In this paper, we calculate the Cabibbo-Kobayashi-Maskawa matrix element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> by the semileptonic decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>ω</mi><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi></math>. For the transition form factors (TFFs) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>V</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>ω</mi></math>, we employ the QCD light-cone sum rules method for calculation, and by constructing the correlation function using left-handed chiral current, we make the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>δ</mi><mn>1</mn></msup></math>-order twist-2 light-cone distribution amplitudes (LCDA) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><mi>ω</mi></mrow><mo stretchy="false">∥</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></math> dominate the contribution, in which the twist-2 LCDA <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><mi>ω</mi></mrow><mo stretchy="false">∥</mo></msubsup><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></math> is constructed by the light-cone harmonic oscillator model. Then, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.20</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.042</mn></mrow><mrow><mo>+</mo><mn>0.049</mn></mrow></msubsup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.20</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>0.042</mn></mrow><mrow><mo>+</mo><mn>0.051</mn></mrow></msubsup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>V</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.25</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.048</mn></mrow><mrow><mo>+</mo><mn>0.058</mn></mrow></msubsup></math> at the large recoil region. Two important ratios of TFFs are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mi>V</mi></msub><mo>=</mo><mn>1.23</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>0.322</mn></mrow><mrow><mo>+</mo><mn>0.425</mn></mrow></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>r</mi><mn>2</mn></msub><mo>=</mo><mn>0.98</mn><msubsup><mn>5</mn><mrow><mo>−</mo><mn>0.274</mn></mrow><mrow><mo>+</mo><mn>0.347</mn></mrow></msubsup></math>. After extrapolating TFFs to the whole physical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> region by a 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>,</mo><mi>t</mi><mo stretchy="false">)</mo></math>-series expansion, we obtain the differential decay width and branching fraction <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><mi>ω</mi><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1.3</mn><msubsup><mn>5</mn><mrow><mo>−</mo><mn>0.03</mn><mo>−</mo><mn>0.66</mn></mrow><mrow><mo>+</mo><mn>0.02</mn><mo>+</mo><mn>1.22</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, which show good agreement with BABAR and Belle Collaborations. Finally, we extract the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> by using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">B</mi><mi>Exp</mi></msub><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>ω</mi><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi><mo stretchy="false">)</mo></math> result from the BABAR Collaboration, which leads to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3.6</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>0.17</mn><mo>−</mo><mn>0.95</mn></mrow><mrow><mo>+</mo><mn>0.12</mn><mo>+</mo><mn>1.26</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math>.

China

Yin-Long Yang

Ya-Ling Song

Fang-Ping Peng

Hai-Bing Fu

Tao Zhong

Samee Ullah

Exploring the exclusive decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mi>ω</mi><msup><mo>ℓ</mo><mo>+</mo></msup><mi>ν</mi></math> with light-cone sum rules

In this paper, we investigate the semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub><mo>ℓ</mo><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo>ℓ</mo></msub></math> using the quantum chromodynamics (QCD) sum rules within the framework of Standard Model (SM). We further explore the potential to probe signatures of new physics (NP) beyond the SM through these decays. First, we derive the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>-moments <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow><mi>n</mi></msubsup><mo stretchy="false">⟩</mo></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math>-meson leading-twist 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>η</mi><mi>c</mi></msub></mrow></msub></math> using the QCD sum rules within the background field theory. Considering contributions from the vacuum condensates up to dimension-six, the first two nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>-moments at the scale of 4 GeV are found to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>ξ</mi></mrow><mrow><mn>2</mn><mo>;</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mn>0.10</mn><msubsup><mrow><mn>3</mn></mrow><mrow><mo>−</mo><mn>0.009</mn></mrow><mrow><mo>+</mo><mn>0.009</mn></mrow></msubsup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow><mn>4</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mn>0.03</mn><msubsup><mn>1</mn><mrow><mo>−</mo><mn>0.003</mn></mrow><mrow><mo>+</mo><mn>0.003</mn></mrow></msubsup></math>. Using these moments, we then fix the Gegenbauer expansion series of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow></msub></math> and apply it to compute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> transition form factors (TFFs) using QCD light cone sum rules. Second, we extrapolate those TFFs to physically allowable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math>-range via a simplified series expansion, and we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>SM</mi></mrow></msub><mo>=</mo><mn>0.30</mn><msubsup><mrow><mn>8</mn></mrow><mrow><mo>−</mo><mn>0.062</mn></mrow><mrow><mo>+</mo><mn>0.084</mn></mrow></msubsup></mrow></math>. Furthermore, we explore the potential impacts of various NP scenarios on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>R</mi><msub><mi>η</mi><mi>c</mi></msub></msub></math>. Specifically, we compute the forward-backward asymmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">A</mi><mi>FB</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, the convexity parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="script">C</mi><mi>F</mi><mi>τ</mi></msubsup><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, and the longitudinal and transverse polarizations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">P</mi><mi>L</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> transitions within both the SM and two types of NP scenarios. Our results contribute to a deeper understanding of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub></math>-meson semileptonic decays and provide insights into the search for the NP beyond the SM.

