A solution of 2D QCD at Finite $N$ using a conformal basis

TL;DR

This work addresses the low-energy spectrum of 2D QCD with a fundamental fermion at finite number of colors $N$ using a conformal-basis method. By constructing a basis from conformal quasi-primary operators and diagonalizing the truncated mass matrix $M^2=2P^+P^-$, the authors demonstrate that effective conformal dominance persists at finite $N$, with high-dimension operators decoupling exponentially and the light states becoming accurately characterized. They show that stable single-particle states have analytic wavefunctions and parton distributions even for $N=3$, the massless sector is captured by a free scalar CFT with a single non-interacting state, and multi-particle states below thresholds align with a non-interacting bosonic description. The approach provides non-perturbative, translation-invariant access to the spectrum without lattice discretization and offers a promising path to extend these results to more complex theories and higher dimensions.

Abstract

We study 2D QCD with a fundamental fermion at small-$N$ using the recently proposed conformal basis approach. We find that effective conformal dominance still holds, namely that the spectrum converges efficiently, with high scaling-dimension operators decoupling exponentially quickly from the stable single-particle states. Consequently, for these stable bound states, accurate, analytic expressions for wavefunctions and parton distribution functions can be given, even for $N=3$.

Figures (5)

• Figure 1: The spectra of the lowest mesonic states at large $N$$(N=1000)$ and small $N$$(N=3 \ \text{solid lines}, \ N= 6 \ \text{dashed lines})$, calculated up to $\Delta_\text{max}=9$. The unit of $M^2$ is $g^2/2\pi$. The lowest three states in all cases are decoupled from the high dimensional operators. The "Regge trajectory" of the spectrum at large $N$ can be seen in the first plot.
• Figure 2: Plots of the weight of states as a function of operator dimension $\Delta$, showing the decoupling of the lowest five single-particle-states. The circles connected by solid lines correspond to the $N=1000$ case, the diamond points - dotted lines to $N=6$ and the square points - dashed lines to $N=3$. We did not include points for which the weight is smaller than $10^{-4}$.
• Figure 3: Matching of the spectrum of multi-particle-states, $|B_1\rangle\otimes n_1 |B_0\rangle$, $|B_2\rangle\otimes n_2 |B_0\rangle$ and $|B_3\rangle\otimes n_3 |B_0\rangle$ ($n_i\geqslant 1$, $n_i\in \mathbb Z$), with the non-interacting multi-meson spectrum, at $N=1000$. States are shown below the threshold of $4M^2_{B_1}\sim 45$. The black dots show the spectrum of the finite $N$ 't Hooft model, which is obtained from a fermonic operator basis, whereas the colored dots indicate the mass eigenvalues obtained from diagonalizing a free Hamiltonian using a bosonic basis. The charge-conjugate $C$-even sector is shown in the first plot (blue: $|B_1\rangle \otimes n_1 |B_0\rangle$, red: $|B_2\rangle \otimes n_2 |B_0\rangle$, gray: $|B_3\rangle\otimes n_3 |B_0\rangle$). The second plot is for the $C$-odd sector (green: $|B_1\rangle\otimes n_1 |B_0\rangle$, orange: $|B_2\rangle\otimes n_2 |B_0\rangle$, purple: $|B_3\rangle\otimes n_3 |B_0\rangle$).
• Figure 4: Matching of the spectrum of multi-particle-states, $|B_1\rangle\otimes n_1 |B_0\rangle$, $|B_2\rangle \otimes n_2 |B_0\rangle$ and $|B_3\rangle \otimes n_3 |B_0\rangle$ ($n_i\geqslant 1$, $n_i\in \mathbb Z$), with the non-interacting multi-meson spectrum, at N=3. States are shown below the threshold of $4M^2_{B_1}\sim 60$. The color code is similar to that of the case of $N=1000$ (Figure \ref{['fig:multiparticle-convergence-N1000']}).
• Figure 5: Density of states of the two-body state of $|B_1\rangle$ and $|B_0\rangle$ at a high cutoff dimension $\Delta_{\text{max}}=200$. The histogram shows the counting of states with bin width $\Delta M^2=5$. It matches with the expected density of states of the two-body continuous spectrum (red line).