Abstract
A naive application of the heavy quark expansion (HQE) yields theory estimates for the decay rate of neutral $D$ mesons that are four orders of magnitude below the experimental determination. It is well known that this huge suppression results from severe GIM cancellations. We find that this mismatch can be solved by individually choosing the renormalisation scale of the different internal quark contributions. For $b$ and $c$ hadron lifetimes, as well as for the decay rate difference of neutral $B$ mesons the effect of our scale setting procedure lies within the previously quoted theory uncertainties, while we get enlarged theory uncertainties for the semi-leptonic CP asymmetries in the $B$ system.
Highlights
An improvement of our theoretical understanding of charm physics is crucial to make use of the huge amount of current and future experimental charm data obtained by LHCb [1], BESIII [2], and Belle II [3]
A naive application of the heavy quark expansion (HQE) leads to Ω 1⁄4 3.4 × 10−5 at leading order (LO)-QCD (6.2 × 10−5 at next-to-leading order (NLO)-QCD), i.e., the HQE prediction of the decay rate difference is more than 4 orders of magnitude smaller than the experimental determination
In Ref. [66], it was shown that violations as small as 20 percent could be sufficient to explain the experimental value of D-mixing. (iii) The HQE is not applicable and we have to rely on different methods, like summing over the exclusive decays channels contributing to the decay rate difference; see e.g., Refs. [84,85,86]
Summary
An improvement of our theoretical understanding of charm physics is crucial to make use of the huge amount of current and future experimental charm data obtained by LHCb [1], BESIII [2], and Belle II [3]. In the case of Bs-mixing and b hadron lifetimes, the HQE predicts values [57,58,59,60,61,62,63] which are in good agreement with the experimental ones [13]. This impressive result when the expansion parameter is Λ=mb (Λ denotes an hadronic scale of the order of ΛQCD) suggests that one might still get reasonably well-behaving estimates moving to the charm system, where the expansion parameter increases by a factor of 3. It is quite surprising that a naive application of the HQE fails completely for D-mixing
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