# High-Precision Thermodynamics and Hagedorn Density of States

###### Abstract

We compute the entropy density of the confined phase of QCD without quarks on the lattice to very high accuracy. The results are compared to the entropy density of free glueballs, where we include all the known glueball states below the two-particle threshold. We find that an excellent, parameter-free description of the entropy density between and is obtained by extending the spectrum with the exponential spectrum of the closed bosonic string.

###### pacs:

12.38.Gc, 12.38.Mh, 25.75.-q^{†}

^{†}preprint: MIT-CTP 4041

## I Introduction

The phase diagram of quantum chromodynamics (QCD) is being actively studied in heavy ion collision experiments as well as theoretically. A form of matter with remarkable properties Muller (2008) has been observed in the Relativistic Heavy Ion Collider (RHIC) experiments Arsene et al. (2005); Back et al. (2005); Adcox et al. (2005); Adams et al. (2005). It appears to be a strongly coupled plasma of quarks and gluons (QGP), but no consensus on a physical picture that accounts for both equilibrium and non-equilibrium properties has been reached yet. On the other hand, below the short interval of temperatures where the transition from the confined phase to the QGP takes place Aoki et al. (2006, 2009); Cheng et al. (2008); Bazavov et al. (2009), it is widely believed that the most prominent degrees of freedom are the ordinary hadrons. From this point of view, the zeroth order approximation to the properties of the system is to treat the hadrons as infinitely narrow and non-interacting. We will refer to this approximation as the hadron resonance gas model (HRG). The HRG predictions were compared with lattice QCD thermodynamics data in Karsch et al. (2003); Cheng et al. (2008), and lately they have been used to extrapolate certain results to zero temperature Bazavov et al. (2009). The HRG is also the basis of the statistical model currently applied to the analysis of hadron yields in heavy ion collisions Andronic et al. (2009), and recently the transport properties of a relativistic hadron gas have been studied in detail Demir and Bass (2008).

Since any heavy ion reaction ends up in the low-temperature phase of QCD, it is important to understand its properties in detail in order to extract those of the high-temperature phase with minimal uncertainty. In this Letter we study whether the HRG model works in the absence of quarks, in other words in the pure SU() gauge theory, where the low-lying states are glueballs. There are reasons to believe that if the HRG model is to work at any quark content of QCD, it is in the zero-flavor case. Firstly, the mass gap in SU(3) gauge theory is very large, . As we shall see, the thermodynamic properties up to quite close to are dominated by the states below the two-particle threshold, which are exactly stable. Furthermore, because of their large mass, neglecting their thermal width should be a good approximation. Secondly, the scattering amplitudes between glueballs are parametrically suppressed while those between mesons are only suppressed Witten (1979). This means that the glueballs should be free to a better approximation than the hadrons of realistic QCD.

An additional motivation to study the thermodynamics of the confined phase of SU(3) gauge theory is that it is a parameter-free theory, simplifying the interpretation of its properties. Its spectrum is known quite accurately up to the two-particle threshold Meyer (2004); Chen et al. (2006). By contrast, in full QCD calculations, lattice data calculated at pion masses larger than in Nature are often compared out of necessity to the HRG model based on the experimental spectrum Cheng et al. (2008); Bazavov et al. (2009). Finally, calculations in the pure gauge theory are at least two orders of magnitude faster, which allows us to reach a high level of control of statistical and systematic errors; in particular, we are able to perform calculations in very large volumes.

## Ii Lattice calculation

We use Monte-Carlo simulations of the Wilson action for SU(3) gauge theory Wilson (1974), where is the plaquette. The lattice spacing is related to the bare coupling through . We calculate the thermal expectation value of , the (anomalous) trace of the energy-momentum tensor , and of . In the thermodynamic limit,

(1) |

Here are respectively the energy density, pressure and entropy density. The operator requires no subtraction, because its vacuum expectation value vanishes. The choice of of and as independent linear combinations is convenient because they both renormalize multiplicatively. We use the ‘HYP-clover’ discretization of the energy-momentum tensor introduced in Meyer and Negele (2008, 2007). The normalization of the operator differs from its naive value by a factor that we parametrize as . The factor is taken from Meyer (2007) and rests on the results of Engels et al. (2000); its accuracy is about one percent. The factor is obtained by calibrating our discretization to the ‘bare plaquette’ discretization in the deconfined phase at Meyer and Negele (2008). We find, for between 5.90 and 6.41, with an accuracy of half a percent. For the lattice beta-function that renormalizes , we use the parametrization Duerr et al. (2007) of the data in Necco and Sommer (2002) and the same calibration method.

