A novel strategy to prove chiral symmetry breaking in QCD-like theories

We demonstrate that chiral symmetry breaking occurs in the confining regime of QCD-like theories with $N_c$ colors and $N_f$ flavors. Our proof is based on a novel strategy, called `downlifting', by which solutions of the 't Hooft anomaly matching and persistent mass conditions for a theory with $N_f-1$ flavors are constructed from those of a theory with $N_f$ flavors, while $N_c$ is fixed. By induction, chiral symmetry breaking is proven for any $N_f\geq p_{min}$ in the confining regime, where $p_{min}$ is the smallest prime factor of $N_c$. The proof can be extended to $N_f <p_{min}$ under the additional assumption on the absence of phase transitions when quark masses are sent to infinity. Our results do not rely on assumptions on the spectrum of massless bound states other than the fact that they are color-singlet hadrons.