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Fluctuations and correlations from NA61/SHINE

Basic ideas, methods and results related to the NA61/SHINE study of event-by-event fluctuations in high energy nuclear collisions are briefly reviewed.


Introduction
Study of event-by-event fluctuations is the focus of the NA61/SHINE programme on strong interactions. Initially the study was motivated by the possibility to discover the critical point of strongly interacting matter and a need to understand how the onset of deconfinement influences event-by-event fluctuations [1]. Recently, results on fluctuations together with data on mean hadron multiplicities allowed to uncover the onset of fireball 1 -the rapid change of hadron production properties that start when moving from Be+Be to Ar+Sc collisions [2].
Fluctuations in high energy collisions are significantly influenced by fluctuations in the amount of matter (volume) and energy involved in a collision, as well as global and local conservation laws. In the search for the critical point and the study of the onset of deconfinement these are unwanted effects [3]. However, the onset of fireball was discovered thanks to the sensitivity of fluctuations to the conservation laws [2]. Moreover, the dynamics of the early stage of the collision can be studied exploiting volume fluctuations [4].
In this contribution volume fluctuations are treated as unwanted effect and methods to remove their influcence are presented in Secs. 2 and 4. The conservation laws are in Sec. 3 treated as a tool to uncover the onset of fireball and in Sec. 4 as to-be-minimized bias in the search for the critical point.

Removing volume fluctuations
It is probably the simplest to introduce fluctuations of the amount of matter involved in a collision and their impact on fluctuations of produced particles using as an example the Wounded Nucleon Model [5]. The model was proposed in 1976 as a late child of the S-matrix period [6]. It assumes that particle production in nucleon-nucleon and nucleus-nucleus collisions is an incoherent superposition of particle production from wounded nucleons (nucleons which interacted inelastically and whose number is calculated using straight line trajectories of nucleons). Properties of wounded nucleons are independent of the size of colliding nuclei, e.g. they are the same in p+p and Pb+Pb collisions at the same collision energy per nucleon. These assumptions are graphically illustrated in Fig. 1.
Let us consider multiplicity (particle number) fluctuations characterized by second moments of multiplicity distributions. Two quantities of relevance are variance, Var[N] = (N − N ) 2 and scaled variance, ω[N] = Var[N]/ N . Here N and N stand for multiplicity and its mean value, respectively. Then for any probability distribution P(W ), the scaled variance calculated within WNM reads [7]: where ω[N] W stands for the scaled variance at any fixed number of wounded nucleons, and W = W P +W T for the sum of the number W P and W T of projectile and target nucleons. Here the first component of Eq. 2.1 is considered to be the wanted one, whereas the second one is unwanted. Similar relations are valid for Statistical Models of an Ideal Boltzmann gas within the Grand Canonical Ensemble SM(IB-GCE) [7]. W T (here W T = 4) target wounded nucleons and W P (here W P = 3) projectile wounded nucleons produce N particles, where N is given by the sum over all wounded nucleons of particle multiplicities n from single wounded nucleons.
In order to limit the unwanted component NA61/SHINE selects collisions with the smallest energy recorded by the Projectile Spectator Detector -the calorimeter that predominantly measures energy of projectile spectators, see Fig. 2. Multiplicity fluctuations for the selected most violent collisions (collisions with the largest W P ) are only weakly increased by W fluctuations. This is demonstrated in Fig. 3 based on simulations performed with the HSD and UrQMD models [8].
Since even for the most violent collision volume fluctuations cannot be fully eliminated, it is important to further minimise their effect by defining suitable fluctuation measures. It appears that using second and first moments of the distribution of two extensive quantities (their first moments are proportional to volume) one can construct fluctuation measures which are, for the WNM and the SM(IB-GCE) models [9,7,10], independent of volume fluctuations.
In particular, one can construct the strongly intensive scaled variance [7],  where

2)
under two conditions N · E P W ≈ N W · E P W and E P W ∼ W that are expected to be fulfilled for violent A+A collisions. The above defined procedures to minimize the effects of volume fluctuations were tested using theoretical models and experimental data. An exmaple of the experimental test performed for Be+Be collisions at 75A GeV/c is presented in Fig. 4. The scaled variance ω increases with increasing percentile of violent collisions selected for the analysis using the measured values of E F . The strongly intensive scaled variance Ω is almost independent of the percentile of selected collisions. The two coincide for the most violent collisions.  The most efficient description of bulk properties of hadron production in high energy heavy ion collisions is given by statistical and hydrodynamical models [12]. Thus, here the impact of conservation laws on mean multiplicity and multiplicity fluctuations is discussed with statistical models. For simplicity consider only material conservation laws (conserved charges: Q, B, S, ...). Then the simplest model is a Statistical Model with Ideal Boltzmann Gas using the Canonical Ensemble, SM(IB-CE). In this model correlations between particles are only due to material conservation laws. Their influence on mean multiplicity and scaled variance is presented in Fig. 5, where results for a gas of positively and negatively charged particles contained in a cluster of volume V C and with total charge equal to zero are shown as a function of mean multiplicity calculated with the SM(IB-GCE). Note that the mean multiplicity in the SM(IB-GCE) is proportional to the cluster volume provided the temperature is independent of V C . The well know "canonical suppression" of the mean multiplicity [13] and a somewhat less well known "canonical enhancement" of the scaled variance [14] are seen. Note that in the large V C limit the results for the mean multiplicity obtained with the SM(IB-CE) and SM(IB-GCE) are equal and for the scaled variance they remain different.

