D_{\varSigma }\)) of the results were then calculated. The statistical uncertainty of \(\varPhi _{p_{T}}\) (\(\varDelta [P_{T},N]\) or \(\varSigma [P_{T},N]\)) is taken to be equal to \(D_{\varPhi }/\sqrt{30}\) (\(D_{\varDelta }/\sqrt{30}\) or \(D_{\varSigma }/\sqrt{30}\)).

For each beam momentum, 5 million events were generated and reconstructed, several times more than the recorded experimental data. Therefore statistical uncertainties arising from the event statistics of the simulation were neglected.

### 5.8 Systematic uncertainties

The uncertainties of corrections for the event selection and feed-down are expected to be mostly due to uncertainties in modeling of p+p interactions, whereas the uncertainties of the remaining corrections (e.g. for the track reconstruction inefficiency) are expected to be mostly due to imperfectness of the detector simulation. Total systematic uncertainties were calculated by adding in quadrature uncertainties calculated in (i) and (ii).

## 6 Results

*accepted*hadrons produced in all inelastic p+p interactions. The accepted hadrons are hadrons produced within the kinematical acceptance selected for the analysis [23]; see also Sect. 5. The results are corrected for event and track losses due to detector inefficiencies, selection criteria and the interaction trigger, as well as contamination of tracks from weak decays and secondary interactions and leptons from primary interactions. The correction procedure is described in detail in Sect. 5. Results are not corrected for the kinematic acceptance. This acceptance should be taken into account when the data are compared with models. Table 5 shows mean multiplicities of negatively and positively charged hadrons within the kinematical acceptance selected for the analysis in this paper.

Mean multiplicities of negatively and positively charged hadrons produced in inelastic p+p interactions at 20, 31, 40, 80 and 158 \(\text{ GeV }\!/\!c\) in the kinematical acceptance used in this paper and in the NA49 publications. For comparison also mean multiplicity of \(\pi ^-\) mesons in full phase space as obtained by NA61/SHINE [20] is shown in the second column

NA61/SHINE | NA49- | NA49- | NA49- | ||||||
---|---|---|---|---|---|---|---|---|---|

kinematical acceptance used in the analysis (see Sect. 5.3) | narrow \(\phi \) acc. common for all energies; \(1.1< y_{\pi } < 2.6\) | no VTPC-1-only tracks; \(1.1< y_{\pi } < y_\mathrm{beam}\) | no VTPC-1-only tracks; \(0< y_{\pi } < y_\mathrm{beam}\) | ||||||

\(p_\mathrm {beam}\)[\(\text{ GeV }\!/\!c\) ] | \(\langle \pi ^{-}\rangle \) [20] | \(\langle h^{-}\rangle \) | \(\langle h^{+}\rangle \) | \(\langle h^{-}\rangle \) | \(\langle h^{+}\rangle \) | \(\langle h^{-}\rangle \) | \(\langle h^{+}\rangle \) | \(\langle h^{-}\rangle \) | \(\langle h^{+}\rangle \) |

20 | 1.01 | 0.34 | 0.91 | 0.04 | 0.18 | 0.10 | 0.32 | 0.18 | 0.50 |

31 | 1.31 | 0.51 | 1.14 | 0.06 | 0.20 | 0.16 | 0.41 | 0.29 | 0.66 |

40 | 1.48 | 0.64 | 1.30 | 0.07 | 0.21 | 0.21 | 0.48 | 0.38 | 0.76 |

80 | 1.94 | 1.04 | 1.78 | 0.10 | 0.23 | 0.41 | 0.74 | 0.66 | 1.11 |

158 | 2.44 | 1.49 | 2.26 | 0.15 | 0.25 | 0.68 | 1.09 | 1.05 | 1.56 |

First, one observes that the results, in general, significantly deviate from the reference values \(\varDelta [P_{T},N] = \varSigma [P_{T},N] = 1\) and \(\varPhi _{p_T} =0\) \(\text{ MeV }/c\), which are expected in the case of independent particle production.

Second, the results for the three charge selections show differences. The effect of long-range correlations, e.g., caused by conservation laws, decreases with decreasing fraction of accepted particles. In particular, many charged hadrons come from decays of resonances into two or more hadrons, e.g., \(\rho \rightarrow \pi ^+ + \pi ^-\). Positively charged hadrons are mostly \(\pi ^{+}\) mesons and protons. Less of them come from resonance decays into two or more positively charged hadrons. The majority of negatively charged hadrons are negatively charged pions and only a small fraction comes from resonance decays into two or more negatively charged hadrons. Thus the correlations due to resonance decay kinematics decrease from all charged hadrons to positively and negatively ones. Other sources of correlations like quantum statistics and Coulomb interactions are also expected to impact fluctuations differently for different charge selections.

Third, the collision energy dependence of \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) are opposite. With increasing collision energy \(\varDelta [P_{T},N]\) decreases, whereas \(\varSigma [P_{T},N]\) increases. The largest deviations from unity for both \(\varDelta [P_{T},N]\) and \(\varSigma [P_{T},N]\) are observed for all charged hadrons at 158 \(\text{ GeV }\!/\!c\). Note that at this energy the fraction of accepted particles is the largest.

