The isospin-singlet scalar meson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> is hypothesized to consist of two energy eigenstates, forming a mixture of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo stretchy="false">(</mo><mover accent="true"><mi>u</mi><mo stretchy="false">¯</mo></mover><mi>u</mi><mo>+</mo><mover accent="true"><mi>d</mi><mo stretchy="false">¯</mo></mover><mi>d</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mover accent="true"><mi>s</mi><mo stretchy="false">¯</mo></mover><mi>s</mi></math>. Building on this framework, we apply the quantum chromodynamics (QCD) sum rules approach in the background field theory to compute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> decay constant, yielding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><msub><mi>f</mi><mn>0</mn></msub></msub><mo stretchy="false">(</mo><msub><mi>μ</mi><mn>0</mn></msub><mo>=</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0.386</mn><mo>±</mo><mn>0.009</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math>. Subsequently, we derive the first two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>-moments of the leading-twist light-cone distribution amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>f</mi><mn>0</mn></msub></mrow></msub></math>. Using QCD light-cone sum rules, we then calculate the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> transition form factor (TFF) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mo>+</mo></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, obtaining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mo>+</mo></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.51</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>0.024</mn></mrow><mrow><mo>+</mo><mn>0.027</mn></mrow></msubsup></math> at the large-recoil region. We extend <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mo>+</mo></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> to the full physical region via a simplified series expansion parametrization, enabling calculations of the differential decay widths and the branching fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mspace linebreak="goodbreak"></mspace><mo stretchy="false">(</mo><mn>1.78</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.189</mn></mrow><mrow><mo>+</mo><mn>0.227</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>. Our theoretical predictions align well with the latest BESIII Collaboration measurements within reasonable errors.

China

Dan-Dan Hu

Xing-Gang Wu

Hai-Jiang Tian

Tao Zhong

Hai-Bing Fu

Light-cone sum rules analysis of the semileptonic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo stretchy="false">→</mo><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo><mspace linebreak="goodbreak"></mspace><mo stretchy="false">(</mo><mo stretchy="false">→</mo><msup><mi>π</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><mo stretchy="false">)</mo><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub></math> decay incorporating the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> mixing state twist-2 distribution amplitudes

In this paper, we fit the $$B\rightarrow D$$ vector transition form factor (TFF) by using the data measured by $\textit{BABAR}$ and Belle Collaborations within Monte Carlo (MC) method. Meanwhile, the $$B\rightarrow D$$ TFF is also calculated by using the QCD light-cone sum rules approach (LCSRs) within right-handed chiral current correlation function. In the TFF, $\textit{D}$-meson leading-twist light-cone distribution amplitude (LCDA) is the most important non-perturbative input parameter, the precise behavior of which is mainly determined by the parameter $$B_{2;D}$$ in the light-cone harmonic oscillator model. Through fitting the TFF, we determine the value $$B_{2;D}=0.445$$ . Then, we present the curve of $\textit{D}$-meson leading-twist LCDA in comparison with other theoretical approaches. Subsequently, the $$B\rightarrow D$$ TFF $$f_{+}^{BD}(q^2)$$ at the large recoil region is $$f_{+}^{BD}(0)=0.648_{-0.063}^{+0.067}$$ , which is compared in detail with theoretical estimates and experimental measurements. Furthermore, we calculated the decay width and branching fractions of the Cabibbo-favored semileptonic decays $$B\rightarrow D\ell ^+\nu _{\ell }$$ , which lead to the results $${\mathcal {B}}(B^0\rightarrow D^-\ell ^+\nu _{\ell }) =(2.10_{-0.38}^{+0.44})\times 10^{-2}$$ and $${\mathcal {B}}(B^+\rightarrow {\bar{D}}^0\ell ^+\nu _{\ell }) =(2.26_{-0.41}^{+0.48})\times 10^{-2}$$ . Finally, we predict the CKM matrix element with two scenarios $$|V_{cb}|_{\textrm{SR}}=41.55_{-4.50}^{+4.91}\times 10^{-3}$$ and $$|V_{cb} |_{\textrm{MC}}=41.47_{-2.66}^{+2.55 }\times 10^{-3}$$ from $$B^0\rightarrow D^-\ell ^+\nu _{\ell }$$ , $$|V_{cb}|_{\textrm{SR}}=40.54_{-3.80}^{+4.33}\times 10^{-3}$$ and $$|V_{cb} |_{\textrm{MC}}=40.46_{-1.38}^{+1.40}\times 10^{-3}$$ from $$B^+\rightarrow {\bar{D}}^0\ell ^+\nu _{\ell }$$ which are in good agreement with theoretical and experimental predictions.

