Beta functions of 2D adjoint QCD

We discuss the long-distance physics of 2D adjoint QCD when it is viewed as an effective field theory and determine the $\beta$ functions for its two classically marginal four-Fermi operators. These four-fermion terms preserve the invertible symmetries of the kinetic terms, and they have important implications at long distances if they are generated at short distances. Our results are likely to be important for future numerical lattice Monte Carlo studies of 2D adjoint QCD.

Figure 1

Feynman rules for the action in Eq. (14). Here, $A,B,C,D,E=1,\dots ,N^2-1$ are adjoint color indices, $r$, $s$, $t$, and $u$ are spinor indices, $i$ is a Lorentz index, $d_{ABC}$ are the totally symmetric structure constants of $\mathrm{S}\mathrm{U}\mathrm{(}\mathrm{N}\mathrm{)}$, and summation over repeated indices is implied.

Figure 2

The one-loop Feynman diagrams needed for our calculation of the $\beta$ functions all take the basic form sketched in this diagram.

Figure 3

One-loop renormalization group flow of the four-fermion couplings ${\lambda }_m$ and ${\lambda }_j$ of 2D $N=3$ QCD[Adj], with the arrows pointing toward the long-distance limit. The ${\lambda }_m$ coupling always grows as we flow to the long-distance limit, while the behavior of ${\lambda }_j$ depends on the starting point in parameter space. While the plot suggests that ${\lambda }_j$ flows to zero if ${\lambda }_j\ll {\lambda }_m$, this conclusion is not reliable, because ${\lambda }_m$ becomes large in the same limit, leading to a breakdown of the one-loop approximation.

Figure 4

A tree-level gluon-exchange diagram from the Wilson term in Eq. (19), which leads to an effective ${\lambda }_j\sim r^2{g}^2{a}^2$.