arXiv subjects

Xu-Chang Zheng

Publications and source records attributed to Xu-Chang Zheng.

At least 19 recordsLinked to original sources

arXiv2026

Next-to-leading order QCD and relativistic corrections to $Z \to J/ψ+Υ(nS)$

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to J/ψ+Υ(nS)$ ($n=1,2,3$) at future super $Z$ factory and at the CEPC/FCC-ee, including both the relativistic and QCD corrections within the framework of nonrelativistic QCD. Both the relativistic and QCD corrections are found to be large and negative. Compared to the leading-order results, the decay widths are significantly reduced by the higher-order corrections due to significant numerical cancellations. Despite the resulting large theoretical uncertainties, sizable event rates from these rare decay channels could still be anticipated at future high-luminosity electron-positron colliders running around the Z-pole. Ultimately, our results provide an essential theoretical baseline, highlighting the necessity of incorporating even higher-order corrections and resummation techniques for future precision phenomenological studies.

hep-ph↗

arXiv2026

EEXICC: An event generator for doubly heavy baryon production at $e^+e^-$ colliders

We present EEXICC, a Monte Carlo event generator designed to simulate the production of doubly heavy baryons ($Ξ_{cc}$, $Ξ_{bc}$, and $Ξ_{bb}$) via $e^+e^-$ annihilation. Based on nonrelativistic QCD effective theory, the generator calculates the process $e^{+}+e^{-}\rightarrow Ξ_{QQ'}+\bar{Q}'+\bar{Q}$ using an improved trace technique at the amplitude level, which greatly improves numerical efficiency compared with traditional squared-amplitude methods. EEXICC is developed in Fortran with a modular structure and is fully compatible with the PYTHIA framework, enabling convenient integration into complete event simulation workflows. The program supports both weighted and unweighted event generation, and its numerical reliability has been verified against existing theoretical results. EEXICC provides a flexible and robust tool for studying the properties of doubly heavy baryons at future high-luminosity and high-energy $e^+e^-$ colliders such as the CEPC and FCC-ee.

hep-ph↗

arXiv2026

Production of doubly heavy quarkonium associated with two heavy quarks via top quark decays

In this paper, we analyze the $1 \rightarrow 4$ decay channel for the production of doubly heavy quarkonium, $(b\bar{c})$ or $(c\bar{c})$, via top-quark decays, $t \to (b\bar{c}) + c + c + \bar{s}$ and $t \to (c\bar{c}) + b + c + \bar{s}$, within the framework of nonrelativistic QCD (NRQCD). The dominant contributions are considered in color-singlet S-wave states, i.e., $(b\bar{c})[^1S_0]$, $(b\bar{c})[^3S_1]$, $(c\bar{c})[^1S_0]$, and $(c\bar{c})[^3S_1]$. Our calculations show that the decay widths for $\bar{B_{c}}$, $\bar{B_{c}^{*}}$, $η_{c}$ and $J/ψ$ production are 0.2251, 0.3099, 0.0537 and 0.0555 MeV, respectively, resulting in ${\cal O}(10^{4}\text{--}10^{6})$ level of $\bar{B}_c^{(*)}$ events and ${\cal O}(10^{3}\text{--}10^{5})$ level of charmonium produced at LHC per year. In particular, we find that the dominant contribution to $η_{c}$ and $J/ψ$ production via top-quark decays arises from this decay channel proposed in this work. Moreover, this multi-body top-quark decay process can serve as a sensitive probe for validating the narrow-width approximation (NWA). Finally, we provide a detailed analysis of theoretical uncertainties and differential distributions to facilitate the corresponding experimental searches. The production of a hadron associated with three quarks contains rich physical information, providing new insights for the LHC to study $B_c$ mesons and charmonia.

hep-ph↗

arXiv2026

Renormalization and Factorization Scale-Invariant Predictions for the Higgs Rare Decay $H\to J/ψ+γ$ via the Principle of Maximum Conformality

