  The Heavy Quark Expansion (HQE) gives an expansion of inclusive decay rates
of $b$-hadrons as a simultaneous series in $\alpha_s$ and $1/m_b$, in terms of
perturbatively defined coefficients and non-perturbative matrix elements.
Spectator effects arise from the dimension-$6$ operators in the HQE in which
along with the heavy quark, a light spectator quark from the hadron
participates in the weak decay. In this work, we provide a lattice
determination of the bare matrix elements that contribute to the spectator
effects for the $B^+$-meson, $B_d$-meson and the $\Lambda_b$-baryon. The
computations were performed on two separate lattice spacings of the RBC-UKQCD
(2+1)-flavor Domain Wall Fermion ensembles. Renormalization and continuum
extrapolation of the matrix elements have not yet been computed, and are left
to future work.
