The light U boson as the mediator of a new force, coupled to a   combination of Q, B, L and dark matter
  A new light gauge boson $U$ may have both vector and axial couplings. In a large class of theories however, the new $U(1)$ current $J^\mu_F$ naturally combines with the weak neutral current $J^\mu_{Z_{\rm sm}}$, both parity-violating, into a vectorial current $J^\mu_U$, combination of the $B,\,L$ and electromagnetic currents with a possible dark matter current.   $U^\mu $ may be expressed equivalently as $\cos\xi \,C^\mu\! + \sin \xi \,Z^\mu_{\rm \,sm}$ ("mixing with the $Z$") or $(1/\cos\chi) \,\hat C^\mu+\tan\chi \,A^\mu$ ("mixing with the photon"), with $\hat C$ coupled to $B,\,L$ and dark matter. The $U$ boson may be viewed as a generalized dark photon, coupled to SM particles through $Q_U=Q+\lambda_B B+\lambda_i L_i$, with strength $\,g"\cos\xi\,\cos^2\theta=e\tan\chi$. "Kinetic mixing" terms, gauge invariant or not, simply correspond to a description in a non-orthogonal field basis (rather than to a new physical effect), with the dark photon in general also coupled to $B$ and $L$.   In a grand-unified theory $\,Q_U^{\rm gut}=Q-2\,(B-L)\,$ at the GUT scale for SM particles, invariant under the $SU(4)$ electrostrong symmetry group, with a non-vanishing $\epsilon = \tan\chi$ already present at the GUT scale, leading to $Q_U\simeq \,Q-1.64\, (B-L)\,$ at low energy. This also applies, for a very light or massless $U$ boson, to a new long-range force, which could show up through apparent violations of the Equivalence Principle. 
Pierre Fayet
hep-ph hep-ex