Estimation of compositeness with correction terms

Almost all hadrons are considered to be $qqq$ or $q\bar{q}$ states in the constituent quark models. However, some hadrons are expected to have an extraordinary structure, and called exotic hadrons. In recent experiments in the heavy quark sector, candidates for exotic hadrons have been observed, as represented by the $X(3872)$ Belle:2003nnu . One possible component of the candidates for the exotic hadrons is the hadronic molecule, which is a weakly bound state of hadrons.

We can quantitatively characterize the internal structure of the state by compositeness whether it is a hadronic molecule dominant (composite dominant) state or not Hyodo:2013nka . The compositeness $X$ is defined as the probability to find the hadronic molecule component in the normalized wavefunction of the bound states $\ket{\Psi}$, $X=|\braket{{\rm molecule}}{\Psi}|^{2}$. Here $\ket{\rm molecule}$ is the schematic notation of the hadronic molecule component. We can determine $X$ by using the weak-binding relation Weinberg:1965zz ; Kamiya:2016oao :

$\displaystyle a_{0}$

$\displaystyle=R\left\{\frac{2X}{1+X}+\mathcal{O}\left(\frac{R_{\rm typ}}{R}\right)\right\},$

(1)

where $a_{0}$ is the scattering length and $R\equiv\sqrt{2\mu B}$ is the radius of the bound state, determined by the binding energy $B$ and the reduced mass $\mu$. Taking into account the range correction to the weak-binding relation Kinugawa:2022fzn , we define $R_{\rm typ}$ as the largest one among the length scale of the interaction $R_{\rm int}$ and those in the effective range expansion except for $a_{0}$:

$\displaystyle R_{\rm typ}=\max\{R_{\rm int},|r_{e}|,|P_{s}/R^{2}|,\cdots\},$

(2)

where $r_{e}$ is the effective range and $P_{s}$ is the shape parameter (for more details, see Sec. III in Ref. Kinugawa:2022fzn ).

When we consider sufficiently shallow bound states with $R\gg R_{\rm typ}$, the correction terms of the weak-binding relation $\mathcal{O}(R_{\rm typ}/R)$ are negligible, and the compositeness $X$ is determined only from the observables $a_{0}$ and $R$. Thanks to this universal feature, the weak-binding relation has been utilized as a model-independent approach to calculate $X$. However, naive application of Eq. (1) without the correction terms sometimes contradicts the definition domain of the compositeness, $0\leq X\leq 1$. For example, the compositeness of the deuteron $d$ is given as $X=1.68$ with $a_{0}=5.42$ fm (taken from CD-Bonn potential Machleidt:2000ge ) and $B=2.22$ MeV (taken from PDG ParticleDataGroup:2020ssz ). This problem is discussed in Refs. Li:2021cue ; Song:2022yvz ; Albaladejo:2022sux in connection with the effective range. To avoid this contradiction, here we propose a reasonable estimation method of the compositeness with the uncertainty which arises from the correction terms $\mathcal{O}(R_{\rm typ}/R)$ in Eq. (1).

Now we estimate the compositeness $X$ of the actual physical systems with the uncertainty estimation discussed in Sec. 2.2. We consider the deuteron, $X(3872)$, $D^{*}_{s0}(2317)$, $D_{s1}(2460)$, $N\Omega$ dibaryon, $\Omega\Omega$ dibaryon, ${}^{3}_{\Lambda}{\rm H}$, and ${}^{4}{\rm He}$ dimer. The deuteron $d$ in the $p$-$n$ scattering is chosen as the typical observed hadron. $X(3872)$ in the $D^{0}$-$\bar{D}^{*0}$ scattering, $D^{*}_{s0}(2317)$ in the $D$-$K$ scattering, and $D_{s1}(2460)$ in the $D^{*}$-$K$ scattering are the candidates for the exotic hadrons which are experimentally observed ParticleDataGroup:2020ssz . $N\Omega$ and $\Omega\Omega$ dibaryons are the states obtained by the lattice QCD calculation HALQCD:2018qyu ; Gongyo:2017fjb . We can apply the weak-binding relation not only to the hadron systems but also to the nuclei and atomic systems. ${}^{3}_{\Lambda}{\rm H}$ in the $\Lambda$-$d$ scattering is an example of nuclei, and ${}^{4}{\rm He}$ dimer which is the weakly bound state of ${}^{4}{\rm He}$ atoms is an example in the atomic systems.

For the estimation of $X$ from the weak-binding relation, we need the scattering length $a_{0}$, the reduced mass $\mu$, the binding energy $B$, the effective range $r_{e}$, and the interaction range $R_{\rm int}$. The radius of the bound state is calculated by $R=\sqrt{2\mu B}$. We tabulate relevant quantities in Tab. 1. We note that $R_{\rm int}$ is not an observable, and therefore it is determined from the theoretical consideration. The procedure to determine these physical quantities is explained in Ref. Kinugawa:2022fzn .

The results of the estimated compositeness with the uncertainty band (7) are shown in the right column in Tab. 1. It is found that the range correction is important for the application to the $X(3872)$ and the $N\Omega$ dibaryon Kinugawa:2022fzn . We find that those bound states are dominated by the composite component because the lower boundaries $\bar{X}_{l}$ are larger than 0.5.

The compositeness $X$ characterizes the internal structure of shallow bound states, especially for the candidates for exotic hadrons. The weak-binding relation is one of the approaches to estimate $X$. When we neglect the correction terms $\mathcal{O}(R_{\rm typ}/R)$, the weak-binding relation becomes completely model-independent. However, if the scattering length $a_{0}$ is larger than the radius of the bound state $R$, the compositeness is overestimated as $X\geq 1$ without the correction terms. To avoid this problem, we discuss the method to evaluate the correction terms $\mathcal{O}(R_{\rm typ}/R)$. We propose the estimation method of $X$ with the uncertainty band, which includes the contribution of the correction terms $\mathcal{O}(R_{\rm typ}/R)$. Our uncertainty estimation provides the compositeness in $0\leq X\leq 1$ which can be interpreted as a probability. We finally perform reasonable estimations of $X$ as shown in Tab. 1, and find that all states which we consider are composite dominant ($\bar{X}_{l}\geq 0.5$).