Searching for minimum in dependence of squared speed-of-sound on collision energy

Experimental results of the rapidity distributions of negatively charged pions produced in proton-proton ($p$-$p$) and beryllium-beryllium (Be-Be) collisions at different beam momentums, measured by the NA61/SHINE Collaboration at the super proton synchrotron (SPS), are described by a revised (three-source) Landau hydrodynamic model. The squared speed-of-sound parameter $c^2_s$ is then extracted from the width of rapidity distribution. There is a local minimum (knee point) which indicates a softest point in the equation of state (EoS) appearing at about 40$A$ GeV/$c$ (or 8.8 GeV) in $c^2_s$ excitation function [the dependence of $c^2_s$ on incident beam momentum (or center-of-mass energy)]. This knee point should be related to the searching for the onset of quark deconfinement and the critical point of quark-gluon plasma (QGP) phase transition.


Introduction
Comparing with the relativistic heavy ion collider (RHIC) in USA [1][2][3] and the large hadron collider (LHC) in Switzerland [4,5], the fixed target experiments performed at the super proton synchrotron (SPS) in Switzerland [6,7] present relatively simple and clean collisions process. The multiplicity in collisions at the SPS is also low comparing with those at the RHIC and LHC. As one of the "first day" measurement quantities, the rapidity (pseudorapidity) distributions of charged particles are reported by experimental collaborations [1][2][3][4][5][6][7]. These distributions give us a chance to analyze longitudinal picture of particle productions. At the same time, based on the rapidity distributions, one can extract other information such as the penetrating (stopping) power of projectile (target) nucleus, energy and rapidity losses of projectile nucleus, energy and particle densities of interacting region, and so forth. As a measurement of particle density and mean free path, the squared speed-of-sound parameter which characterizes partly the formation of matters in interacting region can be extracted from the width of Gaussian rapidity distribution described by the Landau

The model
The revised (three-source) Landau hydrodynamic model used in the present work can be found in our previous work [25,26]. To give a short and clear description of the model, we introduce the main results of the model in the following. The Landau hydrodynamic model [8][9][10][11][12][13][14][15][16][17][18] results approximately in a Gaussian rapidity distribution which does not exactly describe the experimental data [6,19,20]. We have revised the model to three sources: a central source which locates at midrapidity and covers the rapidity range as wide as possible, and a target (projectile) source which locates in the backward (forward) region and revises the rapidity distribution from the central source [25,26]. The experimental rapidity distributions are then described by three Gaussian functions. And based on the three Gaussian functions, the experimental pseudorapidity distributions can be described by a method which makes a distinction between rapidity and pseudorapidity.
According to the Landau hydrodynamic model [8][9][10][11][12][13][14][15][16][17][18] and our revision [25,26], the rapidity distribution, dN ch /dy, of charged particles produced in a given source in high energy collisions can be described by a Gaussian function where σ X , y X , and N 0 denote the width, peak position, and normalization constant, respectively; X = C, T , and P are for the central, target, and projectile sources, respectively. The experimental result is in fact a sum weighted by the three Gaussian functions. The relation between σ X and the squared speed-of-sound c 2 s (X) is where L = ln( √ s N N /2m p ) is the logarithmic Lorentz contraction factor which is independent of X, √ s N N denotes the center-of-mass energy per pair of nucleons in nucleus-nucleus collisions and is simplified to √ s in p-p collisions, and m p denotes the rest mass of a proton.
From Eq. (2), c 2 s (X) is expressed by using σ X to be In the above extractions of σ X and c 2 s (X) from the rapidity distribution, we distinguish accurately the rapidity and pseudorapidity distributions [25,26]. In the case of representing the pseudorapidity distribution in experiment, we also extract σ X and c 2 s (X) from the hidden rapidity distribution. Our treatment ensures the method of extraction being concordant.  [6,7] and the curves are our results fitted by the revised (three-source) Landau hydrodynamic model. Different centrality classes (0-5%, 5-10%, 10-15%, and 15-20%) for 7 Be-9 Be collisions are presented by different symbols marked in the panels. The values of fit parameters, σ C , σ T (= σ P ), y C , rapidity shift ∆y (= y C − y T = y P − y C ), relative contribution k T of the target source (= k P , the relative contribution of the projectile source), and normalization constant N 0 , are listed in Table 1 with the values of χ 2 per degree of freedom (dof). The last two columns in Table 1

