Production of $X(3872)$ Accompanied by a Pion in $B$ Meson Decay

If the $X(3872)$ is a weakly bound charm-meson molecule with flavor structure $(D^{*0}\bar{D}^0 +D^{0}\bar{D}^{*0})/\sqrt2$, it can be produced by the creation of charm meson pairs at short distances followed by the formation of the bound state. It can also be produced by the creation of $D^{*}\bar{D}^*$ at short distances followed by their rescattering into $X \pi$. An effective field theory called XEFT predicts that the rate for producing $X$ accompanied by a soft pion should be roughly comparable to that for an isolated $X$. We use the results of a previous isospin analysis of $B$ meson decays into $K D^{(*)}\bar D^{(*)}$ to estimate the short-distance amplitudes for creating $D^*\bar D^*$. We use XEFT to calculate the amplitudes for rescattering into $X \pi$. The resulting predictions for the branching fractions of $B$ mesons into $K X$ plus a soft $\pi^+$ or $\pi^-$ are consistent with measurements of the branching fractions into $KX\pi^\pm$ by the Belle collaboration. If $K X \pi^\pm$ events in which $K\pi^\pm$ come from the $K^*$ resonance are removed, the remaining events are predicted to come primarily from the region of the Dalitz plot with small $X\pi^\pm$ invariant mass.