Lattice QCD Matrix Elements for the $B_s^0-{\overline{B}}_s^0$ Width Difference beyond Leading Order

Predicting the $B_s^0-{\overline{B}}_s^0$ width difference $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ relies on the heavy quark expansion and on hadronic matrix elements of $\mathrm{\Delta }B=2$ operators. We present the first lattice QCD results for matrix elements of the dimension-7 operators $R_{2,3}$ and linear combinations ${\tilde{R}}_{2,3}$ using nonrelativistic QCD for the bottom quark and a highly improved staggered quark (HISQ) action for the strange quark. Computations use MILC Collaboration ensembles of gauge field configurations with $2+1+1$ flavors of sea quarks with the HISQ discretization, including lattices with physically light up or down quark masses. We discuss features unique to calculating matrix elements of these operators and analyze uncertainties from series truncation, discretization, and quark mass dependence. Finally we report the first standard model determination of $\mathrm{\Delta }{\mathrm{\Gamma }}_s$ using lattice QCD results for all hadronic matrix elements through $\mathcal{O}(1/m_b)$. The main result of our calculations yields the $1/m_b$ contribution $\mathrm{\Delta }{\mathrm{\Gamma }}_{1/m_b}=-0.022(10){\mathrm{ps}}^{-1}$. Adding this to the leading order contribution, the standard model prediction is $\mathrm{\Delta }{\mathrm{\Gamma }}_s=0.092(14){\mathrm{ps}}^{-1}$.

Figure 1

Ratios of unsubtracted and subtracted $R$ matrix elements to $⟨O_1⟩$. Only statistical errors are shown here.

Figure 2

Matrix elements of $R_2$ and $R_3$ plotted against lattice spacing squared. Error bars include systematic uncertainties. Results of fits to Eq. (5) are plotted as stars.