Strangeness Neutrality and Baryon-Strangeness Correlations

PHYSICAL REVIEW D 100, 111501(R) (2019) Rapid Communications

Strangeness neutrality and baryon-strangeness correlations

Wei-jie Fu,1 Jan M. Pawlowski,2 and Fabian Rennecke 3,* 1School of Physics, Dalian University of Technology, Dalian 116024, People’s Republic of China 2Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany 3Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA

(Received 24 October 2018; published 20 December 2019)

We derive a simple relation between strangeness neutrality and baryon-strangeness correlations. In heavy-ion collisions, the former is a consequence of quark number conservation of the strong interactions while the latter are sensitive probes of the character of QCD matter. This relation allows us to directly extract baryon-strangeness correlations from the strangeness chemical potential at strangeness neutrality. The explicit calculations are performed within a low-energy theory of QCD with 2 þ 1 dynamical quark flavors at finite temperature and density. Nonperturbative quark and hadron fluctuations are taken into account within the functional renormalization group. The results show the pronounced sensitivity of baryon-strangeness correlations on the QCD phase transition and the crucial role that strangeness neutrality plays for this observable.

DOI: 10.1103/PhysRevD.100.111501

I. INTRODUCTION Focusing on the three lightest flavors, up u,downd, and strange s, the associated chemical potentials are Extracting the phase diagram of QCD as a function conventionally given by linear combinations of baryon, of temperature T and baryon chemical potential μ from B charge, and strangeness chemical potentials μ , μ , and μ . hadronic final states is a main goal but also a main B Q S With the chemical potential flavor-matrix challenge in ultrarelativistic heavy-ion collisions. A detailed understanding of QCD matter in the hot and dense 1 2 1 1 1 1 μ μ μ μ − μ μ − μ −μ medium is indispensable for the interpretation and pre- ¼ diag 3 B þ3 Q;3 B 3 Q;3 B 3 Q S ; ð1Þ diction of experimental data. The situation is further T complicated by the fact that under the conditions of a their coupling to the quarks, q ¼ðu; d; sÞ ,isgivenby ¯γ μ heavy-ion collision, the phase diagram is not only spanned q 0 q. In this work, we want to focus on strangeness and baryon number and therefore assume μ ¼ 0 for the sake of by T and μB, but for instance also other chemical potentials, Q electromagnetic fields, and various timescales. This is simplicity. This corresponds to isospin symmetric matter. relevant for beam-energy scan experiments aiming at Since the incident nuclei do not carry strangeness, the net- exploring the QCD phase diagram [1]. Fortunately, con- strangeness hSi ∼ hs¯γ0si is fixed from the initial conditions servation laws can help to constrain some of these of the collision. The condition hSi¼0 is called strangeness parameters. neutrality. Due to the peculiar beam-energy dependence of The hadrons reaching the detectors in heavy-ion experi- the net-baryon rapidity spectrum, the net-baryon number ments inherit the properties of the QCD medium at freeze- density hBi ∼ hq¯γ0qi in the quark-gluon plasma at central out. Since the typical freeze-out time is many orders of rapidities depends on the beam energy; see, e.g., [2].We μ magnitude shorter than the timescale of flavor-changing therefore work with the standard assumption that B is a μ weak interactions, the net quark number conservation parameter we may choose freely, while S is fixed through of the strong interactions has to be taken into account. quark number conservation. Consequently, there is a chemical potential associated Strangeness is particularly interesting since strange to the conserved quark number of each quark flavor. particles are only created by collisions in the first place. This makes them valuable probes of the matter created in heavy-ion collisions [3]. In a recent work [4],wehave *[email protected] investigated the effect of imposing strangeness neutrality on the phase structure and thermodynamics of QCD. Published by the American Physical Society under the terms of Strangeness neutrality has a sizable impact on the QCD the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to equation of state and the phase diagram, owing to an the author(s) and the published article’s title, journal citation, intricate interplay of strangeness coupled to meson-, 3 and DOI. Funded by SCOAP . baryon-, and quark dynamics at finite T and μB.

