Can One See the Number of Colors?
Abstract
We formulate the standard model with an arbitrary number of colors Nc. The cancellation of Witten's global SU(2)L anomaly requires Nc to be odd, while the cancellation of triangle anomalies determines the consistent Nc-dependent values of the quark charges. In this theory, the width of the neutral pion decay into two photons is not proportional to (Nc)2. In fact, in the case of a single generation and hence for two quark flavors (Nf = 2), Nc 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 Nf > 2, on the other hand, Nc 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 Nc-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 (Nc)2. By detecting the emerging photon, this process thus allows one to literally see Nc for Nf > 2.
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