Consistency of $\chi$SB in chiral Yang-Mills theory with adiabatic continuity

We study the pattern of chiral symmetry breaking ($\chi$SB) in the $\psi\chi\eta$ model (with the chiral fermion sector containing $ \psi^{\{ij\}}$, $\chi_{[ij]}$, and $\eta_{i}^{A}$, see [1]) on $\mathbb{R}^3 \times S^{1}_{L}$ and derive implications to $\mathbb{R}^4$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L(S^1)\ll \Lambda^{-1}$, i.e. at weak coupling, no phase transitions are expected in passing to large $L(S^1)\gg \Lambda^{-1}$ (here $\Lambda$ is the dynamical Yang-Mills scale). Starting with the small $L$-limit, we find the leading-order nonperturbative corrections in the given theory. The instanton-monopole operators induce the adjoint chiral condensate $\langle \psi^{\{ij\}}\chi_{[jk]}\rangle \neq 0$ at weak coupling i.e. at $L(S^1)\ll \Lambda^{-1}$. Then adiabatic continuity tells us that $\langle \psi^{\{ij\}}\chi_{[jk]}\rangle \neq 0$ exists on $\mathbb{R}^4$, in full accord with the prediction [2]. Simultaneously with $\langle \psi^{\{ij\}}\chi_{[jk]}\rangle \sim \Lambda^3\delta^i_k$ the SU($N_c$) gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.