How to identify the structure of near-threshold states from the line shape

Abstract

We revisit the compositeness theorem proposed by Weinberg in an effective field theory (EFT) and explore criteria which are sensitive to the structure of $S$-wave threshold states. On a general basis, we show that the wave function renormalization constant $Z$, which is the probability of finding an elementary component in the wave function of a threshold state, can be explicitly introduced in the description of the threshold state. As an application of this EFT method, we describe the near-threshold line shape of the $D^{\ast 0}\bar D^0$ invariant mass spectrum in $B\rightarrow D^{\ast 0}\bar D^0 K$ and determine a nonvanishing value of $Z$. It suggests that the $X(3872)$ as a candidate of the $D^{\ast 0}\bar D^0$ molecule may still contain a small $c\bar{c}$ core. This elementary component, on the one hand, explains its production in the $B$ meson decay via a short-distance mechanism, and on the other hand, is correlated with the $D^{\ast 0}\bar D^0$ threshold enhancement observed in the $D^{\ast 0}\bar D^0$ invariant mass distributions. Meanwhile, we also show that if $Z$ is non-zero, the near-threshold enhancement of the $D^{\ast 0}\bar D^0$ mass spectrum in the $B$ decay will be driven by the short-distance production mechanism. This conclusion is still true even if the long-distance production is enhanced by some unexpected mechanism.