<p>In this article, we solve the instantaneous Bethe-Salpeter equation with Cornell potential and Coulomb potential and conduct a meticulous study of the mass spectrum and wave function of toponium. Our investigation reveals that, owing to the exceedingly heavy mass of the top quark, the mass splitting between singlet and triplet states, as well as within the triplet states, is negligible. Consequently, relativistic corrections can be safely disregarded in the study of toponium. As such, we present the nonrelativistic wave functions for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi></math>-wave, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>P</mi></math>-wave, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>D</mi></math>-wave toponia and study the decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">→</mo><mi>g</mi><mi>g</mi></mrow></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Θ</mi><mo stretchy="false">→</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math>.</p>

China

Guo-Li Wang

Tai-Fu Feng

Ya-Qian Wang

Mass spectra and wave functions of toponia

We have studied the impact of CP-violating (CPV) effects on electroweak baryogenesis (EWBG) and the electric dipole moment (EDM) of electron ( $$d_e$$ ), the electric dipole moment (EDM) of mercury neutron ( $$d_n$$ ) and the electric dipole moment (EDM) of ( $$d_{Hg}$$ ) in an extension of the Minimal Supersymmetric Standard Model (MSSM). The model incorporating a (2) triplet with hypercharges of $$\pm 1$$ and a gauge singlet from the standard model that is mutually coupled, is collectively referred to as the next-to-minimal supersymmetric standard model with triplets (TNMSSM). Furthermore, we discuss the strong first-order phase transition (PT) achieved by this model via tree-level effects. The numerical results indicate that the TNMSSM can account for the observed baryon asymmetry. Additionally, the regions favored by EWBG can be compatible with the corresponding electron EDM bounds when different contributions to the $$d_e$$ cancel each other out.

Xiang Yang

Ming-Hui Guo

Jin-Lei Yang

Qing-Hua Li

Electroweak baryogenesis and electron EDM in the TNMSSM

Basing on Mellin-Barnes representations and Miller’s transformation, we present the Gel’fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems of 4-loop vacuum Feynman integrals with arbitrary masses. Through the GKZ hypergeometric systems, the analytical hypergeometric solutions of 4-loop vacuum Feynman integrals with arbitrary masses can be obtained in neighborhoods of origin including infinity. The analytical expressions of Feynman integrals can be formulated as a linear combination of the fundamental solution systems in certain convergent region, which the combination coefficients can be determined by the integral at some regular singularities, the Mellin-Barnes representation of the integral, or some mathematical methods.

Hai-Bin Zhang

GKZ hypergeometric systems of the four-loop vacuum Feynman integrals

<p>In the framework of the minimal supersymmetric model extension with new scalar triplets and singlet (TNMSSM), we analyze the electric dipole moment (EDM) of neutrons (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>n</mi></msub></math>), electrons EDM(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>e</mi></msub></math>), mercury EDM(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mrow><mi>H</mi><mi>g</mi></mrow></msub></math>), and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi></math> quarks (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>b</mi></msub></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>c</mi></math> quarks (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>c</mi></msub></math>) by considering the contributions from the one-loop diagrams, some two-loop diagrams, and the Weinberg operators. The effects of TNMSSM specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math>-violating sources <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>n</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>e</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mrow><mi>H</mi><mi>g</mi></mrow></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>b</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>c</mi></msub></math> are specialized; it is found that they have significant contributions to these EDMs, and the current upper bounds on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>n</mi></msub></math> impose strict constraints on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></math>. The theoretical predictions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>b</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>c</mi></msub></math> can reach about <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>22</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi mathvariant="normal">e</mi><mo>·</mo><mi>cm</mi></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>23</mn></mrow></msup><mtext> </mtext><mtext> </mtext><mi mathvariant="normal">e</mi><mo>·</mo><mi>cm</mi></mrow></math> respectively by taking the upper bounds on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>n</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mi>e</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>d</mi><mrow><mi>H</mi><mi>g</mi></mrow></msub></math> into account, which have great potential to be observed in the future.</p>

Qing-hua Li

Electric dipole moments from the perspective of a scalar triplet and singlet extension of the MSSM: A study of neutrons, electrons, mercury, and b and c quarks

Embedding Feynman integrals in Grassmannians, we can write Feynman integrals as some finite linear combinations of generalized hypergeometric functions. In this paper, we present a general method to obtain Gauss relations among those generalized hypergeometric functions. The hypergeometric expressions of Feynman integral are continued from a connected component to another by the inverse Gauss relations, then continued to the whole domain of definition by the Gauss-Kummer relations. The Laurent series of the Feynman integral around the space-time dimension <math><mi>D</mi><mo>=</mo><mn>4</mn></math> is obtained by the Gauss adjacent relations, where the coefficient of the term with power of <math><mi>D</mi><mo>−</mo><mn>4</mn></math> is given as the linear combinations of hypergeometric functions with integer parameters. As examples, we illustrate how to obtain the expressions for the Feynman integrals of the 1-loop self-energy and a 2-loop massless triangle diagram in the domains of definition.

