Weak G-band (WGB) stars are a rare class of cool luminous stars that present a strong depletion in carbon, but also lithium abundance anomalies that have been little explored in the literature since the first discovery of these peculiar objects in the early 50s. Here we focus on the Li-rich WGB stars and report on their evolutionary status. We explore different paths to propose a tentative explanation for the lithium anomaly. Using archive data, we derive the fundamental parameters of WGB (Teff, log g, log(L/Lsun)) using Hipparcos parallaxes and recent temperature scales. From the equivalent widths of Li resonance line at 6707 {AA}, we uniformly derive the lithium abundances and apply when possible NLTE corrections following the procedure described by Lind et al. (2009). We also compute dedicated stellar evolution models in the mass range 3.0 to 4.5 Msun, exploring the effects of rotation-induced and thermohaline mixing. These models are used to locate the WGB stars in the H-R diagram and to explore the origin of the abundance anomalies. The location of WGB stars in the H-R diagram shows that these are intermediate mass stars of masses ranging from 3.0 to 4.5 Msun located at the clump, which implies a degeneracy of their evolutionary status between subgiant/red giant branch and core helium burning phases. The atmospheres of a large proportion of WGB stars (more than 50%) exhibit lithium abundances A(Li) geq 1.4 dex similar to Li-rich K giants. The position of WGB stars along with the Li-rich K giants in the H-R diagram however indicates that both are well separated groups. The combined and tentatively consistent analysis of the abundance pattern for lithium, carbon and nitrogen of WGB stars seems to indicate that carbon underabundance could be decorrelated from the lithium and nitrogen overabundances.
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