1 Introduction

Angular correlations between two particles have been established as a powerful tool to study the properties of the system created in high energy collisions of hadrons and nuclei [1–16]. These measurements are usually performed in a two dimensional space as a function of \(\Delta \eta \) and \(\Delta \varphi \). Here \(\Delta \eta \) and \(\Delta \varphi \) are the differences in pseudorapidity \(\eta = -{\mathrm {ln}}[\tan (\theta /2)]\) (where \(\theta \) is the polar angle of a particle relative to the beam axis) and in azimuthal angle \(\varphi \) of the two particles.

In heavy-ion collisions at both the Relativistic Heavy Ion Collider (RHIC) [3–11] and at the Large Hadron Collider (LHC) [12–16], these correlations exhibit characteristic structures: (a) a peak at (\(\Delta \eta \),\(\Delta \varphi \)) \(=\) (0, 0), usually referred to as the near-side jet peak, resulting from intra-jet correlations as well as correlation due to decay of resonances and quantum statistics correlations, (b) an elongated structure over \(\Delta \eta \) at \(\Delta \varphi \) \(=\) \(\pi \) originating partially from correlations between particles from back-to-back jets and from collective effects such as anisotropic flow, and (c) a similar component at \(\Delta \varphi \) \(=\) 0 extending to large values of \(\Delta \eta \), usually called the near-side ridge, whose origin was subject of a theoretical debate [17–31]. Although initially the near-side ridge was also attributed to jet–medium interactions [17–20], it is now believed to be associated to the development of collective motion [24–31] and to initial state density fluctuations, including the initial state effects within the framework of the Color Glass Condensate (CGC) [21–23].

Similar structures have recently been reported in two-particle correlation analyses in smaller systems. In particular, the CMS Collaboration, by studying angular correlations between two particles in \(\Delta \eta \) and \(\Delta \varphi \), reported the development of an enhancement of correlations on the near-side (i.e. \(\Delta \varphi \) \(=\) 0) in high- compared to low-multiplicity pp collisions at \(\sqrt{s} = 7\) TeV that persists over large values of \(\Delta \eta \) [32]. In the subsequent data taking periods at the LHC, similar ridge structures were observed on both the near- and the away-side in high-multiplicity p–Pb collisions at \(\sqrt{s_{\mathrm {NN}}}\) \(=\) 5.02 TeV [33–38]. The origin of these effects, appearing in small systems, is still debated theoretically. In particular, it was suggested in [39–41] that in high-multiplicity collisions the small system develops collective motion during a short hydrodynamic expansion phase. On the other hand, in [42–44] the authors suggested that the ridge structure can be understood within the CGC framework.

The ALICE Collaboration also reported a particle mass ordering in the extracted \(v_{2}\) (i.e. the second coefficient of the Fourier expansion of the azimuthal distribution of particles relative to the symmetry plane) values for \(\pi ^{\pm }\), \({\mathrm {K}}^{\pm }\), and p(\(\overline{{\mathrm {p}}}\)) in high-multiplicity p–Pb collisions [45]. This mass ordering becomes evident once the correlations observed in the lowest multiplicity class are subtracted from the ones recorded in the highest multiplicity class. The ordering is less pronounced, yet still present, if this subtraction procedure is not applied. Similar mass ordering in Pb–Pb collisions [46] is usually attributed to the interplay between radial and elliptic flow induced by the collective motion of the system. These observations in p–Pb collisions were reproduced by models incorporating a hydrodynamic expansion of the system [47, 48]. Recently, it was suggested in [49] that the signatures of collective effects observed in experiments could be partially described by models that couple the hot QCD matter created in these small systems, described as an ensemble of non-interacting particles, to a late stage hadronic cascade model. More recently, the CMS Collaboration demonstrated that the effects responsible for the observed correlations in high-multiplicity p–Pb events are of multiparticle nature [50]. This strengthens the picture of the development of collective effects even in these small systems.

The charge-dependent part of two-particle correlations is traditionally studied with the balance function (BF) [51], described in detail in Sect. 4. Such studies have emerged as a powerful tool to probe the properties of the system created in high energy collisions. Particle production is governed by conservation laws, such as local charge conservation. The latter ensures that each charged particle is balanced by an oppositely-charged partner, created at the same location in space and time. The BF reflects the distribution of balancing charges in momentum space. It is argued to be a sensitive probe of both the time when charges are created [51, 52] and of the collective motion of the system [26, 53]. In particular, the width of the balance function is expected to be small in the case of a system consisting of particles that are created close to the end of its evolution and are affected by radial flow [26, 51–53]. On the other hand, a wide balance function distribution might signal the creation of balancing charges at the first stages of the system’s evolution [26, 51–53] and the reduced contribution or absence of radial flow.

In this article, we extend the previous measurements [54] by reporting results on the balance function in pp, p–Pb, and Pb–Pb collisions at \(\sqrt{s_{\mathrm {NN}}} = 7\), 5.02, and 2.76 TeV, respectively. The data were recorded with the ALICE detector [55–57]. The results are presented as a function of multiplicity and transverse momentum (\(p_{{\mathrm {T}}}\)) to investigate potential scaling properties and similarities or differences between the three systems. The article is organized as follows: Sect. 2 briefly describes the experimental setup, while details about the data sample and the selection criteria are introduced in Sect. 3. In Sect. 4, the analysis technique and the applied corrections are illustrated. In Sect. 5, the specifics about the estimation of the systematic uncertainties are described. Section 6 discusses the results followed by a detailed comparison with models to investigate the influence of different mechanisms (e.g. unrelated to hydrodynamic effects) on the balance functions. In the same section, the comparison of the results among the three systems is presented.

2 Experimental setup

ALICE [57] is one of the four major detectors at the LHC. It is designed to efficiently reconstruct and identify particles in the high-particle density environment of central Pb–Pb collisions [58, 59]. The experiment consists of a number of central barrel detectors positioned inside a solenoidal magnet providing a 0.5 T field parallel to the beam direction, and a set of forward detectors. The central detector systems of ALICE provide full azimuthal coverage for track reconstruction within a pseudorapidity window of \(|\eta | < 0.9\). The experimental setup is also optimized to provide good momentum resolution (about \(1~\%\) at \(p_{{\mathrm {T}}}~< 1\) GeV/c) and particle identification (PID) ov