Decays of the X(3872) and $χ_{c1}(2P)$ charmonium
We re-examine the re-scattering mechanism for the X(3872), as a candidate for the 2P charmonium state $χ_{c1}(2P)$, decaying to $J/ψρ(ω)$ through exchanging $D^{(*)}$ mesons between intermediate states $D(\bar{D})$ and $\bar{D}^*(D^*)$. We evaluate the dispersive part, as well as the absorptive one, of the re-scattering amplitude and find that the contribution from the dispersive part is dominant even when X(3872) lies above the threshold of the neutral channel $th_n=m_{D^0}+m_{D^{*0}}$. We predict $R_{ρ/ω}\simeq 1$ for the $m_X$ region scanned by experiments. Meanwhile, we also estimate the rate of $X\to D^0\bar{D}^{0}π^0$. Our results favor a charmonium interpretation of X(3872) when it lies slightly below the threshold of $D^0\bar{D}^{*0}$. Furthermore, we evaluated the width of $X\to J/ψρ$ with the help of a phenomenological effective coupling constant $g_X$, and find the total width of X(3872) to be in the range of 1-2 MeV.