Theoretical progress in describing the B-meson lifetimes

The present status of the theoretical estimates of the difference between the widths of the neutral Bs-mesons and of the B-meson lifetime ratios is reviewed. In particular, the lattice results for the matrix elements of the relevant ∆B = 2 operators are updated and the first lattice QCD results for the matrix elements of ∆B = 0 operators are presented. In both cases, the NLO perturbative QCD corrections in the coefficient functions have been included. The theoretically updated results are: (∆Γ/Γ)Bs = 7±4 %, τ(B)/τ(Bd) = 1.07(3) and τ(Bs)/τ(Bd) = 1.00(2). I discuss the following two topics: ◦ (∆Γ/Γ)Bs , the quantity that recently attracted quite a bit of attention among theorists and for which the experimental upper limit has been set at [1]: ( ∆Γ Γ ) Bs < 0.31 (95% C.L.) . (1) ◦ τ(Bu(s))/τ(Bd) have been measured quite accurately [1] τ(Bu) τ(Bd) = 1.07(2) , τ(Bs) τ(Bd) = 0.95(4) . (2) Important theoretical progress in computing these ratios has been made this year. I will not discuss the theoretical predictions for the ratio τ(Λb)/τ(Bd), where, in my opinion, substantial progress is yet to be made. Theoretical set-up for both of the above topics relies on the hypothesis of the (global and local) quark–hadron duality [2]. The validity of that assumption is not totally clear, although the impressive agreement of many theoretical predictions in τ -physics (for which the duality has been assumed) with the precise experimental data is very encouraging [3]. ∗Speaker. h e p 2 0 0 1 International Europhysics Conference on High Energy Physics DAMIR BECIREVIC 1. WIDTH DIFFERENCE OF THE B S-SYSTEM It has been demonstrated in ref. [4] that in the combined Nc → ∞ and SV limit ∗, the quark–hadron duality for ∆ΓBs indeed works. Out of that limit, however, the quark–hadron duality is again an assumption. The (modern) theoretical expression, based on the operator product expansion (OPE), for ∆ΓBs has been derived in ref. [6]: ∆ΓBs = GFm 2 b 12πmBs |V ∗ cbVcs| { G1(μ)〈B̄s|O1(μ)|Bs〉+G2(μ)〈B̄s|O2(μ)|Bs〉+ δ1/mb } ,(1.1) where the flavour structure of the operators O1,2(μ) is ∆B = 2; δ1/mb contains all the contributions from the 1/mb corrections to the first two terms. Corrections ∝ 1/mn≥2 b are neglected. ♣ Short distance physics is encoded in the functions G1,2(μ) which are the combinations of ∆B = 1 Wilson coefficients. The next-to-leading order (NLO) corrections to these functions have been computed in ref. [7], where the authors also kept the ratio mc/mb 6= 0. The residual scale dependence of ∆B = 1 Wilson coefficients entering the functions G1,2(mb) is estimated to be −20% and +15%; ♣ Long distance QCD dynamics is described by the matrix elements, which are parametrized as 〈B̄s|O1(μ)|Bs〉 ≡ 〈B̄s|(b̄s)V−A(b̄s)V−A|Bs〉 = 8 3 f Bsm 2 BsB1(μ) , 〈B̄s|O2(μ)|Bs〉 ≡ 〈B̄s|(b̄s)S−P (b̄s)S−P |Bs〉 = − 3 ( fBsm 2 Bs mb(μ) +ms(μ) )2 B2(μ) . (1.2) The above parameters B1,2 are equal to unity in the vacuum saturation approximation (VSA). A priori, VSA gives a gross estimate and one has to include the (non-factorizable) non-perturbative QCD effects. QCD simulations on the lattice represent a suitable method for that part of the job. This year progress in reducing uncertainties of the heavy quark extrapolation of the Bparameters (1.2) has been reported in ref. [10]. Besides several ‘minor’ (albeit important) improvements, we combined the static HQET results of ref. [8] with those of ref. [9], where lattice QCD is employed for the mesons of masses, 1.8 GeV . mP . 2.4 GeV. To use the HQ scaling laws we matched the QCD matrix elements with the HQET ones, so that we could actually “interpolate” to the mass of the Bs-meson. The obtained results are then matched back to their full QCD values. This matching HQET↔ QCD (CB(mP ) in fig. ??) is for the first time made at NLO in perturbation theory. For consistency, the computation of the B-parameters is performed precisely in the MS(NDR) scheme in which the functions Gi(μ) have been calculated [7]. We obtain the following results B1(mb) = 0.87(2)(5) , B2(mb) = 0.84(2)(4) , (1.3) ∗SV (Shifman–Voloshin limit) is the limit in which ΛQCD mb − 2mc mb [5].