<p>We present a comprehensive reappraisal of the in-medium properties of the rho meson using the inverse QCD sum rules (QCDSR) formalism, offering a novel, model-independent approach to studying hadronic modifications in nuclear matter. Unlike conventional QCDSR, which rely on a predefined <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>pole</mi><mo>+</mo><mi>continuum</mi></mrow></math> structure, the inverse method reconstructs the spectral function directly from the operator product expansion (OPE), eliminating assumptions about the spectral ansatz. To the best of our knowledge, this is the first application of the inverse QCDSR method to the rho meson in nuclear matter. Our analysis reveals a significant reduction in the rho meson mass, consistent with previous theoretical predictions, and highlights the crucial role of medium-induced modifications, including condensate suppression and factorization-breaking effects. Furthermore, we assess the sensitivity of our results to the factorization assumption and higher-dimensional condensates, demonstrating the necessity of refining nonperturbative contributions for an accurate description of in-medium hadron properties. Our findings establish the inverse QCDSR method as a robust alternative to conventional spectral analysis techniques, providing a systematically controlled framework for exploring strongly interacting matter under extreme conditions. These results offer important theoretical benchmarks for lattice QCD simulations and heavy-ion collision experiments, shedding light on the restoration of chiral symmetry and the evolution of hadronic matter in dense environments.</p>

Türkiye

Halil Mutuk

Reappraisal of the rho meson in nuclear matter by the inverse QCD sum rules method

The $$\Lambda (1405)$$ state has strangeness $$S=-1$$ and isospin $$I=0$$ with quantum number $$J^P=1/2^-$$ . It is located just below the threshold of $$\bar{K}N$$ . In this study, the internal structure of the $$\Lambda (1405)$$ state is investigated based on meson-baryon molecular configuration using one-boson-exchange (OBE) model. We take into account various states ( $$K^-p$$ , $$K^0 n$$ , $$K^-p-K^0 n$$ , $$\pi ^+ \Sigma ^-$$ , $$\pi ^- \Sigma ^+$$ , $$\pi ^0 \Sigma ^0$$ , $$\pi ^+ \Sigma ^- - \pi ^- \Sigma ^+$$ , $$\pi ^+ \Sigma ^- - \pi ^0 \Sigma ^0$$ and $$\pi ^- \Sigma ^+ - \pi ^0 \Sigma ^0$$ and $$\eta \Lambda $$ ) and calculate spectroscopic parameters such as mass and magnetic moment. Some of the predicted masses for the states in this work are below corresponding threshold. The predicted masses of $$K^-p$$ , $$K^0 n$$ , and $$K^-p-K^0 n$$ states are below $$\bar{K}N$$ threshold whereas predicted masses of $$\pi ^+ \Sigma ^-$$ , $$\pi ^- \Sigma ^+$$ , $$\pi ^0 \Sigma ^0$$ , $$\pi ^+ \Sigma ^- - \pi ^- \Sigma ^+$$ , $$\pi ^+ \Sigma ^- - \pi ^0 \Sigma ^0$$ , and $$\pi ^- \Sigma ^+ - \pi ^0 \Sigma ^0$$ states are below $$\pi \Sigma $$ threshold. These states may be bound state candidates. We obtain that all the states of this present work have sizes larger than the confinement scale, $$(1/\Lambda _{\text {QCD}} \sim 1 ~\text {fm})$$ . We also study compositeness criterion for these states. Using compositeness criterion, we calculate scattering length $$a_s$$ and effective range $$r_e$$ . The results reflect that the present states of this work have compositeness dominance, i.e., they are in molecular configuration. The magnetic moment results yield that the magnetic moment is mainly dominated by the corresponding baryon component. Our results of magnetic moments support $$\bar{K}N$$ molecular configuration for $$\Lambda (1405)$$ .

