The BESIII Collaboration recently performed a precise measurement of the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover></math> Born cross sections, and confirmed the <math><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></math> structure reported by BABAR and Belle with high significance. We identify the <math><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></math> as the first <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> molecular resonance. The experimental and theoretical identification of the <math><mi>P</mi></math>-wave dimeson state holds paramount importance in enhancing our comprehension of the nonperturbative QCD and few-body physics. Its existence is firmly established in a unified meson-exchange model that simultaneously depicts the features of the <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></math>, <math><msub><mi>Z</mi><mi>c</mi></msub><mo>(</mo><mn>3900</mn><mo>)</mo></math>, and <math><msub><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow></msub><mo>(</mo><mn>3875</mn><mo>)</mo></math>. This scenario can be directly examined in the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> cross section by seeing whether a resonance exists at the threshold. The credibility of the investigations is also ensured by the fact that the <math><mi>P</mi></math>-wave interaction dominantly arises from the well-known long-range pion exchange. Additionally, thanks to the centrifugal barrier, it is easier to form resonances in <math><mi>P</mi></math>-wave than in <math><mi>S</mi></math>-wave. We extensively calculate all systems up to <math><mi>P</mi></math>-wave with various quantum numbers and predict a dense population of the <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> and <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> states, where the <math><mi>S</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><msup><mi>I</mi><mi>G</mi></msup><mo>(</mo><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup><mo>(</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>−</mo></mrow></msup><mo>)</mo></math>, <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><mo>/</mo><mover><mi>D</mi><mo>¯</mo></mover><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><msup><mi>I</mi><mi>G</mi></msup><mo>(</mo><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup><mo>(</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>)</mo></math>, and <math><mi>P</mi></math>-wave <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> state with <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mn>0</mn><mo>−</mo></msup><mo>)</mo></math> are more likely to be observed in experiments.

China

Zi-Yang Lin

Jun-Zhang Wang

Jian-Bo Cheng

Germany

Lu Meng

Shi-Lin Zhu

Identification of the <math><mrow><mi>G</mi><mo>(</mo><mn>3900</mn><mo>)</mo></mrow></math> Structure as the <math><mi>P</mi></math>-Wave <math><mrow><mi>D</mi><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mo>/</mo><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup></mrow></math> Resonance

We use the leading-order contact interactions and one-pion-exchange potentials to investigate the newly observed double-charm state <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math>. We employ the complex scaling method to search for the poles on the first Riemann sheet with respect to the <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> threshold and the second Riemann sheet with respect to the <math><mi>D</mi><mi>D</mi><mi>π</mi></math> threshold. The <math><mi>D</mi><mi>D</mi><mi>π</mi></math> three-body effect is important in this system since the intermediate states can go on shell. We involve the three-body effect through an energy-dependent one-pion-exchange potential, which results in a unitary cut at the <math><mi>D</mi><mi>D</mi><mi>π</mi></math> three-body threshold. We find a pole corresponding to the <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> on the physical Riemann sheet. Its width is around 80 keV and nearly independent of the choice of the cutoff. Assuming the <math><mi>D</mi><mover><mi>D</mi><mo>¯</mo></mover><mi>π</mi></math> and <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> channels as the main decay channels, we apply the similar calculations to the <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></math>, and find its width is even smaller. The isospin breaking effect is so significant for the <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></math> that it is mainly a molecular state of <math><msup><mi>D</mi><mn>0</mn></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></math>. Furthermore, we introduce a revised Schrödinger equation for unstable particles to include the contribution of the <math><msup><mi>D</mi><mo>*</mo></msup></math> width, which is also revised by the three-body effect.

