I. INTRODUCTION Neutral B -meson mixing is a sensitive probe of quark flavor-changing interactions. When combined with experimental measurements of the B 0 d and B 0 s oscillation frequencies, pre- cise QCD determinations of the B 0 d and B 0 $_{s}$-mixing hadronic matrix elements allow clean determinations of the Cabibbo-Kobayashi-Maskawa (CKM) [1, 2] matrix elements | V$_{td}$ | and | V$_{ts}$ | with all sources of systematic uncertainty under control. In the Standard Model, the mass-difference of the neutral B -meson mass eigenstates Δ m$_{q}$ (often called the oscillation frequency) is given by [3, 4] where q ∈ { d, s } . Both the Inami–Lim function S$_{0}$ ( x$_{t}$ ) with x$_{t}$ = m 2 $_{t}$/m 2 W [5] and the QCD coefficient η$_{B}$ can be computed in perturbation theory [3]. The hadronic contribution to the Δ B = 2 mixing matrix element, must be calculated nonperturbatively from first principles using lattice QCD. The hadronic matrix element M q is conventionally parametrized as where m$_{B}$$_{q}$ is the mass of the B -meson, f$_{B}$$_{q}$ is the B -meson decay constant and B$_{B}$$_{q}$ is the B -meson bag parameter. Many statistical and systematic uncertainties from lattice calculations cancel in the SU (3)-breaking ratio M $_{s}$/ M $_{d}$, which would be one in the limit m$_{d}$ → m$_{s}$ . It is therefore convenient and conventional to introduce the quantity Given improved lattice calculations of the nonperturbative factor ξ , recent experimental measurements of the oscillation frequencies Δ m$_{d}$ and Δ m$_{s}$ to about 1% accuracy [6–9] now allow the possibility of precisely determining the ratio of the CKM matrix elements | V$_{td}$ | / | V$_{ts}$ | (see, for example, Ref. [10]). The ratio | V$_{td}$ | / | V$_{ts}$ | constrains the apex of the CKM unitarity triangle [11, 12]. It is likely that new physics would give rise to new quark-flavor changing interactions and additional 3 ξ ≡ f$_{B}$$_{s}$ √ B$_{B}$$_{s}$ f$_{B}$$_{d}$ B$_{B}$$_{d}$ . (4) M q = 8 3 m 2 $_{B$_{q}$}$f 2 $_{B$_{q}$}$B$_{B}$$_{q}$ , (3) M q = 〈 B 0 q ∣ [ bγ $^{µ}$(1 − γ$_{5}$ ) q ][ bγ$_{µ}$ (1 − γ$_{5}$ ) q ] ∣ B 0 q 〉 , (2) Δ m$_{q}$ = G 2 $_{F}$m 2 W 16 π $^{2}$m$_{B}$$_{q}$ | V ∗ $_{tq}$V$_{tb}$ | $^{2}$S$_{0}$ ( x$_{t}$ ) η$_{B}$ M $_{q}$, (1)