Appendix A: Effective Photon Flux, Target Nucleus and Beam-Energy Dependence In this appendix we summarize the formulas used in Section III for the reduced effective photon flux ˜ χ , and highlight its dependence on the A ′ mass, target nucleus, and beam energy. The effective photon flux χ is obtained as in [78, 79] by integrating electromagnetic form-factors over allowed photon virtualities: For a general electric form factor G$_{2}$ ( t ), pair mass spectrum and search for new gauge bosons A ′ in the mass range 65 MeV < m$_{A}$ ′ < 550 MeV that have weak coupling to the electron. Parametrizing this cou- pling by the ratio α ′ /α that controls the A ′ production cross-section, this experiment would probe α /α as small as ∼ (6 − 8) × 10 − 8 at masses from 65 to 300 MeV, and α ′ /α ∼ (2 − 3) × 10 − 7 at masses up to 525 MeV, making it sensitive to production rates 10–1000 times lower than the best current limits set by measurements of the anomalous muon magnetic moment and by direct searches at BaBar. The experiment uses the JLab elec- tron beam in Hall A at energies of about 1, 2, 3, and 4 GeV incident on a long (50 cm) thin tilted tungsten wire mesh target, and both arms of the High Resolution Spec- trometer at angles between 5 . 0 ◦ and 5 . 5 ◦ relative to the nominal target position. The experiment can determine the mass of an A ′ to an accuracy of ∼ 1–2 MeV. While this paper was motivated by a specific exper- imental proposal for JLab Hall A, very similar experi- ments are possible at other experimental facilities, such as the Mainz Microtron or JLab Hall B. Many of the considerations discussed in this paper are applicable to these other facilities. Constraints on new vector bosons with mass near 50 MeV – 1 GeV are remarkably weak. However, such light force carriers are well motivated theoretically, and sev- eral recent anomalies from terrestrial and satellite exper- iments suggest that dark matter interacting with Stan- dard Model particles has interactions with new vector bosons in precisely this mass range. The proposed ex- periment can probe these hypothetical particles with a sensitivity that is un-rivaled by any existing or planned experiment. We thank Nima Arkani-Hamed, James Bjorken, Dou- glas Finkbeiner, Mathew Graham, John Jaros, Takashi Maruyama, Ken Moffeit, Richard Partridge, Michael Pe- skin, Maxim Pospelov, Allen Odian, Art Snyder, Jay Wacker, and Neal Weiner for many useful discussions and encouragement. We also thank Johan Alwall for an- swering questions about Madgraph. We especially thank Peter Bosted, Kees de Jager, Doug Higinbotham, John LeRose, and Yi Qiang for their help in preparing the pro- posal for JLab Hall A. RE and PS are supported by the US DOE under contract number DE-AC02-76SF00515. RE, PS, and NT would like to thank the Kavli Institute for Theoretical Physics in Santa Barbara for hospital- ity during part of this research. PS and NT also thank the Perimeter Institute for Theoretical Physics for hos- pitality. We also thank the SLAC National Accelerator Laboratory for providing funding for the “Dark Forces” workshop in September 2009. The resulting reduced form factor ˜ χ ( m $^{2}$, E$_{0}$ ) = χ/Z 2 are plotted in the left panel of Figure 12 as a function of e $^{+}$e − mass for various electron energies (1, 2, 3, and 4 GeV) incident on a Tungsten target. The relative ef- ficiency of A ′ production in targets of different composi- tions but the same thickness in radiation lengths is given where the first term parametrizes the inelastic atomic form factor and the second the inelastic nuclear form factor, and where a ′ = 773 Z − 2 / $^{3}$/m$_{e}$ , m$_{p}$ is the proton mass, and µ$_{p}$ = 2 . 79 [78]. This expression is valid when t/ 4 m 2 p is small, which is the case for m$_{A}$ ′ in the range of interest in this paper. At large t the form factors will deviate from these simple parameterizations but can be measured from data. One can show that the contribu- tion from the other inelastic nuclear form factor G$_{1}$ ( t ) is negligible. 15 Acknowledgements G$_{2}$$_{,in}$ ( t ) =$^{(}$ a ′ $^{2}$t 1 + a ′ $^{2}$t$^{)}$$^{2}$$^{(}$ 1 + t 4 m 2 $_{p}$( µ 2 p − 1) (1 + t 0 . 71 GeV $_{2}$) 4 ) 2 Z, (A4) where the first term parametrizes electron screening (the elastic atomic form factor) with a = 111 Z − 1 / $^{3}$/m$_{e}$ , and the second finite nuclear size (the elastic nuclear form fac- tor) with d = 0 . 164 GeV $^{2}$A − 2 / $^{3}$. We have multiplied to- gether the simple parametrizations used for each in [78]. The logarithm from integrating (A1) is large for t$_{min}$ < d , which is true for most of the range of interest. However, for heavy A ′ , the elastic contribution is suppressed and is comparable to an inelastic term, G$_{2}$$_{,el}$ ( t ) =$^{(}$ a $^{2}$t 1 + a $^{2}$t$^{)}$$^{2}$$^{(}$ 1 1 + t/d$^{)}$$^{2}$ Z $^{2}$, (A3) For most energies in question, G$_{2}$ ( t ) is dominated by an elastic component The integral in (A1) receives equal contributions at all t , and so is logarithmically sensitive to t$_{min}$ = ( m 2 A ′ / 2 E$_{0}$ ) 2 and t$_{max}$ = m 2 A ′ . (the other form factor, G$_{1}$ ( t ), contributes only a negligi- ble amount in all cases of interest). Since we are domi- nated by a coherent scattering with G$_{2}$ ∝ Z $^{2}$, it is useful to define a reduced photon flux, χ ≡ ∫ t$_{max}$ t$_{min}$ dt t − t$_{min}$ t 2 G$_{2}$ ( t ) (A1) ˜ χ ≡ χ/Z $^{2}$. (A2)