Theoretical calculations for event shape observables are often determined by using the conventional scale setting; i.e. the procedure defined by setting the renormalization scale to the center-of-mass energy $\textit{μ}$ = $$ \sqrt{s} $$ and evaluating theoretical uncertainties by varying the same scale $\textit{μ}$ in an arbitrary range. Both the event shape distributions and the extracted QCD coupling $\textit{α}$ are plagued by the large renormalization scale uncertainties when using the conventional scale setting. The Principle of Maximum Conformality (PMC) provides a rigorous method to eliminate the renormalization scheme and scale ambiguities in perturbative QCD predictions. In this paper, we perform a detailed analysis of the event shape observables by applying the PMC method together with the use of the physical $\textit{V}$ -scheme. The PMC scales are not simple single-valued functions, but depend with continuity on the value of the unintegrated event shape variable. This reflects the virtuality of the underlying quark and gluon subprocess and yields to a physical behavior of the scale all over the entire range of each observable. Moreover, the PMC scales in the $\textit{V}$ -scheme exhibits a faster increase compared to the $$ \overline{\textrm{MS}} $$ scheme, and a better convergence in the perturbative series can be obtained. Results obtained by the PMC method for the event shape variables, thrust ($\textit{T}$), heavy jet mass $$ \left(\rho ={M}_H^2/s\right) $$ , wide jet broadening ($\textit{B}$ ), total jet broadening ($\textit{B}$ ), C-sparameter ($\textit{C}$), are in agreement with the high precision experimental data, and for the case of the jet transition variable $\textit{Y}$ , we obtain a first improvement in the results to some extent compared with the $$ \overline{\textrm{MS}} $$ scheme.

China

Zhu-Yu Ren

Sheng-Quan Wang

Jian-Ming Shen

Xing-Gang Wu

Italy

Leonardo Giustino

A reanalysis of event shape distributions in electron-positron annihilation

In this paper, we investigate the semileptonic decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub><mo>ℓ</mo><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo>ℓ</mo></msub></math> using the quantum chromodynamics (QCD) sum rules within the framework of Standard Model (SM). We further explore the potential to probe signatures of new physics (NP) beyond the SM through these decays. First, we derive the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>-moments <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow><mi>n</mi></msubsup><mo stretchy="false">⟩</mo></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math>-meson leading-twist distribution amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow></msub></math> using the QCD sum rules within the background field theory. Considering contributions from the vacuum condensates up to dimension-six, the first two nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math>-moments at the scale of 4 GeV are found to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo stretchy="false">⟨</mo><msubsup><mrow><mi>ξ</mi></mrow><mrow><mn>2</mn><mo>;</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mn>0.10</mn><msubsup><mrow><mn>3</mn></mrow><mrow><mo>−</mo><mn>0.009</mn></mrow><mrow><mo>+</mo><mn>0.009</mn></mrow></msubsup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow><mn>4</mn></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mn>0.03</mn><msubsup><mn>1</mn><mrow><mo>−</mo><mn>0.003</mn></mrow><mrow><mo>+</mo><mn>0.003</mn></mrow></msubsup></math>. Using these moments, we then fix the Gegenbauer expansion series of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>η</mi><mi>c</mi></msub></mrow></msub></math> and apply it to compute the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> transition form factors (TFFs) using QCD light cone sum rules. Second, we extrapolate those TFFs to physically allowable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math>-range via a simplified series expansion, and we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></msub><msub><mrow><mo stretchy="false">|</mo></mrow><mrow><mi>SM</mi></mrow></msub><mo>=</mo><mn>0.30</mn><msubsup><mrow><mn>8</mn></mrow><mrow><mo>−</mo><mn>0.062</mn></mrow><mrow><mo>+</mo><mn>0.084</mn></mrow></msubsup></mrow></math>. Furthermore, we explore the potential impacts of various NP scenarios on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>R</mi><msub><mi>η</mi><mi>c</mi></msub></msub></math>. Specifically, we compute the forward-backward asymmetry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">A</mi><mi>FB</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, the convexity parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="script">C</mi><mi>F</mi><mi>τ</mi></msubsup><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>, and the longitudinal and transverse polarizations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">P</mi><mi>L</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">P</mi><mi>T</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub><mo stretchy="false">→</mo><msub><mi>η</mi><mi>c</mi></msub></math> transitions within both the SM and two types of NP scenarios. Our results contribute to a deeper understanding of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>B</mi><mi>c</mi></msub></math>-meson semileptonic decays and provide insights into the search for the NP beyond the SM.

