Solving 2D QCD with an adjoint fermion analytically

TL;DR

This work introduces an analytic framework for 1+1D QCD with an adjoint Majorana fermion by building a basis from conformal quasi-primary operators of the UV free-fermion CFT. By truncating at $Δ_{max}$, the authors demonstrate effective conformal dominance: high-dimension operators decouple from the low-energy spectrum with errors of order $e^{- riangle_{max}}$, enabling accurate spectra without large DLCQ baselines. They construct and diagonalize the mass-squared matrix $M^2$ in this basis at $Δ_{max}=9.5$ (810 states) and reproduce the six lowest single-particle states and several two-particle thresholds, in good agreement with DLCQ results. The approach provides insight into how holography-inspired dominance can persist in non-holographic theories and offers a potentially generalizable scheme for solving strongly coupled gauge theories, with clear implications for higher dimensions and holographic modeling.

Abstract

We present an analytic approach to solving 1+1 dimensional QCD with an adjoint Majorana fermion. In the UV this theory is described by a trivial CFT containing free fermions. The quasi-primary operators of this CFT lead to a discrete basis of states which is useful for diagonalizing the Hamiltonian of the full strongly interacting theory. Working at large-$N$, we find that the decoupling of high scaling-dimension quasi-primary operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. This suggests a scheme, whereby, truncating the basis to operators of dimension below $Δ_{max}$, one can calculate the low-energy spectrum, parametrically to an accuracy of $e^{-Δ_{max}}$ (although the precise accuracy depends on the state). Choosing $Δ_{max} =9.5$ we find very good agreement with the known spectrum obtained earlier by numerical DLCQ methods. Specifically, below the first three-particle threshold, we are able to identify all six single-particle bound-states, as well as several two-particle thresholds.

Figures (5)

• Figure 1: The single particle spectrum of adjoint QCD$_{2A}$.
• Figure 2: The convergence of the spectrum of the low-lying single-particle-states. Here $\Delta_{max}$ is the dimension of the highest quasi-primary operator used to generate a truncated Hilbert space. In the bosonic sector we calculated the spectrum up to $\Delta_{max}=9$, whereas in the fermionic sector the largest $\Delta_{max}$ is equal to 9.5. The second plot demonstrates more clearly the degree of convergence of the single-particle states. All states but the highest one appear to have a similar rate of convergence. The asymptotic masses are taken to be the values at the highest $\Delta_{max}$ calculated. The spectrum appears to converge to the asymptotic values parametrically as $e^{-\Delta_{max}}$.
• Figure 3: The weight in dimension for the six single particle states. In the left and the right column are the bosonic states and the fermionic states, respectively. The color code for each state is the same as that in Fig. \ref{['fig:singleparticle-invdelta-plot']}.
• Figure 4: The convergence of the free two-particle spectrum, as a function of $1/n_{max}$ to the continuum. In the truncated basis, $n_{max}$ is the largest degree of Jacobi polynomials $P_n^{(a,b)}$ used, corresponding to quasi-primary operators below a certain maximum dimension. The construction of the operator basis can be found in Appendix \ref{['freetwopartbasis']}. The red, purple, green and blue lines plot the expected spectra of free two-particle states $F_1\otimes F_1$, $F_1\otimes F_2$, $F_1\otimes B_1$ and $B_1\otimes B_1$, respectively.
• Figure 5: The spectra of the multi-particle states in the four sectors with a given $T$-parity and statistics. They are compared with the free two-particle spectra of states $F_1\otimes F_1$ (red), $F_1\otimes F_2$ (purple), $F_1\otimes B_1$ (green) and $B_1\otimes B_1$ (blue). The $F_1\otimes B_1$ spectrum is not included in the bosonic sectors because there is no obvious counterpart of this state in the QCD$_{2A}$ spectrum.