We investigate the $J/ψ$ direct production mechanism in the rare exclusive Higgs decay $H\to J/ψ+γ$ within nonrelativistic QCD (NRQCD), which provides a clean probe for extracting the charm-quark Yukawa coupling to the Higgs boson. The Principle of Maximum Conformality (PMC) is used to remove conventional renormalization-scheme and scale ambiguities in the next-to-next-to-leading-order (N$^2$LO) perturbative QCD series. Large logarithmic contributions arising from Yukawa coupling renormalization are resummed, providing a reliable foundation for subsequent analyses. Using the experimentally measured leptonic decay width of $J/ψ$ and the N$^2$LO perturbative result, we extract the factorization-scale-dependent long-distance matrix element $\langle J/ψ({\bm ε})|ψ^{\dagger}{\bm σ}\cdot{\bm ε}χ(μ_Λ) |0\rangle$. Combining this with the factorization-scale-dependent short-distance coefficient, we obtain a factorization-scale-invariant decay width for the channel. Compared with earlier predictions in the literature, our fixed-order result for $Γ(H\to J/ψ+γ)$ is more robust and precise, with good convergence and no renormalization- or factorization-scale dependence. We find $Γ(H\to J/ψ+γ) = (6.4574^{+0.3995}_{-0.3995}) \times 10^{-11}$ GeV, where the uncertainty is the quadratic sum of contributions from $Δα_s(m_Z) = \pm 0.0009$, $ΔΓ_{J/ψ\to e^+e^-} = \pm 0.10\ \text{GeV}$, $Δ\overline{m}_c(\overline{m}_c) = \pm 0.0046\ \text{GeV}$, and the estimated magnitude of N$^3$LO contributions from Bayesian analysis. This work demonstrates for the first time how the PMC can be applied to obtain fixed-order perturbative predictions that are invariant under both renormalization and factorization scale variations.

hep-ph↗

arXiv2026

Fragmentation functions for gluon into $P$-wave $B_c$ mesons

We calculate the fragmentation functions for a gluon into $P$-wave $B_c$ mesons within the nonrelativistic QCD factorization framework, incorporating color-singlet and color-octet contributions. Ultraviolet divergences arising from phase-space integrals are removed via operator renormalization in the modified minimal subtraction scheme. The resulting fragmentation functions are presented in both graphical form and as fitted analytic expressions.

hep-ph↗

arXiv2025

Determination of $|V_{\rm cb}|$ using Bayesian analysis of the $B \rightarrow D^{*}\ell {\bar ν}_\ell$ semileptonic decay width with four-loop QCD corrections

The Cabibbo-Kobayashi-Maskawa matrix element $|V_{\rm cb}|$ is an important Standard Model parameter, whose value can be determined by using the semi-leptonic decay $B\rightarrow D^{*}\ell{\bar ν}_\ell$. The perturbative QCD (pQCD) corrections to the $B \to D^{*}$ transform form factor ${\cal F}(w)$ has been known up to the N$^3$LO level, whose magnitude remains sensitive to the choice of renormalization scale $μ_r$. To improve the precision of ${\cal F}(w)$ and hence $|V_{\rm cb}|$, we first apply the single-scale approach of Principle of Maximum Conformality (PMC) to eliminate the conventional (Conv.) renormalization scale dependence of the short-distance parameter $η_A$ and then predict the contribution of its unknown N$^4$LO term via Bayesian analysis. In this paper, we adopt two probabilistic models for Bayesian analysis: the Cacciari-Houdeau model (CH model) and the geometric behavior model (GB model). It is shown that by using the PMC series in combination with Bayesian analysis, one can achieve high degree of reliability in estimating unknown higher-order terms. A more convergent behavior is achieved by applying the PMC, confirmed by the predicted N$^4$LO contributions: for the CH model, $η_A|_{\rm Conv.}^{\rm N^4LO}=\{-0.0032,+0.0052\}$ and $η_A|_{\rm PMC}^{\rm N^4LO}=\{-0.0004,+0.0004\}$; for the GB model, $η_A|_{\rm Conv.}^{\rm N^4LO}=\{-0.0049,+0.0075\}$ and $η_A|_{\rm PMC}^{\rm N^4LO}=\{-0.0007,+0.0007\}$. Comparing with the latest experimental measurements, we obtain $|V_{\rm cb}||_{\rm PMC}=(40.58^{+0.53}_{-0.57})\times10^{-3}$, which is consistent for both CH and GB models and in good agreement with the PDG world average, $|V_{\rm cb}|_{\rm PDG}=(41.1\pm1.2)\times10^{-3}$ within errors.

