Electroweak (EW) amplitudes in the gauge-Goldstone five-component formalism have a distinctive property: gauge symmetry is imprinted in the amplitudes, manifested as the massive Ward identity (MWI) $ k^M{\cal{M}}_M=0 $ . In this study, we used the HELAS package to numerically study gauge symmetry in EW amplitudes. First, we directly tested gauge symmetry by examining the MWI of amplitudes. Second, we modified the couplings within a vertex and among vertices to check if and how the MWI changes. Third, we tested gauge symmetry by considering the couplings modified by operators from the standard model effective field theory (SMEFT). Similar to the standard model, there are relations between different couplings that are protected by gauge symmetry. We observed that, if we modify the couplings to deviate from the relations, the MWI is violated. In contrast, the MWI is restored when the relations between couplings reduce to those in the SMEFT.

China

Wang-Fa Li

Junmou Chen

Qian-Jiu Wang

Zhao-Huan Yu

Numerical study on the gauge symmetry of electroweak amplitudes * Junmou Chen is supported by National Natural Science Foundation of China (12205118), Wang-Fa Li, Qian-Jiu Wang, and Zhao-Huan Yu are supported by the Guangzhou Science and Technology Planning Project (2024A04J4026)

In this article, the electroweak transition form factors of $$\Lambda _b\rightarrow \Lambda _c$$ and $$\Xi _b\rightarrow \Xi _c$$ are analyzed within the framework of three-point QCD sum rules. In phenomenological side, all possible couplings of interpolating current to hadronic states are considered. In QCD side, the perturbative part and the contributions of vacuum condensates up to dimension 8 are also included. With the estimated form factors, we study the decay widths and branching ratios of semileptonic decays $$\Lambda _b\rightarrow \Lambda _cl\bar{\nu }_l$$ and $$\Xi _b\rightarrow \Xi _cl\bar{\nu }_l$$ ( $$l=e,\mu $$ and $$\tau $$ ). Our results for the branching ratios of $$\Lambda _b\rightarrow \Lambda _cl\bar{\nu }_l$$ are comparable with experimental data and the results from other collaborations. In addition, our prediction for the branching ratio of $$\Xi _b\rightarrow \Xi _cl\bar{\nu }_l$$ can provide a valuable reference for future experimental measurements.

Jie Lu

Guo-Liang Yu

Dian-Yong Chen

Zhi-Gang Wang

Bin Wu

Analysis of the semileptonic decays $$\Lambda _b\rightarrow \Lambda _cl\bar{\nu }_l$$ and $$\Xi _b\rightarrow \Xi _cl\bar{\nu }_l$$ in QCD sum rules

This paper presents a systematic study of hyperon non-leptonic two-body decays induced by light quark transitions, particularly the $ s \rightarrow u\bar{u}d $ process, within the framework of $\textit{SU}$(3) flavor symmetry. The effective weak Hamiltonian is decomposed into irreducible $\textit{SU}$(3) representations, including the 27-plet and octet components, and applied to analyze decays of octet and decuplet baryons and charmed baryons. Both the irreducible representation amplitude (IRA) approach and the topological diagrammatic analysis (TDA) are employed to construct decay amplitudes and constrain the parameter space. $\textit{SU}$(3) symmetry-breaking effects arising from the strange quark mass are incorporated systematically. A global fit to current experimental data allows us to extract form factors and predict branching ratios and asymmetry parameters for several decay channels. Our results demonstrate the predictive power of $\textit{SU}$(3) flavor symmetry while highlighting significant symmetry-breaking effects, especially in amplitudes related to the 27-plet. Notably, the $ \Sigma^+ \rightarrow p\pi^0 $ decay channel exhibits a deviation exceeding $ 1\sigma $ from experimental measurements, suggesting the possible presence of new decay mechanisms or contributions beyond the Standard Model. Besides, we also evaluate the size of symmetry breaking. However, the large uncertainty of the experimental data makes it difficult to precisely determine the value of symmetry breaking. We strongly recommend that future experimental efforts aim to reduce the measurement uncertainties, especially for the processes $ \Lambda^0\to p\pi^- $ and $ \Lambda^0\to n\pi^0 $ , which have larger experimental errors compared to other data and play an important role in determining the symmetry breaking parameter. This study provides a systematic framework for future tests of the Standard Model and the search for new physics in hyperon decays.

