We study the $$\Lambda _{b}\rightarrow \Lambda _{c}\bar{D}^{0}K^{-}$$ and $$\Lambda _{b}\rightarrow \Lambda _{c}D^{-}\bar{K}^{0}$$ decays which proceed via a Cabibbo and $$N_c$$ favored process of external emission, and we determine the $$\bar{D}^{0}K^{-}$$ and $$D^{-}\bar{K}^{0}$$ mass distributions close to the $$\bar{D} \bar{K}$$ threshold. For this, we use the tree level contribution plus the rescattering of the meson-meson components, using the extension of the local hidden gauge approach to the charm sector that produces the $$D^*_{s0}(2317)$$ resonance. We observe a large enhancement of the mass distributions close to threshold due to the presence of this resonance below threshold. Next we undertake the inverse problem of extracting the maximum information on the interaction of the $$\bar{D} \bar{K}$$ channels from these distributions, and using the resampling method we find that from these data one can obtain precise values of the scattering lengths and effective ranges, the existence of an $$I=0$$ bound state with a precision of about $$4 \;\mathrm MeV$$ in the mass, plus the $$\bar{D} \bar{K}$$ molecular probability of this state with reasonable precision. Given the fact that the $$\Lambda _{b}\rightarrow \Lambda _{c}\bar{D}^{0}K^{-}$$ decay is already measured by the LHCb collaboration, it is expected that in the next runs with more statistics of the decay, these mass distributions can be measured with precision and the method proposed here can be used to determine the nature of the $$D^*_{s0}(2317)$$ , which is still an issue of debate.

China

Hai-Peng Li

Wei-Hong Liang

Chu-Wen Xiao

Ju-Jun Xie

Spain

Eulogio Oset

Determination of the binding and $\textit{DK}$ probability of the $$D^{*}_{s0}(2317)$$ from the $$(\bar{D}\bar{K})^-$$ mass distributions in $$\Lambda _{b}\rightarrow \Lambda _{c} (\bar{D}\bar{K})^-$$ decays

The roles of the $ \Delta (1232) $ , $ {N}^{*}(1520) $ , and $ {N}^{*}(1650) $ resonances in the $ \gamma p\to{\pi }^{0}{\pi }^{0}p $ reaction near threshold is investigated within an effective Lagrangian approach. The differential cross sections of the $ \gamma p\to{\pi }^{0}{\pi }^{0}p $ reaction was calculated including contributions from the $ \Delta (1232) $ , $ {N}^{*}(1520) $ , and $ {N}^{*}(1650) $ intermediate states decaying into $ \pi^0 p $ via the $\textit{s}$-channel nucleon pole and $\textit{t}$-channel $\textit{ρ}$ exchange. Current experimental measurements were well reproduced. The production of $ \Delta(1232) $ was mainly from the mechanism of the $\textit{s}$-channel nucleon pole, while the $ {N}^{*}(1520) $ and $ {N}^{*}(1650) $ were produced from the mechanism of the $\textit{t}$-channel $\textit{ρ}$ exchange. More experimental data on the $ \gamma p \to \pi^0 \pi^0 p $ reaction could be used to explore the properties of the low-lying excited baryon state.

Meng-Yuan Dai

Si-Wei Liu

Cheng Chen

De-Min Li

En Wang

Roles of ${\boldsymbol\Delta(\bf 1232)}$ , ${\boldsymbol N^*(\bf 1520)}$ , and ${\boldsymbol N^*(\bf 1650)}$ resonances in ${\boldsymbol\gamma \boldsymbol p\to \boldsymbol\pi^{\bf 0} \boldsymbol\pi^{\bf 0}\boldsymbol p}$ reaction within an effective Lagrangian approach * Supported by the National Key R&D Program of China (2023YFA1606703, 2024YFE0105200), the Natural Science Foundation of Henan Province, China (232300421140, 222300420554), and the National Natural Science Foundation of China (12435007, 12361141819, 12475086, 12192263)