Our results for the entropy density from and simulations are shown on Fig. (3). The displayed error bars do not contain the uncertainty on the normalization factor, which is much smaller and would introduce correlation between the points. This factor varies by only over the displayed interval and so to a first approximation amounts to an overall normalization of the curve. Our data is about five times statistically more accurate than that of previous thermodynamic studies Boyd et al. (1996); Namekawa et al. (2001), which were primarily focused on the deconfined phase. Just as importantly, we kept the finite-spatial-volume effects under good control, in particular very close to .

Figure (1) shows the size of finite-volume effects. For instance, at the conventional choice leads to an overestimate of the entropy density by a factor three. The fact that the data fall on the same smooth curve as the is strong evidence that discretization errors are small. We parametrize the volume dependence empirically by a curve, and use it to convert the data to . At , there is no statistically significant difference between and 8 and we do not apply any correction. It is the corrected data that is then displayed on Fig. (3).

In Meyer (2009a), formulas for the leading finite-volume effects on the thermodynamic potentials were derived in terms of the energy gap of the theory defined on a spatial hypertorus. Close to , this gap corresponds to the mass of the ground state flux loop winding around the cycle of length . If , the formula then reads

(2) |

Using the calculation of described in the next section, the predicted asymptotic approach to the infinite-volume entropy density for is displayed on Fig. (1). While the sign is correct, the magnitude of the finite-volume effects is not reproduced for . We conclude that the asymptotic approach to infinite volume sets in for very large values of . Since is only about when , it is not implausible that flux-loop states with high multiplicity dominate the finite-volume effects at that box size.

Next we obtain the correlation length of the order parameter for the deconfining phase transition, the Polyakov loop. The method consists in computing the two-point function of zero-momentum operators, designed to have large overlaps with the ground state flux loop, along a spatial direction. We fit the lattice data for displayed on Fig. (2) with the formula

(3) |

and find, either fitting or setting it to zero,

(4) | |||

(5) |

with in both cases a dof of about 0.3. We remark that the are not far from the Nambu-Goto string Arvis (1983) values 2005) and ( is the tension of the confining string). We extract the ‘Hagedorn’ temperature, defined as in Bringoltz and Teper (2006) by , from the second fit, Lucini et al. (

(6) |

This extraction amounts to assuming mean-field exponents near (it is not clear which universality class should be used Yaffe and Svetitsky (1982)). The result is stable if the fit interval is varied, and also if is fitted with and constrained to the known values of and .

As a check on the normalization of the operators and , we calculate the latent heat in two different ways. The latent heat is the jump in energy density at . Since the pressure is continuous, we obtain it instead from the discontinuity in entropy density or the ‘conformality measure’ . We obtain and on either side of by extrapolating data from the confined (deconfined) phase towards . The result is

(7) |

where the first error is statistical and the second comes from the uncertainty in the extrapolation (taken to be the difference between a linear and quadratic fit). The compatibility between these two estimates of is strong evidence that we control the normalization of our operators. They are in good agreement with previous calculations of the latent heat performed on coarser lattices Beinlich et al. (1997); Lucini et al. (2005). We have also verified more generally that the thermodynamic identity is satisfied within statistical errors.

## Iii Interpretation

In infinite volume the pressure associated with a single non-interacting, relativistic particle species of mass with polarization states reads

(8) |

where is a modified Bessel function. By linearity, the knowledge of the glueball spectrum leads to a simple prediction for the pressure and entropy density , which is expected to become exact in the large- limit. Since only the low-lying spectrum of glueballs is known, it is useful to consider how the density of states might be extended above the two-particle threshold , where is the mass of the lightest (scalar) glueball. The asymptotic closed bosonic string density of states in four dimensions is given by Zwiebach (2004)

(9) |

In the string theory, the Hagedorn temperature is related to the string tension, , corresponding to Lucini et al. (2004). Below we use this value as an alternative to the more direct determination (6).