Xin Wu

Qi Chen

Ye Xing

Zhi-Peng Xing

Ruilin Zhu

$\textit{SU}$(3) flavor symmetry analysis of hyperon non-leptonic two body decays * Ruilin Zhu is supported by NSFC (12322503, 12075124) and the Natural Science Foundation of Jiangsu (BK20211267). Zhi-Peng Xing is supported by NSFC (12375088, 12335003, 12405113). Ye Xing is supported by NSFC (12005294)

In this article, the vector, axial vector and tensor form factors of <italic>B<inf>c</inf></italic> to <italic>D</italic>-wave charmonia <italic>&#968;<inf>J</inf></italic> ( <math altimg='si138.svg'><mrow><mi>J</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn></mrow></math>) and <italic>&#951;</italic> <inf><italic>c</italic>2</inf> are analyzed within the framework of three-point QCD sum rules. With the predicted vector and axial vector form factors, we study the decay widths and branching ratios of semileptonic decays <math altimg='si139.svg'><mrow><msub><mi>B</mi><mi>c</mi></msub><mo>&#8594;</mo><msub><mi>&#968;</mi><mi>J</mi></msub><mi>l</mi><mover accent='true'><msub><mi>&#957;</mi><mi>l</mi></msub><mo>&#175;</mo></mover></mrow></math> and <math altimg='si140.svg'><mrow><msub><mi>B</mi><mi>c</mi></msub><mo>&#8594;</mo><msub><mi>&#951;</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mi>l</mi><mover accent='true'><msub><mi>&#957;</mi><mi>l</mi></msub><mo>&#175;</mo></mover></mrow></math>. Our results indicate that the semileptonic decay branching ratios of <italic>B<inf>c</inf></italic> to <italic>D</italic>-wave charmonia decreases as the total angular momentum of the final state <italic>D</italic>-wave charmonia increases. For decay processes <math altimg='si141.svg'><mrow><msub><mi>B</mi><mi>c</mi></msub><mo>&#8594;</mo><msub><mi>&#968;</mi><mn>1</mn></msub><mi>l</mi><msub><mover accent='true'><mi>&#957;</mi><mo>&#175;</mo></mover><mi>l</mi></msub></mrow></math>, <math altimg='si142.svg'><mrow><msub><mi>B</mi><mi>c</mi></msub><mo>&#8594;</mo><msub><mi>&#968;</mi><mn>2</mn></msub><mrow><mo>[</mo><msub><mi>&#951;</mi><mrow><mi>c</mi><mn>2</mn></mrow></msub><mo>]</mo></mrow><mi>l</mi><msub><mover accent='true'><mi>&#957;</mi><mo>&#175;</mo></mover><mi>l</mi></msub></mrow></math> and <math altimg='si143.svg'><mrow><msub><mi>B</mi><mi>c</mi></msub><mo>&#8594;</mo><msub><mi>&#968;</mi><mn>3</mn></msub><mi>l</mi><msub><mover accent='true'><mi>&#957;</mi><mo>&#175;</mo></mover><mi>l</mi></msub></mrow></math>, the branching ratios can reach the order of magnitude <math altimg='si100.svg'><msup><mn>10</mn><mrow><mo>&#8722;</mo><mn>3</mn></mrow></msup></math>, <math altimg='si144.svg'><msup><mn>10</mn><mrow><mo>&#8722;</mo><mn>4</mn></mrow></msup></math> and <math altimg='si145.svg'><msup><mn>10</mn><mrow><mo>&#8722;</mo><mn>5</mn></mrow></msup></math>, respectively, which can provide a valuable reference for future experimental measurements for <italic>B<inf>c</inf></italic> meson by LHCb collaboration. Furthermore, these results also can provide useful information to study the properties of <italic>B<inf>c</inf></italic> meson and <italic>D</italic>-wave charmonia.

