field theory, it cannot be ruled out by the data. We therefore take the difference between the extrapolated values for f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ obtained with the preferred fit and with a purely linear fit to estimate the systematic error due to the choice of matching point; this leads to an error in f$_{B}$$_{s}$/f$_{B}$$_{d}$ of 6.6% for both the APE and HYP data and in ξ of 6.8% (6.9%) for the APE (HYP) data. We compare the results of the preferred fit with those of the purely linear extrapolation in Fig. 7 for Φ$_{B}$$_{s}$ / Φ$_{B}$$_{l}$ and in Fig. 8 for √ m$_{B}$$_{s}$/m$_{B}$$_{l}$ ξ . Comparison with the linear extrapolation leads to a conservative systematic error estimate since we know that, when the pion mass is sufficiently light, the chiral logarithms will become important and the extrapolation function will begin to curve upward. However, the linear fit provides a clear lower-bound on the possible extrapolated value. In order to estimate the error due to the omission of higher-order terms in the chiral expansion, we compare the result of the linear plus NLO SU (2) HM χ PT fit to a fit sup- plemented by NNLO analytic terms. Once the valence and sea strange quarks have been integrated out of the chiral effective theory, the SU (2) HM χ PT expressions can only depend upon the light up-down quark mass m$_{l}$ and the lattice spacing a . Thus the only possible NNLO analytic terms are those proportional to m 2 $_{l}$, m$_{l}$a $^{2}$, and a $^{4}$. Because we are working at 32 FIG. 7: Chiral extrapolation of Φ$_{B}$$_{s}$ / Φ$_{B}$$_{l}$ = √ m$_{B}$$_{s}$/m$_{B}$$_{l}$ · f$_{B}$$_{s}$/f$_{B}$$_{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 ( m$_{l}$ + m$_{res}$ ) Φ B s / Φ B l linear: Φ$_{B}$$_{s}$ / Φ$_{B}$$_{d}$ =1 . 088(82) linear + SU (2) HM χ PT: Φ$_{B}$$_{s}$ / Φ$_{B}$$_{d}$ =1 . 165(88) a ( m$_{l}$ + m$_{res}$ ) linear: Φ$_{B}$$_{s}$ / Φ$_{B}$$_{d}$ =1 . 077(44) linear + SU (2) HM χ PT: Φ$_{B}$$_{s}$ / Φ$_{B}$$_{d}$ =1 . 153(47)