Nonperturbative fluctuation effects of charged bosonic fields: A quark-diquark model study at nonzero density

TL;DR

The paper develops a nonperturbative functional renormalization group analysis of a two-flavor quark-diquark model at nonzero density, explicitly incorporating full field dependence and charged bosonic fluctuations under Silver-Blaze considerations. It introduces several truncation schemes, with the local potential approximation (LPA) at leading order in the derivative expansion, particularly the LPA$_2$ variant, proving robust for capturing diquark condensation and the interplay between quark- and diquark-induced singularities. The main result is a phase diagram featuring a low-temperature first-order transition to a color-superconducting phase that ends at a tricritical point near $(\mu,T)\approx(188\,\mathrm{MeV}, 2.6\,\mathrm{MeV})$, while a second-order transition persists at higher temperatures; extensions to a relativistic Bose gas and a quark-meson-diquark model demonstrate the framework’s versatility and consistency with lattice data in applicable regimes. Overall, the work provides a nonperturbative, gauge-aware approach to dense, strongly interacting matter and highlights the critical role of bosonic fluctuations in color superconductivity, laying groundwork for more realistic multi-flavor studies.

Abstract

We study the renormalization group flow of the scale-dependent effective potential of a quark-diquark model with full field dependence at nonzero chemical potential. This includes a discussion of approximations in relation to complex bosonic fields and the Silver-Blaze property. The resulting flow equation for the scale-dependent effective potential can in principle be solved down to the infrared limit. For our quark-diquark model, which may serve as a low-energy model for dense strong-interaction matter, we find that a competition between the Bardeen-Cooper-Schrieffer singularity and bosonic fluctuations can trigger a first-order phase transition at low temperatures that turns into a second-order phase transition at a tricritical point as the temperature increases.

Figures (7)

• Figure 1: Derivative of the scale-dependent effective potential in as a function of $\bar{\Delta}$ for various scales for $\mu = 0.5 \,\mathrm{GeV}$ and two values of the temperature, $T = 0$ (solid lines) and $T = 5 \,\mathrm{MeV}$ (dashed lines).
• Figure 2: Diquark propagator functions $G_1$, see \ref{['eq:bosonic-propagator']}, and $G_2$, see \ref{['eq:bosonic-propagator-2']}, for $(F \mu)/k = 1$ and various temperatures with $F=2$. Note that, at $T=0$, the propagator function $G_1$ remains finite in the limit $E/k\to 0$, in contrast to the propagator function $G_2$.
• Figure 3: Phase diagram of the for as defined by the truncation ${\mathds{T}}^{\mathrm{LPA}_2}$. In the left panel, the contour lines are associated with the size of the condensate in (solid lines) and (dashed lines), respectively. The right-hand panel shows a zoom into the first-order region depicted in the left panel. This region opens up at low temperatures and moderate chemical potentials. The red line depicts the first-order phase transition. The red dot is the tricritical point located at $(\mu,T)\approx (188.2\,\mathrm{MeV}, 2.6\,\mathrm{MeV})$. In the shaded area, we find indications for the formation of a scale-dependent effective potential of the type associated with a first-order phase transition but the symmetry is eventually restored in the flow, see, e.g., \ref{['fig:LPA-V1-first-order-flow-mu-180']}.
• Figure 4: Derivative of the scale-dependent effective potential for various scales (see left panel) at $T = 1 \,\mathrm{MeV}$ for $\mu=190\,\mathrm{MeV}$ (left panel, symmetric phase) and $\mu=210\,\mathrm{MeV}$ (right panel, phase associated with ).
• Figure 5: Derivative of the effective potential at $T = 1 \,\mathrm{MeV}$ for various values of $\mu$ near the first-order phase transition, see \ref{['fig:LPA-V1-phase-diagram-contour']}.