Is $B \to X_s γ$ equal to $b \to s γ$ ? Spectator contributions to rare inclusive B decays

In $B \to X_s \gamma $ there are perturbative $QCD$ corrections at order $\alpha_s$ not only to the single quark line process $ b \to s \gamma$ but also coming from a set of diagrams where the weak interaction vertex involves a gluon which interacts with the spectator quark in the B hadron. We discuss the impact of the full set of these diagrams. These can influence the decay rate and also the line shape of the photon spectrum as they favor a softer photon energy than does the pure spectator decay $b \to s \gamma$. A subset of these diagrams generate differences in the decay rates for charged and neutral B's , $B^+ \to X_s \gamma$ vs. $B^0 \to X_s \gamma$ which can be searched for experimentally, although our calculation indicates that the effect will be hard to observe. Bound state effects turn out to be required to give rise to spectator corrections to the decay rate at order $\alpha_s$. The resulting contribution to the total rate is rather small (of about $5 \% $ ), although the effect is somewhat larger on portions of the photon spectrum.