On the Apparent Convergence of Perturbative QCD at High Temperature

TL;DR

The problem is the poor convergence of perturbative QCD for the thermal pressure at high temperature $T$, driven by soft-scale physics. The authors propose separating hard ($2π T$) and soft ($gT$) contributions using dimensional reduction to EQCD, treating hard perturbatively and soft contributions with minimal nonperturbative resummation, including four-loop logs. Their main contributions show that untruncated soft contributions greatly improve convergence and reduce renormalization-scale sensitivity; the three-loop results agree well with lattice data for $T$ above about $3T_c$, and the inclusion of four-loop logarithms keeps the trend, bringing the results close to the lattice down to roughly $2.5T_c$. In the large-$N_f$ limit the untruncated results remain consistently close to the exact solution, validating the approach. Overall, the study supports resummation-based reorganizations of perturbation theory as a viable route to reliable hot QCD thermodynamics and identifies DRSPT and related methods as practical alternatives.

Abstract

The successive perturbative estimates of the pressure of QCD at high temperature T show no sign of convergence, unless the coupling constant g is unrealistically small. Exploiting known results of an effective field theory which separates hard (order 2 pi T) and soft (order gT) contributions, we explore the accuracy of simple resummations which at a given loop order systematically treat hard contributions strictly perturbatively, but soft contributions without truncations. This turns out to improve significantly the two-loop and the three-loop results in that both remain below the ideal-gas value, and the degree of renormalization scale dependence decreases as one goes from two to three loop order, whereas it increases in the conventional perturbative results. Including the four-loop logarithms recently obtained by Kajantie et al., we find that this trend continues and that with a particular sublogarithmic constant the untruncated four-loop result is close to the three-loop result, which itself agrees well with available lattice results down to temperatures of about 2.5 T_c. We also investigate the possibility of optimization by using a variational (``screened'') perturbation theory in the effective theory. At two loops, this gives a result below the ideal gas value, and also closer to lattice results than the recent two-loop hard-thermal-loop-screened result of Andersen et al. While at three-loop order the gap equation of dimensionally reduced screened perturbation theory does not have a solution in QCD, this is remedied upon inclusion of the four-loop logarithms.

Figures (8)

• Figure 1: Strictly perturbative results for the thermal pressure of pure glue QCD normalized to the ideal-gas value, as a function of $T/T_c$ (assuming $T_c/\Lambda_{\overline{\hbox{\scriptsize MS}}}=1.14$). The various gray bands bounded by differently dashed lines show the perturbative results to order $g^2$, $g^3$, $g^4$, and $g^5$, with $\overline{\hbox{MS}}$ renormalization point $\bar{\mu}$ varied between $\pi T$ and $4\pi T$. The thick dark-grey line shows the continuum-extrapolated lattice results from Ref. Boyd:1996bx; the lighter one behind that of a lattice calculation using an RG-improved action Okamoto:1999hi.
• Figure 2: Two-loop pressure in pure-glue QCD with untruncated EQCD contributions when $\bar{\mu}$ is varied between $\pi T$ and $4\pi T$ (broad gray band) in comparison with the lattice result from Ref. Boyd:1996bx (thick dark-grey curve). The narrow darker-grey band above the former is the result of 2-loop DRSPT considered in Sect. \ref{['sec:DRSPT2']}; its lower boundary corresponds to the extremal value when varying $\bar{\mu}$.
• Figure 3: Three-loop pressure in pure-glue QCD with untruncated EQCD contributions when $\bar{\mu}$ is varied between $\pi T$ and $4\pi T$ (medium-gray band); the dotted lines indicate the position of this band when only the leading-order result for $m_E$ is used. The broad light-gray band underneath is the strictly perturbative result to order $g^5$ with the same scale variations. The full line gives the result upon extremalization (PMS) with respect to $\bar{\mu}$ (which does not have solutions below $\sim 1.3T_c$); the dash-dotted line corresponds to fastest apparent convergence (FAC) in $m_E^2$, which sets $\bar{\mu}\approx 1.79\pi T$.
• Figure 4: PMS-extremalized full-three-loop pressure in QCD with $N_f=0$ (full line), $N_f=2$ (dashed line), and $N_f=3$ (dash-dotted line) in comparison with the estimated continuum extrapolation of QCD with 2 light quark flavors of Ref. Karsch:1999vy.
• Figure 5: Four-loop pressure in pure-glue QCD including the recently determined $g^6\ln(1/g)$ contribution of Kajantie:2002wa together with three values for the undetermined constant $\delta$ in Eq. (\ref{['P4tot']}) when evaluated fully with $\bar{\mu}$ varied between $\pi T$ and $4\pi T$ (medium-gray bands). The broad light-gray band underneath is the strictly perturbative result to order $g^6$ corresponding to the central value $\delta=1/3$. The full line gives the untruncated result with $\delta=1/3$ extremalized with respect to $\bar{\mu}$ (which does not have solutions below $\sim 1.9T_c$); the dash-dotted line corresponds to fastest apparent convergence (FAC) in $m_E^2$, which sets $\bar{\mu}\approx 1.79\pi T$.