Sheng-Bo Wu

Yin-Long Yang

Wei Cheng

Footprint in fitting $$B\rightarrow D$$ vector form factor and determination for $\textit{D}$-meson leading-twist LCDA

In the paper, we conduct a detailed investigation of the rare decay processes of charged meson, specifically <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><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>ℓ</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></math> and <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><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math>. These processes involve flavor-changing-neutral-current (FCNC) transitions, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo stretchy="false">→</mo><mi>s</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo stretchy="false">→</mo><mi>s</mi><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math>. The essential components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>K</mi></math> scalar, vector, and tensor transition form factors (TFFs) are calculated by using the QCD light-cone sum rules approach up to next-to-leading order QCD corrections. In which, the kaon twist-2 and twist-3 light-cone distribution amplitudes are calculated from both the QCD sum rules within the framework of background field theory and the light-cone harmonic oscillator model. The TFFs at large recoil point are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>f</mi><mo>+</mo><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>f</mi><mn>0</mn><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.34</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>0.018</mn></mrow><mrow><mo>+</mo><mn>0.021</mn></mrow></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>f</mi><mi mathvariant="normal">T</mi><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.34</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.019</mn></mrow><mrow><mo>+</mo><mn>0.021</mn></mrow></msubsup></math>, respectively. To obtain the behavior of those TFFs in the whole <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> region, we extrapolate them by utilizing the simplified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>-series expansion. Furthermore, we compute the differential branching fractions with respect to the squared dilepton invariant mass for the two different decay channels and present the corresponding curves. Our predictions of total branching fraction are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>6.62</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.709</mn></mrow><mrow><mo>+</mo><mn>0.648</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>6.61</mn><msubsup><mn>2</mn><mrow><mo>−</mo><mn>0.698</mn></mrow><mrow><mo>+</mo><mn>0.638</mn></mrow></msubsup><mspace linebreak="goodbreak"></mspace><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1.79</mn><msubsup><mn>5</mn><mrow><mo>−</mo><mn>0.116</mn></mrow><mrow><mo>+</mo><mn>0.110</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>4.17</mn><msubsup><mn>0</mn><mrow><mo>−</mo><mn>0.440</mn></mrow><mrow><mo>+</mo><mn>0.402</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math>, respectively. Lastly, the observables such as the lepton universality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">R</mi><mi>K</mi></msub></math> and the angular distribution “flat term” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>F</mi><mi mathvariant="normal">H</mi><mo>ℓ</mo></msubsup></math> are given, which show good agreement with the theoretical and experimental predictions.

Ya-Xiong Wang

Rare decay <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><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></math> under the QCD sum rules approach

