Light nuclei production as a probe of the QCD phase diagram Kai-Jia Sun∗ ,1 Lie-Wen Chen† ,1 Che Ming Ko‡ ,2 Jie Pu§ ,1 and Zhangbu Xu¶3, 4
arXiv:1801.09382v1 [nucl-th] 29 Jan 2018
1
School of Physics and Astronomy and Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai 200240, China 2 Cyclotron Institute and Department of Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA 3 Brookhaven National Laboratory, Upton, New York 11973, USA 4 School of Physics & Key Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong University, Jinan, Shandong 250100, China (Dated: January 30, 2018) We argue that there should exist two maxima in the collision energy dependence of the baryon density fluctuation in heavy-ion collisions, i.e., a lower energy one due to the spinodal instability in the first-order phase transition and a higher energy one due to the critical fluctuations in the vicinity of the critical endpoint (CEP) of the first-order phase transition line in the QCD phase diagram. By analyzing the data on the p, d and 3 H yields in central collisions of Au+Au at √ sN N = 4.86 GeV measured at the Brookhaven Alternating Gradient Synchrotron (AGS) and √ Pb+Pb at sN N = 6.3 GeV, 7.6 GeV, 8.8 GeV, 12.3 GeV and 17.3 GeV measured by the NA49 Collaboration at the CERN Super Proton Synchrotron (SPS) within the coalescence model, we extract the relative neutron density fluctuation ∆n = h(δn)2 i/hni2 at kinetic freeze-out and find √ that the ∆n displays a clear peak at sN N = 8.8 GeV and a possible strong enhancement at √ sN N = 4.86 GeV, suggesting that the CEP could have been reached or closely approached and the first-order phase transition could have occurred in these collisions, respectively. PACS numbers: 25.75.-q, 25.75.Dw
1. Introduction.—Understanding the phase diagram of strongly interacting matter is of fundamental importance in nuclear physics, astrophysics and cosmology. Lattice quantum chromodynamics (LQCD) calculations [1] and various effective models [2–4] have suggested that the transition between quark-gluon plasma (QGP) and hadronic matter is a smooth crossover at vanishing baryon chemical potential (µB ), but likely changes to a first-order phase transition at large µB , with an associated critical endpoint (CEP) or a tricritical endpoint [5]. Experimentally, heavy-ion collisions provide a unique tool to probe the QCD phase diagram [6–10]. In particular, to search for the CEP and locate the phase boundary in the QCD phase diagram is the main motivation for heavy-ion collision experiments being carried out in the Beam Energy Scan (BES) program at the Relativistic Heavy Ion collider (RHIC) as well as those planned at the future Facility for Antiproton and Ion Research (FAIR), Nuclotron-based Ion Collider facility (NICA) and the SPS Heavy Ion and Neutrino Experiment (NA61/SHINE).
the first-order phase transition line (due to the spinodal instability [11–16]) or approaches the CEP (because of a rapid increase of correlation length in the critical region [3, 17]). In particular, for heavy-ion collisions at lower energies when the system enters the region of firstorder phase transition, it is expected that the density fluctuation should reach a maximum at a collision energy √ sM at which the system stays the longest time inside the unstable spinodal region [13]. With increasing collision energy, a second maximum in √ the density fluctuation is expected at a collision energy sE when the first-order phase transition ends and the CEP is reached [3, 17].
In heavy-ion collisions, the created matter could develop strong baryon density fluctuations when its evolution trajectory in the QCD phase diagram passes across
∗
[email protected] † Corresponding
author:
[email protected]
‡
[email protected] §
[email protected] ¶
[email protected]
FIG. 1: A schematic drawing about the collision energy dependence of the density fluctuation in heavy-ion collisions together with the corresponding phase regions in the QCD phase diagram. Point ‘D’ indicates the beginning of firstorder phase transition, ‘M’ denotes the maximum caused by the spinodal instability and ‘E’ denotes the maximum due to the CEP.
