In this study, we consider all P-wave   $\Omega_{b}$   states represented by interpolating currents with a derivative and calculate the corresponding masses and pole residues using the QCD sum rule method. Because of the large uncertainties in our calculation compared with the small difference in the masses of the excited   $\Omega_{b}$   states observed by the LHCb collaboration, it is necessary to study other properties of the P-wave   $\Omega_{b}$   states represented by the interpolating currents investigated in the present work to gain a better understanding of the four excited   $\Omega_{b}$   states observed by the LHCb collaboration.

China

Yong-Jiang Xu

Yong-Lu Liu

Ming-Qiu Huang

 -wave ω States: Masses and Pole Residues  

In this paper, we construct <math><msubsup><mover><mi>D</mi><mo>¯</mo></mover><mi>s</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msubsup><msup><mi>D</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup></math>-molecule-type interpolating currents <math><msub><mi>J</mi><mrow><mo>(</mo><mo>±</mo><mo>)</mo><mi>μ</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math>, calculate the corresponding mass and magnetic moment using the QCD sum rule method and its extension in the weak electromagnetic field, and study the processes of <math><msub><mi>Z</mi><mrow><mo>(</mo><mo>±</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></math> to <math><msub><mi>η</mi><mi>c</mi></msub><msup><mi>K</mi><mo>*</mo></msup></math>, <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>K</mi></math>, <math><mover><mi>D</mi><mo>¯</mo></mover><msubsup><mi>D</mi><mi>s</mi><mo>*</mo></msubsup></math>, and <math><msup><mover><mi>D</mi><mo>¯</mo></mover><mo>*</mo></msup><msub><mi>D</mi><mi>s</mi></msub></math> via three-point sum rules. The numerical values are <math><msub><mi>m</mi><msub><mi>Z</mi><mrow><mo>(</mo><mo>±</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></msub><mo>=</mo><mn>3.9</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.14</mn></mrow><mrow><mo>+</mo><mn>0.17</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>, <math><msub><mi>λ</mi><msub><mi>Z</mi><mrow><mo>(</mo><mo>±</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></msub><mo>=</mo><mn>2.0</mn><msubsup><mn>7</mn><mrow><mo>−</mo><mn>0.16</mn></mrow><mrow><mo>+</mo><mn>0.28</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mn>5</mn></msup></math>, and <math><mrow><msub><mrow><mi>μ</mi></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mo>(</mo><mo>±</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></mrow></msub><mo>=</mo><mn>0.1</mn><msubsup><mrow><mn>8</mn></mrow><mrow><mo>−</mo><mn>0.09</mn></mrow><mrow><mo>+</mo><mn>0.16</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><msub><mrow><mi>μ</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> with <math><msub><mi>μ</mi><mi>N</mi></msub></math> the nucleon magneton, <math><msub><mi>Γ</mi><msub><mi>Z</mi><mrow><mo>(</mo><mo>+</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></msub><mo>=</mo><mn>17.4</mn><msubsup><mn>7</mn><mrow><mo>−</mo><mn>8.08</mn></mrow><mrow><mo>+</mo><mn>12.70</mn></mrow></msubsup></math>, and <math><msub><mi>Γ</mi><msub><mi>Z</mi><mrow><mo>(</mo><mo>−</mo><mo>)</mo><mi>c</mi><mi>s</mi></mrow></msub></msub><mo>=</mo><mn>13.8</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>6.51</mn></mrow><mrow><mo>+</mo><mn>10.37</mn></mrow></msubsup></math>. The masses are in agreement with the recently measured value of <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>3985</mn><mo>)</mo></math> by the BESIII Collaboration, <math><msubsup><mi>m</mi><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mi>exp</mi></mrow></msubsup><mo>=</mo><mo>(</mo><msubsup><mn>3982.5</mn><mrow><mo>−</mo><mn>2.6</mn></mrow><mrow><mo>+</mo><mn>1.8</mn></mrow></msubsup><mo>±</mo><mn>2.1</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The widths are compatible with the experimental value <math><msubsup><mi>Γ</mi><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mrow><mi>exp</mi></mrow></msubsup><mo>=</mo><mspace></mspace><mo>(</mo><msubsup><mn>12.8</mn><mrow><mo>−</mo><mn>4.4</mn></mrow><mrow><mo>+</mo><mn>5.3</mn></mrow></msubsup><mo>±</mo><mn>3.0</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>. The magnetic moment and the various decay modes can help us to determine the inner structure of <math><msub><mi>Z</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>(</mo><mn>3985</mn><mo>)</mo></math> when being confronted with experimental data in the future.

