<p>Within the framework of nonrelativistic quantum chromodynamics factorization, we calculate the next-to-leading order (NLO) perturbative QCD corrections to the form factors for the semileptonic decays of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>J</mi><mo>/</mo><mi>ψ</mi></math> via (axial-)vector and (axial-)tensor currents. We obtain the complete analytical results for the form factors up to NLO, and provide their asymptotic expressions in the hierarchical heavy quark limit. The NLO corrections are found to be both significant and convergent in the relatively small squared transfer momentum (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math>) region, while also reducing the dependence on the renormalization scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math>. Finally, the theoretical predictions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> form factors over the full <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> range are provided.</p>

China

Qin Chang

Wei Tao

Zhen-Jun Xiao

Ruilin Zhu

Next-to-leading order QCD corrections to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>B</mi><mi>c</mi><mo>*</mo></msubsup><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi></math> form factors

In this paper, we study the $$B^{0} \rightarrow J/\psi h_{1}$$ decays for the first time by using perturbative QCD approach up to the presently known next-to-leading order accuracy. The vertex corrections present significant contribution to the amplitude. In the calculation, the mixing between two light axial-vector mesons $$h_{1}(1170)$$ and $$h_{1}(1415)$$ are also studied in detail. The observables including the branching ratios, polarization fractions and asymmetries are predicted and discussed explicitly. It is found that the $$B^{0} \rightarrow J/\psi h_{1}$$ decays have relatively large branching fractions, which are generally at the order of $$\mathcal{O}(10^{-6}\sim 10^{-3}),$$ and thus are possible to be observed by the LHCb and Belle-II experiments in the near future. Moreover, they are very sensitive to the mixing angle $$\theta $$ and can be used to test the values of $$\theta .$$ In addition, some ratios between the branching fractions of $$B^{0} \rightarrow J/\psi h_{1}$$ decays can provide much stronger constraints on $$\theta $$ due to their relatively small theoretical errors. The $$B^{0} \rightarrow J/\psi h_{1}$$ decays are generally dominated by the longitudinal polarization contributions, specifically, $$f_{L}(B^{0} \rightarrow J/\psi h_{1})>80\%,$$ except for the case that $$\theta \sim 35^\circ $$ and $$-55^\circ .$$ Unfortunately, the direct asymmetries of $$B^{0} \rightarrow J/\psi h_{1}$$ decays are too small to be observed soon even if the effect of $$\theta $$ is considered. The future precise measurements on $$B^{0} \rightarrow J/\psi h_{1}$$ decays are expected for testing these theoretical findings and exploring the interesting nature of $$h_{1}(1170)$$ and $$h_{1}(1415).$$

De-Hua Yao

Xin Liu

Studying the $$B^{0} \rightarrow J/\psi h_{1}$$ decays with $$h_{1}(1170){-}h_{1}(1415)$$ mixing in the perturbative QCD approach

In this work, we calculated the form factors of the weak decay process $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+$$ , where the final charm baryon represents an excited state with spin-parity $$\frac{1}{2}^-$$ . Utilizing the light-cone QCD sum rules approach, we incorporated the contributions of the lowest two charm baryon states: the ground state $$\varLambda _c$$ with $$J^P=\frac{1}{2}^+$$ and the excited state $$\varLambda _c(2595)^+$$ with $$J^P=\frac{1}{2}^-$$ in the hadronic representation of the $$\varLambda _b \rightarrow \varLambda _c(2595)^+$$ transition correlation function. This approach allows us to extract the form factors of the $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+$$ from $$\varLambda _b^0 \rightarrow \varLambda _c^+$$ transition. During the light-cone QCD sum rules procedure, we employed the light-cone distribution amplitudes (LCDAs) of the $$\varLambda _b$$ baryon. Furthermore, by combining these form factors with the helicity amplitudes of the bottom baryon transition matrix elements, we calculated the differential decay widths for the processes $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+\ell ^-\bar{\nu }_\ell $$ and provided the optimal choice of the interpolating current for $$\varLambda _c$$ in this process. Additionally, within the lifetime of $$\varLambda _b^0$$ , we obtained the absolute branching fractions for the semileptonic decays $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+ \ell ^- \bar{\nu }_\ell $$ . With the branching fractions of $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+ \ell ^- \bar{\nu }_\ell $$ calculated in this work, we also determined the parameter $$\mathcal {R}(\varLambda _c(2595)^+)$$ which tests the lepton flavor universality. This parameter is defined as the ratio of branching fractions $$\mathcal {B}r(\varLambda _b^0 \rightarrow \varLambda _c(2595)^+\tau ^-\bar{\nu }_\tau )$$ and $$\mathcal {B}r(\varLambda _b^0 \rightarrow \varLambda _c(2595)^+\mu ^-\bar{\nu }_\mu )$$ . Our results provide a valuable theoretical test for these decay channels and offer insights into the LCDAs of bottom baryons, paving the way for further in-depth investigations.