2 This double-peak structure of the√density fluctuation as a function of the collision energy s is plotted schematically in Fig. 1 where the corresponding phase regions in the QCD phase diagram are also indicated. To extract the density fluctuations in heavy-ion collisions from experimental observables is a challenging task as only the particle momentum distributions are generally measured. Because of the rapid expansion of the fireball formed in heavy-ion collisions, the enhanced density fluctuations caused by the spinodal instability or critical fluctuation could, however, survive the finalstate interactions and affect observables that are sensitive to nucleon density fluctuations and correlations at kinetic freeze-out (at which particles cease interacting). Besides the Hanbury-Brown-Twiss (HBT) interferometry of identical particles, which can provide information on the space-time structure of the particle emission source [18, 19] and thus the effect of density fluctuations at hadronization [7], these observables also include light nuclei production via nucleon coalescence [20– 28]. Indeed, we have recently shown [26, 27] that the yield ratio Op-d-t = Np N3 H /Nd2 of produced proton (Np ), deuteron (Nd ), and triton (N3 H ) depends on the relative neutron density fluctuation ∆n as well as the neutron and proton density correlation Cnp at kinetic freeze-out. Assuming the ratio of Cnp to ∆n to be independent of the collision energy, we have found [27] from comparing with the measured yield ratio Op-d-t in Pb+Pb collisions by the NA49 Collaboration [29] at the CERN Super Pro√ ton Synchrotron (SPS) that the ∆n as a function of s shows a possible peak structure. In this Letter, we argue that the phase-space volume of the fireball produced in heavy-ion collisions at kinetic freeze-out is related to the entropy per nucleon and can be determined from the temperature and volume at chemical freeze-out (at which chemical equilibrium is reached and particles ratios are fixed). With the well determined chemical freeze-out temperature and volume from the statistical model fit to available experimental data, we can then uniquely determine the collision energy dependence of Cnp and ∆n as well as their ratio in heavy-ion collisions from the yields of p, d and 3 H (3 He), instead of assuming that this ratio is independent of the collision energy as in Ref. [27]. By analyzing the data in central collisions of Au+Au [30, 31] measured at the Brookhaven Alternating Gradient Synchrotron (AGS) and Pb+Pb measured by the NA49 Collaboration [29] at SPS, we find that the ∆n displays a √ clear peak at sN N = 8.8 GeV and a possible strong √ enhancement at sN N = 4.86 GeV, suggesting that the CEP could have been reached or closely approached in √ central Pb+Pb collisions at sN N = 8.8 GeV and the first-order phase transition could have occurred in cen√ tral Au+Au collisions at sN N = 4.86 GeV. 2. Coalescence production of light nuclei and density fluctuations and correlations.—We start by calculat-
ing the deuteron yield (dN/dy at midrapidity) through a quantum mechanical treatment in full phase space. Assuming nucleons (with mass m, proton number Np and neutron number Nn ) are emitted from an isotropic thermalized fireball with temperature T and volume V and uniformly distributed in space, one can obtain the deuteron yield as Nd ≈
3 21/2
2π mT
3/2
Nn Np , V
(1)
which is exactly the same as the formula in Ref. [27] obtained by assuming Bjorken boost invariance for the expanding fireball. For non-uniform distributions, the deuteron yield becomes 3/2 Z 2π 3 dx n(x)np (x), (2) Nd ≈ 1/2 mT 2 where n(x) and np (x) denote the neutron and proton number densities in space, respectively. Similarly, the 3 H yield is given by N3 H
33/2 ≈ 4
2π mT
3 Z
dx n2 (x)np (x).
(3)
To account for the density fluctuations, the neutron and proton densities in the emission source can be expressed as Z 1 n(x)dx + δn(x) = hni + δn(x), (4) n(x) = V Z 1 np (x) = np (x)dx + δnp (x) = hnp i + δnp (x),(5) V R where h·i = V1 dx denotes the average over space and δn(x) (δnp (x)) with hδni = 0 (hδnp i = 0) denotes the neutron (proton) density fluctuation from its average value hni (hnp i). By neglecting the term h(δn)2 δnp i, we can rewrite Eqs. (2) and (3) as Nd ≈ N3 H
3
21/2
33/2 ≈ 4
2π mT
3/2
2π mT
3
Np hni(1 + Cnp ),