The leptonic decay of <math altimg='si1.svg'><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>-meson has attracted significant interest due to its unique characteristics. In this paper, we carry out an investigation into the <math altimg='si1.svg'><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>-meson leptonic decays <math altimg='si2.svg'><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#8467;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#8467;</mi></mrow></msub></math> with <math altimg='si3.svg'><mi>&#8467;</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><mi>e</mi><mo>,</mo><mi>&#956;</mi><mo>,</mo><mi>&#964;</mi><mo stretchy='false'>)</mo></math> by employing the QCD sum rules approach. In which the <math altimg='si1.svg'><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>-meson decay constant <math altimg='si4.svg'><msub><mrow><mi>f</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></mrow></msub></math> is an important input parameter in the process. To enhance the accuracy of our calculations for <math altimg='si4.svg'><msub><mrow><mi>f</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></mrow></msub></math>, we consider the quark propagator and vertex up to dimension-six within the framework of background field theory. Consequently, we obtain the QCD sum rule expression for <math altimg='si4.svg'><msub><mrow><mi>f</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></mrow></msub></math> up to dimension-six condensates, yielding <math altimg='si5.svg'><msub><mrow><mi>f</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>203.0</mn><mo>&#177;</mo><mn>1.5</mn><mspace width='0.25em' /><mrow><mi mathvariant='normal'>MeV</mi></mrow></math>. Our result is in good agreement with BESIII measurements and theoretical predictions. We also present the integrated decay widths for the <math altimg='si1.svg'><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup></math>-meson in three channels <math altimg='si6.svg'><mi mathvariant='normal'>&#915;</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>e</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>5.263</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.075</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.076</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>21</mn></mrow></msup><mspace width='0.25em' /><mrow><mi mathvariant='normal'>GeV</mi></mrow></math>, <math altimg='si7.svg'><mi mathvariant='normal'>&#915;</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#956;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#956;</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>2.236</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.032</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.032</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>16</mn></mrow></msup><mspace width='0.25em' /><mrow><mi mathvariant='normal'>GeV</mi></mrow></math> and <math altimg='si8.svg'><mi mathvariant='normal'>&#915;</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#964;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#964;</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>5.958</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.085</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.086</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>16</mn></mrow></msup><mspace width='0.25em' /><mrow><mi mathvariant='normal'>GeV</mi></mrow></math>. Accordingly, we compute the branching fractions <math altimg='si134.svg'><mi mathvariant='script'>B</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#8467;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#8467;</mi></mrow></msub><mo stretchy='false'>)</mo></math> with the electron, muon and tau channels, which are <math altimg='si10.svg'><mi mathvariant='script'>B</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>e</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>e</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>8.260</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.118</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.119</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>9</mn></mrow></msup></math>, <math altimg='si11.svg'><mi mathvariant='script'>B</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#956;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#956;</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>3.508</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.050</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.051</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>4</mn></mrow></msup></math> and <math altimg='si12.svg'><mi mathvariant='script'>B</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#964;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#964;</mi></mrow></msub><mo stretchy='false'>)</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mo stretchy='false'>(</mo><msubsup><mrow><mn>0.935</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.013</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.013</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>3</mn></mrow></msup></math>. Furthermore, we present our prediction for the CKM matrix element <math altimg='si13.svg'><mo stretchy='false'>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy='false'>|</mo></math> using the branching fraction <math altimg='si135.svg'><mi mathvariant='script'>B</mi><mo stretchy='false'>(</mo><msup><mrow><mi>D</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><mo stretchy='false'>&#8594;</mo><msup><mrow><mi>&#956;</mi></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo></mrow></msup><msub><mrow><mi>&#957;</mi></mrow><mrow><mi>&#956;</mi></mrow></msub><mo stretchy='false'>)</mo></math> obtained from BESIII Collaboration, yielding <math altimg='si15.svg'><mo stretchy='false'>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy='false'>|</mo><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><msubsup><mrow><mn>0.227</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>0.001</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>0.002</mn></mrow></msubsup></math>.

Prospective analysis of CKM element | <italic>V</italic> <inf><italic>cd</italic></inf>| and <italic>D</italic> +-meson decay constant from leptonic decays <italic>D</italic> +&#8239;&#8594;&#8239; <italic>&#8467;</italic> + <italic>&#957;</italic>

