Theore cal perspec ve on strangeness produc on
Che-Ming Ko, Texas A&M University
q Introduc on q Nuclear equa on of state q Deeply subthreshold strangeness produc on q Mean-field poten als q Phi meson q Sta s cal hadroniza on vs recombina on q Lambda polariza on q Exo c hadrons q Summary
Supported by US Department of Energy and the Welch Founda on
1 Nuclear Physics A343 (1980) 519 - 544; @ ~or~~-~o~ia~ Polishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher VOLUME 48, NUMBER 16 PHYSI t" AL REVIEW LETTERS 19 APRIx. 1982 First theore cal studies on strangeness produc on to hold for large values of A. . We have checked A. is presented. We understand that M. Peskin explicitly the agreement of the free energy of and K. Wilson have obtained similar results. the standard and reduced models in lower orders of perturbationKAONtheory PRODU~IONin arbitrary dimensionsIN RELATIVISTIC NUCLEAR COLLISIONS ’
I up to (1/A)']. It is extremely important to check K. Wilson, Phys. Rev. D 10, 2445 (1974); J. Kogut if the equivalence of the models persistsJ. RANDRUPat small and-and C.L. M.Susskind, KO Phys.NPA 343, 519 (1980) Bev. D 11, 395 (1975). values of A. . ~E. Witten, Cargese Lecture Notes, 1979 (unpub- We are gratefulNuclearto Professor Science Division,E. Brdzin Lawrencefor his Berkeleylished); Laboratory,Pu. M. Berkeley,Makeenko CAand 94720,A. A. USAMigdal, Phys. Lett. critical remark on the original version of our 88B, 135 (1979). manuscript. Received 19 FebruaryD. Gross 1980and E.Witten, Phys. Bev. D 21, 446 (1980), S. Wadia, Enrico Fermi Institute Report No. 79/44 (un- Note added. —After the submission of this paper Abstract: Kaon pr~uct~on in relativistic nuclear collisionspublished) is .studied on the basis of a conventional we received a preprint by G. Bhanot, U. Heller, Foerster, Phys. Lett. 87B, 87 (1979); T. Eguchi, multiple-collision model. The input is the differentialD. cross sections for kaon production in and H. Neuberger where some evidence for a Phys. Lett. 87B, 91 (1979); D. Weingarten, Phys. Lett. elementary baryon-baryon collisions, estimated in a simple model. Inclusive kaon spectra are spontaneous breakdown of U(1) symmetry a.t small 87B, 97 (1979). calculated at 2.1 GeV/nucleon for a number of ‘experimental cases. The calculated kaon yield is approximately isotropic in the mid-rapidity frame and extends considerably beyond the nucleon- nucleon kinematical limit.
Strangeness Production in the Quark-Gluon Plasma 1. Introduction Johann Rafelski and Berndt Muller PRL 48, 1066 (1982) Unioetsita-t, D-6000 Franhfu R am Main, Germany InstitutMostfuw Theoretisoheof our knowledgePhysih, Johann about~otfgang atomicGoethe nuclei has been obtained by using (Received 11 January 1982) (predominantly light) nuclear projectiles with relatively low incident energies. With Rates are calculated for the processes gg-ss and uu, dd-ss in highly excited quark- such toolsgluon plasma.we haveFor arrivedtemperature at a Trather~ 160 MeVgoodthe understandingstr~~~eness abundance of nuclearsaturates propertiesduring at normalthe densitylifetime (-and10 low~ sec) excitation.of the plasma Verycreated littlein high-energy knowledgenuclear hascollisions. been accumulatedThe chemical equilibration time for gluons and light quarks is found to be less than 10 sec. so far about the properties at abnormal densities and/or high excitation. On2 PACS numbers: 12.35.Ht, 21.65.+f theoretical grounds it has been suggested that other phases of nuclear matter may Givenexist.the presentThese exoticknowledge phasesabout includethe interac- pion condensationmultistrange ‘),hadrons. density Afterisomerismidentifying *), the-and tions between constituents (quarks and gluons), strangeness-producing mechanisms we compute it appearsquarkalmost matterunavoidable 3). The properthat, atconditionssufficiently for thethe relevantoccurrencerates ofas suchfunctions phenomenaof the energy may den- high energyexist indensity ~trophysicalcaused by sites,compression such asand/ neutron sitystars,("temperature") and our ~de~tandingof the plasma stateof astro-and com- or excitation,physical thephenomenaindividual hadronsis linkeddissolve to our inknowledgepare themof nuclearwith those propertiesfor light scfarand fromd quarks. the a new phase consisting of almost-free quarks and In lowest order in perturbative QCD ss-quark ordinary state. High-energy nuclear beams offer a unique tool for probing exotic gluons. ' This quark-gluon plasma is a highly ex- pairs can be created by annihilation of light quark- cited nuclearstate of hadronicphases inmatter the laboratory.that occupies a antiquark pairs [Fig. 1(a)] and in collisions of two volume Throughlarge as comparedthe last withseveralall characteristicyears, steady progressgluons [Fig. has1(b)]. been Themadeaveraged in thetotal studycross of sec- volume individual color tions for these processes were calculated by lengthhigh-energyscales. Within nuclearthis collisions 4). A wealth of inclusive or semi-incl~ive data has charges exist and propagate in the same manner as theybeendo insideaccumulatedelementary andparticles a hostas ofde- models have been introduced. However, until scribed,now e.nog. , strikinglywithin the Massachusettsexotic ' signal&Institute have appeared, and it has become a)increasingly clear of Technologythat the (MlT)identificationbag model. of possible exotic phenomena is conditionedk) on our ability It is generally agreed that the best way to create a quark-gluonto accountplasma well inforthe thelaboratory dominantis processeswith of more conventionalk2r character. We must collisionsthereforeof heavy try nucleito understandat sufficiently in detailhigh ener- the overall collision dynamics. gy. We Soinvestigate far it hasthe notabundance been possibleof strangeness to obtain a ~ambiguous view of the evolution of a as function of the lifetime and excitation of the k)~~q plasmahigh-energystate. This nuclearinvestigation collision.was motivatedThe one-particle inclusive spectra, which form the by themainobservation part ofthat thesignificant data, havechanges provedin rela-unsuitable for discriminating between widely tive and absolute abundance of strange particles, k2~~ q2 such as A,' could serve as a probe for quark- b) ’ Prepared for the US Department of Energy under contract No. W-740.5-ENG-48. gluon plasma formation. Another interesting sig- FIG. 1. Lowest-order QCD diagrams for ss produc- nature may be the possible creation of exotic 519tion: (a) qq —ss, (b) gg ss.
1066 1982 The American Physical Society Statistical thermodynamics in relativistic particle and ion physics: Canonical or grand canonical?
