<p>In this work, we investigate the semileptonic decays of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub></math> meson to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2536</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2420</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2430</mn><mo stretchy="false">)</mo></math> in the covariant light-front quark model. We combine the helicity amplitudes via the corresponding form factors to obtain the branching ratios of the semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msubsup><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo>*</mo><mo>*</mo></mrow></msubsup><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><mrow><mo>*</mo><mo>*</mo></mrow></msubsup></math> referring to a P-wave excited charmed meson <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2536</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2420</mn><mo stretchy="false">)</mo></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2430</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>ℓ</mo><mo>=</mo><mi>e</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>μ</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>τ</mi></mrow></math>. Furthermore, we also take into account another two physical observables, namely, the longitudinal polarization fraction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>f</mi><mi>L</mi></msub></math> and the forward-backward asymmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>A</mi><mrow><mi>F</mi><mi>B</mi></mrow></msub></math>. Most of our predictions are comparable to the results given by other theoretical approaches and the currently available data. The branching ratios of the semileptonic decay channels <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>s</mi></msub><mo stretchy="false">→</mo><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2420</mn><mo stretchy="false">)</mo><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> are larger than those of the semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>s</mi></msub><mo stretchy="false">→</mo><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2536</mn><mo stretchy="false">)</mo><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2430</mn><mo stretchy="false">)</mo><mo>ℓ</mo><msub><mi>ν</mi><mo>ℓ</mo></msub></math>, respectively. We find that the long-standing “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>1</mn><mo>/</mo><mn>2</mn></math> vs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>3</mn><mo>/</mo><mn>2</mn></math> puzzle” in the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msubsup><mover accent="true"><mi>D</mi><mo stretchy="false">¯</mo></mover><mn>1</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mn>0</mn></mrow></msubsup><msup><mo>ℓ</mo><mrow><mo>′</mo><mo>+</mo></mrow></msup><msub><mi>ν</mi><msup><mo>ℓ</mo><mo>′</mo></msup></msub></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msup><mo>ℓ</mo><mo>′</mo></msup><mo>=</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></math> can be solved by taking some negative mixing angle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>θ</mi><mi>s</mi></msub></math> values within a range from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>30.3</mn><mi>°</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>−</mo><mn>24.9</mn><mi>°</mi></math>, corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>θ</mi></math> of about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>5</mn><mi>°</mi><mo>∼</mo><mn>10.4</mn><mi>°</mi></math>; whereas the Belle Collaboration updated their measurements for the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>D</mi><mn>0</mn><mrow><mo>*</mo><mo>−</mo></mrow></msubsup><msup><mo>ℓ</mo><mrow><mo>′</mo><mo>+</mo></mrow></msup><msub><mi>ν</mi><msup><mo>ℓ</mo><mo>′</mo></msup></msub></math> with only a small upper limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Br</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>D</mi><mn>0</mn><mrow><mo>*</mo><mo>−</mo></mrow></msubsup><msup><mo>ℓ</mo><mrow><mo>′</mo><mo>+</mo></mrow></msup><msub><mi>ν</mi><msup><mo>ℓ</mo><mo>′</mo></msup></msub><mo stretchy="false">)</mo><mo>&lt;</mo><mn>0.44</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>3</mn></mrow></msup></math> obtained, which is much larger than most theoretical predictions and raises a new puzzle.</p>

China

You-Ya Yang

Zhi-Qing Zhang

Hao Yang

Semileptonic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msub></mrow></math> meson decays to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mn>0</mn><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2300</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>D</mi><mrow><mi>s</mi><mn>0</mn></mrow><mo>*</mo></msubsup><mo stretchy="false">(</mo><mn>2317</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2460</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>2536</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2420</mn><mo stretchy="false">)</mo></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>D</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>2430</mn><mo stretchy="false">)</mo></math> within the covariant light-front approach

