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 ${\sum }_i{\delta }_i^{\mathrm{GS}}\le 2N_f$, where $N_f$ stands for the number of families in the SM and ${\delta }_i^{\mathrm{GS}}$ (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 ${\sum }_i{\delta }_i^{\mathrm{GS}}<2N_f$ is required, where $B(L)$ stands for the baryon(lepton) number.

Comments: Title and main arguments are changed. 5 pages