Can one see the number of colors?

Abstract

We formulate the standard model with an arbitrary number of colors N_c. The cancellation of Witten's global SU(2)_L anomaly requires N_c to be odd, while the cancellation of triangle anomalies determines the consistent N_c-dependent values of the quark charges. In this theory, the width of the neutral pion decay into two photons is not proportional to (N_c)^2. In fact, in the case of a single generation and hence for two quark flavors (N_f = 2), N_c does not appear explicitly in the low-energy effective theory of the standard model. Hence, contrary to common lore, it is impossible to see the number of colors in low-energy experiments with just pions and photons. For N_f > 2, on the other hand, N_c explicitly enters the chiral Lagrangian as the quantized prefactor of the Wess-Zumino-Witten term, but the contribution of this term to photon-pion vertices is completely canceled by the N_c-dependent part of a Goldstone-Wilczek term. However, the width of the eta decay into pi^+ pi^- gamma survives the cancellation and is indeed proportional to (N_c)^2. By detecting the emerging photon, this process thus allows one to literally see N_c for N_f > 2.