Compactness is introduced as a new method to search for the onset of the quark matter transition in relativistic heavy ion collisions. That transition supposedly leads to stronger compression and higher compactness of the source in coordinate space. That effect could be observed via pion interferometry. We propose to measure the compactness of the source in the appropriate principal axis frame of the compactness tensor in coordinate space

Brachmann, Jörg

Dumitru, Adrian

Greiner, Walter

Lisa, Michael Annan

Paech, Kerstin

Stöcker, Horst

English

Lisa, Michael A.

Dumitru, Adrian Bogdan

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0010036 v1   11 Oct 2000Observing compact quark matter droplets inrelativistic nuclear collisionsK. Paecha, J. Brachmanna, M. A. Lisab, A. Dumitruc, H. St¨ ockera, W. Greineraa Institut f¨ ur Theoretische Physik, Universit¨ at Frankfurt a.M., Postfach 111932,D-60054 Frankfurt am Main, Germanyb Department of Physics, The Ohio State University, 174 W. 18th Avenue,Columbus, Ohio 43210, USAc Department of Physics, Columbia University,538 W. 120th Street, New York, NY 10027, USAOct. 2000AbstractCompactness is introduced as a new method to search for the onset of the quarkmatter transition in relativistic heavy ion collisions. That transition supposedly leadsto stronger compression and higher compactness of the source in coordinate space.That eﬀect could be observed via pion interferometry. We propose to measure thecompactness of the source in the appropriate principal axis frame of the compactnesstensor in coordinate space.1A variety of signatures for the observation of the quark matter transition in relativisticheavy ion collisions have been proposed, see e.g. [1] and references therein. Nowadays it iswidely believed that irregularities due to such a new state of matter should be seen mostclearly in excitation functions, i.e. in the bombarding energy dependence, or possibly also inthe impact parameter or mass-number dependence of various observables.However, in spite of circumstantial evidence for such a novel state of matter [2], undis-putable “irregularities” in such experimental excitation functions have not been reportedto date. Most probably a convincing argument that quark matter has indeed been foundrequires several independent measurements of distinct observables pointing to the same con-clusion - namely the onset of deconﬁnement at a particular bombarding energy, impactparameter and system size.In the present paper we introduce a new observable for the onset of the phase transition. Itrelies on the measurement of the compactness of the source, which is related to the pressureand density of the system in the compression and expansion stage of the nucleus-nucleuscollision. Observables related to the pressure do not only reﬂect the transient high densitybut rather relate to the order parameter (ﬁeld), specifying in which thermodynamical state [3]the excited matter was created. The compactness can be identiﬁed via interferometry: Theillumination of the baryon source by the pion radiation is subject to experimental scrutinityvia pion interferometry measurements [4].To illustrate the basic idea, let us ﬁrst discuss compression shocks in heavy-ion collisionsemploying the Rankine Hugoniot Taub shock adiabat (RHTA) solution [5, 6] of relativistichydrodynamics,W2 − W02 + P Wρ−W0ρ0!= 0 . (1)It relates the energy per baryon in the local rest frame of the compressed matter, W, toits baryon density ρ and pressure P. W0 and ρ0 are the energy (=923 MeV) and density(=0.16 fm−3) in the ground state, and the compression factor is ρ/ρ0.Solving for P we obtainP = Wρ0γcm − 1/γcm1 − γcmρ0/ρ. (2)Simply speaking, the onset of the transition to quark matter at a given incident energy Ekinlab =2(γcm2 − 1)W0 lets the pressure increase less rapidly with W and/or ρ, and consequentlyhigher compression ρ/ρ0 can be achieved than for the case without transition. Now, asρV ≃ πR2ALρ must equal 2A by virtue of baryon number