Zeitschrift für Physik C Particles and Fields
December 1985, Volume 27, Issue 4, pp 541–551
R. Hagedorn (2) K. Redlich (1)
1. CERN, Geneva 23, Switzerland 2. Institut für Theoretische Physik, Technische Hochschule, Darmstadt, Germany
Article
Z. Phys. C - Particles and Fields 27, 541-551 (1985) ~Phy~kC Received: andFK 6s 10 September 1984 9 Springer-u 1985
DOI (Digital Object Identifier): 10.1007/BF01436508
Cite this article as: Hagedorn, R. & Redlich, K. Z. Phys. C - Particles and Fields (1985) 27: 541. doi:10.1007/BF01436508 Statistical Thermodynamics in Relativistic Particle and Ion Physics: Canonical74 Citations or Grand Canonical? 93 Downloads R. Hagedorn and K. Redlich 1 Zeitschri Fur PhysiK C 27, 541 (1985) AbstractCERN, CH-1211 Geneva 23, Switzerland Received 10 September 1984
We consider relativistic statistical thermodynamics of an ideal Boltzmann gas consisting of the particlesK, N, Λ, Σ and their antiparticles. Baryon number (B) and strangenessAbstract. We ( Sconsider) are conserved. relativistic While statistical any relativisticthermo- gasin is tiny necessarily volumes, grandbut even the applicability of equilib- canonicaldynamics ofwith an idealrespect Boltzmann to particle gas numbers, consisting conservationof the rium laws statistical can be treatedthermodynamics is questionable in particles K, N, A, 22 and their antiparticles. Baryon processes taking place at times of the order of 10-23 s canonicallynumber (B) andor grand strangeness canonically. (S) are Weconserved. construct While the partitionover spacefunction dimensions for of a few fermi and in the canonicalany relativisticB×S conservationgas is necessarily and grand compare canonical it with with the grandpresence canonical of collective one. It motions is found involving relative veloc- thatrespect the to grand particle canonical numbers, partition conservation function laws iscan equivalent be ities to areaching largeB approximationthe order of the velocity of light. Expe- treated canonically or grand canonically. We construct rience in high-energy physics accumulated over almost ofthe the partition canonical function one. for The canonical relative B difference • S conservation between canonicalthree decades and grandshows, canonicalhowever, that it often does make quantitiesand compare seems it with to decrease the grand like canonical const/B one. (two It numerical is sense examples) to assume and approximate from this local a thermal equilibrium simplefound that thumb the rulegrand for canonical computing partition canonical function quantities is fromin such grand processes: canonical models ones combining is collective mo- tions with equilibrium thermodynamics in local rest guessed.equivalent For to aprecise large B calculations,approximation an of integral the canonical representation is given. one. The relative difference between canonical and frames often describe large amounts of experimental grand canonical quantities seems to decrease like data in good overall approximation. Onconst/B leave (two from: numerical Institute examples)for Theoretical and from Physics, this Wroclaw,a Therefore, Poland. it seems reasonable to further develop simple thumb rule for computing canonical quantities equilibrium thermodynamics (of3 relativistic gases) from grand canonical ones is guessed. For precise because it is one of the ingredients of a full--if calculations, an integral representation is given. approximate--phenomenological description of pro- cesses taking place in high-energy particle collisions, in particular in relativistic heavy ion collisions, where the space-time regions and numbers of particles involved 1. Introduction are larger than in collisions of elementary particles and Relativistic statistical thermodynamics is applied in where it is hoped that the phase transition to a quark- some parts of high-energy physics ranging from cos- gluon plasma might be observed experimentally. mology to particle collisions in the laboratory. Of We shall, therefore, study in this paper relativistic present interest is the phase transition from hadronic statistical thermodynamics as if there were no doubts matter to a quark gluon plasma as well as the about its usefulness. In fact, we shall address ourselves description of hadronic matter and the quark-gluon to one particular question which seems not yet to have plasma as such [1]. been dealt with in the otherwise vast literature: While cosmological applications deal with large What are the differences between the canonical and the amounts of matter/radiation in large volumes, where grand canonical treatment of conservations laws? the grand canonical description is the adequate tool, the laboratory situation is much more complicated: Let us illustrate the situation in a simple example not only do we deal with very small amounts of matter making clear one of the essential distinctions between non-relativistic and relativistic statistical thermo- dynamics. In non-relativistic statistical thermo- 1 On leave of absence from: Institute for Theoretical Physics, dynamics particle numbers are--in the absense Wroclaw, Poland. Present address: Institut fiir Theoretische Physik, of chemical reactions--conserved. Therefore the Technische Hochschule, D-6100 Darmstadt, FRG canonical N-particle partition function is a senseful Subthreshold kaon production in high-energy HIC
Aichelin & Ko, PRL 55, 2661 (1985) Fuchs, PRL 86, 1974 (2001)
Nb+Nb @ 700 AMeV
§ Kaon production at subthreshold energy in HI collisions is sensitive to
nuclear EOS, and data are consistent with a soft one. 4 Nuclear symmetry energy Li, Chen & Ko, Phys. Rep. 464, 113 (2008) EOS of asymmetric nuclear matter
2 ρn − ρp E(ρ, δ ) ≈ E(ρ, δ = 0) + Esym(ρ)δ , δ = ρn + ρp 2 L ⎛ ρ −ρ ⎞ Ksym ⎛ ρ −ρ ⎞ E (ρ) E (ρ ) ⎜ 0 ⎟ ⎜ 0 ⎟ Symmetry energy sym = sym 0 + ⎜ ⎟ + ⎜ ⎟ 3 ⎝ ρ0 ⎠ 18 ⎝ ρ0 ⎠ Symmetry energy coefficient from mass formula Esym(ρ0 ) ≈ 30 MeV
∂Esym(ρ) Slope L = 3ρ theoretical values -50 to 200 MeV 0 ∂ρ ρ=ρ0 ∂2E (ρ) Curvature K = 9ρ2 sym theoretical values -700 to 466 MeV sym 0 ∂2ρ ρ=ρ0 K( ) K K 2, K = K - 6L Nuclear matter Incompressibility δ = 0 + asyδ asy sym
Empirically, K0~ 230±10 MeV, Kasy~ -500 ±50 MeV, L~ 88±50 MeV
γ Esym(ρ)~ 32€ (ρ /ρ0 ) with 0.7<γ<1.1 for ρ<1.2ρ0 § Symmetry energy at high densities is practically undetermined ! 5 Theoretical predictions on nuclear symmetry energy Fuchs, JPG 35, 014049 (2008)
§ Large uncertainties at both low and high densities 6 W.-J. Xie et al. / Physics Letters B 718 (2013) 1510–1514 1513
Conflic ng results on symmetry energy from charged pion ra o
IBUU IQMD
Fig. 3. (Color online.) The π /π ratio as a function of the neutron/proton ratio of reaction systems for central 40Ca 40Ca, 96Ru 96Ru, 96Zr 96Zr and 197Au 197Au − + + + + + collisions at 400A MeV. The results are calculated by different stiffness of the symmetry energy using the ImIBL model with the SM (left panel) and different transport theories (right panel). Xiao et al, PRL 102, 062502 (2009) Feng & Jin, PLB 683, 140 (2010)
BL
-1 < x < 0
RVUU
7 Shi et al., PLB 718, 1510 (2013) Song & Ko, PRC 91, 014901 (2015) Fig. 4. (Color online.) Excitation functions of the π /π ratio in central 197Au 197Au collisions for different stiffness of the symmetry energy using the ImIBL model with − + + the SM (left panel) and different transport theories (right panel). energies near the π threshold, the π − is more easy to pro- Acknowledgements duce than the π + in the early stage of the collisions. For the higher incident energies, the π −/π + values are only affected by This work was supported by the National Natural Science Foun- the symmetry energy because the effects of the fluctuations are dation of China under Grants No. 11025524 and No. 11161130520 negligible. and the National Basic Research Program of China under Grant No. We would like to point out that the threshold effects are not 2010CB832903. included in the present work. According to the Ref. [14,15],the threshold effects increase the π − yield through enhancing the pro- References duction of the "− resonances. The in-medium threshold values of producing π and π 0 production are different because of the dif- [1] B.A. Li, L.W. Chen, C.M. Ko, Phys. Rep. 464 (2008) 113. ± [2] H.A. Bethe, Rev. Mod. Phys. 62 (1990) 801. ferent effective Dirac masses of the four isospin states of the " [3] M.B. Tsang, Y. Zhang, P. Danielewicz, M. Famiano, Z. Li, W.G. Lynch, resonance. In the framework of the RBUU model, the symmetry A.W. Steiner, Phys. Rev. Lett. 102 (2009) 122701. energy is related to the ρ-andδ-meson couplings. Because of the [4] W. Reisdorf, M. Stockmeier, A. Andronic, M.L. Benabderrahmane, O.N. Hart- inconsistent treatment of the mean-field potentials