<p>The Weinberg compositeness criterion dictates that a pure shallow bound state is characterized by a large scattering length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo>≫</mo><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>β</mi><mo stretchy="false">)</mo></math> and a positive effective range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mn>0</mn></msub></math> that naturally scales to the size of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>β</mi><mo stretchy="false">)</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><mi>β</mi></math> signifies the interaction range. In constructing the contact-range effective field theory (EFT) up to the next-to-leading order to describe the pentaquarks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4312</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math> observed by the LHCb collaboration in 2019, we match the effective range <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mn>0</mn></msub></math> at single-channel situation for these pentaquarks with the low-energy couplings within the EFT framework. Three different schemes are used to connect the couplings with the effective range. We find positive effective ranges <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mn>0</mn></msub></math> of the natural size of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">O</mi><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mi>β</mi><mo stretchy="false">)</mo></math> for the spin configurations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>−</mo></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math> within the molecular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup><msub><mi mathvariant="normal">Σ</mi><mi>c</mi></msub></math> description. Additionally, predictions from the power counting for low-energy couplings or Wilsonian coefficients suggest that, under heavy quark spin symmetry, the broad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4380</mn><mo stretchy="false">)</mo></math> resonance, discovered by the LHCb collaboration in 2015, when considered as part of the single-channel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msubsup><mi mathvariant="normal">Σ</mi><mi>c</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup></math> molecular system alongside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4312</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math>, has a mass of approximately <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mn>4376</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>.</p>

Fang-Zheng Peng

Li-Sheng Geng

Constraining the hidden-charm pentaquark predictions and discriminating the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>P</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">(</mo><mn>4440</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>P</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>4457</mn><mo stretchy="false">)</mo></math> spins through the effective range expansion

The total cross sections of the process   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $   close to the threshold are calculated within the vector meson dominance model. It is found that the theoretical results can describe the current experimental measurements. The nontrivial near-threshold energy dependence of the total cross sections of the process   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $   can be well reproduced by considering the contributions from charmonium-like states and the Coulomb factor. In particular, the results for the angular distribution parameters about the differential cross section are consistent with the experiments from the BESIII Collaboration. In addition, the relative phase   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $   of the electromagnetic form factors is given, and the spin polarization of   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $   is predicted at center-mass energy   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $  . It is hopeful that this work will provide a new perspective on the characteristics of the charmed baryon   $ e^+ e^-\to \Lambda_c^+ \bar{\Lambda}_c^- $  .

Bing Yan

The electromagnetic form factors and spin polarization of In the Process   

<p>We study from the theoretical point of view the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><mi>η</mi><mi mathvariant="normal">Λ</mi></math> reaction, recently measured by the Belle and BESIII Collaborations, where clear signals are observed for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1670</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi><mo stretchy="false">(</mo><mn>1385</mn><mo stretchy="false">)</mo></math> excitation. By considering the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1670</mn><mo stretchy="false">)</mo></math> as dynamically generated resonances from the meson meson and meson baryon interaction, respectively, we are able to determine their relative production strengths in the reaction, which is also tied to the strength of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>π</mi><mo>+</mo></msup><mi>η</mi><mi mathvariant="normal">Λ</mi></math> tree level contribution. We observe that this latter strength is very big and there are large destructive interferences between the tree level and the rescattering terms where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1670</mn><mo stretchy="false">)</mo></math> are generated. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Σ</mi><mo stretchy="false">(</mo><mn>1385</mn><mo stretchy="false">)</mo></math> contribution is included by means of a free parameter, the only one of the theory, up to a global normalization, when one considers only external emission, and we observe that the spin flip part of this term, usually ignored in theoretical and experimental works, plays an important role determining the shape of the mass distributions. Internal emission is also considered and it is found to play a minor role.</p>

Man-Yu Duan

Wen-Tao Lyu

Dian-Yong Chen

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mi>η</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mi mathvariant="normal">Λ</mi></mrow></math> reaction and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">Λ</mi><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mn>1670</mn><mo stretchy="false">)</mo></mrow></math> contributions

