variations are discussed in greater detail when we estimate the systematic uncertainty due to the chiral extrapolation in Sec. VI A. In the case of the decay constants, we extrapolate the ratio while for the mixing matrix elements we consider From here on, we drop the superscript “ren” for simplicity because we only refer to the renor- malized quantities. The expressions for the heavy-light meson decay constant and mixing matrix element to NLO in the light-quark expansion, but zeroth order in 1 /m$_{b}$ , are given in Appendix D. For completeness, we present formulae for both SU (3) and SU (2) HM χ PT. The SU (3) fit functions depend on the valence and sea light-quark masses ( m$_{s}$,m$_{l}$,m$_{h}$ ) and the lattice spacing a . The SU (2) functions are obtained from the SU (3) expressions by taking the limit ( m$_{l}$ $_{m$_{s}$}$, m$_{l}$ m$_{h}$ ) ≪ 1. Thus they only apply in the region in which the value of the average up-down quark mass is much smaller than the valence and sea strange quark masses. Furthermore, because the strange quark has been integrated out of the SU (2) the- ory, the SU (2) fit functions only depend upon the light-quark mass m$_{l}$ and the lattice spacing a . Therefore the expressions for the SU (3)-breaking ratios at NLO in SU (2) HM χ PT are particularly simple: where m 2 L = 2 B ( m$_{l}$ + m$_{res}$ ) is the tree-level mass-squared of a pseudoscalar meson composed of two quarks with mass m$_{l}$ , B, and f are the leading-order low-energy constants of χ PT, and the quark masses in the analytic terms are expressed in terms of dimensionless ratios to make the coefficients C$_{l}$ and D$_{l}$ of O (1). These functional forms are derived from Eqs. (D16)– (D20) by taking the ratio of the expressions for the valence quark y = s over the expressions for x = l . Because we are working at a single lattice spacing, the analytic terms proportional to a 2 are absorbed into the values of the leading-order coefficients R$_{Φ}$ and R$_{M}$ . Note that 27 √ M$_{B}$$_{s}$ M$_{B}$$_{l}$ = R$_{M}$ 1 + 2 + 3 g 2 B ∗ Bπ (4 πf ) 2 1 2 m 2 $_{L}$ln m 2 L Λ 2 χ + D$_{l}$ Bm$_{l}$ (4 πf ) 2 , (79) Φ$_{B}$$_{s}$ Φ$_{B}$$_{l}$ = R$_{Φ}$ { 1 + 1 + 3 g 2 B ∗ Bπ (4 πf ) 2 ( 3 4 ) m 2 $_{L}$ln ( m 2 L Λ 2 χ ) + C$_{l}$ 2 Bm$_{l}$ (4 πf ) 2 } , (78) √ M ren B$_{s}$ M ren B$_{l}$ = √ m$_{B}$$_{s}$ m$_{B}$$_{l}$ ( f$_{B}$$_{s}$ √ B$_{B}$$_{s}$ f$_{B}$$_{l}$ B$_{B}$$_{l}$ ) . (77) Φ ren B$_{s}$ Φ ren B$_{l}$ = √ m$_{B}$$_{s}$ m$_{B}$$_{l}$ f$_{B}$$_{s}$ f$_{B}$$_{l}$ , (76)