two degenerate light sea quarks is: where we use the superscript “(2)” to distinguish the SU (2) low-energy constants from their SU (3) analogs in the previous section. The NLO expression for M$_{B}$$_{x}$ is similar: These results agree with the continuum partially-quenched calculation of Sharpe and Zhang in the limit a → 0 [23]. The unitary QCD expressions can easily be obtained by the replacement m$_{x}$ → m$_{l}$ . In the SU (2) theory, the effects of the dynamical strange quark are fully contained in the values of the low-energy constants, e.g. φ (2) 0 ( m$_{h}$ ). For simplicity of notation, however, we suppress the functional dependence of the coefficients on m$_{h}$ . Next we consider B$_{s}$ -type mesons composed of a b -quark and a heavy valence quark with mass m$_{y}$ . Because the SU (2) chiral effective Lagrangian includes only two light quark flavors, this requires an extension of SU (2) χ PT to the kaon sector, and the resulting theory is sometimes called Kaon SU (2) Chiral Perturbation Theory (K χ PT) [34, 87]. In this case, we take the limits of Eqs. (D2) and (D3) assuming The resulting NLO expressions for Φ$_{B}$$_{y}$ and M$_{B}$$_{y}$ are: where we use the superscript “(s)” to distinguish the coefficients from those in Eqs. (D16) and (D17). Because the valence quark has been integrated out, there are no longer any Φ$_{B}$$_{x}$ = φ (2) 0 { 1 − 1 + 3 ( g (2) B ∗ Bπ ) 2 (4 πf $^{(2)}$) 2 ℓ ( m 2 $_{xl}$) + 1 + 3 ( g (2) B ∗ Bπ ) 2 (4 πf $^{(2)}$) 2 ( 1 4 )[ ( m 2 L − m 2 $_{X}$) ˜ ℓ ( m 2 $_{X}$) + ℓ ( m 2 $_{X}$) ] + c (2) $_{sea}$m$_{l}$ + c (2) $_{val}$m$_{x}$ + c (2) a a 2 } , (D16) M$_{B}$$_{x}$ = β (2) 0 { 1 − 1 + 3 ( g (2) B ∗ Bπ ) 2 (4 πf $^{(2)}$) 2 2 ℓ ( m 2 $_{xl}$) − 1 − 3 ( g (2) B ∗ Bπ ) 2 (4 πf $^{(2)}$) 2 ℓ ( m 2 $_{X}$) + 1 (4 πf $^{(2)}$) 2 [ ( m 2 L − m 2 $_{X}$) ˜ ℓ ( m 2 $_{X}$) + ℓ ( m 2 $_{X}$) ] + d (2) $_{sea}$m$_{l}$ + d (2) $_{val}$m$_{x}$ + d (2) a a 2 } . (D17) m$_{l}$ m$_{y}$ , m$_{l}$ m$_{h}$ ≪ 1 . (D18) Φ$_{B}$$_{y}$ = φ ( s ) 0 { 1 + c ( s ) $_{sea}$m$_{l}$ + c ( s ) a a 2 } , (D19) M$_{B}$$_{y}$ = β ( s ) 0 1 + d ( s ) $_{sea}$m$_{l}$ + d ( s ) a a 2 , (D20) 56