  A small fraction of red giants are known to be lithium (Li) rich, in
contradiction with expectations from stellar evolutionary theory. A possible
explanation for these atypical giants is the engulfment of a Li-rich planet or
brown dwarf by the star. In this work, we model the evolution of Li abundance
in canonical red giants including the accretion of a sub-stellar mass
companion. We consider a wide range of stellar and companion masses, Li
abundances, stellar metallicities, and planetary orbital periods. Based on our
calculations, companions with masses lower than 15 M_J dissolve in the
convective envelope and can induce Li enrichment in regimes where extra mixing
does not operate. Our models indicate that the accretion of a substellar
companion can explain abundances up to A(Li)~2.2, setting an upper limit for
Li-rich giants formed by this mechanism. Giants with higher abundances need
another mechanism to be explained. For reasonable planetary distributions, we
predict the Li abundance distribution of low-mass giants undergoing planet
engulfment, finding that between 1% to 3% of them should have A(Li)>1.5. We
show that depending on the stellar mass range, this traditional definition of
Li-rich giants is misleading, as isolated massive stars would be considered
anomalous while giants engulfing a companion would be set aside, flagged as
normal. We explore the detectability of companion engulfment, finding that
planets with masses higher than ~7 M_J produce a distinct signature, and that
descendants of stars originating in the Li-dip and low luminosity red giants
are ideal tests of this channel.