in the RBUU mann, N. Herrmann, K.D. Hildenbrand, Y.J. Kim, M. Kiš, P. Koczon,´ T. Kress, Y. Leifels, X. Lopez, M. Merschmeyer, A. Schüttauf, V. Barret, Z. Basrak, and ImIBL approaches, the predicted dependence of π −/π + ra- N. Bastid, R. Caplar,ˇ P. Crochet, P. Dupieux, M. Dželalija, Z. Fodor, Y. Grishkin, tio on the symmetry energy is opposite. From the left panel of B. Hong, T.I. Kang, J. Kecskemeti, M. Kirejczyk, M. Korolija, R. Kotte, A. Lebe- Fig. 4 we can see that the π −/π + yield at lower incident energies dev, T. Matulewicz, W. Neubert, M. Petrovici, F. Rami, M.S. Ryu, Z. Seres, is more sensitive to the symmetry energy and can provide more B. Sikora, K.S. Sim, V. Simion, K. Siwek-Wilczynska,´ V. Smolyankin, G. Stoicea, information of the symmetry energy, especially at beam energies Z. Tyminski,´ K. Wisniewski,´ D. Wohlfarth, Z.G. Xiao, H.S. Xu, I. Yushmanov, A. Zhilin, FOPI Collaboration, Nucl. Phys. A 781 (2007) 459. smaller than the π threshold. Further experimental works are very [5] Z.G. Xiao, B.A. Li, L.W. Chen, G.C. Yong, M. Zhang, Phys. Rev. Lett. 102 (2009) necessary to constrain the symmetry energy and test our findings 062502. here. [6] Z. Kohley, M. Colonna, A. Bonasera, L.W. May, S. Wuenschel, M. Di Toro, In summary, the π production in central heavy ion collisions S. Galanopoulos, K. Hagel, M. Mehlman, W.B. Smith, G.A. Souliotis, S.N. Soisson, B.C. Stein, R. Tripathi, S.J. Yennello, M. Zielinska-Pfabe, Phys. Rev. C 85 (2012) at the incident energies from 250 to 1200A MeV is studied in 064605. the framework of the ImIBL model. It is found that the calculated [7] Z.Q. Feng, G.M. Jin, Phys. Lett. B 683 (2010) 140. results of π multiplicity with the SM are close to the experimen- [8] H. Stöcker, W. Greiner, Phys. Rep. 137 (1986) 277; tal data. The π multiplicity is very sensitive to the fluctuations Q. Li, Z. Li, S. Soff, Marcus Bleicher, Horst Stöcker, J. Phys. G: Nucl. Part. Phys. 32 at energies smaller than the π threshold. The dependence of the (2006) 151. [9] G.F. Bertsch, H. Kruse, S. Das Gupta, Phys. Rev. C 29 (1984) 673. π −/π + ratio on N/Z of reaction systems at 400A MeV and the [10] J. Aichelin, A. Rosenhauer, G. Peilert, H. Stoecker, W. Greiner, Phys. Rev. Lett. 58 197 197 excitation functions of π /π ratios for central Au Au (1987) 1926. − + + collisions are compared with the SM, but different stiffness of sym- [11] B.A. Li, Phys. Rev. Lett. 88 (2002) 192701. metry energy and different transport theories. Our results support [12] B.A. Li, G.C. Yong, W. Zuo, Phys. Rev. C 71 (2005) 014608. [13] R. Stock, Phys. Rep. 135 (1986) 259. the view of the BUU model that the calculations with the supersoft [14] G. Ferini, T. Gaitanos, M. Colonna, M. Di Toro, H.H. Wolter, Phys. Rev. Lett. 97 symmetry energy approach the FOPI data. (2006) 202301. Symmetry energy effect on K+/K0 ratio 6 Zhi-Gang Xiao et al.: Title Suppressed Due to Excessive Length Lopez et al. (FOPI Collaboration) , Phys. Rev. C75, 011901(R) (2007) the EOS of nuclear matter at high densities [58]. Earlier, the kaon yield has been used successfully in constraining
the incompressibility of isospin symmetric nuclear mat- Zr ) 2 ter. For instance, transport model analyses of the Kaos 0 E=1.528A GeV /K data suggest that the EOS of symmetric matter is soft + Open: Infinite nuclear ma er [59,60,61]. For the isospin effect, Q. Li et al has noticed 1.8 in 132Sn+132Sn at 1.5 GeV/u that the K+/K0 ratio / (K Full: Heavy ion collisions Ru depends on the symmetry potential [8]. Later in [9], an ) + 0 0 1.6
excitation function of K /K in Au+Au collisions with /K different symmetry potential calculated and confirmed the +
+ 0 (K dependence of the K /K ratio on the isovector poten- 1.4 tial. It has been realized that the isovector part of the in- medium interaction affects the kaon production via two mechanisms: (i) a symmetry potential effect, i.e., a larger 1.2 neutron repulsion in n-rich systems, and (ii) a threshold effect, due to the change in the self-energies of the particles involved in inelastic processes. Thus the K+/K0 ratio was 1 proposed as another, probably better, probe of the high- 30 75 100 density behavior of nuclear symmetry energy [9]. Similar 0.8 Esym (MeV, ρB=2.5ρ0) to the π−/π+ observable, the sensitivity of the K+/K0 to the variation of symmetry energy is more pronounced at DATA THERM. NL NLρ NLρδ beam energies near the kaon production threshold [8,9]. Data Vs Models Fig. 7. (Color Online) The double ratio of K+/K0measured § K+/K0 ratioin Ru+Ru only and Zr+Zrincreases systems slightly (red star) in with comparison the withstiffness of The first attempt to extract the stiffness of symmetry thermal model calculations ( open cross in left panel) and rela- energy from K+/K0 observable was done by the symmetry FOPI tivistic energy mean field transport model (right panel) for infinitenu- 8 collaboration [62]. To reduce the systematic uncertainties, clear system (open) and realistic collisions (solid), takenfrom the double ratio of K+/K0 in Zr+Zr and Ru+Ru sys- Ref. [62]. tems with same mass but characterized by different N/Z composition at 1.528 GeV/u is used. The experimental result is compared with thermal model calculation and a 3.3 η production and effects of nuclear symmetry relativistic mean field transport model using two different energy collision scenarios (the infinite nuclear matter and the re- alistic finite nuclear collision) and under different assump- This subsection is a brief summary of the work originally tions on the stiffness of the symmetry energy, as presented published in ref. [16]. Given the inconsisteny of the high in Fig. 7. Thermal model calculation reproduces the data. density Esym(ρ) among various transport model analy- For the transport model calculations, the sensitivity of the ses, searching for more sensitive new probes of Esym(ρ)at double ratio K+/K0 on the stiffness of symmetry energy high-densities is still ongoing. Compared to pion or kaon, differs in the two scenarios. In the former case, it is evident massive mesons generally probe the Esym(ρ) at higher that the double ratio K+/K0 increases rapidly if one goes densities if the effects of the final state interactions do not from NL (symmetry energy only containing kinetic con- smear the signal. The η meson, as the massive member tribution) via NLρ (including the isovector-vector field ρ) in the nonet of pseudoscalar Goldstone mesons with mass 2 to NLρδ (including both ρ and δ effective field), or equiv- 547.853 MeV/c [63], is a preferred probe to Esym(ρ)for alently with increasing the stiffness of Esym(ρ)from50to further good reasons: (1) The η meson experiences weaker 100 MeV at 2.5ρ0. While in the realistic nuclear collisions final state interactions compared to pion due to its hid- of Ru/Zr nuclei, the sensitivity is less pronounced and the den strangeness (the ss¯ component). And because the net double ratio of K+/K0 differs by about 5% between NL strangeness content is zero in η mesons, they can be pro- and NLρδ parameterizations. This reduced sensitivity is duced without another strange partner in the final state mainly attributed to two dynamical effects that tend to and thus require less energies. In addition, it has signifi- weaken the contribution of the isovector interaction, i.e., cant photon and dilepton decay branch ratio, which pro- the fast neutron emission and the transformation of neu- vides a clean electromagnetic probe to Esym(ρ). (2) The tron into proton in inelastic channels. It indicates that to η meson is sensitive to the number of p-n collisions while obtain experimentally accessible sensitivity, one shall use the π−/π+ ratio is determined by the ratio of n-n and p-p the lower beam energy in heavier system with larger N/Z colliding pairs, η and the π−/π+ ratio provide comple- asymmetry. On the experimental side, however, measur- mentary information about the symmetry energy. (3) The ing a sample of K0 and K+ with sufficiently high statis- η meson is produced from the proton-neutron collision for tics and high precision at subthreshold energies will be a which the relative momentum is determined mainly by the considerable challenge and calls for great effort in future gradient of the isovector potential. However, we shall keep experiment. in mind that the elementary η production cross sections Symmetry energy effect on Σ-/Σ+ ratio
Li, Li, Zhao & Gupta, PRC 71, 054907 (2005)
§ Soft symmetry energy (F ) leads to a smaller Σ-/Σ+ ratio − + − + 2 132 132 FIG. 7: The ratios π /π (left) and Σ /Σ (right) for the collisions Sn + Sn (Eb =19 .5A, 112 112 2.5A,and3.5A GeV; b =2fm)and Sn + Sn (Eb =3.5A GeV; b =2fm),calculatedwith
γ=1 a=3 the different symmetry potentials F1 and F2 .
