UV-Complete Models for a Light Axial Gauge Boson

We present new anomaly free gauge models where the gauge field only has axial vector couplings to both quarks and leptons. We use the left-right symmetric universal seesaw models as the basis for this construction with an extra $U(1)_a$ as the axial gauge group. We present three main versions of the model, denoted as models A, B and C (and their variations), with different properties depending on the way the gauge anomaly is canceled. We show how the models can accommodate small neutrino masses. The models allow for a new Dirac fermion coupled via the $U(1)_a$ gauge portal to the SM fields which can be the dark matter. The models A and its variation have the novel property that there is an upper limit on the $U(1)_a$ gauge coupling $g_a$, due to the fact that the standard model Higgs doublet shares the $U(1)_a$ quantum number. For models B and C, we discuss the phenomenological constraints on the gauge coupling $g_a$ and gauge boson mass $m_{\cal A}$ from current low energy observations where, unlike in models A, $g_a$ depends on $m_{\mathcal{A}}$ through a single vacuum expectation value.