<p>Different from the previous treatment in a two-body framework, we introduce the dimeson distribution amplitudes (DAs) to describe the strong dynamics between the S-wave resonances <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo>,</mo><mn>1450</mn><mo stretchy="false">)</mo></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>K</mi><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">(</mo><mi>π</mi><mi>η</mi><mo stretchy="false">)</mo></mrow></math> pair, where the Gegenbauer coefficient required is determined from the experimental data on the timelike form factors involved. The branching ratios and direct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> asymmetries of the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msubsup><mi>h</mi><mo stretchy="false">→</mo><mi>K</mi><mover accent="true"><mi>K</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">(</mo><mi>π</mi><mi>η</mi><mo stretchy="false">)</mo><mi>h</mi></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>′</mo></mrow></msubsup><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>1450</mn><mo stretchy="false">)</mo></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>h</mi></math> referring to a pion or a kaon, are then calculated in the perturbative QCD (PQCD) approach. We find that the branching ratios of the corresponding quasi-two-body decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msubsup><mi>K</mi></math> obtained with the narrow width approximation are closer to those predicted in the QCD factorization (QCDF) approach compared to the previous PQCD calculations, no matter a three-body or a two-body framework is assumed. Furthermore, all our predictions for these <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo></mrow></msubsup><mi>K</mi></math> decays are below the current experimental upper limits except for those of decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>−</mo></mrow></msubsup><msup><mi>K</mi><mo>+</mo></msup></math>, which are (slightly) larger than the upper limits. Under the narrow width approximation, the branching ratios of the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo></mrow></msubsup><msup><mi>π</mi><mn>0</mn></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo></mrow></msubsup><msup><mi>π</mi><mo>−</mo></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mi>B</mi></mrow><mrow><mn>0</mn></mrow></msup><mo stretchy="false">→</mo><msubsup><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mn>0</mn></mrow></msubsup><msup><mrow><mi>π</mi></mrow><mrow><mn>0</mn></mrow></msup></mrow></math> are comparable to or agree well with the previous PQCD and the QCDF calculations. While for the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mn>0</mn></mrow></msubsup><msup><mi>π</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>−</mo></mrow></msubsup><msup><mi>π</mi><mo>+</mo></msup></math>, their branching ratios are predicted to be unexpectedly large, for example, the obtained branching ratio of decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mn>0</mn></msubsup><msup><mi>π</mi><mo>+</mo></msup></math> is even higher than the current experimental upper limit. For the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>−</mo></mrow></msubsup><msup><mi>K</mi><mo>+</mo></msup><mo>,</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mn>0</mn></mrow></msubsup><msup><mi>π</mi><mn>0</mn></msup><mo>,</mo><msubsup><mi>a</mi><mn>0</mn><mrow><mo stretchy="false">(</mo><mo>′</mo><mo stretchy="false">)</mo><mo>+</mo></mrow></msubsup><msup><mi>π</mi><mo>−</mo></msup></math>, where the tree and the penguin amplitudes are comparable in magnitude, the direct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> asymmetries are sensitive to the strong phases, which are very different obtained by other theoretical predictions for some channels. Therefore, it is very important to accurately determine the strong phases of these decays, which are crucial to precisely predict the direct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>C</mi><mi>P</mi></mrow></math> asymmetries and further clarify the inner structures of the resonances <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>980</mn><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mn>1450</mn><mo stretchy="false">)</mo></math>.</p>

Si-Yang Wang

Zhi-Jie Sun

Jian Chai

Peng Li

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>B</mi><mo stretchy="false">→</mo><mi>K</mi><mover accent="true"><mrow><mi>K</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover><mo stretchy="false">(</mo><mi>π</mi><mi>η</mi><mo stretchy="false">)</mo><mi>h</mi></mrow></math> decays in the presence of isovector scalar resonances <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>980</mn><mo>,</mo><mn>1450</mn><mo stretchy="false">)</mo></mrow></math>

