2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/49274In contrast to common belief, the chirally symmetric Nambu--Jona-Lasinio model does not contain massless pions, due to strong chiral fluctuations. Although quarks acquire spontaneously a nonzero constituent mass $M$, pions have a nonzero mass equal to the mass of $ \s $-mesons, both being of the order of $M$. This result is found in several cutoff schemes. Our derivation is nonperturbative, but involves a simple approximation (London limit) which should, however, receive only quantitative, no qualitative corrections.Version 2 contains several improvements. Author Information under http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of paper also at http://www.physik.fu-berlin.de/~kleinert/296High Energy Physics - Phenomenologytext