  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.
