a ( m$_{l}$ + m$_{res}$ ) √ M B s /M B l · ξ linear: √ √$^{m$_{B}$$_{s}$}$ $_{m$_{B}$$_{d}$}$ξ = 1 . 064(67) linear + SU (2) HM χ $_{PT:}$√ √ √$^{m$_{B}$$_{s}$}$ $_{m$_{B}$$_{d}$}$ξ = 1 . 142(72) a ( m$_{l}$ + m $_{res}$) $_{linear:}$√ √ √$^{m$_{B}$$_{s}$}$ m$_{B}$$_{d}$ ξ = 1 . 065(50) linear + SU (2) HM χ $_{PT:}$√ √ √$^{m$_{B}$$_{s}$}$ m$_{B}$$_{d}$ ξ = 1 . 144(54) FIG. 8: Chiral extrapolation of √ M$_{B}$$_{s}$/M$_{B}$$_{l}$ = √ m$_{B}$$_{s}$/m$_{B}$$_{l}$ ( f$_{B}$$_{s}$ √ B$_{B}$$_{s}$/f$_{B}$$_{l}$ √ B$_{B}$$_{l}$ ) using a linear fit matched onto NLO SU (2) HM χ PT at the lightest data point (solid band) and a linear extrapolation all the way to the chiral limit (hatched band). Errors shown are statistical only. The left plot shows the comparison for the APE data, while the right plot shows the same comparison for they HYP- smeared data. a single lattice spacing, the contribution proportional to a 4 is indistinguishable from the LO normalization factors R$_{Φ}$ and R$_{M}$ in Eqs. (78) and (79) and the contribution proportional to m$_{l}$a 2 is indistinguishable from the NLO terms C$_{l}$m$_{l}$ and D$_{l}$m$_{l}$ . Thus there is only one ad- ditional free parameter at NNLO. In practice, we implement the “NNLO fit” by performing a quadratic fit in m$_{l}$ to our data and then matching to the NLO SU (2) HM χ PT expressions supplemented by the NNLO analytic term proportional to m 2 l at the value of our lightest data point. When we include the term quadratic in m$_{l}$ , however, the extrapolated values for f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ have significantly larger statistical errors (about 20 − 40%) than those obtained with the preferred fit (about 4 − 8%). Therefore we cannot draw any meaningful conclusion about the change in the central values of f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ , since the outcome of the NNLO extrapolation is consistent with the NLO extrapolation within the large statistical errors. This increase in statistical errors is to be expected since we have introduced an extra free parameter, and it is difficult to constrain two parameters with only three data points. Thus we do not use the “NNLO fit” as a way to estimate the chiral extrapolation error in this work, but leave it as a way to estimate the uncertainty due to the omission of higher-order terms in future analyses when we have more data points and smaller statistical 33