<math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> and <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>3872</mn><mo>)</mo></math> with the complex scaling method and <math><mi>D</mi><mi>D</mi><mo>(</mo><mover><mi>D</mi><mo>¯</mo></mover><mo>)</mo><mi>π</mi></math> three-body effect

A new approach to partial widths is proposed through the analytical extension of the wave function in momentum space. By including the residue of the wave function, the Schrödinger equation is extended to the second Riemann sheet. As a result, the partial width is associated with the pole of the wave function. The resonance wave function is convergent in momentum space and can be used to evaluate other observables. This approach is applied to a coupled-channel analysis for <math><msub><mi>P</mi><mi>c</mi></msub></math> states, involving the contact interactions and one-pion-exchange potential with the three-body effects. Under the reasonable assumption that the off-diagonal contact interactions are small, the <math><msup><mi>J</mi><mi>P</mi></msup></math> quantum numbers of the <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mn>4440</mn><mo>)</mo></mrow></math> and the <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4457</mn><mo>)</mo></math> are <math><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup></math> and <math><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup></math> respectively. The low energy constants are fitted using the experimental masses and widths as input. The <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math> is found to decay mainly to <math><mrow><msub><mrow><mi>Λ</mi></mrow><mrow><mi>c</mi></mrow></msub><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup></mrow></math>, while the branching ratios of the <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4440</mn><mo>)</mo></math> and <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4457</mn><mo>)</mo></math> in different channels are comparable. Three additional <math><msub><mi>P</mi><mi>c</mi></msub></math> states at 4380 MeV, 4504 MeV and 4516 MeV, together with their branching ratios, are predicted. Additionally, a deduction for the revised one-pion-exchange potential involving the on-shell three-body intermediate states is provided.

Bo-Lin Huang

Partial widths from analytical extension of the wave function: <math><msub><mi>P</mi><mi>c</mi></msub></math> states

We investigate the  $$Z_b$$  <math> <msub> <mi>Z</mi> <mi>b</mi> </msub> </math>  ,  $$Z_c$$  <math> <msub> <mi>Z</mi> <mi>c</mi> </msub> </math>   and  $$Z_{cs}$$  <math> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> </math>   states within the chiral effective field theory framework and the S-wave single channel molecule picture. With the complex scaling method, we accurately solve the Schrödinger equation in momentum space. Our analysis reveals that the  $$Z_b(10610)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>b</mi> </msub> <mrow> <mo>(</mo> <mn>10610</mn> <mo>)</mo> </mrow> </mrow> </math>  ,  $$Z_b(10650)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>b</mi> </msub> <mrow> <mo>(</mo> <mn>10650</mn> <mo>)</mo> </mrow> </mrow> </math>  ,  $$Z_c(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$Z_c(4020)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>4020</mn> <mo>)</mo> </mrow> </mrow> </math>   states are the resonances composed of the S-wave  $$(B\bar{B}^{*}+B^{*}\bar{B})/\sqrt{2}$$  <math> <mrow> <mrow> <mo>(</mo> <mi>B</mi> <msup> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>)</mo> </mrow> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math>  ,  $$B^{*}\bar{B}^*$$  <math> <mrow> <msup> <mi>B</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <msup> <mrow> <mover> <mrow> <mi>B</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> </mrow> </math>  ,  $$(D\bar{D}^{*}+D^{*}\bar{D})/\sqrt{2}$$  <math> <mrow> <mrow> <mo>(</mo> <mi>D</mi> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>)</mo> </mrow> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math>   and  $$D^{*}\bar{D}^*$$  <math> <mrow> <msup> <mi>D</mi> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> </mrow> </math>  , respectively. Furthermore, although the  $$Z_{cs}(3985)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>3985</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$Z_{cs}(4000)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>4000</mn> <mo>)</mo> </mrow> </mrow> </math>   states exhibit a significant difference in width, these two resonances may originate from the same channel, the S-wave  $$(D_{s}\bar{D}^{*}+D_{s}^{*}\bar{D})/\sqrt{2}$$  <math> <mrow> <mrow> <mo>(</mo> <msub> <mi>D</mi> <mi>s</mi> </msub> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msup> <mo>+</mo> <msubsup> <mi>D</mi> <mrow> <mi>s</mi> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>)</mo> </mrow> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </math>  . Additionally, we find two resonances in the S-wave  $$D_s^*\bar{D}^*$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>∗</mo> </msubsup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> </mrow> </math>   channel, corresponding to the  $$Z_{cs}(4123)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>4123</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$Z_{cs}(4220)$$  <math> <mrow> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>4220</mn> <mo>)</mo> </mrow> </mrow> </math>   states that await experimental confirmation.

