Isospin properties of the X state near the D {\bar D}^{*} threshold

The $D{\bar{D}}^{\ast }$ scattering amplitude and the production of the final states ${\pi }^{+}{\pi }^{-}J\mathrm{/}\psi$ and ${\pi }^{+}{\pi }^{-}{\pi }^{0}J\mathrm{/}\psi$ near the $D^0 {\bar D}^{*0}$ threshold are discussed following the recent suggestion that the observed peaks X(3872) and X(3875) in the decays $B\to XK$ are due to a virtual state $X$ in the $D^0{\bar{D}}^{\ast 0}$ channel. The strong interaction is treated using the small interaction radius approximation. It is shown that the mass difference between the charged and neutral $D^{(\ast )}$ mesons results in a distinctive behavior of the relevant isotopic amplitudes. In particular, the shape of the peak in the ${\pi }^{+}{\pi }^{-}J\mathrm{/}\psi$ channel should be significantly narrower than in the ${\pi }^{+}{\pi }^{-}{\pi }^{0}J\mathrm{/}\psi$ channel, which property can be used for an experimental test of the virtual state hypothesis.