hep-ph↗

arXiv2025

Production of doubly heavy baryon at the Muon-Ion Collider

This study forecasts the production of doubly heavy baryons, $Ξ_{cc}$, $Ξ_{bc}$, and $Ξ_{bb}$, within the nonrelativistic QCD framework at the Muon-Ion Collider (MuIC). It examines two production mechanisms: photon-gluon fusion ($γ+ g \to (QQ')[n] +\bar{Q} +\bar{Q'}$) and extrinsic heavy quark channels ($γ+ Q \to (QQ')[n] + \bar{Q'}$), where $Q$ and $Q'$ denote heavy quarks ($c$ or $b$) and $(QQ')[n]$ represents a diquark in specific spin-color configurations. The diquark fragments into $Ξ_{QQ'}$ baryons with high probability. For $Ξ_{cc}$ and $Ξ_{bb}$, the relevant configurations are $[^1S_0]_{\textbf{6}}$ (spin-singlet and color-sextuplet) and $[^3S_1]_{\bar{\textbf{3}}}$ (spin-triplet and color-antitriplet). For $Ξ_{bc}$, the configurations are $[^1S_0]_{\bar{\textbf{3}}}$, $[^1S_0]_{\textbf{6}}$, $[^3S_1]_{\bar{\textbf{3}}}$, and $[^3S_1]_{\textbf{6}}$. The study compares total and differential cross-sections for these channels, highlighting their uncertainties. The results indicate that the extrinsic heavy quark channel, particularly the $[^3S_1]_{\bar{\textbf{3}}}$ configuration, dominates $Ξ_{QQ'}$ production, though other diquark states also contribute significantly. Using quark masses $m_c = 1.80 \pm 0.10$ GeV and $m_b = 5.1 \pm 0.20$ GeV, the study estimates annual event yields at MuIC ($\sqrt{s} = 1$ TeV, luminosity ${\mathcal L}\simeq 40$ ${\rm fb}^{-1}$) of $(3.67^{+1.29}_{-0.91}) \times 10^9$ for $Ξ_{cc}$, $(2.24^{+0.28}_{-0.20}) \times 10^8$ for $Ξ_{bc}$, and $(3.00^{+0.64}_{-0.56}) \times 10^6$ for $Ξ_{bb}$. These findings suggest that MuIC will significantly enhance our understanding of doubly heavy baryons.

hep-ph↗

arXiv2024

QCD corrections of $e^+e^- \to J/ψ+c+\bar{c}$ using the principle of maximum conformality

In this paper, we compute the total and differential cross sections for $e^+e^- \to J/ψ+c+\bar{c}$ at the $B$ factories up to next-to-leading order (NLO) corrections within the framework of nonrelativistic QCD factorization theory. We then obtain improved pQCD series of those cross sections by using the Principle of Maximum Conformality (PMC). We show that the PMC can be applied for any pQCD calculable observable at the total and differential levels via a self-consistent way in perturbation theory. We observe that a more precise prompt total cross section at the NLO level can be achieved after applying the PMC, e.g. $σ|_{\rm prompt}^{\rm PMC}= 0.565^{+0.144}_{-0.125}~\text{pb}$. Here the uncertainty is the squared average of those from the $α_s$ fixed-point uncertainty $Δα_s(M_Z)$, the uncertainty of charm quark mass $Δm_c$, and an estimated contribution of the uncalculated NNLO-terms as predicted by the Padé approximation approach. The differential cross sections $dσ/dP_{J/ψ}$, $dσ/d|\cos θ|$, and $dσ/dz$ for $e^+e^- \to J/ψ+c+\bar{c}$ are further examined. Those results show that by further considering the feed-down contributions, the PMC predictions show better agreement with the Belle measurements.

hep-ph↗

arXiv2024

Precise determination of the bottom-quark on-shell mass using its four-loop relation to the $\overline{\rm MS}$-scheme running mass

In this paper, we explore the properties of the bottom-quark on-shell mass ($M_b$) by using its relation to the $\overline{\rm MS}$ mass (${\overline m}_b$). At present, this $\overline{\rm MS}$-on-shell relation has been known up to four-loop QCD corrections, which however still has a $\sim 2\%$ scale uncertainty by taking the renormalization scale as ${\overline m}_b({\overline m}_b)$ and varying it within the usual range of $[{\overline m}_b({\overline m}_b)/2, 2 {\overline m}_b({\overline m}_b)]$. The principle of maximum conformality (PMC) has been adopted to achieve a more precise $\overline{\rm MS}$-on-shell relation by eliminating such scale uncertainty. As a step forward, we also estimate the magnitude of the uncalculated higher-order terms by using the Padé approximation approach. Numerically, by using the $\overline{\rm MS}$ mass ${\overline m}_b({\overline m}_b)=4.183\pm0.007$ GeV as an input, our predicted value for the bottom-quark on-shell mass becomes $M_b\simeq 5.372^{+0.091}_{-0.075}$ GeV, where the uncertainty is the squared average of the ones caused by $Δα_s(M_Z)$, $Δ{\overline m}_b({\overline m}_b)$, and the estimated magnitude of the higher-order terms.

