Solving 2D QCD with an adjoint fermion analytically

TL;DR

This paper introduces an analytical method for solving 1+1 dimensional QCD with an adjoint fermion, using a basis of quasi-primary operators from a free fermion CFT to accurately compute the low-energy spectrum at large N.

Contribution

The authors develop an exponential truncation scheme based on operator scaling dimensions, enabling analytical calculation of the low-energy spectrum of adjoint QCD in 2D, matching previous numerical results.

Findings

Accurate low-energy spectrum obtained with operator dimension cutoff

Identification of six single-particle bound states below three-particle threshold

Exponential convergence of spectrum calculations with increasing cutoff

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 $\Delta_{max}$, one can calculate the low-energy spectrum, parametrically to an accuracy of $e^{-\Delta_{max}}$ (although the precise accuracy depends on the state). Choosing $\Delta_{max} =9.5$ 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.