continuum calculation of Sharpe and Zhang in the limit of three degenerate sea quarks [23]. We note that one is free to multiply the above expressions by arbitrary powers of the heavy- light meson mass m$_{B}$$_{x}$ without modifying the chiral logarithms. This is because the chiral logarithm contributions to heavy-light meson masses are suppressed by 1 /m$_{b}$ and therefore of higher-order than we consider [84]. In the above expressions for Φ$_{B}$$_{x}$ and M$_{B}$$_{x}$ , the functions ℓ and ˜ ℓ are one-loop chiral logarithms: The function R [ n,k ] j is due to single poles in the partially-quenched propagator: and the sets of flavor-singlet masses that appear in the residue functions are where m 2 X is the mass-squared of a meson composed of two x valence quarks, m 2 L ( H ) is the mass-squared of a meson composed of two l ( h ) sea quarks, and m 2 η = ( m 2 L + 2 m 2 $_{H}$) / 3 for 2+1 flavors of sea quarks. For completeness, we also include the expressions for the decay constant and mixing ma- trix element at the unitary points. In the full QCD and isospin limits, the above expression for Φ$_{x}$ becomes 54 ℓ ( m $^{2}$) ≡ m $^{2}$ln ( m 2 Λ 2 χ ) , (D4) ˜ ℓ ( m $^{2}$) ≡ − ln m 2 Λ 2 χ − 1 . (D5) R [ n,k ] j ( { m } , { µ } ) ≡ ∏ k a $_{=1}$( µ 2 a − m 2 $_{j}$) n i =1 ,i = $_{j}$( m 2 i m 2 $_{j}$) (D6) { µ } = { m 2 $_{L}$, m 2 H } , (D7) { M$_{X}$ } = { m 2 $_{X}$, m 2 η } , (D8) Φ$_{B}$$_{l}$ = φ$_{0}$ { 1 − 1 16 π $^{2}$f 2 ( 1 + 3 g 2 B ∗ Bπ 2 ) [ 3 2 ℓ ( m 2 $_{π}$) + ℓ ( m 2 $_{K}$) + 1 6 ℓ ( m 2 $_{η}$) ] + c$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + c$_{val}$m$_{l}$ + c$_{a}$a 2 } , (D9) Φ$_{B}$$_{h}$ = φ$_{0}$ { 1 − 1 16 π $^{2}$f 2 ( 1 + 3 g 2 B ∗ Bπ 2 ) [ 2 ℓ ( m 2 $_{K}$) + 2 3 ℓ ( m 2 $_{η}$) ] + c$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + c$_{val}$m$_{h}$ + c$_{a}$a 2 } , (D10)