The decays of the $X(3872)$ to $P$-wave quarkonia are calculated under the assumption that it is a shallow bound state of neutral charmed mesons. The $X(3872)$ is described using an effective theory of nonrelativistic $D$ mesons and pions (X-EFT). We calculate $X(3872)$ decays by first matching heavy hadron chiral perturbation theory ($\mathrm{HH}\chi \mathrm{PT}$) amplitudes for $D^0{\overline{D}}^{*0}\to {\chi }_{cJ}({\pi }^0,\pi \pi )$ onto local operators in X-EFT, and then using these operators to calculate the $X(3872)$ decays. This procedure reproduces the factorization theorems for $X(3872)$ decays to conventional quarkonia previously derived using the operator product expansion. For single pion decays, we find nontrivial dependence on the pion energy from $\mathrm{HH}\chi \mathrm{PT}$ diagrams with virtual $D$ mesons. This nontrivial energy dependence can potentially modify heavy-quark symmetry predictions for the relative sizes of decay rates. At leading order, decays to final states with two pions are dominated by the final state ${\chi }_{c1}{\pi }^0{\pi }^0$, with a branching fraction just below that for the decay to ${\chi }_{c1}{\pi }^0$. Decays to all other final states with two pions are highly suppressed.

• Received 1 August 2008

DOI:https://doi.org/10.1103/PhysRevD.78.094019