We perform a theoretical study on the spectrum of the low-lying hidden-bottom ( $$q\bar{q}b\bar{b}$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>b</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   with  $$q=$$  <math> <mrow> <mi>q</mi> <mo>=</mo> </mrow> </math>   u, d and s) and double-bottom ( $$q\bar{q}bb$$  <math> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>b</mi> <mi>b</mi> </mrow> </math>  ) tetraquark states within a nonrelativistic quark model, in which the instanton-induced interaction is taken as the residual spin-dependent hyperfine interaction between quarks. All the model parameters are fixed by fitting the spectrum of the ground hadron states. The numerical results indicate that masses of several  $$X_{q\bar{q}}$$  <math> <msub> <mi>X</mi> <mrow> <mi>q</mi> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </msub> </math>  ,  $$Z_{b}$$  <math> <msub> <mi>Z</mi> <mi>b</mi> </msub> </math>  , and  $$Z_{bs}$$  <math> <msub> <mi>Z</mi> <mrow> <mi>bs</mi> </mrow> </msub> </math>   tetraquark states are below and near thresholds of corresponding meson-meson channels, thus these states may form components of exotic meson states with reasonable probabilities. Especially, in present model, masses of several obtained states with quantum number  $$I^G(J^P)=1^+(1^{+})$$  <math> <mrow> <msup> <mi>I</mi> <mi>G</mi> </msup> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> <mrow> <mo>(</mo> <msup> <mn>1</mn> <mo>+</mo> </msup> <mo>)</mo> </mrow> </mrow> </math>   are close to  $$Z_b^\pm (10610)$$  <math> <mrow> <msubsup> <mi>Z</mi> <mi>b</mi> <mo>±</mo> </msubsup> <mrow> <mo>(</mo> <mn>10610</mn> <mo>)</mo> </mrow> </mrow> </math>  , so one may expect these states to be non-negligible components of the experimentally observed  $$Z_{b}$$  <math> <msub> <mi>Z</mi> <mi>b</mi> </msub> </math>   states. Concerning to the double-bottom tetraquark states, the present results are in general consistent with other previous works. Two possible stable  $$T_{bb}$$  <math> <msub> <mi>T</mi> <mrow> <mi>bb</mi> </mrow> </msub> </math>   states with quark content  $$bb\bar{n}\bar{n}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>n</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>n</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   ( $$n=u$$  <math> <mrow> <mi>n</mi> <mo>=</mo> <mi>u</mi> </mrow> </math>   or d quark) lying at energies 10558 MeV and 10650 MeV are found, and one possible stable  $$bb\bar{n}\bar{s}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>n</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   state is found, whose energy is  $$\sim 10687$$  <math> <mrow> <mo>∼</mo> <mn>10687</mn> </mrow> </math>   MeV.

