On the Spectrum of QCD(1+1) with SU(N_c) Currents

TL;DR

This paper advances the understanding of QCD$_2$ with SU($N_c$) currents by deriving a DLCQ-based Hamiltonian $P^-$ that depends on the harmonic resolution $K$ and the ratio $\lambda=N_f/N_c$, enabling a detailed study of the fermionic spectrum across the Veneziano limit. By recovering the 't Hooft and large-$N_f$ limits and classifying states by their single-particle content, the work reveals that fermionic momentum becomes $\lambda$-dependent in the discrete formulation and provides evidence for a two-Regge-trajectory structure in adjoint QCD$_2$, albeit with an imperfect vacuum and unresolved complete single-particle identification. The results validate the universality of the massive spectrum in two dimensions and offer a framework to interpret multi-particle states and their couplings at finite $K$, with implications for extrapolations toward the continuum and for understanding adjoint theories. Overall, the paper sets up a practical, current-based approach to disentangle single- and multi-particle states in 1+1 dimensional QCD and motivates future work to achieve a complete adjoint-spectrum solution and to connect with continuum formulations.

Abstract

Extending previous work, we calculate in this note the fermionic spectrum of two-dimensional QCD (QCD_2) in the formulation with SU(N_c) currents. Together with the results in the bosonic sector this allows to address the as yet unresolved task of finding the single-particle states of this theory as a function of the ratio of the numbers of flavors and colors, λ=N_f/N_c, anew. We construct the Hamiltonian matrix in DLCQ formulation as an algebraic function of the harmonic resolution K and the continuous parameter λ. Amongst the more surprising findings in the fermionic sector chiefly considered here is that the fermion momentum is a function of λ. This dependence is necessary in order to reproduce the well-known 't Hooft and large N_f spectra. Remarkably, those spectra have the same single-particle content as the ones in the bosonic sectors. The twist here is the dramatically different sizes of the Fock bases in the two sectors, which makes it possible to interpret in principle all states of the discrete approach. The hope is that some of this insight carries over into the continuum. We also present some new findings concerning the single-particle spectrum of the adjoint theory.

Figures (6)

• Figure 1: Fermionic spectrum in the large $N_f$ limit. Left: (a) $Z_2$ even sector. Right: (b) $Z_2$ odd sector. Solid (dash-dotted) lines connect associated single(multi)-particle eigenvalues at different $K$. Dotted lines connect analytically calculable eigenvalues. Dashed lines are extrapolations to the continuum limit. Note that masses are in units $g^2N_f/\pi$.
• Figure 2: Fermionic spectra of the 't Hooft limit in the $Z_2$ even (a) and odd (b) sectors. Solid (dash-dotted) lines connect associated single(multi)-particle eigenvalues at different $K$. Dotted lines connect analytically calculable eigenvalues. Dashed lines are extrapolations to the continuum limit. Masses are in units $g^2N_c/\pi$.
• Figure 3: Fermionic spectra of the theory with adjoint fermions in the $Z_2$ even (a) and odd (b) sectors. Solid (dash-dotted) lines connect conjectured single(multi)-particle eigenvalues at different $K$. Dotted lines connect analytically calculable eigenvalues. Dashed lines are extrapolations to the continuum limit. Masses are in units $g^2N_c/\pi$.
• Figure 4: The spectrum of two-dimensional QCD in all sectors of the theory as a function of $\lambda$. Plotted are the lowest 100 eigenvalues in the fermionic sectors (top row) and the bosonic sectors (bottom row). Left column: $Z_2$ even sectors, reduced eigenvalues $\hat{M}^2\equiv M^2/(1+\lambda)$ vs. $\lg\lambda$. Right column: $Z_2$ odd sectors, actual eigenvalues vs. $\lambda$.
• Figure 5: The wavefunctions of four adjoint states at $K=25/2$: (a) $M^2=5.6021$, (b) $M^2=16.4261$, (c) $M^2=21.8688$, (d) $M^2=32.6060$ [from bottom to top]. Plotted are the amplitudes $\psi_n$ vs. $\bar{n}=n/2^{K-3/2}$. The third state is the only multi-particle state in this plot and its eigenfunction has clearly a different shape. The number of currents in a basis state changes at the dashed lines.