Rare decays $B \to Xd + \gamma$ in the standard model

We present an estimate of the inclusive decay rate and photon energy- and hadron mass-spectrum for the CKM-suppressed radiative rare decays $B → Xd + γ$, based on perturbative QCD and a phenomenological model for the B-meson wave function (here Xd denotes nonstrange hadrons). Present constraints on $|V_{td}|$ are used to predict BR($B → Xd + γ$) = (0.8−4)× 10$^{−5}$ for the top quark mass in the range 100 ⩽ mt ⩽ 200 GeV. From the inclusive invariant hadron-mass spectrum and using vector meson dominance in the region m($X_d$) ⩽ 1 GeV, we estimate the branching ratio for the exclusive rare decay mode $B → ϱ + γ$ and the ratio $Γ(B→ ϱ + γ)/Γ(B→K∗+ γ)$. The importance of measuring these decays in determining the CKM matrix element $|V_{td}|$ is emphasized