2470-0010=2019=100(11)=111501(6) 111501-1 Published by the American Physical Society FU, PAWLOWSKI, and RENNECKE PHYS. REV. D 100, 111501 (2019)

Furthermore, this interplay also leads to the observation where V is the spatial volume. We drop the superscript of that a finite μS is required to enforce hSi¼0. Hence, we are the susceptibilities if the corresponding subscript is zero. led to the implicitly defined function, The baryon-strangeness correlation CBS [6] relevant for the present work reads at strangeness neutrality μ μ μ μ S0ðT; BÞ¼ SðT; BÞjhSi¼0: ð2Þ χBS μ μ μ μ −3 hBSi −3 11 ðT; B; S0Þ μ μ CBSðT; B; S0Þ¼ 2 ¼ S : ð5Þ Since B couples to all quark flavors equally, increasing B hS i χ2ðT;μB; μS0Þ also increases the number of strange quarks over anti- μ strange quarks. To ensure strangeness neutrality, a finite S Since the pressure is a function of T, μB, and μS, requiring is necessary for compensation; see also, e.g., [5]. In the strangeness neutrality implicitly fixes one of these variables hadronic phase, the dominant degrees of freedom (d.o.f.) as function of the others. Here, we choose the strangeness are either open strange mesons or strange baryons, depend- chemical potential, giving rise to Eq. (2). hSi¼0 implies ing on μB. But while both couple to μS, only the latter μ μ couple to B. Thus, it is intuitively clear that S0 is a d S 0 ¼ χ1ðT;μB; μS0Þ nontrivial function that is intimately tied to the nature of dμˆ B QCD matter. ∂μˆ χBS μ μ χS μ μ S0 In this paper, we will demonstrate that this deep con- ¼ 11 ðT; B; S0Þþ 2ðT; B; S0Þ ∂μˆ : ð6Þ nection between strangeness neutrality, which is a conse- B quence of quark number conservation, and the dynamical This simple equation implicitly defines μS0 and we arrive at interplay of hadrons and quarks, which is interweaved with our central result, the phase structure of QCD, can be made explicit. This is achieved by establishing an exact relation between μS0 and ∂μS0ðT;μBÞ 1 C T;μ ; μ 0 : 7 the baryon-strangeness correlation CBS; see (5). This ∂μ ¼ 3 BSð B S Þ ð Þ correlation has been introduced in [6] as a sensitive probe B of the nature of QCD matter. We will exploit said relation to Hence, together with the initial condition μ 0ðμ ¼ 0Þ¼0, compute C at various T and μ and carve out the S B BS B μ 0 can be extracted directly from C at strangeness important role that strangeness neutrality plays for this S BS neutrality. By integrating the experimentally measured CBS quantity. To this end, we employ a Polyakov loop enhanced over the beam energy, one can extract the strangeness 2 1 quark-meson model (PQM) with þ dynamical quark chemical potential at the freeze-out for isospin symmetric flavors as a low-energy effective theory of QCD. matter. Conversely, C is given by the slope of μ 0ðμ Þ. Nonperturbative quantum, thermal and density fluctuations BS S B Most strikingly, CBS has been introduced in [6] as a are taken into account with the functional renormalization diagnostic for the nature of QCD matter. As argued in group. Within this approach, quark-meson models are this work, this can be understood by explicitly examining naturally embedded in QCD [7]. CBS at strangeness neutrality; see (5). Assuming that the system is deep in the deconfined regime, all strangeness is II. STRANGENESS NEUTRALITY AND carried by s and s¯ and there is a strict relation between the BARYON-STRANGENESS CORRELATIONS baryon number carried by strange particles, Bs, and 3 Generalized susceptibilities of conserved charges play a strangeness, Bs ¼ S= . Furthermore, due to asymptotic central role for theoretical and experimental studies of the freedom, there is no correlation between different quark QCD phase structure. This is due to the fact that the closely flavors in this regime; it is a system of dilute current quarks. 1 related cumulants of particle number distributions are Equation (5) then implies CBS ¼ . directly sensitive to the growing correlation length at the The situation is drastically different in the confined phase transition [8]. In the present context at μQ ¼ 0, the phase. Baryons can carry both baryon number and strange- generalized susceptibilities are defined as chemical poten- ness, while mesons can only carry strangeness. Thus, the S tial derivatives of the pressure p, denominator in Eq. (5), χ2, includes both open strange BS mesons and baryons, while the numerator, χ11 , only ∂iþjp T;μ ; μ =T4 includes strange baryons. Hence, one generally finds C ≠ χBS μ μ ð B SÞ BS ij ðT; B; SÞ¼ ; ð3Þ 1 1 ∂μˆ i ∂μˆ j in the confined hadronic phase. For CBS < , the system B S is dominated by the fluctuations of open strange mesons 1 with μˆ ¼ μ=T. Baryon number and strangeness are then and for CBS > it is dominated by strange baryons. Since given by cumulants of net particle numbers are experimental accessible, CBS indeed serves as a sensitive probe of the − χB 3 composition of QCD matter, in particular of its strangeness hBi¼hNB NB¯ i¼ 1 VT ; content [9,10]. It is remarkable that C can be directly S 3 BS hSi¼hN ¯ − NSi¼χ1VT ; ð4Þ S related to μS0 via strangeness neutrality with Eq. (7) for any

