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The $U$ boson, interpolating between a generalized dark photon or dark $Z$, an axial boson and an axionlike particle

A light boson $U$ from an extra $U(1)$ interpolates between a generalized dark photon coupled to $Q,\ B$ and $L_i$ (or $B-L$), plus possibly dark matter, a dark $Z$ coupled to the $Z$ current, and one axially coupled to quarks and leptons. We identify the corresponding $U(1)_F$ symmetries, with $F= γ_Y Y+γ_B B+γ_{L_i}L_i+γ_A F_A+γ_{F'}F'\!+γ_d F_d$, $F_d$ acting in a dark sector and $F'$ on possible semi-inert BEH doublets uncoupled to quarks and leptons. The $U$ current is obtained from the $U(1)_F $ and $Z$ currents, with a mixing determined by the spin-0 BEH fields. The charge $Q_U$ of chiral quarks and leptons is a combination of $Q,\ B,\ L_i$ and $T_{3L}$ with the axial $F_A$. It involves in general isovector and isoscalar axial terms, in the presence of two BEH doublets. A longitudinal $U$ with axial couplings has enhanced interactions, and behaves much as an axionlike particle. Its axial couplings $g_A$, usually restricted to $< 2\times 10^{-7} m_U$(MeV), lead to effective pseudoscalar ones $g_P= g_A\times 2m_{q,l}\,/m_U=2^{1/4}\, G_F^{1/2}\, m_{q,l} \ A_\pm $. $\,A_\pm$ is proportional to an invisibility parameter $r=\cosθ_A$ induced by a singlet v.e.v., possibly large and allowing the $U$ to be very weakly interacting. This allows for a very small gauge coupling, expressed with two doublets and a singlet as $g"\!/4\simeq 2\times 10^{-6} m_U$(MeV) $ r/\sin 2β$. We discuss phenomenological implications for meson decays, neutrino interactions, atomic-physics parity violation, naturally suppressed $π^0\toγU$ decays, etc.. The $U$ boson fits within the grand-unification framework, in symbiosis with a $SU(4)_{\rm es }$ electrostrong symmetry broken at the GUT scale, with $Q_U$ depending on $Q,\ B-L,\ F_A$ and $T_{3A}$ through three parameters $γ_Y,\ γ_A$ and $η$.