γ=1 a=3 with F1 and F2 reduces strongly. As the beam energy increases further, at Eb =3.5A GeV the Σ−/Σ+ ratio falls further but the difference between the Σ−/Σ+ ratios calculated γ=1 a=3 − + with F1 and F2 appears again, the Σ /Σ ratio with soft symmetry potential now becoming higher than that with the stiff one. In sequel, it might be interesting to investigate why the behavior of Σ−/Σ+ and π−/π+ ratios is so different, as far as the influence of the density dependence of the symmetry potential is concerned. One basic difference is that, like nucleons, Σ± hyperons are under the influence of the mean field produced by the surrounding nucleons, as soon as they are produced. The symmetry potential of hyperons also play an important dynamic role and results in a strong effect on the ratio of the negatively to positively charged Σ hyperons. Thus, we further investigate the Σ−/Σ+ and π−/π+ production ratios when the symmetry potential of Σ hyperons and resonances, except nucleons, is switched off.Averysmall difference for the π−/π+ ratio, but a large difference for Σ−/Σ+ ratio is found with the switching on and off of the symmetry potential of Σ and resonances. Our results for Σ−/Σ+
14 Subthreshold production of Cascade (Ξ)
Li, Chen, Ko & Lee, PRC 85, 064902 (2012) § Using cross sections calculated from meson-exchange model σ(Κbar Λ→πΞ) ~ 5-10 mb σ(Κbar Σ→πΞ) ~ 5-10 mb σ(ΛΛ→NΞ) ~ 40 mb σ(ΛΣ→NΞ) ~ 40-60 mb σ(ΣΣ→NΞ) ~ 15-30 mb σ(KN→Kbar Ξ) ~ 0.2 mb
→ Ξ-/(Λ+Σ0) ~ 3.4x10-3 compared with HADES data of 5.6x10-3 [Agakishiev et al., PRL 103, 132301 (2009)] § Sensitivity of Ξ-/Ξ0 to nuclear symmetry energy? 10 Role of high mass baryon resonances J. Steiheimer and M. Bleicher, J. Phys. G43, 015104 (2016)
§ Good agreement with data a er including in UrQMD decays of N*(1990), N*(2080), N*(2190), N*(2220), and N*(2250) to ΞKK
with a branching ra o of 10% and to Nφ with a BR of 0.2%. 11 Hyperon produc on at AGS Pal, Ko, Alexander, Chung, and Lacey, PLB 595, 158 (2004)
§ Based on hadronic transport model ART § Strangeness-exchange reac ons
KN¯ ⇤(⌃)⇡ § Au+Au @ 6A GeV ! K¯ ⇤(⌃) ⌅⇡ ! § Reproduce reasonable E895 data § Ξ does not reach chemical equilibrium
12 16 Hadronic poten als in nuclear medium KO, KOCH & LI Ko & Li, JPG 22, 1673 (1996); IN-MEDIUM EFFECTS Ko, 17 Koch & Li, ARNPS 47, 505 (1997) § Kaons and an kaons: Chiral effec ve Lagrangian → repulsive poten al for kaons tikaons [96] are shown. It is seen that the data on subthreshold K− pro- and a rac ve poten al for an kaons duction [99] support the existence of an attractive antikaonmean-field potential. For the K− flow, there only exist very preliminary data from 2 3 2 3 2 2 a 0.22 GeV fmthe, FOPIa collaboration0.45 GeV (J.fm Ritman, personal communication), which U = ω − ω , ω = m + p K = K = K ,K K ,K 0 0 K seem to show that the K−’s have a positive flow rather an antiflow, 2 3 2 2 2 bK = 0.33 GeV fmagain consistent with an attractive antikaon mean-field potential. ωK ,K = mK + p − aK .K ρs +(bK ρB ) ± bK ρB
RAPID COMMUNICATIONS U 20 MeV,U 120 MeV at 0.16 fm -3 ⇒ K = ANTIFLOWK = − OF KAONS IN RELATIVISTICρ 0 HEAVY= ION . . . PHYSICAL REVIEW C 62 061903͑R͒ € €
€
K+ K+ K+ K-
FIG. 3. The transverse momentum dependence of the directed
FIG. 2. The kaon directed transverse flow ͗px͘ as a function of flow v1 for midrapidity protons and kaons without and with a kaon normalized rapidity y/y c.m. in AuϩAu collisions at Ebeam scalar-vector potential. The results are for Au(6A GeV)ϩAu colli- • Experimental data on spectrum and directed flow are consistent with ϭ6A GeV. The filled circles are the E895 data while the other sions at an impact parameter of 6 fm. curves correspond to ART model calculations (bՇ7 fm and pt repulsive kaon and a rac ve р700an kaon MeV/c) without poten als. kaon mean field potential and with kaon effectively repel the K0’s away13 from the nucleons. This sug- dispersion relations obtained from impulse approximationFigure and 4:gests Antikaon that at the AGS yield energies, (left kaon panel) flow may and provide flow infor- (right panel) inheavyion Figure 3: Kaon yield (left panel) and flow (right panel) inscalar-vector heavy ion potential. For the latter kaon potential, results with mation about the kaon dispersion relation for densities below zero kaon formation time K are also shown. The insetcollisions. shows the The solid and dotted curves are results from transport model collisions. The solid and dotted curves are results from transport model fo ϳ40. Since the kaons produced in the ultradense stage calculations with and without kaon mean-field potential, respectively.density dependence of the kaon potential UK at zerocalculations momentum mostly with travel with and nucleons, without the kaon kaon flow is expectedmean-field to be potential, respectively. obtained in the impulse approximation and scalar-vector[The potential. datasensitive are from to its formation ref. [99].] time. This effect is illustrated in [The data are from refs. [89, 90].] Fig. 2 for the scalar-vector potential where the kaon forma- K 0 stage of the reaction at time tՇ5 fm/c and the rescattering tion time fo is set to zero. The early production of K then of the kaons with the nucleons in the dense matter thus leads to larger rescattering in the initial stage of the reaction. We conclude this section on0 kaons by pointing out that the properties causes them to flow in the direction of the nucleons.of kaonsConsequently, in matter the are antiflow qualitatively of K ’s is slightly suppressed described es- in chiral perturbation We shall now demonstrate the effect of inclusion of kaon pecially near midrapidity. It is worth mentioning that the potentials on the K0 flow. Using the kaon potentialtheory. deter- Somescalar-vector of the potential effects has been comes successful from in reproducing the Weinberg-Tomozawa vector mined from the impulse approximation, the K0’s aretype repelled interaction,several data at which GSI energies contributes of 1 – 2 A GeV ͓ because9͔. contrary to thepionthe from the nucleons resulting in antiflow with respect to the Additional insight into the origin of the transverse flow kaon hasmay only be gained one by light considering quark. the azimuthal To explain angular distri- the experimental data on nucleons ͑see Fig. 2͒. However, the ͗px͘ for kaons is found to underestimate the data as in Ref. ͓28͔. In fact,kaon theoretical subthresholdbution dN/d of production the kaons at midrapidity and flow by means in of heavy the ion collisionsrequires calculations ͓29͔ have shown that the simple impulsean additional ap- Fourier attractive expansion ͓30,31 scalar͔ interaction. The latter is related to the proximation also underestimates experimental kaon-nucleus dN scattering data. The scalar-vector potential clearly has a ϭv ͓1ϩ2v cos ϩ2v cos͑2͒ϩ ͔. ͑4͒ stronger density dependence compared to the impulse ap- d 0 1 2 ••• proximation ͑see inset of Fig. 2͒ since the vector potential dominates over the scalar potential at densities above the Here v0 is the normalization constant, and the Fourier coef- normal nuclear matter value. The kaon flow predicted from ficients v1ϭ͗cos ͘ϭ͗px /pt͘ represent the directed flow and 2 2 2 the scalar-vector potential is found to have strikingly good v2ϭ͗cos(2)͘ϭ͗(pxϪpy)/pt ͘, referred to as the elliptic flow 0 agreement with the Ks flow data as seen in Fig. 2. Although ͓32͔, reflects the azimuthal asymmetry of the emitted par- the final kaon flow is opposite to the proton flow, the primor- ticles. While v2Ͼ0 indicates in-plane enhancement, v2Ͻ0 dial kaons flow with the nucleons up to the maximum com- suggests a squeeze-out orthogonal to the reaction plane. pression stage of the collision, and their ͗px͘ values are In Fig. 3 we depict the pt dependence of v1, dubbed as 0 nearly identical irrespective of the kaon potential employed. differential flow, for midrapidity K (͉y/y c.m.͉Ͻ0.3) without This stems from the fact that at the early stage of the colli- any kaon mean field and with the inclusion of scalar-vector sion the density gradient inside the participant matter is kaon potentials. The results are for semicentral (bϭ6 fm) rather small and as a consequence the force acting on the AuϩAu collisions at 6A GeV which can describe equally kaon during its propagation is negligible. The potential ͑or well the flow data. In the absence of a kaon potential, v1 is baryonic density͒ gradient is substantial only during the ex- found to be almost zero at low pt but has a large positive pansion of the system so that the kaon potential could then value at high pt . The low-pt kaons are produced mostly in
061903-3 STAR results from beam energy scan program 3
0.20 STAR Preliminary0.20 0.20 (0-80%) Au+Au
7.7 GeV η-sub EP 11.5 GeV 39 GeV 0.15 0.15 0.15 π+ - 0.2 STAR Preliminary0.2 0.2 π 2 0.10 (0-80%) Au+Au v 0.10 0.10