Hui-Hui Duan

Yong-Lu Liu

Ming-Qiu Huang

Form factors of $$\varLambda _b^0 \rightarrow \varLambda _c(2595)^+$$ within light-cone QCD sum rules

In this paper, the nonfactorizable photonic corrections to the effective four-fermion vertex for the $$\overline{B}$$ $${\rightarrow }$$ $$D^{({*})}$$ $$+$$ $${\ell }^{-}$$ $$+$$ $$\bar{\nu }_{\ell }$$ decays are considered at the quark level within SM, where $${\ell }$$ $$=$$ , $${\mu }$$ and $${\tau }$$ . It is found that the QED corrections are closely related with the mass of the charged lepton. The QED contributions will enhance branching ratios, which results in the slight reduction of ratios $$R(D^{(*)})$$ and the relationship $$R(D^{(*)})_{e}$$ < $$R(D^{(*)})_{\mu }$$ .

Yueling Yang

Liting Wang

Jiazhi Li

Junfeng Sun

The QED nonfactorizable correction to the semileptonic charmed three-body

The recently discovered stellar system, Ursa Major III/UNIONS 1, may be the faintest and densest dwarf spheroidal satellite galaxy of the Milky Way. Owing to its close proximity and substantial dark matter (DM) component, Ursa Major III emerges as a highly promising target for DM indirect detection. It is known that electrons and positrons originating from DM annihilation can generate a broad radio spectrum through the processes of synchrotron radiation and inverse Compton scattering within galaxies. In this study, we investigate the potential of the Square Kilometre Array (SKA) in detecting radio signatures arising from DM annihilation in Ursa Major III over a 100 hour observation period. Our analysis indicates that the SKA has strong capabilities in detecting these signatures. For instance, the SKA sensitivity to the DM annihilation cross section is estimated to reach <math><mrow><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>30</mn></mrow></msup><mo>)</mo><mo>−</mo><mi>O</mi><mo>(</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>28</mn></mrow></msup><mo>)</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>cm</mi></mrow><mrow><mn>3</mn></mrow></msup><mtext> </mtext><msup><mrow><mi>s</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math> in the DM mass range from several GeV to <math><mo>∼</mo><mn>100</mn><mtext> </mtext><mtext> </mtext><mi>GeV</mi></math> for the <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> and <math><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup></math> annihilation channels. The precise results are significantly influenced by various astrophysical factors, such as the strength of magnetic field, the diffusion coefficient, and the DM density profile in the dwarf galaxy. We discuss the impact of the uncertainties associated with these factors, and find that the SKA sensitivities have the potential to surpass the current constraints, even when considering these uncertainties.

Peng-Long Zhang

Xiao-Jun Bi

Peng-Fei Yin

Yi Zhao

SKA sensitivity to potential radio emission from dark matter annihilation in Ursa Major III

Based on the fitting results of the LHCb collaboration on the contributions of various intermediate resonances to the  $$B^{+}\rightarrow \pi ^{+}\pi ^{+}\pi ^{-}$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   decay, we calculate the branching fractions and localized CP asymmetries of the quasi-two-body  $$B^{+}\rightarrow \pi ^{+} \left( \rho (770), \omega (782), \rho (1450), f_{2}\left( 1270\right) \rightarrow \right) \pi ^{+}\pi ^{-}$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mfenced> <mi>ρ</mi> <mrow> <mo>(</mo> <mn>770</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>ω</mi> <mrow> <mo>(</mo> <mn>782</mn> <mo>)</mo> </mrow> <mo>,</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <mn>1450</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mfenced> <mn>1270</mn> </mfenced> <mo>→</mo> </mfenced> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   decays within the perturbative QCD (PQCD) approach systematically. Our theoretical predictions for the branching fractions agree well with the data within the errors. In order to further test the framework of the three-body B meson decays in PQCD and the wave functions of  $$\pi \pi $$  <math> <mrow> <mi>π</mi> <mi>π</mi> </mrow> </math>   pair, we also calculate the branching fractions of the corresponding two-body  $$B^{+}\rightarrow \pi ^{+}\rho (770)$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ρ</mi> <mrow> <mo>(</mo> <mn>770</mn> <mo>)</mo> </mrow> </mrow> </math>  ,  $$B^{+}\rightarrow \pi ^{+}\omega (782)$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mi>ω</mi> <mrow> <mo>(</mo> <mn>782</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$B^{+}\rightarrow \pi ^{+}f_{2}\left( 1270\right) $$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <msub> <mi>f</mi> <mn>2</mn> </msub> <mfenced> <mn>1270</mn> </mfenced> </mrow> </math>   decays under the narrow-width-approximation, which are in consistence with the experimental data. The direct CP asymmetries of  $$B^{+}\rightarrow \pi ^{+}(\rho (770/1450)\rightarrow ) \pi ^{+}\pi ^{-}$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <mi>ρ</mi> <mrow> <mo>(</mo> <mn>770</mn> <mo>/</mo> <mn>1450</mn> <mo>)</mo> </mrow> <mo>→</mo> <mo>)</mo> </mrow> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   decays are found to be very small because these decay modes are tree-dominated. However, due to the large penguin contributions from the chiral enhanced annihilation diagrams,  $$B^{+}\rightarrow \pi ^{+} (f_{2}\left( 1270\right) \rightarrow )\pi ^{+}\pi ^{-}$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mn>2</mn> </msub> <mfenced> <mn>1270</mn> </mfenced> <mo>→</mo> <mo>)</mo> </mrow> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   decay has a large direct CP asymmetry, which is also discovered by LHCb collaboration. A relatively large CP asymmetry is also expected in the  $$B^+\rightarrow \pi ^+ (\omega \rightarrow )\pi ^+\pi ^-$$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mrow> <mo>(</mo> <mi>ω</mi> <mo>→</mo> <mo>)</mo> </mrow> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mrow> </math>   decay occurring via  $$\rho -\omega $$  <math> <mrow> <mi>ρ</mi> <mo>-</mo> <mi>ω</mi> </mrow> </math>   mixing, which would be measured LHCb and Belle-II experiments in future.