In this paper, we investigate the quasi-two-body decays  $$B_c \rightarrow K_0^{*}(1430,1950) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>,</mo> <mn>1950</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   within the perturbative QCD (PQCD) framework. The S-wave two-meson distribution amplitudes (DAs) are introduced to describe the final state interactions of the  $$K\pi $$  <math> <mrow> <mi>K</mi> <mi>π</mi> </mrow> </math>   pair, which involve the time-like form factors and the Gegenbauer polynomials. In the calculations, we adopt two kinds of parameterization schemes to describe the time-like form factors: one is the relativistic Breit–Wigner (RBW) formula, which is usually more suitable for the narrow resonances, and the other is the LASS line shape proposed by the LASS Collaboration, which includes both the resonant and nonresonant components. We find that the branching ratios and the direct CP violations for the decays  $$B_c \rightarrow K_0^{*}(1430) D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   obtained from those of the quasi-two-body decays  $$B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   under the narrow width approximation (NWA) can be consistent well with the previous PQCD results calculated in the two-body framework by assuming that  $$K^*_0(1430)$$  <math> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>   is the lowest lying  $$\bar{q} s$$  <math> <mrow> <mover> <mrow> <mi>q</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> <mi>s</mi> </mrow> </math>   state, which is the so-called scenario II (SII). We conclude that the LASS parameterization is more suitable to describe the  $$K_0^{*}(1430)$$  <math> <mrow> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>   than the RBW formula, and the nonresonant components play an important role in the branching ratios of the decays  $$B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>  . In view of the large difference between the decay width measurements for the  $$K_0^{*}(1950)$$  <math> <mrow> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </math>   given by BaBar and LASS collaborations, we calculate the branching ratios and the CP violations for the quasi-two-body decays  $$B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   by using two values,  $$\Gamma _{K^*_0(1950)}=0.100\pm 0.04$$  <math> <mrow> <msub> <mi>Γ</mi> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.100</mn> <mo>±</mo> <mn>0.04</mn> </mrow> </math>   GeV and  $$\Gamma _{K^*_0(1950)}=0.201\pm 0.034$$  <math> <mrow> <msub> <mi>Γ</mi> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.201</mn> <mo>±</mo> <mn>0.034</mn> </mrow> </math>   GeV, besides the two kinds of parameterizations for the resonance  $$K^*_0(1950)$$  <math> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </math>  . We find that the branching ratios and the direct CP violations for the decays  $$B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   have not as large difference between the two parameterizations as the case of decays involving the  $$K^*_0(1430)$$  <math> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>  , especially for the results with  $$\Gamma _{K^*_0(1950)}=0.201\pm 0.034$$  <math> <mrow> <msub> <mi>Γ</mi> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </msub> <mo>=</mo> <mn>0.201</mn> <mo>±</mo> <mn>0.034</mn> </mrow> </math>   GeV. The effect of the nonresonant component in the  $$K^*_0(1950)$$  <math> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> </mrow> </math>   may be not so serious as that in the  $$K^*_0(1430)$$  <math> <mrow> <msubsup> <mi>K</mi> <mn>0</mn> <mo>∗</mo> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>  . The quasi-two-body decays  $$B^+_c \rightarrow K^{*+}_0(1430) D^{0} \rightarrow K^0 \pi ^+ D^{0}$$  <math> <mrow> <msubsup> <mi>B</mi> <mi>c</mi> <mo>+</mo> </msubsup> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> <mo>+</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msup> <mi>D</mi> <mn>0</mn> </msup> <mo>→</mo> <msup> <mi>K</mi> <mn>0</mn> </msup> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>D</mi> <mn>0</mn> </msup> </mrow> </math>   and  $$B^+_c \rightarrow K^{*0}_0(1430) D^{+} \rightarrow K^+ \pi ^- D^{+}$$  <math> <mrow> <msubsup> <mi>B</mi> <mi>c</mi> <mo>+</mo> </msubsup> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> <mn>0</mn> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msup> <mi>D</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>K</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> <msup> <mi>D</mi> <mo>+</mo> </msup> </mrow> </math>   have large branching ratios, which can reach to the order of  $$10^{-4}$$  <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </math>   and are most likely to be observed in the future LHCb experiments. Furthermore, the branching ratios of the quasi-two-body decays  $$B_c \rightarrow K_0^{*}(1950) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1950</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   are about one order smaller than those of the corresponding decays  $$B_c \rightarrow K_0^{*}(1430) D_{(s)} \rightarrow K \pi D_{(s)}$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>  .

