We consider a low-energy effective theory for the $X(3872)$ (XEFT) that can be used to systematically analyze the decay and production of the $X(3872)$ meson, assuming that it is a weakly bound state of charmed mesons. In a previous paper, we calculated the decays of $X(3872)$ into ${\chi }_{cJ}$ plus pions using a two-step procedure in which Heavy Hadron Chiral Perturbation Theory ($\mathrm{HH}\chi \mathrm{PT}$) amplitudes are matched onto XEFT operators and then $X(3872)$ decay rates are calculated using these operators. The procedure leads to IR divergences in the three-body decay $X(3872)\to {\chi }_{cJ}\pi \pi$ when virtual $D$ mesons can go on shell in tree level $\mathrm{HH}\chi \mathrm{PT}$ diagrams. In previous work, we regulated these IR divergences with the $D^{*0}$ width. In this work, we carefully analyze $X(3872)\to {\chi }_{cJ}{\pi }^0$ and $X(3872)\to {\chi }_{cJ}\pi \pi$ using the operator product expansion in XEFT. Forward scattering amplitudes in $\mathrm{HH}\chi \mathrm{PT}$ are matched onto local operators in XEFT, the imaginary parts of which are responsible for the decay of the $X(3872)$. Here we show that the IR divergences are regulated by the binding momentum of the $X(3872)$ rather than the width of the $D^{*0}$ meson. In the operator product expansion, these IR divergences cancel in the calculation of the matching coefficients so the correct predictions for the $X(3872)\to {\chi }_{c1}\pi \pi$ do not receive enhancements due to the width of the $D^{*0}$. We give updated predictions for the decay $X(3872)\to {\chi }_{c1}\pi \pi$ at leading order in XEFT.

• Received 6 October 2011

DOI:https://doi.org/10.1103/PhysRevD.85.014016