conservation, the longitudinalthickness L of the compressed matter is proportional to 1/ρ. Thus, a transition to quarkmatter leads to a more compact system, just as compact stars with a quark matter core are2more compact than pure neutron stars [7]. Of course, in heavy-ion collisions that expectationis based on the behavior of relativistic compression shocks rather than hydrostatic andgravitational equilibrium.To investigate quantitatively the experimental observable, we perform 3-dimensional (1-ﬂuid) calculations of relativistic hydrodynamics for the compression as well as for the sub-sequent expansion stage of the reaction. That is, we solve numerically the continuity equa-tions for the energy-momentum tensor, ∂µT µν = 0, and the net baryon current, ∂µNµB = 0.Detailed discussions of (3+1)-d numerical solutions for hydrodynamical compression andexpansion can be found e.g. in [8]. We shall employ two diﬀerent equations of state (EoS)P(W,ρ): i) a relativistic mean ﬁeld (RMF) hadron ﬂuid [9] corresponding to baryons andantibaryons interacting via exchange of massive scalar and vector bosons, plus free thermalpions; the parameters of the lagrangian are ﬁtted to the ground state of inﬁnite nuclearmatter, in particular the nuclear saturation density, the energy per nucleon, and the incom-pressibility. ii) the same EoS as in i) for the low-density phase, supplemented by the BagModel equation of state with bag constant B1/4 = 235 MeV for the quark-gluon (QG) phase.The phase coexistence region corresponding to this ﬁrst-order transition is constructed em-ploying Gibbs’ condition of phase equilibrium, pRMF(T, B) = pQG(T, B), where T and  Bdenote the temperature and the baryon-chemical potential, respectively. For example, forρ = 0 we ﬁnd TC ≈ 170 MeV, while at T = 0 phase coexistence sets in at ρ ≈ 4.6ρ0. A moredetailed discussion of these EoS can be found in [6].As an example, we study the compactness in the reactions Au+Au at impact parameterb = 3 fm and energy Ekinlab = 8A GeV, where the transition to quark matter does occur(within the present model). We deﬁne the conﬁguration space sphericity tensor, also called“compactness tensor” below, as the second moment of the net baryon current. On ﬁxed timehypersurfaces we haveFij(t) =Zd3x xi xj N0B(t,  x) Θ(ρ(t,  x) − ρcut) . (3)We apply an additional density cut ρ > ρcut in the integral to discard spectator matter.In the future the cuts and the hypersurface will have to be adapted to the experimentalconditions. However, that is not crucial for understanding the eﬀect.The three eigenvalues fn are the solutions of the cubic equation det(Fij − fδij) = 0, andthe eigenvectors   en follow from solving the linear system of equations (Fij − fnδij)ejn = 0.Let   e1,   e2 be the eigenvectors deﬁning the reaction plane,   e1 corresponding to the biggerof the eigenvalues f1 and f2. To simplify the discussion we shall assume that the matterdistribution is symmetric with respect to the reaction plane.We can now rotate the coordinate frame around   e3 by an angle cosΘ =   e1    ez, where   ez3deﬁnes the longitudinal (beam-) direction in the lab frame. The rotated compactness tensorF ∗ can be written in terms of the eigenvalues fn and orthogonal eigenvectors   e∗n asF∗ = f1  e∗1 ⊗  e∗1 + f2  e∗2 ⊗  e∗2 + f3  e∗3 ⊗  e∗3 . (4)In diagonal form, F ∗ speciﬁes an ellipsoid in conﬁguration space with principal axis along  e∗n and radii√fn. For example, cigar patterns, oriented along the z-axis, would correspondto f1 > f2 = f3,   e∗1 =   ez,   e∗2 =   ex,   e∗3 =   ey. The “compactness” can now be deﬁned as theratio of the in-plane radii,qf2/f1.Figure 1: Contour plot of the time-like component of the net baryon four-current in planestransverse to the longitudinal axis   e∗1 of the compactness tensor (Au+Au collision at Ekinlab =8 AGeV, b = 3 fm). Left: Relativistic mean-ﬁeld EoS without phase transition. Right: withphase transition to quark matter.Fig. 1 shows typical density distributions in the transverse planes of the coordinate