Chun-Yu Cui

<math><msubsup><mover><mi>D</mi><mo>¯</mo></mover><mi>s</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msubsup><msup><mi>D</mi><mrow><mo>(</mo><mo>*</mo><mo>)</mo></mrow></msup></math> molecular state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mn>1</mn><mo>+</mo></msup></math>

In this paper, we tentatively assign the  $$P_{c}(4312)$$  <math> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>4312</mn> <mo>)</mo> </mrow> </mrow> </math>   to be a  $$\bar{D}\Sigma _{c}$$  <math> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <msub> <mi>Σ</mi> <mi>c</mi> </msub> </mrow> </math>   molecular state with quantum number  $$J^{P}=\frac{1}{2}^{-}$$  <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>  , and calculate its magnetic moment using the QCD sum rule method in external weak electromagnetic field. Starting with the two-point correlation function in external electromagnetic field and expanding it in power of the electromagnetic interaction Hamiltonian, we extract the magnetic moment from the linear response to the external electromagnetic field. The numerical value of the magnetic moment of  $$P_{c}(4312)$$  <math> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>4312</mn> <mo>)</mo> </mrow> </mrow> </math>   is  $$\mu _{P_{c}}=1.75^{+0.15}_{-0.11}$$  <math> <mrow> <msub> <mi>μ</mi> <msub> <mi>P</mi> <mi>c</mi> </msub> </msub> <mo>=</mo> <mn>1</mn> <mo>.</mo> <msubsup> <mn>75</mn> <mrow> <mo>-</mo> <mn>0.11</mn> </mrow> <mrow> <mo>+</mo> <mn>0.15</mn> </mrow> </msubsup> </mrow> </math>  .

The magnetic moment of  $$P_{c}(4312)$$  <math> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>4312</mn> <mo>)</mo> </mrow> </mrow> </math>   as a  $$\bar{D}\Sigma _{c}$$  <math> <mrow> <mover> <mrow> <mi>D</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <msub> <mi>Σ</mi> <mi>c</mi> </msub> </mrow> </math>   molecular state

In this article, we study the ground states and the first radial excited states of the flavor antitriplet heavy baryon states   $\Lambda_Q$   and   $\Lambda_Q$   with the spin-parity   $\Lambda_Q$   by carrying out operator product expansion up to vacuum condensates of dimension   $\Lambda_Q$   in a consistent way. We observe for the first time that the higher dimensional vacuum condensates play an important role, and obtain very stable QCD sum rules with variations of the Borel parameters for the heavy baryon states. The predicted masses   $\Lambda_Q$  ,   $\Lambda_Q$  , and   $\Lambda_Q$   for the first radial excited states   $\Lambda_Q$  ,   $\Lambda_Q$  , and   $\Lambda_Q$  , respectively, are in excellent agreement with the experimental data and support assigning   $\Lambda_Q$  ,   $\Lambda_Q$  , and   $\Lambda_Q$   to be the first radial excited states of   $\Lambda_Q$  ,   $\Lambda_Q$  , and   $\Lambda_Q$  , respectively. The predicted mass   $\Lambda_Q$   for   $\Lambda_Q$   can be confirmed using experimental data in the future.

Analysis of the 1 And 2 States of λ And ξ with QCD sum rules  

In this paper, we tentatively assign  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   to be an axialvector molecular state, and calculate its magnetic moment using the QCD sum rule method in an external weak electromagnetic field. Starting with the two-point correlation function in the external electromagnetic field and expanding it in powers of the electromagnetic interaction Hamiltonian, we extract the mass and pole residue of the  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   state from the leading term in the expansion and the magnetic moment from the linear response to the external electromagnetic field. The numerical values are  $$m_{Z_{c}}=3.97\pm 0.12\,\text{ GeV }$$  <math> <mrow> <msub> <mi>m</mi> <msub> <mi>Z</mi> <mi>c</mi> </msub> </msub> <mo>=</mo> <mn>3.97</mn> <mo>±</mo> <mn>0.12</mn> <mspace width="0.166667em"></mspace> <mspace width="0.333333em"></mspace> <mtext>GeV</mtext> <mspace width="0.333333em"></mspace> </mrow> </math>   in agreement with the experimental value  $$m^{exp}_{Z_{c}}=3899.0\pm 3.6\pm 4.9\text{ MeV }$$  <math> <mrow> <msubsup> <mi>m</mi> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mi>exp</mi> </mrow> </msubsup> <mo>=</mo> <mn>3899.0</mn> <mo>±</mo> <mn>3.6</mn> <mo>±</mo> <mn>4.9</mn> <mspace width="0.333333em"></mspace> <mtext>MeV</mtext> <mspace width="0.333333em"></mspace> </mrow> </math>  ,  $$\lambda _{Z_{c}}=2.1\pm 0.4\times 10^{-2}\,\text{ GeV}^{5}$$  <math> <mrow> <msub> <mi>λ</mi> <msub> <mi>Z</mi> <mi>c</mi> </msub> </msub> <mo>=</mo> <mn>2.1</mn> <mo>±</mo> <mn>0.4</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mspace width="0.166667em"></mspace> <mspace width="0.333333em"></mspace> <msup> <mtext>GeV</mtext> <mn>5</mn> </msup> </mrow> </math>   and  $$\mu _{Z_{c}}=0.19^{+0.04}_{-0.01}\mu _{N}$$  <math> <mrow> <msub> <mi>μ</mi> <msub> <mi>Z</mi> <mi>c</mi> </msub> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>19</mn> <mrow> <mo>-</mo> <mn>0.01</mn> </mrow> <mrow> <mo>+</mo> <mn>0.04</mn> </mrow> </msubsup> <msub> <mi>μ</mi> <mi>N</mi> </msub> </mrow> </math>  .