Long Zeng

Dan-Dan Hu

Yu-Jie Zhang

Hai-Bing Fu

Tao Zhong

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub></math> leading-twist distribution amplitude and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>B</mi></mrow><mrow><mi>c</mi></mrow></msub><mo stretchy="false">→</mo><msub><mrow><mi>η</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>ℓ</mo><msub><mrow><mover accent="true"><mrow><mi>ν</mi></mrow><mrow><mo stretchy="false">¯</mo></mrow></mover></mrow><mrow><mo>ℓ</mo></mrow></msub></mrow></math> semileptonic decays using QCD sum rules

We propose a novel method to determine the strong coupling of quantum chromodynamics (QCD) and fix its running behavior at all scales by using the Bjorken sum rules (BSR). The BSR defines an effective coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>g</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow></math> which includes the nonperturbative higher-twist corrections and perturbative QCD (pQCD) corrections to the leading-twist part. For the leading-twist part of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>α</mi><mi>s</mi><msub><mi>g</mi><mn>1</mn></msub></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></math>, we adopt the infinite-order scale-setting procedure of the principle of maximum conformality (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>PMC</mi><mi>∞</mi></msub></math>) to deal with its pQCD corrections, which reveals the intrinsic conformality of the series and eliminates conventional renormalization scheme-and-scale ambiguities. Using the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>PMC</mi><mi>∞</mi></msub></math> approach, we not only eliminate the first kind of residual scale dependence due to uncalculated higher-order terms, but also resolve the previous “self-consistency problem.” The light-front holographic QCD model is used for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>α</mi><mi>s</mi><msub><mi>g</mi><mn>1</mn></msub></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></math> in the infrared region, which also reveals a conformal behavior at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Q</mi><mo stretchy="false">→</mo><mn>0</mn></math>. As a combination, we obtain a precise <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>α</mi><mi>s</mi><msub><mi>g</mi><mn>1</mn></msub></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></math> at all scales, which matches well with the known experimental data with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>-value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>∼</mo><mn>98</mn><mo>%</mo></math>, and determines the strong coupling constant at the scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><mi>Z</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">(</mo><msub><mi>M</mi><mi>Z</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0.122</mn><msubsup><mn>0</mn><mrow><mo>−</mo><mn>0.0012</mn></mrow><mrow><mo>+</mo><mn>0.0011</mn></mrow></msubsup><mo stretchy="false">(</mo><mi mathvariant="normal">Δ</mi><mi>κ</mi><msubsup><mo stretchy="false">)</mo><mrow><mo>+</mo><mn>0.0007</mn></mrow><mrow><mo>−</mo><mn>0.0046</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>the</mi><mo>.</mo></mrow><mo stretchy="false">)</mo></math>, where the first error comes from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="normal">Δ</mi><mi>κ</mi></math> of LFHQCD model and the second error is from the second kind of residual scale dependence.

Qing Yu

Hua Zhou

Determination of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>α</mi><mi>s</mi></msub><mo stretchy="false">(</mo><msub><mi>M</mi><mi>Z</mi></msub><mo stretchy="false">)</mo></math> via a high-precision effective coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>α</mi><mi>s</mi><msub><mi>g</mi><mn>1</mn></msub></msubsup><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></math>

