<p>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.</p>

China

Long Zeng

Xing-Gang Wu

Dan-Dan Hu

Yu-Jie Zhang

Hai-Bing Fu

Tao Zhong

<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 work, gravitational lensing in the weak and strong field limits is investigated for black hole spacetime within the framework of Hu–Sawicki ( ) gravity. We employ the Ishihara et al. approach for weak lensing and adopt Bozza’s method for strong lensing to explore the impact of Hu–Sawicki model parameters on lensing phenomenon. The deflection angles are computed and analyzed in both the field limits. Our investigation in the weak as well as the strong lensing reveals that in the case of Hu–Sawicki black holes, photons exhibit divergence at smaller impact parameters for different values of the model parameters compared to the Schwarzschild scenario and the photon experiences negative deflection angle when impact parameter moves towards the larger impact parameter values. Additionally, by calculating strong lensing coefficients we study their behavior with model parameters. The strong lensing key observables associated with the lensing effect viz. the angular position $$\vartheta _{\infty }$$ , angular separation and relative magnification $$r_\text {mag}$$ are estimated numerically by extending the analysis to supermassive black holes $$\text {SgrA}^*$$ and $$\text {M87}^*$$ and analyzed their behavior concerning the parameters for each black hole. The analysis shows that $$\text {SgrA}^*$$ demonstrates larger values of $$\vartheta _{\infty }$$ and relative to $$\text {M87}^*$$ .

India

Gayatri Mohan

Nashiba Parbin

Umananda Goswami

Investigating the effects of gravitational lensing by Hu-Sawicki $$\varvec{f(R)}$$ gravity black holes

<p>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.</p>

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

We study the propagation of massless fermionic fields, implementing a family of special functions: Heun functions, in solving the wave equation in three three-dimensional backgrounds, including the BTZ black hole in string theory and Lifshitz black hole solutions in conformal gravity and Hu–Sawicki ( ) theory. The main properties of the selected black hole solutions is that their line elements are Weyl related to that of a homogeneous spacetime, whose spatial part possesses Lie symmetry, described by Lobachevsky-type geometry with arbitrary negative Gaussian curvature. Using the Weyl symmetry of massless Dirac action, we consider the perturbation equations of fermionic fields in relation to those of the homogeneous background, which having definite singularities, are transformed into Heun’s equation. We point out the existence of quasinormal modes labeled by the accessory parameter of the Heun function. The distribution of the quasinormal modes has been clarified to satisfy the boundary conditions that require ingoing and decaying waves at the event horizon and conformal infinity, respectively. It turned out that the procedure based on the Heun function, beside reproducing the previously known results obtained via hypergeometric function for the BTZ and Lifshitz black hole solution in conformal gravity, brings up new families of quasinormal frequencies, which can also contain purely imaginary modes. Also, the analysis of the quasinormal modes shows that with the negative imaginary part of complex frequencies $$\omega =\omega _{Re}+i\omega _{Im}$$ , the fermionic perturbations are stable in this background.

Iran

F. Naderi

A. Rezaei-Aghdam

Quasinormal modes of three $$(2+1)$$ -dimensional black holes in string theory, conformal gravity, and Hu–Sawicki

We derive novel black hole solutions in a modified gravity theory, namely the Hu–Sawicki model of f(R) gravity. After obtaining the black hole solution, we study the horizon radius of the black hole from the metric and then analyse the dependence of the model parameters on the horizon. We then use the 6th order WKB method to study the quasinormal modes of oscillations (QNMs) of the black hole perturbed by a scalar field. The dependence of the amplitude and damping part of the QNMs are analysed with respect to variations in model parameters and the error associated with the QNMs are also computed. After that we study some thermodynamic properties associated with the black hole such as its thermodynamic temperature as well as greybody factors. It is found that the black hole has the possibility of showcasing negative temperatures and is thermodynamically unstable for feasible values of model parameters. Then we analyse the geodesics and derive the photon sphere radius as well as the shadow radius of the black hole. The photon radius is independent of the model parameters while shadow radius showed fair amount of dependence on the model parameters. We tried to constrain the parameters with the help of Keck and VLTI observational data and obtained some bounds on m and  $$c_{2}$$  <math> <msub> <mi>c</mi> <mn>2</mn> </msub> </math>   parameters.