Systematic analysis of the transition form factors of <italic>B<inf>c</inf></italic> to <italic>D</italic>-wave charmonia and corresponding semileptonic decays

In the framework of the relativized quark model, the mass spectra of the doubly heavy baryons are rigorously calculated in the three-quark system under the heavy-quark dominance mechanism, by using the Gaussian expansion method and the infinitesimally-shifted Gaussian basis functions. With the obtained mass spectra of all doubly heavy baryon families, the contribution of each Hamiltonian term to the energy levels is analyzed. It is found that the spin splitting is mainly determined by the spin-dependent interactions associated with the light quark. Moreover, it is shown that the spin splitting evolves regularly with the mass of heavy quarks by the evolution of the spectral structure, which is consistent with the heavy quark symmetry. Meanwhile, the orbital excitation is dominated by the $$\rho $$ -mode, which is different from that of the singly heavy baryons. At last, our analysis indicates that the $$\Xi _{cc}^{+}(3520)$$ state should not exist truly and the $$\Xi _{cc}^{++}(3621)$$ should be the true ground state with $$J^{P}$$ = $$\frac{1}{2}^{+}$$ . It is recommended to design the corresponding experiments to search for the $$\Xi _{cc}^{*}$$ in the energy range from 3694 to 3714 MeV.

Zhen-Yu Li

Jian-Zhong Gu

Hong-Tao Shen

Mass spectra of doubly heavy baryons in the relativized quark model with heavy-quark dominance

The rigorous calculation of the spin-orbit terms in a three-quark system is realized based on the Gaussian expansion method and the infinitesimally-shifted Gaussian basis functions in the framework of the relativized quark model, by ignoring the mixing between different excited states. Then, the complete mass spectra of the singly heavy baryons are obtained rigorously, under the heavy-quark dominance mechanism. Accordingly, systematical analyses are performed for the reliability and predictive power of the model, the fine structure of the singly heavy baryon spectra, the assignments of the excited baryons, and some important topics about heavy baryon spectroscopy, such as the missing states, 'spin-orbit puzzle,' and clustering effect. The results confirm that, under the heavy-quark dominance mechanism, the relativized quark model can describe the excitation spectra and the fine structures of the singly heavy baryons correctly and precisely.

Mass spectra of singly heavy baryons in the relativized quark model with heavy-quark dominance * Supported by the Open Project of Guangxi Key Lab of Nuclear Physics and Technology (NLK2023-04), the Central Government Guidance Funds for Local Scientific and Technological Development in China (Guike ZY22096024), the Natural Science Foundation of Guizhou Province-ZK[2024](General Project)650, the National Natural Science Foundation of China (11675265, 12175068), the Continuous Basic Scientific Research Project (WDJC-2019-13) and the Leading Innovation Project (LC 192209000701)

In this study, we extend our previous study on the $ D^*\bar{D}^* $ molecular states with $ J^{PC}=0^{++} $ , $ 1^{+-} $ , and $ 2^{++} $ to investigate their two-body strong decays via the QCD sum rules based on rigorous quark-hadron duality. We obtain the partial decay widths and, therefore, the total widths of the ground states with $ J^{PC}=0^{++} $ , $ 1^{+-} $ , and $ 2^{++} $ , which indicate that it is reasonable to assign $ X_2(4014) $ as the $ D^*\bar{D}^* $ tetraquark molecular states with $ J^{PC}=2^{++} $ .

Tao Hong

Xiao-Song Yang

Two-body strong decays of the hidden-charm tetraquark molecular states via QCD sum rules * Supported by the National Natural Science Foundation (12175068)

This study is devoted to calculating the form factors of $B_c \to D^{*}$ , $B_c \to D$ , $B_c \to D_s^{*}$ , and $B_c \to D_s$ transitions in the framework of three-point QCD sum rules. At the QCD side, the contributions of $\langle\overline{q}q\rangle$ , $\langle\overline{q}g_{s}\sigma Gq\rangle$ , $\langle g_{s}^{2}G^{2}\rangle$ , $\langle g^{3}f_{abc}G^{3}\rangle$ , and $\langle\overline{q}q\rangle \langle g_{s}^{2}G^{2}\rangle$ are taken into account. With the obtained form factors, the decay widths and branching ratios of several two-body nonleptonic decay processes, $B_c \to \eta_c D^{*}$ , $\eta_c D$ , $ J/\psi D^{*}$ , $ J/\psi D$ , $\eta_c D_s^{*}$ , $\eta_c D_s$ , $J/\psi D_s^{*}$ , and $J/\psi D_s$ , are predicted. These results on the form factors and decay properties of the $B_c$ meson provide useful information for us to study the heavy-quark dynamical behavior.