111501-2 STRANGENESS NEUTRALITY AND BARYON-STRANGENESS … PHYS. REV. D 100, 111501 (2019) Z Z μ β T and B. This establishes a direct connection between 3 ¯ ¯ Γk ¼ dx0 d xfqðγνDν þ γνCνÞq þ hqΣ5q quark number conservation and the phases of QCD. 0 We note that the situation becomes a little more ¯ Σ ¯ Σ† ˜ Σ ¯ þ trðDν · Dν ÞþUkð ÞþUglueðL; LÞg: ð9Þ complicated at μQ ≠ 0. In this case, Eq. (3) trivially gene- χBSQ χS 0 ralizes to ijk . In addition to 1 ¼ , quark number Quantum, thermal, and density fluctuations of modes with conservation also implies, for instance, χQ=χB ¼ r, where Euclidean momenta k ≤ jpj ≲ 1 GeV have been integrated 1 1 ¯ r is a constant. This also implicitly defines the function out. The gauge covariant derivative is Dν ¼ ∂ν − igδν0A0. μQ0ðT;μBÞ and leads to a generalized form of Eq. (7), The scalar and pseudoscalar mesons are encoded in the a a flavor matrix Σ ¼ T ðσa þ iπaÞ, where the T generate Σ a σ γ π QS UðNfÞ, and 5 ¼ T ð a þ i 5 aÞ; see, e.g., [13]. The ∂μ 0 1 χ ∂μ 0 S ¼ C − 11 Q ; ð8aÞ couplings of quarks and mesons to the chemical potential ∂μ 3 BS χS ∂μ B 2 B μ in Eq. (1) is achieved by formally introducing the vector source Cν ¼ δν0μ and defining the covariant derivative ¯ with acting on the meson fields DνΣ ¼ ∂νΣ þ½Cν; Σ, [14]. Spontaneous chiral symmetry breaking is captured by ˜ the meson effective potential UkðΣÞ, which consists of a BS SQ BS S BQ B ∂μ 0 χ11 ðχ11 − rχ11 Þ − χ2ðχ11 − rχ2 Þ Q ¼ : ð8bÞ fully UðNfÞL × UðNfÞR symmetric part plus pieces that ∂μ S Q BQ SQ SQ BS B χ2ðχ2 − rχ11 Þ − χ11 ðχ11 − rχ11 Þ explicitly break chiral symmetry through finite current 1 quark masses and Uð ÞA through the axial anomaly. The deconfinement phase transition is captured statistically by The dependence on T and μ of all quantities above is B including an effective potential for the gluon background implied. The right-hand side of these equations is given by ¯ ratios of different baryon, strangeness, and charge corre- field UglueðL; LÞ, formulated in terms of the order para- lations and therefore can be interpreted in an analogous meter fields for deconfinement, the Polyakov loops ¯ ¯ ¯ † manner to the discussion above, cf. [11]. Furthermore, L ¼ trc expðigA0=TÞ, L ¼ trc½expðigA0=TÞ . The strategy since the involved susceptibilities can be measured, our of Polyakov loop enhanced effective models is to use a main conclusion is not altered for the isospin-asymmetric potential that is fitted to the lattice equation of state of case. It is worth noting that Eq. (8) generalizes the relations Yang-Mills theory and to include the effects of dynamical used on the lattice to implement the freeze-out conditions in