Lei Yang

Zhi-Tian Zou

Ying Li

Study of the  $$B^{+}\rightarrow \pi ^{+}\left( \pi ^{+}\pi ^{-}\right) $$  <math> <mrow> <msup> <mi>B</mi> <mo>+</mo> </msup> <mo>→</mo> <msup> <mi>π</mi> <mo>+</mo> </msup> <mfenced> <msup> <mi>π</mi> <mo>+</mo> </msup> <msup> <mi>π</mi> <mo>-</mo> </msup> </mfenced> </mrow> </math>   decay in PQCD approach

Kinematics restricts the ability of rare charm decays to explore the charged lepton-flavor-violation processes mediated by the quark-level <math><mi>c</mi><mo>→</mo><mi>u</mi><mo>ℓ</mo><mi>τ</mi></math> transition. To fill the gap, we propose exploring new physics (NP) through the quasielastic scattering process <math><msup><mi>e</mi><mo>−</mo></msup><mi>p</mi><mo>→</mo><msup><mi>τ</mi><mo>−</mo></msup><msub><mi>Λ</mi><mi>c</mi></msub></math> and the polarization of the <math><mi>τ</mi></math> lepton. As analyzing modes for the <math><mi>τ</mi></math> polarization, we consider the decays <math><msup><mi>τ</mi><mo>−</mo></msup><mo>→</mo><msup><mi>π</mi><mo>−</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></math>, <math><msup><mi>τ</mi><mo>−</mo></msup><mo>→</mo><msup><mi>ρ</mi><mo>−</mo></msup><msub><mi>ν</mi><mi>τ</mi></msub></math>, and <math><msup><mi>τ</mi><mo>−</mo></msup><mo>→</mo><msup><mo>ℓ</mo><mo>−</mo></msup><msub><mover><mi>ν</mi><mo>¯</mo></mover><mo>ℓ</mo></msub><msub><mi>ν</mi><mi>τ</mi></msub></math>, and show that the <math><mi>τ</mi></math> polarization components can be extracted from analyzing the kinematics of the <math><mi>τ</mi></math> visible decay products. In the framework of a general low-energy effective Lagrangian, we then perform a detailed analysis of the polarization components in various aspects and scrutinize possible NP signals. With one upcoming experimental setup, we finally demonstrate promising event rate can be expected for the cascade process and, even in the worst-case scenario—no signals are observed at all—it can still provide a competitive potential for constraining the NP, compared with those from the high-<math><msub><mi>p</mi><mi>T</mi></msub></math> dilepton invariant mass tails at high-energy colliders.