Ze Zhou

Analysis of the form factors of B$_{c}$ → $\textit{D}$ $^{(*)}$, $\textit{D}$ $\textit{s}$ $^{(*)}$ and relevant nonleptonic decays * Supported by the National Natural Science Foundation (12175068) and Natural Science Foundation of HeBei Province (A2024502002)

In this article, we firstly analyze the mass and pole residue of negative parity nucleon $$N^*(1535)$$ within the two-point QCD sum rules. Basing on these results, we continuously study the strong coupling constants of vertices $$\Lambda _cDN^*$$ , $$\Lambda _cD^*N^*$$ , $$\Lambda _bBN^*$$ and $$\Lambda _bB^*N^*$$ in the framework of three-point QCD sum rules. At hadron side, all possible couplings of interpolating current to hadronic states are considered. At QCD side, the contributions of vacuum condensate terms $$\langle \bar{q}q\rangle $$ , $$\langle g_s^2GG\rangle $$ , $$\langle \bar{q} g_s\sigma Gq\rangle $$ , $$\langle \bar{q}q\rangle ^2$$ and $$g_s^2\langle \bar{q}q\rangle ^2$$ are also considered. By setting the four momentum of $$D^{(*)}[B^{(*)}]$$ mesons off-shell, the strong coupling constants in deep space-like regions ( $$Q^2=-q^2\gg \Lambda _{QCD}^2$$ ) are obtained. Then, the momentum dependent coupling constants in space-like regions are fitted into analytical function $$G(Q^2)$$ and are extrapolated into time-like regions ( $$Q^2<0$$ ). Finally, the on-shell values of strong coupling constants are obtained by taking $$Q^{2}=-m_{D^{(*)}[B^{(*)}]}^2$$ . The results are $$G_{\Lambda _cDN^*}(Q^2=-m_D^2)=4.06^{+0.96}_{-0.75}$$ , $$f_{\Lambda _cD^*N^*}(Q^2=-m_{D^*}^2)=3.73^{+0.68}_{-0.16}$$ , $$g_{\Lambda _cD^*N^*}(Q^2=-m_{D^*}^2)=9.22^{+3.16}_{-0.36}$$ , $$G_{\Lambda _bBN^*}(Q^2=-m_B^2)=9.11^{+1.54}_{-1.61}$$ , $$f_{\Lambda _bB^*N^*}(Q^2=-m_{B^*}^2)=8.55^{+2.69}_{-2.21}$$ and $$g_{\Lambda _bB^*N^*}(Q^2=-m_{B^*}^2)=-\,0.25^{+0.16}_{-0.01}$$ .

Analysis of the strong vertices $$\Lambda _cD^{(*)}N^*(1535)$$ and $$\Lambda _bB^{(*)}N^*(1535)$$ in QCD sum rules

We analyzed two-loop planar contributions to a three-body form factor at next-to-leading power in the high-energy limit, where the masses of the external particles are much smaller than their energies. Calculations were performed by exploiting differential equations for the expansion coefficients, both to facilitate linear relations among them and to derive their analytic expressions. The results are expressed in terms of generalized polylogarithms involving a few simple symbol letters. Our method can readily be applied to calculations of non-planar contributions as well. Our results provide crucial information for establishing sub-leading factorization theorems for massive scattering amplitudes in the high-energy limit.

Yishuai Guo

Zhi-Feng Liu

Mingming Lu

Tianya Xia

Li-Lin Yang

A three-body form factor at sub-leading power in the high-energy limit: planar contributions * Supported in part by the National Natural Science Foundation of China Sunder (12375097, 12347103) and the Fundamental Research Funds for the Central Universities. Zhi-Feng Liu was supported by the China Postdoctoral Science Foundation (2023M733123, 2023TQ0282) and the Postdoctoral Fellowship Program of China Postdoctoral Science Foundation