Matrix elements of $\Delta B=0$ operators in heavy hadron chiral perturbation theory

We study the light-quark mass and spatial volume dependence of the matrix elements of $\Delta B=0$ four-quark operators relevant for the determination of $V_{ub}$ and the lifetime ratios of single-$b$ hadrons. To this end, one-loop diagrams are computed in the framework of heavy hadron chiral perturbation theory with partially quenched formalism for three light-quark flavors in the isospin limit; flavor-connected and -disconnected diagrams are carefully analyzed. These calculations include the leading light-quark flavor and heavy-quark spin symmetry breaking effects in the heavy hadron spectrum. Our results can be used in the chiral extrapolation of lattice calculations of the matrix elements to the physical light-quark masses and to infinite volume. To provide insight on such chiral extrapolation, we evaluate the one-loop contributions to the matrix elements containing external $B_d$, $B_s$ mesons and $\Lambda_b$ baryon in the QCD limit, where sea and valence quark masses become equal. In particular, we find that the matrix elements of the $\lambda_3$ flavor-octet operators with external $B_d$ meson receive the contributions solely from connected diagrams in which current lattice techniques are capable of precise determination of the matrix elements. Finite volume effects are at most a few percent for typical lattice sizes and pion masses.