In this paper, we explore the properties of the Ellis-Jaffe Sum Rule (EJSR) by employing the Principle of Maximum Conformality (PMC) approach to address its perturbative part up to next-to-next-to-next-to-leading order ( <math altimg='si1.svg'><msup><mrow><mi mathvariant='normal'>N</mi></mrow><mrow><mn mathvariant='normal'>3</mn></mrow></msup><mi mathvariant='normal'>LO</mi></math>) QCD contributions. By applying the PMC, we achieve a precise perturbative QCD prediction for the EJSR, free from conventional ambiguities associated with the renormalization scale choices. Considering the presence of the <math altimg='si2.svg'><msub><mrow><mi>&#945;</mi></mrow><mrow><mi>s</mi></mrow></msub></math> Landau pole near the asymptotic scale, we incorporate the low-energy <math altimg='si2.svg'><msub><mrow><mi>&#945;</mi></mrow><mrow><mi>s</mi></mrow></msub></math> model based on analytic perturbation theory (APT) to refine the EJSR behavior in the infrared region. By combining the PMC approach with the low-energy APT model, the agreement between theoretical calculations and experimental measurements of EJSR is significantly improved, as evidenced by the reduced discrepancy from <math altimg='si3.svg'><msup><mrow><mi>&#967;</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy='false'>/</mo><mi>d</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>f</mi><msub><mrow><mo stretchy='false'>|</mo></mrow><mrow><mi mathvariant='normal'>Conv</mi><mo>.</mo></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>1.86</mn></math> to <math altimg='si4.svg'><msup><mrow><mi>&#967;</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy='false'>/</mo><mi>d</mi><mo>.</mo><mi>o</mi><mo>.</mo><mi>f</mi><msub><mrow><mo stretchy='false'>|</mo></mrow><mrow><mi mathvariant='normal'>PMC</mi></mrow></msub><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>1.19</mn></math>, thereby validating the effectiveness of our approach.

Updated determination of Ellis-Jaffe sum rules up to N 3LO QCD corrections

The next-to-next-to-leading-order (NNLO) perturbative QCD (pQCD) predictions for both the decay width and the transition form factor in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> process, based on nonrelativistic QCD (NRQCD), deviate from precise experimental measurements. These significant discrepancies have cast doubt on the applicability of NRQCD to charmonium processes. In this paper, we analyze the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> process by applying the principle of maximum conformality (PMC), a systematic method for eliminating renormalization scheme-and-scale ambiguities. The PMC renormalization scales are determined by absorbing the nonconformal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>β</mi></math>-terms that govern the behavior of the QCD running coupling via the renormalization group equation. We obtain the PMC scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Q</mi><mo>⋆</mo></msub><mo>=</mo><mn>4.49</mn><msub><mi>m</mi><mi>c</mi></msub></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> decay width. Even after using the PMC method, the convergence of the pQCD series is still poor, which indicates the importance of uncalculated next-to-next-to-next-to-leading-order and higher-order terms. The resulting value for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Γ</mi><mrow><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></mrow></msub></math> is in agreement with the Particle Data Group’s reported value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="normal">Γ</mi><mrow><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></mrow></msub><mo>=</mo><mn>5.1</mn><mo>±</mo><mn>0.4</mn><mtext> </mtext><mtext> </mtext><mi>keV</mi></math> within the bounds of uncertainties. Moreover, the transition form factor obtained using the PMC is also in good agreement with precise experimental measurements. The application of the PMC suggests a potential resolution to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> puzzle and supports the applicability of NRQCD to charmonium processes.

Leonardo Di Giustino

United States

Stanley J. Brodsky

Self-consistent analysis for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>η</mi><mi>c</mi></msub><mo stretchy="false">→</mo><mi>γ</mi><mi>γ</mi></math> process