quarks through the coupling to the gluonic background in the quark covariant derivative. For a recent review, see [15]. an expansion about μB=T ¼ 0 to any T and μB [12]. We use the parametrization of the Polyakov loop potential put forward in [16], since it captures the lowest-order III. LOW-ENERGY EFFECTIVE THEORY Polyakov loop susceptibilities which directly contribute to AND FLUCTUATIONS the particle number susceptibilities [17]. In the following, we compute baryon-strangeness corre- Owing to the intricate interplay of meson, baryon, and lations with the help of Eq. (7). The impact of strangeness quark dynamics that contribute to strangeness neutrality neutrality at finite baryon chemical potential is studied and, as a consequence of Eq. (7), also to CBS, accounting within a low-energy effective theory of QCD as initiated for fluctuations of these d.o.f. is indispensable. In [4],we in [4]. In order to capture the main features of strangeness, demonstrated that open strange meson fluctuations are quantum, thermal, and density fluctuations of open strange crucial for strangeness neutrality, exacting a treatment mesons, strange baryons and quarks have to be taken into beyond mean-field. Here, this is achieved by solving the account. Since kaons are pseudo-Goldstone bosons of renormalization group flow equation for the effective action Γ spontaneous chiral symmetry breaking, they are the most k using the functional renormalization group (FRG) [18], relevant strange d.o.f. in the mesonic sector. Moreover, 1 chiral symmetry dictates that if kaons are included as ð2Þ −1 ∂tΓk ¼ Tr½ðΓ ½ΦþRkÞ ∂tRk; ð10Þ effective low-energy d.o.f., all other mesons in the lowest 2 k scalar and pseudoscalar meson nonets have to be included Λ Γð2Þ Φ as well. By coupling quarks to a uniform temporal gluon with t ¼ lnðk= Þ. k ½ is the matrix of second functional ¯ ¯ ð3Þ 3 ¯ ð8Þ 8 c background field A0 ¼ A t þ A t , with t ∈ SUð3Þ, derivatives of the effective action with respect to the fields 0 0 ¯ (statistical) confinement is taken into account. Below the Φ ¼ðq; q;¯ Σ;L;LÞ. Rk implements infrared-regularization 2 2 deconfinement transition temperature Td predominantly at momenta p ≈ k , and the trace involves the integration three-(anti-)quark states contribute and baryons, instead over loop-momenta, the color-, flavor-, and spinor-traces of quarks, are the prevailing fermionic d.o.f. below Td.In as well as the sum over different particle species. total, this gives rise to a 2 þ 1 flavor PQM with the Solving Eq. (10) amounts to successively integrating out β 1 Γ Euclidean effective action ( ¼ =T) fluctuations starting from the initial action k¼Λ, with

