Bond stretching phonon softening and angle-resolved kinks in the photoemission spectra of optimally doped Bi$_2$Sr$_{1.6}$La$_{0.4}$Cu$_2$O$_{6+delta}$ superconductors

We report the first measurement of the Cu-O bond stretching phonon dispersion in optimally doped Bi2Sr1.6La0.4Cu2O6+? using inelastic x-ray scattering. We found a softening of this phonon at q=(?0.25,0,0) from 76 to 60 meV, similar to the one reported in other cuprates. A comparison with angle-resolved photoemission data on the same sample revealed an excellent agreement in terms of energy and momentum between the angle-resolved photoemission nodal kink and the soft part of the bond stretching phonon. Indeed, we find that the momentum space where a 63±5??meV kink is observed can be connected with a vector q=(?,0,0) with ??0.22, corresponding exactly to the soft part of the bond stretching phonon.