7.7 GeV 0.15 η-sub0.15 EP 11.5 GeV 0.15 39 GeV 0.05 0.05 0.05 p p 0.00 0.00 0.00
2 0.1 0.1 0.1 v ) + 1.2 π 1.2 1.2 ( 0.05 0.05 0.05 2 )/v - π ( 1.0
2 1.0 1.0 v 0 0 0 0.8 0.8 0.8 012301230123 012301230123 p (GeV/c) p (GeV/c) T T Mean-field effects on particlep (GeV/c) and antiparticle elliptic flows 4
Xu, Chen, Ko & Lee, PRC 85, 045905 (2012) antiparticle is found to decrease with increasing collision energy as a result of decreasing baryon chemical potential 80 STAR Preliminary STAR Preliminary + K0 Λ-Λ 0.1 π S Λ of the hadronic matter. Although our results are qual- p-p 11.5 GeV p - Au+Au (0-80%) φ Ξ itatively consistent with the experimental observations, + - 0.08 60 K -K + - they somewhat underestimated the relative elliptic flow π -π − (P) (%) 0.06
2 difference between p andp ¯ as well as that between π and 40 π+ and overestimated that between K+ and K−.Inour
/ncq 0.04 ))/v 2 studies, we have, however, not included other effects that P v ( 2 20 0.02 may affect the v2 difference between particles and their 0 antiparticles. For example, we may have overestimated
(P)-v Au+Au (0-80%), η-sub EP the annihilation between baryons and antibaryons as this
2 0 -0.02 could be screened by other particles in the hadronic mat- (v 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 ter [34]. Including the screening effect would increase the -20 2 0 20 40 60 duration of the attractive potential acting on antibaryons (m -m0)/ncq (GeV/c ) T and thus reduces their elliptic flow, leading therefore to sNN (GeV) ± an increase in the difference between the elliptic flows Figure 2. (Color online) Top panel: v2 of p,¯p and π vs. pT in minimumof bias baryons and antibaryons. Also, the different elliptic § Hadronic mean fields lead to splittingFIG. 4:of (Colorparticle online) and antiparticle Relative elliptic elliptic flow diflowsfference between Au+Au collisions at √sNN = 7.7, 11.5,+ 39− GeV.+ Bottom− left panel: Percentageflows between particles and their antiparticles are as- in baryon-rich matter, diminish withp and increasingp ¯, K and collisionK ,and energies,π and similarπ with to and without sumed in the present study to come entirely from the difference between v2 of particles and anti-particles. Bottom right panel: v2/ncq vs. experimental observations by STARhadronic potentials U at three different BES energies from hadronic mean-field potentials. As shown in Ref. [35], the string melting AMPT model. Results for different species § Expect(mT additionalm0)/ncq effects for 0-80% from partonic Au+Au mean-fields collisions at 11.5 GeV [12]. the collective flow of partons can also be affected by their − are slightly shifted in energy to facilitate the presentation. 14 mean-field potentials in the partonic matter. If quarks and antiquarks have different mean-field potentials in the K−,respectively[4].Similartotheexperimentaldata, partonic matter, this would then lead to different ellip- + − tic flows for particles and their antiparticles in the initial for minimum bias (0-80%) Au+Au collisionsthe relative atv2√disffNNerence= between 7.7, 11.5π and andπ 39is negative GeV [12]. To at all energies after including their potentials, although stage of the hadronic phase after hadronization. It will avoid autocorrelations we calculated theours event have smaller plane magnitudes. (EP) in two We have sep alsoarated found hemispheres that, be of great interest to include in future studies these ef- as seen in the experiments [4], the relative v difference fects as well as the effect due to different elliptic flows with an η-gap of 0.05 on each side. Differences are observed in the v2 2of particles and between Λ hyperons and Λ¯ is smaller than that between between produced and transported partons [6] and the anti-particles, which increases as the beamp andp ¯ energy,becausethe decreases.Λ(Λ¯)potentialisonly2 Thepercentagedi/3ofthefferencechiral magnetic effect [7] in order to understand more p(¯p)potential. quantitatively the different elliptic flows between parti- relative to v2 of the particle is also shownTo in summarize, Fig. 2 as we a have function studied theof √ ellipticsNN.Wefindthat flows of cles and their antiparticles observed in relativistic heavy + + ion collisions. v2(anti-baryons)
Xu, Song, Li & Ko, PRL 112, 012301 (2014)
§ Finite partonic vector mean field with Gv/G=0.5 -1.1 is needed to describe STAR data. 15 第 2 期 XU Jun et al：Mean-field Effects on Particle and Antiparticle Elliptic Flows in the Beam-energy · 151 · ······ mass, corresponding to the phase where chiral symme- at zero temperature but has an additional thermal po- try is restored. A smooth change from the first phase tential at finite temperature[5] to the second phase represents a cross-over chiral phase a/T 2 3 3 Ω = bT 54e− l¯l+ln[1 6l¯l 3(l¯l) +4(l +¯l )] , transition, while a sudden change represents a first- Polyakov − · { − − } order phase transition. The critical point is exactly (5) where the cross-over transition changes to a first-order where the parameters a and b are fitted to make the one. One sees from the susceptibility that the criti- quark condensate and the polyakov loop l cross each other at about T = 200 MeV at zero quark chemical cal point moves to lower temperatures with increasing [5] vector coupling constant, similar to that observed in potential . The phase structure with the contribution of the polyakov loop is displayed in Fig. 6, where the Refs. [4–7]. This is not surprising because the equa- [5] tion of state of baryon-rich matter becomes stiff with deconfinement phase boundary is defined as l =1/2 . The deconfinement transition is always a crossover as larger values of GV , so that the phase transition is more likely to be a crossover rather than a first-order a result of finite dynamical quark mass in the pNJL one. model. On the other hand, the critical point moves to Defining the phase boundary where the chiral con- higher temperatures after including the polyakov loop, as already discussed in Ref. [5]. However, the conclu- densate is half of the value at (µq =0,T =0)as the phase boundary[5], the phase diagram with chiral sion that the critical point disappears or is located at lower temperatures with larger values of RV doesn’t phase transition with different RV is shown in Fig. 5. One should keep in mind that the phase boundary is change even with the inclusion of the polyakov loop. only accurate for the first-order phase transition but It would be of great interest to included gluon con- is artificially defined for a crossover, as shown by the tribution in both the transport model study and the dashed lines in Fig. 5.Effect of It is thuspartonic clearly seen vector interac on on QCD phase diagram that phase diagram calculation in the future study. based on the same NJL Lagrangian from the trans- port model study and the phase diagram calculation, the constraint of the vector coupling R =0.5 1.1 V ∼ leads to the conclusion that the critical point may dis- appear or is located at extremely low temperatures.
§ Loca on of cri cal point depends strongly on G ; moving to lower temperature Fig. 6 TheV chiral phase boundary, the correspond- and larger baryon chemical poten al as GV increases. ing critical point (labeled as stars), and the § Cri cal point disappears for G V > 0.6 G in NJL and somewhat larger in PNJL. deconfinement phase boundary based16 on the Polyakov-looped Nambu-Jona-Lasinio model in the (µq,T) plane from different relative vector Fig. 5 The chiral phase boundary and the corre- coupling constant RV = GV /G. sponding critical point (labeled as stars) based on the Nambu-Jona-Lasinio model in the (µq, 6 Conclusions and outlook T ) plane from different relative vector coupling We have reviewed our recent studies on the ef- constant RV = GV /G. fects of mean-field potentials on the elliptic flow split- The above NJL Lagrangian only includes the con- ting between particles and their antiparticles in the tribution of quarks and antiquarks and can not de- beam-energy scan program at RHIC. By including the scribe the deconfinement phase transition. In Ref. [35] hadronic mean-field potentials in the ART model of a [36] Fukushima included the polyakov loop represent- multiphase transport model, the v2 splitting is qualita- ing the gluon contribution, and the poyakov loop can tively consistent with the experimental results, while be taken as the order parameter of the deconfinement the magnitude of the relative v2 difference between + phase transition. The polyakov-looped Nambu-Jona- protons and antiprotons as well as π and π− are un- + Lasinio (pNJL) model is identical to the NJL model derestimated, and that between K and K− is overes- Phi meson absorption and production cross sections