## 7 Comparison with models and central Pb+Pb collisions

Figure 8 shows a comparison of the measured fluctuations with predictions of two commonly used models: Epos1.99 [24] and UrQMD [27]. The predictions were calculated for the NA61/SHINE acceptance [23].

The Epos1.99 predictions agree quantitatively with results on \(\omega [N]\), They exhibit the same trend with increasing collision energy as the experimental results on \(\varSigma [P_{T},N]\) and \(\varPhi _{p_T}\), but there is qualitative disagreement with results on \(\varDelta [P_{T},N]\). Note that the Epos1.99 model agrees reasonably well with single particle spectra of identified hadrons produced in the same inelastic p+p interactions [28].

Essentially all UrQMD predictions disagree with the data. In many cases even qualitative disagreement is observed. Note that UrQMD also fails to describe single particle spectra in p+p reactions [28]. These disagreements are probably due to problems in modeling of hadron production via resonance decays [29].

NA61/SHINE records data at beam momenta per nucleon identical to or close to those at which NA49 [4] performed measurements of central Pb+Pb collisions. This allows for a direct study of the system size dependence of various hadron production properties in the SPS energy range.

The NA61/SHINE results presented in this paper are obtained in a phase space acceptance [23] which is larger than the acceptances used by NA49 to obtain the corresponding results for central Pb+Pb collisions [5, 6, 7, 8]. Thus, in order to compare the NA61/SHINE measurements with the NA49 data, the more restrictive NA49 cuts were applied to the NA61/SHINE data.

The narrowest acceptance (referred as to the NA49-*N* acceptance ) was used in the NA49 study of collision energy dependence of (transverse momentum)-multiplicity fluctuations [8]. The NA49-*N* acceptance is limited to the rapidity range \(1.1< y_{\pi } < 2.6\), where \(y_{\pi }\) is the rapidity calculated in the cms assuming the pion mass, and selects particles in a common narrow azimuthal angle wedge at all beam momenta.

*N*acceptance. As expected, the restriction of the acceptance strongly reduces the values of the fluctuation measure.

*N*acceptance. No significant difference is observed between the results for the two reactions. Moreover, neither shows a structure in the collision energy dependence which could be attributed to the onset of deconfinement or the critical point.

*M*(top row) and NA49-

*B*(bottom row) acceptances [6]. The NA49 medium (NA49-

*M*) acceptance includes all particles well measured by the NA49 detector within the rapidity range \(1.1< y_{\pi } < y_\mathrm{beam}\) and the NA49 broad acceptance (NA49-

*B*) extends the range to \(0< y_{\pi } < y_\mathrm{beam}\). Significant differences are observed between the results for p+p and Pb+Pb reactions; see below for a discussion. However, neither shows a structure in the collision energy dependence which could be attributed to the onset of deconfinement or the critical point.

The scaled variance is significantly larger for inelastic p+p interactions at 158 \(\text{ GeV }\!/\!c\) than for the 1% most central Pb+Pb collisions at 158\(A\,\text{ GeV }\!/\!c\). In the following this difference will be discussed within the Wounded Nucleon Model (WNM) [13] and the Statistical Model (SM) [30] of particle production. These models are selected because they are simple and play a special role in analyzing the physics of heavy-ion collisions. The discussion will be focused on the results for negatively charged hadrons as they are weakly influenced by decays of resonances [31].

*W*is the number of wounded nucleons and \(\omega ^*[N]\) denotes the scaled variance calculated for any fixed value of

*W*. The number of wounded nucleons (protons) in p+p interactions is fixed to two and the measured scaled variance for p+p interactions can be used as \(\omega ^*[N]\). The second component of the sum Eq. 11 vanishes or is positive. Consequently the WNM predicts that the scaled variance in nucleus–nucleus collisions has to be equal to or larger than the one in inelastic proton–proton interactions. The NA61/SHINE and NA49 results presented in Fig. 12 clearly contradict this prediction.

Note that at SPS energies multiplicity distributions of negatively charged hadrons in inelastic p+p, n+p and n+n interactions are different [32]. The largest difference is observed between the distributions in p+p and n+n interactions. Thus the prediction of the WNM for Pb+Pb collisions which takes into account the isospin effects estimated using the Epos model ranges between the results for p+p (the measured NA61/SHINE data) and n+n interactions (estimated based on the Epos predictions and the NA61/SHINE data). This range is bounded in Fig. 12 (*left*) by the horizontal dashed lines.