Deciphering $$\Lambda (1405)$$ state via mass, compositeness and magnetic moment

<p>This study reexamines the spectroscopic parameters of light-flavor diquarks within the framework of quantum chromodynamics sum rules (QCDSR) using the inverse matrix method. Conventional QCDSR analyses are based on assumptions such as quark-hadron duality and continuum models, which introduce a degree of systematic uncertainty. The inverse matrix method circumvents these assumptions by reformulating the problem as an inverse integral equation and expanding the unknown spectral density using orthogonal Laguerre polynomials. This method allows for a direct determination of spectral densities, thereby enhancing the precision of predictions regarding resonance masses and decay constants. By employing this methodology with regard to light-flavor diquarks (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>s</mi><mi>q</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>u</mi><mi>d</mi></math>), it is possible to extract the associated masses and decay constants. The results indicate that the masses of diquarks with quantum numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>0</mn><mo>−</mo></msup></math> are nearly degenerate. We compare our results regarding masses and decay constants with those of other theoretical predictions, which could prove a useful complementary tool in interpretation. Our results are consistent with those in the literature and can be shown as evidence for the consistency of the method. The results achieved in this study highlight the potential of the inverse matrix method as a robust tool for exploring nonperturbative QCD phenomena and elucidating the internal structure of exotic hadronic systems.</p>

Revisiting light-flavor diquarks in the inverse matrix method of QCD sum rules

We study systematically magnetic moments of hiddden-bottom pentaquark states with quantum numbers  $$J^P=\frac{1}{2}^{\pm }$$  <math> <mrow> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>=</mo> <msup> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>±</mo> </msup> </mrow> </math>  ,  $$J^P=\frac{3}{2}^{\pm }$$  <math> <mrow> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>=</mo> <msup> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>±</mo> </msup> </mrow> </math>  , and  $$J^P=\frac{5}{2}^{\pm }$$  <math> <mrow> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>=</mo> <msup> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> <mo>±</mo> </msup> </mrow> </math>   with molecular, diquark–diquark–antiquark, and diquark–triquark models. The numerical results show that magnetic moments are different within the same model according to same quantum numbers and spin-orbit couplings. The results are also different when different models are taken into account with the same angular momentum. The magnetic moments encode valuable information about inner structures. We believe that our results may be helpful for experimental studies.

Magnetic moments of hidden-bottom pentaquark states

One of the celebrated tools in explaining the hydrogen atom is Born-Oppenheimer approximation. The resemblance of <math><mi>Q</mi><mi>Q</mi><mover><mi>q</mi><mo>¯</mo></mover><mover><mi>q</mi><mo>¯</mo></mover></math> tetraquarks to hydrogen atom within quantum chromodynamics implies usage of Born-Oppenheimer approximation for these multiquark states. In this work, we use dynamical diquark model to calculate mass spectra and sizes of doubly charmed and charged tetraquark states denoted as <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msubsup></math>. Our results for mass spectra indicate some bound-state candidates with respect to corresponding two-meson thresholds. Calculation of expectation values of <math><msqrt><mrow><mo>⟨</mo><msup><mi>r</mi><mn>2</mn></msup><mo>⟩</mo></mrow></msqrt></math> reflects that doubly charmed and charged tetraquark states are compact.

Doubly-charged <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mrow><mo>+</mo><mo>+</mo></mrow></msubsup></math> states in the dynamical diquark model

The dynamical diquark model assumes that exotic hadrons can be formed from colored diquarks. This model asserts a multiquark exotic state composed of a compact diquark <italic>&#948;</italic> and antidiquark <math altimg='si35.svg'><mover accent='true'><mrow><mi>&#948;</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> for a tetraquark interacting through a gluonic field of finite extent. Using Born-Oppenheimer (BO) approximation and BO potential calculated numerically on the lattice, we study mass spectra of <italic>S</italic>-wave <math altimg='si1.svg'><mi>c</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math>, <math altimg='si122.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math>, and <math altimg='si3.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> tetraquark systems in the basis of diquark spins. We assume that color-antitriple diquark and color-triplet antidiquark form tetraquark state. The predicted mass spectrum for ground state of <math altimg='si1.svg'><mi>c</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> is found to be lower then their corresponding two-meson thresholds. This system may be a candidate for bound state. Masses of <math altimg='si122.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> and <math altimg='si3.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> tetraquark states are found to be above than the corresponding two-meson thresholds.

Germany

Spectrum of <math altimg='si1.svg'><mi>c</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math>, <math altimg='si122.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>c</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math>, and <math altimg='si3.svg'><mi>b</mi><mi>c</mi><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover><mover accent='true'><mrow><mi>b</mi></mrow><mrow><mo stretchy='false'>&#175;</mo></mrow></mover></math> tetraquark states in the dynamical diquark model

Motivated by the discovery of hidden-charm strange pentaquarks, we conduct a systematic study of the magnetic moments of the hidden-bottom strange pentaquarks in molecular picture. We calculate magnetic moments of hidden-bottom strange pentaquarks with strangeness-1 and 2. Magnetic moment gives valuable information about the inner structure and shape of the hadron. The obtained results may be helpful to determine the inner structure of these hypothetical states.