Exploiting conservation laws
In the following the qualitative expectations derived from the SM(IB-CE) will be confronted with experimental data. Figure 6 shows example plots on the system size dependence of the ratio of K + and π + yields at mid-rapidity and of the scaled variance of multiplicity distributions. The Be+Be results are close to p+p independently of collision energy. Moreover, the data show a jump between light (p+p, Be+Be) and intermediate, heavy (Ar+Sc, Pb+Pb) systems.
Here one recalls the following: 1. The K + /π + ratio in p+p interactions is below the predictions of statistical models. However, the ratio in central Pb+Pb collisions is close to statistical model predictions for large volume systems. For detail see e.g. Ref.
[15] 2. In p+p interactions, and thus also in Be+Be collisions, multiplicity fluctuations are larger than predicted by statistical models. However, they are close to statistical model predictions for large volume systems in central Ar+Sc and Pb+Pb collisions, for detail see Ref. [16].
Thus the observed rapid change of hadron production properties that start when moving from Be+Be to Ar+Sc collisions can be interpreted as the beginning of creation of large clusters of strongly interacting matter -the onset of fireball [2]. We note that non-equilibrium clusters produced in p+p and Be+Be collisions seem to have similar properties at all beam momenta studied here. This is well seen in Fig. 7 where scaled variance and strongly intensive scaled variance are plotted as a function of collision energy. Finally we recall that hadron production properties in heavy ion collisions were found to change rapidly with increasing collision energy in the low SPS energy domain, √ s NN ≈ 10 GeV (for a recent review see Ref. [17]). The NA61/SHINE results shown in Fig. 8 indicate that this is also the case in inelastic p+p interactions and probably also in Be+Be collisions. The phenomenon is labelled as the onset of deconfinement and interpreted as the beginning of creation of quark-gluon plasma with increasing collision energy [18].
Consequently the two-dimensional scan conducted by NA61/SHINE by varying collision energy and nuclear mass number of colliding nuclei indicates four domains of hadron production properties separated by two thresholds: the onset of deconfinement and the onset of fireball. The sketch presented in Fig. 9 illustrates this conclusion.

Search for critical point
A characteristic feature of a second order phase transition (the critical point or line) is the divergence of the correlation length. The system becomes scale invariant. This leads to large fluctuations in particle multiplicity. Moreover these fluctuations have specific characteristics [19,20]. Also other properties of the system should be sensitive to the vicinity of the critical point [21]. Thus when scanning the phase diagram a region of increased fluctuations may signal the critical point or the critical line.
In the experimental search for the critical point one would like to minimize influence of both volume fluctuations and conservation laws. Thus it is recommended to use quantities which are insensitive (in the WNM and the SM(IB-GCE)) to these. It appears that strongly intensive quantities composed of suitably selected extensive quantities have this property. For example these are [7,22]: and where P T is the sum of the absolute values of transverse momenta p T . The quantity ω[p T ] is the scaled variance of the inclusive p T distribution (summation runs over all particles and all events)  Results refer to negatively charged hadrons at forward rapidity (0 < y π < y beam ) and p T < 1.5 GeV/c.
In the grand canonical ensemble the correlation length ξ diverges at the critical point (or second order phase transition line) and the system becomes scale invariant [19,23]. This leads to large multiplicity fluctuations with special properties. They can be conveniently exposed using scaled factorial moments F r (M) [24] of rank (order) r: where M = ∆/δ is the number of the subdivision intervals of size δ of the momentum phase space region ∆. N i refers to particle multiplicity in the interval i and ... indicates averaging over the analysed collisions.
In the SM(IB-GCE) one gets F r (M) = 1 for all values of r and M provided the mean particle multiplicity is proportional to δ . The latter condition is trivially obeyed for a subdivision in configuration space where the particle density is uniform throughout the gas volume. For the case of subdivision in momentum space the subdivision should be performed using so-called cumulative kinematic variables [25] in which the particle density is uniform.
At the second order phase transition the matter properties strongly deviate from the ideal gas. The system is a simple fractal and F r (M) possess a power law dependence on M: (4.5) Moreover the exponent (intermittency index) φ r satisfies the relation: with the anomalous fractal dimension d r being independent of r [20]. Note, that F r (M) is sensitive to to both volume fluctuations and material conservation laws. A formulation of a new method to study intermittency using strongly intensive quantities is needed. NA61/SHINE results and NA49 results on F 2 (M) for protons are presented in Fig. 11. The NA49 [26] results for violent Si+A collisions at 158A GeV are consistent with about 1% of protons obeying critical fluctuations with φ 2 ≈ 1. The NA61/SHINE [2] results for violent Si+A collisions at 150A GeV/c establish the upper limit for protons obeying critical fluctuations to be about 0.3%. For details see Ref. [27].