QCD chiral transition temperature in a Dyson-Schwinger-equation context

We analyze the chiral phase transition with the help of the QCD gap equation. Various models for the effective interaction in rainbow truncation are contrasted with regard to the resulting chiral transition temperatures. In particular, we investigate possible systematic relations of the details of the effective interaction and the value of $T_c$. In addition, we quantify changes to the transition temperature beyond the rainbow truncation.

Figure 1

The QCD gap equation in pictorial form.

Figure 2

Plot of the different forms for the effective interaction used in this work as simple functions of a momentum squared $s$. The corresponding parameters can be found in Table . Note the different behavior of the interactions on all momentum ranges. In particular, the reader should keep in mind that the $\delta$ function present in MR, Eq. (12) is invisible in the double-logarithmic presentation as well as the MN interaction, Eq. (9), which is zero for $s\ne 0$.

Figure 3

Plot of the order parameter $B_0$ versus temperature $T$, shown exemplarily for model MT2.

Figure 4

Chiral susceptibilities ${\chi }_T(T)$ (upper panel) and ${\chi }_h(T)$ (lower panel) for the MT1 model plotted versus temperature $T$ for various values of $h$. The curves between the data points were obtained by interpolation using cubic splines.

Figure 5

Peak heights of the chiral susceptibilities ${\chi }_T^{\mathrm{pc}}$ and ${\chi }_h^{\mathrm{pc}}$ plotted versus the reduced mass $h$ on log-log scale, for the MT1 model. The lines are linear fits through the calculated points which are used to obtain the critical exponents according to Eqs. (17, 18).

Figure 6

From left to right, the four panels (Figs. 6a, 6b, 6c, 6d) show the results for $T_c$ as a function of $\omega$, left-most panel (a); $\omega D$, left-middle panel (b); $D$, right-middle panel (c); and the chiral condensate, right-most panel (d). The different colors and symbols on the lines denote the corresponding interactions, namely: black (circles)—AWW; red (X)—MT; and blue (diamonds)—MR. The result for MN (single green square) is only shown in panels (c) and (d), since it does not have a value of $\omega$ associated with it. The insert in panel (d) enlarges the region of interest for the $\omega$-dependent interactions.