Advances in High Energy Physics

Volume 2019, Article ID 4651908, 6 pages

https://doi.org/10.1155/2019/4651908

## Effective-Range-Expansion Study of Near Threshold Heavy-Flavor Resonances

^{1}Department of Physics and Hebei Advanced Thin Films Laboratory, Hebei Normal University, Shijiazhuang 050024, China^{2}College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China^{3}Departamento de Física, Universidad de Murcia, E-30071 Murcia, Spain

Correspondence should be addressed to Zhi-Hui Guo; nc.ude.utbeh@oughz

Received 26 December 2018; Revised 5 April 2019; Accepted 15 May 2019; Published 16 June 2019

Academic Editor: Burak Bilki

Copyright © 2019 Rui Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

In this work, we study the resonances near the thresholds of the open heavy-flavor hadrons using the effective-range-expansion method. The unitarity, analyticity, and compositeness coefficient are also taken into account in our theoretical formalism. We consider the , , , , and . The scattering lengths and effective ranges from the relevant elastic -wave scattering amplitudes are determined. Tentative discussions on the inner structures of the aforementioned resonances are given.

#### 1. Introduction

Since the discovery of the [1], the study of the exotic hidden heavy-flavor hadrons has become one of the most important and active research topics in particle physics. Up to now more than twenty of the so-called hadrons are observed by experiments [2] and we refer to [3, 4] for recent comprehensive reviews on this subject. One of the most important features of the newly observed hadrons is that they are usually close to the nearby thresholds of the open heavy-flavor states. As a result typically one needs to properly take into account the threshold effects when studying those possible exotic states. The effective range expansion (ERE), which is based on the three-momentum expansion near the threshold, provides a useful tool to address the dynamics in the energy region around the relevant threshold in question [5, 6].

By combining the ERE, unitarity, analyticity, and the compositeness coefficients developed in [7, 8], we have successfully analyzed several non-ordinary hadronic states that lie close to the thresholds of the underlying two-particle states, including the [9], , and [10], and the newly observed pentaquark candidates and [11]. The essential idea of the formalism is that we construct the elastic unitarized partial-wave amplitude using the ERE as the kernel, which includes the scattering length and effective range as free parameters. The latter are determined by reproducing the values of the mass and width of the observed resonance. We always obtain real values consistently with the assumption of only one-channel scattering. Then the residue of the resonance pole, which corresponds to the coupling strength of the resonance to the interacting two-particle state, can be obtained as well. With all of these ingredients, we can apply the compositeness formalism to infer the probability to find the two-particle state inside the resonance. In this work we first briefly recapitulate the essentials of the theoretical formalism. Then we tentatively generalize this formalism to other newly observed hadronic states, including the near the (denoted shortly as in the following) threshold, the near the threshold, the near the (denoted as in the following) threshold, the near the (denoted shortly as in the following) threshold, and the near the threshold [2]. Historically, there is also a state named that we identify with the resonance as in the PDG [2] and [12–14].

#### 2. Effective Range Expansion and Compositeness Coefficients of Resonances

The basic staring point of our theoretical formalism is the ERE up to the next-to-leading order:with the scattering length, the effective range, and the three-momentum in the center of mass (CM) frame. At a given CM energy around the threshold, one can write the nonrelativistic three-momentum aswhere the reduced mass of the system with masses and is and the threshold is given by .

We mention that a more general expression to write the scattering amplitude near threshold is to include the so-called Castillejo-Dalitz-Dyson (CDD) poles. The standard ERE in (1) can be obtained by expanding the full expression with CDD poles [9, 10]. However when the CDD pole happens to be near the threshold, the expansion in terms of will be invalid, or at least quite limited to a tiny region. One of the prominent features in this situation is the huge effective range , which usually becomes much larger than its standard value around 1 fm. If this is the case, one has to work explicitly with the CDD pole in the full expression, as done in [9, 10, 18].

