====== C1b: $\mathbf{B - \overline{B}}$ mixing, CP violation, and Lifetimes ======

^Principal Investigators ^^
|[[https://www.ttp.kit.edu/memberpages/nierste | Prof. Ulrich Nierste ]] | Karlsruhe Institute of Technology |
| [[https://www.ttp.kit.edu/memberpages/steinhauser | Prof. Matthias Steinhauser ]] |Karlsruhe Institute of Technology|


==== Subject ====

The project addresses lifetimes differences between heavy
hadrons containing a heavy quark $Q=b,c$ and a light quark $q=u,d,s$ and
semileptonic CP asymmetries of $B_d$ and $B_s$ decays. The lifetime (or
width) difference between the two mass eigenstates of the neutral $B_q$
meson, $q=d,s$, and the semileptonic CP asymmetry
$a_{\rm CP} (B_q\to X \ell \nu)$ both originate from $\mathbf{\overline{B}_q}$ and
involve the decay matrix element $\mathbf{\Gamma_{12}^q}$.
The Heavy Quark Expansion (HQE) expresses these quantities as an
expansion in $\Lambda_{QCD}/m_Q$ and $\alpha_s(m_Q)$, where $m_Q$ (with
$Q=b,c$) is the heavy quark mass.
For instance, the width difference $\Delta \Gamma$ between different
$Q$-flavoured hadrons has the schematic form
$$
  \displaystyle \Delta \Gamma \propto \frac{1}{m_Q^3} \sum_j
  \left[\frac{\alpha_s}{4\pi} \right]^j \Gamma_3^{(j)} \, +\,
  \frac{1}{m_Q^4} \sum_j
  \left[\frac{\alpha_s}{4\pi} \right]^j \Gamma_4^{(j)} \, +\, \ldots
  \,. \nonumber % \label{hqe}
$$

**Key features of studied quantities:**

  * same technique (HQE) but different sensitivity to new physics
{{ :research:dega_nnlo.jpg?200|}}
  * simultaneously test formalism and probe new physics
  * theory uncertainties larger than experimental errors 
  * NLO perturbative uncertainty larger than or comparable to hadronic uncertainty 

In this project we aim at the calculation of NNLO (three-loop)
corrections to $\Gamma_3^{(j)}$ and of NLO (two-loop) corrections to
$\Gamma_4^{(j)}$
for the mentioned lifetimes differences
and CP asymmetries.


==== Project Topics ====

  - $\mathbf{\Gamma_{12}^q}$ to NNLO for $\mathbf{m_c=0}$
    * will reduce perturbative uncertainty of leading-power contribution to $\Delta \Gamma_s$ from $\sim$10% to $\sim$3%  
    * requires solution of new three-loop integral families
  - $\mathbf{\Gamma_{12}^q}$ at order $\mathbf{\alpha_s/m_b}$
    * needed to  reduce overall uncertainty in $\Delta \Gamma_q/\Delta M_q$ well below 10\%
  - $\mathbf{\tau(B^+)/\tau(B_d)}$ and $\mathbf{\tau(\Xi_b^0)/\tau(\Xi_b^-)}$ at orders $\mathbf{\alpha_s^2}$ and $\mathbf{\alpha_s/m_b}$
    * tests the formalism (HQE and lattice calculations)
  - Charm hadron lifetimes
    * tests the HQE at order $1/m_Q$
  - $\mathbf{\tau(B_s)/\tau(B_d)}$ and $\mathbf{\tau(\Lambda_b)/\tau(\Xi_b^0)}$
    * probes new physics in penguin coefficients 
  - $\mathbf{\Gamma_{12}^q}$ to NNLO at order $\mathbf{m_c^2/m_b^2}$
    * NNLO accuracy for $a_{\text{fs}}^d$, which probes new physics in e.g. $b\to d \bar{q} q$ decays
  - Phenomenological studies
    * devise judicious combinations to reduce hadronic uncertainties 
    * propose new observables