Vector Dark Matter Detection using the Quantum Jump of Atoms

The hidden sector U(1) vector bosons created from inflationary fluctuations can be a substantial fraction of dark matter if their mass is around $10^{-5}$eV. The creation mechanism makes the vector bosons' energy spectral density $\rho_{cdm}/\Delta E$ very high. Therefore, the dark electric dipole transition rate in atoms is boosted if the energy gap between atomic states equals the mass of the vector bosons. By using the Zeeman effect, the energy gap between the 2S state and the 2P state in hydrogen atoms or hydrogen like ions can be tuned. The $2S$ state can be populated with electrons due to its relatively long life, which is about $1/7$s. When the energy gap between the semi-ground $2S$ state and the 2P state matches the mass of the cosmic vector bosons, induced transitions occur and the 2P state subsequently decays into the 1S state. The $2P\to1S$ decay emitted Lyman-$\alpha$ photons can then be registered. The choices of target atoms depend on the experimental facilities and the mass ranges of the vector bosons. Because the mass of the vector boson is connected to the inflation scale, the proposed experiment may provide a probe to inflation.Comment: 5 pages, 3 figures; references added; matches version published in PL

Matrix resolvent and the discrete KdV hierarchy

Based on the matrix-resolvent approach, for an arbitrary solution to the discrete KdV hierarchy, we define the tau-function of the solution, and compare it with another tau-function of the solution defined via reduction of the Toda lattice hierarchy. Explicit formulae for generating series of logarithmic derivatives of the tau-functions are then obtained, and applications to enumeration of ribbon graphs with even valencies and to the special cubic Hodge integrals are considered.Comment: It was added a remark after Theorem 1; added Appendix A; added references; two statements moved to Introduction; corrected typos. 26 page