$$Z_{cs}$$  <math> <msub> <mi>Z</mi> <mrow> <mi>cs</mi> </mrow> </msub> </math>  ,  $$Z_c$$  <math> <msub> <mi>Z</mi> <mi>c</mi> </msub> </math>   and  $$Z_b$$  <math> <msub> <mi>Z</mi> <mi>b</mi> </msub> </math>   states under the complex scaling method

We investigate the double-charm and hidden-charm hexaquarks as molecules in the framework of the one-boson-exchange potential model. The multichannel coupling and <math><mi>S</mi><mo>−</mo><mi>D</mi></math> wave mixing are taken into account carefully. We adopt the complex scaling method to investigate the possible quasibound states, whose widths are from the three-body decay channel <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mi>Λ</mi><mi>c</mi></msub><mi>π</mi></math> or <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub><mi>π</mi></math>. For the double-charm system of <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>1</mn><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo></math>, we obtain a quasibound state, whose width is 0.50 MeV if the binding energy is <math><mo>−</mo><mn>14.27</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, and the <math><mi>S</mi></math>-wave <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mi>Σ</mi><mi>c</mi></msub></math> and <math><msub><mi>Λ</mi><mi>c</mi></msub><msubsup><mi>Σ</mi><mi>c</mi><mo>*</mo></msubsup></math> components give the dominant contributions. For the <math><mn>1</mn><mo>(</mo><msup><mn>0</mn><mo>+</mo></msup><mo>)</mo></math> double-charm hexaquark system, we do not find a pole. We find more poles in the hidden-charm hexaquark system. We obtain one pole as a quasibound state in the <math><msup><mi>I</mi><mi>G</mi></msup><mo>(</mo><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup><mo>(</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>)</mo></math> system, which only has one channel <math><mo>(</mo><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Σ</mi><mo>¯</mo></mover><mi>c</mi></msub><mo>+</mo><msub><mi>Σ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub><mo>)</mo><mo>/</mo><msqrt><mn>2</mn></msqrt></math>. Its width is 1.72 MeV with a binding energy of <math><mo>−</mo><mn>5.37</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, but we do not find a pole for the scalar <math><msup><mn>1</mn><mo>−</mo></msup><mo>(</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>)</mo></math> system. For the vector <math><msup><mn>1</mn><mo>−</mo></msup><mo>(</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>)</mo></math> system, we find a quasibound state. Its energies, widths, and constituents are very similar to those of the <math><mn>1</mn><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo></math> double-charm case. In the vector <math><msup><mn>1</mn><mo>+</mo></msup><mo>(</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>)</mo></math> system, we get two poles—a quasibound state and a resonance. The quasibound state has a width of 0.38 MeV with a binding energy of <math><mo>−</mo><mn>16.79</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. For the resonance, its width is 4.06 MeV with an energy of 60.78 MeV relative to the <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Σ</mi><mo>¯</mo></mover><mi>c</mi></msub></math> threshold, and its partial width from the two-body decay channel <math><mo>(</mo><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Σ</mi><mo>¯</mo></mover><mi>c</mi></msub><mo>−</mo><msub><mi>Σ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub><mo>)</mo><mo>/</mo><msqrt><mn>2</mn></msqrt></math> is apparently larger than the partial width from the three-body decay channel <math><msub><mi>Λ</mi><mi>c</mi></msub><msub><mover><mi>Λ</mi><mo>¯</mo></mover><mi>c</mi></msub><mi>π</mi></math>. In particular, the <math><msup><mn>1</mn><mo>+</mo></msup><mo>(</mo><msup><mn>0</mn><mrow><mo>−</mo><mo>−</mo></mrow></msup><mo>)</mo></math> and <math><msup><mn>1</mn><mo>−</mo></msup><mo>(</mo><msup><mn>1</mn><mrow><mo>−</mo><mo>+</mo></mrow></msup><mo>)</mo></math> hidden-charm hexaquark molecular states are very interesting. These isovector mesons have exotic <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup></math> quantum numbers which are not accessible to the conventional <math><mi>q</mi><mover><mi>q</mi><mo>¯</mo></mover></math> mesons.

