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Rare top quark decay t --> u_1 \bar{u}_2 u_2 in the standard model

The one-loop induced top quark decay $t\to u_1\bar{u}_2u_2$ ($u_i=u,c$) is calculated in the context of the standard model. The dominant contribution to this top quark decay arises from the Feynman diagrams induced by the off-shell $tu_1g^*$ vertex, whereas the box diagrams are negligibly small. In contrast with the on-shell $tu_1g$ vertex, which only gives rise to a pure dipolar effect, the off-shell $tu_1g^*$ coupling also involves a monopolar term, which gives a larger contribution than the dipolar one. It is found that the branching ratio for the three-body decay $t\to u_1\bar{u}_2u_2$ is about of the same order of magnitude of the two-body decay $t\to u_1 g$, the latter being slightly larger, which stems from the fact that the three-body decay is dominated by a monopolar term.