In this paper, we calculate the semileptonic decays <math><mrow><mi>D</mi><mo>→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo><mo>ℓ</mo><msub><mrow><mi>ν</mi></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> with <math><mo>ℓ</mo><mo>=</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math> induced by <math><mi>c</mi><mo>→</mo><mi>s</mi><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> transition. For the key component, <math><mi>D</mi><mo>→</mo><msubsup><mi>K</mi><mn>0</mn><mo>*</mo></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo></math> transition form factors (TFFs) <math><msub><mi>f</mi><mo>±</mo></msub><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> are calculated within the framework of QCD light cone sum rule. Then, we consider two scenarios for <math><msubsup><mi>K</mi><mn>0</mn><mo>*</mo></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo></math>-meson twist-2 distribution amplitude. For the scenario 1 (S1), we take the truncated form based on Gegenbauer polynomial series. Meanwhile, we also consider the scenario 2 (S2) constructed by light cone harmonic oscillator model, where the model parameters are fixed by the <math><msubsup><mi>K</mi><mn>0</mn><mo>*</mo></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo></math>-meson twist-2 distribution amplitude tenth-order <math><mi>ξ</mi></math> moments calculated by using the background field theory. For the TFFs at a large recoil point, we have <math><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.597</mn></mrow><mrow><mo>−</mo><mn>0.121</mn></mrow><mrow><mo>+</mo><mn>0.122</mn></mrow></msubsup></mrow></math> and <math><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mo>−</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msubsup><mrow><mo>−</mo><mn>0.136</mn></mrow><mrow><mo>−</mo><mn>0.035</mn></mrow><mrow><mo>+</mo><mn>0.023</mn></mrow></msubsup></mrow></math>, <math><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>2</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.663</mn></mrow><mrow><mo>−</mo><mn>0.134</mn></mrow><mrow><mo>+</mo><mn>0.135</mn></mrow></msubsup></mrow></math>, and <math><mrow><msubsup><mrow><mi>f</mi></mrow><mrow><mo>−</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>2</mn><mo>)</mo></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msubsup><mrow><mo>−</mo><mn>0.202</mn></mrow><mrow><mo>−</mo><mn>0.046</mn></mrow><mrow><mo>+</mo><mn>0.026</mn></mrow></msubsup></mrow></math>. After extrapolating TFFs to the whole physical <math><msup><mi>q</mi><mn>2</mn></msup></math> region, we calculate the branching fractions of <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo><mo>+</mo></mrow></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo><msup><mrow><mo>ℓ</mo></mrow><mrow><mo>−</mo></mrow></msup><msub><mrow><mover><mrow><mi>ν</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> and <math><msup><mi>D</mi><mo>+</mo></msup><mo>→</mo><msubsup><mi>K</mi><mn>0</mn><mrow><mo>*</mo><mn>0</mn></mrow></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math>, which at <math><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>-order level for the S1 and S2 cases. Meanwhile, we predict the CKM matrix <math><mrow><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>s</mi></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msup><mo>=</mo><mrow><mn>0.97</mn><msubsup><mrow><mn>3</mn></mrow><mrow><mo>−</mo><mn>0.183</mn></mrow><mrow><mo>+</mo><mn>0.259</mn></mrow></msubsup></mrow></mrow></math>, <math><mrow><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>s</mi></mrow></msub><msup><mrow><mo>|</mo></mrow><mrow><mo>(</mo><mi>S</mi><mn>2</mn><mo>)</mo></mrow></msup><mo>=</mo><mrow><msubsup><mrow><mn>0.880</mn></mrow><mrow><mo>−</mo><mn>0.165</mn></mrow><mrow><mo>+</mo><mn>0.234</mn></mrow></msubsup></mrow></mrow></math>, and lepton flavor universality <math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msubsup><mo>=</mo><msubsup><mrow><mn>0.768</mn></mrow><mrow><mo>−</mo><mn>0.368</mn></mrow><mrow><mo>+</mo><mn>0.560</mn></mrow></msubsup></mrow></math>, <math><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup></mrow><mrow><mo>(</mo><mi>S</mi><mn>2</mn><mo>)</mo></mrow></msubsup><mo>=</mo><msubsup><mrow><mn>0.764</mn></mrow><mrow><mo>−</mo><mn>0.365</mn></mrow><mrow><mo>+</mo><mn>0.555</mn></mrow></msubsup></mrow></math>. Finally, we discuss the angular observables of forward-backward asymmetries, lepton polarization asymmetries, and <math><msup><mi>q</mi><mn>2</mn></msup></math>-differential flat terms for this decay.