The Statistical Model of multi-particle production was initiated by Fermi in 1950 [30]. Its basic assumption states that all possible micro-states of the macroscopic system created in a collision are equally probable. For a non-interacting (ideal) gas of Boltzmann particles in the grand canonical ensemble (IB-GCE) the multiplicity distribution is Poissonian (\(\omega [N] = 1\)) independent of the (fixed) system volume (upper dashed line in Fig. 12 (right)). Resonance decays and Bose effects increase the scaled variance from 1 to 1.1, whereas the conservation laws reduce it if the system volume is sufficiently large [31]. The combined effect is demonstrated by the lower dashed line in Fig. 12 (right). In fact the NA49 result for the 1% most central Pb+Pb collisions at 158\(A\,\text{ GeV }\!/\!c\) is well described by the hadron gas model in the micro–canonical ensemble (HG-MCE) [31]. Within the statistical models a scaled variance significantly larger than one as measured in inelastic p+p interactions at high collision energies (top SPS and higher) can be understood as a result of volume and/or energy fluctuations [33].

Multiplicity and transverse momentum fluctuations quantified using strongly intensive measures were studied in a number of theoretical papers. In particular, the influence of resonance decays [17, 34], quantum statistics [18, 34] and a dependence of the mean transverse momentum on multiplicity [17, 34] was considered. These studies are important for a qualitative understanding of experimental data and predictions of complicated Monte Carlo models. However, the obtained results cannot be directly compared to the measurements as they did not include important effects. In particular, the limited experimental acceptance is difficult to take into account in simple models.

## 8 Summary

This paper presents measurements of multiplicity and transverse momentum fluctuations of negatively, positively and all charged hadrons produced in inelastic p+p interactions at 20, 31, 40, 80 and 158 \(\text{ GeV }\!/\!c\) beam momentum. Values for the scaled variance of multiplicity distributions and three strongly intensive measures of multiplicity–transverse momentum fluctuations \(\varDelta [P_{T},N]\), \(\varSigma [P_{T},N]\) and \(\varPhi _{p_T}\) were obtained. These results were calculated in the NA61/SHINE acceptance which has to be taken into account in a quantitative comparison with models and other results. For the first time the results on fluctuations are fully corrected for experimental biases, in particular, for the losses of inelastic events due to the trigger and analysis event selections as well as for the contamination of particles from weak decays and secondary interactions.

The measurements of multiplicity and transverse momentum fluctuations significantly deviate from expectations for independent particle production (\(\varDelta [P_{T},N] = \varSigma [P_{T},N] = 1\), \(\varPhi _{p_T} =0\) \(\text{ MeV }/c\)). They also depend on the charges of the selected hadrons. The scaled variances of the multiplicity distributions increase with increasing collision energy and for all charged hadrons at 158 \(\text{ GeV }\!/\!c\) reach the value of 2.

The string-resonance Monte Carlo models Epos and Urqmd do not describe the data well. In several cases the collision energy dependence predicted by the models even shows a trend qualitatively different from the measurements.

The scaled variance of multiplicity distributions is significantly higher in inelastic p+p interactions than in the 1% most central Pb+Pb collisions measured by NA49 at the same energy per nucleon. The largest difference is observed at 158\(A\,\text{ GeV }\!/\!c\). This result is in qualitative disagreement with the predictions of the Wounded Nucleon Model. The low level of multiplicity fluctuations measured in central Pb+Pb collisions is well reproduced by the statistical model in the micro-canonical ensemble formulation. Within the statistical framework the enhanced multiplicity fluctuations in inelastic p+p interactions can be interpreted as due to event-by-event fluctuations of the fireball energy and/or volume.

## Notes

### Acknowledgements

We would like to thank the CERN PH, BE, and EN Departments for the strong support of NA61. This work was supported by the Hungarian Scientific Research Fund (Grants OTKA 68506 and 71989), the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, the Polish Ministry of Science and Higher Education (Grants 667/N-CERN/2010/0, NN 202 48 4339 and NN 202 23 1837), the Polish National Center for Science (Grants 2011/03/N/ST2/03691, 2013/11/N/ST2/03879, 2014/13/N/ST2/02565, 2014/14/E/ST2/00018 and 2015/18/M/ST2/00125), the Foundation for Polish Science – MPD program, co-financed by the European Union within the European Regional Development Fund, the Federal Agency of Education of the Ministry of Education and Science of the Russian Federation (SPbSU research grant 11.38.242.2015), the Russian Academy of Science and the Russian Foundation for Basic Research (Grants 08-02-00018, 09-02-00664 and 12-02-91503-CERN), the Ministry of Education, Culture, Sports, Science and Technology, Japan, Grant-in-Aid for Scientific Research (Grants 18071005, 19034011, 19740162, 20740160 and 20039012), the German Research Foundation (Grants GA 1480/2-2), the EU-funded Marie Curie Outgoing Fellowship, Grant PIOF-GA-2013-624803, the Bulgarian Nuclear Regulatory Agency and the Joint Institute for Nuclear Research, Dubna (bilateral contract No. 4418-1-15/17), Ministry of Education and Science of the Republic of Serbia (Grant OI171002), Swiss Nationalfonds Foundation (Grant 200020117913/1), ETH Research Grant TH-01 07-3 and the U.S. Department of Energy.

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