  We use functional integral methods for an estimate of the three-body loss in
a three-component $^6$Li ultracold atom gas. We advocate a simple picture where
the loss proceeds by the formation of a three-atom bound state, the trion. In
turn, the effective amplitude for the trion formation from three atoms is
estimated from a simple effective boson exchange process. The energy gap of the
trion and other key quantities for the loss coefficient are computed in a
functional renormalization group framework.

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