111501-3 FU, PAWLOWSKI, and RENNECKE PHYS. REV. D 100, 111501 (2019)

Λ ¼ 900 MeV in our case, down to the full quantum than 1=3. At larger μB, strange baryons become dominant Γ 1 3 effective action k¼0. The FRG provides a nonperturbative resulting in a slope larger than = . At large temperatures, regularization and renormalization scheme for the resum- the system undergoes a crossover to the deconfined phase mation of an infinite class of Feynman diagrams; see, e.g., with a pseudocritical temperature of Td ≈ 155 MeV in our [19] for QCD-related reviews. model. Asymptotically, μS0ðμBÞ approaches the dashed It was shown in [4] that this approach leads to a very black line. However, since the Polyakov loops are still good agreement with the results of lattice QCD for the smaller than one at T ¼ 250 MeV in our computations, equation of state at vanishing μB and finite μB=T within the implying that the system is not fully deconfined, this region accessible on the lattice. For further technical details asymptotic limit is not fully reached here. Still μS0ðμBÞ on the model and on the RG flow equations, we refer to this is approximately linear already at T ≈ 180 MeV, with a work. Since the main qualitative features relevant for slope only slightly smaller than 1=3. strangeness dynamics are captured by this approach, we We extract the baryon-strangeness correlations CBS from will use it to compute the strangeness chemical potential at our result in Fig. 1 via Eq. (7). This is shown in Fig. 2.We strangeness neutrality, μS0ðT;μBÞ and then use Eq. (7) to find good agreement with the results of lattice QCD at extract the baryon-strangeness correlation CBS. vanishing μB, highlighting that we capture the relevant effects quite accurately. Following our discussion above, μ IV. NUMERICAL RESULTS we see that with increasing B in the hadronic phase the baryon-strangeness correlations change from being domi- From the solution of the RG flow equation (10),we nated by the dynamics of open strange mesons to being obtain the full quantum effective action Γ0½Φ. With μ ≳ 420 dominated by strange baryons. At B MeV, CBSðTÞ the solution of the quantum equation of motion develops a nonmonotonicity. It first grows with T since ðδΓ0½Φ=δΦÞjΦ Φ ¼ 0, the pressure is given by ¼ EoM with increasing temperature more strange baryons can be p ¼ −Γ0½Φ =βV. With this we are in the position to EoM excited and a larger μS has to be chosen in order to enforce extract the generalized susceptibilities according to Eq. (3). strangeness neutrality. The resulting increasing slope of μS0 First, we compute the strangeness number hSi in Eq. (3) as then directly translates to a rising C through Eq. (7).For μ μ μ BS a function of T, B, and S. It turns out that for any T and B temperatures above the pseudocritical transition temper- μ μ it is always possible to find a S ¼ S0 such that ature, quark dynamics eventually take over, driving the μ μ μ 0 μ ≤ hSiðT; B; S ¼ S0Þ¼ . We restrict ourselves to B system toward its asymptotic limit. 675 MeV since our model fails to capture important As a result of this dynamical interplay, C ðTÞ shows μ BS qualitative features of the theory at larger B, cf. [4]. nonmonotonous behavior and develops a pronounced μ The transition is a crossover in this range. The resulting S0 maximum already at moderate μ . The sharper the cross- μ μ B is shown in Fig. 1. The characteristic shape of S0ð BÞ can over, the stronger this effect becomes. The maximum is be understood qualitatively from our discussion of Eq. (7). At small μB strangeness is dominated by open strange 1 μ μ mesons, so CBS < and hence S0ð BÞ has a slope smaller

FIG. 2. Baryon-strangeness correlation CBS as a function of temperature T for different baryon chemical potential μB at strangeness neutrality. μB increases from 0 to 675 MeV from FIG. 1. Strangeness chemical potential at strangeness neutral- bottom to top. We compare our results to the hadron resonance ity, μS0, as a function of the baryon chemical potential μB for gas (HRG) containing only experimentally observed resonances various temperatures T (solid lines). T is increasing from bottom [20].AtμB ¼ 0 we also compare to the result of lattice QCD to top from 100 to 250 MeV. The dashed line corresponds to the [9,10,21]. The thin black line indicates the free quark limit. The 1 asymptotic limit of free quarks. In this limit, one finds CBS ¼ errors reflect the 95% confidence level of a cubic spline which leads to μS0 ¼ μB=3 according to Eq. (7). interpolation of our numerical data.