According to the renormalization group invariance, any physical observable should be invariant under different renormalization schemes and scales. Nevertheless, a fixed-order pQCD approximant cannot inherently satisfy this requirement, giving rise to the conventional scheme-and-scale ambiguities. The principle of maximum conformality (PMC) offers a process-independent approach to address these two types of ambiguities. In practice, it is necessary to handle the $\textit{n}$ -terms in the pQCD series with caution to obtain accurate PMC predictions. In this paper, a novel method via using the characteristic operator (CO) $$ {\hat{D}}_{n_{\gamma },{n}_{\beta }} $$ is proposed to extend the applicability of PMC, which is a theoretical generalization of previous PMC single-scale setting approach. The CO framework not only streamlines derivations for complex scenarios, yielding simplified procedures and more compact expressions, but also achieves a scheme-and-scale invariant pQCD series by fixing the correct effective magnitude of $\textit{α}$ and the running mass simultaneously. Both are well matched with the expansion coefficients of the series, leading to the wanted scheme-and-scale invariant conformal series. As an example, we show the achievement of scale-invariant N LO total decay width Γ($\textit{H →}$ $$ b\overline{b} $$ ) under the $$ \overline{\textrm{MS}} $$ -scheme. Using the CO framework, its effective coupling $\textit{α}$ ($\textit{Q}$ ) and effective $\textit{b}$-quark $$ \overline{\textrm{MS}} $$ -mass $$ {\overline{m}}_b\left({Q}_{\ast}\right) $$ are determined by absorbing all non-conformal {$\textit{β}$ }-terms from the renormalization group equations for either $\textit{α}$ or $$ {\overline{m}}_b $$ simultaneously. The PMC scale is fixed up to N LL-accuracy, $\textit{Q}$ = 55$\textit{.}$2916 GeV and a scale-invariant total decay width is obtained, $$ \Gamma \left(H\to b\overline{b}\right)={2.3819}_{-0.0231}^{+0.0230} $$ MeV, whose errors are squared averages of the ones associated with ∆$\textit{α}$ ($\textit{M}$ ) = ±0$\textit{.}$0009, ∆$\textit{M}$ = 0$\textit{.}$11 GeV, $$ \Delta {\overline{m}}_b\left({\overline{m}}_b\right) $$ = ±0$\textit{.}$007 GeV, and the uncalculated N LO contributions ∆Γ = ±0$\textit{.}$0001 MeV predicted via Bayesian analysis with the degree-of-belief DoB = 95$\textit{.}$5%.

Jiang Yan

Xu-Dong Huang

Zhi-Fei Wu

Scale-invariant total decay width Γ($\textit{H →}$ $$ b\overline{b} $$ ) using the novel method of characteristic operator

In the paper, we conduct a detailed investigation of the rare decay processes of charged meson, specifically <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>ℓ</mo><mo>=</mo><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math>. These processes involve flavor-changing-neutral-current (FCNC) transitions, namely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo stretchy="false">→</mo><mi>s</mi><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo stretchy="false">→</mo><mi>s</mi><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover></math>. The essential components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><mi>K</mi></math> scalar, vector, and tensor transition form factors (TFFs) are calculated by using the QCD light-cone sum rules approach up to next-to-leading order QCD corrections. In which, the kaon twist-2 and twist-3 light-cone distribution amplitudes are calculated from both the QCD sum rules within the framework of background field theory and the light-cone harmonic oscillator model. The TFFs at large recoil point are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>f</mi><mo>+</mo><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>f</mi><mn>0</mn><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.34</mn><msubsup><mn>4</mn><mrow><mo>−</mo><mn>0.018</mn></mrow><mrow><mo>+</mo><mn>0.021</mn></mrow></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>f</mi><mi mathvariant="normal">T</mi><mrow><mi>B</mi><mi>K</mi></mrow></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0.34</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.019</mn></mrow><mrow><mo>+</mo><mn>0.021</mn></mrow></msubsup></math>, respectively. To obtain the behavior of those TFFs in the whole <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> region, we extrapolate them by utilizing the simplified <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>z</mi><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math>-series expansion. Furthermore, we compute the differential branching fractions with respect to the squared dilepton invariant mass for the two different decay channels and present the corresponding curves. Our predictions of total branching fraction are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>6.62</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.709</mn></mrow><mrow><mo>+</mo><mn>0.648</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msup><mi>μ</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>6.61</mn><msubsup><mn>2</mn><mrow><mo>−</mo><mn>0.698</mn></mrow><mrow><mo>+</mo><mn>0.638</mn></mrow></msubsup><mspace linebreak="goodbreak"></mspace><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>+</mo></msup><msup><mi>τ</mi><mo>−</mo></msup><mo stretchy="false">)</mo><mo>=</mo><mn>1.79</mn><msubsup><mn>5</mn><mrow><mo>−</mo><mn>0.116</mn></mrow><mrow><mo>+</mo><mn>0.110</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>7</mn></mrow></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mn>4.17</mn><msubsup><mn>0</mn><mrow><mo>−</mo><mn>0.440</mn></mrow><mrow><mo>+</mo><mn>0.402</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup></math>, respectively. Lastly, the observables such as the lepton universality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="script">R</mi><mi>K</mi></msub></math> and the angular distribution “flat term” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>F</mi><mi mathvariant="normal">H</mi><mo>ℓ</mo></msubsup></math> are given, which show good agreement with the theoretical and experimental predictions.

