Probing new $U(1)$ gauge symmetries via exotic $Z \to Z' \gamma$ decays

New $U(1)$ gauge theories involving Standard Model (SM) fermions typically require additional electroweak fermions for anomaly cancellation. We study the non-decoupling properties of these new fermions, called anomalons, in the $Z-Z'-\gamma$ vertex function, reviewing the connection between the full model and the effective Wess-Zumino operator. We calculate the exotic $Z \to Z' \gamma$ decay width in $U(1)_{B-L}$ and $U(1)_B$ models, where $B$ and $L$ denote the SM baryon and lepton number symmetries. For $U(1)_{B-L}$ gauge symmetry, each generation of SM fermions is anomaly free and the exotic $Z \to Z_{BL}' \gamma$ decay width is entirely induced by intragenerational mass splittings. In contrast, for $U(1)_B$ gauge symmetry, the existence of two distinct sources of chiral symmetry breaking enables a heavy, anomaly-free set of fermions to have an irreducible contribution to the $Z \to Z_B' \gamma$ decay width. We show that the current LEP limits on the exotic $Z \to Z'_B \gamma$ decay are weaker than previously estimated, and low-mass $Z'_B$ dijet resonance searches are currently more constraining. We present a summary of the current collider bounds on $U(1)_B$ and a projection for a TeraZ factory on the $Z \to Z_B' \gamma$ exotic decay, and emphasize how the $Z \to Z' \gamma$ decay is emblematic of new anomalous $U(1)$ gauge symmetries.