Abstract
The calculation of the decay $X(3872)\to D^0 \bar{D}^0 \pi^0$ in effective field theory is revisited to include final state $\pi^0 D^0$, $\pi^0 \bar{D}^0$and $D^0\bar{D}^0$ rescattering diagrams. These introduce significant uncertainty into the prediction for the partial width as a function of the binding energy. The differential distribution in the pion energy is also studied for the first time. The normalization of the distribution is again quite uncertain due to higher order effects but the shape of the distribution is unaffected by higher order corrections. Furthermore the shape of the distribution and the location of the peak are sensitive to the binding energy of $X(3872)$. The shape is strongly impacted by the presence of virtual $D^{*0}$ graphs which highlights the molecular nature of the $X(3872)$. Measurement of the pion energy distribution in the decay $X(3872)\to D^0 \bar{D}^0 \pi^0$ can reveal interesting information about the binding nature of the $X(3872)$.
Highlights
The Xð3872Þ [1] is the first of many exotic charmonium and bottomonium states found in various high energy experiments since 2003
D0D 0 rescattering diagrams in studying the partial decay width of Xð3872Þ → D0D 0π0, we review the power counting of Feynman diagrams in Fig. 1 that was already discussed in detail in Ref. [21] and Ref. [7]
In this paper we revisited the XEFT calculation of Xð3872Þ → D0D 0π0 first performed in Ref. [21]
Summary
The Xð3872Þ [1] is the first of many exotic charmonium and bottomonium states found in various high energy experiments since 2003. The ERT prediction for Xð3872Þ → D0D 0π0 emerges as the leading order in XEFT, and corrections from pion loops, range corrections, and higher dimension operators in the effective Lagrangian can be treated systematically. D0D 0 rescattering diagrams in studying the partial decay width of Xð3872Þ → D0D 0π0, we review the power counting of Feynman diagrams in Fig. 1 that was already discussed in detail in Ref. If D0D 0 rescattering becomes nonperturbative, C0D scales as Q−1 which is similar to the scaling of C0 discussed above and which gives even more significant contribution to the decay of Xð3872Þ With these basic scaling rules in hand, it is straightforward to check, in Fig. 1, that LO diagram (a) scales as Q−1 and NLO diagrams (b)–(f) scale as Q0. The loop integral IðpÞ is as that defined in Appendix A, where m1, m2 and m3 should be replaced by the mass of DÃ0, D0 and D0, respectively
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