Hadron-hadron interactions, being nonperturbative in nature, play a significant role in addressing phenomenological questions in particle physics. Femtoscopy is a powerful tool in heavy-ion collision experiments, enabling the extraction of hadron-hadron interactions via momentum-correlation functions (CFs). These CFs are typically expressed as a convolution of source functions and hadron-hadron wave functions, with the latter encoding information about the interactions. However, source functions remain poorly constrained and are commonly approximated by a Gaussian form. Reconstructing source functions from experimental correlation data constitutes an 'inverse problem.' To address this, we propose a toy model based on Tikhonov regularization. Using a square-well potential with four distinct strengths, we calculate the CFs for inputs of single Gaussian source functions and mixed Gaussian source functions. The resulting CFs are then used to reconstruct the source functions via Tikhonov regularization. Our results show that the Gaussian source function can be successfully reconstructed, highlighting the potential of this approach for extracting realistic source functions from hadron pairs of interest.

China

Ao-Sheng Xiong

Qi-Wei Yuan

Ming-Zhu Liu

Fu-Sheng Yu

Zhi-Wei Liu

Li-Sheng Geng

Solving the inverse source problem in femtoscopy with a toy model * Supported in part by the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (JYB2025XDXM204) and the National Natural Science Foundation of China (W2543006, 12435007). Ming-Zhu Liu acknowledges support from the National Natural Science Foundation of China (12575086). Zhi-Wei Liu acknowledges support from the National Natural Science Foundation of China (12405133) and from the Shenzhen Science and Technology Program (ZDSYS20230626091501002). Fu-Sheng Yu and Ao-Sheng Xiong acknowledge support from the Scientific Research Innovation Capability Support Project for Young Faculty (ZYGXQNJSKYCXNLZCXM-P2) and from the Fundamental Research Funds for the Central Universities (lzujbky-2023-stlt01, lzujbky-2024-oy02, lzujbky-2025-eyt01)

In this work, we first present a systematic investigation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow></msub></math>-type charmed-strange molecular tetraquark candidates composed of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> meson and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math>-doublet anticharmed meson using the one-boson-exchange model, which exhibit exotic flavor content <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mi>q</mi><mi>q</mi></math>. Our results suggest that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>1</mn></msub></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>0</mn><mo>−</mo></msup><mo>,</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>2</mn><mo>*</mo></msubsup></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo>,</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> represent the most promising candidates of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow></msub></math>-type charmed-strange molecular tetraquarks, while the coupled <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>1</mn></msub><mo>/</mo><msup><mi>K</mi><mo>*</mo></msup><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>1</mn></msub><mo>/</mo><msup><mi>K</mi><mo>*</mo></msup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>2</mn><mo>*</mo></msubsup></math> system with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>1</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> and the coupled <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>K</mi><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>2</mn><mo>*</mo></msubsup><mo>/</mo><msup><mi>K</mi><mo>*</mo></msup><msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>1</mn></msub><mo>/</mo><msup><mi>K</mi><mo>*</mo></msup><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>2</mn><mo>*</mo></msubsup></math> system with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math> can only be regarded as the possible candidates of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mover accent="true"><mi>c</mi><mo stretchy="false">¯</mo></mover><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow></msub></math>-type charmed-strange molecular tetraquarks. We further extend our analysis to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msub><mi>D</mi><mn>1</mn></msub><mo>/</mo><msup><mi>K</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup><msubsup><mi>D</mi><mn>2</mn><mo>*</mo></msubsup></math> systems, where our results suggest a series of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow></msub></math>-type charmed-strange molecular tetraquark candidates. These findings provide a comprehensive picture of the molecular spectrum in the charmed-strange tetraquark sector composed of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave kaons and (anti)charmed mesons in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math>-doublet and can be tested in future experimental studies.

Fu-Lai Wang

Si-Qiang Luo

Xiang Liu

Predicting charmed-strange molecular tetraquarks with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>T</mi></math>-doublet charmed or anticharmed meson

We investigate the dynamically generated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub></math> state with spin-parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math> and a mass around 1790 MeV, arising from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> interaction within the chiral unitary approach. The partial decay widths into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo><mi>π</mi></math> channels are calculated via a triangular-loop mechanism. In this mechanism, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub></math> state couples to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math>, and final-state interactions between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> proceed through pseudoscalar-meson exchange, leading to the final states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>π</mi></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></math> [or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math>]. We also present the invariant-mass distributions of a vector meson and a pseudoscalar meson originating from the decays of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math>, along with the corresponding decay widths. Our results show that these decay widths are all of the order of a few MeV. We hope that future experiments can test the predictions presented here, thereby helping to identify this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub></math> state.

