In search for signals of the $D\bar{D}$ bound state $X(3700)$ from study of the $B^{+}\to D^{+}{D}^{-}{K}^{+}$, $B^0\to D^{+}{D}^{-}{K}^0$ and ${\Lambda }_b\to D^{+}{D}^{-}\Lambda$ reactions

Abstract

We perform a theoretical study of the $B^{+}\to D^{+}{D}^{-}{K}^{+}$, $B^0\to D^{+}{D}^{-}{K}^0$ and ${\Lambda }_b\to D^{+}{D}^{-}\Lambda$ reactions by looking at the production mechanisms, with special emphasis in the final state interaction of the charmed mesons, which gives rise to the $X(3930)$, coupling strongly to $D_s{\bar{D}}_s$, and another state that we call $X(3700)$, coupling strongly to $D\bar{D}$ that qualifies as a $D\bar{D}$ bound state. The combined study of all these reactions shows that the final state interaction responsible for the production of these resonances is more important in the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ reaction than in the ${\Lambda }_b\to D^{+}{D}^{-}\Lambda$ one. We have taken this into account and shown that normalizing the $D^{+}{D}^{-}$ mass distributions to the same value at the peak of the $\psi (3770)$ production, the two mass distributions are quite different in a range of 10 MeV above the $D^{+}{D}^{-}$ threshold, with a value about an order of magnitude bigger for the $B^{+}\to D^{+}{D}^{-}{K}^{+}$ reaction than for the ${\Lambda }_b\to D^{+}{D}^{-}\Lambda$ one. The $D^{+}{D}^{-}$ mass distribution by itself also shows an enhancement of about one order of magnitude in the presence of the $X(3700)$ with respect to a calculation where this resonance is absent. We make a call to measure these magnitudes in the coming LHCb upgrades, which would bring great support to the existence of the bound $D^{+}{D}^{-}$ state.

Comments: 15 pages, 23 figures, 1 table