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 \mathrm { 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 ) \approx 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 \mathrm { A ( Li ) } \geq 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 \sim 7 \mathrm { 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 .