Deciphering the phases of QCD matter with ﬂuctuations and correlations of conserved charges Anar Rustamov12

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Deciphering the phases of QCD matter with ﬂuctuations

and correlations of conserved charges

Anar Rustamov1,2,∗

1GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany

2National Nuclear Research Center, Baku, Azerbaijan

Abstract. A review is given on recent experimental and theoreti-

cal/phenomenological developments regarding the phase structure of the

strongly interacting matter. Speciﬁcally, evolution with the collision energy

of net-proton number ﬂuctuations as measured by several experiments are pre-

sented and their implications for the QCD phase diagram are outlined. In ad-

dition, theoretical calculations on correlations between conserved charges are

presented and prospects for their experimental explorations are addressed.

1 Introduction

The quest for the phase structure of strongly interacting matter, hereinafter referred to as

Quantum Chromodynamics (QCD) matter, remains at the focus of contemporary theoretical

and experimental investigations. By know it is well established that, in nuclear collisions at

cm energies from several TeV down to about 12 GeV per nucleon pair the matter created in

head-on collisions of heavy ions freezes out close to the chiral phase transition line, taking

place at a pseudo-critical temperature values of about 156 MeV [1, 2]. This opens unique

experimental opportunities to probe the strongly interacting matter close to the phase bound-

ary, such as experimental veriﬁcation of its Equation of State (EoS). In this energy range the

lattice QCD (lQCD) calculations predict crossover chiral transition [2], which still awaits ex-

perimental conﬁrmations. Contrary to the high energy region, ﬁrst principle calculations at

low energies (high net-baryon densities) are still not available, however a number of eﬀective

models predict, that at reasonably large net-baryon densities the QCD matter undergoes a

ﬁrst order chiral phase transition [3–7]. This lends support to the existence of the chiral Crit-

ical End Point (CEP) at which the anticipated ﬁrst order phase transition line terminates and

smooth crossover transition sets in. Hence at the CEP the system would undergo a second

order phase transition. One of the main goals of experiments at lower energies is to locate the

position of the chiral CEP in the QCD phase diagram.

Fluctuations of conserved charges, such as baryon number, electric charge, strangeness

and charm, as well as correlations between them are identiﬁed as promising tools to probe the

chiral criticality. The objective is to characterise the response of the system to external pertur-

bations, such as inﬁnitesimal changes in the chemical potentials. In theoretical calculations,

within the Grand Canonical Ensemble (GCE) of statistical mechanics, for a system of ﬁxed

volume Vand temperature Tthis task is performed by evaluating the nth order susceptibilities

∗e-mail: a.rustamov@gsi.de

arXiv:2210.14810v1 [hep-ph] 26 Oct 2022

χn

q, i.e., derivatives of the thermodynamic pressure Pwith respect to the chemical potentials

µqresponsible for the conservation of the corresponding charge qon average, evaluated at

vanishing values of chemical potentials

ˆχq

n=∂nˆ

∂ˆµn



~

µ=0

=κn(Nq)

VT 3,(1)

where ˆχq

n,ˆ

P=P(T,V,~

µ)/T4and ˆµq=µq/Tdenote reduced susceptibility, reduced pres-

sure and reduced chemical potential, respectively. In experiments the cumulants are directly

measured in a given detector acceptance either by ﬁrst reconstructing the multiplicity distri-

butions of the corresponding net-charge number (Nq) or by using probabilistic approaches

based on the detector response functions for particle identiﬁcation (cf. [8, 9] and references

therein). The so obtained experimental cumulants κn(Nq) can be directly compared to those

computed via derivatives of thermodynamic pressure (cf. Eq. 1), provided that the system

under consideration is in thermal equilibrium and all necessary instrumental eﬀects and non-

critical signals, such as those stemming from detection eﬃciencies, participant ﬂuctuations,

suppression of ﬂuctuations due to e-by-e conservation laws and resonance decays are prop-

erly accounted for [10–13].

2 Experimental Results

In this section I brieﬂy discuss experimental measurements related to crossover and critical

point studies, provide their comparison with the model calculations and implications for the

QCD phase diagram.

2.1 Search for a critical point

Fig. 1 shows the evolution with collision energy of the fourth to seconder order cumulants

of net-proton distributions1as measured by the STAR [14, 15] (black star symbols) and

HADES [16] (magenta box) collaborations. The data points are contrasted with calculations

within two diﬀerent ensembles of statistical mechanics [13, 17, 18]: (i) Grand Canonical En-

semble (GCE) with the ideal gas EoS in the Boltzmann limit, represented with the dashed

line along unity; (ii) Canonical Ensemble (CE), implemented in the full phase space, also

using the ideal gas EoS. In the following these are referred to as the GCE or Hadron Reso-

nance Gas (HRG) baseline and the CE baseline, respectively. Such treatment of cumulants

within CE ensures that in each event the baryon number (number of baryons minus num-

ber of antibaryons) is exactly conserved in the full phase space, inducing strict correlations

between baryon and anti-baryon numbers. As a consequence, since there are essentially no

antibaryons at low energies, the baryon number does not ﬂuctuates in the full phase space,

leading to a binomial distributions in the ﬁnite acceptance at low energies (for more details

see the Appendix C of Ref. [13])2. This, however, apples if wounded nucleons [19] do not

ﬂuctuate from event-to-event, while in practice they do [11, 20]. Thus, non-critical contribu-

tions to the ﬁnally measured (net-)proton cumulants, stemming from ﬂuctuations of wounded

nucleons, need to be corrected for. This is in particular essential at low energies because,

in addition to the above mentioned arguments regarding binomial multiplicity distributions,

most of these contributions scale with the net-proton number. For example, at LHC energies,

where net-proton number at mid-rapidity is practically zero, contributions from wounded nu-

cleon ﬂuctuations are immaterial for the measured second order cumulants of net-protons,

1At low collisions energies these are essentially cumulants of proton number.

2At high energies baryons and anti-baryons ﬂuctuate in the full phase space, only net-baryon number does not.

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DecipheringthephasesofQCDmatterwithuctuationsandcorrelationsofconservedchargesAnarRustamov1;2;1GSIHelmholtzzentrumfürSchwerionenforschung,Darmstadt,Germany2NationalNuclearResearchCenter,Baku,AzerbaijanAbstract.Areviewisgivenonrecentexperimentalandtheoreti-cal/phenomenologicaldevelopmentsregardingt...

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Deciphering the phases of QCD matter with ﬂuctuations and correlations of conserved charges Anar Rustamov12

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