China

Xian-Wei Kang

Unveiling the structure of hidden-bottom strange pentaquarks via magnetic moments

We present mass spectrum and magnetic moments of the  $$\bar{n}\bar{n}QQ$$  <math> <mrow> <mover> <mrow> <mi>n</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>n</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>Q</mi> <mi>Q</mi> </mrow> </math>   states, where  $$n=u,d,s$$  <math> <mrow> <mi>n</mi> <mo>=</mo> <mi>u</mi> <mo>,</mo> <mi>d</mi> <mo>,</mo> <mi>s</mi> </mrow> </math>   and  $$Q=c,b.$$  <math> <mrow> <mi>Q</mi> <mo>=</mo> <mi>c</mi> <mo>,</mo> <mi>b</mi> <mo>.</mo> </mrow> </math>   We solve four-body Schrödinger equation with a quark potential model by using diffusion Monte Carlo (DMC) method. The quark potential is based on the Coulomb, confinement and spin–spin interaction terms. We find mass and magnetic moment of the  $$T_{cc}^+$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> </math>   state as  $$M_{T_{cc}^+}=3892 ~\text {MeV}$$  <math> <mrow> <msub> <mi>M</mi> <msubsup> <mi>T</mi> <mrow> <mi>cc</mi> </mrow> <mo>+</mo> </msubsup> </msub> <mo>=</mo> <mn>3892</mn> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>   and  $$\mu =0.28 \mu _N,$$  <math> <mrow> <mi>μ</mi> <mo>=</mo> <mn>0.28</mn> <msub> <mi>μ</mi> <mi>N</mi> </msub> <mo>,</mo> </mrow> </math>  respectively. We also find the mass and magnetic moment of  $$T_{bb}^-$$  <math> <msubsup> <mi>T</mi> <mrow> <mi>bb</mi> </mrow> <mo>-</mo> </msubsup> </math>   as  $$M_{T_{bb}^-}=10338 ~\text {MeV}$$  <math> <mrow> <msub> <mi>M</mi> <msubsup> <mi>T</mi> <mrow> <mi>bb</mi> </mrow> <mo>-</mo> </msubsup> </msub> <mo>=</mo> <mn>10338</mn> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>   and  $$\mu =-0.32 \mu _N,$$  <math> <mrow> <mi>μ</mi> <mo>=</mo> <mo>-</mo> <mn>0.32</mn> <msub> <mi>μ</mi> <mi>N</mi> </msub> <mo>,</mo> </mrow> </math>   respectively. We find some bound state candidates of doubly heavy tetraquark systems with  $$I(J^P)=0(1)^+$$  <math> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>    $$nn {\bar{b}} {\bar{b}},$$  <math> <mrow> <mi>n</mi> <mi>n</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>,</mo> </mrow> </math>    $$I(J^P)=0(0)^+$$  <math> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <msup> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>    $$nn {\bar{c}} {\bar{b}},$$  <math> <mrow> <mi>n</mi> <mi>n</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>,</mo> </mrow> </math>    $$I(J^P)=0(1)^+$$  <math> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>    $$nn {\bar{c}} \bar{b},$$  <math> <mrow> <mi>n</mi> <mi>n</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>,</mo> </mrow> </math>   and  $$I(J^P)=1/2(1)^+$$  <math> <mrow> <mi>I</mi> <mrow> <mo>(</mo> <msup> <mi>J</mi> <mi>P</mi> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> <mo>+</mo> </msup> </mrow> </math>    $$ns {\bar{b}} {\bar{b}}.$$  <math> <mrow> <mi>n</mi> <mi>s</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mo>.</mo> </mrow> </math>   We compare our results with other approaches in the literature.

