The light U boson as the mediator of a new force, coupled to a combination of Q, B, L and dark matter

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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 JμF naturally combines with the weak neutral current JμZ sm, both parity-violating, into a vectorial current Jμᵤ, combination of the B,L and electromagnetic currents with a possible dark matter current. Uμ may be expressed equivalently as ξCμ + ξZμ sm ("mixing with the Z") or (1/χ) Cμ+χAμ ("mixing with the photon"), with C 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+λB B+λᵢ Lᵢ, with strength g"ξ²θ=eχ. "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 Qᵤ gut=Q-2(B-L) at the GUT scale for SM particles, invariant under the SU(4) electrostrong symmetry group, with a non-vanishing ε= 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.