Hai-Jiang Tian

Ya-Xiong Wang

Rare decay <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msup><mi>K</mi><mo>+</mo></msup><msup><mo>ℓ</mo><mo>+</mo></msup><msup><mo>ℓ</mo><mo>−</mo></msup><mo stretchy="false">(</mo><mi>ν</mi><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo></math> under the QCD sum rules approach

We study the transverse momentum ( $p_T$ ) differential anisotropic flow and flow fluctuation in Pb+Pb collisions at $\sqrt{s_{_{NN}}}$ =5.02 TeV at the LHC. A (3+1)-dimensional CLVisc hydrodynamics framework with fluctuating TRENTO (or AMPT) initial conditions is utilized to simulate the space-time evolution of the quark-gluon plasma (QGP) medium. The effects of shear viscosity and the sub-nucleon structure on anisotropic flow and flow fluctuation are analyzed. Our result shows that shear viscosity tends to suppress both flow coefficients ( ${v_2\{2}\}$ , ${v_2\{4\}}$ , and ${\langle v_2\rangle}$ ) and flow fluctuation ( ${\sigma_{v_2}}$ ) owing to its smearing effect on local density fluctuation. The flow coefficients appear to be insensitive to the sub-nucleon structure, whereas the flow fluctuation ${\sigma_{v_2}}$ tends to be suppressed by the sub-nucleon structure in central collisions but enhanced in peripheral collisions. After taking into account the sub-nucleon structure effect, our numerical result can qualitatively describe the relative flow fluctuations ( ${v_2\{4\}/v_2\{2\}}$ , $F({v_2})$ ) measured by the ALICE Collaboration at the LHC. We further investigate the effects of shear viscosity, sub-nucleon structure, and initial condition model on the flow angle and flow magnitude decorrelations ( ${A_2^f}$ , ${M_2^f}$ ) using the four-particle correlation method. We find that the flow decorrelation effect is typically stronger in central collisions than in peripheral collisions. The flow angle decorrelation is found to be insensitive to the shear viscosity and sub-nucleon structure, whereas the flow magnitude decorrelation shows quite different behaviors when using the TRENTO and AMPT initial condition models. Our study sheds light on the anisotropic flow, transport properties, and initial structure of the QGP created in high-energy nuclear collisions.

Germany

Jie Zhu

Canada

Xiang-Yu Wu

Guang-You Qin

Anisotropic flow, flow fluctuation, and flow decorrelation in relativistic heavy-ion collisions: the roles of sub-nucleon structure and shear viscosity * Supported in part by the Natural Science Foundation of China (12225503, 11890710, 11890711, 11935007). Zhu Jie is supported in part by the China Scholarship Council (202306770009). Wu Xiang-Yu is supported in part by the National Science Foundation within the framework of the JETSCAPE collaboration (OAC-2004571) and in part by the Natural Sciences and Engineering Research Council of Canada. Some of the calculations were performed in the Nuclear Science Computing Center at Central China Normal University (NSC$^{3}$), Wuhan, Hubei, China.

