Calculation of the continuum--lattice HQET matching for the complete basis of four--fermion operators: reanalysis of the $B^{0}$-$\bar{B}^{0}$ mixing

In this work, we find the expressions of continuum HQET four-fermion operators in terms of lattice operators in perturbation theory. To do so, we calculate the one--loop continuum--lattice HQET matching for the complete basis of $\Delta B=2$ and $\Delta B=0$ operators (excluding penguin diagrams), extending and completing previous studies. We have also corrected some errors in previous evaluations of the matching for the operator $O_{LL}$. Our results are relevant to the lattice computation of the values of unknown hadronic matrix elements which enter in many very important theoretical predictions in $B$--meson phenomenology: $B^{0}$-$\bar{B}^{0}$ mixing, $\tau_{B}$ and $\tau_{B_{s}}$ lifetimes, SUSY effects in $\Delta B=2$ transitions and the $B_{s}$ width difference $\Delta \Gamma_{B_{s}}$. We have reanalyzed our lattice data for the $B_{B}$ parameter of the $B^{0}$-$\bar{B}^{0}$ mixing on 600 lattices of size $24^{3}\times 40$ at $\beta=6.0$ computed with the SW-Clover and HQET lattice actions. We have used the correct lattice--continuum matching factors and boosted perturbation theory with tadpole improved heavy--light operators to reduce the systematic error in the evaluation of the renormalization constants. Our best estimate of the renormalization scale independent $B$--parameter is $\hat{B}_{B} = 1.29 \pm 0.08 \pm 0.06$, where the first error is statistical and the second is systematic coming from the uncertainty in the determination of the renormalization constants. Our result is in good agreement with previous results obtained by extrapolating Wilson data. As a byproduct, we also obtain the complete one--loop anomalous dimension matrix for four--fermion operators in the HQET.