imation, then rescaled to the cross-section for the full- interference process. The contribution from the radiative diagrams (Figure 8(a)) alone is also useful as a guide to the behavior of A ′ signals at various masses. Indeed, the kinematics of the A ′ signal events is identical to the distribution of radiative trident events restricted in an invariant mass window near the A ′ mass. Moreover, the rate of the A ′ signal is simply related to the radiative trident cross- section within the spectrometer acceptance and a mass window of width δm by [3] FIG. 8: Sample diagrams of (a) radiative trident ( γ $^{∗}$) and (b) Bethe-Heitler trident reactions that comprise the primary QED background to A ′ → ℓ $^{+}$ℓ − search channels. dσ ( e − Z → e − Z ( A ′ → ℓ $^{+}$ℓ − )) dσ ( e − Z e − Z ( γ ∗ ℓ $^{+}$ℓ − )) =$^{(}$ 3 πϵ 2 2 N$_{eff}$α$^{)}$$^{(}$ m$_{A}$ ′ δm $^{)}$, (18) a safe level for a tungsten foil target [83], so we expect that the wire mesh target will perform quite well. where N$_{eff}$ counts the number of available decay products and is defined below equation (15). This exact analytic formula was also checked with a MC simulation of both the A ′ signal and the radiative tridents background re- stricted to a small mass window δm , and we find nearly perfect agreement. Thus, the radiative subsample can be used to analyze the signal, which simplifies the analysis considerably. The stark kinematic differences between QED trident backgrounds and the A ′ signal are the primary consid- erations in determining the momentum settings of the spectrometers. As we will show in § VI, QED tridents dominate the final event sample after offline rejection of accidentals, so we consider their properties in some detail here. V. SIGNAL AND TRIDENT KINEMATICS It is instructive to compare kinematic features of the radiative and Bethe-Heitler distributions, as the most sensitive experiment maximizes acceptance of radiative events and rejection of Bethe-Heitler tridents. Although the Bethe-Heitler process has a much larger total cross- sections than either the signal or the radiative trident background, it can be significantly reduced by exploit- ing its very different kinematics. In particular, the A ′ carries most of the beam energy (see discussion in § III), while the recoiling electron is very soft and scatters to a wide angle. In contrast, the Bethe-Heitler process is not enhanced at high pair energies. Moreover, Bethe-Heitler processes have a forward singularity that strongly favors asymmetric configurations with one energetic, forward electron or positron and the other constituent of the pair much softer. These properties are discussed further in the Appendix of [3], and illustrated in Figure 9, which shows a scatter- plot of the energy of the positron and the higher-energy electron for the signal (red crosses) and Bethe-Heitler background (black dots). The signal electron-positron pairs are clearly concentrated near the kinematic limit, E ( e $^{+}$) + E ( e − ) ≈ E$_{beam}$ . Background rejection is opti- mized in symmetric configurations with equal angles for the two spectrometers and momentum acceptance of each spectrometer close to half the beam energy (blue box). While the signal over background ( S/B ) can be sig- nificantly improved with a judicious choice of kinematic cuts, the final S/B in a small resolution limited mass window is still very low, ∼ 1%. A “bump-hunt” for a small signal peak over the continuous background needs to be performed. This requires an excellent mass reso- lution, which has an important impact on target design and calls for a target that is tilted with respect to the beam line (see Appendix B for a discussion of the mass The continuum trident background was simulated in- cluding the full interference effects between the diagrams in Figure 8. In addition, a “reduced-interference” ap- proximation simplifies the analysis and is much less com- putationally intensive. In this approximation, we treat the recoiling e − and the e − from the produced pair as dis- tinguishable. Furthermore, we separate trident processes into the radiative diagrams (Figure 8(a)) and the Bethe- Heitler diagrams (Figure 8(b)), and we calculate the cross-section for both of these diagrams separately. This approximation under-estimates the background rates by a factor of about 2–3 in the range of A ′ masses and beam energies considered in this paper and [1]. For the reach analysis discussed below, we have used differential distri- butions computed in the “reduced-interference” approx- 11 We have simulated the production of these continuum trident background events in QED using the nuclear elas- tic and inelastic form-factors in [78]. The simulation is done using MadGraph and MadEvent [84] to compute the matrix elements for e − Z → e − ( e $^{+}$e − ) Z exactly, but neglecting the effect of nuclear excitations on the kinematics in inelastic processes. The MadEvent code was modified to properly account for the masses of the incoming nucleus and electron in event kinematics, and the nucleus is assumed to couple with a form-factor G$_{2}$ defined in Appendix A. The irreducible background rates are given by the di- agrams shown in Figure 8. These trident events can be usefully separated into “radiative” diagrams (Figure 8(a)), and “Bethe-Heitler” diagrams (Figure 8(b)), that are separately gauge-invariant. ℓ + ℓ − ℓ + ℓ − e − Z Z e − ( a ) ( b )