tions. This leads to a small amount of chiral symmetry breaking which can be parameterized in terms of an additive shift to the bare domain-wall quark mass. At the value L$_{s}$ = 16 used in our simulations we obtain a residual quark mass of am$_{res}$ = 0 . 00315(2) [34]. Because mixing between heavy-light four-fermion operators of different chiralities is proportional to the value of am$_{res}$ , this indicates that the size of errors from mixing with wrong-chirality operators is negligible. We use the Iwasaki gauge action for the gluons [39]: where β ≡ 6 /g 2 0 and g$_{0}$ is the bare lattice coupling. U$_{P}$ is the path-ordered product of gauge links around the 1 × 1 plaquette P and U$_{R}$ is the path-ordered product of gauge links around the 1 × 2 rectangle R . The constant c$_{1}$ is set to − 0 . 331 in the Iwasaki action and we use β = 2 . 13 in our simulations. As was shown in Refs. [40, 41] for the quenched approximation, the use of the Iwasaki action in combination with domain-wall valence quarks leads to improved chiral symmetry and a smaller residual quark mass than for the Wilson gauge action [42]. In the case of N$_{f}$ = 2 + 1 dynamical domain-wall simulations, the use of the Iwasaki action also allows frequent tunneling between topological sectors [43]. We use an improved static action for the b -quarks in the 2-point and 3-point correlation functions needed to compute the decay constants and matrix elements. We build upon the original lattice formulation of the static effective action that was constructed by Eichten and Hill [44]: where h ( x ) is the static quark field at site x , U$_{0}$ ( x ) is the gauge link in the temporal direction and ̂ 0 denotes the unit vector along the temporal direction. The projectors P$_{±}$ = 1 $_{2}$(1 ± γ$_{0}$ ) select the parity even and odd components of h ( x ), which we denote by h $^{(+)}$( x ) and h ( − $^{)}$( x ). 8 S$_{static}$ = a 3 ∑ x,y ( h ( x )[ δ$_{x,y}$ − U † $_{0}$( y ) δ$_{x}$$_{−}$$_{̂}$ 0 $_{,y}$] P$_{+}$h ( y ) − h ( x )[ δ$_{x,y}$ − U$_{0}$ ( x ) δ$_{x}$$_{+}$$_{̂}$ 0 $_{,y}$] P$_{−}$h ( y ) ) , (12) S$_{gauge}$ = − β 3 ( (1 − 8 c$_{1}$ ) ∑ P Re Tr [ U$_{P}$ ] + c$_{1}$ ∑ R ReTr[ U$_{R}$ ] ) , (11) B. Gluon action C. Heavy-quark action