Long Zeng

Xing-Gang Wu

Dan-Dan Hu

Yu-Jie Zhang

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> leading-twist distribution amplitude and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">→</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>ℓ</mo><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> semileptonic decays using QCD sum rules

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

Hai-Jiang Tian

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

In the paper, we derive the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math> moments <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mi>n</mi><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo></math> of the longitudinal leading-twist distribution amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">∥</mo></msubsup></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow></math>-state <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> meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension six, its first two nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math> moments at the scale 1 GeV are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mn>1</mn><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mn>0.64</mn><msubsup><mn>7</mn><mrow><mo>−</mo><mn>0.113</mn></mrow><mrow><mo>+</mo><mn>0.118</mn></mrow></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mn>3</mn><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mn>0.32</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.052</mn></mrow><mrow><mo>+</mo><mn>0.055</mn></mrow></msubsup></math>, respectively. Using those moments, we then fix the Gegenbauer expansion series of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">∥</mo></msubsup></math> and apply it to calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math> transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> range via the simplified series expansion. As for the branching fractions, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>e</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.17</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.422</mn></mrow><mrow><mo>+</mo><mn>0.553</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</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><mn>0</mn></msup><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mo>−</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.16</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>0.415</mn></mrow><mrow><mo>+</mo><mn>0.544</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</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><mspace linebreak="goodbreak"></mspace><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.35</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.456</mn></mrow><mrow><mo>+</mo><mn>0.597</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</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><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.33</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.448</mn></mrow><mrow><mo>+</mo><mn>0.587</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, respectively.

Longitudinal leading-twist distribution amplitude of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow></math>-state <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> meson and its implications on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> decays

