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 ${\mathbb{R}}^3\times S_L^1$ 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 $L(S^1)\ll {\mathrm{\Lambda }}^{-1}$, i.e. at weak coupling, no phase transitions are expected in passing to large $L(S^1)\gg {\mathrm{\Lambda }}^{-1}$ (here $\mathrm{\Lambda }$ 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 $⟨{\psi }^{\{ij\}}{\chi }_{[jk]}⟩\mathrm{\ne }0$ at weak coupling i.e. at $L(S^1)\ll {\mathrm{\Lambda }}^{-1}$. Then adiabatic continuity tells us that $⟨{\psi }^{\{ij\}}{\chi }_{[jk]}⟩\mathrm{\ne }0$ exists on ${\mathbb{R}}^4$, in full accord with the prediction [2]. Simultaneously with $⟨{\psi }^{\{ij\}}{\chi }_{[jk]}⟩\sim {\mathrm{\Lambda }}^3{\delta }_k^i$ the SU($N_c$) gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.