(6)
Np hni2 (1 + ∆n + 2Cnp ), (7)
where Cnp = hδnδnp i/(hnihnp i) and ∆n = h(δn)2 i/hni2 characterize the neutron and proton density correlation and the relative neutron density fluctuation, respectively. We would like to emphasize that both Cnp and ∆n are defined to be dimensionless to eliminate possible effects due to the collision energy dependence of average nucleon densities. It is seen from Eqs. (6) and (7) that the deuteron yield depends on Cnp but not ∆n, while the triton yield depends on both. In addition, one sees
3 from Eqs. (6) and (7) that Cnp and ∆n can be uniquely determined to be Cnp ≈ gp-d Rnp Vph Op-d − 1, 2
∆n ≈ gp-d-t (1 + Cnp ) Op-d-t − 2Cnp − 1, 1/2
(8) (9)
2 3/2 ≈ with gp-d = 3(2π) 3 ≈ 0.0019, gp-d-t = 9/4 × (4/3) 2 3.5, Op-d = Nd /Np and Op-d-t as defined earlier in the Introduction. The ratio Rnp = Np /Nn = hnp i/hni denotes the isospin factor and can be estimated from the pion yield ratio as Rnp = (π + /π − )1/2 , and Vph = (2πmT )3/2 V is the effective phase-space volume of nucleons in the fireball at kinetic freeze-out [26]. The phase-space volume Vph is directly related to the entropy per nucleon (S/N ), which is given by the SackurTetrode equation [32] S/N = 5/2 + ln(Vph /N ) in the non-relativistic Boltzmann approximation. If we assume 3/2 T 3/2 V = λTch Vch with the chemical freeze-out temperature Tch and volume Vch , λ is then unity if a gas of constant number of nucleons expands isentropically after chemical freeze-out. A recent microscopic transport model study [33] has shown, however, that it is the entropy per particle that remains a constant after chemical freeze-out in heavy-ion collisions, which is dominated by pion production if the collision energy is high, and that all particle ratios remain essentially unchanged from the chemical to the kinetic freeze-out in these collisions as assumed in the statistical model for particle production. Based on the time evolution of T and V after chemical freeze-out from this study [33], we find that the λ value ranges from 1.5 at 7.7 GeV to 1.7 at 200 GeV, suggesting that although T 3/2 V increases as fireball expands, λ has a very weak dependence on the collision energy. Therefore, we can uniquely determine the values of Cnp and ∆n by using a constant λ = 1.6. It should be mentioned that light nuclei (like d and 3 H) are formed from nucleons in a very restricted phasespace volume of ∆x ∼2 fm and ∆p ∼0.1 GeV, and their production in heavy-ion collisions can thus be used to probe the local nucleon density fluctuations. In contrast to the studies [9, 34–36] that are mainly focusing on the event-by-event fluctuations of conserved quantities within a specific window in momentum space, the ∆n in Eq. (9) has the advantage that it directly measures the spacial density fluctuation. Mapping the fluctuations in momentum space to those in coordinate space is highly non-trivial, especially at SPS and AGS energies where the longitudinal boost invariance is not well satisfied. With Cnp and ∆n, we can further investigate the fluctuation of neutron and proton density difference (h(δn − δnp )2 i) which is closely related to the isospin density fluctuation as neutron and proton have opposite isospin quantum numbers. While the fluctuations of baryonic (B), electric (Q) and strange (S) charges have been extensively investigated [9, 34], the fluctuation of isospin is less studied in heavy-ion collisions. Recently it has been
shown [37] that the isospin effect at BES energies can dramatically change the critical behavior of baryon and charge number fluctuations, suggesting that the isospin effects would be strong in heavy-ion collisions at SPS and AGS energies. Defining the isospin density fluctuation ∆nI as ∆nI = =
h(δn − δnp )2 i (hni + hnp i)2 2 Rnp ∆np ∆n 2Rnp Cnp + − ,(10) (1 + Rnp )2 (1 + Rnp )2 (1 + Rnp )2