The magnetic moment of  $$Z_{c}(3900)$$  <math> <mrow> <msub> <mi>Z</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>3900</mn> <mo>)</mo> </mrow> </mrow> </math>   as an axial-vector molecular state

The existence of light sterile neutrinos is a long-standing question in particle physics. Several experimental “anomalies” might be explained by introducing eV mass scaled light sterile neutrinos. Many experiments are actively searching for such light sterile neutrinos through neutrino oscillation. For long baseline experiments, the matter effect should be treated carefully for precise calculation of the neutrino oscillation probabilities. However, this is usually time-consuming or analytically complex. In this manuscript, we adopt a Jacobi-like method to diagonalize the Hermitian Hamiltonian matrix and derive analytically simplified neutrino oscillation probabilities for 3 (active) + 1 (sterile)-neutrino mixing for a constant matter density. These approximations can reach a considerably high numerical accuracy while retaining their analytical simplicity and fast computing speed. This would be useful for current and future long baseline neutrino oscillation experiments.

Bao-Biao Yue

Wei Li

Jia-Jie Ling

Fan-Rong Xu

A compact analytical approximation for a light sterile neutrino oscillation in matter Jiajie Ling acknowledges the support from National Key R&D program of China (2018YFA0404103) and National Natural Science Foundation of China (11775315). F. Xu is supported partially by NSFC (11605076), as well as the FRFCU (Fundamental Research Funds for the Central Universities in China) (21616309)

In the present work, the partial decay widths of <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math> to <math><msub><mi>η</mi><mi>c</mi></msub><mi>p</mi></math> and <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>p</mi></math> are investigated with the QCD sum rule method under the assumption that <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math> is a <math><mover><mi>D</mi><mo>¯</mo></mover><msub><mi>Σ</mi><mi>c</mi></msub></math> molecular state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup></math>. In the analysis, the pole residue of <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math>, one of the input parameters for the calculations of the strong decay constants, is calculated first. With the numerical values of the strong decay constants, the partial decay widths to <math><msub><mi>η</mi><mi>c</mi></msub><mi>p</mi></math> and <math><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>p</mi></math> are estimated to be <math><mi>Γ</mi><mo>(</mo><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo><mo>→</mo><msub><mi>η</mi><mi>c</mi></msub><mi>p</mi><mo>)</mo><mo>=</mo><mn>5.5</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>0.5</mn></mrow><mrow><mo>+</mo><mn>0.75</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> and <math><mi>Γ</mi><mo>(</mo><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo><mo>→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mi>p</mi><mo>)</mo><mo>=</mo><mn>1.6</mn><msubsup><mn>7</mn><mrow><mo>−</mo><mn>0.56</mn></mrow><mrow><mo>+</mo><mn>0.92</mn></mrow></msubsup><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, respectively, which are compatible with the measured total width of <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math>. The results suggest that it is reasonable to assign <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math> to be a <math><mover><mi>D</mi><mo>¯</mo></mover><msub><mi>Σ</mi><mi>c</mi></msub></math> molecular state with <math><msup><mi>J</mi><mi>P</mi></msup><mo>=</mo><msup><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>−</mo></msup></math>.

Partial decay widths of <math><msub><mi>P</mi><mi>c</mi></msub><mo>(</mo><mn>4312</mn><mo>)</mo></math> as a <math><mover><mi>D</mi><mo>¯</mo></mover><msub><mi>Σ</mi><mi>c</mi></msub></math> molecular state