Renormalization of HQET $Delta B=2$ operators: O($a$) improvement and $1/m$ matching with QCD

We determine a basis of dimension-7 operators which arise at O($a$) in the Symanzik expansion of the $Delta B=2$ operators with static heavy quarks. We consider both Wilson-like and Ginsparg-Wilson light quarks. Exact chiral symmetry reduces the number of these O($a$) counterterms by a factor of two. Only a subset of these operators has previously appeared in the literature. We then extend the analysis to the O($1/m$) operators contributing beyond the static approximation.