Research Article

Scale for the Phase Diagram of Quantum Chromodynamics

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Science  24 Jun 2011:
Vol. 332, Issue 6037, pp. 1525-1528
DOI: 10.1126/science.1204621

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Matter described by quantum chromodynamics (QCD), the theory of strong interactions, may undergo phase transitions when its temperature and the chemical potentials are varied. QCD at finite temperature is studied in the laboratory by colliding heavy ions at varying beam energies. We present a test of QCD in the nonperturbative domain through a comparison of thermodynamic fluctuations predicted in lattice computations with the experimental data of baryon number distributions in high-energy heavy ion collisions. This study provides evidence for thermalization in these collisions and allows us to find the crossover temperature between normal nuclear matter and a deconfined phase called the quark gluon plasma. This value allows us to set a scale for the phase diagram of QCD.

Quantum chromodynamics (QCD) is the theory of strong interactions—one of the four fundamental interactions occurring in nature and an essential part of the standard model of particle physics. It describes interactions between quarks and gluons, which are the ultimate constituents of the majority of the visible mass of the universe (1, 2). In the short-distance regime in which the momentum exchange between quarks and gluons is large, the strong coupling constant becomes small through the mechanism of asymptotic freedom. In this perturbative region, QCD is very successful in explaining various processes observed in experiments involving electron-positron, proton-proton, and proton-antiproton collisions (3). In the nonperturbative regime, tests of the theory were related to the computation of hadron properties (4). In other regimes of long-distance nonperturbative physics, the theory is yet to be tested. Here, we test the thermodynamics of bulk strongly interacting matter.

Experimental tests of nonperturbative QCD in the bulk can be carried out by colliding heavy ions (such as U, Pb, Au, and Cu) at different center-of-mass energies, sNN (58). Several experimental programs have been launched or are in the planning stage at facilities such as the Large Hadron Collider (LHC), the Relativistic Heavy Ion Collider (RHIC), the Super Proton Synchrotron (SPS), the Facility for Anti-proton and Ion Research (FAIR), and the Nuclotron-based Ion Collider fAcility (NICA), where the essential features of the QCD phase diagram can be studied.

In QCD, there are conserved quantities such as the net-baryon number B, the net-electric charge Q, and the net-strangeness S. The term “net” means the algebraic sum of the quantum numbers, where those of anti-particles are the negatives of the corresponding particles. As a result, the thermodynamics of the bulk can be characterized by the corresponding chemical potentials (energy needed to add or remove one unit of the conserved quantity to or from the system, respectively) μB, μQ, and μS in addition to the temperature T, conjugate to the conserved energy of a bulk system. In experimental studies of particle ratios measured in heavy-ion collisions, it is observed that the relevant values of μQ and μS are small compared with μB. For example, in Au ion collisions within rapidity range of ±0.1 unit at sNN = 200 GeV (with impact parameter less than 3 fm) one finds that μB = 22 ± 4.5 MeV, whereas μS = 3.9 ± 2.6 MeV, and μQ is still smaller (9).

The lattice formulation of QCD is a nonperturbative approach from first principles for obtaining the predictions of QCD. Space-time is replaced by a lattice; quarks occupy the sites, and gluons occupy the links between the sites. The lattice spacing a is the inverse of the cutoff required to regulate any interacting quantum field theory. The theory is solved numerically at several values of a. The extrapolation to the continuum (a = 0) can then be made through the renormalization group equations. In QCD, there is a conventional temperature, Tc, that is an intrinsic scale of bulk hadronic matter. We follow the definition that it is the temperature at the peak of a susceptibility related to the confinement-deconfinement order parameter (called the Polyakov loop susceptibility, χL) at μB = 0 (1013). Lattice QCD computations show that this peak is finite, which corresponds to a crossover (14, 15). The temperature at which χL peaks, of course, changes with μB. However, once Tc is known such shifts as a function of μB can be quantified. This is similar to saying that the Celsius scale of temperature is defined by the boiling point of water at normal pressure P, and that the boiling point changes with P.