Ronit Karmakar

Quasinormal modes, thermodynamics and shadow of black holes in Hu–Sawicki  $$\varvec{f(R)}$$  <math> <mrow> <mi>f</mi> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </math>   gravity theory

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>

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>.

Hai-Jiang Tian

Yin-Long Yang

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

The  $$\eta ^{(\prime )}$$  <math> <msup> <mi>η</mi> <mrow> <mo>(</mo> <mo>′</mo> <mo>)</mo> </mrow> </msup> </math>  -mesons in the quark-flavor basis are mixtures of two mesonic states  $$|\eta _{q}\rangle =|{\bar{u}} u+{\bar{d}} d\rangle /\sqrt{2}$$  <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>η</mi> <mi>q</mi> </msub> <mrow> <mo>⟩</mo> <mo>=</mo> <mo>|</mo> <mover> <mrow> <mi>u</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>u</mi> <mo>+</mo> <mover> <mrow> <mi>d</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>d</mi> <mo>⟩</mo> <mo>/</mo> </mrow> <msqrt> <mn>2</mn> </msqrt> </mrow> </math>   and  $$|\eta _{s}\rangle =|{\bar{s}} s\rangle .$$  <math> <mrow> <mrow> <mo>|</mo> </mrow> <msub> <mi>η</mi> <mi>s</mi> </msub> <mrow> <mo>⟩</mo> <mo>=</mo> <mo>|</mo> <mover> <mrow> <mi>s</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>s</mi> <mo>⟩</mo> <mo>.</mo> </mrow> </mrow> </math>   In previous work, we have made a detailed study on the  $$\eta _{s}$$  <math> <msub> <mi>η</mi> <mi>s</mi> </msub> </math>   leading-twist distribution amplitude by using the  $$D^+_s$$  <math> <msubsup> <mi>D</mi> <mi>s</mi> <mo>+</mo> </msubsup> </math>   meson semileptonic decays. As a sequential work, in the present paper, we fix the  $$\eta _q$$  <math> <msub> <mi>η</mi> <mi>q</mi> </msub> </math>   leading-twist distribution amplitude by using the light-cone harmonic oscillator model for its wave function and by using the QCD sum rules within the QCD background field to calculate its moments. The input parameters of  $$\eta _q$$  <math> <msub> <mi>η</mi> <mi>q</mi> </msub> </math>   leading-twist distribution amplitude  $$\phi _{2;\eta _q}$$  <math> <msub> <mi>ϕ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>η</mi> <mi>q</mi> </msub> </mrow> </msub> </math>   at the initial scale  $$\mu _0\sim 1$$  <math> <mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> <mo>∼</mo> <mn>1</mn> </mrow> </math>   GeV are fixed by using those moments. The QCD sum rules for the  $$0_{\textrm{th}}$$  <math> <msub> <mn>0</mn> <mtext>th</mtext> </msub> </math>  -order moment can also be used to fix the magnitude of  $$\eta _q$$  <math> <msub> <mi>η</mi> <mi>q</mi> </msub> </math>   decay constant, giving  $$f_{\eta _q}=0.141\pm 0.005$$  <math> <mrow> <msub> <mi>f</mi> <msub> <mi>η</mi> <mi>q</mi> </msub> </msub> <mo>=</mo> <mn>0.141</mn> <mo>±</mo> <mn>0.005</mn> </mrow> </math>   GeV. As an application of  $$\phi _{2;\eta _q},$$  <math> <mrow> <msub> <mi>ϕ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>η</mi> <mi>q</mi> </msub> </mrow> </msub> <mo>,</mo> </mrow> </math>   we calculate the transition form factors  $$B(D)^+ \rightarrow \eta ^{(\prime )}$$  <math> <mrow> <mi>B</mi> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>η</mi> <mrow> <mo>(</mo> <mo>′</mo> <mo>)</mo> </mrow> </msup> </mrow> </math>   by using the QCD light-cone sum rules up to twist-4 accuracy and by including the next-to-leading order QCD corrections to the leading-twist part, and then fix the related CKM matrix element and the decay width for the semi-leptonic decays  $$B(D)^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell .$$  <math> <mrow> <mi>B</mi> <msup> <mrow> <mo>(</mo> <mi>D</mi> <mo>)</mo> </mrow> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>η</mi> <mrow> <mo>(</mo> <mo>′</mo> <mo>)</mo> </mrow> </msup> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> <mo>.</mo> </mrow> </math>

Zai-Hui Wu

Properties of the  $$\eta _q$$  <math> <msub> <mi>η</mi> <mi>q</mi> </msub> </math>   leading-twist distribution amplitude and its effects to the  $$B/D^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$  <math> <mrow> <mi>B</mi> <mo>/</mo> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>η</mi> <mrow> <mo>(</mo> <mo>′</mo> <mo>)</mo> </mrow> </msup> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>   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>.