In this paper, we investigate the rare decay $$B^0\rightarrow K_0^*(1430)\ell ^+\ell ^-$$ with $$\ell =(e,\mu ,\tau )$$ and $$B^0\rightarrow K_0^*(1430)\nu \bar{\nu }$$ induced by the flavor changing neutral current transition of $$b\rightarrow s\ell ^+\ell ^-(\nu \bar{\nu })$$ . Firstly, the $$B^0\rightarrow K_0^*(1430)$$ transition form factors (TFFs) are calculated by using the QCD light-cone sum rule approach up to next-to-leading order accuracy. In which the $$K_0^*(1430)$$ -meson twist-2 and twist-3 LCDAs have been calculated both from the SVZ sum rule in the background field theory framework and light-cone harmonic oscillator model. Then, we obtain the three TFFs at large recoil point, i.e., $$f_+^{B^0\rightarrow K_0^*}(0)= 0.470_{-0.101}^{+0.086}$$ , $$f_-^{B^0\rightarrow K_0^*}(0)= -0.340_{-0.068}^{+0.068}$$ , and $$f_\textrm{T}^{B^0\rightarrow K_0^*}(0)= 0.537^{+0.112}_{-0.115}$$ . Meanwhile, we extrapolate TFFs to the whole physical $$q^2$$ -region by using the simplified $$z(q^2)$$ -series expansion. Furthermore, we calculate the $$B^0\rightarrow K_0^*(1430)\ell ^+\ell ^-(\nu \bar{\nu })$$ decay widths, branching fractions, and longitudinal lepton polarization asymmetries of $$B^0\rightarrow K_0^*(1430)\ell ^+\ell ^-$$ , which lead to $$\mathcal{B}(B^0\rightarrow K_0^*(1430)e^+e^-) = (6.65^{+2.52}_{-2.42})\times 10^{-7}$$ , $$\mathcal{B}(B^0\rightarrow K_0^*(1430)\mu ^+\mu ^-)=(6.62^{+2.51}_{-2.41})\times 10^{-7}$$ , $$\mathcal{B}(B^0\rightarrow K_0^*(1430)\tau ^+\tau ^-)=(1.88^{+1.10}_{-0.97})\times 10^{-8}$$ , $$\mathcal{B}(B^0\rightarrow K_0^*(1430)\nu \bar{\nu })= 3.85^{+1.55}_{-1.48}\times 10^{-6}$$ and the integrated longitudinal lepton polarization asymmetries $$\langle A_{P_L} \rangle = (-0.99, -0.96, -0.03)$$ for the cases $$\ell =(e, \mu , \tau )$$ respectively.

Ya-Lin Song

Scrutinizing $$B^0$$ -meson flavor changing neutral current decay into scalar $$K_0^*(1430)$$ meson with $$b\rightarrow s \ell ^+\ell ^-(\nu \bar{\nu })$$ transition

The measurements in $$b\rightarrow s$$ penguin-dominated decays are widely recognized as a powerful test for searching for New Physics by studying the deviation from theoretical estimations within the Standard Model. We examine the final-state rescattering effects on the decay $$B_s\rightarrow K^{*0}\bar{K}^{*0}$$ and provide estimations of the branching ratio and longitudinal polarization of $$B_s\rightarrow K^{*0}\bar{K}^{*0}$$ , which is consistent with experimental observations. Our conclusion is that both short- and long-distance interactions contribute significantly in this decay. The small longitudinal polarization in $$B\rightarrow VV$$ modes may not be a signal for New Physics.

Yao Yu

Han Zhang

Bai-Cian Ke

A phenomenological estimate of rescattering effects in $$B_s\rightarrow K^{*0}\bar{K}^{*0}$$