Qing-Hua Shen

Ju-Jun Xie

Dynamically generated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>h</mi><mn>1</mn></msub></math> state by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>K</mi><mo>*</mo></msup><msup><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> interaction and its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>K</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1270</mn><mo stretchy="false">)</mo><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo><mi>π</mi></math> decays

This study investigates the ultraviolet realization of semileptonic four-fermion operator $ O_{ledq}^{\alpha \beta 11} $ that incorporates Majorana dark matter (DM) in both lepton-flavor-conserving (LFC) and lepton-flavor-violating (LFV) scenarios at the one-loop level via box diagram, thereby effectively alleviating the lower bounds on the new physics scale. In addition, we investigate the interplay between model-independent constraints on the Wilson coefficients and DM direct detection, relic density, and collider searches in the context of fermion portal DM model with two mediators. We found that both the projected future constraint on the LFC Wilson coefficient $ C_{ledq}^{2211}/\Lambda^2 \lt $ $ (12.3\; \text{TeV})^{-2} $ from the measurements of neutrino non-standard interaction in the next-generation neutrino oscillation experiments, and the LFV constraint $ C_{ledq}^{1211}/\Lambda^2 \lt \left(2.2\times 10^3\; \text{TeV} \right)^{-2} $ from ongoing charged-lepton-flavor-violation searches, provide a complementary exploration of the parameter space encompassing the DM and scalar masses. With a colored mediator mass of approximately $ 2\; \text{TeV} $ , the sensitivity of the indirect constraints on the four-fermion operator can surpass those of collider searches and DM direct detection, in scenarios where the masses of the DM and scalar are close. By ensuring the correct DM relic density, however, we find that the collider searches and DM direct detection are more sensitive to the electroweak-scale DM and scalar than indirect constraints.

Yuxuan He

Gang Li

Jia Liu

Xiao-Ping Wang

Xiang Zhao

Lepton flavor of four-fermion operator and fermion portal dark matter * Gang Li and Xiang Zhao are supported by the National Natural Science Foundation of China (12347105, 12505127), the Guangdong Basic and Applied Basic Research Foundation (2024A1515012668). The work of Jia Liu is supported by the National Science Foundation of China (12235001, 12475103) and State Key Laboratory of Nuclear Physics and Technology (NPT2025ZX11). The work of Xiao-Ping Wang is supported by National Science Foundation of China (12375095) and the Fundamental Research Funds for the Central Universities

The discovery of hidden-charm pentaquarks and open-flavor tetraquarks motivates the search for even more exotic hadron configurations. In this work, we investigate genuinely exotic molecular pentaquark candidates comprising five different flavors, focusing on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>b</mi></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mi>B</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> systems. Employing the one-boson-exchange model with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>S</mi><mtext>−</mtext><mi>D</mi></mrow></math> wave mixing and coupled-channel dynamics, we identify the most promising molecular pentaquark candidates comprising five different flavors. These include the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>b</mi></msub><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>b</mi><mo>′</mo></msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><mi>B</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><mi>B</mi></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>b</mi></msub><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>b</mi><mo>′</mo></msubsup><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Ξ</mi><mi>c</mi></msub><msup><mi>B</mi><mo>*</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>′</mo></msubsup><msup><mi>B</mi><mo>*</mo></msup></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>,</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>b</mi><mo>*</mo></msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><mi>B</mi></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math>, as well as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>b</mi><mo>*</mo></msubsup><msup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mo>*</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mo>*</mo></msubsup><msup><mi>B</mi><mo>*</mo></msup></math> states with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>I</mi><mo stretchy="false">(</mo><msup><mi>J</mi><mi>P</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>,</mo><mn>3</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo>,</mo><mn>5</mn><mo>/</mo><msup><mn>2</mn><mo>−</mo></msup><mo stretchy="false">)</mo></math>. Importantly, these loosely bound states exhibit pronounced spin splittings across different total angular momentum configurations after incorporating the spin-dependent interactions or the channel couplings. In addition, we identify several possible isovector molecular pentaquark candidates within the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>b</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><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></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="normal">Ξ</mi><mi>c</mi><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mi>B</mi><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></math> systems. Our predictions provide clear targets for experimental searches at facilities such as LHCb and Belle II, where the unique five-flavor quark configuration offers a distinctive experimental signature.

Five-flavor molecular pentaquarks in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>b</mi></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mover accent="true"><mrow><mi>D</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msubsup><msup><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></msup></mrow></math> systems

Inspired by the recent discovery of a pseudoscalar enhancement structure near the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mover accent="true"><mi>t</mi><mo stretchy="false">¯</mo></mover></math> threshold reported by the CMS and ATLAS Collaborations, this work investigates the mass spectra of single topped hadrons—including both topped mesons and topped baryons—based on a relativistic potential model. Using the same parameters obtained from the fit to mesons and baryons, we provide predictions for the mass spectra of ground and low-lying orbitally excited single topped mesons and baryons. In addition, we point out that the precise measurement of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mover accent="true"><mi>t</mi><mo stretchy="false">¯</mo></mover></math> mass could test the limitation of the potential model. Given the extremely large mass of the topped quark, we discuss spectroscopic properties of topped hadrons in the approximation of an ideal heavy-quark limit.

