Decay of the X(3872) into χ _cJ and the operator product expansion in effective field theory

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 χ _cJ plus pions using a two-step procedure in which Heavy Hadron Chiral Perturbation Theory (HHχPT) 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)→χ _cJππ when virtual D mesons can go on shell in tree level HHχPT diagrams. In previous work, we regulated these IR divergences with the D ^*0 width. In this work, we carefully analyze X(3872)→χ _cJπ0 and X(3872)→χ _cJππ using the operator product expansion in XEFT. Forward scattering amplitudes in HHχPT 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)→χ _c1ππ do not receive enhancements due to the width of the D ^*0. We give updated predictions for the decay X(3872)→χ _c1ππ at leading order in XEFT.