Du-xin Zheng

Double-charm and hidden-charm hexaquark states under the complex scaling method

The LHCb Collaboration discovered a double-charm tetraquark <math><mrow><msubsup><mrow><mi>T</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></math> with a very small width. We investigate the <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> as a <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> molecule with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> in the framework of the one-boson-exchange potential model. The isospin breaking effect and <math><mi>S</mi><mo>−</mo><mi>D</mi></math> wave coupling are taken into account carefully. We adopt the complex scaling method to study the <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> system and obtain a quasibound state corresponding to the <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math>. Its binding energy relative to the <math><msup><mi>D</mi><mn>0</mn></msup><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></math> and width are <math><mo>−</mo><mn>354</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> and 61 keV respectively. The isospin breaking effect is found to be enormous, and the <math><mrow><mi>S</mi></mrow></math>-wave <math><msup><mi>D</mi><mn>0</mn></msup><msup><mi>D</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup></math> and <math><msup><mi>D</mi><mo>+</mo></msup><msup><mi>D</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup></math> components give dominant contributions with the probabilities of 72.1% and 27.1% respectively. In addition, we do not find any resonances in the <math><mi>D</mi><msup><mi>D</mi><mo>*</mo></msup></math> system. As a by-product, we study the <math><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></math> as a <math><mrow><mo>(</mo><mi>D</mi><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><mo>−</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover><mo>)</mo><mo>/</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></math> molecule with <math><msup><mi>J</mi><mrow><mi>P</mi><mi>C</mi></mrow></msup><mo>=</mo><msup><mn>1</mn><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>. We also find a quasibound state corresponding to the <math><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></math>. Its binding energy relative to the <math><msup><mi>D</mi><mn>0</mn></msup><msup><mover><mi>D</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup></math> threshold and width are <math><mo>−</mo><mn>111</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> and 26 keV, respectively. The <math><mrow><mi>S</mi></mrow></math>-wave <math><mrow><mo>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo><mn>0</mn></mrow></msup><mo>−</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo><mn>0</mn></mrow></msup><msup><mrow><mover><mrow><mi>D</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mn>0</mn></mrow></msup><mo>)</mo><mo>/</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></math> component dominates this state with the probability of 92.7%.

Double-charm tetraquark under the complex scaling method

We have systematically investigated the mass spectrum and rearrangement decay properties of the exotic tetraquark states with four different flavors using a color-magnetic interaction model. Their masses are estimated by assuming that the <math><mrow><mi>X</mi><mo>(</mo><mn>4140</mn><mo>)</mo></mrow></math> is a <math><mi>c</mi><mi>s</mi><mover><mi>c</mi><mo>¯</mo></mover><mover><mi>s</mi><mo>¯</mo></mover></math> tetraquark state and their decay widths are obtained by assuming that the Hamiltonian for decay is a constant. According to the adopted method, we find that the most stable states are probably the isoscalar <math><mi>b</mi><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> and <math><mi>c</mi><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup></math> and <math><msup><mn>1</mn><mo>+</mo></msup></math>. The width for most unstable tetraquarks is about tens of MeVs, but that for unstable <math><mi>c</mi><mi>u</mi><mover><mi>s</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> and <math><mi>c</mi><mi>s</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> can be around 100 MeV. For the <math><mi>X</mi><mo>(</mo><mn>5568</mn><mo>)</mo></math>, our method cannot give consistent mass and width if it is a <math><mi>b</mi><mi>u</mi><mover><mi>s</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> tetraquark state. For the <math><mi>I</mi><mo>(</mo><msup><mi>J</mi><mi>P</mi></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mn>0</mn><mo>+</mo></msup><mo>)</mo><mo>,</mo><mn>0</mn><mo>(</mo><msup><mn>1</mn><mo>+</mo></msup><mo>)</mo></math> double-heavy <math><msub><mi>T</mi><mrow><mi>b</mi><mi>c</mi></mrow></msub><mo>=</mo><mi>b</mi><mi>c</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover></math> states, their widths can be several MeVs.

