The decay of the X(3872) into χcJ and the Operator Product Expansion in XEFT

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XEFT is a low energy effective theory for the X(3872) 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 χcJ plus pions using a two-step procedure in which Heavy Hadron Chiral Perturbation Theory (HH) amplitudes are matched onto XEFT operators and then X(3872) decay rates are then calculated using these operators. The procedure leads to IR divergences in the three-body decay X(3872) → χcJ ππwhen virtual D mesons can go on-shell in tree level HH diagrams. In previous work, we regulated these IR divergences with the D*⁰ width. In this work, we carefully analyze X(3872) → χcJ π⁰ and X(3872) → χcJ ππusing the operator product expansion (OPE) in XEFT. Forward scattering amplitudes in HH 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 OPE, these IR divergences cancel in the calculation of the matching coefficients so the correct predictions for the X(3872) → χc₁ ππdo not receive enhancements due to the width of the D*⁰. We give updated predictions for the decay X(3872) → χc₁ ππat leading order in XEFT.