528 S. Pal et al. / Nuclear Physics A 707 (2002) 525–539 Pal, Ko & Lin, NPA 707, 525 (2002)
Besides φ ↔ K K , phi meson can be produced and absorbed via various hadronic reactions, calculable by
€ § Meson-exchange model Chung, Li, and Ko, NPA 625, 347 (97)
§ Chiral Lagrangian with hidden local symmetry Averez-Ruso and Koch, PRC 65, 054901 (02)
Fig. 1. Phi meson scattering cross sections by nucleons (upper left panel), mesons (upper right panel), kaons (lower left panel), and K∗ (lower right panel). 17 For phi meson interactions with baryons, we include the absorption reactions given by the inverse reactions of phi meson production from meson–baryon interactions given above. The corresponding cross sections are obtained using the detailed balance relations. The cross section for phi meson elastic scattering with a nucleon is taken to be 0.56 mb as extracted from the data on phi-meson photoproduction using the vector meson dominance model [22]. These cross sections are shown in the upper left panel of Fig. 1. Phi mesons can also be scattered by mesons. Using effective hadronic Lagrangians, where coupling constants were determined from experimental partial decay rates, the total collisional width of φ due to the reactions φπ KK , φρ KK, φK φK and → ∗ → → φφ KK was found to be less than 35 MeV [23] (where K denotes either a kaon or → an antikaon as appropriate). However, recent calculations [24] based on a Hidden Local Symmetry Lagrangian shows that the collisional rates of φ with pseudoscalar (π, K)and vector (ρ, ω, K∗, φ)mesonsareappreciablylarge,especiallyfortheK∗,resultingin amuchsmallermeanfreepath,ofabout2.4fminahadronicmatterattemperatureT> 170 MeV, compared to the typical hadronic system size of 10–15 fm created in heavy ion ∼ Phi meson rapidity532 distributionS. at Pal SPS et al. / Nuclear Physics A 707 (2002) 525–539 534 S. Pal et al. / Nuclear Physics A 707 (2002) 525–539 534 S. Pal et al. / Nuclear Physics A 707 (2002) 525–539
Fig. 3. Rapidity distribution of phi meson reconstructed from K+K− pairs (solid curves) and from µ+µ− Fig. 3. Rapidity distribution of phi meson reconstructed from K+K− pairs (solid curves) and from µ+µ− channel (dashed curves) for Pb Pb collisions at 158 A GeV at an impact parameter of b ! 3.5fmintheAMPT + channel (dashed curves) for Pb Pb collisions at 158 A GeV at an impact parameter of b ! 3.5fmintheAMPT model. The results are for without (thin curves) and with (thick+ curves) in-medium mass modifications. The model. The results are for without (thin curves) and with (thick curves) in-medium mass modifications. The dotted curve corresponds to phi mesons from the dimuon channel with in-medium masses and with the phi meson dotted curve corresponds to phi mesons from the dimuon channel with in-medium masses and with the phi meson number from HIJING increased by a factor of two. The solid circles are the NA49 experimental data [7] from the number from HIJING increased by a factor of two. The solid circles are the NA49 experimental data [7] from the K+K− channel. K+K− channel. 18 seen that at low mT the phi meson from K+K− channel is suppressed due to rescattering. seen that at low mT the phi meson from K+K− channel is suppressed due to rescattering. Since the transverse momentumSince the of atransverse particle increases momentum due of to a increasing particle increases number due of to increasing number of scattering and due to pressurescattering build-up and insidedue to the pressure system, build-up the decayed inside kaons the system, at the the decayed kaons at the early stages, which are predominantlyearly stages, scattered, which are thus predominantly have low transverse scattered, momenta. thus have The low transverse momenta. The transverse mass spectra can be approximately fitted by exp( mT /T).Theinverseslope( m /T) transverse mass spectra can be approximately− fitted by exp 2 T .Theinverseslope parameter T reported by NA50 [8] from the µ+µ− channel for 1.7
Fig. 5. The rapidity distribution (top panel) and the transverse mass spectra (bottom panel) for midrapidity ( y < 0.5) phi mesons reconstructed from K K pairs (solid curves) and from µ µ channel (dashed curves) | | + − + − for Au Au collisions at RHIC energy of √s 130 A GeV at an impact parameter of b 5.3fmintheAMPT + = ! model. The solid circles are the STAR experimental data [41] for 0–11% central collisions for φ reconstructed19 from K+K− decay. The transverse mass spectra of φ meson from the two channels are shown in the bottom panel of Fig. 5, and compared with the STAR data [41] for the K+K− channel. Compared to a slope parameter of T 379 50 MeV in the data, the AMPT model predicts a smaller = ± value of T 335 MeV in the kaonic channel in the range 0 5. Summary and conclusions In summary, we have investigated in a multiphase transport (AMPT) model the production of phi mesons reconstructed from K+K− and µ+µ− decays. Considering Energy dependence of width of rapidity distribu on Pb+Pb @ SPS § Increases with energy § Larger for light hadron § Larger for K+ than K-? § Phi behaves anomalously, like Λ at lower energy but like pion at high energy. 20 Phi flow from AMPT Y. G. Ma et al., EPJ A29, 11 (2006) Phi meson v2 is similar to that of kaon → quark number scaling 21 Dynamical quark coalescence model Chen & Ko, PRC 73, 044903 (06) Based on the phase-space distribution of strange quarks from AMPT and including quark spatial and momentum distributions in hadrons Although scaled phi and Omega satisfy constituent quark number scaling, they are smaller than the strange quark elliptic flow. 22 The escape mechanism: flavour dependence Flavor dependence of escaping contribu on to v2 dAu@200GeV pPb@5TeV AuAu@200GeV [email protected] b=0 fm b=0 fm b=6.6-8.1 fm b=8 fm u/d 93%(all quarks) 72.9% 65.6% 42.5% s 59.1% 47.4% 26.5% c 56.8% 21.8% 8.5% v2 of charm quarks in AuAu@RHIC-200GeV & PbPb@LHC: mostly comes from collective flow (not the escape mechanism). èheavy quarks are more sensitive probes of collective flow & the medium. Esha, Md. Nasim & Huang, JPG44 (2017) v2 of light quarks: escape mechanism is more important for AuAu@RHIC, pPb@LHC and smaller/lower-energy systems; hydro-type collective flow is more important for PbPb@LHC although with significant contribution from the escape mechanism.23 Neutron rela ve density fluctua on from yield ra o of light nuclei Sun, Chen, Ko & Xu, arXiv: 1702.07620 [nucl-th] N3HNp p-d-t = 2 O Nd 1+(1+2↵) n = g (1 + ↵ n)2 ( n)2 n = h i n 2 h i n n n = ↵h pi ( n)2 h pi n h i h i α: correla on factor TC ≈ 144 MeV μC ≈ 385 MeV pK0 § Expect a similar behavior for from u-quark density fluctua on. ⇡+⇤ 24 Chemical freeze-out in rela vis c heavy ion collisions Jun Xu & CMK, PLB 772, 290 (2017) 35 20 28 16 21 12 14 8 7 4 0 0 0 10 20 0 10 20 30 § Both ra o of effec ve par cle numbers and entropy per par cle remain essen ally constant from chemical to kine c freeze-out. 25 10 the K+/π+ ratio. We mention that the maximum in the K+/π+ ratio is not so pronounced in the PHSD cal- culations as in the data since the pion production is still overestimated by the PHSD. We speculate that this over- estimation might be due to the complex pion dynamics in the nuclear medium where the pion separates into a ’pion’ and ’∆-hole’ branch [86]. IV. SUMMARY We have studied effects from chiral symmetry restora- tion (CSR) in relativistic heavy-ion collisions in the en- ergy range from 4 to 158 A GeV by using the Parton- Hadron-String Dynamics (PHSD) approach [53] that has been extended essentially in the hadronic phase to also describe CSR apart from a deconfinement transition to dynamical quarks, antiquarks and gluons in the QGP. We have assumed that effects of chiral symmetry restoration for the scalar quark condensate § Quarks follow chiral vor cal equa ons of mo on 1+ ! x˙ = p 1+3 ! pˆ p · p˙ =0 1 ! = v 2r⇥ :helicity=1 ± v :velocityfield § Quarks and an quarks of opposite helici es undergo helicity flip sca ering. § Polarized Lambda formed from polarized quarks via coalescence. § Lambda polariza on decreases with energy due to decreasing vor city. § Similar to results based on sta s cal model based on final vor city from hydro transport model. § Seemingly larger Lambda than an -Lambda polariza ons in data is not understood. 