Femtoscopy can tell whether $Z_c(3900)$ and $Z_{cs}(3985)$ are resonances or virtual states

TL;DR

The paper demonstrates that femtoscopy can distinguish whether near-threshold states like $Z_c(3900)$ and $Z_{cs}(3985)$ are resonances, virtual states, or bound states by analyzing the $D^0D^{*-}$ and $D^0D_s^{*-}$ correlation functions. It builds a near-threshold S-wave framework with $V(k)=a+b k^2$ and computes the T-matrix via $T(\sqrt{s})=(1-VG)^{-1}V$, using a loop function $G$ regulated by a sharp cutoff $q_{\max}$ and extracting poles to characterize the three scenarios. The femtoscopy observable $C(k)$ is obtained from the Koonin-Pratt formula with a Gaussian source, $C(k)=1+\mathcal{F}_1\sin^2\delta+\mathcal{F}_2\sin\delta\cos\delta$, where $\delta$ encodes the near-threshold dynamics. The results show clear differences in the low-momentum region among the three scenarios, with additional high-momentum distinctions in small systems, and they explore coupled-channel effects and LL-model benchmarks to establish a robust, experimentally testable method to clarify the nature of these exotic states and extract hadron-hadron interactions from correlation data.

Abstract

There have been extended and heated discussions on the nature of the two exotic states, $Z_c(3900)$ and $Z_{cs}(3985)$, particularly whether they are near-threshold resonances, virtual states, or bound states. In this work, we demonstrate for the first time that the femtoscopic technique can be employed to distinguish between these three scenarios. More concretely, based on the Koonin-Pratt formula with a Gaussian source, we show that the low-momentum $D^0D^{*-}$/$D^0D_s^{*-}$ correlation functions significantly differ in the three scenarios. The high-momentum results exhibit distinct characteristics in the resonant and virtual state scenarios, especially in small collision systems of 1 fm, as produced in $pp$ collisions at the LHC. We hope that these discoveries will stimulate further experimental studies and help clarify the nature of the many exotic states that have been discovered.

Figures (6)

• Figure 1: $D^0D^{*-}$ (upper) and $D^0D_s^{*-}$ (lower) $|T|^2$, $\rho|T|^2$ as a function of the c.m. energy $\sqrt{s}$, and the phase shifts as a function of the relative momentum $k$. The bands reflect the variation of the sharp cutoff in the range of $q_{\rm max}=0.6 - 1.4$ GeV.
• Figure 2: $D^0D^{*-}$ (upper) and $D^0D_s^{*-}$ (lower) correlation functions as a function of the relative momentum $k$ for different source sizes $R = 1$, $2$, and $5$ fm. The bands reflect the variation of the sharp cutoff in the range of $q_{\rm max}=0.6 - 1.4$ GeV.
• Figure 3: Coupled-channel $D^0D^{*-}$ correlation functions for $R = 1$ fm as a function of the relative momentum $k$. The dashed and solid lines denote the elastic contribution and full coupled-channel result, respectively. The sharp cutoff $q_{\rm max}$ is fixed at $1$ GeV.
• Figure 4:
• Figure 5: Pole-position dependence of the $D^0D_s^{*-}$ correlation functions. The source size is set at $R = 1$ fm, and the hard cutoff $q_{\rm max}$ is fixed at $1$ GeV.