Locating the critical endpoint of QCD: Mesonic backcoupling effects

We study the effects of pion and sigma meson backcoupling on the chiral order parameters and the QCD phase diagram and determine their effect on the location of the chiral critical endpoint. To this end, we solve a coupled set of truncated Dyson-Schwinger equations for Landau gauge quark and gluon propagators with $N_{\mathrm{f}}=2+1$ dynamical quark flavors and explicitly backcoupled mesons. The corresponding meson bound-state properties and the quark-meson Bethe-Salpeter vertices are obtained from their homogeneous Bethe-Salpeter equation. We find chiral-restoration effects of the pion and/or sigma meson backcoupling and observe a (small) shift of the critical endpoint towards smaller chemical potentials. The curvature of the chiral crossover line decreases. Our results indicate that the location of the critical endpoint in the phase diagram is mainly determined by the microscopic degrees of freedom of QCD (in contrast to its critical properties).

Figure 1

Truncated DSEs for the quark propagator (top) and gluon propagator (bottom left), and truncated Bethe-Salpeter equation (BSE) (bottom right). Quark, gluon, and meson propagators are denoted as solid, curly, and dashed lines, respectively. The intersection of two quarks and a gluon or a meson represents a quark-gluon or a Bethe-Salpeter vertex, respectively. Dressed quantities are indicated by big full dots; the remaining ones are bare. The signs and prefactors are absorbed into the diagrams.

Figure 2

Left: Separation of the full quark-gluon vertex into nonhadronic (NH) contributions (first term) and lowest-order mesonic contributions resulting from a resonance expansion of the quark-antiquark scattering kernel. For the mesonic contribution, a one-meson exchange (second term) is considered. Right: Approximation of the meson-backcoupling quark self-energy resulting from the insertion of the separation of the full quark-gluon vertex into the quark DSE. The blue dot with a white center represents an effective Bethe-Salpeter vertex, which will be discussed in the text. The remaining components are defined in the same way as in Fig. 1.

Figure 3

Dynamical quark mass (left) and corresponding quark wave-function renormalization function $Z_{\mathrm{F}}^f(p)=1/A_f(p)$ (right) for light quarks and different numbers of backcoupled mesons in vacuum. The results are obtained with a fixed strength parameter $d_1^{\mathrm{q}}=12.85{\mathrm{GeV}}^2$ [38] in order to make the effects of the additional diagrams visible.

Figure 4

Vacuum-normalized regularized quark condensate of the truncated DSE calculation at ${\mu }_{\mathrm{B}}=0$ without meson backcoupling (solid, gray line, taken from Ref. [38]) and with pion (dashed-dotted, black line) as well as pion and sigma (dashed, blue line) backcoupling compared to continuum-extrapolated lattice results [43] (solid, red circles). The parameters for the meson backcoupling data were rescaled as detailed in the text.

Figure 5

QCD phase diagram for the setups explained in the text: no meson contributions to the quark-gluon interaction (black), taking into account only the pion backcoupling (red) and both pion and sigma (blue). We further compare with results from Ref. [6] (gray). Dashed lines correspond to crossover transitions while solid lines represent a first-order spinodal. The big dots show the location of the second-order CEP of the corresponding truncation. The shaded area represents the coexistence region within the physical first-order phase transition takes place.