Yang Zhou

Gauss relations in Feynman integrals

In an extension of MSSM with two triplets and a singlet, called the TNMSSM, there are seven neutralinos which can enrich the study of cold dark matter if one expects that the weakly interacting massive particle (WIMP) is responsible for the observation of Planck satellite. Such a model, compared to the MSSM, can naturally offer a solution to the  $$\mu $$  <math> <mi>μ</mi> </math>   problem, and its lightest neutralino, which is bino-like, can also provide a correct relic density by using the coannihilation mechanism due to the newly added triplinos. Taking into account the related experimental measurements, such as the bound on the SM-like Higgs mass, the B meson rare decays, the anomalous magnetic moment of the muon  $$a_\mu $$  <math> <msub> <mi>a</mi> <mi>μ</mi> </msub> </math>  , the Large Hadron Collider (LHC) measurements and the latest dark matter direct detection experiment LUX-ZEPLIN (LZ), the TNMSSM parameter space can be strictly limited. In respect to all the constraints mentioned above, we find that a bino-like neutralino with a mass in the region  $$[100, 450]~\textrm{GeV}$$  <math> <mrow> <mo>[</mo> <mn>100</mn> <mo>,</mo> <mn>450</mn> <mo>]</mo> <mspace width="3.33333pt"></mspace> <mtext>GeV</mtext> </mrow> </math>   can successfully account for the correct dark matter relic density. Additionally, most of the viable parameter space can be tested in the near future experiments such as the Xenon-nT experiment or LHC.

Zhong-Jun Yang

Shu-Min Zhao

Xing-Gang Wu

Neutralino dark matter in the extension of MSSM with two triplets and singlet

The precise measurement of the W boson mass is closely related to the contributions of new physics (NP), which can significantly constrain the parameter space of NP models, particularly those with an additional   $ U(1) $   local gauge group. The inclusion of a new Z' gauge boson and gauge couplings in these models can contribute to the oblique parameters S, T, and U and the W boson mass at tree level. Considering the effects of kinetic mixing, in this study, we calculate and analyze the oblique parameters S, T, and U and the W boson mass in such NP models. We find that the kinetic mixing effects can make significant contributions to the W boson mass, and these contributions can be eliminated by redefining the gauge boson fields by removing neutral currents with charged leptons if the leptonic Yukawa couplings are invariant under the extra   $ U(1) $   local gauge group, even with nonzero kinetic mixing effects.

Zhao-Feng Ge

Xiu-Yi Yang

Sheng-Kai Cui

 boson mass in NP models with an extra (1) Gauge Group  

In the standard model, the charged lepton-flavor violation (CLFV) processes are forbidden, so the observation of CLFV represents a clear signal of new physics that goes beyond the standard model. In this work, we focus on the CLFV processes <math><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mi>γ</mi></mrow></math> and <math><mrow><msubsup><mrow><mi>l</mi></mrow><mrow><mi>j</mi></mrow><mrow><mo>−</mo></mrow></msubsup><mo>→</mo><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>−</mo></mrow></msubsup><msubsup><mrow><mi>l</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo>+</mo></mrow></msubsup></mrow></math> in the left-right supersymmetric standard model (LRSSM). Considering the constraints of the updated experimental data, the numerical results show that the new contributions of <math><mi>S</mi><mi>U</mi><msub><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow><mi>R</mi></msub></math> gauge interaction and a large number of other new particles, such as the <math><msup><mi>Z</mi><mo>′</mo></msup></math> boson and double-charged Higgs bosons in LRSSM can make significant contributions to the CLFV processes <math><msubsup><mi>l</mi><mi>j</mi><mo>−</mo></msubsup><mo>→</mo><msubsup><mi>l</mi><mi>i</mi><mo>−</mo></msubsup><mi>γ</mi></math> and <math><msubsup><mi>l</mi><mi>j</mi><mo>−</mo></msubsup><mo>→</mo><msubsup><mi>l</mi><mi>i</mi><mo>−</mo></msubsup><msubsup><mi>l</mi><mi>i</mi><mo>−</mo></msubsup><msubsup><mi>l</mi><mi>i</mi><mo>+</mo></msubsup></math>, which is much different from the ones predicted in other supersymmetry models. This work provides a theoretical base for finding the lepton-flavor violation phenomena in new physics.