framewhere F ∗ is diagonal, eq. (4). The polar angle relative to the lab frame as well as theeccentricityǫ =f3 − f2f3 + f2(5)4are shown as well. As already indicated in the introduction, we ﬁnd very diﬀerent eigenvaluesfor the two equations of state, the one without and the other with a ﬁrst order phasetransition included. The calculation with transition to quark matter corresponds to highercompactness of the baryon distribution. Because the incompressibility ∂p/∂ρ or ∂p/∂(Wρ)is smaller in the presence of the coexistence phase, the compactness tensor is much ﬂatter(nearly a factor of two !) in model ii) than in model i). Moreover, our (3+1)-dimensionalexpansion solutions show that the ratio of the in-plane radiiqf2/f1 remains much smallerin the case with phase transition, cf. Fig. 2.2 3 4 5 6 7 8 9 10 11 12t [fm/c]0.00.10.20.30.40.50.60.7(f2/f1)1/2cut = 2 0 cut = 0/2with p.t.no p.t.Figure 2: Ratio of the in-plane radiiqf2/f1 as it evolves during the compression and ex-pansion stages, for the relativistic mean-ﬁeld EoS without phase transition and for the casewith transition to quark matter (Au+Au, Ekinlab = 8 AGeV, b = 3 fm).This allows to measure directly the density increase in the high density stage of thereaction, if a phase transition occurs. Care must be taken that the investigated range ofimpact parameters constitutes a moderately small bin of centrality values. One should keepin mind that the compression factor is aﬀected by the incompressibility evaluated on theRHTA (1), but not along a path of ﬁxed speciﬁc entropy, because a large amount of entropyis being produced in the compression process.The predicted change in compactness due to the reduced incompressibility associated witha ﬁrst-order phase transition can be observed experimentally via the novel interferometryanalysis developed recently by Lisa et al. [4]. The proposed method is quite robust and5incorporates other interesting information as the conﬁguration space tilt angles [10], whichcomplement the momentum-space ﬂow angles. It avoids cuts in tilted ellipsoids, which arenot analysed in the appropriate rotated frame, and were the eccentricity and the RMS-radiiare much less distinct for models i) and ii).Acknowledgements: This work is supported by BMBF, GSI, DFG, and Graduiertenkol-leg “Theoretische und Experimentelle Schwerionenphysik”. A.D. acknowledges support fromthe DOE Research Grant, Contract No. De-FG-02-93ER-40764. We would like to thank Ste-fan Scherer, Sven Soﬀ and Steﬀen Bass for stimulating discussions.References[1] Proceedings of the 14th Intern. Conference on Ultrarelativistic Nucleus-Nucleus Colli-sions (Quark Matter 99), Torino, Italy, 10–15 May 1999, Nucl. Phys. A661, 1 (1999).[2] U. Heinz and M. Jacob, nucl-th/0002042.[3] T. D. Lee and G. C. Wick, Phys. Rev. D9, 2291 (1974).[4] M. A. Lisa, U. Heinz, and U. A. Wiedmann, nucl-th/0003022; M. A. Lisa, talkgiven at the Utah Winter Workshop RHIC 2000, http://www.star.bnl.gov/∼lisa/tilt/,http://www.nscl.msu.edu/∼bauer/.[5] H. G. Baumgardt et al., Z. Phys. A273, 359 (1975); M. Gyulassy, K. A. Frankel andH. St¨ ocker, Phys. Lett. B110, 185 (1982).[6] D. H. Rischke, Y. P¨ urs¨ un and J. A. Maruhn, Nucl. Phys. A595, 383 (1995).[7] N. K. Glendenning, “Compact stars: Nuclear physics, particle physics, and generalrelativity,” New York, USA: Springer (1997) 390 p.[8] D. H. Rischke, S. Bernard and J. A. Maruhn, Nucl. Phys. A595, 346 (1995); C. Nonaka,E. Honda and S. Muroya, hep-ph/0007187.[9] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1 (1986).[10] J. Brachmann, S. Soﬀ, A. Dumitru, H. St¨ ocker, J. A. Maruhn, W. Greiner, L. V. Bravinaand D. H. Rischke, Phys. Rev. C61, 024909 (2000).6

Compact stars: Nuclear physics, particle physics, and general relativity,”

talk given at the Utah Winter Workshop RHIC

Observing compact quark matter droplets in relativistic nuclear collisions

Observing compact quark matter droplets in relativistic nuclear collisions

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