QCD-Like Theories with Different Color Numbers

TL;DR

By varying $N_c$, this work uses the $1/N_c$ expansion to organize QCD-like theories across hadronic, hot, and dense regimes, linking meson/baryon dynamics to phase transitions and EOS behavior. It emphasizes how confinement, chiral dynamics, and many-body forces rearrange in the large-$N_c$ limit, and how quarkyonic matter provides a unified picture of dense QCD where a quark Fermi sea coexists with confining gluodynamics. Two-color QCD and isospin QCD are highlighted as sign-problem-free laboratories that corroborate large-$N_c$ intuition and illuminate the dense-matter EOS, including a robust peak in the sound speed beyond the conformal limit. The synthesis connects ChPT, Skyrmion-like descriptions, and perturbative QCD with lattice results to bridge the gap between $N_c=3$ QCD and theoretical limits, offering avenues to understand the dense QCD landscape and its astrophysical implications.

Abstract

Quantum chromodynamics (QCD) with a general number of colors, $\Nc$, provides a powerful theoretical laboratory to explore the dynamics of non-Abelian gauge theories. Although $\Nc =3$ does not look a large number, the $1/\Nc$ expansion provides us with a very useful classification and book-keeping scheme for hadronic processes and sharpens conceptions otherwise obscured in real-world QCD with $\Nc = 3$. Important applications are dense QCD matter where the first principle methods for QCD are not available and many conceptual issues remain to be clarified. In this chapter we first review hadrons at large $\Nc$ from the viewpoint of quark-gluon dynamics, and then extend the discussions to hot/dense matter, focusing on confinement-deconfinement aspects. We emphasize how the large-$\Nc$ limit provides a unified organizing principle for hadronic and quark degrees of freedom in regimes where first-principle methods are limited. Two-color and isospin QCD, for which lattice simulations at finite density can be performed for a special reason, is reviewed.

Figures (8)

• Figure 1: Double-line representation and large-$N_{\rm c}$ counting rules. (a)--(c) Fundamental interaction vertices. (d) Planar gluon loop giving a factor $N_{\rm c}$ with $g_s^2 N_{\rm c} \sim 1$ fixed. (e) Non-planar contraction suppressed as $g_s^2 \sim 1/N_{\rm c}$.
• Figure 2: Large $N_{\rm c}$ matching between quark-gluon and hadronic graphs. (a) Matching of the meson two-point function, which fixes the state-operator coupling $\langle M | \bar{q}\Gamma q | 0 \rangle \sim N_{\rm c}^{1/2}$. (b) Meson three-point interaction whose vertex scales as $N_{\rm c}^{-1/2}$. (c) Quark-meson coupling determined to be $N_{\rm c}^{-1/2}$. (d) Baryon-meson coupling scaling as $N_{\rm c}^{1/2}$ or $N_{\rm c}^{-1/2}$, depending on whether quark--meson graphs add constructively or destructively.
• Figure 3: Schematic nuclear forces and their large $N_{\rm c}$ implications. (a) Meson-exchange picture of the nucleon-nucleon interaction. (b) Qualitative shape of the nuclear potential at $N_{\rm c}=3$, showing a hard core, intermediate-range attraction, and long-range pion exchange, leading to a shallow bound state. (c) Extrapolation to large $N_{\rm c}$, where the axial coupling $g_A \sim N_{\rm c}$ enhances the long-range pion-exchange interaction, leading to a deeply bound two-nucleon state. As a consequence, nuclear matter is expected to form a crystal at large $N_{\rm c}$.
• Figure 4: Observational constraints on neutron star $M$-$R$ relations and the corresponding inference which suggests that the EOS stiffens rapidly around 2-3$n_0$ ($n_0\simeq 0.16\,{\rm fm}^{-3}$: nuclear saturation density $\simeq$ nucleon density in typical nuclei) and approaches the quark matter behavior with $P \simeq \varepsilon/3$.
• Figure 5: Large-$N_{\rm c}$ scaling of many-body forces in mesonic and baryonic systems. For mesons, the $N$-body vertices scale as $\sim N_{\rm c}^{-N/2}$ and are increasingly suppressed for larger $N$, implying that hot mesonic matter is dominated by two-body interactions. For baryons, in contrast, the $N$-body vertices can remain as large as the two-body vertices, scaling as $\sim N_{\rm c}$. As a result, many-body forces play a crucial role in dense baryonic matter, in sharp contrast to mesonic matter.