Chiral and U(1)_A restorations high in the hadron spectrum, semiclassical approximation and large N_c

In quantum systems with large $n$ (radial quantum number) or large angular momentum the semiclassical (WKB) approximation is valid. A physical content of the semiclassical approximation is that the quantum fluctuations effects are suppressed and vanish asymptotically. The chiral as well as $U(1)_A$ breakings in QCD result from quantum fluctuations. Hence these breakings must be suppressed high in the spectrum and the spectrum of high-lying hadrons must exhibit $U(2) \times U(2)$ symmetry of the classical QCD Lagrangian. This argument can be made precise for mesons in the large $N_c$ limit. In this limit all mesons are stable against strong decays and the spectrum is infinite. Hence, one can excite mesons of arbitrary large size with arbitrary large action, in which case the semiclassical limit is manifest.