One of the most basic questions to ask about bulk hadronic matter is the value of Tc. This can be represented as a link in a “circle of reasoning” that encompasses all the regimes of nonperturbative QCD (Fig. 1). So far, the strategy to find Tc has been indirect: First, lattice QCD computations are performed at both T = 0 and T > 0 in order to determine a ratio Tc/m, where m is a typical hadronic scale [Fig. 1, step (b)]. Then one replaces the scale m, determined on the lattice, with an experimental measurement [Fig. 1, step (a)]. The temperature at eachsNN extracted from models of particle yields (16, 17) is step (d) of the circle of reasoning. From such models, one finds that the fireball of bulk nuclear matter created in heavy ion collisions, which is initially out of equilibrium, evolves to a state of thermal equilibrium at chemical freeze-out. The models do not give Tc; however, they allow the extraction of T and μB at freeze-out. We show that predictions of lattice computations of finite temperature QCD (18), taken in conjunction with determinations of Tc in step (b) (1013), agree well with experimental measurements on bulk hadronic matter (19). This allows us to invert the reasoning and extract Tc directly from the experimental measurements in heavy ion collisions [Fig. 1, step (c)]. The agreement of the temperature from steps (c) and (d) along with the agreement of Tc extracted from steps (a) and (b) with that from (c) show the complete compatibility of a single theory of hadron properties and of bulk QCD matter, that is of all nonperturbative regimes of the strong interactions. This approach may present a new domain of tests of the standard model of particle physics.

Fig. 1

Illustration of the chain of reasoning for testing QCD in the nonperturbative domains of the strong interactions and obtaining the scale, Tc, of the QCD phase diagram.

The conjectured phase diagram of QCD. In the current conjectures for the parts of the phase diagram that is accessible with heavy ion collisions (Fig. 2) (20), calculations within simplified models that mimic QCD show that at large μB there is a first-order hadron–quark–gluon plasma (QGP) phase transition. This phase boundary is expected to end in a critical point at finite μB because lattice computations (1013) agree with general symmetry arguments (21), which indicate that at μB = 0 there is neither a first-order nor a second-order phase transition but only a crossover at Tc. The determination of Tc sets the scale of the QCD phase diagram. Current best estimates of the position of the critical point (22) are reflected in the position indicated in Fig. 2. Currently, the experimental focus is on an attempt to locate the critical point and the line of phase coexistence (23, 24).

Fig. 2

Current conjectures for the QCD phase diagram. The phase boundary (solid line) between the normal low-temperature hadronic phase of bulk QCD matter and the high-temperature partonic phase is a line of first-order phase transitions that begins at large μB and small T and curves toward smaller μB and larger T. This line ends at the QCD critical point, whose probable position, derived from lattice computations, is marked by a square. At even smaller μB, there are no phase transitions, only a line of cross-overs (dashed line). The red-yellow dotted line corresponds to the chemical freeze-out line from the evolution of the bulk QCD matter produced in high-energy heavy-ion collisions. The solid point at T = 0 and μB = 938 MeV represents nuclear matter in the ground state. At large μB and low T is the color superconductor phase (CSC) (35).

By changing sNN, one traces out a line of chemical freeze-out in the phase diagram, as shown in Fig. 2. This line is parameterized through a hadron resonance gas model (16, 17). Because this work focuses on making a connection between QCD thermodynamic calculations and observables measured in experimental facilities, we also show in Fig. 2 the range of μB values covered by various experiments as one traverses the chemical freeze-out line by changing sNN. The solid point around μB = 938 MeV is the location of ordinary nuclear matter (25), the best characterized point on the phase diagram.

