<p>We study the internal structure of exotic hadrons, especially focusing on the relation between the compositeness and physical observables. Defined as the probability of finding hadronic molecular components in the wave function, compositeness serves as a quantitative measure of the internal structure of exotic hadrons. We utilize the coupled-channel potential model incorporating both quark and hadron degrees of freedom, which naturally generate the “bare state” responsible for the elementary component as the bound state in the quark channel. The behavior of the compositeness under the variation of the model parameters is investigated by using the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> as an example. In particular, we analyze the associated scattering phase shifts and the bound-state wave functions to discuss the relation between the compositeness and the scattering observables for a shallow bound state. As a phenomenological application of the present framework, the compositeness of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow><mo>,</mo><mo> </mo><mrow><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mrow><mo>(</mo><mn>3875</mn><mo>)</mo></mrow></mrow><mo>,</mo><mo> </mo><mrow><msub><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow></msub><mrow><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mrow><mo>(</mo><mn>2460</mn><mo>)</mo></mrow></mrow></math> is discussed.</p>

Japan

Ibuki Terashima

Tetsuo Hyodo

Compositeness and wave function of shallow bound states in relation to scattering observables

<p>We discuss the general behavior of the scattering amplitude with channel couplings near the two-body threshold. It is known that the Flatté amplitude, which is often used in the analysis of experimental data involving exotic hadrons, has some constraint in the near-threshold energy region. While the <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>M</mi></mrow></math>-matrix gives the general expression of the scattering amplitude, it is not smoothly connected to the Flatté amplitude, due to the property of the determinant of the amplitude in channel space. In this paper, based on the effective field theory, we propose new parametrization of the scattering amplitude which gives the general expression near the threshold and has a well-defined limit reproducing the Flatté amplitude. We show that the nonresonant background contribution exists in the general amplitude even in the first order in the momentum. Finally, we quantitatively evaluate the cross sections by changing the strength of the background contribution. We find that the interference with the background term may induce a dip structure of the cross section near the threshold, in addition to the peak and threshold cusp structures.</p>

Katsuyoshi Sone

General amplitude of near-threshold hadron scattering for exotic hadrons

The compositeness of weakly bound states is discussed using the effective field theory from the viewpoint of the low-energy universality. We introduce a model with coupling of the single-channel scattering to the bare state, and study the compositeness of the bound state by varying the bare state energy. In contrast to the naive expectation that the near-threshold states are dominated by the molecular structure, we demonstrate that a noncomposite state can always be realized even with a small binding energy. At the same time, however, it is shown that a fine tuning is necessary to obtain the noncomposite weakly bound state. In other words, the probability of finding a model with the composite dominant state becomes larger with the decrease of the binding energy, in accordance with the low-energy universality. For the application to exotic hadrons, we then discuss the modification of the compositeness due to decay and coupled-channels effects. We quantitatively show that these contributions suppress the compositeness, because of the increase of the fraction of other components. Finally, as examples of near-threshold exotic hadrons, the structures of <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> and <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> are studied by evaluating the compositeness. We find the importance of the coupled-channels and decay contributions for the structures of <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> and <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math>, respectively.

Tomona Kinugawa

Compositeness of <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub></math> and <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> by considering decay and coupled-channels effects

We study the <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> resonance near the <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> threshold in the light of recent experimental constraints. The Belle Collaboration have found a resonance peak of <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> slightly below the <math><mrow><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup><mi>Λ</mi></mrow></math> threshold in the <math><mrow><msup><mi>π</mi><mo>+</mo></msup><msup><mi>Ξ</mi><mo>−</mo></msup></mrow></math> invariant mass spectrum, and the ALICE Collaboration have determined the <math><mrow><msup><mi>K</mi><mo>−</mo></msup><mi>Λ</mi></mrow></math> scattering length from the measurement of the momentum correlation functions in heavy ion collisions. Using the effective range expansion, we classify the nature of the pole of the near-threshold eigenstate in terms of the scattering length, in the presence of the decay channel. It is shown that the quasibound state below the threshold can be described by only the scattering length, while the description of the resonance above the threshold requires the contribution from the effective range. Based on the chiral unitary approach, we construct a theoretical model which generates the pole of <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> below the <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> threshold with relatively narrow width, as reported by the Belle Collaboration. It is quantitatively demonstrated that the spectrum of the <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> quasibound state is distorted by the effect of the nearby <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> threshold. We then construct another model which reproduces the <math><mrow><msup><mi>K</mi><mo>−</mo></msup><mi>Λ</mi></mrow></math> scattering length by the ALICE Collaboration. In this case, the eigenstate pole does not appear in the physically relevant Riemann sheet, and the spectrum shows a cusp structure at the <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> threshold. We finally examine the compatibility of the value of the <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> scattering length and the subthreshold pole of <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> including the experimental uncertainties.

