Abstract
We consider the effects of a non-vanishing strange-quark mass in the determination of the full basis of dimension six matrix elements for Bs mixing, in particular we get for the ratio of the V − A Bag parameter in the Bs and Bd system: {overline{B}}_{Q_1}^s/{overline{B}}_{Q_1}^d={0.987}_{-0.009}^{+0.007} . Combining these results with the most recent lattice values for the ratio of decay constants {f}_{B_s}/{f}_{B_d} we obtain the most precise determination of the ratio xi ={f}_{B_s}sqrt{{overline{B}}_{Q_1}^s}/{f}_{B_d}sqrt{{overline{B}}_{Q_1}^d}={1.2014}_{-0.0072}^{+0.0065} in agreement with recent lattice determinations. We find ΔMs = (18.5− 1.5+ 1.2)ps− 1 and ΔMd = (0.547− 0.046+ 0.035)ps− 1 to be consistent with experiments at below one sigma. Assuming the validity of the SM, our calculation can be used to directly determine the ratio of CKM elements |Vtd/Vts| = 0.2045− 0.0013+ 0.0012, which is compatible with the results from the CKM fitting groups, but again more precise.
Highlights
We consider the effects of a non-vanishing strange-quark mass in the determination of the full basis of dimension six matrix elements for Bs mixing, in particular we get for the ratio of the V − A Bag parameter in the Bs and Bd system: B s Q1
Our paper is organised as follows: in section 2 we set up the sum rule for the Bag parameter and determine the ms corrections, in section 3 we present a numerical study of the sum rules and we perform a phenomenological analysis
Compared to the absolute Bag parameters we reduce the intrinsic sum rule error to 0.005, the condensate error to 0.002 and the uncertainty due to power corrections to 0.002 since the respective uncertainties cancel to a large extend in FNAL/MILC'16
Summary
While our HQET basis is defined as Q1 = h{i (+)γμ(1 − γ5)si h(j−)}γμ(1 − γ5)sj , Q4 = h{i (+)(1 − γ5)si h(j−)}(1 + γ5)sj, Q2 = h{i (+)(1 − γ5)si h(j−)}(1 − γ5)sj, Q5 = h{i (+)(1 − γ5)sj h(j−)}(1 + γ5)si,. We use the same basis of evanescent operators. As mentioned in [15], the HQET evanescent operators are defined up to 3 constants ai with i = 1, 2, 3 in order to gauge the scheme dependence. The QCD bag parameters BQs are defined through [20]. Where MBs denotes the Bs meson mass, mq corresponds to quark pole masses and the Bs meson decay constant fBs is defined by 0| ̄bγμγ5s|Bs(p) = −ifBspμ. The HQET bag parameters are defined through. From our sum rule analysis we determine the HQET bag parameters BQs. Using (2.4), (2.5), (2.8), and (2.12) we arrive at the relation. O(1/mb), which allows us to match the values of BQsonto their QCD counterparts
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