where m 2 π = m 2 L and m 2 K = ( m 2 L + m 2 $_{H}$) / 2. Similarly, the expression for M$_{x}$ becomes The expressions for Φ$_{l}$ and Φ$_{h}$ agree with those derived by Goity in Ref. [85]. One can account for lattice finite volume effects at NLO in χ PT by turning the one-loop integrals to sums. This yields an additive correction to the chiral logarithms [86]: where δ F V i ( m L) is the finite volume correction to the infinite volume result and K$_{0}$ and K$_{1}$ are modified Bessel functions of imaginary argument. The NLO SU (2) HM χ PT expressions can easily be obtained from the SU (3) results in the previous subsection by integrating out the strange valence and sea quarks. After this procedure, however, the expressions for the decay constant and mixing matrix element differ for B$_{d}$ -type mesons and B$_{s}$ -type mesons. First we consider B$_{d}$ -type mesons composed of a b -quark and a light valence quark with mass m$_{x}$ . In this case, we take the limits of Eqs. (D2) and (D3) assuming The resulting expression for Φ$_{B}$$_{x}$ at NLO in the partially-quenched domain-wall theory with 55 m$_{x}$ m$_{h}$ , m$_{l}$ m$_{h}$ ≪ 1 . (D15) ˜ ℓ ( m $^{2}$) = − ln m 2 Λ 2 χ + 1 ) + δ F V 3 ( m L) , δ F V 3 ( m L) = 2 ⃗ r =0 K$_{0}$ ( | ⃗ r | m L) , (D14) ℓ ( m $^{2}$) = m 2 ( ln m 2 Λ 2 χ + δ F V 1 ( m L) ) , δ F V 1 ( m L) = 4 m L ∑ ⃗r =0 K$_{1}$ ( | ⃗ r | m L) ⃗ r , (D13) M$_{B}$$_{h}$ = β$_{0}$ { 1 − 1 + 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 [ 2 ℓ ( m 2 $_{K}$) + ℓ ( m 2 $_{S}$) ] − 1 − 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 ℓ ( m 2 $_{S}$) + 1 24 π $^{2}$f 2 [ 3 ℓ ( m 2 $_{S}$) − 2 ℓ ( m 2 $_{η}$) ] + d$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + d$_{val}$m$_{h}$ + d$_{a}$a 2 } . (D12) M$_{B}$$_{l}$ = β$_{0}$ { 1 − 1 + 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 [ 2 ℓ ( m 2 $_{π}$) + ℓ ( m 2 $_{K}$) ] − 1 − 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 ℓ ( m 2 $_{π}$) + 1 48 π $^{2}$f 2 [ 3 ℓ ( m 2 $_{π}$) − ℓ ( m 2 $_{η}$) ] + d$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + d$_{val}$m$_{l}$ + d$_{a}$a 2 } , (D11) 2. SU (2) HM χ PT expressions