Takuma Nishibuchi

Analysis of the <math><mrow><mi>Ξ</mi><mo>(</mo><mn>1620</mn><mo>)</mo></mrow></math> resonance and <math><mrow><mover><mi>K</mi><mo>¯</mo></mover><mi>Λ</mi></mrow></math> scattering length with a chiral unitary approach

We study the properties of the hadron-hadron potentials and quark-antiquark potentials from the viewpoint of the channel coupling. We demonstrate that, for finite quark masses, the coupling to the two-hadron continuum induces the imaginary part of the quark-antiquark potential, in contrast to the string-breaking phenomena in the static limit. It is also shown that the elimination of the different degrees of freedom induces the nonlocality and energy dependence of the effective potentials. For the obtained nonlocal potentials, we apply two methods of the local approximation proposed previously, the formal derivative expansion and the derivative expansion in the HAL QCD method, by carefully examining the energy dependence of the potential. As an application, we construct a coupled-channel model of <math><mrow><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover></mrow></math> and <math><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></mrow></math> to describe <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math>, and discuss the property of the effective <math><mrow><msup><mi>D</mi><mn>0</mn></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></mrow></math> potentials. We confirm that the local approximation by the HAL QCD method works better than the formal derivative expansion also for the energy-dependent potential. At the same time, we show that, in the HAL QCD method, the resulting phase shift is sensitive to the choice of the wave function to construct the local potential when the system has a shallow bound state such as <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math>.

Hadron-hadron potentials coupled to quark degrees of freedom for exotic hadrons

The composite nature of a shallow bound state is studied by using the weak-binding relation, which connects the compositeness of the bound state with observables. We first show that the previous weak-binding relation cannot be applied to the system with a large effective range. To overcome this difficulty, we introduce the finite-range correction by redefining the typical length scale in the weak-binding relation. A method to estimate the uncertainty of the compositeness is proposed. It is numerically demonstrated that the range correction enlarges the applicable region of the weak-binding relation. Finally, we apply the improved weak-binding relation to the actual hadrons, nuclei, and atomic systems [deuteron, <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow><mo>,</mo><mo> </mo><mrow><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mrow><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></mrow><mo>,</mo><mo> </mo><mrow><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mrow><mo>(</mo><mn>2460</mn><mo>)</mo></mrow></mrow><mo>,</mo><mo> </mo><mrow><mi>N</mi><mi>Ω</mi></mrow></math> dibaryon, <math><mrow><mi>Ω</mi><mi>Ω</mi></mrow></math> dibaryon, <math><mmultiscripts><mi>H</mi><mprescripts></mprescripts><mi>Λ</mi><mn>3</mn></mmultiscripts></math>, and <math><mmultiscripts><mi>He</mi><mprescripts></mprescripts><none></none><mn>4</mn></mmultiscripts></math> dimer] to discuss their internal structure from the compositeness. We present a reasonable estimation of the compositeness of the deuteron by properly taking into account the uncertainty. The results of <math><mrow><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></mrow></math> and the <math><mrow><mi>N</mi><mi>Ω</mi></mrow></math> dibaryon show that the range correction is important to estimate the compositeness of physical states.