Comparison of experimental measurements with lattice QCD predictions. Lattice QCD computations leave open the question of a scale and yield dimensionless predictions—for example, for P/T4 as a function of T/Tc and μB/T. Here, we discuss the nonlinear susceptibilities (NLSs) of baryons, χB(n), of order n (26). These are the Taylor coefficients in the expansion of P with respect to μB at fixed T in the usual dimensionless formTn-4χB(n)(TTc,μBT)=1T4n(μB/T)nP(TTc,μBT)|T/Tc(1)Lattice measurements of the series expansion of the NLS in powers of μB/T are resummed by using PadÉ approximants in order to give predictions for the above quantities (18). They are of interest because they are related to cumulants of the fluctuations of the baryon number in thermal and chemical equilibrium in a grand canonical ensemble.

The nth cumulant of such fluctuations, [Bn], is given by[Bn]=VT3Tn-4χB(n)(TTc,μBT)(2)where V is the volume of the observed part of the fireball. Because observed hadrons are in thermal and chemical equilibrium at the freeze-out, this relation should hold for cumulants of the observed event-by-event distribution of net-baryon number in heavy ion collisions. The cumulants are often reported as the variance σ2 = [B2], the skewness S = [B3]/[B2]3/2, and the Kurtosis κ = [B4]/[B2]2. It is clear from these definitions that the V-dependence in Eq. 2 gives the correct volume scaling expected from the central limit theorem. This leads to the classic extraction of the susceptibility from fluctuations in the grand canonical ensemble (27, 28).

There is one remaining subtlety in comparing lattice computations with experimental data. Most experiments are designed to measure event-by-event net-protons. The data discussed in the current work is from the STAR experiment at RHIC (19), which identifies protons and anti-protons by measuring the specific ionization energy loss of these particles in the gas of a time projection chamber. These measurements miss neutrons, the other dominant part of the baryon distribution. This may impose limitations on our measurement of fluctuations. However, the effect of isospin fluctuations on the shape of the net-baryon distributions is small (29). Hence, we proceeded under the assumption that the shape of the net-proton distributions reflects the net-baryon distributions up to distortions smaller than the estimated errors in measurements of the cumulants.

We are unable to exploit Eq. 2 directly in heavy-ion experiments because the volume, V, is hard to determine precisely experimentally. However, the ratios(m1):Sσ=[B3][B2]=TχB(3)χB(2),(m2):κσ2=[B4][B2]=T2χB(4)χB(2),(m3):κσS=[B4][B3]=TχB(4)χB(3)(3)do not contain the volume and therefore provide a direct and convenient comparison of experiment and theory (30). The above equations are written in a form that emphasizes this connection: The left hand side can be measured in an experiment, whereas the right hand side can be predicted with lattice QCD. We use the notation m1,2,3 generically to refer to either side.

We now discuss the comparison of m1 and m2 from experiment and theory (Fig. 3). The experimental measurements (19) were made by using the number of protons (p) and anti-protons (–p) produced in the collision of Au ions around 90° to the beam axis with the impact parameter of the collisions being less than 3 fm (31). p and –p are in the range of 400 MeV/c to 800 MeV/c, where c is the speed of light. This choice of momentum range is designed to obtain the purest sample of p and p¯. A large fraction of p and p¯ is contained in this kinematic range. The effect of finite reconstruction efficiency of p and p¯ has been shown to be negligible (19). The experimental values of Sσ and κσ2 are shown as a function of sNN.

Fig. 3

Comparison of lattice QCD and experimental data for (A) m1 and (B) m2. Experimentally measured ratios of cumulants of net-proton distributions, m1 = Sσ and m2 = κσ2, are shown as a function of Embedded Image for impact parameter values of less than 3 fm for Au+Au collisions at RHIC (19). Also plotted on the top scale are the chemical freeze-out values of μB and T corresponding to Embedded Image as obtained from a hadron resonance gas model, which assumes the system to be in chemical and thermal equilibrium at freeze-out (16, 17). The prediction of such a model for m1 (33) is shown as the dashed red line. The lattice predictions for these quantities are drawn from a computation with lattice cutoff of 1/a ≅ 960 to 1000 MeV and converted to the dimensional scale of T and μ by using Tc = 175 MeV.

