Partial breakdown of center symmetry in large-N QCD with adjoint Wilson fermions
Barak Bringoltz
TL;DR
This paper analyzes the one-loop effective potential for large-$N$ QCD with adjoint Wilson fermions on a lattice that discretizes $R^3\times S^1$ with an anisotropic lattice spacing ratio $\xi=a_s/a_t$. By expressing the potential as $V(\Omega)=2\sum_{r=1}^\infty V_r\,|\mathrm{tr}\,\Omega^r|^2+{\rm const}$ and deriving $V_r$ from lattice dispersion relations, it shows that for $0<\xi<2$ the ground state is $Z_N$ symmetric at $m=0$, and as $m$ grows a cascade to smaller center subgroups $Z_K$ occurs, with $K$ depending on $m$, $\xi$, and $N_f$. The results reproduce the continuum behavior of center-symmetry breaking found by Kovtun-Ünsal-Yaffe, Myers-Ogilvie, and Hollowood-Myers in the appropriate limits, while highlighting lattice UV sensitivities that require counterterms and caution when extrapolating to nonperturbative simulations. The study emphasizes the connection between the Polyakov-loop winding (r) and Euclidean distance in color space, offering a physical picture for the cascade via the embedding of spacetime into color space. Overall, the work clarifies how center symmetry in large-$N$ gauge theories with adjoint fermions responds to mass and lattice anisotropy, linking lattice results to established continuum results and informing interpretation of lattice simulations at finite $N$ and weak coupling.
Abstract
We study the one-loop potential of large-N QCD with adjoint Dirac fermions. Space-time is a discretization of R^3 x S^1 where the compact direction consists of a single lattice site. We use Wilson fermions with different values of the quark mass m and set the lattice spacings in the compact and non-compact directions to be a_t and a_s respectively. Extending the results of JHEP 0906:091,2009, we prove that if the ratio xi = a_s/a_t obeys 0<xi<2, then the minimum of the one-loop lattice potential for one or more Dirac flavors is Z_N symmetric at the chiral point. For xi=0 our formulas reduce to those obtained in a continuum regularization of the R^3, and our proof holds in that case as well. As we increase m from zero, we find a cascade of transitions where Z_N breaks to Z_K. For very small masses, K ~ 1/(a_t m) >> 1, while for large masses K ~ O(1). Despite certain UV sensitivities of the lattice one-loop potential, this phase structure is similar to the one obtained in the continuum works of Kovtun-Unsal-Yaffe, Myers-Ogilvie, and Hollowood-Myers. We explain the physical origin of the cascade of transitions and its relation to the embedding of space-time into color space.