Structure of exotic hadrons by a weak-binding relation with finite-range correction

We study the nature of the scalar <math><msubsup><mi>D</mi><mrow><mn>0</mn></mrow><mo>*</mo></msubsup></math> meson from the viewpoint of chiral symmetry. With the linear representation of chiral symmetry, we construct the <math><mrow><mi>D</mi><mi>π</mi></mrow></math> scattering amplitude satisfying the chiral low-energy theorem, in which the <math><msubsup><mi>D</mi><mrow><mn>0</mn></mrow><mo>*</mo></msubsup></math> meson appears as an <math><mi>s</mi></math>-wave resonance. We show that the properties of the <math><msubsup><mi>D</mi><mrow><mn>0</mn></mrow><mo>*</mo></msubsup></math> meson can be successfully reproduced as the chiral partner of the <math><mi>D</mi></math> meson coupled with the <math><mrow><mi>D</mi><mi>π</mi></mrow></math> scattering states. At the same time, we find that the spectral function and the pole position of <math><msubsup><mi>D</mi><mrow><mn>0</mn></mrow><mo>*</mo></msubsup></math> are not very sensitive to the reduction of the chiral condensate, indicating the importance of the dressing of the bare state by the <math><mrow><mi>D</mi><mi>π</mi></mrow></math> molecular component.

Takumi Sugiura

Nature of the <math><msubsup><mi>D</mi><mrow><mn>0</mn></mrow><mo>*</mo></msubsup></math> meson in <math><mrow><mi>D</mi><mi>π</mi></mrow></math> scattering with chiral symmetry

The <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo><mi>D</mi></math> and <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo><msup><mi>D</mi><mo>*</mo></msup></math> heavy meson systems can exchange a kaon that is emitted in the S-wave due to the opposite intrinsic parity of the <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>)</mo></mrow></math> and <math><mrow><mi>D</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>)</mo></mrow></math> mesons. As a consequence of the mass difference of the <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>)</mo></mrow></math> and <math><mrow><mi>D</mi><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>)</mo></mrow></math> mesons, the range of the kaon exchange potential will be significantly longer than expected, corresponding to an effective mass of about 200 MeV. The potential will be very strong: the strength of the interaction is proportional to <math><mrow><mo>(</mo><msub><mrow><mi>m</mi></mrow><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow></msub></mrow></msub><mo>−</mo><msub><mrow><mi>m</mi></mrow><mrow><mi>D</mi></mrow></msub><msup><mrow><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msubsup><mrow><mi>f</mi></mrow><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></math> and <math><mo>(</mo><msub><mi>m</mi><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub></msub><mo>−</mo><msub><mi>m</mi><msup><mi>D</mi><mo>*</mo></msup></msub><msup><mo>)</mo><mn>2</mn></msup><mo>/</mo><msubsup><mi>f</mi><mi>π</mi><mn>2</mn></msubsup></math>. This combination of range and strength almost guarantees the existence of <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo><mi>D</mi></math> and <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo><msup><mi>D</mi><mo>*</mo></msup></math> bound states with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup></math> and <math><mrow><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo>=</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow></math>, <math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow></math>, respectively. Concrete calculations indicate a binding energy of 5–15 MeV independently of <math><msup><mi>J</mi><mi>P</mi></msup></math>. The <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo><mi>D</mi></math> and <math><msubsup><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>*</mo></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo><msup><mi>D</mi><mo>*</mo></msup></math> molecules have manifestly exotic flavor quantum numbers: <math><mi>C</mi><mo>=</mo><mn>2</mn></math>, <math><mrow><mi>S</mi><mo>=</mo><mn>1</mn></mrow></math>, and <math><mi>I</mi><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math>. We expect the existence of bottom counterparts composed of the <math><mi>B</mi><msub><mi>B</mi><mrow><mi>s</mi><mn>0</mn></mrow></msub></math> and <math><msup><mi>B</mi><mo>*</mo></msup><msubsup><mi>B</mi><mrow><mi>s</mi><mn>1</mn></mrow><mo>*</mo></msubsup></math> mesons, which will be more bound and have a richer spectrum that might include a shallow P-wave state and an excited S-wave state.