one sees that a negative Cnp can increase ∆nI . Although the isospin in high energy heavy-ion collisions is mostly carried by pions, the ∆nI defined in terms of nucleons may still carry important information on the isospin density fluctuation due to the frequent interactions between pions and nucleons. 3. Results and discussions.—Shown in Fig. 2 is the collision energy dependence of Cnp , ∆n and ∆nI . The results are obtained from Eqs. (8)-(10) using the data on the p, d and 3 H yields together with the yield ra√ tio π + /π − from central collisions of Pb+Pb at sN N = 6.3 GeV, 7.6 GeV, 8.8 GeV, 12.3 GeV and 17.3 GeV measured by the NA49 Collaboration at the CERN SPS [29, 38, 39]. It is seen from Fig. 2 (a) that the Cnp shows a non-monotonic behavior with a valley lo√ cated at sN N =8.8 GeV. Besides, the extracted Cnp at SPS energies are all negative, indicating a strong negative correlation between neutron and proton densities. From Fig. 2 (b), one sees that ∆n and ∆nI show very similar non-monotonic behaviors with peaks also located √ at sN N = 8.8 GeV. The obtained peak structure of ∆n is consistent with that obtained in Ref. [27], where the ratio α = Cnp /∆n is fixed at some constant values (e.g., α = −0.2, −0.1, 0, 0.1 and 0.2), but is much pronounced. From the extracted values of ∆n and Cnp , we find that their ratio α has the values −1.339±0.022, −1.282±0.018, −1.235 ± 0.022, −1.339 ± 0.032 and −1.342 ± 0.029 for √ sN N = 6.3, 7.6, 8.8, 12.3 and 17.3 GeV, respectively. Although the α value only has a weakly non-monotonic dependence on the collision energy, justifying its constancy assumed in Ref. [27], its magnitude is significantly larger, leading to a much more pronounced peak of ∆n √ at sN N = 8.8 GeV than that found in Ref. [27]. The similar behavior of ∆n and ∆nI with both peaked √ at sN N = 8.8 GeV could be due to the same underlying physics, namely, the critical fluctuation in the vicinity of CEP. According to the universality of critical behavior, the singular parts of ∆n and ∆nI in the second-order phase transition both scale with the correlation length l as l2−η , where η denotes the critical exponent and is zero in the mean-field approximation, and diverge at the CEP in the QCD phase diagram. Due to the effects of critical slowing down [17] and dynamical expansion in heavy-ion collisions, only modest but simi-
4 -0 .6 0
0 .8 ∆n
-0 .6 5
0 .7
-0 .7 0
0 .6
-0 .7 5
0 .5
C
n p
I
D e n s ity flu c tu a tio n
∆n
C e n tr a l P b + P b c o llis io n s (y = 0 )
(b )
(a ) -0 .8 0
0 .4 5
1 0
1 5
5
(s
N N
)
1 0 1 /2
1 5
2 0
(G e V )
FIG. 2: Collision energy dependence of the neutron and proton density correlation Cnp (a) and density fluctuations ∆n and ∆nI (b) in central Pb+Pb collisions at SPS energies.
lar enhancements of ∆n and ∆nI can be developed. As a result, the non-monotonic behaviors shown in Fig. 2 are consistent with the scenario that the CEP is reached or closely approached in the produced QGP during its time evolution in central Pb+Pb collisions at around √ √ sN N = 8.8 GeV. For both lower (e.g., sN N = 6.3 √ and 7.6 GeV) and higher (e.g., sN N = 12.3 and 17.3 GeV) energies, both ∆n and ∆nI would decrease because the correlation length l quickly decreases as the evolution trajectory moves away from the CEP. From √ sN N = 6.3 GeV to 8.8 GeV, the correlation length l effectively increases by about 13% if we assume η 1. This implies that the fourth-order baryon number cumulant h(δNB )4 ic ∼ l7−2η [35] would be increased by a √ factor of about 2.35 from sN N = 6.3 GeV to 8.8 GeV. From the parametrization in Ref. [40] for the chemical freeze-out conditions based on the statistical model fit to available experimental data of hadron yields, the √ temperature at sN N = 8.8 GeV is estimated to be T CEP ∼ 144 MeV with a corresponding baryon chemical potential µCEP ∼ 385 MeV, which is close to the B predicted CEP from the LQCD [1], the Dyson-Schwinger equation (DSE) [41] and the hadronic bootstrap approach [42], but is much larger than that (∼ 95 MeV) inferred from a finite size scaling (FSS) analysis of twopion correlations [43]. It is very likely that the criti√ cal region is reached within the energy region sN N = 6.3 ∼ 17.3 GeV. With the temperature and baryon chemical potential of about 131 ∼ 159 MeV and 481 ∼ 229 MeV [40], respectively, as determined from the statistical model, one can estimate the size of critical region to be ∆T /T CEP ≈ 0.1 and ∆µB ≈ 0.1 GeV, which is consistent with the effective model calculations [4]. The critical exponents, however, can not be determined from present data. In the scenario depicted in Fig. 1, signals from the firstorder phase transition and the CEP are connected, because the CEP stays well on the top of the spinodal unstable region in the T -µB plane of the QCD phase diagram.