111501-4 STRANGENESS NEUTRALITY AND BARYON-STRANGENESS … PHYS. REV. D 100, 111501 (2019)

Finally, we want to explore the relevance of strangeness conservation on baryon-strangeness correlations. To this end, we compute CBS also at vanishing strangeness chemical potential, μS ¼ 0. Since there is no connection μ χBS between S and CBS in this case, we have to compute 11 S and χ2 separately from the pressure. The result in comparison to the one at strangeness neutrality is shown in Fig. 3. We find a sizable enhancement of CBS if strangeness neutrality is not taken into account, emphasizing the important role it plays here.

V. SUMMARY We have shown that there is an intimate relation between FIG. 3. Comparison between C computed at strangeness BS particle number conservation and the QCD phase structure. neutrality, i.e., μS ¼ μS0 (solid lines), and at μS ¼ 0 (dashed This has been achieved by deriving a direct relation lines), where strangeness conservation is violated at finite μB. between baryon-strangeness correlations and strangeness neutrality. We explicitly demonstrated the sensitivity of located exactly in the crossover region. We therefore find a C on the phase structure and the relevance of strangeness distinct sensitivity of baryon-strangeness correlations to the BS neutrality for this observable. The study of the critical chiral phase transition at finite μ . This is potentially B behavior of C is deferred to future work. Since it is given relevant for experimental measurements of C : if the BS BS by a ratio of second-order cumulants, it may not be freeze-out is close to the phase transition, we predict a steep sensitive to criticality at all. We emphasize that the rise of CBS with decreasing beam-energy. As seen in Fig. 2, the HRG predicts monotonous sensitivity of CBS to the phase transition observed here μ is due to the change from hadronic to partonic dynamics, behavior of CBSðTÞ for any B within the range of which is more pronounced at larger μ due to the specific temperatures studied here. At large μB, the nonmonotonous B behavior we find leads to a significant enhancement over dynamics that drive CBS. A meaningful comparison the HRG predictions at temperatures around the chiral between our theoretical prediction and experimental mea- crossover transition. This could be related to long-range surements may require, among other things, the description strangeness dynamics close to the phase transition. In of net-kaons and protons instead of conserved charges, as general, the HRG is in good agreement with our results well as nonequilibrium effects. well below the phase transition temperature. At μB ¼ 300 MeV and 420 MeV, we find good agreement also ACKNOWLEDGMENTS for larger temperatures. Whether this is physical or coinci- dental is not clear to us. In [10], it was argued that the We thank Robert D. Pisarski for discussions and discrepancy between the lattice and the HRG at μB ¼ 0 Swagato Mukherjee for providing us with the lattice could be due to yet undiscovered strange resonances which data for Fig. 2. F. R. is supported by the Deutsche are not taken into account in the conventional HRG. So Forschungsgemeinschaft (DFG) through Grant No. RE perhaps the effects of missing open strange mesons are 4174/1-1. W. F. is supported by the National Natural compensated by baryon fluctuations at intermediate den- Science Foundation of China under Contract sities. In any case, this leads us to another prediction for the No. 11775041. This work is supported by the ExtreMe measurement of CBS: it should be significantly enhanced Matter Institute. It is part of and supported by the DFG over the HRG prediction at small beam-energies [22]. This Collaborative Research Centre “SFB 1225 (ISOQUANT)”. would indicate a sharp crossover transition.

[1] M. Gazdzicki, Z. Fodor, and G. Vesztergombi No. CERN-SPSC-2006-034, 2006, revised version submit- (NA49 Collaboration), Study of hadron production in ted on 2006-11-06; STAR Collaboration, Star note 598: hadron-nucleus and nucleus-nucleus collisions at the CERN Studying the phase diagram of QCD matter at RHIC (2014); SPS, CERN Technical Reports No. SPSC-P-330 and B. Friman, C. Hohne, J. Knoll, S. Leupold, J. Randrup,

111501-5 FU, PAWLOWSKI, and RENNECKE PHYS. REV. D 100, 111501 (2019)

