Phenomenology of quarkyonic percolation at FAIR

Abstract: We will give an introduction to the concept of quarkyonic matter, presenting an overview of what is meant by this term in the literature. We will then argue that the quarkyonic phase, as defined in the original paper, is a percolation-type phase transition whose phase transition line is strongly curved in $ρ_B-N_c$ space, where $N_c$ is the number of colors and $ρ_B$ the baryon density. With a toy model estimate, we show that it might be possible to obtain a percolating but confined phase at $N_c=3,N_f=2$ at densities larger than one baryon per one baryon size. We conclude by discussing how this phase can be observed at FAIR.

• The paper introduces a percolation mechanism for quarkyonic matter, showing that quark wavefunction delocalization leads to a confined but percolating phase.
• It employs percolation theory to map the phase boundary in the ρB-Nc diagram, contrasting large-Nc scaling with physical QCD (Nc=3, Nf=2).
• It predicts unique experimental signatures in dilepton spectral functions, suggesting observable dips corresponding to density-dependent momentum scales.

Phenomenology of Quarkyonic Percolation at FAIR

Conceptual Framework and Motivation

The paper develops a phenomenological approach to the concept of "quarkyonic matter," a phase in QCD proposed for baryon-rich systems at intermediate temperatures ($T < T_c$) but high quark chemical potentials ($\mu_Q \sim \Lambda_{QCD}$). In this regime, the baryon density exceeds one baryon per baryon volume, implying close packing of nucleons. The exploration of this region is motivated by the theoretical inaccessibility of dense QCD via standard expansions: effective hadron models become unreliable, and lattice QCD is hindered by the sign problem at large chemical potential.

The essential notion of quarkyonic matter, as introduced by McLerran and Pisarski (2007), is that confinement survives at densities well above nuclear matter, but the Fermi surface is populated densely enough for perturbative QCD to ostensibly operate in partonic degrees of freedom. This led to speculation about exotic QCD regimes where features such as confinement, chiral symmetry restoration, and percolation of quark wavefunctions coexist.

Percolation Transition as the Mechanism

A central claim of the work is that the quarkyonic phase boundary corresponds, not to deconfinement as classically defined, but to a percolation transition of quark wavefunctions in densely packed baryonic matter. This is formalized using percolation theory, considering the spread of quark wavefunctions across the system as baryon density and the number of colors ($N_c$) vary. The phase boundary in the $\rho_B-N_c$ diagram is strongly curved: for small $N_c$, deconfinement precedes percolation, precluding a true quarkyonic phase, while for large $N_c$, percolation can occur in the confined phase, yielding an extended region where quarkyonic matter exists.

The model constructs an effective interaction probability between quarks in different baryons, scaling as $\lambda/N_c$, with a range set by the baryon size. The percolation threshold is determined by finding for which $(N_c,\, \rho_B)$ a cluster of connected quark states spans the system, i.e., where the quark wavefunctions delocalize macroscopically even as baryons retain identity. The analysis shows that with $N_f = 2$ and $N_c = 3$, a percolating but confined phase is possible for baryon density above normal nuclear matter, although the region is predicted to be parametrically smaller than in the large-$N_c$ limit.

Large-$N_c$ Scaling and Real-World QCD

The large-$N_c$ expansion is extensively used in QCD phenomenology. In large $N_c$, features such as weakly interacting mesons, classical Skyrmion-like baryons, and the first-order deconfinement transition are obtained. However, real-world QCD with $N_c = 3$ deviates qualitatively: for example, the deconfinement transition is a crossover at physical quark masses, and nuclear matter has a binding energy much smaller than the baryon mass.

This analysis incorporates those subtleties and emphasizes that while the large-$N_c$ paradigm predicts a wide quarkyonic region, at physical $N_c$, the percolation and deconfinement curves cross differently, potentially restricting quarkyonic matter. However, the percolation-based calculation yields a nonzero quarkyonic region even at $N_c = 3$, especially for $N_f = 2$. The model highlights that, for strange quark masses of order $\Lambda_{QCD}$, $N_f = 2$ is more appropriate when considering the number of "active" quark flavors in this density regime.

Experimental Signatures and Theoretical Implications

The work discusses observable consequences, proposing that the percolating quarkyonic phase would manifest in distinctly non-hadronic spectral functions for probe processes. In particular, the dilepton production rate in the medium,

$$\Gamma(M) = \int \mathcal{M}((p_1-p_2)^2) f(p_1) f(p_2) d^3p_1 d^3p_2 \delta((p_1^\mu-p_2^\mu)^2 - M^2),$$

is expected to reflect the nontrivial structure of quark wavefunctions in the mean baryonic field. Unlike hadron gases, which show resonance peaks, or deconfined QGPs, which yield rather flat spectral functions, quarkyonic matter should display dips at invariant masses corresponding to the density-dependent momentum scale where delocalization sets in. The paper argues that these distinctive patterns could be observable via dilepton yields in heavy-ion collisions, such as those studied at the FAIR facility.

The approach also sets the stage for quantitative analyses by integrating semiclassical baryon dynamics (e.g., as calculated in UrQMD) with Dirac equation solutions for quark modes in realistic baryon configurations and folding these with pQCD matrix elements. Such work is proposed as ongoing and necessary to establish robust experimental discriminants.

Broader Impact and Future Directions

This percolation framework offers an alternative to the conventional chiral/deconfinement paradigm for QCD phase structure at high density. It bridges perturbative and nonperturbative regimes, suggesting a concrete mechanism (i.e., quark wavefunction percolation) for the onset of quarkyonic matter. The model advances the theoretical understanding of the interplay between confinement, compositeness, and many-body dynamics at high baryon density, and generates testable predictions for collider programs at intermediate energies.

Notably, the analysis emphasizes the sensitivity to the number of colors and flavors, indicating that the real-world relevance of the quarkyonic phase at $N_c = 3$, $N_f = 2$ is not excluded, though it is constrained.

Conclusion

The paper delivers an in-depth exploration of quarkyonic percolation as the underlying mechanism for a novel phase of dense QCD matter, substantiated within a controlled toy model. It asserts that a percolating but confined phase is plausible in QCD at physical values of $N_c$ and $N_f$, and this region may be probed experimentally via dilepton spectral function analysis at FAIR. The theoretical approach provides a model-based route for addressing the baryon-rich sector of the QCD phase diagram while highlighting the need for further dynamical modeling and experimental verification.