Yong-Kang Huang

Lepton-flavor violation in the left-right supersymmetric standard model

In the U(1) X SSM, we delve into the quark flavor violation of h → bs, where h is identified as the SM-like Higgs boson discovered at the LHC. As the U(1) extension of the minimal supersymmetric standard model (MSSM), the U(1) X SSM has new super fields such as right-handed neutrinos and three Higgs singlets. We conduct a thorough analysis of the underlying mechanisms and parameter dependencies of h → bs in the U(1) X SSM. In the U(1) X SSM, we discover that the Br(h → bs) for the Higgs decay to bs could significantly differ from the expectation in the standard model (SM), depending on the values of the new parameters introduced in the model. Our research not only contributes to a deeper understanding of Higgs physics within the U(1) X SSM, but also provides valuable guidance for new physics (NP).

Song Gao

Ming-Yue Liu

Xing-Yu Han

Xi Wang

The Higgs boson decay h → bs in the U(1) X SSM

The average values of the powers of <math><mrow><mover><mrow><mo>|</mo><mover><mrow><mi>q</mi></mrow><mrow><mo>→</mo></mrow></mover><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>¯</mo></mrow></mover><mo>≡</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> and <math><mrow><mover><mrow><mo>|</mo><mover><mrow><mi>v</mi></mrow><mrow><mo>→</mo></mrow></mover><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>¯</mo></mrow></mover><mo>≡</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> are commonly used physical quantities, where <math><mover><mi>q</mi><mo>→</mo></mover></math> and <math><mover><mi>v</mi><mo>→</mo></mover></math> are the three-dimensional momentum and velocity of a quark inside a meson, respectively. For example, the average kinetic energy <math><msubsup><mi>μ</mi><mi>π</mi><mn>2</mn></msubsup><mo>=</mo><msup><mi>q</mi><mn>2</mn></msup></math> is an important quantity in the heavy quark effective theory, and the expansion of <math><msup><mi>v</mi><mi>n</mi></msup></math> is commonly used in the nonrelativistic quantum chromodynamics. In this paper, based on the instantaneous Bethe-Salpeter equation method, we calculate the average values <math><msup><mi>q</mi><mi>n</mi></msup></math> and <math><msup><mi>v</mi><mi>n</mi></msup></math> (<math><mi>n</mi><mo>=</mo><mn>1</mn></math>, 2, 3, 4) of a heavy quark inside <math><mi>S</mi></math> wave and <math><mi>P</mi></math> wave heavy-light mesons. We obtain the kinetic energy <math><mrow><msubsup><mrow><mi>μ</mi></mrow><mrow><mi>π</mi></mrow><mn>2</mn></msubsup><mo>=</mo><mn>0.455</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></mrow></math> for the <math><mi>B</mi></math> meson, which is consistent with the experimental result <math><mrow><mn>0.464</mn><mo>±</mo><mn>0.076</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></mrow></math>. For the <math><msub><mi>B</mi><mi>s</mi></msub></math>, <math><mi>D</mi></math>, and <math><msub><mi>D</mi><mi>s</mi></msub></math> mesons, the <math><msubsup><mi>μ</mi><mi>π</mi><mn>2</mn></msubsup></math> are 0.530, 0.317, and <math><mrow><mn>0.369</mn><mtext> </mtext><mtext> </mtext><msup><mi>GeV</mi><mn>2</mn></msup></mrow></math>, respectively, and <math><msup><mi>v</mi><mn>2</mn></msup><mo>=</mo><mn>0.0185</mn></math>, 0.0215, 0.121, and 0.140 for <math><mi>B</mi></math>, <math><msub><mi>B</mi><mi>s</mi></msub></math>, <math><mi>D</mi></math>, and <math><msub><mi>D</mi><mi>s</mi></msub></math>, respectively. We obtain some relationships, for example, <math><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mi>S</mi><mo>)</mo><mo>≈</mo><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>−</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mi>S</mi><mo>)</mo></mrow></math>, <math><mrow><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mi>P</mi><mo>)</mo><mo>≈</mo><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo><mo>′</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><msup><mrow><mi>P</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo><mo>&gt;</mo><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mi>P</mi><mo>)</mo><mo>≈</mo><msubsup><mrow><mi>q</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mo>+</mo></mrow></msup></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mi>P</mi><mo>)</mo></mrow></math>, and <math><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>m</mi><mi>S</mi><mo>)</mo><mo>&lt;</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>m</mi><mi>P</mi><mo>)</mo></mrow></math> (<math><mi>m</mi><mo>=</mo><mn>1</mn></math>, 2, 3), etc.

Wei Li

Tianhong Wang

Chao-Hsi Chang

Kinetic energy and speed powers <math><mrow><msup><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math> of a heavy quark inside <math><mi>S</mi></math> wave and <math><mi>P</mi></math> wave heavy-light mesons