Sheng-Quan Wang

Dong Huang

Probing <math><mo>|</mo><msub><mi>V</mi><mrow><mi>c</mi><mi>s</mi></mrow></msub><mo>|</mo></math> and lepton flavor universality through <math><mrow><mi>D</mi><mo>→</mo><msubsup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>1430</mn><mo>)</mo><mo>ℓ</mo><msub><mrow><mi>ν</mi></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> decays

In this paper, we carry out an investigation into the semileptonic decays <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> with <math><mi>ℓ</mi><mo>=</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math> by employing the QCD light-cone sum rules approach. The vector transition form factor (TFF) <math><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msubsup><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> for <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></math> decay is calculated while considering its next-to-leading order contribution. Subsequently, we briefly introduce the twist-2, 3 kaon distribution amplitudes, which are calculated by using QCD sum rules within the framework of the background field theory. At the large recoil point, the TFF has <math><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msubsup><mrow><mn>0.692</mn></mrow><mrow><mo>−</mo><mn>0.026</mn></mrow><mrow><mo>+</mo><mn>0.027</mn></mrow></msubsup></math>. Then, we extrapolate <math><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msubsup><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> to the whole physical <math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math>-region via the simplified <math><mi>z</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>t</mi><mo>)</mo></math>-series expansion, and the behavior of TFF <math><msubsup><mrow><mi>f</mi></mrow><mrow><mo>+</mo></mrow><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></msubsup><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math> is exhibited in the numerical results part, including the theoretical and experimental predictions for comparison. In addition, we compute the differential branching fraction <math><mi>B</mi><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>ℓ</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math> with the electron and muon channels, which are expected to be <math><mi>B</mi><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mrow><mn>3.379</mn></mrow><mrow><mo>−</mo><mn>0.275</mn></mrow><mrow><mo>+</mo><mn>0.301</mn></mrow></msubsup><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> and <math><mi>B</mi><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>μ</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>μ</mi></mrow></msub><mo>)</mo><mo>=</mo><msubsup><mrow><mn>3.351</mn></mrow><mrow><mo>−</mo><mn>0.273</mn></mrow><mrow><mo>+</mo><mn>0.299</mn></mrow></msubsup><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> as well as contained other results for comparison. Our results show good agreement with the BESIII measurements and theoretical predictions. Furthermore, we present our prediction with respect to the CKM matrix element <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>d</mi></mrow></msub><mo>|</mo></math> by using the <math><mi>B</mi><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub><mo>)</mo></math> result from BESIII Collaboration, yielding <math><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>c</mi><mi>d</mi></mrow></msub><mo>|</mo><mo>=</mo><msubsup><mrow><mn>0.221</mn></mrow><mrow><mo>−</mo><mn>0.010</mn></mrow><mrow><mo>+</mo><mn>0.008</mn></mrow></msubsup></math>. Finally, we provide the ratio between <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub></math> and <math><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mi>η</mi><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msub><mrow><mi>ν</mi></mrow><mrow><mi>e</mi></mrow></msub></math> channels, i.e. <math><msubsup><mrow><mi>R</mi></mrow><mrow><msup><mrow><mi>K</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>/</mo><mi>η</mi></mrow><mrow><mi>e</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mn>0.144</mn></mrow><mrow><mo>−</mo><mn>0.020</mn></mrow><mrow><mo>+</mo><mn>0.028</mn></mrow></msubsup></math>.

Searching for | | Through the Exclusive Decay within QCD sum rules

In this paper, we carried out a systematic investigation for the excited doubly heavy baryons production in Z-boson decays within the NRQCD factorization approach. Our investigation accounts for all the P-wave intermediate diquark states, i.e. <math><mo>〈</mo><mi>c</mi><mi>c</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mover><mrow><mn>3</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math>, <math><mo>〈</mo><mi>c</mi><mi>c</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mi>J</mi></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mn>6</mn></mrow></msub></math>, <math><mo>〈</mo><mi>b</mi><mi>c</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mover><mrow><mn>3</mn></mrow><mrow><mo>¯</mo></mrow></mover><mo>/</mo><mn>6</mn></mrow></msub></math>, <math><mo>〈</mo><mi>b</mi><mi>c</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mi>J</mi></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mover><mrow><mn>3</mn></mrow><mrow><mo>¯</mo></mrow></mover><mo>/</mo><mn>6</mn></mrow></msub></math>, <math><mo>〈</mo><mi>b</mi><mi>b</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mover><mrow><mn>3</mn></mrow><mrow><mo>¯</mo></mrow></mover></mrow></msub></math>, and <math><mo>〈</mo><mi>b</mi><mi>b</mi><mo>〉</mo><msub><mrow><mo>[</mo><mmultiscripts><mrow><mi>P</mi></mrow><mrow><mi>J</mi></mrow><none></none><mprescripts></mprescripts><none></none><mrow><mn>3</mn></mrow></mmultiscripts><mo>]</mo></mrow><mrow><mn>6</mn></mrow></msub></math> with <math><mi>J</mi><mo>=</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>. The results show that contributions from all diquark states in P-wave were 7%, 8%, and 3% in comparing with S-wave for the production of <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> and <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> via Z-boson decay, respectively. Based on these results, we predicted about <math><mn>0.539</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> events for <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, <math><mn>1.827</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> events for <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>c</mi></mrow></msub></math>, and <math><mn>0.036</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> events for <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> can be produced annually at the LHC (CEPC). Additionally, we plot the differential decay widths of <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>c</mi><mi>c</mi></mrow></msub></math>, <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>c</mi></mrow></msub></math> and <math><msub><mrow><mi>Ξ</mi></mrow><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> as a function of the invariant mass <math><msub><mrow><mi>s</mi></mrow><mrow><mn>23</mn></mrow></msub></math> and energy function z distributions, and analyze the theoretical uncertainties in decay width arising from the mass parameters of heavy quark.