France

M. Sanchez Sanchez

China

Li-Sheng Geng

Jun-Xu Lu

M. Pavon Valderrama

Exotic doubly charmed <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo><mi>D</mi></mrow></math> and <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2460</mn><mo>)</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> molecules

Based on a baryon-baryon interaction model with meson exchanges, we investigate the origin of the strong attraction in the <math><mrow><mi>N</mi><mi>Ω</mi><mo>(</mo><mmultiscripts><mi>S</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mo>)</mo></mrow></math> interaction, which was indicated by recent lattice QCD simulations. The long-range part of the potential is constructed by the conventional mechanisms, the exchanges of the <math><mi>η</mi></math> meson and of the correlated two mesons in the scalar-isoscalar channel, denoted by “<math><mi>σ</mi></math>” in the literature. The short-range part is represented by the contact interaction. We find that the meson exchanges do not provide sufficient attraction. This means that most of the attraction is attributed to the short-range contact interaction. We then evaluate the effect of the coupled channels to the <math><mrow><mi>N</mi><mi>Ω</mi><mo>(</mo><mmultiscripts><mi>S</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mo>)</mo></mrow></math> interaction. We find that, while the <math><mi>D</mi></math>-wave mixing of the <math><mrow><mi>N</mi><mi>Ω</mi></mrow></math> channel is negligible, the inelastic <math><mrow><mi>Λ</mi><mi>Ξ</mi></mrow></math>, <math><mrow><mi>Σ</mi><mi>Ξ</mi></mrow></math>, and <math><mrow><mi>Λ</mi><mi>Ξ</mi><mo>(</mo><mn>1530</mn><mo>)</mo></mrow></math> channels via the <math><mi>K</mi></math> meson exchange give the attraction of the <math><mrow><mi>N</mi><mi>Ω</mi><mo>(</mo><mmultiscripts><mi>S</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mo>)</mo></mrow></math> interaction to the same level with the elastic meson exchanges. Although the elimination of these channels induces the energy dependence of the single-channel <math><mrow><mi>N</mi><mi>Ω</mi></mrow></math> interaction, this effect is not significant. With the present model parameters fit to reproduce the scattering length of the HAL QCD result of the nearly physical quark masses, we obtain the <math><mrow><mi>N</mi><mi>Ω</mi><mo>(</mo><mmultiscripts><mi>S</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mo>)</mo></mrow></math> quasibound state with its eigenenergy <math><mrow><mn>2611.3</mn><mo>−</mo><mn>0.7</mn><mi>i</mi><mspace width="4.pt"></mspace><mtext>MeV</mtext></mrow></math>, which corresponds to the binding energy <math><mrow><mn>0.1</mn><mspace width="4.pt"></mspace><mtext>MeV</mtext></mrow></math> and width <math><mrow><mn>1.5</mn><mspace width="4.pt"></mspace><mtext>MeV</mtext></mrow></math> for the decay to the <math><mrow><mi>Λ</mi><mi>Ξ</mi></mrow></math> and <math><mrow><mi>Σ</mi><mi>Ξ</mi></mrow></math> channels. From the analysis of the spatial structure and the compositeness, the quasibound state is shown to be the molecular state of <math><mrow><mi>N</mi><mi>Ω</mi></mrow></math>. We also construct an equivalent local potential for the <math><mrow><mi>N</mi><mi>Ω</mi><mo>(</mo><mmultiscripts><mi>S</mi><mn>2</mn><none></none><mprescripts></mprescripts><none></none><mn>5</mn></mmultiscripts><mo>)</mo></mrow></math> system which is useful for various applications.

Takayasu Sekihara

Yuki Kamiya

<math><mrow><mi>N</mi><mi>Ω</mi></mrow></math> interaction: Meson exchanges, inelastic channels, and quasibound state

We discuss the relation between the analytic structure of the scattering amplitude and the origin of an eigenstate represented by a pole of the amplitude. If the eigenstate is not dynamically generated by the interaction in the channel of interest, the residue of the pole vanishes in the zero coupling limit. Based on the topological nature of the phase of the scattering amplitude, we show that the pole must encounter with the Castillejo-Dalitz-Dyson (CDD) zero in this limit. It is concluded that the dynamical component of the eigenstate is small if a CDD zero exists near the eigenstate pole. We show that the line shape of the resonance is distorted from the Breit-Wigner form as an observable consequence of the nearby CDD zero. Finally, studying the positions of poles and CDD zeros of the <math><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><mi>N</mi><mtext>−</mtext><mi>π</mi><mi>Σ</mi></mrow></math> amplitude, we discuss the origin of the eigenstates in the <math><mrow><mi>Λ</mi><mo>(</mo><mn>1405</mn><mo>)</mo></mrow></math> region.