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Pure dephasing is widely used in the literature to explain experimentalobservations on quantum dots in cavities. In many cases, its use is not enoughand extra terms need to be "fictitiously" added to accomplish with the observeddata as it is the case of cavity pumping of an unknown source. Here wecontrovert the validity of the pure dephasing mechanism as a source ofdecoherence and present a theoretical study based on the phonon-mediatedcoupling that can explain the emission spectrum and photon auto- andcross-correlation results in recent experiments without the need of anyartificial assumptions. We also demonstrated that the phonon-mediated couplingaccounts for unexplained features recently reported in measurements of photonauto- and cross-correlation functions. Our work illuminates many of the debatesin this field and opens up new possibilities for experimental verification andtheoretical predictions.

Pure dephasing vs. Phonon mediated off-resonant coupling in a quantum-dot-cavity system