In the paper, we derive the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math> moments <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mi>n</mi><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo></math> of the longitudinal leading-twist distribution amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">∥</mo></msubsup></math> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow></math>-state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math> meson by using the QCD sum rules under the background field theory. Considering the contributions from the vacuum condensates up to dimension six, its first two nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ξ</mi></math> moments at the scale 1 GeV are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mn>1</mn><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mn>0.64</mn><msubsup><mn>7</mn><mrow><mo>−</mo><mn>0.113</mn></mrow><mrow><mo>+</mo><mn>0.118</mn></mrow></msubsup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">⟨</mo><msubsup><mi>ξ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mrow><mn>3</mn><mo>;</mo><mo stretchy="false">∥</mo></mrow></msubsup><mo stretchy="false">⟩</mo><mo>=</mo><mo>−</mo><mn>0.32</mn><msubsup><mn>8</mn><mrow><mo>−</mo><mn>0.052</mn></mrow><mrow><mo>+</mo><mn>0.055</mn></mrow></msubsup></math>, respectively. Using those moments, we then fix the Gegenbauer expansion series of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>ϕ</mi><mrow><mn>2</mn><mo>;</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">∥</mo></msubsup></math> and apply it to calculate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math> transition form factors (TFFs) that are derived by using the QCD light-cone sum rules. Those TFFs are then extrapolated to the physically allowable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> range via the simplified series expansion. As for the branching fractions, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mover accent="true"><mi>B</mi><mo stretchy="false">¯</mo></mover><mn>0</mn></msup><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><msub><mover accent="true"><mi>ν</mi><mo stretchy="false">¯</mo></mover><mi>e</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.17</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.422</mn></mrow><mrow><mo>+</mo><mn>0.553</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mn>0</mn></msup><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mo>−</mo></msup><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.16</mn><msubsup><mn>6</mn><mrow><mo>−</mo><mn>0.415</mn></mrow><mrow><mo>+</mo><mn>0.544</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><mspace linebreak="goodbreak"></mspace><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup><msup><mi>e</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>e</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.35</mn><msubsup><mn>3</mn><mrow><mo>−</mo><mn>0.456</mn></mrow><mrow><mo>+</mo><mn>0.597</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi mathvariant="script">B</mi><mo stretchy="false">(</mo><msup><mi>B</mi><mo>+</mo></msup><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><msup><mo stretchy="false">)</mo><mn>0</mn></msup><msup><mi>μ</mi><mo>+</mo></msup><msub><mi>ν</mi><mi>μ</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>2.33</mn><msubsup><mn>9</mn><mrow><mo>−</mo><mn>0.448</mn></mrow><mrow><mo>+</mo><mn>0.587</mn></mrow></msubsup><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>4</mn></mrow></msup></math>, respectively.

Longitudinal leading-twist distribution amplitude of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mmultiscripts><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mprescripts></mprescripts><none></none><mrow><mn>1</mn></mrow></mmultiscripts></mrow></math>-state <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo></math> meson and its implications on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>B</mi><mo stretchy="false">→</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mn>1235</mn><mo stretchy="false">)</mo><msup><mo>ℓ</mo><mo>+</mo></msup><msub><mi>ν</mi><mo>ℓ</mo></msub></math> decays

We investigate the Klein-Gordon (KG) oscillators in a global monopole (GM) spacetime in Eddington-inspired Born-Infeld (EiBI) gravity and a Wu-Yang magnetic monopole (WYMM). We discuss the gravitational effects in the presence of Ricci scalar curvature <math altimg='si1.svg'><mi>R</mi><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><msubsup><mrow><mi>R</mi></mrow><mrow><mi>&#965;</mi></mrow><mrow><mi>&#965;</mi></mrow></msubsup></math>. It is observed that the presence of the Ricci scalar curvature, effectively and manifestly, introduces a force field that makes the corresponding quantum mechanical repulsive core more repulsive. Similar effect is also observed for the EiBI-gravitational field. We reiterate and report that the corresponding bosonic KG-oscillator quantum mechanical system admits a solution in the form of confluent Heun functions, the truncation of which into a physically admissible polynomial is shown to be associated with some parametric correlations/conditions. The use of such conditions/correlations is mandatory and yields a set of allowed/restricted quantum mechanical orbital <italic>&#8467;</italic>-excitations, for all radial quantum numbers <math altimg='si2.svg'><msub><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>&#8805;</mo><mn>0</mn></math>. Our procedure is shown to be quite handy, in the sense that it allows one to retrieve results for KG-oscillators in GM-spacetime in different EiBI-gravity and Ricci scalar curvature settings.