We present a comprehensive study of the phonon dispersion in an underdoped, superconducting Ca 2−x CuO 2 Cl 2 crystal. We interpret the results using lattice dynamical calculations based on a shell model, and we compare the results, to other hole-doped cuprates, in particular to the ones isomorphic to La 2−x Sr x CuO 4 (LSCO). We found that an anomalous dip in the Cu-O bond stretching dispersion develops in oxychlorides with a simultaneous marked broadening of the mode. The broadening is maximum at ≈ (π/(2a) 0 0) that corresponds to the charge-modulations propagation vector. Our analysis also suggests that screening effects in calculations may cause an apparent cosine-shaped bending of the Cu-O bond-stretching dispersion along both the (q 0 0) and (q q 0) directions, that is not observed on the data close to optimal doping. This observation suggests that the discrepancy between experimental data and ab-initio calculations on this mode originates from an overestimation of the doping effects on the mode. PACS numbers: 74.72.Gh, 63.20.D-, 63.20.kd, 78.70.Ck 1 arXiv:1305.6719v3 [cond-mat.str-el] 5 Jun 2013 I. INTRODUCTIONThe discovery of high temperature superconductivity (HTS) in cuprates was originally motivated by the search for superconductors with strong electron-phonon coupling 1 . The first HTS compounds, La 2−x Ba x CuO 4 (LBCO) and LSCO, do indeed display a very large isotopic effect for oxygen, although the doping dependence of this effect turns out to be very anomalous 2 . Later studies pointed out that electron-phonon coupling may not be the main origin of the Cooper pairing 3 , in particular for optimal doping. But its role across the phase diagram of cuprates is still controversial, in particular in its under-doped region.An important point is that single electron band approximations, such as Density Functional Theory (DFT) find a small electron-phonon coupling 4,5 , but they possibly miss many body effects, as it was suggested that large couplings may exist in the presence of strong electronelectron correlations 6 .An important point stressed in DFT based investigations on electron-phonon coupling in cuprates is that these calculations seem to reproduce well the marked softening with a cosine dispersion, that develops under doping for longitudinal Cu-O bond-stretching mode along the direction (q 0 0) of such bonds (modes with ∆ 1 character). At the same time, large discrepancies between those calculations for the same mode, but along the diagonal direction (q q 0) (modes with Σ 1 character) are found 4,7 . It has been suggested that the physical origin of the anomaly in the dispersion of the Cu-O bond-stretching modes is different for the two directions 4,5 .Underdoped cuprates exhibit a static, or quasi-static, modulation of the charge, characterized by a stripe pattern parallel to the Cu-O bonds, which can give clear diffraction peaks both in the charge 8,9 and spin channel, for doping close to δ ∼ 1/8 with a propagation vector of ≈ π/(2a), i.e. a periodicity of ≈ 4a, where a is the lattice parameter...