Charmonium options for theX(3872)

In this paper we consider all possible $1D$ and $2P$ $c\overline{c}$ assignments for the recently discovered $X\left(3872\right).$ Taking the experimental mass as input, we give numerical results for the $E1\mathrm{}$ radiative widths as well as the three principal types of strong decays; open-charm, $c\overline{c}$ annihilation and closed-charm hadronic transitions. We find that many assignments may be immediately eliminated due to the small observed total width. The remaining viable $c\overline{c}$ assignments are $1{}^3{D}_3,$ $1{}^3{D}_2,$ $1{}^1{D}_2,$ $2{}^3{P}_1$ and $2{}^1{P}_1.$ A search for the mode $J/\psi {\pi }^0{\pi }^0$ can establish the C parity of the $X\left(3872\right),$ which will eliminate many of these possibilities. Radiative transitions can then be used to test the remaining assignments, as they populate characteristic final states. The $1{}^3{D}_2$ and $1{}^1{D}_2$ states are predicted to have large (ca. 50%) radiative branching fractions to ${\chi }_{c1\mathrm{}}\gamma$ and $h_c\gamma ,$ respectively. We predict that the $1{}^3{D}_3$ will also be relatively narrow and will have a significant (ca. 10%) branching fraction to ${\chi }_{c2\mathrm{}}\gamma ,$ and should also be observable in B decay. Tests for non-$c\overline{c}X\left(3872\right)\mathrm{}$ assignments are also discussed.