10 GLYPH<Minus>$_{2}$ 10 GLYPH<Minus>$_{1}$ 1 10 GLYPH<Minus>$_{10}$ 10 GLYPH<Minus>$_{9}$ 10 GLYPH<Minus>$_{8}$ 10 GLYPH<Minus>$_{7}$ 10 GLYPH<Minus>$_{6}$ 10 GLYPH<Minus>$_{5}$ 10 GLYPH<Minus>$_{2}$ 10 GLYPH<Minus>$_{1}$ 1 10 GLYPH<Minus>$_{10}$ 10 GLYPH<Minus>$_{9}$ 10 GLYPH<Minus>$_{8}$ 10 GLYPH<Minus>$_{7}$ 10 GLYPH<Minus>$_{6}$ 10 GLYPH<Minus>$_{5}$ m$_{A}$$_{'}$ GLYPH<LParen1> GeV GLYPH<RParen1> Α ' GLYPH<Slash1> Α E137 E141 E774 a$_{Μ}$ a$_{e}$ GLYPH<CapUpsilon> GLYPH<LParen1> 3 S GLYPH<RParen1> FIG. 1: An ticipated 2 σ sensitivit y in α $^{′}$/α = ϵ 2 for the A ′ exp erim en t (APEX) at Hall A in J Lab (thic k blue line), with existing constrain ts on an A ′ from electron and muon anoma- lous magnetic moment measurements, a$_{e}$ and a$_{µ}$ (see [27]), the BaBar search for Υ(3 S ) → γµ $^{+}$µ − [28], and three beam dump experiments, E137, E141, and E774 [29–31] (see [3]). The a$_{µ}$ and Υ(3 S ) limits assume equal-strength couplings to electrons and muons. The red region indicates the region of greatest theoretical interest, as described in the text. The gray dashed line indicates the scale used for other plots in this paper. The irregularity of the reach is an artifact of com- bining several different run settings (see Table II). The precise mass range probed by this type of experiment can be varied by changing the spectrometer angular settings and/or the beam energies. We stress this point as other experimental facilities may be able to perform experiments similar to APEX, but targeting complementary regions of parameter space. Smaller ϵ are expected if nature has enhanced sym- metry at high energies. For example, it has been con- jectured that the strong and electroweak gauge groups of the Standard Model are embedded in a grand unified These quantum effects are significant regardless of the mass m$_{X}$ of the particle in question, which could be well above the TeV scale (or even at the Planck scale) and thus evade detection. 3 ϵ ∼ 10 − 3 − 10 − 2 ( α ′ /α ∼ 10 − 6 − 10 − $^{4}$) . (3) As noted in [2], a new gauge boson A ′ that does not couple to Standard Model matter at a classical level can still couple through quantum-mechanical corrections. For example, lo ops of any particle X that couples to both the A ′ and Standar d Model hypercharge generates mix- ing of the form (2), with where F ′ µν = ∂$_{µ}$A ′ ν − ∂$_{ν}$A ′ µ is the field strength of the A ′ gauge b oson, and similarly F µν Y is the hypercharge field strength. This effect is generic, ensures that the A ′ interactions respect parity, and (as we discuss below) naturally pro duces small g ′ and A ′ masses near the GeV scale. This mixing is equivalent in low-energy interac- tions to assigning a charge ϵeq$_{i}$ to Standard Model parti- cles of electromagnetic charge q$_{i}$ , where ϵ = ϵ$_{Y}$ / (cos θ$_{W}$ ) and θ$_{W}$ is the Weinberg mixing angle. The A ′ couplings to neutrinos an d parity-violating couplings are negligible compared to Z -mediated effects (see e.g. [13]). δ L = ϵ$_{Y}$ 2 F ′ $_{µν}$F µν Y , (2) A ′ couplings to Standard Model matter with the lat- ter structure can be induced by ordinary electromagnetic interactions through the kinetic mixing interaction pro- posed by Holdom [2], to a new gauge boson A ′ , which is quite poorly con- strained for small g ′ (see Figure 1)[3]. Similar couplings between the A ′ and other Standard Model fermions are also allow ed, with relations between their couplings (anomaly cancellation) required for the A ′ gauge symme- try to be quan tum-mechanically consistent. For example, the A ′ can couple only to electrons and muons, with op- posite charges g ′ e = − g ′ µ ( a U $^{(1)$_{e}$}$− µ boson), or can have couplings prop ortional to the electromagnetic charges q$_{i}$ of each fermion, g$_{i}$ = ϵeq$_{i}$ . δ L = g ′ A ′ µ ¯ ψγ $^{µ}$ψ (1) New light vector particles, matter states, and their as- sociated interactions are ubiquitous in extensions of the Standard Model [2, 32–40]. However, the symmetries of the Standard Model restrict the interaction of ordinary matter with such new states. Indeed, most interactions consistent with Standard Model gauge symmetries and Lorentz inv ariance have couplings suppressed by a high mass scale. One of the few unsuppressed interactions is the coupling of charged Standard Model particles ψ (as mentioned previously, similar considerations apply to pseudo-vectors, scalars, and pseudo-scalars with sub- GeV mass that couple to electrons). It is useful to param- eterize the coupling g ′ of the A ′ to electrons by a dimen- sionless ϵ ≡ g ′ /e , where e is the electron charge. Cross- sections for A production then scale as α ′ /α = ϵ $^{2}$, where α ′ = g ′ $^{2}$/ (4 π ) and α = e $^{2}$/ (4 π ) are the fine-structure con- stants for the dark photon and ordinary electromagnetic interactions, respectively. This experiment will search for A ′ bosons with mass m$_{A}$ ′ ∼ 65 MeV – 550 MeV and α ′ /α ≳ 6 × 10 − $^{8}$, which can be produced by a reaction analogous to photon bremsstrahlung (see § III) and de- cays promptly to e $^{+}$e − or other charged particle pairs. We refer the reader to Figure 1 for a summary of the reach of this experiment. A. Motivation for New Physics Near the GeV Scale