========================================================================== Summary of global analysis of heavy-light decay constants and B-parameters ========================================================================== [Hartmut Wittig, DESY, 11/7/03] (1) D-sector: decay constants ----------------------------- Juttner & Rolf (ALPHA Collab.) have computed f_Ds in the continuum limit in the quenched approximation [hep-lat/0302016]. The Ds does not suffer from any uncertainties coming from extrapolations in the quark masses, and is thus an ideal starting point. Juttner & Rolf have shown that discretisation & renormalisation effects are controlled at the level of 4%. Their results, as well as a compilation of results for f_Ds show that the central value may vary between 225 and 255 MeV, depending on the choice of scale. Thus we have (1.1) f_Ds = 240(10)(15) MeV, Nf=0 (quenched approximation), where the first error is Gaussian (statistics, extrapolation), and the second (flat) error reflects the quenching error due to the scale ambiguity. Dynamical simulations usually report an increase in the estimates for decay constants, but are not yet as mature as the quenched calculations. Rather than quoting results from simulations with Nf=2 dynamical flavours, we can estimate the ratio (f_Ds)_{Nf=2}/(f_Ds)_{Nf=0} There are several collaborations [CP-PACS, MILC, JLQCD] who have quoted values for this ratio, either determined for D-mesons or B-Mesons. The results indicate that (f_Pq)_{Nf=2}/(f_Pq)_{Nf=0}, P=D,B q=s,d is rather insensitive to different heavy quark masses. There is also only a weak dependence on the light flavour. The results are represented well by (1.2) (f_Pq)_{Nf=2}/(f_Pq)_{Nf=0}=1.10(6), P=D,B q=s,d which we will take to estimate the shift in the central value of decay constants due to unquenching. We get (1.3) f_Ds = 264(11)(17)(14) MeV, Nf=2 first error: statistical second error: scale uncertainty (quenching) third error: uncertainty in eq. (1.2) where the last two errors can be combined to yield the total quenching error. To get the decay constant f_D one needs to multiply (1.3) with the SU(3)-flavour breaking ratio f_Ds/f_D. For Nf=2 this has been quoted as f_Ds/f_D=1.12(4) at Lattice 2001 by Ryan. In the meantime, there has been much debate as to the influence of chiral logarithms [Kronfeld+Ryan: hep-ph/0206058, Becirevic et al: hep-ph/0211271, JLQCD Collab.: hep-ph/0307039]. The upshot is that f_Ps/f_Pd for P=B,D could be underestimated by 10%, due to the appearance of chiral logarithms in the light quark mass dependence of f_Pq. This argument is supported by the well-known failure to reproduce the experimental value of f_K/f_pi in lattice simulations. In this study I take (1.4) f_Ds/f_Dd = 1.12(4)(+0.11), Nf=2 which is the result based on the naive extrapolation plus an additional upward error reflecting the uncertainty due to the chiral logarithm. This gives (1.5) f_D = 236(13)(20)(-21) MeV, Nf=2 first error: statistical + first error of (1.4) second error: combined quenching error third error: uncertainty due to chiral logarithm I have NOT made an attempt to estimate the result for Nf=3, since there is no direct determination so far, and since there is no argument that these quantities are a simple, linear function of Nf. (2) B-sector: decay constants ----------------------------- Calculations in the B-sector either involve an extrapolation in the heavy quark mass, or a non-relativistic approximation. Several complementary formulations of the heavy quark actions are typically compared to extract global estimates. similarly to the D-sector, f_Bs has the advantage that no chiral extrapolation in the light quark mass is involved. Unlike f_Bs, the continuum limit of f_Bs has not been corroborated as it was done for f_Ds. One may still start with a 'representation' of all available data for f_Bs in the quenched approximation, i.e. (2.1) f_Bs = 200(20) MeV, Nf=0, where the error is obtained from the spread of results from different simulations, which may also use different formulations of the heavy quark. The quoted error is thus a convolution of different effects, such as statistics, discretisation & quenching effects, etc. In order to obtain the result for Nf=2, I combine (2.1) with (1.2), which gives (2.2) f_Bs = 220(25) MeV, Nf=2, where the error is the combination of the uncertainties in (1.2) and (2.1). Finally, I multiply (2.2) with f_Bs/f_Bd, which is estimated as (2.3) f_Bs/f_Bd = 1.15(5)(+0.12) where again the issue of chiral logs gives rise to the asymmetric error. This gives (2.4) f_Bd = 191(23)(-0.18) MeV where the last error is exclusively due to the chiral log issue. Again, I have made no attempt to estimate the effect when going to Nf=3. (3) B-sector: B-parameter and xi -------------------------------- There are far fewer results for the B-parameter - especially with dynamical quarks [see JLQCD, hep-ph/0307039 for a recent calculation with Nf=2]. The published values for B_B are remarkably stable, given the wide variation of procedures. In particular: - the dependence on the lattice spacing is weak - details of the treatment of heavy quarks matter little - quenching effects appear to be small Recent determinations of the B-parameter are all well represented by (3.1) B_Bd(m_b) = 0.85(8) (3.2) B_Bs/B_Bd = 1.00(3) Converting the result of (3.1) into the renormalisation group invariant (RGI) B-parameter at NLO yields (3.3) B_Bd(RGI) = 1.34(12) In this analysis I have not distinguished between quenched and unquenched, partly because of the fact that still little is known about dynamical quark effects, and partly because these effects appear to be quite small. (4) Summary of 'global estimates' --------------------------------- By suitably combining the above results I arrive at f_Ds = 264(11)(17)(14) MeV f_D = 236(13)(20)(-21) MeV f_Ds/f_Dd = 1.12(4)(+0.11) f_Bs = 220(25) MeV f_Bd = 191(23)(-0.18) MeV f_Bs/f_Bd = 1.15(5)(+0.12) B_Bd(m_b) = 0.85(8), B_Bd(RGI) = 1.34(12) B_Bs/B_Bd = 1.00(3) f_Bd*sqrt{B_Bd(RGI)} = 221(28)(-21) MeV xi = 1.15(5)(+0.12)