errors. We can also estimate the systematic uncertainty in the chiral extrapolation by comparing the values of f$_{B}$$_{s}$/f$_{B}$$_{d}$ and ξ obtained by matching onto SU (2) HM χ PT with those obtained by matching onto SU (3) HM χ PT. Although the SU (2) chiral logarithms are a subset of the SU (3) chiral logarithms, the SU (2) and SU (3) theories have different series expansions and different degrees-of-convergence within their ranges of applicability. Therefore the compar- ison with NLO SU (3) HM χ PT provides another means of estimating the error due to the use of NLO SU (2) HM χ PT. The expressions for SU (3)-breaking ratios at NLO in SU (3) HM χ PT are, schematically, where “chiral logs” indicate non-analytic functions of the pseudo-Goldstone meson masses, e.g. m 2 $_{L}$log( m 2 $_{L}$/ Λ 2 $_{χ}$). These are derived from Eqs. (D2) and (D3) by taking the ratio of the expressions for the valence quark x = s over the expressions for x = l . Because the strange quark is treated in the same manner as the up and down quarks in the SU (3) chiral effective theory, the low-energy constants are the same in the numerator and denominator. Thus the overall normalizations cancel in the expressions in Eqs. (80) and (81) and the ratios are constrained to unity in the limit m$_{l}$ → m$_{s}$ . Therefore the expressions for the SU (3)- breaking ratios have one free parameter each, instead of two as in the SU (2) HM χ PT case. This means that the SU (3) HM χ PT expressions cannot be matched smoothly onto the linear fit of the heavy data. We choose to make the value of the extrapolation function continuous, while leaving a discontinuity in the slope at the matching point. The difference between the linear plus NLO SU (2) HM χ PT fit and the linear plus NLO SU (3) HM χ PT fit leads to a difference in f$_{B}$$_{s}$/f$_{B}$$_{d}$ of 2.3% (2.4%) for the APE (HYP) data and in ξ of 2.6% (2.5%) for APE (HYP). In the preferred linear plus NLO SU (2) HM χ PT fit, we set the leading-order pseudoscalar meson decay constant f equal to the experimentally-measured value of f$_{π}$ . This fixes the coefficient of the chiral logarithms and improves the convergence of the chiral expansion [63, 64]. At NLO in χ PT, however, it is equally consistent to use the pseudoscalar decay constant in the SU (2) chiral limit f$_{0}$ or the kaon decay constant f$_{K}$ because the different choices only 34 Φ$_{B}$$_{s}$ Φ$_{B}$$_{l}$ = 1 + “chiral logs” + 2 B (4 πf ) 2 ˜ c$_{val}$ ( m$_{s}$ − m$_{l}$ ) , (80) B √ M$_{B}$$_{s}$ M$_{B}$$_{l}$ = 1 + “chiral logs” + B (4 πf ) 2 d$_{val}$ ( m$_{s}$ − m$_{l}$ ) , (81)