D2: Non-perturbative parameter within the HQE

The project will increase the theoretical precision in the description of lifetimes of heavy mesons, containing a heavy $b$ or $c$ quark and light $u, d, s$ quarks. The total decay rate of such a hadron can be expressed as

$$ \displaystyle \Gamma = \Gamma_0 + \Gamma_2 \frac{ \langle {\cal O}_5 \rangle }{m_b^2} + \Gamma_3 \frac{\langle {\cal O}_6 \rangle}{m_b^3} + ... + 16 \pi^2 \Big{[} \tilde{\Gamma}_3 \frac{\langle \tilde{\mathcal{O}}_6 \rangle}{m_b^3} + \tilde{\Gamma}_4 \frac{\langle \tilde{\mathcal{O}}_7\rangle}{m_b^4} + ... \Big{]}. \label{eq:HQE} $$

We plan to determine the non-perturbative matrix elements $\langle {\cal O}_6 \rangle$, $\langle \tilde{\mathcal{O}}_6 \rangle$ and $\langle \tilde{\mathcal{O}}_7\rangle$ with sum rules using the Heavy Quark Effective Theory (HQET) for different observables, like $\tau (B_s)/ \tau (B_d)$, $\tau (D_s^+)/ \tau (D_0)$ and the decay rate difference $\Delta \Gamma_s$. All these quantities are experimentally well-known and they will be measured even more precisely in the future.

Key features of studied quantities:

1. Determination of the matrix element of the Darwin operator for $D$ mesons
   • will give one of the dominant contributions to the lifetime ratio $\tau (D_s^+)/ \tau (D^0)$ which is measured very precisely
   • will allow a convergence test of the HQE in the charm sector
2. Determination of strange quark mass corrections to the Bag parameter of four quark operators triggering the $D_s^+$ lifetime
   • will allow a convergence test of the HQE in the charm sector
3. Determination of dimension seven Bag parameter of four quark operators triggering the $B_s$ mixing
   • tests the formalism HQE
   • reduces the dominant uncertainty in $\Delta \Gamma_s$
4. Determination of dimension seven Bag parameter of four quark operators contribution to the $B_s$ lifetime
   • will reduce the theory uncertainty in the lifetime ratio $\tau (B_s) / \tau (B_d) $ which is measured to a precision of four per mille
   • will allow a precision test of the HQE
   • can be used to look for invisible decay channels of the $B_s$ meson