Indeed, it has been estimated [12] that the temperature (T M ) at point ‘M’ is related to T CEP by T M /T CEP ≈ 1 1 3 − 2 . From the parametrization in Ref. [40] for the chemical freeze-out conditions, one √ can roughly estimate that T M ∼ 50 − 70 MeV and sM ∼ 2 − 3 GeV, which is much smaller than the energies available at SPS, indicating that the signals from the first-order phase transition and the CEP are well separated from each other. Hence, the scenario depicted in Fig. 1 allows us to unambiguously identify the signals from both the first-order phase transition and the CEP. In fact, according to Ref. [13], the enhanced density fluctuation due to spinodal instabil√ ity happens at around 2-4 AGeV ( sN N ∼ 2.3 − 3 GeV) and is in nice agreement √ with our estimate. Of course, the definitive value of sM should depend on the specific equation of state (EOS) adopted in the calculations, which is still largely unknown. √ With decreasing collision energy to sN N ∼ 2−6 GeV, the ∆n is expected to rise again and reach a second maximum as a result of the spinodal instability as shown in Fig. 1. However, the collision energy dependence of ∆nI √ and ∆n at sN N ∼ 2 − 6 GeV might not coincide in this unstable spinodal region, due to the fact that the effect of spinodal instability in the isospin density is not as strong as that in the baryon density for heavy-ion collisions at AGS/SPS energies. To our best knowledge, the only existing data on the p, d, and 3 H yields in the energy region √ sN N ∼ 2 − 6 GeV are from central Au+Au collisions at √ 11.6 AGeV/c ( sN N = 4.86 GeV), namely, 63.2 ± 1.7 for proton [30], 5.3±0.6 for deuteron [30] and 0.264±0.049 for 3 He [31], from which we can obtain Cnp = −0.246±0.094, ∆n = 0.572 ± 0.328 and ∆nI = 0.409 ± 0.171 (with Rnp ≈ (0.84)1/2 , Tch = 0.117 GeV [40], Vch = 1980 fm3 [44] and a 15% weak decay contribution correction to the proton yield [45]), indicating a possible strong increase of ∆n when the collision energy is decreased √ from sN N = 6.3 GeV to 4.86 GeV. In contrast, the ∆nI (Cnp ) continues to decrease (increase). However, the √ statistical uncertainties of ∆n and ∆nI at sN N = 4.86 GeV are large, and more precise measurements are extremely important to confirm the present conclusion. Some effects can potentially affect the above numerical results. These include the uncertainties in the value of λ associated with the entropy of the expanding fireball as well as the chemical freeze-out temperature Tch and volume Vch . Also, the finite size of the fireball and the use of non-relativistic approximation for the nucleon momentum distributions [26, 27] can affect the extracted values of Cnp , ∆n and ∆nI . However, these effects on the above dimensionless quantities are expected to have a weak dependence on the collision energy, and the non-monotonic behaviors shown in Fig. 2 should remain qualitatively similar. 4. Conclusion.—We have proposed that there should exist two separated maxima in the collision energy dependence of the baryon density fluctuation in heavy-ion
5 collisions, with the lower energy one from the spinodal instability due to the first-order phase transition and the higher energy one induced by the second-order phase transition at the CEP. This double-peak structure seems to be supported by the collision energy dependence of the relative neutron density fluctuation ∆n = h(δn)2 i/hni2 at kinetic freeze-out extracted from analyzing the measured yields of p, d and 3 H in central heavy-ion collisions at AGS and SPS energies within the coalescence model. In particular, we have found the ∆n to display a clear √ peak at sN N = 8.8 GeV and a possible strong enhance√ ment at sN N = 4.86 GeV, suggesting that the CEP could have been reached or closely approached in central √ Pb+Pb collisions at sN N = 8.8 GeV and the first-order phase transition could have occurred in central Au+Au √ collisions at sN N = 4.86 GeV. It is particularly interesting to verify the present conclusion by using microscopic transport model simulations as well as hydrodynamics calculations with the proper treatment of the EOS (with and without the CEP) and critical fluctuations. Future measurements of light nuclei production at BES/RHIC, FAIR, NICA and NA61/SHINE are extremely useful to confirm the present observations and to more precisely determine the structure of the QCD phase diagram. Acknowledgments.—The authors thank Vadim Kolesnikov and Peter Seyboth for providing the experimental data. This work was supported in part by the National Natural Science Foundation of China under Grant No. 11625521, the Major State Basic Research Development Program (973 Program) in China under Contract No. 2015CB856904, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, Key Laboratory for Particle Physics, Astrophysics and Cosmology, Ministry of Education, China, the Science and Technology Commission of Shanghai Municipality (11DZ2260700), the US Department of Energy under Contract No. de-sc0015266 and No. de-sc0012704, as well as the Welch Foundation under Grant No. A-1358 and Shandong University.
