A necessary condition for sphaleron process in the presence of anomalous $U(1)$ symmetry

Abstract

We argue that, in the presence of anomalous $U(1)$ symmetries, the invariance of Lagrangian including the standard model (SM) under the axionic shift symmetries requires a necessary condition $\sum_i\delta^{\rm GS}_{i}\leq2N_f$, where $N_f$ stands for the number of families in the SM and $\delta^{\rm GS}_i$ (Green-Schwarz parameter) characterizes the coupling of the anomalous $U(1)$ gauge boson to the corresponding axion. In turn, we show that in order for the usual $B+L$ violating sphaleron process to be valid a necessary condition $\sum_i\delta^{\rm GS}_{i}<2N_f$ is required, where $B(L)$ stands for the baryon(lepton) number.