Xuan Luo

Further study on the production of -wave Doubly Heavy Baryons From -boson Decays

In this paper, the semileptonic decays <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> with <math><mo>ℓ</mo><mo>=</mo><mo>(</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math> are investigated using the light cone sum rule approach. Firstly, the neutral meson mixing scheme between <math><msup><mi>π</mi><mn>0</mn></msup></math>, <math><mi>η</mi></math>, <math><msup><mi>η</mi><mo>′</mo></msup></math> and pseudoscalar gluonium <math><mi>G</mi></math> is discussed in a unified way, which leads to the direct connection between two different channels for <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> and <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><mi>η</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> by the <math><msup><mi>π</mi><mn>0</mn></msup><mo>−</mo><mi>η</mi></math> mixing angle. Then we calculate the <math><msub><mi>D</mi><mi>s</mi></msub><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup></math> transition form factors (TFFs) within the QCD light cone sum rule approach up to next-to-leading order correction. At the large recoil point, we have <math><msubsup><mi>f</mi><mo>+</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><msup><mi>π</mi><mn>0</mn></msup></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.011</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.0019</mn></mrow><mrow><mo>+</mo><mn>0.0024</mn></mrow></msubsup></math> and <math><msubsup><mi>f</mi><mo>−</mo><mrow><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><msup><mi>π</mi><mn>0</mn></msup></mrow></msubsup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0.002</mn><msubsup><mn>0</mn><mrow><mo>−</mo><mn>0.0009</mn></mrow><mrow><mo>+</mo><mn>0.0008</mn></mrow></msubsup></math>. Furthermore, the TFFs are extrapolated to the whole physical <math><msup><mi>q</mi><mn>2</mn></msup></math>-region by using the simplified <math><mi>z</mi><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math>-series expansion. The behaviors of TFFs and related three angular coefficient functions <math><msub><mi>a</mi><msub><mi>θ</mi><mo>ℓ</mo></msub></msub><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math>, <math><msub><mi>b</mi><msub><mi>θ</mi><mo>ℓ</mo></msub></msub><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> and <math><msub><mi>c</mi><msub><mi>θ</mi><mo>ℓ</mo></msub></msub><mo>(</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> are given. The differential decay widths for <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> with respect to <math><msup><mi>q</mi><mn>2</mn></msup></math> and <math><mi>cos</mi><msub><mi>θ</mi><mo>ℓ</mo></msub></math> are presented, and also lead to the branching fractions <math><mi>B</mi><mo>(</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub><mo>)</mo><mo>=</mo><mn>2.6</mn><msubsup><mn>0</mn><mrow><mo>−</mo><mn>0.51</mn></mrow><mrow><mo>+</mo><mn>0.57</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></math> and <math><mi>B</mi><mo>(</mo><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mo>)</mo><mo>=</mo><mn>2.5</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.51</mn></mrow><mrow><mo>+</mo><mn>0.56</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></math>. These results show good agreement with the recent BESIII measurements and theoretical predictions. Then the differential distributions and integrated predictions for three angular observables, i.e. forward-backward asymmetries, <math><msup><mi>q</mi><mn>2</mn></msup></math>-differential flat terms and lepton polarization asymmetries are given separately. Lastly, we estimate the ratio for different decay channels, <math><msubsup><mi>R</mi><mrow><msup><mi>π</mi><mn>0</mn></msup><mo>/</mo><mi>η</mi></mrow><mo>ℓ</mo></msubsup><mo>=</mo><mn>1.10</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.071</mn></mrow><mrow><mo>+</mo><mn>0.039</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>.

