Chiral dynamics in the low-temperature phase of QCD

We investigate the low-temperature phase of QCD and the crossover region with two light flavors of quarks. The chiral expansion around the point $(T,m=0)$ in the temperature vs quark-mass plane indicates that a sharp real-time excitation exists with the quantum numbers of the pion. An exact sum rule is derived for the thermal modification of the spectral function associated with the axial charge density; the (dominant) pion pole contribution obeys the sum rule. We determine the two parameters of the pion dispersion relation using lattice QCD simulations and test the applicability of the chiral expansion. The time-dependent correlators are also analyzed using the maximum entropy method, yielding consistent results. Finally, we test the predictions of the chiral expansion around the point $(T=0,m=0)$ for the temperature dependence of static observables.

Figure 1

Sketch of the domain of validity of the chiral effective field theory in the quark mass vs temperature plane. The expansion is represented by the blue arrowed vertical line. The quark mass on the vertical axis is understood to be ${\overline{m}}^{\overline{\mathrm{MS}}}$. The value of the critical temperature at the chiral limit $T_c(0)\simeq 170$ is taken from [16].

Figure 2

Renormalized quark mass in physical units in the $\overline{\mathrm{MS}}$ scheme for both temperature scans (C1 and D1). For the C1 scan, $T_c=211(5)\mathrm{MeV}$, and for D1, $T_c=193(7)\mathrm{MeV}$.

Figure 3

Midpoints of the renormalized correlators $G_{\mathrm{A}}(x_0)$ and $G_{\mathrm{P}}(x_0)$. The former was renormalized via multiplication with $Z_{\mathrm{A}}^2$, the latter via multiplication with $Z_{\mathrm{P}}^2$ as well as the conversion factor from the SF to the $\overline{\mathrm{MS}}$ scheme at the scale $\mu =2\mathrm{GeV}$. All data from the C1 temperature scan.

Figure 4

Example of an effective-mass plot showing $m_{\text{cosh}}(x_3+a/2)$ for the pseudoscalar density two-point function in the $x_3$ direction in the C1 scan at $T=150\mathrm{MeV}$. The result of the fit to the correlation function is represented by a ($1\sigma$) band. Here the chosen fit window was 26, which corresponds to ignoring the three points closest to each operator, and the (uncorrelated) ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}$ amounts to 0.05.

Figure 5

Inverse screening mass $l_{\pi }\equiv m_{\pi }^{-1}$ (left) and screening pion “decay constant” (right) in the C1 scan, divided by the same quantity at $T\simeq 0$ extracted from the A5 ensemble. The displayed error bars represent the statistical errors originating from the ensembles of the C1 scan and from ensemble A5.

Figure 6

Effective chiral condensate defined from the GOR relation, divided by its $T=0$ counterpart, in the temperature scan C1. In addition, the predictions of [19] both for the infinite volume limit and for our finite lattice volume are displayed. The temperature is given in units of the zero-temperature decay constant $f_{\pi ,\mathbf{0}}$.

Figure 7

Left: The two estimators of the pion velocity in the C1 scan. Right: Ratio of the estimators, which serves as a test of the chiral prediction (37).

Figure 8

Temperature dependence of the pion screening mass and the associated decay constant, for two quark masses. The temperature is given in units of the pseudocritical temperature at the corresponding quark mass.

Figure 9

The MEM reconstruction of the vacuum $P$ (left) and $A_0$ (right) channel spectral functions. In both cases a free theory inspired default model $m_{\mathrm{A}/\mathrm{P}}^{\text{free}}(\omega )$ is used as prior information for the MEM analysis. The black error bars indicate the half-maximum width and midpoint, which we define to give $m_{\pi }$ and its error. The broad peak structure at large frequencies is understood as lattice artifact. The quoted error bands represent the spread of spectral functions obtained in a jackknife analysis.

Figure 10

Reconstruction of the spectral function ${\rho }_{\mathrm{A}}$ of the axial charge at finite temperature and vanishing spatial momentum. The notation ${\rho }_{\mathrm{A}}={\rho }_T^A(\omega ,m(\omega ))$ emphasizes the default-model dependence. Left: The three input default models used for the reconstruction of the thermal spectral function. Middle: The MEM reconstruction of the spectral function on the $6/g_0^2=5.20$ ensemble of the C1 scan for all three default models. The quoted error bands represent the spread of spectral functions obtained in a jackknife analysis. Right: The corresponding MEM results for the $6/g_0^2=5.30$ ensemble of the C1 scan.

Figure 11

Left: The area $\mathcal{A}(\mathrm{\Lambda },m(\omega ))$ for different values of the cut-parameter $\mathrm{\Lambda }$ on the $6/g_0^2=5.20$ ensemble of the C1 scan obtained from a MEM reconstruction using all three default models $m(\omega )$. We observe a clear plateau and therefore separation region. Right: The area $\mathcal{A}(\mathrm{\Lambda },m(\omega ))$ for all available lattice ensembles in units of $T^2$, compared to the results of Sec. 3 for $f_{\pi }^2/(T^2{u}_f^2)$, which make use of static quantities.

Figure 12

The autocorrelation function $\mathrm{\Gamma }(t)$ of the observable $G_{\mathrm{A}}(\beta /2)$ for six ensembles of the temperature scan C1 as a function of the configuration number. Consecutive configurations are separated by 10 units of molecular dynamics time. The uncertainty on $\mathrm{\Gamma }(t)$ was estimated using the method described in Appendix E of Ref. [56].

Figure 13

Effect on the estimated uncertainty of the observable $G_{\mathrm{A}}(\beta /2)$ of binning the data before applying the jackknife method.