Radioactivity Backgrounds in ZEPLIN-III

We examine electron and nuclear recoil backgrounds from radioactivity in the ZEPLIN-III dark matter experiment at Boulby. The rate of low-energy electron recoils in the liquid xenon WIMP target is 0.75$\pm$0.05 events/kg/day/keV, which represents a 20-fold improvement over the rate observed during the first science run. Energy and spatial distributions agree with those predicted by component-level Monte Carlo simulations propagating the effects of the radiological contamination measured for materials employed in the experiment. Neutron elastic scattering is predicted to yield 3.05$\pm$0.5 nuclear recoils with energy 5-50 keV per year, which translates to an expectation of 0.4 events in a 1-year dataset in anti-coincidence with the veto detector for realistic signal acceptance. Less obvious background sources are discussed, especially in the context of future experiments. These include contamination of scintillation pulses with Cherenkov light from Compton electrons and from $\beta$ activity internal to photomultipliers, which can increase the size and lower the apparent time constant of the scintillation response. Another challenge is posed by multiple-scatter $\gamma$-rays with one or more vertices in regions that yield no ionisation. If the discrimination power achieved in the first run can be replicated, ZEPLIN-III should reach a sensitivity of $\sim 1 \times 10^{-8}$ pb$\cdot$year to the scalar WIMP-nucleon elastic cross-section, as originally conceived.