the physical quark masses and the continuum. Although we label the formulae with the subscript “ B ”, we note that these functions can also be used to extrapolate D -meson decay constants and mixing matrix elements with the caveat that the low-energy constants (except for the light-light meson tree-level parameters f and B ) are different for the case of B -mesons and D -mesons. We first show the SU (3) HM χ PT formulae in Sec. D 1; we then take the appropriate limits of the SU (3) expressions to obtain those in SU (2) HM χ PT in Sec. D 2. The tree-level mass-squared of a meson composed of two domain-wall valence quarks with flavors x and y is where B is a continuum low-energy constant and m$_{res}$ is the residual quark mass. The NLO result for Φ$_{B}$$_{x}$ = f$_{B}$$_{x}$ √ m$_{B}$$_{x}$ in the partially-quenched domain-wall theory with 2+1 flavors of sea quarks is [81, 82]: where f ≈ 130 . 4 MeV is the tree-level pion decay constant. The NLO expression for M$_{B}$$_{x}$ = 8 / 3 m$_{B}$$_{x}$f 2 $_{B$_{x}$}$B$_{B}$$_{x}$ is similar [81, 83]: In both the decay constant and the mixing matrix element, the only effect of the nonzero lattice spacing is a new analytic term proportional to a $^{2}$. These results agree with the 53 M$_{B}$$_{x}$ = β$_{0}$ { 1 − 1 + 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 ∑ f = l,l,h ℓ ( m 2 $_{xf}$) − 1 − 3 g 2 B ∗ Bπ 16 π $^{2}$f 2 ℓ ( m 2 $_{X}$) + 1 24 π $^{2}$f 2 [ R [2 , 2] X ( { M$_{X}$ } ; { µ } ) ˜ ℓ ( m 2 $_{X}$) − ∑ j ∈{ M$_{X}$ } ∂ ∂m 2 X ( R [2 , 2] j ( { M$_{X}$ } ; { µ } ) ) ℓ ( m 2 $_{j}$) ] + d$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + d$_{val}$m$_{x}$ + d$_{a}$a 2 } . (D3) Φ$_{B}$$_{x}$ = φ$_{0}$ { 1 − 1 16 π $^{2}$f 2 1 + 3 g 2 B ∗ Bπ 2 ∑ f = l,l,h ℓ ( m 2 $_{xf}$) + 1 16 π $^{2}$f 2 1 + 3 g 2 B ∗ Bπ 6 [ R [2 , 2] X ( { M$_{X}$ } ; { µ } ) ˜ ℓ ( m 2 $_{X}$) − ∑ j ∈{ M$_{X}$ } ∂ ∂m 2 X ( R [2 , 2] j ( { M$_{X}$ } ; { µ } ) ) ℓ ( m 2 $_{j}$) ] + c$_{sea}$ (2 m$_{l}$ + m$_{h}$ ) + c$_{val}$m$_{x}$ + c$_{a}$a 2 } , (D2) m 2 xy = B ( m$_{x}$ + m$_{y}$ + 2 m$_{res}$ ) , (D1) 1. SU (3) HM χ PT expressions