Higher order corrections to the lifetime of heavy hadrons

In this work we discuss the theoretical status for the study of the lifetime of heavy hadrons.
After presenting some introductory topics like the effective weak Hamiltonian and the heavy quark effective theory (HQET), we describe the construction of the heavy quark expansion (HQE), which constitutes the theoretical framework to systematically compute the total decay width of heavy hadrons, in terms of an expansion in inverse powers of the heavy quark mass. The structure of the HQE is discussed in detail, and the computation of the lowest dimensional contributions, explicitly outlined. Particular emphasis is put in describing the expansion of the quark propagator in the external gluon field using the Fock-Schwinger (FS) gauge, which represents a fundamental ingredient of the calculation. Moreover, the main result is the computation of the dimension-six contribution due to the Darwin operator, only recently determined and found to have a sizeable effect. Finally, we consider two phenomenological applications of the HQE in the charm sector, namely the study of the lifetime of charmed mesons and the analysis of the Glashow-Iliopoulos-Maiani (GIM) cancellations in neutral $D$-meson mixing. By comparing our results with recent measurements performed by the LHCb, Belle-II and BESIII collaborations, we conclude that the HQE is able to reproduce, within large theoretical uncertainties, the experimental pattern for the lifetimes of charmed mesons and we discuss a potential solution for the discrepancy of previous theoretical determinations of $D$-mixing with data.