The lattice calculations (18) were carried out by using two flavors of staggered quarks in QCD. The lattice cutoff 1/a ≅ 960 to 1000 MeV and the bare quark mass were tuned to give a pion mass of about 230 MeV (32). These computations were performed at μB = 0, and the Taylor series coefficients of P/T4 were used to extrapolate m1 and m2 to the freeze-out conditions by using appropriate order Padé approximants to resum the series expansions. Because lattice results are obtained in terms of T/Tc and μB/T (Eq. 1), their extrapolation to the freeze-out conditions required the input of Tc. The lattice values were obtained by using Tc = 175 MeV, which is compatible with indirect determinations of Tc (1013).

On the upper scales of Fig. 3, we also show the μB and T values at chemical freeze-out that correspond to the varioussNN. For this, we used the functional relationship between these values from the hadron resonance gas model using the yields of hadrons discussed in (16, 17). The model predictions for m1 (33) are also shown. The hadron resonance gas model predictions can be reproduced if baryon and anti-baryon numbers are independently Poisson distributed. Having established a connection between sNN and (T, μB), we compare the experimental data on fluctuations with those predicted from lattice QCD. Excellent agreement is seen between lattice QCD predictions and experimental measurements for all three beam energies. This marks the first successful direct test of QCD against experimental data in the nonperturbative context of bulk hadronic matter. The agreement with the data are yet another nontrivial indication that the fireball produced in heavy ion collisions is in thermal and chemical equilibrium at chemical freeze-out.

Setting the scale of bulk QCD. Lattice QCD results for m1,2,3 are obtained for dimensionless arguments T/Tc and μB/T, as shown in Eq. 2. For a given value ofsNN, the experimental observations are realized at the corresponding chemical freeze-out, characterized by T and μB. Thus, a comparison of the two requires a choice of the scale, Tc. By varying this scale to obtain the best fit between the QCD predictions and experimental measurements, we determined Tc for the first time through an observable connected to strongly interacting bulk matter. The results are, of course, subject to all the caveats expressed in the previous section. The observable that we choose for comparison is m3. The lattice computation of this quantity has the smallest systematic uncertainties among the three explored here and thus is the best quantity to use to constrain Tc.

The comparison of m3 between experimental results from Au ion collisions and lattice QCD predictions is shown in Fig. 4A. This is an extension of Fig. 3, which shows a comparison with m1 and m2. In this analysis, the results of m1, m2, and m3 are consistent, as required in Eq. 3. The new information here is that we show lattice predictions obtained with different values of Tc. The errors on the experimental data points are statistical (lines) and systematic (brackets) errors (19). The errors bars on the lattice predictions are statistical errors, with a cutoff of 1/a ≅ 960 to 1000 MeV. The lattice spacing effects and the effect of tuning the bare quark mass are the main sources of remaining uncertainties in the predictions. These are not parameterized as systematic uncertainties. However, it is known that their effect is small at the two highest values of sNN (18).

Fig. 4

Comparison of m3 from experiment and lattice predictions, and the extraction of Tc. (A) κσ/S of net-proton distribution measured in collisions of Au ions at varying Embedded Image and with an impact parameter of less than 3 fm. This is compared with lattice QCD predictions with cutoff 1/a ≅ 960 to 1000 MeV for the corresponding ratio of susceptibilities extrapolated to the freeze-out conditions by using different values of Tc. The lattice results at each Embedded Image are slightly shifted for clarity in presentation. (B) The comparison of experimental data and lattice QCD predictions, shown through χ2 as a function of Tc by using the definition given in Eq. 4. This yields the estimate of Tc, and its errors are as discussed in the text.