Investigating the <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>π</mi><mn>0</mn></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> decay process within the QCD sum rule approach

Z-boson decay provides a good opportunity to search for the   $ \Xi_{bQ} $   baryon because a large number of its events can be collected at high energy colliders. In this paper, we perform a complete investigation of the indirect production of the   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $   baryons via the Z-boson decay process   $ \Xi_{bQ} $   with a   $ \Xi_{bQ} $   quark under the NRQCD factorization approach. After considering the contribution from the diquark states   $ \Xi_{bQ} $  ,   $ \Xi_{bQ} $  ,   $ \Xi_{bQ} $  , and   $ \Xi_{bQ} $  , the calculated branching fractions are   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $  . Furthermore, the   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $   production events are predicted to be of the order of   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $   at the LHC collider, and the order of   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $   for the CEPC collider. We then estimate the production ratio for   $ \Xi_{bQ} $   with   $ \Xi_{bQ} $   in Z-boson decay, that is,   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $  , respectively. Finally, we present the differential decay widths of   $ \Xi_{bQ} $   and   $ \Xi_{bQ} $   with respect to   $ \Xi_{bQ} $   and z distributions and analyze the uncertainties.

Investigation of -boson Decay Into And baryons within the NRQCD factorization approach  

The first search for the doubly heavy   $ {{{{\varOmega}_{bc}^{0}}}} $   baryon and a search for the   $ {{{{\varOmega}_{bc}^{0}}}} $   baryon are performed using   $ {{{{\varOmega}_{bc}^{0}}}} $   collision data collected via the   $ {{{{\varOmega}_{bc}^{0}}}} $   experiment from 2016 to 2018 at a centre-of-mass energy of   $ {{{{\varOmega}_{bc}^{0}}}} $  , corresponding to an integrated luminosity of 5.2   $ {{{{\varOmega}_{bc}^{0}}}} $  . The baryons are reconstructed via their decays to   $ {{{{\varOmega}_{bc}^{0}}}} $   and   $ {{{{\varOmega}_{bc}^{0}}}} $  . No significant excess is found for invariant masses between 6700 and 7300   $ {{{{\varOmega}_{bc}^{0}}}} $  , in a rapidity range from 2.0 to 4.5 and a transverse momentum range from 2 to 20   $ {{{{\varOmega}_{bc}^{0}}}} $  . Upper limits are set on the ratio of the   $ {{{{\varOmega}_{bc}^{0}}}} $   and   $ {{{{\varOmega}_{bc}^{0}}}} $   production cross-section times the branching fraction to   $ {{{{\varOmega}_{bc}^{0}}}} $   (   $ {{{{\varOmega}_{bc}^{0}}}} $  ) relative to that of the   $ {{{{\varOmega}_{bc}^{0}}}} $   (   $ {{{{\varOmega}_{bc}^{0}}}} $  ) baryon, for different lifetime hypotheses, at 95% confidence level. The upper limits range from   $ {{{{\varOmega}_{bc}^{0}}}} $   to   $ {{{{\varOmega}_{bc}^{0}}}} $   for the   $ {{{{\varOmega}_{bc}^{0}}}} $   (   $ {{{{\varOmega}_{bc}^{0}}}} $  ) decay, and from   $ {{{{\varOmega}_{bc}^{0}}}} $   to   $ {{{{\varOmega}_{bc}^{0}}}} $   for the   $ {{{{\varOmega}_{bc}^{0}}}} $   (   $ {{{{\varOmega}_{bc}^{0}}}} $  ) decay, depending on the considered mass and lifetime of the   $ {{{{\varOmega}_{bc}^{0}}}} $   (   $ {{{{\varOmega}_{bc}^{0}}}} $  ) baryon.