Large-N volume reduction of lattice QCD with adjoint Wilson fermions at weak-coupling

TL;DR

This work investigates large-$N$ volume independence for lattice QCD with adjoint Wilson fermions in a single-site reduced direction, using a weak-coupling one-loop analysis of the holonomy potential $V(\{\theta_a\})$ to determine regimes with unbroken $Z_N$ center symmetry. It combines a gauge-field and adjoint-fermion contribution to $V(\theta)$, maps lattice bare parameters to continuum-like variables, and treats the reduced model as a non-renormalizable EFT that requires specific counterterms to stabilize center symmetry. The study shows that $Z_N$-invariant vacua are favored across a broad range of quark masses ($N_f=0.5,1,2$) and anisotropies, supporting large-$N$ volume independence at weak coupling, including for relatively heavy quarks. It further clarifies apparent tensions with continuum EFT results (BBCS) by highlighting the need for regulator-dependent counterterms ($b_1$, $b_2$) in the EFT and demonstrates that lattice regularization with Wilson fermions can realize volume reduction consistent with large-$N$ equivalence. The findings provide actionable guidance for lattice simulations and illuminate the role of UV regularization in reduced models, with companion non-perturbative results reinforcing the perturbative conclusions for certain parameter regimes.

Abstract

We study the large-N volume reduction of QCD with adjoint quarks regularized on the lattice. Specifically, we use Wilson fermions, and while our d-dimensional lattice has (d-1) infinite dimensions, the remaining direction is reduced to a point. We perform a weak-coupling one-loop calculation of the free energy as a function of the holonomy in the reduced direction, and map the regimes in the bare lattice parameter space where the holonomy averages to zero and a Z_N-center symmetric configuration is the ground state. For d=4 and N_f=1/2,1 and 2 Dirac flavors we see that the center symmetry is intact in a generous regime of the phase diagram that includes the chiral point. Thus we see that large-N volume independence of lattice QCD with adjoint Wilson quarks works at weak coupling. Interestingly, we find that this is true even if the quark mass is quite large, and this opens a path to study the volume reduced large-N pure gauge theory. Finally, we analyze in detail the UV sensitivity of the one-loop potential and show that treating the reduced theory as a (d-1)-dimensional effective field theory requires the introduction of certain relevant operators that are a subset of those suggested by Unsal and Yaffe to stabilize the center symmetry. This means that different regularizations of the volume-reduced theory can be compared only if one includes these terms in the action.

Figures (12)

• Figure 1: The vertex of a term of the form of Eq. (\ref{['vertex']}).
• Figure 2: The bubble diagram that gives rise to the term $|{\text{t}r}\, P|^2$ with $L_t=8$. The numbers next to the vertices denote the $t$ index of the $\Omega$ and $\Omega^\dag$ fields on the external legs, while the numbers on the gluon lines denote the $t$ indices of the $A$ fields that flow in the loop.
• Figure 3: A map of the phase diagram symmetric lattice spacing and $N_f=1/2$ (a single Majorana fermion in the continuum of the $Z_N$ invariant phase) as a function of $\kappa$ ($\kappa=1/8$ is the chiral point and it has $Z_N$ symmetry intact).
• Figure 4: As in Fig. \ref{['sym05']} but for $N_f=1$.
• Figure 5: As in Fig. \ref{['sym1']} but zooming on the regime of $\kappa \in [0,0.05]$.