Lin-Juan Jiang

Chun-Sheng An

Cheng-Rong Deng

Gang Li

Theoretical study on low-lying hidden-bottom and double-bottom tetraquark states

The Belle collaboration recently reported a pro-mising candidate for the spin-2  $$D^*\bar{D}^*$$  <math> <mrow> <msup> <mi>D</mi> <mo>∗</mo> </msup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> </mrow> </math>   partner of the X(3872), called the  $$X_2$$  <math> <msub> <mi>X</mi> <mn>2</mn> </msub> </math>   for short, having a mass of  $$(4014.3 \pm 4.0 \pm 1.5)~\textrm{MeV}$$  <math> <mrow> <mo>(</mo> <mn>4014.3</mn> <mo>±</mo> <mn>4.0</mn> <mo>±</mo> <mn>1.5</mn> <mo>)</mo> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>   and a width of  $$(4 \pm 11 \pm 6)~\textrm{MeV} $$  <math> <mrow> <mo>(</mo> <mn>4</mn> <mo>±</mo> <mn>11</mn> <mo>±</mo> <mn>6</mn> <mo>)</mo> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>  . In present work, we assume the  $$X_2$$  <math> <msub> <mi>X</mi> <mn>2</mn> </msub> </math>   as a pure molecule of the  $$D^*\bar{D}^*$$  <math> <mrow> <msup> <mi>D</mi> <mo>∗</mo> </msup> <msup> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> <mo>∗</mo> </msup> </mrow> </math>   under four cases, i.e., pure neutral components ( $$\theta = 0$$  <math> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>  ), isospin singlet ( $$\theta = \pi /4$$  <math> <mrow> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </math>  ), neutral components dominant ( $$\theta = \pi /6$$  <math> <mrow> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>6</mn> </mrow> </math>  ), and purely charged components ( $$\theta = \pi /2$$  <math> <mrow> <mi>θ</mi> <mo>=</mo> <mi>π</mi> <mo>/</mo> <mn>2</mn> </mrow> </math>  ), where  $$\theta $$  <math> <mi>θ</mi> </math>   is a phase angle describing the proportion of neutral and charged constituents. Using an effective Lagrangian approach, we calculated the partial widths of  $$X_2\rightarrow VV$$  <math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>→</mo> <mi>V</mi> <mi>V</mi> </mrow> </math>   and  $$X_2 \rightarrow PP$$  <math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>→</mo> <mi>P</mi> <mi>P</mi> </mrow> </math>   (V and P stand for light vector and pseudoscalar mesons, respectively). The predicted decay widths of  $$X_2 \rightarrow VV$$  <math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>→</mo> <mi>V</mi> <mi>V</mi> </mrow> </math>   can reach a few hundreds of  $$\textrm{keV}$$  <math> <mtext>keV</mtext> </math>  , while the decay widths of  $$X_2 \rightarrow PP$$  <math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>→</mo> <mi>P</mi> <mi>P</mi> </mrow> </math>   are about several tens of  $$\textrm{keV}$$  <math> <mtext>keV</mtext> </math>  . In addition, the effects from the proportion of neutral and charged constituent on the decay widths of  $$X_2\rightarrow VV$$  <math> <mrow> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo>→</mo> <mi>V</mi> <mi>V</mi> </mrow> </math>   and PP are also investigated. We hope that the present calculations will be checked experimentally in the future.

Zu-Xin Cai

Zhao-Sai Jia

Shi-Dong Liu

Charmless decays of the spin-2 partner of X(3872)

Motivated by the recent experimental measurements, we have investigated the Cabibbo-favored process <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>p</mi><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup><mi>η</mi></math>, where the <math><mi>N</mi><mo>(</mo><mn>1535</mn><mo>)</mo></math> resonance is dynamically generated from the <math><mi>S</mi></math>-wave pseudoscalar meson-octet baryon interactions within the chiral unitary approach. The contributions from the intermediate <math><mi>N</mi><mo>(</mo><mn>1650</mn><mo>)</mo></math> and the predicted low-lying baryon <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math> are also considered. In addition, a Breit-Wigner amplitude for the <math><mi>N</mi><mo>(</mo><mn>1535</mn><mo>)</mo></math> resonance is checked. By comparing with the measured <math><mi>η</mi><mi>p</mi></math>, <math><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup><mi>η</mi></math>, and <math><mi>p</mi><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup></math> invariant mass squared distributions, our results support the interpretation of <math><mi>N</mi><mo>(</mo><mn>1535</mn><mo>)</mo></math> as a dynamically generated state. Furthermore, we demonstrate that, with the contribution from <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math> taken into account, the calculated invariant mass distributions agree with the Belle measurements. Future precise measurements of the <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>p</mi><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup><mi>η</mi></math> process can further elucidate the existence of the low-lying baryon <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math>.

Ying Li

Theoretical study of <math><mi>N</mi><mo>(</mo><mn>1535</mn><mo>)</mo></math> and <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math> in the Cabibbo-favored process <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>p</mi><msup><mover><mi>K</mi><mo>¯</mo></mover><mn>0</mn></msup><mi>η</mi></math>