Zi-Yu Zhang

Ming-Xuan Xie

Ming-Yang Li

Hong-Xia Guo

Branching ratios and CP asymmetries of the quasi-two-body decays  $$B_c \rightarrow \ K^{*}_0(1430,1950) D_{(s)} \rightarrow K \pi D_{(s)} $$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mspace width="4pt"></mspace> <mmultiscripts> <mi>K</mi> <mn>0</mn> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </mmultiscripts> <mrow> <mo>(</mo> <mn>1430</mn> <mo>,</mo> <mn>1950</mn> <mo>)</mo> </mrow> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> <mo>→</mo> <mi>K</mi> <mi>π</mi> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   in the PQCD approach

In this work, we investigate the form factors of the transitions <math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msub><mo>→</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2300</mn><mo>)</mo><mo>,</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></math>, <math><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>2460</mn><mo>)</mo></math>, and <math><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>2536</mn><mo>)</mo></math> in the covariant light-front quark model (CLFQM), where these final states are considered as P-wave excited charmed mesons. In order to obtain the form factors for the physical transition processes, we extend these form factors from the spacelike region to the timelike region. The <math><msup><mi>q</mi><mn>2</mn></msup></math>-dependence for each transition form factor is also plotted. Then, combined with those form factors, the branching ratios of the two-body nonleptonic decays <math><mrow><msub><mrow><mi>B</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msub><mo>→</mo><msubsup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mi>s</mi><mo>)</mo><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2300</mn><mo>,</mo><mn>2317</mn><mo>)</mo><mi>M</mi></mrow></math>, <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>2460</mn><mo>,</mo><mn>2536</mn><mo>)</mo><mi>M</mi></mrow></math> with <math><mi>M</mi></math> being a light pseudoscalar (vector) meson or a charmed meson are calculated by considering the quantum chromodynamics (QCD) radiative corrections to the hadronic matrix elements within the QCD factorization approach. Most of our predictions are comparable to the results given by other theoretical approaches and the present available data.

Study of <math><msub><mi>B</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msub></math> meson decays to <math><mrow><msubsup><mrow><mi>D</mi></mrow><mn>0</mn><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2300</mn><mo>)</mo></mrow></math>, <math><mrow><msubsup><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>0</mn></mrow><mrow><mo>*</mo></mrow></msubsup><mo>(</mo><mn>2317</mn><mo>)</mo></mrow></math>, <math><mrow><msub><mrow><mi>D</mi></mrow><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>2460</mn><mo>)</mo></mrow></math> and <math><msub><mi>D</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>(</mo><mn>2536</mn><mo>)</mo></math> within the covariant light-front approach

