QCD on a small circle

TL;DR

This work demonstrates controlled analytic access to QCD-like theories on a tiny circle, revealing a rich spectrum of hadronic bound states in the small-$L$ regime. By combining center-stabilized adiabatic compactification with a 3D non-relativistic EFT for heavy degrees of freedom and a dual-photon light sector, the authors derive glueball, meson, and baryon spectra, including an exponentially growing Hagedorn density of states and nonperturbative energy scales that are iterated exponentials of the inverse coupling. They classify states by center charge and flavor, compute two-body bound-state energies through 2D logarithmic quantum mechanics, and analyze decay processes (radiative and annihilation) within the NR EFT. The results illuminate adiabatic continuity to large circle size, reveal distinctive large-$N$ scaling, and suggest potential lattice tests and implications for QCD thermodynamics and multi-baryon physics. Overall, the small-$L$, center-stabilized framework provides a quantitatively tractable window into confinement, chiral symmetry breaking, and the full hadron spectrum of QCD-like theories.

Abstract

QCD-like theories can be engineered to remain in a confined phase when compactified on an arbitrarily small circle, where their features may be studied quantitatively in a controlled fashion. Previous work has elucidated the generation of a non-perturbative mass gap and the spontaneous breaking of chiral symmetry in this regime. Here, we study the rich spectrum of hadronic states, including glueball, meson, and baryon resonances. We find an exponentially growing Hagedorn density of states, as well as the emergence of non-perturbative energy scales given by iterated exponentials of the inverse Yang-Mills coupling $g^2$.

Figures (9)

• Figure 1: Characteristic length scales in the static potential $V(r)$ for a heavy quark and antiquark separated by a distance $r$ in Yang-Mills theory on $\mathbb{R}^4$ (top), and in adiabatically compactified YM theory on $\mathbb{R}^3 \times S^1_L$ (bottom). On $\mathbb{R}^4$ there is only one intrinsic length scale $\Lambda^{-1}$ which separates the short ($V \sim 1/r$) and long ($V \sim r$) distance regimes. In the small-$L$ regime of adiabatically compactified YM theory on $\mathbb{R}^3 \times S^1_L$, there is a parametrically large intermediate regime, $m_{\rm W}^{-1} \ll r \ll m_\gamma^{-1}$, in which the potential is logarithmic, $V \sim \ln r$. Here $m_{\rm W}^{-1} \sim NL$ and $m_{\gamma}^{-1} \sim NL \, \eta^{-11/6}$ with $\eta \equiv N L \Lambda \ll 1$. (See Eq. (\ref{['eq:dYMphoton_mass']}) below for details.)
• Figure 2: Examples of glueball states when $N \,{=}\, 4$. Filled circles represent the charged $W$-bosons, with larger circles indicating more massive constituents. Lines connecting the constituents indicate attractive logarithmic interactions (of relative strength 1).
• Figure 3: Examples of meson states (with $N \ge 3$). Filled circles represent the charged constituents. Solid lines connecting constituents indicate attractive logarithmic interactions of relative strength 1, dashed lines represent attractive interactions of strength $1{-}\frac{1}{N}$, and dotted lines represent repulsive logarithmic interactions of strength $1/N$.
• Figure 4: Examples of baryon states when $N \,{=}\, 4$. Filled circles represent the charged constituents, with larger circles indicating more massive constituents. Dotted lines represent attractive logarithmic interactions of strength $1/N$, dashed lines represent repulsive interactions of strength $1{-}\frac{1}{N}$, and solid lines show attractive interactions of strength 1. In the single flavor example (left), each quark constituent has a different mass due to their differing Cartan indices. The multi-flavor example (right) shows the special case with $n_{\rm f} \,{=}\, 4$ where all constituents have equal mass.
• Figure 5: Energy spectrum of the radial Schrödinger equation (\ref{['eq:radial']}).