Abstract
We study local quark-hadron duality and its violation for the {D}^0-{overline{D}}^0 , {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings in the ’t Hooft model, offering a laboratory to test QCD in two-dimensional spacetime together with the large-Nc limit. With the ’t Hooft equation being numerically solved, the width difference is calculated as an exclusive sum over two-body decays. The obtained rate is compared to inclusive one that arises from four-quark operators to check the validity of the heavy quark expansion (HQE). In view of the observation in four-dimensions that the HQE prediction for the width difference in the {D}^0-{overline{D}}^0 mixing is four orders of magnitude smaller than the experimental data, in this work we investigate duality violation in the presence of the GIM mechanism. We show that the order of magnitude of the observable in the {D}^0-{overline{D}}^0 mixing is enhanced in the exclusive analysis relative to the inclusive counterpart, when the 4D-like phase space function is used for the inclusive analysis. By contrast, it is shown that for the {B}_d^0-{overline{B}}_d^0 and {B}_s^0-{overline{B}}_s^0 mixings, small yet non-negligible corrections to the inclusive result emerge, which are still consistent with what is currently indicated in four-dimensions.
Highlights
(2) due to Glashow-Iliopoulous-Miani (GIM) mechanism [16], observables undergo severe cancellation unlike the milder one for b quark
We study local quark-hadron duality and its violation for the D0 −D 0, Bd0 −Bd0 and Bs0 − Bs0 mixings in the ’t Hooft model, offering a laboratory to test QCD in twodimensional spacetime together with the large-Nc limit
We have studied local quark-hadron duality and its violation in the heavy quark mixings on the basis of one certain dynamical mechanism
Summary
2.1 D0 − D0, Bd0 − Bd0 and Bs0 − Bs0 mixings For the D0 − D0 mixing, we introduce mass eigenstates denoted by |D1,2 that diagonalize the Schrödinger equations [145] in the CP-conserving limit, where |D1 (|D2 ) coincides with a CP-even (odd) state. The off-diagonal element of the mixing matrix is given by, M2(1D0). The width difference between the two CP states defined by ∆ΓD = Γ(1D0) − Γ(2D0) can be expressed in terms of the off-diagonal element of the mixing matrix,. |B1 (|B2 ) is a CP-even (odd) state, and define ∆ΓBq = Γ(1Bq) − Γ(2Bq). For this case, the following notation similar to one for the D0 − D0 mixing is introduced, M1(2Bq0).
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