Faddeev description of baryons in two-dimensional QCD with $N_c=3$. I. Chiral spectrum, isospin, and strangeness

Technical Summary: Faddeev Description of Baryons in Two-Dimensional QCD with $N_c = 3$

Problem Statement
The relativistic three-body problem remains a central non-perturbative challenge in quantum field theory. In four-dimensional QCD, baryons are strongly correlated three-quark states, making their description difficult. Two-dimensional QCD ($QCD_2$) retains linear confinement while simplifying dynamical content, serving as an ideal laboratory for testing non-perturbative many-body methods. Despite previous successes in meson spectroscopy (e.g., the 't Hooft model) and baryon studies via Discretized Light-Cone Quantization (DLCQ) and bosonization, a complete, exact light-front Faddeev formulation for baryons at finite $N_c=3$—specifically one that clarifies the relationship between the full three-body problem and the widely used quark–diquark approximation—has been lacking.

Methodology
The authors develop an exact light-front Faddeev decomposition for baryons in $QCD_2$ at $N_c=3$, restricted to the valence three-quark sector. The approach proceeds through several stages:

1. Faddeev Decomposition: The three-body baryon equation is reformulated on the light front. For identical quarks, the wavefunction is decomposed into three components ($\Psi_{ij}$) related by cyclic permutations. This reduces the problem to a single exact equation for a component $\Phi$, with the full wavefunction reconstructed via permutation operators.
2. Kernel Specification: The light-front Coulomb kernel, appropriate for $QCD_2$, is defined in terms of pair variables (intrinsic fraction $z$ and spectator fraction $s$). The kernel coefficient is fixed by matching to the equal-time string tension.
3. Exact Coupled-Channel Reduction: By diagonalizing the intrinsic pair Hamiltonian (the 't Hooft meson Hamiltonian), the authors derive an exact coupled-channel representation. This expansion expresses the Faddeev component as a sum over intrinsic pair eigenmodes ($\phi_n$), transforming the integral equation into a system of coupled equations for the spectator functions $\chi_n(s)$.
4. Variational Solution: Instead of integrating the coupled equations directly, the authors solve the equivalent variational problem for the full symmetric wavefunction on the physical simplex ($x_1+x_2+x_3=1$). They employ a basis of elementary symmetric polynomials, orthogonalized via the Löwdin method, to ensure the massless chiral mode is retained and the kernel quadratic form is manifestly self-adjoint.
5. Flavor Extension: The framework is extended to include flavor degrees of freedom. This involves enforcing Fermi statistics on the combined flavor-spatial wavefunction, allowing for the study of isospin (two degenerate flavors) and strangeness (adding a massive strange quark).

Key Contributions and Results

• Chiral Spectrum and the Massless Baryon: In the chiral limit (massless quarks), the calculation yields an exactly massless lightest baryon ($M^2=0$). This state is not imposed but emerges variationally as the constant mode ($\phi_0 = 1$). The excitation tower reproduces Webber's permutation-symmetric states state-by-state, establishing a valence gap of $1.371 M^*_{mes}$ (where $M^*_{mes}$ is the lightest massive 't Hooft meson).
• Channel Content and the Quark–Diquark Approximation: A central structural result is the analysis of channel content. The massless baryon is a pure single-channel state ($n=0$, the constant mode). In contrast, all excited (gapped) states are genuinely multi-channel, with significant weights in higher pair modes (e.g., the first gapped state has $w_0 \approx 0.55$, $w_2 \approx 0.45$). The authors prove that a single-channel truncation (the quark–diquark approximation) possesses no massive chiral solutions; it can only describe the massless state. Thus, the quark–diquark picture fails to describe the excited baryon spectrum in this framework.
• Regge Trajectories: The baryon trajectories are classified. The "leading" trajectory (starting from the massless state) has a slope distinct from the meson trajectory, while "daughter" and higher trajectories run parallel to the meson trajectory with slopes within 1–5% of the meson slope.
• Isospin and Flavor Structure:
  • Two Flavors: The massless baryon is identified as the $I=3/2$ ($\Delta$-like) quartet. The $I=1/2$ ($N$-like) doublet is gapped ($M_N \approx 0.977 M^*_{mes}$) and is an exact 50:50 superposition of "good" ($I=0$) and "bad" ($I=1$) diquark channels.
  • Strangeness: Introducing a massive strange quark yields hyperon spectroscopy. A key finding is the inversion of the $\Sigma$–$\Lambda$ ordering: $\Sigma$-like states fall below $\Lambda$-like states, contrary to four-dimensional expectations. This is attributed to the internal structure of the diquarks: the $\Lambda$'s "good" diquark carries the internal energy of the lightest massive 't Hooft mode, while the $\Sigma$'s "bad" diquark is internally massless.
  • Mass Relations: The chiral thresholds for hyperons follow a boundary-density law, where the mass squared of a state with $k$ strange quarks approaches $2k$ times the kaon mass squared in the chiral limit. The Gell-Mann–Okubo relation is found to be protected in mass squared, holding to within 0.2% for small strange masses.
• Isospin Breaking: With non-degenerate light quark masses ($m_u \neq m_d$), the proton-like state falls below the neutron-like one. The $\Sigma^0$–$\Lambda$ mixing appears, and the Coleman–Glashow relation holds at the percent level.
• Quantitative Agreement: The results are anchored at both ends of the coupling range. The meson-to-baryon mass ratio agrees with strong-coupling bosonization predictions to within 0.3–4.3%. At weak coupling, the valence masses reproduce full DLCQ results to within 0.04% (mesons) and 0.15% (baryons).

Significance and Claims
The paper claims to provide the first exact light-front Faddeev description of baryons in $QCD_2$ at $N_c=3$ that resolves the three-body problem into exact coupled channels. Its primary significance lies in:

1. Structural Insight: Demonstrating that the quark–diquark picture is a single-channel truncation that is exact only for the massless chiral baryon and fundamentally inadequate for describing the excited spectrum, which requires coherent multi-channel mixing.
2. Spectroscopic Precision: Providing a high-precision benchmark for the valence sector that matches existing DLCQ and bosonization results, while offering explicit wavefunctions and channel weights previously unavailable.
3. Flavor Dynamics: Establishing the hyperon spectroscopy of $QCD_2$ at the valence level for the first time, revealing the inversion of the $\Sigma$–$\Lambda$ ordering and the specific mechanism (diquark internal energy) responsible for it.
4. Methodological Utility: The framework serves as a benchmark for Hamiltonian, tensor-network, and quantum-simulation studies of $QCD_2$, offering a controlled variational description of the interacting valence sector.

The authors emphasize that while the excited spectrum is a variational upper bound subject to corrections from higher Fock sectors, the massless chiral baryon is exact and protected by unitarity, surviving in the full theory. The work does not propose new experimental applications but rather solidifies the theoretical understanding of baryon structure in a solvable relativistic setting.