Nonperturbative contributions to the QCD pressure

We summarize the most important arguments for why a perturbative description of finite-temperature QCD is unlikely to be possible and review various well-established approaches to deal with this problem. Then, using a recently proposed method, we investigate nonperturbative contributions to the QCD pressure and other observables (such as energy, anomaly, and bulk viscosity) obtained by imposing a functional cutoff at the Gribov horizon. Finally, we discuss how such contributions fit into the picture of consecutive effective theories, as proposed by Braaten and Nieto, and give an outline of the next steps necessary to improve this type of calculation.

Figure 1

A “ladder diagram” contributing at order $g^6$ to the free energy for $\ell \ge 3$.

Figure 2

The scales of perturbative QCD (pQCD), electrostatic QCD (EQCD), magnetostatic QCD (MQCD) and the different contributions $f_E$, $f_M$ and $f_G$ to the free energy. The coefficients $a_k$ and $b_k$ are polynomials in logarithms of ratios of scales $a_k=P_k(\ln \frac{T}{{\Lambda }_E})$, $b_k=Q_k(\ln \frac{{\Lambda }_E}{gT},\ln \frac{gT}{{\Lambda }_M})$. While the coefficients $a_k$ and $b_k$ can be determined, at least in principle, in perturbation theory, this is not possible for $c_k$.

Figure 3

Rescaled pressure of SU(3) lattice gauge theory, from 38, where $T_c$ is the transition temperature, and $N_{\tau }=4,6,8$.

Figure 4

Rescaled anomaly of SU(3) lattice gauge theory, from 38.

Figure 5

The convergence of an (optimized) perturbation series for “long-distance contribution” to the pressure, from 57. The order of the expansion is characterized by the dimensionless parameter $y\sim \frac{g^2{T}^2}{g_3^4}$, where $g_3^2$ denotes the gauge coupling in the effective three-dimensional theory. The perturbative contribution $F$ to the free energy contains a factor $(\frac{g_3^2}{T}{)}^3$; the inclusion of lower powers of $y$ corresponds to higher orders in perturbation theory.

Figure 6

Graph of the renormalization scale $\mu (T)$: We show the piecewise linear form ${\mu }_{\mathrm{a}}$ from (28) and the exponential form ${\mu }_{\mathrm{e}}$ from (29) together with the asymptotics ${\mu }_{0,\mathrm{l}}=2\pi {\Lambda }_{\mathrm{QCD}}$ and ${\mu }_{0,\mathrm{h}}=2\pi T$.

Figure 7

The rescaled Gribov mass $m^{*}=\frac{m}{T}$: (a) solid red line—numerical solution, dotted line—asymptotic expression from (35); (b) relative deviation $\Delta m_{\mathrm{rel}}^{*}=(m_{\mathrm{num}}^{*}-m_{\mathrm{asy}}^{*})/m_{\mathrm{asy}}^{*}$.

Figure 8

The rescaled reduced free energy $w_r-w_{\mathrm{r},\mathrm{SB}}$: (a) solid red line—numerical solution, dotted line—asymptotic expression from (38); (b) relative deviation $\Delta w_{\mathrm{r},\mathrm{rel}}=(w_{\mathrm{r},\mathrm{num}}-w_{\mathrm{r},\mathrm{asy}})/(w_{\mathrm{r},\mathrm{asy}}-w_{\mathrm{r},\mathrm{SB}})=(w_{\mathrm{num}}-w_{\mathrm{asy}})/(w_{\mathrm{asy}}-w_{\mathrm{SB}})$.

Figure 9

The rescaled anomaly $A_r$: (a) solid red line—numerical solution, dotted line—asymptotic expression from (48); (b) relative deviation $\Delta A_{\mathrm{r},\mathrm{rel}}=(A_{\mathrm{r},\mathrm{num}}-A_{\mathrm{r},\mathrm{asy}})/A_{\mathrm{r},\mathrm{asy}}$.

Figure 10

The rescaled bulk viscosity: (a) solid red line—numerical solution ${\zeta }_{\mathrm{r},\mathrm{num}}$, dotted line—asymptotic expression ${\zeta }_{\mathrm{r},\mathrm{asy}}$ from (48); (b) relative deviation $\Delta {\zeta }_{\mathrm{r},\mathrm{rel}}=({\zeta }_{\mathrm{r},\mathrm{num}}-{\zeta }_{\mathrm{r},\mathrm{asy}})/{\zeta }_{\mathrm{r},\mathrm{asy}}$.

Figure 11

The rescaled free energy in the low-temperature region [solid red line—numerical solution, dotted line—asymptotic expression from (48)].

Figure 12

The rescaled anomaly $\frac{e-3p}{T^4}$ in the low-temperature region [solid red line—numerical solution, dotted line—asymptotic expression from (48)].

Figure 13

The rescaled bulk viscosity in the low-temperature region (solid red line—numerical solution, dotted line—asymptotic expression from (48)].