The light U U boson as the mediator of a new force, coupled to a combination of Q,\ B,\ L 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^μ F naturally combines with the weak neutral current J^μ Zₛm , both parity-violating, into a vectorial current J^μ U , combination of the $$B,\,L$$ and electromagnetic currents with a possible dark-matter current. U^μ may be expressed equivalently as cos ξ \,C^μ + sin ξ \,Z^μ _\,sm\, (“mixing with the Z”) or (1/cos χ ) \,Ĉ^μ + tan χ \,A^μ \, (“mixing with the photon”), with Ĉ coupled to $$B,\,L$$ and dark matter. The U boson may be viewed as a generalized dark photon, coupled to SM particles through QU\!=Q+λ B B+λ ᵢ Lᵢ , with strength g''\!cos ξ cos ²θ =etan χ . “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 QUᵍut\!=Q-2\,(B\!-\!L) at the GUT scale for SM particles, invariant under the SU(4) electrostrong symmetry group, with a non-vanishing ε \!= tan χ already present at the GUT scale, leading to QU \,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.