Context: A small number of K-type giants on the red giant branch (RGB) is known to be very rich in lithium (Li). This fact is not accounted for by standard stellar evolution theory. The exact phase and mechanism of Li enrichment is still a matter of debate. Aims: Our goal is to probe the abundance of Li along the RGB, from its base to the tip, to confine Li-rich phases that are supposed to occur on the RGB. Methods: For this end, we obtained medium-resolution spectra with the FLAMES spectrograph at the VLT in GIRAFFE mode for a large sample of 401 low-mass RGB stars located in the Galactic bulge. The Li abundance was measured in the stars with a detectable Li 670.8 nm line by means of spectral synthesis with COMARCS model atmospheres. A new 2MASS (J-K) - Teff calibration from COMARCS models is presented in the Appendix. Results: Thirty-one stars with a detectable Li line were identified, three of which are Li-rich according to the usual criterion ($logepsilon({rm Li})>1.5$). The stars are distributed all along the RGB, not concentrated in any particular phase of the red giant evolution (e.g. the luminosity bump or the red clump). The three Li-rich stars are clearly brighter than the luminosity bump and red clump, and do not show any signs of enhanced mass loss. Conclusions: We conclude that the Li enrichment mechanism cannot be restricted to a clearly defined phase of the RGB evolution of low-mass stars ($Msim1M_{sun}$), contrary to earlier suggestions from disk field stars.
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