We investigated the pionic transitions between the <math><msub><mi>X</mi><mn>2</mn></msub></math> [spin-2 partner of the <math><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></math>] and <math><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math> using a nonrelativistic effective field theory. The <math><msub><mi>X</mi><mn>2</mn></msub></math> is assumed to be a bound state of the <math><msup><mi>D</mi><mo>*</mo></msup></math> and <math><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> mesons and to decay through several kinds of loops, including the bubble, triangle, and box loops. Within the present model, the widths for the single-pion decays <math><msub><mi>X</mi><mn>2</mn></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> are predicted to be about 3–30 keV. For the dipion decays, the widths are a few keVs. These widths yield a branching fraction of <math><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup><mi>–</mi><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></math>. The ratio <math><msub><mi>R</mi><mrow><mi>c</mi><mn>0</mn></mrow></msub><mo>=</mo><mi>Γ</mi><mo>(</mo><msub><mi>X</mi><mn>2</mn></msub><mo>→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>/</mo><mi>Γ</mi><mo>(</mo><msub><mi>X</mi><mn>2</mn></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>π</mi><mn>0</mn></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub><mo>)</mo><mo>≃</mo><mn>1.6</mn></math>, which is a bit smaller than the expected value of 2, and <math><msub><mi>R</mi><mn>21</mn></msub><mo>=</mo><mi>Γ</mi><mo>(</mo><msub><mi>X</mi><mn>2</mn></msub><mo>→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>)</mo><mo>/</mo><mi>Γ</mi><mo>(</mo><msub><mi>X</mi><mn>2</mn></msub><mo>→</mo><mi>π</mi><mi>π</mi><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo>)</mo><mo>≃</mo><mn>0.85</mn></math>. These ratios are nearly independent of the <math><msub><mi>X</mi><mn>2</mn></msub></math> mass and the coupling constants, which might be a good quantity for the experiments. Moreover, the invariant mass spectra of the <math><msup><mi>π</mi><mn>0</mn></msup><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> final state for the dipion processes are presented, showing a cusp structure at the <math><mi>D</mi><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup></math> threshold enhanced and narrowed by the nearby triangle singularity.

Fan Wang

Xiao-Hai Liu

Pionic transitions of the spin-2 partner of <math><mi>X</mi><mo>(</mo><mn>3872</mn><mo>)</mo></math> to <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math>

Motivated by the Belle measurements of the process <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>η</mi><msup><mi>π</mi><mo>+</mo></msup><mi>Λ</mi></math>, we investigate this process by considering the contributions from the <math><mi>Λ</mi><mo>(</mo><mn>1670</mn><mo>)</mo></math>, <math><msub><mi>a</mi><mn>0</mn></msub><mo>(</mo><mn>980</mn><mo>)</mo></math>, and <math><mi>Σ</mi><mo>(</mo><mn>1385</mn><mo>)</mo></math>. In addition, we also consider the predicted low-lying baryon <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math>. Our results involving the <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math> are favored by fitting to the Belle data of the <math><mi>η</mi><mi>Λ</mi></math> and <math><msup><mi>π</mi><mo>+</mo></msup><mi>Λ</mi></math> invariant mass distributions. Furthermore, we predict the <math><mi>η</mi><msup><mi>π</mi><mo>+</mo></msup></math> invariant mass distribution and the angular distribution <math><mi>d</mi><mi>Γ</mi><mo>/</mo><mi>d</mi><mi>cos</mi><mi>θ</mi></math>, which are significantly different depending on whether or not the contribution from the <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math> is considered. Finally, we show that, with the contribution from the <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math>, the calculated Dalitz plot agrees with the Belle measurements. Future precise measurements of the process <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>η</mi><msup><mi>π</mi><mo>+</mo></msup><mi>Λ</mi></math> could shed further light on the existence of the low-lying <math><msup><mi>Σ</mi><mo>*</mo></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>)</mo></math>.

Sheng-Chao Zhang

Guan-Ying Wang

Jia-Jun Wu

Evidence of the low-lying baryon <math><mrow><msup><mrow><mi>Σ</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>(</mo><mn>1</mn><mo>/</mo><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo></mrow></msup><mo>)</mo></mrow></math> in the process <math><msubsup><mi>Λ</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><mi>η</mi><msup><mi>π</mi><mo>+</mo></msup><mi>Λ</mi></math>