Do $1^- c \bar c g$ hybrid meson exist, do they mix with charmonium ?

Hybrid $c \bar c g$ of quantum numbers $1^{--}$ are considered in a quark model with constituant quarks and gluon. The lowest $J^P=1^-$ states may be built in two ways, $l_g=1$ (gluon excited) corresponding to an angular momentum between the gluon and the $c \bar c$ system, while $l_{c\bar c}=1$ (quark excited) corresponds to an angular momentum between the $c$ and the $\bar c$. The lowest lying hybrid $J^P=1^-$ state in the flux tube model is similar to the $l_g=1$ in the quark-gluon model. In particular it verifies the selection rule that it cannot decay into two fundamental mesons. The $l_{q\bar q}=1$ hybrid may decay into two fundamental mesons, but with decay widths larger than 1 GeV, which tells that they do not really exist as resonant states. Using a chromoharmonic potential, we find no mixing between the $l_g=1$ and $l_{c\bar c}=1$. More realistic potentials might induce a strong mixing between them, implying that no hybrid meson exist. If, on the contrary, such a strong mixing does not occur, we find, in agreement with the flux tube model, that only the $l_g=1$ appears as a real resonant state. In such a case, hybrid mesons may exist as resonances only if they are decoupled from the ground state channels, which explains the difficulty to observe them experimentally. We reconsider accordingly the Ono-Close-Page scenario of mixing between charmonium and charmed-hybrid to explain the anomalies around 4.1 GeV. We find a very small mixing between radially excited charmonium and hybrid mesons, which forbids considering the $\psi(4.040)$ and $\psi(4.160)$ as combinations of 3S charmonium and $l_g=1$ hybrid meson with a large mixing.