In addition to the strong and electromagnetic decay modes, the  $$\psi (1S,2S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   and  $$\eta _{c}(1S,2S)$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </math>   can also decay via the weak interaction. Such weak decays can be detected by the high-luminosity heavy-flavor experiments. At present, some of the semileptonic and nonleptonic  $$J/\Psi $$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>Ψ</mi> </mrow> </math>   weak decays have been measured at BESIII. Researching for these charmonium weak decays to  $$D_{(s)}$$  <math> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </math>   meson can provide a platform to check of the standard model (SM) and probe new physics (NP). So we investigate the semileptonic and nonleptonic weak decays of  $$\psi (1S,2S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   and  $$\eta _{c}(1S,2S)$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </math>   to  $$D_{(s)}$$  <math> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </math>   within the covariant light-front quark model (CLFQM). With form factors of the transitions  $$\psi (1S,2S)\rightarrow D_{(s)}$$  <math> <mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>→</mo> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   and  $$\eta _{c}(1S,2S)\rightarrow D_{(s)}$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>→</mo> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </mrow> </math>   calculated under the CLFQM, we predict and discuss some physical observables, such as the branching ratios, the longitudinal polarizations  $$f_{L}$$  <math> <msub> <mi>f</mi> <mi>L</mi> </msub> </math>   and the forward–backward asymmetries  $$A_{FB}$$  <math> <msub> <mi>A</mi> <mrow> <mi>FB</mi> </mrow> </msub> </math>  . One can find that the Cabibbo-favored semi-leptonic decay channels  $$\psi (1\,S,2\,S)\rightarrow D_{s}^{-}\ell ^{+}\nu _{\ell }$$  <math> <mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mspace width="0.166667em"></mspace> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mspace width="0.166667em"></mspace> <mi>S</mi> <mo>)</mo> </mrow> <mo>→</mo> <msubsup> <mi>D</mi> <mrow> <mi>s</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>ℓ</mi> <mo>+</mo> </msup> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>   with  $$\ell =e,\mu $$  <math> <mrow> <mi>ℓ</mi> <mo>=</mo> <mi>e</mi> <mo>,</mo> <mi>μ</mi> </mrow> </math>   and the nonleptonic decay modes  $$\psi (1S,2S)\rightarrow D_{s}^{-}\rho ^{+}$$  <math> <mrow> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>→</mo> <msubsup> <mi>D</mi> <mrow> <mi>s</mi> </mrow> <mo>-</mo> </msubsup> <msup> <mi>ρ</mi> <mo>+</mo> </msup> </mrow> </math>   have relatively large branching ratios of the order  $$\mathcal {O}(10^{-9})$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>9</mn> </mrow> </msup> <mo>)</mo> </mrow> </math>  , which are most likely to be accessible at the future high-luminosity experiments.

Semileptonic and nonleptonic weak decays of  $$\psi (1S,2S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   and  $$\eta _{c}(1S,2S)$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </math>   to  $$D_{(s)}$$  <math> <msub> <mi>D</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> </msub> </math>   in the covariant light-front approach

In this work, we investigate the semileptonic decays of  $$B_{c}$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   meson to  $$\eta _{c}(1S,2S,3S)$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </math>  ,  $$\psi (1S,2S,3S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   and X(3872) within the framework of covariant light-front quark model (CLFQM). We combine the helicity amplitudes via the corresponding form factors to obtain the branching ratios of the semileptonic decays  $$B_{c}\rightarrow \eta _{c}(1S, 2S, 3S)\ell \nu _{\ell }$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> <mi>ℓ</mi> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>  ,  $$B_{c}\rightarrow \psi (1S, 2S, 3S))\ell \nu _{\ell }$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mrow> <mo>→</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>)</mo> <mi>ℓ</mi> </mrow> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>   and  $$B_{c}\rightarrow X(3872)\ell \nu _{\ell }$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <mi>X</mi> <mrow> <mo>(</mo> <mn>3872</mn> <mo>)</mo> </mrow> <mi>ℓ</mi> <msub> <mi>ν</mi> <mi>ℓ</mi> </msub> </mrow> </math>  , with  $$\ell =e,\mu ,\tau $$  <math> <mrow> <mi>ℓ</mi> <mo>=</mo> <mi>e</mi> <mo>,</mo> <mi>μ</mi> <mo>,</mo> <mi>τ</mi> </mrow> </math>  . In view of the  $$R_{J/\Psi }$$  <math> <msub> <mi>R</mi> <mrow> <mi>J</mi> <mo>/</mo> <mi>Ψ</mi> </mrow> </msub> </math>   anomaly released by the LHCb collaboration, it is necessary to systematically calculate the ratios  $$R_X$$  <math> <msub> <mi>R</mi> <mi>X</mi> </msub> </math>   with  $$X=\psi (1S,2S,3S),\eta _c(1S,2S,3S),X(3872)$$  <math> <mrow> <mi>X</mi> <mo>=</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mi>S</mi> <mo>,</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>,</mo> <mi>X</mi> <mrow> <mo>(</mo> <mn>3872</mn> <mo>)</mo> </mrow> </mrow> </math>  , which are helpful for checking the lepton flavor universality (LFU). We also take into account another two physical observables, the longitudinal polarization fraction  $$f_{L}$$  <math> <msub> <mi>f</mi> <mi>L</mi> </msub> </math>   and the forward–backward asymmetry  $$A_{FB}$$  <math> <msub> <mi>A</mi> <mrow> <mi>FB</mi> </mrow> </msub> </math>  , which can provide new clues for understanding the  $$R_{J /\Psi }$$  <math> <msub> <mi>R</mi> <mrow> <mi>J</mi> <mo>/</mo> <mi>Ψ</mi> </mrow> </msub> </math>   anomaly. Such theoretical predictions are necessary and interesting, and can be tested in future LHCb experiments.

