Remark on the theoretical uncertainty in B^0-B^0 mixing [rapid communication]

We re-examine the theoretical uncertainty in the Standard Model expression for B-B̄ mixing. We focus on lattice calculations of the ratio ξ, needed to relate the oscillation frequency of B s -B̄ 0 s mixing to the poorly known CKM element Vtd. We replace the usual linear chiral extrapolation with one that includes the logarithm that appears in chiral perturbation theory. We find a significant shift in the ratio ξ, from the conventional 1.15 ± 0.05 to ξ = 1.32 ± 0.10. It is anticipated that the oscillation frequency of B s -B̄ 0 s mixing will be measured during Run 2 of the Tevatron [ 1]. It is thus timely to assess the measurement’s impact on tests of the Cabibbo-Kobayashi-Maskawa (CKM) picture of flavor and CP violation. The CKM interpretation is limited by the poorly known hadronic matrix elements for B s ↔ B̄ s and B d ↔ B̄ d transitions. In this paper we re-examine lattice calculations of these matrix elements, focusing on the chiral extrapolation. We find that the range usually quoted is probably incorrect. In the Standard Model, the theoretical expression for the oscillation frequency is ∆mq = ( G2F m 2 W S0 16πmBq ) |V ∗ tqVtb|ηBMq, (1) where q ∈ {d, s}, S0 is an Inami-Lim function, ηB is a short-distance QCD correction, and Mq is the hadronic matrix element for B q ↔ B̄ q transitions. In Eq. (1), the parentheses consists of accurately known quantities, and |V ∗ tqVtb| is the CKM factor. The hadronic matrix element Mq = 〈B̄ q |[b̄γ(1 − γ)q][b̄γμ(1 − γ)q]|B q 〉 (2) and ηB depend on the renormalization scheme, but the product ηBMq does not. The renormalization-group invariant value of the short-distance factor is η̂B = 0.55. One should keep in mind that non-Standard physics at short distances can modify Eq. (1). For convenience we shall couch the discussion as using ∆mq and the hadronic matrix element to determine |Vtq|. The resulting value of |Vtq| can then be compared to other CKM determinations to test for deviations from the Standard Model. Mq must be computed with a non-perturbative method, such as lattice gauge theory. For historical reasons one usually writes Mq = 8 3 m2Bqf 2 Bq BBq (3) and focuses on the decay constants fBq and the bag parameters BBq . But lattice QCD gives Mq directly (and fBq separately from 〈0|b̄γμγq|B q 〉). The separation does, however, turn out to be useful, as we shall see below, when considering the dependence of fBq and BBq on the masses of the light quarks. At present the uncertainty in the matrix elements is large. A recent review [ 2] of lattice calculations quotes fBs = 230 ± 30 MeV, B̂Bs = 1.34 ± 0.10, (4) fBd = 198 ± 30 MeV, B̂Bd = 1.30 ± 0.12. (5) These estimates take into account the first (partially) unquenched calculations of fBq [ 3, 4, 5, 6, 7], several quenched calculations of BBq and preliminary results suggesting that BBq changes little when the quenched approximation is removed [ 6]. The raw Monte Carlo data in lattice calculations are generated with the light quark mass mq in the range 0.2–0.5 < mq/ms < 1, and the physical matrix elements are obtained by extrapolating mq to the down quark’s mass md. This method of reaching physically light quarks is called the chiral extrapolation, and it plays an important role in our analysis below. The frequency for B d-B̄ 0 d mixing has been measured precisely, ∆md = 0.494 ± 0.007 ps [ 8]. With Eqs. (1) and (5) the uncertainty on |Vtd| is limited to 15% by fBd √ BBd . The precision on |Vtd| will not improve until better (unquenched) lattice calculations have been carried out. The frequency for B s -B̄ 0 s mixing is known to be high, ∆ms > 15 ps −1 [ 8]. But details of the way ∆ms is extracted from the data mean that the first measurement will immediately have a precision at the percent level [ 1]. Thus, it is interesting to form the ratio