On Fig. 3, we show the entropy contribution of the glueballs lying below the two-particle threshold . The curve is just about consistent with the smallest temperature lattice data point, but clearly fails to reproduce the strong increase in entropy density as . The figure also illustrates that the two lowest-lying states, the scalar and tensor glueballs, account for about three quarters of the stable glueballs’ contribution. We have used the continuum-extrapolated lattice spectrum Meyer (2004, 2009b).

Adding the Hagedorn spectrum contribution, Eq. (9) with given by Eq. (6), leads to the solid curve on Fig. 3. It describes the direct calculation of the entropy density surprisingly well, particularly close to . The curve tends to underestimate somewhat the entropy density at the lower temperatures. This is likely to be a cutoff effect. Indeed, at fixed lower temperatures correspond to a coarser lattice spacing, and the scalar glueball mass in physical units is known to be smaller on coarse lattices with the Wilson action Lucini and Teper (2001). If we use the stable glueball spectrum calculated at instead of the continuum spectrum, the agreement of the non-interacting glueball + Hagedorn spectrum with the lattice data at the lower four temperatures is again excellent. This difference provides an estimate for the size of lattice effects.

To summarize, we have computed to high accuracy the entropy of the confined phase of QCD without quarks. The low-lying states of the theory are therefore bound states called glueballs, and their spectrum is well determined Meyer (2004); Chen et al. (2006). If the size of the gauge group is increased, the interactions of the glueballs are expected to be suppressed Witten (1979). To what extent the glueballs really are weakly interacting at is not known precisely. Some evidence for the smallness of their low-energy interactions was found some time ago by looking at the finite-volume effects on their masses Meyer (2005). But it seems unlikely that glueballs well above the two-particle threshold would have a small decay width. We have nevertheless compared the entropy density data to the entropy density of a gas of non-interacting glueballs. While restricting the spectral sum to the stable glueballs leads to an underestimate by at least a factor two of the entropy density near , extending the spectral sum with an exponential spectrum , suggested long ago by Hagedorn Hagedorn (1965), leads to a prediction in excellent agreement with the lattice data for the entropy density (Fig. 3). This is remarkable, since the analytic form of the asymptotic spectrum is completely predicted by free bosonic string theory, including its overall normalization (Eq. 9). Therefore, since we also separately computed the temperature (identified with ) where the flux loop mass vanishes, no parameter was fitted in the comparison with the thermodynamic data. By contrast, the entropy density is not nearly as well described if the Nambu-Goto value of is used, see Fig. (3).

The success of the non-interacting string density of states in reproducing the entropy density suggests that once the Hagedorn temperature has been determined directly from the divergence of the flux-loop correlation length, the residual effects of interactions on the thermodynamic potentials are small. It may be that thermodynamic properties in general are not strongly influenced by interactions when a large number of states are contributing. A well-known example is provided by the super-Yang-Mills theory, whose entropy density at very strong coupling is only reduced by a factor 3/4 with respect to the free theory Gubser et al. (1996). In this interpretation, the main effect of interactions among glueballs on thermodynamic properties is to slightly shift the value of the Hagedorn temperature with respect to its free-string value. A possible mechanism is that the string tension that effectively determines is an in-medium string tension that is lower than at .

Returning to full QCD, our results lend support to the idea that the hadron resonance gas model can largely account for the thermodynamic properties of the low-temperature phase. Whether the open string density of states reproduces the entropy calculated on the lattice can also be tested at quark masses not necessarily as light as in Nature using a simple open string model Selem and Wilczek (2006).

###### Acknowledgements.

I thank B. Zwiebach for a discussion on the bosonic string density of states. The simulations were done on the Blue Gene L rack and the desktop machines of the Laboratory for Nuclear Science at M.I.T. This work was supported in part by funds provided by the U.S. Department of Energy under cooperative research agreement DE-FG02-94ER40818.## References

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