Differences and similarities between fundamental and adjoint matters in SU(N) gauge theories

TL;DR

The paper investigates how fundamental and adjoint matter differ in $SU(N)$ gauge theories, focusing on the role of $Z_N$ symmetry. By imposing a flavor-dependent twist boundary condition (FTBC) on fundamental fermions, the authors restore $Z_N$ symmetry for all fermion masses and compare the resulting theory to the adjoint-mominated case using the PNJL model. They show that FTBC fundamental fermions exhibit confinement/deconfinement dynamics closely mirroring those of adjoint fermions, with chiral observables related by a simple scaling and the sign problem context discussed. In addition, they explore high-energy gauge symmetry breaking via the Hosotani mechanism by evaluating one-loop effective potentials on $R^3\times S^1$, finding that FTBC-PB can induce gauge symmetry breaking to $SU(2)\times U(1)$ (distinct from the adjoint case which tends to yield $U(1)\times U(1)$), though such GB phenomena may require large flavor numbers and can push the theory beyond asymptotic freedom. Overall, the work highlights $Z_N$ symmetry as the key differentiator between fundamental and adjoint matter and opens avenues for dynamical gauge-Higgs unification scenarios via GB in FTBC settings.

Abstract

We investigate differences and similarities between fundamental fermions and adjoint fermions in SU(N) gauge theories. The gauge theory with fundamental fermions possesses ZN symmetry only in the limit of infinite fermion mass, whereas the gauge theory with adjoint fermions does have the symmetry for any fermion mass. The flavor-dependent twisted boundary condition (FTBC) is then imposed on fundamental fermions so that the theory with fundamental fermions can possess ZN symmetry for any fermion mass. We show similarities between FTBC fundamental fermions and adjoint fermions, using the Polyakov-loop extended Nambu--Jona-Lasinio (PNJL) model. In the mean-field level, the PNJL model with FTBC fundamental fermions has dynamics similar to the PNJL model with adjoint fermions for the confinement/deconfinement transition related to ZN symmetry. The chiral property is somewhat different between the two models, but there is a simple relation between chiral condensates in the two models. As an interesting high-energy phenomenon, a possibility of the gauge symmetry breaking is studied for FTBC fundamental fermions.

Figures (13)

• Figure 1: $T$ dependence of order parameters in the PNJL model of FTBC fermion with $N=N_{F,fund}=3$ and that of ADJ fermion with $N=3$ and $N_{F,adj}=1$. In the both cases, the boundary condition $\varphi =\pi$ is taken. Three panels correspond to (a) $\Phi$, (b) $\phi_c$, and (c) $M/3$ and $M_f$, respectively.
• Figure 2: $T$ dependence of constituent quark masses in different types of fermions and boundary conditions. Four cases of FTBC-APB, FTBC-PB, ADJ-APB and ADJ-PB are taken, where FTBC and ADJ stand for kinds of fermions while PB and APB correspond to kinds of boundary conditions. Here we set $N=N_{F,fund}=3$ and $N_{F,adj}=1$. For FTBC-APB and FTBC-PB, constituent quark masses depend on flavor in the deconfinement phase at higher temperature, so the average vales are shown in the cases.
• Figure 3: Contour plot of ${\cal V}_{f}L^4$ in the limit $mL\to 0$ for the case of FD-APB. Here, $q_3$ is given by $-q_1-q_2$.
• Figure 4: The same figure as Fig. \ref{['FD_APB']} but for the case of FD-PB.
• Figure 5: The same figure as Fig. \ref{['FD_APB']} but for the case of ADJ-APB.