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A microscopic theory of optical transitions in quantum dots with carrier-phonon interaction is developed. Virtual transitions into higher confined states with acoustic phonon assistance add a quadratic phonon coupling to the standard linear one, thus extending the independent Boson model. Summing infinitely many diagrams in the cumulant, a numerically exact solution for the interband polarization is found. Its full time dependence and the absorption lineshape of the quantum dot are calculated. It is the quadratic interaction which gives rise to a temperature-dependent broadening of the zero-phonon line, being here calculated for the first time in a consistent scheme.