We present Li and Fe abundances for 87 stars in the GC M4,obtained with GIRAFFE high-resolution spectra. The targets range from the TO up to the RGB Bump. The Li abundance in the TO stars is uniform, with an average value A(Li)=2.30+-0.02 dex,consistent with the upper envelope of Li content measured in other GCs and in the Halo stars,confirming also for M4 the discrepancy with the primordial Li abundance predicted by WMAP+BBNS. The iron content of M4 is [Fe/H]=-1.10+-0.01 dex, with no systematic offsets between dwarf and giant stars.The behaviour of the Li and Fe abundance along the entire evolutionary path is incompatible with models with atomic diffusion, pointing out that an additional turbulent mixing below the convective region needs to be taken into account,able to inhibit the atomic diffusion.The measured A(Li) and its homogeneity in the TO stars allow to put strong constraints on the shape of the Li profile inside the M4 TO stars. The global behaviour of A(Li) with T_{eff} can be reproduced with different pristine Li abundances, depending on the kind of adopted turbulent mixing.One cannot reproduce the global trend starting from the WMAP+BBNS A(Li) and adopting the turbulent mixing described by Richard et al.(2005) with the same efficiency used by Korn et al.(2006) to explain the Li content in NGC6397. Such a solution is not able to well reproduce simultaneously the Li abundance observed in TO and RGB stars.Otherwise, theWMAP+BBNS A(Li) can be reproduced assuming a more efficient turbulent mixing able to reach deeper stellar regions where the Li is burned. The cosmological Li discrepancy cannot be easily solved with the present,poor understanding of the turbulence in the stellar interiors and a future effort to well understand the true nature of this non-canonical process is needed.
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