Semileptonic decays  $$B_{c}$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   meson to S-wave charmonia and X(3872) within the covariant light-front approach

In this work, we study the quasi-two-body decays <math><msub><mi>B</mi><mi>c</mi></msub><mo>→</mo><msup><mi>K</mi><mo>*</mo></msup><mi>h</mi><mo>→</mo><mi>K</mi><mi>π</mi><mi>h</mi><mo>(</mo><mi>h</mi><mo>=</mo><mi>D</mi><mo>,</mo><msub><mi>D</mi><mi>s</mi></msub><mo>,</mo><mi>K</mi><mo>,</mo><mi>π</mi><mo>,</mo><mi>η</mi><mo>,</mo><msup><mi>η</mi><mo>′</mo></msup><mo>)</mo></math> in the perturbative QCD (PQCD) approach. The two-meson distribution amplitudes <math><msubsup><mi>Φ</mi><mrow><mi>K</mi><mi>π</mi></mrow><mtext>P-wave</mtext></msubsup></math> are introduced to describe the final-state interactions of the <math><mi>K</mi><mi>π</mi></math> pair, which involve the timelike form factors <math><msub><mi>F</mi><mrow><mi>K</mi><mi>π</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo></math> parametrized by the relativistic Breit-Wigner function and the Gegenbauer polynomials. We calculate the branching ratios for these quasi-two-body decays, from which one can obtain the branching ratios for the corresponding two-body decays under the narrow width approximation relation. We find that <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>D</mi><mn>0</mn></msup></math> and <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup><msup><mi>D</mi><mo>+</mo></msup></math> have the largest branching ratios, which can reach up to <math><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math>, while the branching ratios for other two-body decays are very small and only about <math><msup><mn>10</mn><mrow><mo>−</mo><mn>8</mn></mrow></msup><mo>−</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>. As we expected, the branching ratios of the pure annihilation decays are usually small, while in our considered such type of decays, the channel <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mover><mi>K</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup><msup><mi>K</mi><mo>+</mo></msup></math> has the largest branching ratio, which is near <math><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math>. These results are consistent with the previously PQCD calculations obtained in the two-body framework, which can be tested by future LHCb experiments. For the decays <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>D</mi><mn>0</mn></msup><mo>→</mo><msup><mi>K</mi><mn>0</mn></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>D</mi><mn>0</mn></msup></math>, <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mrow><mo>*</mo><mn>0</mn></mrow></msup><msup><mi>D</mi><mo>+</mo></msup><mo>→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>π</mi><mo>−</mo></msup><msup><mi>D</mi><mo>+</mo></msup></math>, and <math><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mover><mi>K</mi><mo>¯</mo></mover><mrow><mo>*</mo><mn>0</mn></mrow></msup><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mo>−</mo></msup><msup><mi>π</mi><mo>+</mo></msup><msubsup><mi>D</mi><mi>s</mi><mo>+</mo></msubsup></math>, we also calculate their direct <math><mi>C</mi><mi>P</mi></math> violations and find that <math><msub><mi>A</mi><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo>(</mo><msubsup><mi>B</mi><mi>c</mi><mo>+</mo></msubsup><mo>→</mo><msup><mi>K</mi><mrow><mo>*</mo><mo>+</mo></mrow></msup><msup><mi>D</mi><mn>0</mn></msup><mo>→</mo><msup><mi>K</mi><mn>0</mn></msup><msup><mi>π</mi><mo>+</mo></msup><msup><mi>D</mi><mn>0</mn></msup><mo>)</mo><mo>=</mo><mo>(</mo><mo>−</mo><msubsup><mn>14.6</mn><mrow><mo>−</mo><mn>1.1</mn></mrow><mrow><mo>+</mo><mn>9.2</mn></mrow></msubsup><mo>)</mo><mo>%</mo></math> is the largest one, which is possible measured by the present LHCb experiments. For the pure annihilation type decays, there is no <math><mi>C</mi><mi>P</mi></math> violation, because only the tree operators are involved. Furthermore, we give the differential distributions of the branching ratios and the direct <math><mi>C</mi><mi>P</mi></math> violations in the <math><mi>K</mi><mi>π</mi></math> invariant mass <math><mi>ω</mi></math> for the decays <math><msub><mi>B</mi><mi>c</mi></msub><mo>→</mo><msup><mi>K</mi><mo>*</mo></msup><msub><mi>D</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msub><mo>→</mo><mi>K</mi><mi>π</mi><msub><mi>D</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></msub></math>.

