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Contribution of the Darwin operator to non-leptonic decays of heavy quarks

We compute the Darwin operator contribution ($1/m_b^3$ correction) to the width of the inclusive non-leptonic decay of a $B$ meson ($B^+$, $B_d$ or $B_s$), stemming from the quark flavour-changing transition $b \to q_1 \bar q_2 q_3$, where $q_1,q_2 = u, c$ and $q_3 = d, s$. The key ideas of the computation are the local expansion of the quark propagator in the external gluon field including terms with covariant derivative of the gluon field strength tensor and the standard technique of the Heavy Quark Expansion (HQE). We confirm the previously known expressions of the $1/m_b^3$ contributions to the semi-leptonic decay $b \to q_1 \ell \bar ν_\ell$, with $\ell = e, μ, τ$ and of the $1/m_b^2$ contributions to the non-leptonic modes. We find that this new term can give a sizeable correction of about $- 4 \, \%$ to the non-leptonic decay width of a $B$ meson. For $B_d$ and $B_s$ mesons this turns out to be the dominant correction to the free b-quark decay, while for the $B^+$ meson the Darwin term gives the second most important correction - roughly 1/2 to 1/3 of the phase space enhanced Pauli interference contribution. Due to the tiny experimental uncertainties in lifetime measurements the incorporation of the Darwin term contribution is crucial for precision tests of the Standard Model.