Consistency of chiral symmetry breaking in chiral Yang-Mills theory with adiabatic continuity

Abstract

We study the pattern of chiral symmetry breaking in the $\ensuremath{\psi}\ensuremath{\chi}\ensuremath{\eta}$ model [with the chiral fermion sector containing ${\ensuremath{\psi}}^{{ij}}$, ${\ensuremath{\chi}}_{[ij]}$, and ${\ensuremath{\eta}}_{i}^{A}$; see Armoni and Shifman Phys. Rev. D 85, 105003 (2012)] on ${\mathbb{R}}^{3}\ifmmode\times\else\texttimes\fi{}{S}_{L}^{1}$ and derive implications forcing ${\mathbb{R}}^{4}$ physics. Center-symmetric vacua are stabilized by a double-trace deformation. With the center symmetry maintained at small $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$, i.e., at weak coupling, no phase transitions are expected in passing to large $L({S}^{1})\ensuremath{\gg}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$ (here, $\mathrm{\ensuremath{\Lambda}}$ 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 $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ at weak coupling, i.e., at $L({S}^{1})\ensuremath{\ll}{\mathrm{\ensuremath{\Lambda}}}^{\ensuremath{-}1}$. Then adiabatic continuity tells us that $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\ne}0$ exists on ${\mathbb{R}}^{4}$, in full accord with the prediction from Bolognesi et al. [Phys. Rev. D 97, 094007 (2018)]. Simultaneously with $⟨{\ensuremath{\psi}}^{{ij}}{\ensuremath{\chi}}_{[jk]}⟩\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{3}{\ensuremath{\delta}}_{k}^{i}$, the $\mathrm{SU}({N}_{c})$ gauge symmetry is spontaneously broken at strong coupling down to its maximal Abelian subgroup.