Quasi-two-body decays <math><msub><mi>B</mi><mi>c</mi></msub><mo>→</mo><msup><mi>K</mi><mo>*</mo></msup><mi>h</mi><mo>→</mo><mi>K</mi><mi>π</mi><mi>h</mi></math> in perturbative QCD

In this work, we investigate the quasi-two-body decays   $ B_c\to D^*h\to D\pi h $   with   $ B_c\to D^*h\to D\pi h $   using the perturbative QCD (PQCD) approach. The description of final state interactions of the   $ B_c\to D^*h\to D\pi h $   pair is achieved through the two-meson distribution amplitudes (DAs), which are normalized to the time-like form factor. The PQCD predictions on the branching ratios of the quasi-two-body decays   $ B_c\to D^*h\to D\pi h $   show an obvious hierarchy:   $ B_c\to D^*h\to D\pi h $     $ B_c\to D^*h\to D\pi h $     $ B_c\to D^*h\to D\pi h $   and   $ B_c\to D^*h\to D\pi h $  . From the invariant mass   $ B_c\to D^*h\to D\pi h $  -dependence of the decay spectrum for each channel, one can find that the branching fraction is concentrated in a narrow region around the   $ B_c\to D^*h\to D\pi h $   pole mass. Thus, one can obtain the branching ratios for the corresponding two-body decays   $ B_c\to D^*h\to D\pi h $   under the narrow-width approximation. We find that the branching ratios of the decays   $ B_c\to D^*h\to D\pi h $   are consistent with the previous PQCD calculations within errors. These predictions will be tested in future experiments.