[1] Z. Fodor and S. D. Katz, JHEP 0404, 050 (2004). [2] M. Asakawa and K. Yazaki, Nucl. Phys. A504, 668 (1989). [3] M. A. Stephanov, K. Rajagopal, and E. V. Shuryak, Phys. Rev. Lett. 81, 4816 (1998). [4] Y. Hatta and T. Ikeda, Phys. Rev. D 67, 014028 (2003). [5] H. -T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod. Phys. E 24, 1530007 (2015). [6] E. Shuryak, Prog. Part. Nucl. Phys. 62, 48 (2009). [7] L. W. Chen, C. M. Ko, W. Liu, and B. W. Zhang, PoS (CPOD 2009) 034, (2009).
[8] P. Braun-Munzinger, V. Koch, T. Sch¨ afer, and J. Stachel, Phys. Rep. 621, 76 (2016). [9] X. F. Luo and N. Xu, Nucl. Sci. Tech. 28, 112 (2017). [10] K. A. Bugaev et al., arXiv:1709.0541 (2017). [11] I. C. Arsene et al., Phys. Rev. C 75, 034902 (2007). [12] J. Randrup, Phys. Rev. C 79, 054911 (2009). [13] J. Steinheimer and J. Randrup, Phys. Rev. Lett. 109, 212301 (2012). [14] J. Steinheimer, J. Randrup, and V. Koch, Phys. Rev. C 89, 034901 (2014). [15] C. Herold, M. Nahrgang, I. Mishustin, and M. Bleicher, Nucl. Phys. A925, 14 (2014). [16] F. Li and C. M. Ko, Phys. Rev. C 93, 035205 (2016); Phys. Rev. C 95, 055203 (2017). [17] B. Berdnikov and K. Rajagopal, Phys. Rev. D 61, 105017 (2000). [18] S. Pratt, Phys. Rev. Lett. 53, 1219 (1984); Phys. Rev. D 33, 1314 (1986). [19] G. F. Bertsch, M. Gong, and M. Tohyama, Phys. Rev. C 37, 1896 (1988). [20] H. Sato and K. Yazaki, Phys. Lett. B98, 153 (1981). [21] L. P. Csernai and J. I. Kapusta, Phys. Rep. 131, 223 (1986). [22] C. B. Dover, U. Heinz, E. Schnedermann, and J. Zimanyi, Phys. Rev. C 44, 1636 (1991). [23] L. W. Chen, C. M. Ko, and B. A. Li, Phys. Rev. C 68, 017601 (2003); Nucl. Phys. A729, 809 (2003). [24] N. -U. Bastian et al., Eur. Phys. J. A 52, 244 (2016). [25] S. Mr` owczy` nski, Acta Phys. Polon. B48, 707 (2017). [26] K. J. Sun and L. W. Chen, Phys. Rev. C 95, 044905 (2017). [27] K. J. Sun, L. W. Chen, C. M. Ko, and Z. Xu, Phys. Lett. B774, 103 (2017). [28] Y. B. Ivanov and A. A. Soldatov, arXiv:1703.05040 (2017). [29] T. Anticic et al. (NA49 Collaboration), Phys. Rev. C 94, 044906 (2016). [30] L. Ahle et al., Phys. Rev. C 60, 064901 (1999). [31] S. Albergo et al., Phys. Rev. C 65, 034907 (2002). [32] W. Grimus, arXiv:1112.3748 (2013). [33] J. Xu and C. M. Ko, Phys. Lett. B772, 290 (2017). [34] V. Koch, arXiv:0810.2520 (2008). [35] M. A. Stephanov, Phys. Rev. Lett. 102 032301 (2009). [36] M. Asakawa and M. Kitazawa, Prog. Part. Nucl. Phys. 90, 299 (2016). [37] H. Liu and J. Xu, arXiv:1709.05178 (2017). [38] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 73, 044910 (2006). [39] C. Alt et al. [NA49 Collaboration], Phys. Rev. C 77, 024903 (2008). [40] J. Cleymans, H. Oeschler, K. Redlich, and S. Wheaton, Phys. Rev. C 73, 034905 (2006). [41] X. Y. Xin, S. X. Qin, and Y. X. Liu, Phys. Rev. D 90, 076006 (2014). [42] N. G. Antoniou and A. S. Kapoyannis, Phys. Lett. B563, 165 (2003). [43] R. A. Lacey, Phys. Rev. Lett. 114, 142301 (2015). [44] A. Andronic, Int. J. Mod. Phys. A 29, 1430047 (2014). [45] A. Andronic, P. Braun-Munzinger, and J. Stachel, Nucl. Phys. A772, 167 (2006).
