NNLO hard-thermal-loop thermodynamics for QCD

doi:10.1016/j.physletb.2010.12.070

Physics Letters B

Volume 696, Issue 5, 14 February 2011, Pages 468-472

https://doi.org/10.1016/j.physletb.2010.12.070 Get rights and content

Abstract

We calculate the thermodynamic functions of a quark–gluon plasma for general $N_{c}$ and $N_{f}$ to three-loop order using hard-thermal-loop perturbation theory. At this order, all the ultraviolet divergences can be absorbed into renormalizations of the vacuum, the HTL mass parameters, and the strong coupling constant. We show that at three loops, the results for the pressure and trace anomaly are in very good agreement with recent lattice data down to temperatures $T \sim 2 T_{c}$ .

Introduction

The ultrarelativistic heavy-ion collision experiments at Brookhaven National Labs (RHIC), and CERN (LHC) allow the experimental study of matter at energy densities exceeding that required to create a quark–gluon plasma. At RHIC, the initial temperatures were up to twice the critical temperature for deconfinement,¹ $T_{c} \sim 170 MeV$ . This corresponds to a strong coupling constant of $α_{s} = g_{s}^{2} / 4 π \sim 0.3$ . Theoretically, one expected that this state of matter could be described in terms of weakly interacting quasiparticles; however, data from RHIC suggest that the state of matter created behaves more like a strongly coupled fluid with a small viscosity [2]. This has inspired work on strongly-coupled formalisms based on e.g. the AdS/CFT correspondence.

In the upcoming heavy-ion collisions at LHC, the energy densities and therefore the initial temperatures will be higher than those at RHIC. One expects temperatures up to $4 – 6 T_{c}$ and due to asymptotic freedom of QCD, this corresponds to a smaller coupling constant. An important question is then whether the matter generated can be described in terms of weakly interacting quasiparticles at these higher temperatures. Lattice simulations of QCD provide a clean testing ground for the quasiparticle picture and in this Letter we compare new next-to-next-to-leading order (NNLO) results for thermodynamic functions of QCD with lattice data [3], [4] and with previous results obtained at leading order (LO) and next-to-leading order (NLO) [5]. The calculation is based on hard-thermal-loop perturbation theory (HTLpt) which is a reorganization of finite-temperature perturbation theory. In HTLpt one expands around an ideal gas of massive gluonic and quark quasiparticles where screening effects and Landau damping are built in. Our results indicate that the lattice data are consistent with a quasiparticle picture down to temperatures of $T \sim 2 T_{c}$ .

The calculation of thermodynamic functions for quantum field theories at weak coupling has a long history. The free energy of QCD is now known up to order $α_{s}^{3} \log α_{s}$ [6], [7]. Unfortunately, a straightforward application of perturbation theory is of no quantitative use at phenomenologically relevant temperatures. The problem is that the weak-coupling expansion oscillates wildly and shows no sign of convergence unless the temperature is astronomically high. For example, if one compares the $g_{s}^{3}$ -contribution to the QCD free energy with three quark flavors to the $g_{s}^{2}$ -contribution, the former is smaller only if $α_{s} ⩽ 0.07$ , which corresponds to $T \sim 10^{5} GeV$ or $T \sim 5 \times 10^{5} T_{c}$ .

There are several ways of reorganizing the perturbative series at finite temperature [8] and they are all based on a quasiparticle picture where one is perturbing about an ideal gas of massive quasiparticles, rather than an ideal gas of massless quarks and gluons. In scalar $ϕ^{4}$ -theory the basic idea is to add and subtract a thermal mass term from the bare Lagrangian and to include the added piece in the free part of the Lagrangian. The subtracted piece is then treated as an interaction on the same footing as the quartic term [9]. In gauge theories, however, simply adding and subtracting a local mass term, violates gauge invariance [10]. Instead, one adds and subtracts an HTL improvement term, which dresses the propagators and vertices self-consistently so that the reorganization is manifestly gauge invariant [11].