27 PHYSICAL REVIEW LETTERS week ending PRL 97, 132301 (2006) 29 SEPTEMBER 2006 to the fact that the higher mass particles come from events Au Au suggest that rescattering dominates regeneration with average multiplicities a factor of 2 or more higher in the hadronic medium after chemical freezeout. than those for the minimum bias events. The increase in the A thermal model using an additional pure rescattering pT and the larger event multiplicities imply that these phase, which depends on the respective momenta of the hresonancesi come from mini-jet-like events [20]. The re- resonance decay products, after chemical freezeout at T scattering and regeneration is expected to change the pT 160 MeV, can be fit to the data. The fit yields a hadronic h i 10 in Au Au collisions. However, it is surprising that the lifetime of the source of 95 fm=c from the = 1:5 ÿ p of in p p and Au Au collisions are in agree- and 2:5 fm=c from the K =K ratio [9,22]. The h Ti 1 ment within their uncertainties. small difference ÿ between the time spans can be explained The ratios of yields of resonances to stable particles as a by an enhanced regeneration cross section for the K in the function of the charged particle multiplicity are presented medium. This theory is supported by the null suppression in Fig. 3. The ratios are normalized to unity in p p of the =. The smaller lifetime of the compared to collisions to study variations in Au Au relative to p the should lead to a larger signal loss due to rescatter- p. We measure a suppression for = when comparing ing, thus the lack of suppression requires an enhanced central Au Au with minimum bias p p. K=Kÿ [17] regeneration probability of the . Based on the same seems to show a smaller suppression while the =, and argument the K regeneration cross section needs to be =Kÿ [21] ratios are consistent with unity. In a thermal larger than that of the due to the observed smaller model, the measured ratios of resonance to nonresonant suppression and shorter lifetime of the K (i.e., defining particles with identical valence quarks are particularly R as the ratio of regeneration to rescattering cross section, sensitive to the chemical freezeout temperature, as all of we find RK p Thermal model Thermal discussed predominantly, other methods to describe the 1.4 data have been proposed. For example, in a sudden freeze- 1.2 1 out scenario, where the time between the chemical and 0.8 kinetic freezeout is negligible, the = suppression in Au Au with respect to p p can be explained by the resonance/non-resonance 0.6 0.4 influence of the dense medium on the production of . Even though the valence quarks of the are in a L 1 0.2 ÿ 0 state, it must decay through a relative angular momentum -100 0 100 200 300 400 500 600 700 L 2 process in order to conserve isospin [25]. The high dNch /dy partial wave component of the in a dense medium can § Measured Λ(1520)/Λ(1115) ratio is significantly smaller than thermal modelsuppress its decay phase space. FIG. 3 (color online). Resonance to stable particle ratios for prediction,p butp andcan beAu explainedAu collisions. by the coalescence The ratios aremodel normalized after taking to into We have presented the first measurements of and account unitythe p-wave in p statepand of strang compared quark to in thermal Λ(1520) and (Kanada UrQMD-En model’yo and production in p p and Au Au collisions at psNN Meullerpredictions, PRC 74, 061901(R) for central (2006).Au Au [8,12]. Statistical and system- 20028 GeV.Thelarge pT of the and measurements atic uncertainties are included in the error bars. in p p collisions suggestsh i that the heavy particle pro- 132301-5 8 TABLE IV: List of exotic hadrons discussed in this paper. Shown are the mass (m), degeneracy (g), isospin (I), spin and parity (J P ), the quark structure (2q/3/q/6q and 4q/5q/8q), molecular configuration (Mol.) and corresponding oscillator frequency (ωMol.), and decay mode of a hadron. For the ωMol.,itisfixedbythebindingenergiesBofhadrons(ω 6 B, marked (B)) Exotic mesons,2 2 baryons and dibaryons ≃ × or their mean square distances r (ω 3/2µ ˙r ¸,marked(R)).Inthecaseofthree-bodymolecularconfigurations for exotic ⟨ ⟩ ≃ ∗) Cho, dibaryons,Furumoto we adopt, Hyodo the, ωJidoMol., Ko, Lee, Nielsen, Ohnishi, as that for the subsystem,Sekihara as marked, Yasui (T)., and FurtherYazaki marked [ExHIC by Collabora on], PRL are undetermined quantum 106, 212001 (2011); PRC 84, 064910 (2011); PPNP 95, 279 (2017)numbers of existing particles, by ‡) particles which are not yet established, and by †) particles which are newly predicted by theoretical models. m ωMol. decay Particle g I P 2q/3q/6q 4q/5q/8q Mol. (MeV) J (MeV) mode Mesons + f0(980) 980 1 0 0 qq,¯ ss¯ (L =1) qqs¯ s¯ KK¯ 67.8(B) ππ (strong decay) + a0(980) 980 3 1 0 qq¯ (L =1) qqs¯ s¯ KK¯ 67.8(B) ηπ (strong decay) K(1460) 1460 2 1/2 0− qs¯ qqq¯ s¯ KKK¯ 69.0(R) Kππ (strong decay) + Ds(2317) 2317 1 0 0 cs¯ (L =1) qqc¯ s¯ DK 273(B) Dsπ (strong decay) 1 †) + ∗ + − + − − Tcc 3797 3 0 1 — qqc¯c¯ D¯D¯ 476(B) K π + K π + π X(3872) 3872 3 0 1+,2−∗) cc¯ (L =2) qqc¯ c¯ DD¯ ∗ 3.6(B) J/ψππ (strong decay) + ‡) −∗) ∗ Z (4430) 4430 3 1 0 — qqc¯ c¯ (L =1) D1D¯ 13.5(B) J/ψπ (strong decay) 0 †) + + − + − Tcb 7123 1 0 0 — qqc¯¯b DB¯ 128(B) K π + K π Baryons Λ(1405) 1405 2 0 1/2− qqs (L =1) qqqsq¯ KN¯ 20.5(R)-174(B) πΣ (strong decay) Θ+(1530) ‡) 1530 2 0 1/2+ ∗) — qqqqs¯ (L =1) — — KN (strong decay) KKN¯ †) 1920 4 1/2 1/2+ — qqqss¯ (L =1) KKN¯ 42(R) KπΣ, πηN (strong decay) DN¯ †) 2790 2 0 1/2− — qqqqc¯ DN¯ 6.48(R) K+π−π− + p D¯ ∗N †) 2919 4 0 3/2− — qqqqc¯ (L =2) D¯ ∗N 6.48(R) D¯ + N (strong decay) †) + + − Θcs 2980 4 1/2 1/2 — qqqsc¯ (L =1) — — Λ + K π BN †) 6200 2 0 1/2− — qqqq¯b BN 25.4(R) K+π−π− + π+ + p B∗N †) 6226 4 0 3/2− — qqqq¯b (L =2) B∗N 25.4(R) B + N (strong decay) Dibaryons H †) 2245 1 0 0+ qqqqss — ΞN 73.2(B) ΛΛ (strong decay) KNN¯ ‡) 2352 2 1/2 0−∗) qqqqqs (L =1) qqqqqq sq¯ KNN¯ 20.5(T)-174(T) ΛN (strong decay) ΩΩ †) 3228 1 0 0+ ssssss — ΩΩ 98.8(R) ΛK− + ΛK− ++ †) + − + + Hc 3377 3 1 0 qqqqsc — ΞcN 187(B) ΛK π π + p DNN¯ †) 3734 2 1/2 0− — qqqqqq qc¯ DNN¯ 6.48(T) K+π− + d, K+π−π− + p + p BNN †) 7147 2 1/2 0− — qqqqqq q¯b BNN 25.4(T) K+π− + d, K+π− + 29p + p coupled channels. A later study in Ref. [46] con- 61], D K mixing [62], four-quark state [63–66] or sidered the three-body coupled-channel dynamics amixtureoftwo-mesonandfour-quarkstates[67].− of KKK¯ in a Faddeev approach and found a very 1 similar kaonic excitation. A non-relativistic poten- 4. Tcc(udc¯c¯): The set of tetraquarks with two heavy tial model for the KKK¯ system was also used in quarks were first considered in Ref. [68]. The struc- Ref. [46], and a quasi-bound state of binding energy ture with [ud] diquark is expected to be particu- 21 MeV and root mean square radius 1.6 fm was larly stable [69, 70] and could be bound against the obtained. Thus, the K(1460) could be understood strong decay into D1D.Thequantumnumberis as a KKK¯ hadronic molecular state. J P =1+ with I =0;hencethedecayintoDD is forbidden due to the angular momentum conserva- 3. DsJ (2317): This state predicted in Refs. [47, 48] tion. Estimates based on the simple color-spin in- was first observed by the BaBar Collaboration [2] teraction suggest the mass to be 3796 MeV [14, 15]. + 0 + − ∗− ¯ 0 through the Ds π channel in inclusive e e anni- The hadronic decay mode of Tcc is D D if its hilation. Its measured mass is approximately 160 mass is above the threshold and D¯ 0D0π− if be- MeV below the prediction of the very successful low. If the Tcc is strongly bound, it can then decay quark model for the charmed meson [49]. Due to weakly to D∗−K+π− with a lifetime similar to that its low mass, the structure of DsJ (2317) has been of the D¯ meson. A molecular state with the same under extensive debate. It has been interpreted as quantum number was also predicted to exist within a cs¯ state [50–54], two-meson molecular state [55– the pion-exchange model [71]. The production of Ratio of exotic hadron yields from coalescence and SUNGTAEstatistical CHO et al. models PHYSICAL REVIEW C 84,064910(2011) Coalescence / Statistical model ratio at RHIC Coalescence / Statistical model ratio at LHC f (980) 2q/3q/6q f (980) 2q/3q/6q 0 4q/5q/8q 0 4q/5q/8q a0(980) a0(980) K(1460) Mol K(1460) Mol Ds(2317) Ds(2317) 1 1 Tcc Tcc X(3872) X(3872) Z+(4430) Z+(4430) 0 0 Tcb Tcb Λ(1405) Λ(1405) + + Θ (1530) Θ (1530) KbarKN KbarKN DbarN DbarN D*barN D*barN Θcs Θcs BN BN B*N B*N H H KbarNN KbarNN ΩΩ ΩΩ ++ ++ Hc Hc DbarNN DbarNN BNN BNN 10-2 10-1 100 101 102 10-2 10-1 100 101 102 § FIG.Mul quark hadrons are suppressed while hadronic molecules are enhanced 1. (Color online) Ratio of the yield of an exotic hadron in the coalescence model to that of the statistical model. in coalescence model, compared to the sta s cal model predic ons more abundant production of !(1405). However, for the larger despite the difference in the coalescence temperatures30T C and ω value, which corresponds to the case that the pole position TF .Thiscanbeattributedtothelargersizeofthemolecular of the S-matrix for the two-hadron interaction is around configuration. Assuming other factors are similar, the s-wave 1405 MeV, the !