Fierz-complete four-quark interactions and the QCD phase diagram

Technical Summary: Fierz-complete four-quark interactions and the QCD phase diagram

Problem and Motivation
The location and existence of the critical end point (CEP) in the Quantum Chromodynamics (QCD) phase diagram remain open questions, particularly at high baryon chemical potential where lattice QCD simulations are hindered by the sign problem. Previous functional renormalization group (fRG) studies of QCD at finite temperature and density, such as the work by Fu et al. [22], typically employed a truncation that considered only a single scalar-pseudoscalar channel for four-quark interactions. While this channel dominates in the vacuum, there is a theoretical expectation that other tensor structures (e.g., vector or diquark channels) may become increasingly relevant as the baryon chemical potential increases. This paper investigates the dynamics of Fierz-complete four-quark interactions to address the ambiguity arising from projecting different tensor structures and to quantify their influence on the phase boundary and the location of the CEP.

Methodology
The authors employ the functional renormalization group (fRG) approach to QCD at finite temperature and densities. The core methodological advancement is the extension of the truncation scheme from a single scalar-pseudoscalar channel to a Fierz-complete basis for the light ($u, d$) quarks. This basis includes ten distinct tensor channels:
$$F = \{ \sigma, \pi, a, \eta, (V \pm A), (V - A)_{\text{adj}}, (S \pm P)_{\text{adj}}^-, (S + P)_{\text{adj}}^+ \}$$
The study utilizes the technique of dynamical hadronization. In this framework, the resonant dynamics of the scalar ($\sigma$) and pseudoscalar ($\pi$) channels are transferred from the four-quark sector to the quark-meson sector. Consequently, the four-quark couplings in the $\sigma$ and $\pi$ channels are set to zero at all RG scales, replaced by the exchange of $\sigma$ and $\pi$ mesons mediated by Yukawa couplings ($h_\sigma, h_\pi$). Crucially, unlike previous approximations where $h_\sigma = h_\pi$, this work computes the flows of these couplings separately.

The flow equations for the effective action include contributions from gluon, ghost, quark, and meson propagators. The four-quark vertex flows are analyzed by sorting diagrams into three classes based on energy scales: high-energy quark-gluon interactions, low-energy meson-mediated interactions, and mixing diagrams. The authors calculate the running of four-quark couplings, Yukawa couplings, and strong couplings across the phase diagram, specifically examining four representative points: the vacuum, a high-temperature point, and two points near the phase boundary (one near the CEP).

Key Results

1. Channel Dominance: In the vacuum ($T=0, \mu_B=0$), the pion and sigma channels play an overwhelmingly dominant role in the four-quark dynamics, rendering all other channels negligible.
2. CEP Proximity: As the system approaches the critical end point, the magnitude of four-quark couplings in the sub-dominant channels (specifically $\eta, a, (S+P)_{\text{adj}}^-, (S-P)_{\text{adj}}^-, (V-A), (V-A)_{\text{adj}}$) increases sizably, reaching approximately 10% of the magnitude of the pion and sigma channels. This indicates that these channels become non-negligible near the CEP.
3. Phase Boundary Curvature: The inclusion of Fierz-complete interactions results in a slight increase in the curvature of the chiral crossover phase boundary. The quadratic coefficient is calculated as $\kappa_2 = 0.0151(1)$, which is marginally larger than the value $\kappa_2 = 0.0142(2)$ obtained in the single-channel study [22].
4. CEP Location: The calculated location of the CEP is $(T_{\text{CEP}}, \mu_{B,\text{CEP}}) = (102, 647)$ MeV. Compared to the single-channel result of $(107, 635)$ MeV, the inclusion of complete four-quark interactions shifts the CEP to a slightly lower temperature and a larger baryon chemical potential.
5. Coupling Dynamics: The study confirms that the Yukawa couplings for $\sigma$ and $\pi$ distinguish themselves in the infrared, and their behavior near the CEP is influenced by the massless nature of the sigma mode. The strong couplings for light and strange quarks deviate from the perturbative regime at lower scales ($k \lesssim 3 \sim 7$ GeV) due to the emergence of a gluon mass gap.

Significance and Claims
The authors claim that this work constitutes a systematic analysis of the fRG approach to first-principles QCD at finite temperature and density. By extending the truncation to a Fierz-complete basis, the study removes technical ambiguities associated with projecting tensor structures and provides a more robust assessment of the QCD phase diagram.

The paper modestly concludes that while the dynamics of Fierz-complete four-quark interactions do alter the phase boundary curvature and the CEP location, these changes are relatively small and remain within the error margins arising from the truncations inherent to the fRG approach, particularly in the regime of large baryon chemical potential ($\mu_B/T \gtrsim 4$). The primary contribution is the demonstration that while the scalar-pseudoscalar channel dominates in the vacuum, the inclusion of other channels is necessary for a complete description near the critical end point, even if their quantitative impact on the CEP location is moderate.