The decayb→sggin the standard model and a nonsoft bremsstrahlung extension of the low-energy theorem

An extension to the low-energy theorem for a flavor-changing two-gluon decay is discussed. It is shown that in cases where external momenta are much smaller than $M_W$ but not small compared with the mass of the decaying particle, contributions with the lowest powers of external momenta are only proportional to the corresponding one-gluon decay dipole-moment form factor. In particular, contributions from the color charge-radius form factor must cancel by gauge invariance. As an illustration, an explicit calculation for the decay $b\to \mathrm{sgg}$ is carried out in the standard model. The result of the calculation agrees with our theorem. We estimate that the branching ratio $B(b\to sgg)$ is roughly ≲ ${10}^{-4\mathrm{}}$ and hence completely negligible in the study of nonleptonic inclusive charmless $B$ decays. This differs from other estimates.