Masses and magnetic moments of doubly heavy tetraquarks via diffusion Monte Carlo method

We study the S-wave mass spectra of flavor exotic triply-heavy tetraquark states  $$cc\bar{c}\bar{q}$$  <math> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$cc\bar{b}\bar{q}$$  <math> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$bb\bar{c}\bar{q}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   and  $$bb\bar{b}\bar{q}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  . We adopt a diquark–antidiquark scheme to solve Schrödinger equation. The calculations are carried out in a nonrelativistic quark model with a color interaction described by a potential computed in AdS/QCD. The AdS/QCD potential model consists of a central potential which reflects short distance and large distance behaviour of QCD, spin dependent term for hyperfine splitting and a constant term. We find stable state candidates in the  $$cc\bar{c}\bar{q}$$  <math> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   sector whereas in the  $$cc\bar{b}\bar{q}$$  <math> <mrow> <mi>c</mi> <mi>c</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>  ,  $$bb\bar{c}\bar{q}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>c</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   and  $$bb\bar{b}\bar{q}$$  <math> <mrow> <mi>b</mi> <mi>b</mi> <mover> <mrow> <mi>b</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   sectors all the states lie above corresponding S-wave meson-meson thresholds.

Flavor exotic triply-heavy tetraquark states in AdS/QCD potential

We study  $$D_s^+ D_s^-$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   and  $$D \bar{D}$$  <math> <mrow> <mi>D</mi> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   states assuming that they are hadronic molecules with  $$J^{PC}=0^{++}$$  <math> <mrow> <msup> <mi>J</mi> <mrow> <mi>PC</mi> </mrow> </msup> <mo>=</mo> <msup> <mn>0</mn> <mrow> <mo>+</mo> <mo>+</mo> </mrow> </msup> </mrow> </math>   quantum number. We use two-point QCD sum rule formalism and extract the mass and decay constant values of these states. We take into account contributions of various quark, gluon, and mixed vacuum condensates up to dimension eight. The extracted mass and decay constant values of  $$D \bar{D}$$  <math> <mrow> <mi>D</mi> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   and  $$D_s^+ D_s^-$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   states read as  $$M_{D \bar{D}} = 3795^{+85}_{-82}~{\textrm{MeV}} $$  <math> <mrow> <msub> <mi>M</mi> <mrow> <mi>D</mi> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </msub> <mo>=</mo> <msubsup> <mn>3795</mn> <mrow> <mo>-</mo> <mn>82</mn> </mrow> <mrow> <mo>+</mo> <mn>85</mn> </mrow> </msubsup> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>  ,  $$f_{D \bar{D}} = 1.70^{+0.33}_{-0.29} \times 10^{-2} ~{\textrm{GeV}}^5$$  <math> <mrow> <msub> <mi>f</mi> <mrow> <mi>D</mi> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <msubsup> <mn>70</mn> <mrow> <mo>-</mo> <mn>0.29</mn> </mrow> <mrow> <mo>+</mo> <mn>0.33</mn> </mrow> </msubsup> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="3.33333pt"></mspace> <msup> <mrow> <mtext>GeV</mtext> </mrow> <mn>5</mn> </msup> </mrow> </math>  , and  $$M_{D_s^+ D_s^-} = 3983^{+93}_{-88} ~{\textrm{MeV}}$$  <math> <mrow> <msub> <mi>M</mi> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </msub> <mo>=</mo> <msubsup> <mn>3983</mn> <mrow> <mo>-</mo> <mn>88</mn> </mrow> <mrow> <mo>+</mo> <mn>93</mn> </mrow> </msubsup> <mspace width="3.33333pt"></mspace> <mtext>MeV</mtext> </mrow> </math>  ,  $$f_{D_s^+ D_s^-} = 2.52^{+0.64}_{-0.54} \times 10^{-2} ~{\textrm{GeV}}^5$$  <math> <mrow> <msub> <mi>f</mi> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>.</mo> <msubsup> <mn>52</mn> <mrow> <mo>-</mo> <mn>0.54</mn> </mrow> <mrow> <mo>+</mo> <mn>0.64</mn> </mrow> </msubsup> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="3.33333pt"></mspace> <msup> <mrow> <mtext>GeV</mtext> </mrow> <mn>5</mn> </msup> </mrow> </math>  , respectively. The predicted mass of  $$D_s^+ D_s^-$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   state is in good agreement with the recent LHCb observation and supports quantum number and molecular picture assignments. A possible observation of  $$D \bar{D}$$  <math> <mrow> <mi>D</mi> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   state would help for establishing the lowest four-quark state in charmonium sector.

Molecular interpretation of X(3960) as  $$D_s^+ D_s^-$$  <math> <mrow> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> <msubsup> <mi>D</mi> <mi>s</mi> <mo>-</mo> </msubsup> </mrow> </math>   state