In order to arrive at a quantitative estimate of the scale parameter Tc, we perform a standard statistical analysis. For each value of Tc, we computeχ2(Tc)=sNN[m3expt(sNN)-m3QCD(sNN,Tc)]Errorexpt2+ErrorQCD22(4)where the errors in the experimental and lattice QCD quantities are obtained as explained above. The lattice predictions are calculated for the grid of Tc values (Fig. 4). The minimum of χ2, corresponding to the most probable value of the parameter being estimated, occurs at Tc = 175 MeV. The standard errors on the parameter are the values of Tc for which χ2 exceeds the minimum value by unity. It is clear from Fig. 4B that this is bounded by +5 and −10 MeV. A piecewise linear interpolation between the grid points yields the more reliable error estimate, +1 and −7 MeV. By comparing different interpolation schemes, we found that the error estimate is stable. As a result, we conclude thatTc=1757+1MeV(5)The error estimates include systematic and statistical errors on experimental data but only statistical errors on the lattice QCD computations.

The result in Eq. 5 is compatible with current indirect estimates of Tc that come from setting the scale of thermal lattice QCD computations via hadronic observables. Furthermore, this gives a scale for temperatures that is compatible with the resonance gas model, as shown in Fig. 3. As we discussed in the introduction, this closes a circle of inferences that shows that phenomena obtained in heavy ion collisions are fully compatible with hadron phenomenology and provides a first check in bulk hot and dense matter for the standard model of particle physics.

Conclusions and outlook. We have performed a direct comparison between experimental data from high-energy heavy ion collisions on net-proton number distributions and lattice QCD calculations of net-baryon number susceptibilities. The agreement between experimental data, lattice calculations, and a hadron resonance gas model indicates that the system produced in heavy ion collisions attained thermalization during its evolution. The comparison further enables us to set the scale for nonperturbative, high-temperature lattice QCD by determining the critical temperature for the QCD phase transition to be 175-7+1MeV.

This work reveals the rich possibilities that exist for a comparative study between theory and experiment of QCD thermodynamics and phase structure. In particular, the current work can be extended to the search for a critical point. In a thermal system, the correlation length (ξ) diverges at the critical point. ξ is related to various moments of the distributions of conserved quantities, such as net-baryons, net-charge, and net-strangeness. Finite size and dynamical effects in heavy ion collisions put constraints on the values of ξ (34). The lattice calculations discussed here and several QCD-based models have shown that moments of net-baryon distributions are related to baryon number susceptibilities and that the ratio of cumulants m2 = κσ2, which is related to the ratio of fourth-order to second-order susceptibilities, shows a large deviation from unity near the critical point. Experimentally, κσ2 can be measured as a function of sNN (or T and μB) in heavy ion collisions. A nonmonotonic variation of κσ2 as a function ofsNN would indicate that the system has evolved in the vicinity of the critical point and thus could be taken as evidence for the existence of a critical point in the QCD phase diagram.

References and Notes

  1. μB of bulk nuclear matter is quoted in the usual convention adopted in relativistic treatments because anti-baryon production also needs to be accounted for. An alternative (nonrelativistic) definition that takes nucleon number to be fixed would give much smaller chemical potential. However, this does not correspond to the physics of baryon number fluctuations that we examine.
  2. The impact parameter is determined through a Glauber Monte Carlo procedure. The selected events correspond to the most central 0 to 5% and 0 to 10% events of the total hadronic cross section in Au+Au collisions for sNN = 200, 62.4, and 19.6 GeV, respectively.
  3. The change in the radius of convergence in going from mπ/mρ = 0.33 to 0.2 is likely to be between 10 and 15% (22). The corresponding effect on m3 is about 2% or less at the two highest energies and less than 20% at an energy of 19.6 GeV.
  4. Acknowledgments: We thank Z. Fodor, R. V. Gavai, F. Karsch, D. Keane, V. Koch, B. Mueller, K. Rajagopal, K. Redlich, H. Satz, and M. Stephanov for enlightening discussions. We acknowledge the Indian Lattice Gauge Theory Initiative for computational support, the Department of Atomic Energy–Board of Research in Nuclear Sciences through the project sanction 2010/21/15-BRNS/2026, the U.S. Department of Energy under contract DE-AC03-76SF00098, and the Chinese Ministry of Education.
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