The concept of \textit{optical} exciton - a photo-excited bound electron-hole pair within a crystal - is routinely used to interpret and model a wealth of excited-state phenomena in semiconductors. Beside originating sub-band gap signatures in optical spectra, optical excitons have also been predicted to condensate, diffuse, recombine, relax. However, all these phenomena are rooted on a theoretical definition of the excitonic state based on the following simple picture: "excitons" are actual particles that both appear as peaks in the linear absorption spectrum and also behave as well-defined quasiparticles. In this paper we show, instead, that the electron-phonon interaction decomposes the initial optical excitons into \textit{elemental} excitons, the latter being a different kind of bound electron-hole pairs lacking the effect caused by the induced, classical, electric field. This is demonstrated within a many-body perturbation theory approach starting from the interacting electronic Hamiltonian including both electron-phonon and electron-hole interactions. We then apply the results on two realistic systems, monolayer MoS$_2$ (where the lowest-bound exciton is optically inactive) and monolayer MoSe$_2$ (where it is optically active), using first-principles methods to compute the exciton-phonon coupling matrix elements. Among the consequences of optical-elemental decomposition, we point to a homogeneous broadening of absorption peaks occurring even for the lowest-bound optical exciton, we demonstrate this by computing exciton-phonon transition rates. More generally, our findings suggest that the optical excitons gradually lose their initial structure and evolve as elemental excitons. These states can be regarded as the real intrinsic excitations of the interacting system, the ones that survive when the external perturbation and the induced electric fields have vanished.

18 pages, 8 figures