(1405) can be regarded as a deeply bound factors involved in the coalescence at TF are similar to those state and thus has a smaller size, and hence its yield becomes at TC as long as the relevant molecular size is related to the 1/3 smaller. hadron size as σC (VC /VF ) σF as can be inferred from Eq. (25)afterneglectingthetemperaturedependenceinthe= denominator. If we additionally assume that the volume scales 3 V. DISCUSSIONS as V T − ,wefindthattheconditionforthemolecular coalescence∝ to be similar to two-quark coalescence is that Our results based on the coalescence model for hadron the molecular size scales as σF σC TC /TF ,whichismoreor production in relativistic heavy ion collisions have indicated less satisfied by some exotic hadrons= considered here, such as that their yields are strongly dependent on their structures. the Ds (2317). Therefore, measuring the yields of exotic hadrons allows us CS Our study also shows the interesting result that the ratio Rh to infer the internal configuration of exotic hadrons [20,21]. of the yield in the coalescence model to that in the statistical For example, we have mentioned in Sec. IV Athataspossible model is almost the same at RHIC and LHC. This similarity configurations for f0(980), quark-antiquark pairs ( ss¯, uu¯, comes from the universal feature of the QCD phase transition; ∼ and dd¯), a tetraquark state, and a KK¯ hadronic molecule have the common critical temperature and the common volume been proposed. To confirm its structure, we refer to preliminary ratio VC /VH .Inthenonrelativisticapproximationshownin data from the STAR Collaboration for the production yield Eq. (3), it is possible to rewrite the statistical model yield in 0 ratios of f0(980), π,andρ [133]. From these results we the coalescencelike form, find that the measured yield of f0(980) is close to 8, which means that it is more probable for the f0(980) to be produced 3/2 3/2 as a hadronic molecule state than a tetraquark state (see ghVH (mhTH ) Ni,H (2π) N stat eB/TH , (31) the order-of-magnitude difference between the yield in the h = (2π)3/2 g V (m T )3/2 i i H i H 4q/5q/8q column and that in the Mol. column in Table V). ! Therefore, we conclude that the STAR data seem to rule out adominanttetraquarkconfigurationforthef0(980). Further where we consider the hadron h to be composed of several experimental efforts to reduce the error bar are thus highly constituents, B mi mh is the binding energy, and Ni,H = i − desirable. represents the yield of the ith constituent at the volume VH . For some exotic hadrons, our results show that the yields This relation holds" because the fugacity of a particle is equal are similar for the hadronic and the molecular configuration, to the product of the constituent fugacities, and the particle 064910-12 Diomega prodcution in HIC According to the chiral quark model of Zhang et al. (PRC 61, 065204 (2000)), diomega (ΩΩ)0+ is bound by ~ 116 MeV with a root-mean-square radius ~ 0.84 fm and lifetime ~ 10-10 sec. § Cross sections for I-VI are ~ 2-25 µb and thus unimportant. § Production is dominated by two-step processes through VII, VIII (~ 50-175 µb) and IX (~ 20-50 mb). § Yield is order of magnitude smaller than in statistical or coalescence model Pal, Ko & Zhang, PLB 624, 210 (05) (neglect of three-body process?) 31 + + Hidden-charm pentaquark hadrons: Pc (4380), Pc (4450) Wang, Song, Sun, Chen, Li, & Shao, PRC 94, 044913 (2016) WANG, SONG, SUN, CHEN, LI, AND SHAO PHYSICAL REVIEW C 94,044913(2016) 105 0.01 P – / D0 D0×100 (b) uudcc 103 0.008 Λ+ – 0 c Puuscc / D ] – 0.006 -1 Puudcc×10 10 – Puuscc 0.004 10-1 0.002 0 -3 0.04 dy) [(GeV/c) 10 – + T Puudcc / Λc 0.03 (c) P – / + 10-5 uuscc Λc N/(dp 2 d 0.02 10-7 0.01 (a) 10-9 0 0 2.5 5 7.5 10 12.5 15 0 2.5 5 7.5 10 12.5 15 pT (GeV/c) pT (GeV/c) 0 0 0 FIG. 1. (a) pT spectra of D mesons, !c+ baryons, and hidden-charm pentaquark states Puudcc¯ and Puuscc¯; (b) ratios Puudcc¯/D and Puuscc¯/D ; (c) ratios Puudcc¯/!+ and Puuscc¯/!+ at midrapidity in central Pb Pb collisions at √sNN 2.76 TeV. Filled symbols represent the experimental c c + = data from Ref. [45], and different lines show our results. Solid lines show results under the equal pT combination scenario; dashed lines, results 0 under the equal vT combination scenario. In (a) pT distributions of D and Puudcc¯ are scaled by 100 and 10, respectively, for clarity. (n) normal baryon-to-meson ratios. These are the characteristics Nqi ,Nq¯i )fc¯ (pT5 ; Nqi ,Nq¯i ). In our previous work [29] on normal light, strange, and charm hadrons, we have of the quark combination. The peaks and widths of these humps extracted the distributions of light and strange quarks, a are larger in the equal vT case than in the equal pT case. These32 interesting behaviors can be used to test the quark combination two-component parameterized pattern, dNl /(pT dpT ) i ∝ 2 2 pT Sl production of hidden-charm pentaquark states. exp( √p m /Tl ) Rl (1 )− i ,theexponential − T + li i + i + 5 GeV item for thermal quarks at a low pT and the power-law item IV. SUMMARY AND DISCUSSION for shower quarks with a high pT .Parameters(Tli ,Sli ,Rli )are (0.31 GeV, 8.0, 0.02) for light quarks and (0.36 GeV, 8.6, 0.04) With the basic ideas of the QCM and a few assumptions for strange quarks. Also, we have obtained that the extracted and/or simplifications based on symmetry and general prin- charm quark pT distribution under the equal pT combination ciples, we have derived the yield formulas and studied the 2 pT 8 is dNc/(pT dpT ) (pT 0.4GeV)(1 )− ,while yield ratios of different species of hidden-charm pentaquark ∝ + + 2.2 GeV that under the equal vT combination is dNc/(pT dpT ) states in heavy ion collisions at ultrarelativistic collision en- 2 pT 6.32 ∝ (pT 0.99 GeV) (1 )− .Withtheseinputquark ergies. We found some interesting relations between different + + 0.87 GeV pT distributions, we can predict the pT spectra of different hidden-charm pentaquark states. These results are properties hidden-charm pentaquark states. of pentaquark state production in the QCM under these Figure 1(a) shows the calculated pT spectra of hidden- assumptions such as factorization of the kernel function and the charm pentaquark states in two combination scenarios. We same γ for all species of hidden-charm pentaquark states. Pl1l2l3cc¯ here take Puudcc¯ and Puuscc¯ as two examples. Note that They are independent of the particular quark combination Puudcc¯ includes hidden-charm pentaquark states of quark models, so they can be used to test the quark combination contents (uudcc¯)withdifferentquantumnumbers,J P production mechanism of the exotic resonances. We have also = (1/2)±,(3/2)±,(5/2)±,andthesamecaseholdsforPuuscc¯. calculated the yields of normal identified light, strange, and 0 The pT spectra of D mesons and !c+ baryons are also charm hadrons with the SDQCM, and on this basis we have given as references. Filled symbols are the experimental shown how to estimate the yields of different hidden-charm 0 data on D mesons from Ref. [45]. All solid lines represent pentaquark states. We have predicted the yields and pT spectra results computed under the equal pT combination and dashed of hidden-charm pentaquark states in the most central Pb Pb + lines show the equal vT combination case. We find that the collisions at √sNN 2.76 TeV. All of this can help to search = results in these two combination scenarios are nearly the for the partners of the observed Pc+(4380) and Pc+(4450) and same except for Puudcc¯ and Puuscc¯ at very low pT ,below2 shed light on the understanding of the production mechanism GeV/c.Figures1(b) and 1(c) show the pT dependence of of exotic pentaquark states in heavy ion collisions. 0 0 the ratios Puudcc¯/D , Puuscc¯/D ,andPuudcc¯/!c+,Puuscc¯/!c+, Note that all the above derivations and results are obtained respectively. Solid lines represent results computed under the in the case of considering the observed Pc+(4380), Pc+(4450), equal pT combination, and dashed lines show the equal vT and their partners as compact pentaquark configurations. case. All these ratios have humps in the intermediate pT range, However, the other internal configuration candidate for which is similar to the behaviors of the pT dependence of these hidden-charm exotic states is loosely bound hadronic 044913-10 Summary § Are K0/K+ and Σ-/Σ+ good probes of symmetry energy at high densi es? § How important are mean-field poten als in BES and FAIR? § Does phi meson sca er frequently in expanding hadronic ma er? § Where can one see quark recombina on or coalescence for strange hadrons? § Why does hadronic evolu on not affect strange hadron abundances? § What is the cause of “horn” in strange/pion ra o? § Can the ra o provide informa on on u-quark density fluctua on? pK0 ⇡+⇤ § What is the chance to find exo c hadrons in HIC such as diomega + + as well as Pc (4380) and Pc (4450)? 33