The universe seems to be filled with a mysterious substance called Dark Matter, which appears to interact gravitationally, with a density of about 0.3 GeV/cm3. Since we cannot see the dark matter, it cannot have electric charge. On the other hand, it must have some non-grav- itational interactions to match our models of early universe cosmology. The leading candidates for what dark matter might be are Weakly Interacting Massive Particles, or WIMPs. For example, in supersymmetry, a natural WIMP appears as the Bino, which is the Fermionic superpartner of the Z-boson. 1. Direct Detection One way to look for WIMPs is though their weak interactions with matter. For example, the LUX experiment (which Professor Morii works on), looks for WIMP scat- tering off of Xenon nuclei. Call the WIMP ψ (a Fermion) and the Xenon φ (a scalar). Then a appropriate Lagrangian describing some of the interactions of ψ, φ and the Z boson is L = − 1 ( F Z ) 2 + m 2 Z 2 − 1 φ ( ﰀ + m 2 ) φ + ψ ̄ ( i ∂ + M ) ψ ( 1 ) 4μν Zμ2 +igZμψ ̄γμψ+iNgZμ[φ⋆∂μφ−φ∂μφ⋆] (2) Note the factor of N = 131 (Xenon’s atomic mass number) which results from the Z interacting with all of the nucleons in Xe coherently. LUX looks for the scatting process ψφ → ψφ by measuring the recoil energy of the φ. a) Draw the relevant Feynman diagram(s) for ψφ → ψφ. Calculate |M|2 for this process, averaging over initial spins and summing over final spins. b) Calculate the total cross section for a WIMP of energy E and mass M to scatter off a Xenon nucleus, in the rest frame of the Xenon. c) WIMPs are expected to be highly energetic, with an average velocity in our neck of the galaxy of 230km . How much Xenon would you need to see 10 scattering events in 1 year for a WIMP mass of 100 GeV? You can take g = 0.3 and mZ = 90 GeV.