Results and discussion
Be- 9 Be, 0−5% 5−10% 10−  Table 1 and the line segments for two cases are given for guiding the eyes. Different symbols correspond to different types (centralities) of collisions.
spectively. The results corresponding to p-p collisions and 7 Be-9 Be collisions with different centrality classes are displayed by different symbols, which reflect a part of parameter values listed in Table 1. The line segments for p-p collisions and for 7 Be-9 Be collisions with centrality class 15-20% are given for guiding the eyes. One can see that  (Table 1) and the results for p-p, proton-antiproton (p-p), copper-copper (Cu-Cu), gold-gold (Au-Au), and lead-lead (Pb-Pb) collisions at higher energies (≥ 19.6 GeV) are taken from our previous works [25,26]. Different symbols represent different collisions, which are marked in the panels. The symbols with large size denote central collisions, and the symbols with small size denote non-central collisions. The centrality classes for Cu-Cu collisions are from 0-3% to 50-55% [27]; for Au-Au collisions are from 0-3% to 45-50% (maximum 40-45% at 19.6 GeV) [27]; and for Pb-Pb collisions are from 0-5% to 20-30% [28]. The line segments for p-p collisions at SPS and higher energies are given for guiding the eyes, and the two dotted lines in Figure 3 Table 1 (for the energy range ≤ 17.3 GeV), and taken from our previous work [25,26] (for the energy range ≥ 19.6 GeV). The line segments for p-p collisions at SPS and higher energies are given for guiding the eyes, and the two dotted lines in Figure  3 The softest point (8.8 GeV) obtained in the present work is compatible with the previous works [29,30] which used the Landau hydrodynamic model and the ultra-relativistic quantum molecular dynamics hybrid approach and indicated the softest point locating in the energy range from 4 to 9 GeV. Other works which study dependences of ratio of numbers of positive kaons and pions (K + /π + ) [20,31,32], chemical freeze-out temperature (T ch ) [31,32], mean transverse mass minus rest mass ( m T − m 0 ) [31], and ratio of widths of experimental negative pion rapidity distribution and Landau hydrodynamic model prediction [σ y (π − )/σ y (hydro)] [32] on √ s N N show knee point around 7-8 GeV. A wiggle in the excitation function of a specific reduced curvature of the net-proton rapidity distribution at midrapidity is expected in the energy range from 4 to 8 GeV [33,34]. However, the searching for the onset of quark deconfinement and the critical point of QGP phase transition is a complex process [35].    19.6 GeV, we explain it as the result of changing the mean free path of produced particles, if the statistical fluctuation is excluded. At the SPS energies, the interacting system has a small density due to low collision energy and the produced particles have a large mean free path which results from a gas-like state and results in a small c 2 s . The situation at the RHIC or LHC energies is opposite, where the interacting system has a large density due to high collision energy and the produced particles have a small mean free path which results from a liquid-like state and results in a large c 2 s . Let D denote the dimensionality of space. According to Refs. [36,37], we have the relation of c 2 s = 1/D for massless particles. The particles stay at the gas-like state have a larger probability to appear in three-dimensional space, which results in the maximum c 2 s being 1/3 which is the situation at the SPS energies. The particles stay at the liquid-like state have a larger probability to appear in two-dimensional space, which results in the maximum c 2 s being 1/2 which is the situation at the RHIC and LHC energies.
To see the relations of σ C and other parameters at the SPS energies which show the softest point in the EoS, Figures 4(a), 4(b), 4(c), and 4(d) present the dependences of σ C on y C , ∆y, k T , and N 0 , respectively, where the closed circles, open circles, closed squares, open squares, and stars correspond to the collision energy being 6.3, 7.7, 8.8, 12.3 (11.9), and 17.3 (16.8) GeV, respectively, which are taken from Table 1. One can see that σ C does not show an obvious dependence on y C and k T , though k T shows somehow a saturation. σ C increases with increases of ∆y and N 0 due to the latter two increasing with increase of √ s N N . The relation between σ C and √ s N N is the most important one among all the relations. Other relations such as the relations between σ C and y C , ∆y, k T , as well as N 0 are less important. To present the relations between c 2 s (C) and y C , ∆y, k T , as well as N 0 is trivial due to c 2 s (C) being calculated from σ C . To study the most important relation between σ C and √ s N N in detail, Figure 5(a) displays a few examples of linear relations between σ C and ln √ s N N which reflect approximately the main area of parameter points in Figure 3(a). The solid lines corresponded to cases A, B, C, and D can be expressed by σ C = 0.5 ln √ s N N + 0.3, σ C = 0.3 ln √ s N N + 0.5, σ C = 0.2 ln √ s N N + 0.6, and σ C = 0.1 ln √ s N N + 0.7, respectively. After conversion by Eq.
(3), the four similar lines of σ C −ln √ s N N show very different relations of c 2 s (C) −ln √ s N N given correspondingly in Figure 5(b). Cases A and B show obvious minimums, while cases C and D do not show. Decreasing the intercept in case A from 0.3 to 0.2 and 0.1 respectively, the corresponding results presented respectively by the dashed and dotted lines in Figure 5(a) have a small change, while the results presented respectively by the dashed and dotted curves in Figure 5(b) have a large change. Decreasing the slope in case B from 0.3 to 0.29 and 0.28 respectively, the corresponding results presented respectively by the dashed and dotted lines (curves) in Figure 5(a) [5(b)] have a small change. The results presented in Figure 5 show that the minimum in Eq. (3) appearing in some special conditions. Table 1. Values of free parameters, normalization constants, and χ 2 /dof corresponding to the curves in Figure 1.