Xuan Luo

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

In the paper, we investigate the moments  $$\langle \xi _{2;a_1}^{\Vert ;n}\rangle $$  <math> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mi>n</mi> </mrow> </msubsup> <mo>⟩</mo> </mrow> </math>   of the axial-vector  $$a_1(1260)$$  <math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> </mrow> </math>  -meson distribution amplitude by using the QCD sum rules approach under the background field theory. By considering the vacuum condensates up to dimension-six and the perturbative part up to next-to-leading order QCD corrections, its first five moments at an initial scale  $$\mu _0=1~{\mathrm{GeV}}$$  <math> <mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> <mspace width="3.33333pt"></mspace> <mi>GeV</mi> </mrow> </math>   are  $$\langle \xi _{2;a_1}^{\Vert ;2}\rangle |_{\mu _0} = 0.223 \pm 0.029$$  <math> <mrow> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mn>2</mn> </mrow> </msubsup> <mo>⟩</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> </msub> <mo>=</mo> <mn>0.223</mn> <mo>±</mo> <mn>0.029</mn> </mrow> </math>  ,  $$\langle \xi _{2;a_1}^{\Vert ;4}\rangle |_{\mu _0} = 0.098 \pm 0.008$$  <math> <mrow> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mn>4</mn> </mrow> </msubsup> <mo>⟩</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> </msub> <mo>=</mo> <mn>0.098</mn> <mo>±</mo> <mn>0.008</mn> </mrow> </math>  ,  $$\langle \xi _{2;a_1}^{\Vert ;6}\rangle |_{\mu _0} = 0.056 \pm 0.006$$  <math> <mrow> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mn>6</mn> </mrow> </msubsup> <mo>⟩</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> </msub> <mo>=</mo> <mn>0.056</mn> <mo>±</mo> <mn>0.006</mn> </mrow> </math>  ,  $$\langle \xi _{2;a_1}^{\Vert ;8}\rangle |_{\mu _0} = 0.039 \pm 0.004$$  <math> <mrow> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mn>8</mn> </mrow> </msubsup> <mo>⟩</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> </msub> <mo>=</mo> <mn>0.039</mn> <mo>±</mo> <mn>0.004</mn> </mrow> </math>   and  $$\langle \xi _{2;a_1}^{\Vert ;10}\rangle |_{\mu _0} = 0.028 \pm 0.003$$  <math> <mrow> <mrow> <mo>⟨</mo> <msubsup> <mi>ξ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mrow> <mo>‖</mo> <mo>;</mo> <mn>10</mn> </mrow> </msubsup> <mo>⟩</mo> </mrow> <msub> <mrow> <mo>|</mo> </mrow> <msub> <mi>μ</mi> <mn>0</mn> </msub> </msub> <mo>=</mo> <mn>0.028</mn> <mo>±</mo> <mn>0.003</mn> </mrow> </math>  , respectively. We then construct a light-cone harmonic oscillator model for  $$a_1(1260)$$  <math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> </mrow> </math>  -meson longitudinal twist-2 distribution amplitude  $$\phi _{2;a_1}^{\Vert }(x,\mu )$$  <math> <mrow> <msubsup> <mi>ϕ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mo>‖</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </math>  , whose model parameters are fitted by using the least squares method. As an application of  $$\phi _{2;a_1}^{\Vert }(x,\mu )$$  <math> <mrow> <msubsup> <mi>ϕ</mi> <mrow> <mn>2</mn> <mo>;</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> </mrow> <mo>‖</mo> </msubsup> <mrow> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </math>  , we calculate the transition form factors (TFFs) of  $$D\rightarrow a_1(1260)$$  <math> <mrow> <mi>D</mi> <mo>→</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> </mrow> </math>   in large and intermediate momentum transfers by using the QCD light-cone sum rules approach. At the largest recoil point ( $$q^2=0$$  <math> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math>  ), we obtain  $$ A(0) = 0.130_{ - 0.013}^{ + 0.015}$$  <math> <mrow> <mi>A</mi> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>130</mn> <mrow> <mo>-</mo> <mn>0.013</mn> </mrow> <mrow> <mo>+</mo> <mn>0.015</mn> </mrow> </msubsup> </mrow> </math>  ,  $$V_1(0) = 1.898_{-0.121}^{+0.128}$$  <math> <mrow> <msub> <mi>V</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>.