  We demonstrate a method for calculating the neutral B-meson decay constants
and mixing matrix elements in unquenched lattice QCD with domain-wall light
quarks and static b-quarks. Our computation is performed on the "2+1" flavor
gauge configurations generated by the RBC and UKQCD Collaborations with a
lattice spacing of a approx 0.11 fm (a^-1 = 1.729 GeV) and a lattice spatial
volume of approximately (1.8 fm)^3. We simulate at three different light sea
quark masses with pion masses down to approximately 430 MeV, and extrapolate to
the physical quark masses using a phenomenologically-motivated fit function
based on next-to-leading order heavy-light meson SU(2) chiral perturbation
theory. For the b-quarks, we use an improved formulation of the Eichten-Hill
action with static link-smearing to increase the signal-to-noise ratio. We also
improve the heavy-light axial current used to compute the B-meson decay
constant to O(alpha_s p a) using one-loop lattice perturbation theory. We
present initial results for the SU(3)-breaking ratios f_{B_s}/f_{B_d} and xi =
f_{B_s} sqrt{B_{B_s}}/f_{B_d} sqrt{B_{B_d}}, thereby demonstrating the
viability of the method. For the ratio of decay constants, we find
f_{B_s}/f_{B_d} = 1.15(12) and for the ratio of mixing matrix elements, we find
xi = 1.13(12), where in both cases the errors reflect the combined statistical
and systematic uncertainties, including an estimate of the size of neglected
O(1/m_b) effects.
