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

TL;DR

This paper revisits Weinberg's compositeness theorem within an effective field theory framework to analyze the structure of near-threshold states, applying it to the $X(3872)$ and revealing a small elementary component.

Contribution

It introduces a wave function renormalization constant $Z$ in EFT to quantify the elementary component of near-threshold states, providing a new criterion for their structure analysis.

Findings

Determined a non-zero $Z$ for $X(3872)$, indicating a small $c\bar{c}$ core.

Showed near-threshold enhancement can be driven by short-distance production if $Z$ is non-zero.

Applied EFT to line shape analysis of $D^{*0}\bar D^0$ spectrum in $B$ decay.

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.