Productions of $X(3872)$, $Z_c(3900)$, $X_2(4013)$, and $Z_c(4020)$ in $B_{(s)}$ decays offer strong clues on their molecular nature

The exotic states $X(3872)$ and $Z_c(3900)$ have long been conjectured as isoscalar and isovector $\bar{D}^*D$ molecules. In this work, we first propose the triangle diagram mechanism to investigate their productions in $B$ decays as well as their heavy quark spin symmetry partners, $X_2(4013)$ and $Z_c(4020)$. We show that the large isospin breaking of the ratio $\mathcal{B}[B^+ \to X(3872) K^+]/\mathcal{B}[B^0 \to X(3872) K^0] $ can be attributed to the isospin breaking of the neutral and charged $\bar{D}^*D$ components in their wave functions. For the same reason, the branching fractions of $Z_c(3900)$ in $B$ decays are smaller than the corresponding ones of $X(3872)$ by at least one order of magnitude, which naturally explains its non-observation. A hierarchy for the production fractions of $X(3872)$, $Z_c(3900)$, $X_2(4013)$, and $Z_c(4020)$ in $B$ decays, consistent with all existing data, is predicted. Furthermore, with the factorization ansatz we extract the decay constants of $X(3872)$, $Z_c(3900)$, and $Z_c(4020)$ as $\bar{D}^*D^{(*)}$ molecules via the $B$ decays, and then calculate their branching fractions in the relevant $B_{(s)}$ decays, which turn out to agree with all existing experimental data. The mechanism we proposed is useful to elucidate the internal structure of the many exotic hadrons discovered so far and to extract the decay constants of hadronic molecules,which can be used to predict their production in related processes.