Cumulants vs correlation functions and the QCD phase diagram at low energies

We discuss the relation between particle number cumulants and genuine correlation functions. It is argued that measur- ing multi-particle correlation functions could provide cleaner information on possible non-trivial dynamics in heavy-ion collisions.


Introduction
The search for structures in the QCD phase diagram is one of the most exciting topics of strong interaction physics, see e.g., a recent overview by Luo and Xu [1].
In this paper we focus on the measurement of net-proton cumulants [2][3][4], K n , performed by the STAR Collaboration at RHIC [1,5]. By definition K 1 ≡ N ; K 2 ≡ (δN) 2 ; K 3 ≡ (δN) 3 ; K 4 ≡ (δN) 4 − 3 (δN) 2 where δN = N − N and N is the net-proton number. At low energies the number of produced anti-protons is negligible and N can be very well approximated by the proton number. STAR measured K 4 /K 2 and K 3 /K 2 in Au+Au collisions for energies ranging from 7.7 GeV to 200 GeV. The most intriguing preliminary results are (i) a large value of K 4 /K 2 ≈ 3.5, with rather large error bars, at √ s = 7.7 GeV and (ii) a small value of K 4 /K 2 ≈ 0.2 at energies close to 19 GeV. The interpretation of the STAR data is challenging since the cumulant ratios are sensitive to various sources of fluctuations, see, e.g., [1].

Results
In a system of protons (we neglect antiprotons) without any correlations the cumulants are given by the average number of protons, K n = N , which at 7.7 GeV, for a given STAR acceptance |y| < 0.5 and 0.4 < p t < 2 GeV, is roughly 40. Consequently, any nontrivial physics related to correlations between protons might not be clearly visible. It seems natural to measure the genuine multi-proton correlation functions. Performing straightforward calculations [6,7] we obtain where C n is the n-proton genuine correlation function. C n = 0 for a proton system with no correlations. Using the preliminary STAR data we can straightforwardly extract C n . This is presented in Fig. 1. We observe that a large value of K 4 /K 2 ≈ 3.5 at 7.7 GeV is driven by a positive genuine four-proton correlation. This is in contrast to 19.6 GeV, where the signal is driven by a negative two-proton correlation.
To put the STAR numbers in perspective let us consider a simple model. Suppose we have clusters, with Poisson multiplicity distribution, which decay into a fixed number of proton, m. In this case C n = N cl m!/(m − n)! where N cl is the average number of proton clusters. Taking m = 4 (four-proton clusters) we obtain C 4 = 24 N cl . In order to get C 4 ∼ 170 we need to assume N cl = 6 − 8. It means that 24 − 32 protons, out of 40 measured protons, should originate from such clusters. This is a rather large fraction, indeed.
One natural source of multi-proton correlations are the fluctuations of the number of wounded nucleons, N part , which we also call volume fluctuation. We verified that this contribution is way too small to explain the preliminary STAR data. In particular, the obtained value of C 4 is smaller by roughly three orders of magnitude. This is demonstrated in Fig. 2. 1 In this model we assume only two natural sources of correlations namely (i) baryon conservation and (ii) volume fluctuation. More details can be found in Ref. [8].
It seems clear that in order to understand the preliminary STAR data, in particular the large value of four-proton correlation function, we need a nontrivial source of strong correlations between protons. For example, in Ref. [8] the preliminary STAR data could be qualitatively reproduced if assuming the collective stopping of four-proton clusters or, equivalently, eight-nucleon clusters. It remains to be seen whether the observed strong correlations indeed originate from the collective stopping of protons (currently not understood), or perhaps we witness the first evidence of proton clustering related to a nontrivial structure of the QCD phase diagram. This problem requires further study.