Yan-Chao Zhao

Qiu-Bo Meng

Quasi-two-body decays in perturbative QCD  

In this work, we investigate the form factors of  $$B_c$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   decays into  $$J/\Psi , \psi (2S,3S),\eta _c,$$  <math> <mrow> <mi>J</mi> <mo>/</mo> <mi>Ψ</mi> <mo>,</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>η</mi> <mi>c</mi> </msub> <mo>,</mo> </mrow> </math>    $$ \eta _c(2\,S,3\,S), \chi _{c0}, \chi _{c1}, h_c$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mspace width="0.166667em"></mspace> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mspace width="0.166667em"></mspace> <mi>S</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>χ</mi> <mrow> <mi>c</mi> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>χ</mi> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mi>c</mi> </msub> </mrow> </math>  , and X(3872) mesons in the covariant light-front quark model (CLFQM). For the purpose of the branching ratio calculation, the form factors of  $$B_c\rightarrow D^{(*)}, D^{(*)}_s$$  <math> <mrow> <msub> <mi>B</mi> <mi>c</mi> </msub> <mo>→</mo> <msup> <mi>D</mi> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> </mrow> </msup> <mo>,</mo> <msubsup> <mi>D</mi> <mi>s</mi> <mrow> <mo>(</mo> <mrow></mrow> <mo>∗</mo> <mo>)</mo> </mrow> </msubsup> </mrow> </math>   transitions are also included. In order to obtain the form factors for the physical transition processes, we need to extend these form factors from the space-like region to the time-like region. The  $$q^2$$  <math> <msup> <mi>q</mi> <mn>2</mn> </msup> </math>   dependence for each transition form factor is also plotted. Then, using the factorization method, we calculate the branching ratios of 80  $$B_c$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   decay channels with a charmonium involved in each mode. Most of our predictions are comparable to the results given by other approaches. As to the decays with the radially excited-state S-wave charmonia involved, such as  $$\psi (2S,3S)$$  <math> <mrow> <mi>ψ</mi> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> </math>   and  $$\eta _c(2S,3S)$$  <math> <mrow> <msub> <mi>η</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>3</mn> <mi>S</mi> <mo>)</mo> </mrow> </mrow> </math>  , two sets of parameters for their light-front wave functions, corresponding to scenario I (SI) and scenario II (SII), are adopted to calculate the branching ratios. By comparing with the future experimental data, one can discriminate which parameters are more favored.

Covariant light-front approach for  $$B_c$$  <math> <msub> <mi>B</mi> <mi>c</mi> </msub> </math>   decays into charmonium: implications on form factors and branching ratios

We study the   $ B_{c,u,d}\to X(3872)P $   decays in the perturbative QCD (PQCD) approach, involving the puzzling resonance   $ B_{c,u,d}\to X(3872)P $  , where P represents a light pseudoscalar meson (K or π). Assuming   $ B_{c,u,d}\to X(3872)P $   to be a   $ B_{c,u,d}\to X(3872)P $   charmonium state, we obtain the following results. (a) The branching ratios of the   $ B_{c,u,d}\to X(3872)P $   and   $ B_{c,u,d}\to X(3872)P $   decays are consistent with the results predicted by the covariant light-front approach within errors; however, they are larger than those given by the generalized factorization approach. (b) The branching ratio of the   $ B_{c,u,d}\to X(3872)P $   decay is predicted as   $ B_{c,u,d}\to X(3872)P $  , which is smaller than the previous PQCD calculation result but still slightly larger than the upper limits set by Belle and BaBar. Hence, we suggest that the  $ B_{c,u,d}\to X(3872)P $   decays should be precisely measured by the LHCb and Belle II experiments to help probe the inner structure of   $ B_{c,u,d}\to X(3872)P $  . (c) Compared with the   $ B_{c,u,d}\to X(3872)P $  decays, the   $ B_{c,u,d}\to X(3872)P $   decays have significantly smaller branching ratios, which drop to values as low as   $ B_{c,u,d}\to X(3872)P $  . (d) The direct CP violations of these considered decays are small (  $ B_{c,u,d}\to X(3872)P $  ) because the penguin contributions are loop suppressed compared to the tree contributions. The mixing-induced CP violation of the   $ B_{c,u,d}\to X(3872)P $   decay is highly consistent with the current world average value   $ B_{c,u,d}\to X(3872)P $  %. Experimentally testing the results for the branching ratios and CP violations, including the implicit   $ B_{c,u,d}\to X(3872)P $   and isospin symmetries of these decays, helps probe the nature of   $ B_{c,u,d}\to X(3872)P $  .