Qi Huang

Investigating topped hadrons to probe the boundaries of the potential model

Motivated by the observations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2900</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2900</mn><msup><mo stretchy="false">)</mo><mrow><mo>+</mo><mo>+</mo></mrow></msup></math>, we propose to search for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mrow><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2900</mn><mo stretchy="false">)</mo></mrow><mn>0</mn></msup></math> in the cleaner process <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>−</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>D</mi><mn>0</mn></msup><msup><mi>K</mi><mn>0</mn></msup></math>. In the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mo>*</mo></msup><msup><mi>K</mi><mo>*</mo></msup></math> molecular picture, our estimates suggest that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2900</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup></math> should contribute significantly to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>D</mi><mn>0</mn></msup><msup><mi>K</mi><mn>0</mn></msup></math> invariant mass distribution in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>−</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>D</mi><mn>0</mn></msup><msup><mi>K</mi><mn>0</mn></msup></math>, as reported by the Belle II Collaboration. The corresponding fit fraction is estimated to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>9.72</mn><mo>±</mo><mn>3.92</mn><mo stretchy="false">)</mo><mo>%</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mn>7.09</mn><mo>±</mo><mn>5.88</mn><mo stretchy="false">)</mo><mo>%</mo></math> in different fitting schemes. Further precise measurements of this process at Belle II and LHCb could be helpful for clarifying the nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>T</mi><mrow><mi>c</mi><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>2900</mn><mo stretchy="false">)</mo></math>.

Man-Yu Duan

Guan-Ying Wang

Yun Liang

En Wang

Dian-Yong Chen

Finding the clean process <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>−</mo></mrow></msup><mo stretchy="false">→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> to probe absolutely exotic four-quark states

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

Brazil

Ireland

Y.F. Adans

United States

H. Aratyn

C.P. Constantinidis

J.F. Gomes

G.V. Lobo

T.C. Santiago

New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations

In this work, we conduct a systematic investigation of the electromagnetic properties, specifically the magnetic moments and the M1 radiative decay behavior, of the predicted $$\Xi _c^{(',*)}D^{(*)}_s$$ -type double-charm hidden-strangeness molecular pentaquarks. The study is carried out within the framework of the constituent quark model to evaluate these electromagnetic observables, and our analysis incorporates three distinct scenarios: single-channel analysis, $\textit{S}$–$\textit{D}$ wave mixing analysis, and coupled-channel analysis. The calculated magnetic moments reveal characteristic patterns that reflect their underlying constituent configurations and provide sensitive probes for their quantum number assignments. Furthermore, we identify several M1 radiative decay channels with sizable widths that may offer promising signatures for future experimental detection. These M1 transitions also act as sensitive probes into their inner structures, displaying distinctive features that help differentiate between their constituent configurations and quantum number assignments. We anticipate that this study will stimulate experimental interest in exploring the electromagnetic properties of the $$\Xi _c^{(',*)}D^{(*)}_s$$ molecular states, thereby advancing our structural understanding of these exotic hadronic states.

Sheng-He Zhu

Electromagnetic characteristics as probes into the inner structures of the predicted $$\Xi _c^{(',*)}D^{(*)}_s$$ molecular states

The Born-Oppenheimer approximation is widely used to investigate the properties of hydrogenlike systems and doubly heavy hadrons. However, the extent to which this approximation captures the features of such systems within potential models remains an open question. In this work, we adopt the results obtained with the Gaussian expansion method as a benchmark to assess the validity of the Born-Oppenheimer approximation within potential models for hadronic systems. We also investigate the dependence of the Born-Oppenheimer approximation results on the choice of trial wave functions. A comprehensive study of the Born-Oppenheimer approximation is carried out by performing calculations using Slater-type functions and Gaussian-type functions as trial wave functions, and by comparing the resulting predictions with those obtained from the Gaussian expansion method. We find that the calculations performed within the Born-Oppenheimer approximation are close to those obtained with the Gaussian expansion method when the heavy-quark mass is relatively small. However, as the heavy-quark mass increases, calculations employing Slater-type functions yield larger values than those from the Gaussian expansion method, whereas those using Gaussian-type functions lead to smaller ones. The use of Slater-type functions generally leads to an enhanced binding energy. The underestimation observed in Born-Oppenheimer approximation calculations with Gaussian-type functions primarily stems from the neglect of nonadiabatic corrections. This comparative study provides deeper insight into the structure of doubly heavy hadrons and helps clarify the applicability and limitations of the Born-Oppenheimer treatment within potential models.

Zi-Long Man

Hao Zhou

Assessing the validity of the Born-Oppenheimer approximation in potential models for doubly heavy hadrons