Section snippets

Hard-thermal-loop perturbation theory

The Lagrangian density for an $SU (N_{c})$ Yang–Mills theory with $N_{f}$ fermions in Minkowski space is $L_{QCD} = - \frac{1}{2} Tr [G_{μ ν} G^{μ ν}] + i \bar{ψ} γ^{μ} D_{μ} ψ + L_{gf} + L_{gh} + Δ L_{QCD},$ where the field strength is $G^{μ ν} = \partial^{μ} A^{ν} - \partial^{ν} A^{μ} - i g_{s} [A^{μ}, A^{ν}]$ and the covariant derivative is $D^{μ} = \partial^{μ} - i g_{s} A^{μ}$ . $Δ L_{QCD}$ contains the counterterms necessary to cancel the ultraviolet divergences. The ghost term $L_{gh}$ depends on the gauge-fixing term $L_{gf}$ . In this Letter we choose the class of covariant gauges where the gauge-fixing term is $L_{gf} = - Tr [{(\partial_{μ} A^{μ})}^{2}] / ξ$ .

HTLpt is, by

Thermodynamic potential

In this section, we present the final results for the thermodynamic potential Ω at orders $δ^{0}$ (LO), δ (NLO), and $δ^{2}$ (NNLO). The LO and NLO results were first obtained in Ref. [5] and they are listed here for completeness. At LO, the thermodynamic potential was calculated exactly, while at NLO and NNLO the resulting expressions for the diagrams are too complicated. To make the calculations tractable, the thermodynamic potential is therefore evaluated approximately by additional expansions in

Results

In Fig. 1, we show the normalized pressure for $N_{c} = 3$ and $N_{f} = 2 + 1$ (left panel), and $N_{c} = 3$ and $N_{f} = 2 + 1 + 1$ (right panel) as a function of T. The results at LO, NLO, and NNLO use the BN mass given by Eq. (7) as well as $m_{q} = 0$ . For the strong coupling constant $α_{s}$ , we used three-loop running [16] with $Λ_{\bar{MS}} = 344 MeV$ which for $N_{f} = 3$ gives $α_{s} (5 GeV) = 0.2034$ [17]. The central line is evaluated with the renormalization scale $μ = 2 π T$ which is the value one expects from effective field theory calculations [7], [18] and

Summary and outlook

We have presented results for the LO, NLO, and NNLO thermodynamic functions for $SU (N_{c})$ Yang–Mills theory with $N_{f}$ fermions using HTLpt. We compared our predictions with lattice data for $N_{c} = 3$ and $N_{f} \in {3, 4}$ and found that HTLpt is consistent with available lattice data down to $T \sim 2 T_{c}$ for the pressure and the trace anomaly. This is in line with expectations since one is expanding about the trivial vacuum $A_{μ} = 0$ and therefore neglects the approximate center symmetry $Z (N_{c})$ . Close to the deconfinement

Acknowledgements

The authors would like to thank S. Borsanyi for providing us with the latest lattice data of the Wuppertal–Budapest collaboration. We thank S. Borsanyi and Z. Fodor for useful discussions. N. Su was supported by the Frankfurt International Graduate School for Science and Helmholtz Graduate School for Hadron and Ion Research. N. Su thanks the Department of Physics at NTNU for kind hospitality. M. Strickland was supported by the Helmholtz International Center for FAIR LOEWE program.

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NNLO hard-thermal-loop thermodynamics for QCD

Abstract

Introduction

Section snippets

Hard-thermal-loop perturbation theory

Thermodynamic potential

Results

Summary and outlook

Acknowledgements

Nucl. Phys. A

Nucl. Phys. A

Nucl. Phys. A

Nucl. Phys. A

Nucl. Phys. A

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Sov. Phys. JETP

Zh. Eksp. Teor. Fiz.

Nucl. Phys. B

Int. J. Theor. Phys.

Int. J. Theor. Phys.

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Phys. Lett. B

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Phys. Rev. D

Nature

Phys. Rev. Lett.

Phys. Rev. D