</mo> <msubsup> <mn>898</mn> <mrow> <mo>-</mo> <mn>0.121</mn> </mrow> <mrow> <mo>+</mo> <mn>0.128</mn> </mrow> </msubsup> </mrow> </math>  ,  $$V_2(0) = 0.228_{-0.021}^{ + 0.020}$$  <math> <mrow> <msub> <mi>V</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>228</mn> <mrow> <mo>-</mo> <mn>0.021</mn> </mrow> <mrow> <mo>+</mo> <mn>0.020</mn> </mrow> </msubsup> </mrow> </math>  , and  $$V_0(0) = 0.217_{ - 0.025}^{ + 0.023}$$  <math> <mrow> <msub> <mi>V</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>.</mo> <msubsup> <mn>217</mn> <mrow> <mo>-</mo> <mn>0.025</mn> </mrow> <mrow> <mo>+</mo> <mn>0.023</mn> </mrow> </msubsup> </mrow> </math>  . By applying the extrapolated TFFs to the semi-leptonic decay  $$D^{0(+)} \rightarrow a_1^{-(0)}(1260)\ell ^+\nu _\ell $$  <math> <mrow> <msup> <mi>D</mi> <mrow> <mn>0</mn> <mo>(</mo> <mo>+</mo> <mo>)</mo> </mrow> </msup> <mo>→</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mrow> <mo>-</mo> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>  , we obtain  $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) e^+\nu _e) = (5.261_{-0.639}^{+0.745}) \times 10^{-5}$$  <math> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mo>-</mo> </msubsup> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>.</mo> <msubsup> <mn>261</mn> <mrow> <mo>-</mo> <mn>0.639</mn> </mrow> <mrow> <mo>+</mo> <mn>0.745</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mrow> </math>  ,  $${\mathcal {B}}(D^+\rightarrow a_1^0(1260) e^+\nu _e) = (6.673_{-0.811}^{+0.947}) \times 10^{-5}$$  <math> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>e</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>.</mo> <msubsup> <mn>673</mn> <mrow> <mo>-</mo> <mn>0.811</mn> </mrow> <mrow> <mo>+</mo> <mn>0.947</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mrow> </math>  ,  $${\mathcal {B}}(D^0\rightarrow a_1^-(1260) \mu ^+ \nu _\mu )=(4.732_{-0.590}^{+0.685}) \times 10^{-5}$$  <math> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mo>-</mo> </msubsup> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>μ</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>.</mo> <msubsup> <mn>732</mn> <mrow> <mo>-</mo> <mn>0.590</mn> </mrow> <mrow> <mo>+</mo> <mn>0.685</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mrow> </math>  ,  $${\mathcal {B}}(D^+ \rightarrow a_1^0(1260) \mu ^+ \nu _\mu )=(6.002_{-0.748}^{+0.796}) \times 10^{-5}$$  <math> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msubsup> <mi>a</mi> <mn>1</mn> <mn>0</mn> </msubsup> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>μ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>μ</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>.</mo> <msubsup> <mn>002</mn> <mrow> <mo>-</mo> <mn>0.748</mn> </mrow> <mrow> <mo>+</mo> <mn>0.796</mn> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </mrow> </math>  .

$$a_1(1260)$$  <math> <mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> </mrow> </math>  -meson longitudinal twist-2 distribution amplitude and the  $$D\rightarrow a_1(1260)\ell ^+\nu _\ell $$  <math> <mrow> <mi>D</mi> <mo>→</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>1260</mn> <mo>)</mo> </mrow> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>   decay processes