where E$_{n}$ is the energy of the state | n 〉 , a state with unit normalization. Because of the translational invariance of either the wall source operator A W 0 or the sum of the local operator A L 0 over a temporal hyperplane, the intermediate state | n 〉 must have zero momentum. There is then a single, finite-volume state, | B$_{q}$ ( ⃗ p = 0)$_{V}$ 〉 , separated from the other excited states by a non-zero energy gap, which will dominate as the time separation t − t$_{0}$ becomes large. Thus, in the limit of large t − t$_{0}$ , Eqs. (C1) and (C2) become: where we have used translational symmetry to replace the sum over ⃗ x in Eq. (C1) with the factor L $^{3}$. Finally, Eq. (63) follows by taking the ratio of Eq. (C3) and the square root of Eq. (C4) to remove the unwanted factor 〈 0 | A L $_{0}$(0) | B$_{q}$ ( ⃗ p = 0) 〉 and recognizing that in the limit of large volume the unit-normalized state | B$_{q}$ ( ⃗ p = 0)$_{V}$ 〉 in Eq. (C3) and the covariant, delta-function normalized state | B$_{q}$ ( ⃗ p = 0) 〉 in Eq. (57) are related by: This discussion is standard and has been repeated here to provide a familiar background for the derivation of Eq. (72) where a new approach, special to the static approximation, is required. The complication in Eq. (72) arises because of the use of a box source which is not translationally invariant and which produces a superposition of B -meson states with various momenta. In the static limit, all of these states are degenerate since their energy no longer depends on their momenta. Thus, we cannot assume that the large time limit, t$_{f}$ ≫ t ≫ t$_{0}$ will project onto a unique ground state. However, as is worked out below, we can use an additional symmetry of the static approximation to show that the normalization of the box source cancels between the numerator and denominator of Eq. (72) [57]. Fortunately, this large set of degenerate states resulting from the momentum indepen- dence of the energy of the heavy-light meson, can be distinguished by a new conservation law which becomes exact in the static limit: the local conservation of heavy-quark num- ber. The absence of spatial derivatives in heavy-quark actions shown in Eqs. (12) and (16) implies that the total number of heavy quarks at each spatial site is separately conserved. This conservation law results from the invariance of the heavy-quark action under the phase 50 √ 2 m$_{B}$$_{q}$L $^{3}$| B$_{q}$ ( ⃗ p = 0)$_{V}$ 〉 → | B$_{q}$ ( ⃗ p = 0) 〉 . (C5) C $^{WW}$( t, t$_{0}$ ) = |〈 B$_{q}$ ( ⃗ p = 0)$_{V}$ | A W 0 (0) $^{†}$| 0 〉| $^{2}$e − m ∗ B $_{q}$( t − t$_{0}$ $^{)}$, (C4) C $^{LW}$( t, t$_{0}$ ) = L $^{3}$〈 0 | A L $_{0}$(0) | B$_{q}$ ( ⃗ p = 0)$_{V}$ 〉〈 B$_{q}$ ( ⃗ p = 0)$_{V}$ | A W 0 (0) $^{†}$| 0 〉 e − m ∗ B $_{q}$( t − t$_{0}$ ) (C3)