e − e − Z A ′ γ which is parametrically smaller than the opening angle of the A ′ decay products, ∼ m$_{A}$ ′ /E$_{0}$ . Al- though this opening angle is small, the backgrounds mimicking the signal (discussed in § VI) dominate at even smaller angles. Energy: A ′ bremsstrahlung is sharply peaked at x ≈ 1, where U ( x, 0) is minimized. When an A ′ is produced, it carries nearly the entire beam en- ergy — in fact the median value of (1 − x ) is ∼ max ( m$_{e}$ m$_{A}$ $_{′}$, m$_{A}$ ′ E$_{0}$ $^{)}$. FIG. 4: A ′ production by bremsstrahlung off an incoming electron scattering off protons in a target with atomic number Z . The latter two properties are quite important in improv- ing signal significance, and are discussed further in § VI. where we have neglected phase-space corrections, and N$_{eff}$ counts the number of available decay products. If the A ′ couples only to electrons, N$_{eff}$ = 1. If the A ′ mixes kinetically with the photon, then N$_{eff}$ = 1 for m$_{A}$ ′ < 2 m$_{µ}$ when only A ′ → e $^{+}$e − decays are pos- sible, and 2 + R ( m$_{A}$ ′ ) for m$_{A}$ ′ ≥ 2 m$_{µ}$ , where R = σ ( e $^{+}$e $^{−}$→ hadrons ; E = m$_{A}$ ′ ) σ ( e $^{+}$e µ $^{+}$µ $^{−}$; E = m$_{A}$ ′ ) [81]. For the ranges of ϵ and m$_{A}$ ′ probed by this experiment, the mean decay length ℓ$_{0}$ ≲ 250 µ m is not significant, but the ability to cleanly reconstruct vertices displaced forward by a few cm would open up sensitivity to considerably lower values of ϵ . The total number of A ′ produced when N$_{e}$ electrons scatter in a target of T ≪ 1 radiation lengths is is the virtuality of the intermediate electron in initial- state bremsstrahlung, and ˜ χ ∼ 0 . 1 − 10 is the Weizs¨ acker- Williams effective photon flux, with an overall factor of Z 2 removed. The form of ˜ χ and its dependence on the A ′ mass, beam energy, and target nucleus are discussed in Appendix A. The above results are valid for where Z is the atomic number of the target atoms, α ≃ 1 / 137, θ$_{A}$ ′ is the angle in the lab frame between the emitted A ′ and the incoming electron, m$_{A}$ ′ with energy E$_{A}$ ′ ≡ xE$_{0}$ is dσ dxd cos θ$_{A}$ ′ ≈ 8 Z $^{2}$α$^{3}$ϵ$^{2}$E2 $_{0}$x U 2 ˜ χ × [ (1 − x + x 2 2 ) − x (1 − x ) m 2 A $^{$_{′}$(}$E 2 $_{0}$x θ 2 A $_{′}$) U 2 ] (10) ℓ$_{0}$ ≡ γcτ ≃ 3 E$_{A}$ ′ N$_{eff}$m 2 A ′ αϵ 2 ≃ 0 . 8cm N$_{eff}$ ( E$_{0}$ 10GeV$^{)(}$ 10 − 4 ϵ ) 2 $^{(}$100 MeV m$_{A}$ ′ ) 2 , (15) U ( x, θ$_{A}$ ′ ) = E 2 $_{0}$xθ 2 A ′ + m 2 A ′ 1 − x x + m 2 $_{e}$x (11) N ∼ N$_{e}$ N$_{0}$X$_{0}$ A T Z $^{2}$α$^{3}$ϵ2 m 2 A ′ ˜ χ ∼ N$_{e}$ C T ϵ 2 m 2 e m 2 A ′ , (16) m$_{e}$ ≪ m$_{A}$ ′ ≪ E$_{0}$, x θ 2 A ′ ≪ 1 . (12) dσ dx ≈ 8 Z $^{2}$α$^{3}$ϵ$^{2}$x m 2 A ′ ( 1 + x 2 3(1 − x )$^{)}$ ˜ χ. (13) N C ∼ 10 $^{6}$˜ χ$^{(}$ T 0 . 1$^{)}$$^{(}$ ϵ 10 − 4 ) 2 $^{(}$100 MeV m$_{A}$ ′ ) 2 . (17) θ$_{A}$ ′ max ∼ max$^{(}$ √ m$_{A}$ ′ m$_{e}$ E$_{0}$ , m 3 / 2 A ′ E 3 / 2 0 ) , (14) 8 The spectrometer efficiency can be estimated from Monte Carlo simulation of the signal, discussed in § V. It is quite low in APEX, but of course depends on the pre- cise spectrometer settings. For example, for m$_{A}$ ′ = 200 MeV, E$_{0}$ = 3 . 056 GeV, an angular acceptance window of θ$_{x}$ = 0 . 055 − 0 . 102 rad and | θ$_{y}$ | ≤ 0 . 047 rad (corre- sponding to an HRS central angle of 4 . 5 ◦ ) and a momen- tum acceptance of E = 1 . 452 − 1 . 573 GeV for both the where X$_{0}$ is the radiation length of the target in g/cm $^{2}$, N$_{0}$ ≃ 6 × 10 23 $_{mole}$− 1 is Avogadro’s number, and A is the target atomic mass in g/mole. The numerical factor C ≈ 5 is logarithmically dependent on the choice of nucleus (at least in the range of masses where the form-factor is only slowly varying) and on m$_{A}$ ′ , because, roughly, X$_{0}$ ∝ A Z 2 (see [3] and [81]). For a Coulomb of incident electrons, the total number of A ′ ’s produced is given by Angle: A ′ emission is dominated at angles θ$_{A}$ ′ such that U ( x, θ$_{A}$ ′ ) ≲ 2 U ( x, 0) (beyond this point, wide- angle emission falls as 1 /θ 4 A ′ ). For x near its median value, the cutoff emission angle is Rate: The total A ′ production rate is controlled by α $^{3}$ϵ2 m 2 A ′ . Therefore, it is suppressed relative to pho- ton bremsstrahlung by ∼ ϵ 2 m 2 e m 2 A $_{′}$. Additional sup- pression from small ˜ χ occurs for large m$_{A}$ ′ or small E$_{0}$ . The rate and kinematics of A ′ radiation differ from mass- less bremsstrahlung in several important ways: For m$_{A}$ ′ ≫ m$_{e}$ , the angular integration gives