On the $N_c$-insensitivity of QCD$_{2A}$

Abstract

The spectrum of two-dimensional adjoint QCD is surprisingly insensitive to the number of colors $N_c$ of its gauge group. It is argued that the cancellation of finite $N_c$ terms is rather natural and a consequence of the singularity structure of the theory. In short, there are no finite $N_c$ contraction terms, hence there cannot be any finite $N_c$ singular terms, since the former are necessary to guarantee well-behaved principal value integrals. We evaluate and categorize the matrix elements of the theory's light-cone Hamiltonian to show how terms emerging from finite $N_c$ contributions to the anti-commutator cancel against contributions from the purely finite $N_c$ term of the Hamiltonian. The cancellation is not complete; finite terms survive and modify the spectrum, as is known from numerical work. Additionally we show that several parton-number changing finite $N_c$ matrix elements vanish. In particular, there is only one trace-diagonal finite $N_c$ correction. It seems therefore that considering matrix elements rather than individual contributions can provide substantial simplifications when computing the spectrum of a theory with a large symmetry group.