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 the present paper, we study the properties of <math><mi>ϕ</mi></math>-meson longitudinal leading-twist light-cone distribution amplitude <math><mrow><msubsup><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn><mo>;</mo><mi>ϕ</mi></mrow><mrow><mo>∥</mo></mrow></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></math> by starting from a light-cone harmonic oscillator model for its wave function. To fix the input parameters, we derive the first ten order <math><mi>ξ</mi></math> moments of <math><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><mi>ϕ</mi></mrow><mo>∥</mo></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math> by using the QCD sum rules approach under the background field theory. The curves of <math><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><mi>ϕ</mi></mrow><mo>∥</mo></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi><mo>)</mo></math> tend to be a single-peak behavior, which is consistent with the latest lattice QCD result. To show how the twist-3 light-cone distribution amplitudes (LCDAs) affect the results, we consider two scenarios for the <math><mi>ϕ</mi></math>-meson chiral twist-3 LCDAs <math><msubsup><mi>ϕ</mi><mrow><mn>3</mn><mo>;</mo><mi>ϕ</mi></mrow><mo>⊥</mo></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math><msubsup><mi>ψ</mi><mrow><mn>3</mn><mo>;</mo><mi>ϕ</mi></mrow><mo>⊥</mo></msubsup><mo>(</mo><mi>x</mi><mo>)</mo></math>, i.e., the ones using the Wandzura-Wilczek approximation with <math><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><mi>ϕ</mi></mrow><mo>∥</mo></msubsup><mo>(</mo><mi>x</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math> (S1) and the ones using self-consistent conformal expansion with second-order Gegenbauer moments <math><msubsup><mi>a</mi><mrow><mn>2</mn><mo>;</mo><mi>ϕ</mi></mrow><mn>2</mn></msubsup></math> in this work (S2). As an application, we derive the <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><mi>ϕ</mi></math> transition form factors (TFFs) by using the QCD light-cone sum rules. The TFFs at large recoil point for those two scenarios are given separately. As for the two TFF ratios <math><msub><mi>γ</mi><mi>V</mi></msub></math> and <math><msub><mi>γ</mi><mn>2</mn></msub></math>, we obtain <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>V</mi></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msubsup><mo>=</mo><mn>1.75</mn><msubsup><mrow><mn>5</mn></mrow><mrow><mo>−</mo><mn>0.005</mn></mrow><mrow><mo>+</mo><mn>0.008</mn></mrow></msubsup></mrow></math>, <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>(</mo><mi>S</mi><mn>1</mn><mo>)</mo></mrow></msubsup><mo>=</mo><mn>0.85</mn><msubsup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mn>0.133</mn></mrow><mrow><mo>+</mo><mn>0.135</mn></mrow></msubsup></mrow></math>, <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mi>V</mi></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>2</mn></mrow><mo>)</mo></mrow></msubsup><mo>=</mo><mn>1.72</mn><msubsup><mrow><mn>3</mn></mrow><mrow><mo>−</mo><mn>0.021</mn></mrow><mrow><mo>+</mo><mn>0.023</mn></mrow></msubsup></mrow></math>, and <math><mrow><msubsup><mrow><mi>γ</mi></mrow><mrow><mn>2</mn></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>2</mn></mrow><mo>)</mo></mrow></msubsup><mo>=</mo><mn>0.78</mn><msubsup><mrow><mn>5</mn></mrow><mrow><mo>−</mo><mn>0.104</mn></mrow><mrow><mo>+</mo><mn>0.100</mn></mrow></msubsup></mrow></math>. After extrapolating those TFFs to the physically allowable region, we then obtain the transverse, longitudinal, and total decay widths for semileptonic decay <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><mi>ϕ</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math>. Then the branching fractions are <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>1</mn></mrow><mo>)</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mi>ϕ</mi><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><mo>(</mo><mn>2.34</mn><msubsup><mrow><mn>7</mn></mrow><mrow><mo>−</mo><mn>0.191</mn></mrow><mrow><mo>+</mo><mn>0.342</mn></mrow></msubsup><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>1</mn></mrow><mo>)</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mi>ϕ</mi><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><mo>(</mo><mn>2.33</mn><msubsup><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mn>0.190</mn></mrow><mrow><mo>+</mo><mn>0.341</mn></mrow></msubsup><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>2</mn></mrow><mo>)</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mspace></mspace><mi>ϕ</mi><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><mo>(</mo><mn>2.36</mn><msubsup><mrow><mn>7</mn></mrow><mrow><mo>−</mo><mn>0.132</mn></mrow><mrow><mo>+</mo><mn>0.256</mn></mrow></msubsup><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, and <math><mrow><msup><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mrow><mi>S</mi><mn>2</mn></mrow><mo>)</mo></mrow></msup><mo>(</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo>→</mo><mi>ϕ</mi><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><mo>(</mo><mn>2.34</mn><msubsup><mrow><mn>9</mn></mrow><mrow><mo>−</mo><mn>0.132</mn></mrow><mrow><mo>+</mo><mn>0.255</mn></mrow></msubsup><mo>)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>3</mn></mrow></msup></mrow></math>, which show good agreement with the data issued by the BESIII, CLEO, and BABAR Collaborations. We finally calculate <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><mi>ϕ</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> polarization and asymmetry parameters, which can be measured and tested in future experiments.

Improved light-cone harmonic oscillator model for the <math><mrow><mi>ϕ</mi></mrow></math>-meson longitudinal leading-twist light-cone distribution amplitude and its effects to <math><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><mi>ϕ</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math>