Abstract
We calculate the bag parameters for neutral $B$-meson mixing in and beyond the Standard Model, in full four-flavour lattice QCD for the first time. We work on gluon field configurations that include the effect of $u$, $d$, $s$ and $c$ sea quarks with the Highly Improved Staggered Quark (HISQ) action at three values of the lattice spacing and with three $u/d$ quark masses going down to the physical value. The valence $b$ quarks use the improved NRQCD action and the valence light quarks, the HISQ action. Our analysis was blinded. Our results for the bag parameters for all five operators are the most accurate to date. For the Standard Model operator between $B_s$ and $B_d$ mesons we find: $\hat{B}_{B_s}=1.232(53)$, $\hat{B}_{B_d}=1.222(61)$. Combining our results with lattice QCD calculations of the decay constants using HISQ quarks from the Fermilab/MILC collaboration and with experimental values for $B_s$ and $B_d$ oscillation frequencies allows determination of the CKM elements $V_{ts}$ and $V_{td}$. We find $V_{ts} = 0.04189(93)$, $V_{td} = 0.00867(23)$ and $V_{ts}/V_{td} = 0.2071(27)$. Our results agree well (within $2\sigma$) with values determined from CKM unitarity constraints based on tree-level processes (only). Using a ratio to $\Delta M$ in which CKM elements cancel in the Standard Model, we determine the branching fractions ${\text{Br}}(B_s\rightarrow \mu^+\mu^-) = 3.81(18) \times 10^{-9}$ and ${\text{Br}}(B_d\rightarrow \mu^+\mu^-) = 1.031(54) \times 10^{-10}$. We also give results for matrix elements of the operators $R_0$, $R_1$ and $\tilde{R}_1$ that contribute to neutral $B$-meson width differences.
Highlights
The Standard Model (SM) description of neutral Bd and Bs oscillations requires knowledge of hadronic parameters derived from the matrix elements of 4-quark operators between Bq and Bq states
These 4-quark operators come from the effective electroweak Lagrangian at energy scales appropriate to B physics, and the matrix elements can only be determined by lattice QCD calculations, which are able to include the full impact of QCD on such hadronic
Details about our analysis are contained in four appendixes: Appendix A on the lattice QCD correlators that we calculate and how we fit them to obtain matrix elements and bag parameters; Appendix B on the chiral perturbation theory fits we use to combine results at physical and unphysical light quark masses; Appendix C on correlations in the uncertainties for our final results and last, with more general applications beyond this analysis, Appendix D on singular value decomposition (SVD) cuts and fitting correlators
Summary
The Standard Model (SM) description of neutral Bd and Bs oscillations requires knowledge of hadronic parameters derived from the matrix elements of 4-quark operators between Bq and Bq states. A very recent result in [9] uses domain-wall quarks on gluon field configurations including u, d, and s in the sea and extrapolates in heavy quark mass to the b quark mass for Bs to Bd ratios for SM mixing matrix elements and decay constants. Details about our analysis are contained in four appendixes: Appendix A on the lattice QCD correlators that we calculate and how we fit them to obtain matrix elements and bag parameters; Appendix B on the chiral perturbation theory fits we use to combine results at physical and unphysical light quark masses; Appendix C on correlations in the uncertainties for our final results and last, with more general applications beyond this analysis, Appendix D on singular value decomposition (SVD) cuts and fitting correlators
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