Shi-Yuan Li

Yan-Rui Liu

Yu-Nan Liu

Zong-Guo Si

Tao Yao

Spectrum and rearrangement decays of tetraquark states with four different flavors

In a chromomagnetic model, we analyze the properties of the newly observed <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4457</mn><msup><mo>)</mo><mo>+</mo></msup></math>, <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4440</mn><msup><mo>)</mo><mo>+</mo></msup></math>, and <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><msup><mo>)</mo><mo>+</mo></msup></math> states. We estimate the masses of the <math><mrow><mo>(</mo><mi>u</mi><mi>u</mi><mi>d</mi><msub><mrow><mo>)</mo></mrow><mrow><msub><mrow><mn>8</mn></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><mo>(</mo><mi>c</mi><mover><mrow><mi>c</mi></mrow><mrow><mo>¯</mo></mrow></mover><msub><mrow><mo>)</mo></mrow><mrow><msub><mrow><mn>8</mn></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub></mrow></math> and <math><mo>(</mo><mi>u</mi><mi>d</mi><mi>s</mi><msub><mo>)</mo><msub><mn>8</mn><mi>c</mi></msub></msub><mo>(</mo><mi>c</mi><mover><mi>c</mi><mo>¯</mo></mover><msub><mo>)</mo><msub><mn>8</mn><mi>c</mi></msub></msub></math> pentaquark states by considering the isospin breaking effects. Their values are determined by calculating mass distances from the <math><msubsup><mi>Σ</mi><mi>c</mi><mrow><mo>+</mo><mo>+</mo></mrow></msubsup><msup><mi>D</mi><mo>−</mo></msup></math> and <math><msubsup><mi>Ξ</mi><mi>c</mi><mrow><mo>′</mo><mo>+</mo></mrow></msubsup><msup><mi>D</mi><mo>−</mo></msup></math> thresholds, respectively. It is found that the isospin breaking effects on the spectrum are small. From the uncertainty consideration and the rearrangement decay properties in a simple model, we find that it is possible to assign the <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4457</mn><msup><mo>)</mo><mo>+</mo></msup></math>, <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4440</mn><msup><mo>)</mo><mo>+</mo></msup></math>, and <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><msup><mo>)</mo><mo>+</mo></msup></math> as <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, <math><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math>, and <math><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup></math> pentaquark states, respectively. The assignment in the molecule picture can be different, in particular for the <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><msup><mo>)</mo><mo>+</mo></msup></math>. The information from open-charm channels, e.g., <math><mi>B</mi><mo>[</mo><msub><mi>P</mi><mi>c</mi></msub><mo>→</mo><msubsup><mi>Σ</mi><mi>c</mi><mrow><mo>+</mo><mo>+</mo></mrow></msubsup><msup><mi>D</mi><mo>−</mo></msup><mo>]</mo><mo>/</mo><mi>B</mi><mo>[</mo><msub><mi>P</mi><mi>c</mi></msub><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>p</mi><mo>]</mo></math>, will play an important role in distinguishing the inner structures of the <math><msub><mi>P</mi><mi>c</mi></msub></math> states. Discussions and predictions based on the calculations are also given.

<math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4457</mn><msup><mo>)</mo><mo>+</mo></msup></math>, <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mn>4440</mn><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>, and <math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mn>4312</mn><msup><mrow><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>: Molecules or compact pentaquarks?