Fermiophobic Z' Bosons in 3-4-1 Models: Classification and Collider Constraints

We investigate the fermion spectra and collider phenomenology of models based on the $SU(3)_C \otimes SU(4)_L \otimes U(1)_X$ gauge symmetry, focusing on two realizations of the electric charge operator characterized by $b=c=1$ and $b=1,\ c=-2$. For each configuration, we classify anomaly-free fermion sets and identify five minimal spectra with at least three SM-like quark and lepton families for each configuration, distinguishing algebraically irreducible structures from reducible spectra that remain minimal under the three-family requirement. We derive the chiral couplings of the additional neutral gauge boson $Z'$ to Standard Model fermions and show that distinct anomaly-free fermion spectra can lead to identical low-energy neutral-current interactions. We then investigate the corresponding collider constraints using the ATLAS dilepton resonance search at $\sqrt{s}=13$ TeV with an integrated luminosity of $139\,\mathrm{fb}^{-1}$, assuming negligible $Z-Z'$ mixing and kinematically inaccessible exotic fermions. A particularly interesting result emerges for the $b=c=1$ configuration, where all $Z'$ couplings to Standard Model fermions are proportional to $\sinψ$, with $ψ$ denoting the additional neutral gauge-boson mixing angle. Consequently, these couplings vanish simultaneously at $ψ=nπ$, giving rise to a fermiophobic limit in which conventional Drell-Yan dilepton constraints disappear. In contrast, the $b=1,\ c=-2$ configuration does not exhibit a universal fermiophobic limit. Our results demonstrate that collider constraints on additional neutral gauge bosons depend sensitively on the fermion embeddings and neutral gauge-boson mixing structure, and suggest the possibility of comparatively light $Z'$ bosons in suitable realizations of extended electroweak gauge theories.