Fluctuation-induced modifications of the phase structure in ($2+1$)-flavor QCD

The low-energy sector of QCD with $N_f=2+1$ dynamical quark flavors at nonvanishing chemical potential and temperature is studied with a nonperturbative functional renormalization group method. The analysis is performed in different truncations in order to explore fluctuation-induced modifications of the quark-meson correlations as well as quark and meson propagators on the chiral phase transition of QCD. Depending on the chosen truncation, significant quantitative implications on the phase transition are found. In the chirally symmetric phase, the quark flavor composition of the pseudoscalar $(\eta ,{\eta }^{\prime })$-meson complex turns out to be drastically sensitive to fluctuation-induced modifications in the presence of the axial $U(1{)}_A$ anomaly. This has important phenomenological consequences for the assignment of chiral partners to these mesons.

Figure 1

The different types of diagrams that contribute to the running of the Yukawa couplings (a), the quark anomalous dimensions (b), and the meson anomalous dimensions (c). The solid lines are the quark propagators, and the dashed lines are the meson propagators. The crossed circle illustrates the insertion of the scale derivative of the regulator. The black dots denote the Yukawa couplings, and the gray dots denote the three-meson interactions. Note that the vertices and propagators are fully dressed here and the diagram with a quark regulator insertion vanishes for the anomalous dimension.

Figure 2

The phase boundaries of the chiral transition in the ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$ and LPA truncations. The crossover transitions correspond to the width of the light $\partial {\overline{\sigma }}_{L,k=0}/\partial T$ at 80% of its peak height. The dots mark the critical end points, and the dotted lines mark the first-order transitions.

Figure 3

The light and strange quark masses in the IR as a function of temperature for the three truncations at $\mu =0$ and $\mu =200\mathrm{MeV}$.

Figure 4

Meson, light, and strange quark wave function renormalizations as a function of the temperature at $\mu =0$ and $\mu =200\mathrm{MeV}$. The strange $Z_{s,k=0}$ is almost temperature independent since it varies only about 0.4% in the shown temperature region.

Figure 5

Light and strange quark Yukawa couplings ${\overline{h}}_{l,0}$ and ${\overline{h}}_{s,0}$ as a function of the temperature at $\mu =0$ and $\mu =200\mathrm{MeV}$.

Figure 6

Meson masses as a function of the temperature at $\mu =0$ in ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$ (red lines) in comparison to the LPA result (thinner gray lines). Note, in ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$, the $a_0$ meson degenerates with the $\eta$ meson, while in LPA, the $a_0$ meson and ${\eta }^{\prime }$ meson degenerate.

Figure 7

The masses of $\eta$, ${\eta }^{\prime }$, and $a_0$ meson at large temperatures. While the $a_0$ meson (solid lines) degenerates with the ${\eta }^{\prime }$ meson in LPA (thin dotted-dashed gray line), it degenerates with the $\eta$ meson in ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$ (dashed red dark line).

Figure 8

The scalar and pseudoscalar mixing angles ${\phi }_{p,s}$ as a function of temperature at vanishing density in ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$ (red lines) in comparison to the LPA results (gray lines). The $\eta$ meson becomes a purely light state at large temperatures, and the ${\eta }^{\prime }$ meson is a purely strange state.

Figure 9

Temperature dependence of the quartic meson coupling, which enters the pseudoscalar mixing angle, at vanishing density in ${\mathrm{LPA}}^{\prime }+\mathrm{Y}$ (solid red line) and in LPA (dashed gray line).