Does confinement imply chiral symmetry breaking in QCD-like theories
In plain words
In QCD, quark confinement comes with the spontaneous breaking of chiral symmetry (the symmetry rotating left- and right-handed quarks separately), which makes pions nearly massless. Whether confinement always forces this breaking, for any number of colors and quark types, is not proven.
Precise statement
In 4D $\mathrm{SU}(N_{c})$ gauge theory with $N_{f}$ massless Dirac quarks in the fundamental representation, assume confinement in the sense that all massless bound states are color singlets. Prove that $\mathrm{SU}(N_{f})_{L}$ x $\mathrm{SU}(N_{f})_{R}$ breaks to $\mathrm{SU}(N_{f})_{V}$ for every $N_{c} \ge 3$ and every $N_{f}$ below the conformal window, using 't Hooft anomaly matching without further dynamical assumptions, or exhibit a consistent chirally symmetric confining spectrum.
What would settle it
A proof from anomaly matching and the confinement assumption alone covering every $N_{c} \ge 3$ and every $N_{f}$ below the conformal window, or a consistent massless-baryon spectrum that matches all anomalies.
Status in the literature
Unverified note
Ciambriello, Contino, Luzio, Romano and Xu (2024, arXiv:2404.02967) proved breaking for $N_{f} \ge p_{\mathrm{min}}$, the smallest prime factor of $N_{c}$; smaller $N_{f}$ requires assuming no phase transition as quark masses are sent to infinity.