R. Rapp, and P. Senger, Lect. Notes Phys. 814, 11 (2011); [13] F. Rennecke and B.-J. Schaefer, Phys. Rev. D 96, 016009 V. D. Kekelidze, R. Lednicky, V. A. Matveev, I. N. (2017). Meshkov, A. S. Sorin, and G. V. Trubnikov, Eur. Phys. J. [14] J. B. Kogut and D. Toublan, Phys. Rev. D 64, 034007 A 52, 211 (2016); T. Galatyuk, Nucl. Phys. A931,41 (2001). (2014); H. Sako et al., Nucl. Phys. A931, 1158 (2014). [15] K. Fukushima and V. Skokov, Prog. Part. Nucl. Phys. 96, [2] L. Adamczyk et al. (STAR Collaboration), Phys. Rev. C 96, 154 (2017). 044904 (2017). [16] P. M. Lo, B. Friman, O. Kaczmarek, K. Redlich, and C. [3] P. Koch, B. Muller, and J. Rafelski, Phys. Rep. 142, 167 Sasaki, Phys. Rev. D 88, 074502 (2013). (1986). [17] W.-j. Fu, J. M. Pawlowski, F. Rennecke, and B.-J. Schaefer, [4] W.-j. Fu, J. M. Pawlowski, and F. Rennecke, arXiv: Phys. Rev. D 94, 116020 (2016). 1808.00410. [18] C. Wetterich, Phys. Lett. B 301, 90 (1993). [5] J. Letessier, A. Tounsi, U. W. Heinz, J. Sollfrank, and J. [19] J. Berges, N. Tetradis, and C. Wetterich, Phys. Rep. 363, Rafelski, Phys. Rev. D 51, 3408 (1995). 223 (2002); J. M. Pawlowski, Ann. Phys. (Amsterdam) 322, [6] V. Koch, A. Majumder, and J. Randrup, Phys. Rev. Lett. 95, 2831 (2007); H. Gies, Lect. Notes Phys. 852, 287 (2012); 182301 (2005). B.-J. Schaefer and J. Wambach, Phys. Part. Nucl. 39, 1025 [7] M. Mitter, J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D (2008); J. Braun, J. Phys. G 39, 033001 (2012). 91, 054035 (2015); J. Braun, L. Fister, J. M. Pawlowski, and [20] P. Braun-Munzinger, K. Redlich, and J. Stachel, in F. Rennecke, Phys. Rev. D 94, 034016 (2016); F. Rennecke, Quark-Gluon Plasma, edited by R. C. Hwa (World Scien- – Phys. Rev. D 92, 076012 (2015); A. K. Cyrol, M. Mitter, tific Publishing, Singapore, 2003), Vol. 3, pp. 491 599; M. J. M. Pawlowski, and N. Strodthoff, Phys. Rev. D 97, Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 054006 (2018). 030001 (2018). [8] M. A. Stephanov, Phys. Rev. Lett. 102, 032301 (2009). [21] A. Bazavov et al., Phys. Rev. D 95, 054504 (2017);A. [9] A. Bazavov et al., Phys. Rev. Lett. 111, 082301 (2013). Bazavov et al. (HotQCD Collaboration), Phys. Lett. B 795, [10] A. Bazavov et al., Phys. Rev. Lett. 113, 072001 (2014). 15 (2019); S. Mukherjee (private communication). [11] A. Majumder and B. Muller, Phys. Rev. C 74, 054901 [22] We start seeing a pronounced peak forming at μ ≳ 550 (2006). B MeV. If we take the scales of our computation [12] A. Bazavov et al., Phys. Rev. Lett. 109, 192302 (2012);S. atffiffiffi face value, this corresponds to beam energies of p ≲ 5 Borsanyi, Z. Fodor, S. D. Katz, S. Krieg, C. Ratti, and K. K. s GeV. This would be beyond the reach of current Szabo, Phys. Rev. Lett. 111, 062005 (2013); A. Bazavov beam-energy scan experiments, but within the range et al. (HotQCD Collaboration), Phys. Rev. D 96, 074510 covered by future experiments, e.g., at FAIR, NICA, or (2017). J-PARC.

111501-6