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

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

• 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.

1. $\mathbf{\Gamma_{12}^q}$ 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
2. $\mathbf{\Gamma_{12}^q}$ at order $\mathbf{\alpha_s/m_b}$
   • needed to reduce overall uncertainty in $\Delta \Gamma_q/\Delta M_q$ well below 10\%
3. $\mathbf{\tau(B^+)/\tau(B_d)}$ 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)
4. Charm hadron lifetimes
   • tests the HQE at order $1/m_Q$
5. $\mathbf{\tau(B_s)/\tau(B_d)}$ and $\mathbf{\tau(\Lambda_b)/\tau(\Xi_b^0)}$
   • probes new physics in penguin coefficients
6. $\mathbf{\Gamma_{12}^q}$ 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
7. Phenomenological studies
   • devise judicious combinations to reduce hadronic uncertainties
   • propose new observables