Xin-Shuai Yan

Liang-Hui Zhang

Ya-Dong Yang

Probing new physics with polarization components of the tau lepton in quasielastic <math><msup><mi>e</mi><mo>−</mo></msup><mi>p</mi><mo>→</mo><msub><mi>Λ</mi><mi>c</mi></msub><msup><mi>τ</mi><mo>−</mo></msup></math> scattering process

Inspired by the brilliant prospects of the ongoing B meson experiments, the hadronic charmless B  $${\rightarrow }$$  <math> <mo>→</mo> </math>   SS decays are studied by considering the next-to-leading (NLO) contributions with the QCD factorization approach, where S denotes the scalar mesons  $$K_{0}^{*}(1430)$$  <math> <mrow> <msubsup> <mi>K</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$a_{0}(1450)$$  <math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1450</mn> <mo>)</mo> </mrow> </mrow> </math>  . Branching ratios and CP violating asymmetries are estimated with the updated values of hadronic parameters obtained from a covariant light-front quark model, for two scenarios where the scalar mesons are the  $$1^{3}P_{0}$$  <math> <mrow> <msup> <mn>1</mn> <mn>3</mn> </msup> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math>   and  $$2^{3}P_{0}$$  <math> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> <msub> <mi>P</mi> <mn>0</mn> </msub> </mrow> </math>   states. It is found that the NLO contributions are very important for the B  $${\rightarrow }$$  <math> <mo>→</mo> </math>   SS decays. For the B  $${\rightarrow }$$  <math> <mo>→</mo> </math>    $$a_{0}(1450)K_{0}^{*}(1430)$$  <math> <mrow> <msub> <mi>a</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mn>1450</mn> <mo>)</mo> </mrow> <msubsup> <mi>K</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>   and  $$B_{s}$$  <math> <msub> <mi>B</mi> <mi>s</mi> </msub> </math>    $${\rightarrow }$$  <math> <mo>→</mo> </math>    $$K_{0}^{*}(1430)\overline{K}_{0}^{*}(1430)$$  <math> <mrow> <msubsup> <mi>K</mi> <mrow> <mn>0</mn> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <msubsup> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mrow> <mn>0</mn> </mrow> <mrow> <mrow></mrow> <mo>∗</mo> </mrow> </msubsup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> </mrow> </math>   decays, branching ratios can reach up to the order of  $${\mathcal {O}}(10^{-5})$$  <math> <mrow> <mi>O</mi> <mo>(</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> <mo>)</mo> </mrow> </math>   by assuming that the scalar mesons are the 1P states, and should first be investigated in the future experiments.

Lili Chen

Mengfei Zhao

Yueyang Kang

Study of the nonleptonic charmless  $$B \rightarrow SS$$  <math> <mrow> <mi>B</mi> <mo>→</mo> <mi>S</mi> <mi>S</mi> </mrow> </math>   decays with the QCD factorization approach

The ground vector   $ B_{c}^{\ast} $   meson has not yet been experimentally discovered until now. Besides the dominant electromagnetic decays, nonleptonic weak decays provide another choice to search for the mysterious   $ B_{c}^{\ast} $   mesons. Inspired by the potential prospects of   $ B_{c}^{\ast} $   mesons in future high-luminosity colliders, nonleptonic   $ B_{c}^{\ast} $   weak decays induced by bottom and charm quark decays are studied within the SM by using a naive factorization approach. It is found that for   $ B_{c}^{\ast} $     $ B_{c}^{\ast} $     $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $  ,   $ B_{c}^{\ast} $   and   $ B_{c}^{\ast} $   decays, a few hundred and even thousands of events might be observable in CEPC, FCC-ee and LHCb@HL-LHC experiments.

Jinshu Huang

Possibility of experimental study on nonleptonic Weak Decays  

The current knowledge on the <math><msup><mi>D</mi><mo>*</mo></msup></math> mesons is still inadequate. Encouraged by the positive development prospects of high-luminosity and high-precision experiments, the Cabibbo-favored nonleptonic <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>→</mo><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>, <math><msup><mover><mi>K</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mi>π</mi><mo>+</mo></msup></math>, <math><mover><mi>K</mi><mo>¯</mo></mover><msup><mi>ρ</mi><mo>+</mo></msup></math> weak decays are studied with the naive factorization approach. It is found that branching ratios of these processes can reach up to <math><mi>O</mi><mo>(</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>10</mn></mrow></msup><mo>)</mo></math> or more, and can be accessible at STCF, CEPC, FCC-ee, and LHCb@HL-LHC experiments in the future. It might even be possible to search for the <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo><mn>0</mn></mrow></msup><mo>→</mo><msup><mrow><mi>K</mi></mrow><mrow><mo>*</mo><mo>−</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> and <math><msup><mi>K</mi><mo>−</mo></msup><msup><mi>ρ</mi><mo>+</mo></msup></math> decays at the running SuperKEKB experiments.

Kang Li

Zhenglin Li

Feasibility of searching for the Cabibbo-favored <math><mrow><msup><mrow><mi>D</mi></mrow><mrow><mo>*</mo></mrow></msup><mo>→</mo><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>, <math><mrow><msup><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mo>*</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math>, <math><mrow><mover><mrow><mi>K</mi></mrow><mrow><mo>¯</mo></mrow></mover><msup><mrow><mi>ρ</mi></mrow><mrow><mo>+</mo></mrow></msup></mrow></math> decays