The partial-wave amplitude given in (1) corresponds to the physical one in the first Riemann sheet (RS). Its expression in the second RS is given byThe resonance poles only appear in the second RS. Comparing with (1) and (3), there is a change of sign for the term. Notice that, in the conventions of (1) and (3), the three-momentum should be evaluated with .

Let us now consider a resonance whose pole position in the unphysical RS is located at , with . For a conventional narrow-width resonance, one can identify and as its mass and width, respectively. The corresponding three-momentum at the resonance pole is then given byFor later convenience, we further defineOne has to be careful when evaluating in terms of the threshold and the resonance parameters and , specially distinguishing the sign of . Detailed discussions can be found in [10].

The resonance pole corresponds to the zero of the denominator of ; i.e., at the pole position one hasBy a straightforward algebraic manipulation, one can solve and in terms of and :

Substituting (7) and (8) into (3), one can write the partial-wave amplitude around the resonance pole aswhere the ellipses stand for the neglected terms when expanding the denominator of (3) in terms of . From (9), we can identify the residue at the pole in the variable , which turns out to besince . Alternatively one can also expand the partial-wave amplitude around the pole asThe residue is related to as

In [7], a probabilistic interpretation for the compositeness of the two-particle state inside the resonance is derived. The value of can be calculated once the resonance pole position and the corresponding residues are provided. Around the two-particle threshold, the probability reduces to the following [10]:However, we point out that the probabilistic interpretation of is restricted to resonances with [7]. In [15–17, 19–24], other approaches to generalize the Weinberg’s compositeness of bound states [25] to the resonances are discussed. We compare below our results with some of them in Section 3.

In the next section, we proceed to study several near-threshold resonances within the present ERE approach. Let us notice that if this type of ERE study is applied to a near-threshold resonance which is composite of the nearby channel (the so-called elastic one) then . From here it also follows that if we apply this type of ERE study to a near-threshold resonance and it results that , then one can conclude for sure that this resonance is not a composite one of the elastic channel. On the other hand, if it results that then the interpretation of this resonance as a composite one of the elastic channel is favored (In a similar sense that a cloudy sky favors rainfall.).

#### 3. Phenomenological Discussions

Before entering the detailed discussions, we stress that the theoretical approach developed is based on the elastic -wave two-body scattering. Strictly speaking, the present formalism applies to the scattering of two stable hadrons. A rigorous approach to handle the presence of unstable hadrons in the scattering process is to perform the study within three- or even few-body scattering [26], which is clearly beyond the scope of the present work. Another indirect way to understand the role of the width in the unstable hadrons is to use a complex mass () in the expression for the three-momentum, Eq. (4), which then readsIn this way, one can take into account the self-energy effects of the decaying channels. This is applicable, as compared with a full three-body study, if is much smaller than the difference between the mass of the resonance and the threshold of the decay channel, as indicated by the results from [26] concerning the , and scattering and the resonance. This is the case for the hadron, and the numerical results obtained with a complex mass are also indicated below. Another interesting point is that unstable hadrons could introduce additional crossed-channel contributions, comparing with stable ones; see, e.g., [26].

The crossed-channel effects, such as the light-flavor hadron exchanges, are neglected in (1), which is strictly valid in the near-threshold region before any other branch-point singularity, associated with the onset of crossed channels, sets in. Nonetheless, the theoretical formalism presented here can be used also to study the systems in which the crossed-channel dynamics can be treated perturbatively. In [9, 10], the resonances , and have been successfully addressed within this formalism. In the following, we tentatively generalize the discussions to the , , , and , which may be composed by some specific -wave open-charm two-body states and lie close to their thresholds.

The masses and widths of the , , , , and and the thresholds of the nearby two-body states are collected in Table 1. The spin and parity of the , , , and given in the PDG [2] are compatible with the -wave elastic scattering of , , and , respectively. For the , its spin and parity are not confirmed by experiments yet. By assuming the -wave molecular nature of , a possible quantum number of the would be .