Dissecting the moat regime at low energies I: Renormalization and the phase structure

TL;DR

The paper tackles the problem of understanding the moat regime in dense QCD-like matter using a two-flavor QM model. It develops a renormalization framework to treat momentum-dependent meson self-energies within RPA, ensuring RG consistency by enforcing $Z^rac_\perp(p=0;T=0,\mu=0)=1$ and employing a vacuum-subtracted MS scheme to avoid large-momentum artifacts. The main findings show that moat behavior at large $T$ is dominated by particle-antiparticle (CA) contributions and that, at finite density, particle-hole (PH) fluctuations drive the moat near the CEP; crucially, in-medium modifications of the Yukawa coupling $h_\pi(T,\mu)$ can suppress CA effects and relocate the moat regime toward larger $\mu$ near the CEP, in qualitative agreement with QCD expectations. Overall, the work clarifies how proper renormalization and in-medium vertex corrections influence the presence and location of the moat regime, with implications for interpreting inhomogeneous or oscillatory phases in low-energy QCD-like models and in related many-body systems.

Abstract

Dense QCD matter can feature a moat regime, where the static energy of mesons is minimal at nonzero momentum. Valuable insights into this regime can be gained using low-energy models. This, however, requires a careful assessment of model artifacts. We therefore study the effects of renormalization and in-medium modifications of quark-meson interaction on the moat regime. To capture the main effects, we use a two-flavor quark-meson model at finite temperature and baryon density in the random phase approximation. We put forward a convenient renormalization scheme to account for the nontrivial momentum dependence of meson self-energies and discuss the role of renormalization conditions for renormalization group consistent results on the moat regime. In addition, we demonstrate and that its extent in the phase diagram critically depends on the interaction of quarks and mesons.

Figures (8)

• Figure 1: The full meson propagator, denoted by the gray dot, as computed in this work. The blue double-lines and the black lines stand for free meson and quark propagators, respectively.
• Figure 2: Spatial pion wave function renormalization at vanishing momentum as function of temperature for different renormalization scales $M=300$, $400$ and $500$ MeV for $\bar{C} = 0$ in Eq. (\ref{['eq:Zpi_re']}). The inset gives the renormalized result using the condition in Eq. (\ref{['eq:Zcond']}).
• Figure 3: Pion self-energy as function of the spatial momentum in vacuum (left) and at $\mu$=400 MeV, T=10 MeV (right) using the conventional $\overline{\rm MS}$ scheme (blue solid line) and the scheme with additional vacuum subtraction defined in Eq. (\ref{['eq:two_point_re2']}) (dark red solid line). In both cases, the renormalization condition in Eq. (\ref{['eq:Zcond']}) is used, rendering the self-energy independent of the renormalization scale $M$. The black dotted line shows the result without renormalization at $M=500$ MeV.
• Figure 4: Spatial pion wave function renormalization $Z_\pi^\perp(0)$ at vanishing momentum as function of temperature for various chemical potentials.
• Figure 5: Spatial pion wave function renormalization $Z_\pi^\perp(0)$ at large $T$, $\mu=0$ (left panel) and $\mu=320$ MeV (right panel). The dark blue solid lines are the full results. The dashed red line is the asymptotic behavior from the analytic large-$T$ limit in Eq. (\ref{['eq:ZpilargeT']}). The light blue solid, peach dot-dashed and brown dotted lines are the vacuum, creation-annihilation (CA) and particle-hole (PH) contributions respectively.