Fierz-complete NJL model study. II. Toward the fixed-point and phase structure of hot and dense two-flavor QCD

Nambu-Jona-Lasinio-type models are often employed as low-energy models for the theory of the strong interaction to analyze its phase structure at finite temperature and quark chemical potential. In particular, at low temperature and large chemical potential, where the application of fully first-principles approaches is currently difficult at best, this class of models still plays a prominent role in guiding our understanding of the dynamics of dense strong-interaction matter. In this work, we consider a Fierz-complete version of the Nambu-Jona-Lasinio model with two massless quark flavors and study its renormalization group flow and fixed-point structure at leading order of the derivative expansion of the effective action. Sum rules for the various four-quark couplings then allow us to monitor the strength of the breaking of the axial $U_{\mathrm{A}}(1)$ symmetry close to and above the phase boundary. We find that the dynamics in the ten-dimensional Fierz-complete space of four-quark couplings can only be reduced to a one-dimensional space associated with the scalar-pseudoscalar coupling in the strict large-$N_{\mathrm{c}}$ limit. Still, the interacting fixed point associated with this one-dimensional subspace appears to govern the dynamics at small quark chemical potential even beyond the large-$N_{\mathrm{c}}$ limit. At large chemical potential, corrections beyond the large-$N_{\mathrm{c}}$ limit become important, and the dynamics is dominated by diquarks, favoring the formation of a chirally symmetric diquark condensate. In this regime, our study suggests that the phase boundary is shifted to higher temperatures when a Fierz-complete set of four-quark interactions is considered.

Figure 1

The two classes of 1PI diagrams contributing to the RG flow of the four-quark couplings.

Figure 2

Phase boundary associated with the spontaneous breakdown of at least one of the fundamental symmetries of our NJL-type model as obtained from a one-channel, two-channel, and Fierz-complete study of the Ansatz (6); see the main text for details.

Figure 3

Renormalized (dimensionful) four-quark couplings as a function of the RG scale $k$ at $T=0$ and $\mu =0$ as obtained from our Fierz-complete study. Note that the Fierz-complete basis is effectively composed of only six channels in the vacuum limit since $\mathcal{C}$ invariance is intact and the Euclidean time direction is not distinguished. In particular, we have ${\overline{\lambda }}_{(V+A{)}_{\parallel }^{\mathrm{adj}}}\equiv 0\equiv {\overline{\lambda }}_{(V-A{)}_{\perp }^{\mathrm{adj}}}$ in this limit.

Figure 4

Scale dependence of the various renormalized (dimensionful) four-quark couplings at $\mu =0$ and $T/k_0\simeq T_{\mathrm{cr}}(\mu =0)/k_0\approx 0.391$ (left panel) as well as at $\mu /k_0\approx 1.1$ and $T/k_0\simeq T_{\mathrm{cr}}(\mu )/k_0\approx 0.196$ (right panel).

Figure 5

Scale dependence of the explicit $U_{\mathrm{A}}(1)$ breaking as measured by the functions $R_1$ (dashed lines) and $R_2$ (solid lines) at $\mu =0$ (left panel) and at $\mu /k_0\approx 1.1$ (right panel) for three values of the temperature for each of the two cases.

Figure 6

Phase boundary associated with the spontaneous breakdown of at least one of the fundamental symmetries of our NJL-type model as obtained from a manifestly $U_{\mathrm{A}}(1)$-symmetric Fierz-complete study of the Ansatz (6) (red lines) and from a Fierz-complete study with broken $U_{\mathrm{A}}(1)$ symmetry (blue lines); see the main text for details.

Figure 7

RG flow of the two-channel approximation at zero temperature and chemical potential in the plane spanned by the scalar-pseudoscalar coupling and the CSC coupling. The black dot represents the Gaussian fixed point, whereas the blue dot represents the real-valued non-Gaussian fixed point; see Eq. (41). The pink dot depicts our choice for the initial condition. The RG trajectory starting at this point describes four-quark couplings diverging at a finite scale $k_0=k_{\mathrm{cr}}$, while approaching a separatrix (red solid line) as indicated by the arrows. The dominance of the scalar-pseudoscalar channel is illustrated by the position of the corresponding separatrix relative to the bisectrix (dashed back line). The different domains separated by the separatrices (red solid lines) are labeled ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, and ${\mathcal{D}}_3$.

Figure 8

RG flow of the two-channel approximation at $T/k=0.4$ and $\mu =0$ in the plane spanned by the scalar-pseudoscalar coupling and the CSC coupling. The black dot represents the Gaussian fixed point. The blue dot represents the real-valued non-Gaussian fixed point. The pink dot depicts our choice for the initial condition. The dashed black line is the bisectrix of the bottom right quadrant. The different domains separated by the separatrices (red solid lines) are labeled ${\mathcal{D}}_2$, and ${\mathcal{D}}_3$; see also Fig. 7. ${\mathcal{D}}_1$ is not shown. The black arrows indicate the shift of the real-valued non-Gaussian fixed point together with the domains ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ when $T/k$ is increased; see the main text for details. The dashed red lines depict the position of the separatrices in the vacuum limit; see also Fig. 7.

Figure 9

RG flow of the two-channel approximation at $T=0$ for different values of $\mu /k$: $\mu /k=0$ (top left panel; same as Fig. 7), $\mu /k=(\mu /k{)}_0\approx 0.298$ (top right panel), $\mu /k=0.4$ (bottom left panel), and $\mu /k=0.9$ (bottom right panel) in the plane spanned by the scalar-pseudoscalar coupling and the CSC coupling. The black dot represents the Gaussian fixed point. Blue dots represent real-valued non-Gaussian fixed points. The pink dot depicts our choice for the initial condition. The dashed black line is the bisectrix of the bottom right quadrant. Different domains ${\mathcal{D}}_i$ are separated by separatrices (red solid lines). The blue arrows in the bottom left panel indicate the shift of the real-valued non-Gaussian fixed points when $\mu /k$ is increased; see the main text for details.