hep-ph↗

arXiv2024

Improved analysis of double $J/ψ$ production in $Z$-boson decay

In this paper, we present an improved calculation for the decay rate of the rare $Z$-boson decay into $J/ψ+ J/ψ$. This decay is dominated by the photon fragmentation mechanism, i.e., the transition $Z\to J/ψ+ γ^{*}$ followed by the fragmentation $γ^{*}\to J/ψ$. In our calculation, the amplitude of $γ^{*}\to J/ψ$ is extracted from the measured value of $Γ(J/ψ\to e^+ e^-)$, and the amplitude of $Z\to J/ψ+ γ^{*}$ is calculate through the light-cone approach. The higher-order QCD and relativistic corrections in the amplitude of $γ^{*}\to J/ψ$ and the large logarithms of $m_{_Z}^2/m_c^2$ that appear in the amplitude of $Z\to J/ψ+ γ^{*}$ are resummed in our calculation. Besides, the non-fragmentation amplitude is calculated based on the NRQCD factorization, and the next-to-leading order QCD and relativistic corrections are included. The obtained branching fraction for this $Z$ decay channel is $8.66 ^{+1.48} _{-0.69}\times 10^{-11}$.

hep-ph↗

arXiv2024

Precise determination of the top-quark on-shell mass $M_t$ via its scale-invariant perturbative relation to the top-quark $\overline{\rm MS}$ mass ${\overline m}_t({\overline m}_t)$

It has been shown that the principle of maximum conformality (PMC) provides a systematic way to solve conventional renormalization scheme and scale ambiguities. The scale-fixed predictions for physical observables using the PMC are independent of the choice of renormalization scheme -- a key requirement of renormalization group invariance. In the paper, we derive new degeneracy relations based on the renormalization group equations that involve both the usual $β$-function and the quark mass anomalous dimension $γ_m$-function, respectively. These new degeneracy relations lead to an improved PMC scale-setting procedures, such that the correct magnitudes of the strong coupling constant and the $\overline{\rm MS}$-running quark mass can be fixed simultaneously. By using the improved PMC scale-setting procedures, the renormalization scale dependence of the $\overline{\rm MS}$-on-shell quark mass relation can be eliminated systematically. Consequently, the top-quark on-shell (or $\overline{\rm MS}$) mass can be determined without conventional renormalization scale ambiguity. Taking the top-quark $\overline{\rm MS}$ mass ${\overline m}_t({\overline m}_t)=162.5^{+2.1}_{-1.5}$ GeV as the input, we obtain $M_t\simeq 172.41^{+2.21}_{-1.57}$ GeV. Here the uncertainties are combined errors with those also from $Δα_s(M_Z)$ and the approximate uncertainty stemming from the uncalculated five-loop terms predicted through the Padé approximation approach.

hep-ph↗

arXiv2023

$Z$-boson decays into $S$-wave quarkonium plus a photon up to ${\cal O}(α_{s} v^2)$ corrections

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to H+ γ$ up to ${\cal O}(α_{s} v^2)$ accuracy within the framework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium $η_c$, $J/ψ$, $η_b$ or $Υ$, respectively. To compare with the leading-order terms, those corrections show good perturbative behavior as expected. It is found that contributions from the next-to-leading order QCD correction ${\cal O}(α_{s}v^0)$, the relativistic correction ${\cal O}(α^{0}_{s}v^2)$ and their joint correction ${\cal O}(α_{s} v^2)$ are sizable and comparable to each other, especially for the charmonium case. Thus we need to take all of them into consideration for a sound estimation. For a high luminosity electron-positron collider running around the $Z$-pole, due to $Z$-boson resonance effect, sizable events could be produced from those rare decay channels.

hep-ph↗

arXiv2023

Photoproduction of doubly heavy baryons at future $e^+e^-$ colliders

The photoprodution of doubly heavy baryon ($Ξ_{cc},Ξ_{bb},Ξ_{bc}$) is investigated in the context of future high-energy and high-luminosity $e^+e^-$ colliders. The study incorporates two sources of initial photons, namely the LBS photon and the WWA photon. Alongside the direct photoproduction via the sub-process $γ+γ\rightarrow Ξ_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$ ($Q^{(')}=c,b$), the resolved photoproduction channels are specifically considered, encompassing the sub-processes $γ+ g \rightarrow Ξ_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$, $g + g \rightarrow Ξ_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$, and $q + \bar{q} \rightarrow Ξ_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$ with $q=u,d,s$. Within the framework of non-relativistic QCD, two $(cc(bb))$-diquark configurations, ${}_{\bar{\textbf{3}}}[{}^3S_1]$ and ${}_{\textbf{6}}[{}^1S_0]$, and four $(bc)$-diquark configurations, $(bc)_{\bar{\textbf{3}}}[{}^3S_1]$, $(bc)_{\textbf{6}}[{}^1S_0]$, $(bc)_{\textbf{6}}[{}^3S_1]$ and $(bc)_{\bar{\textbf{3}}}[{}^1S_0]$, are considered in the calculations. Numerical results show that the single resolved photoproduction processes provide dominant contributions under certain collision configuration. At the future $e^+e^-$ colliders, the doubly heavy baryon generated via the photoproduction mechanism is promisingly observable and can be well studied.

hep-ph↗

arXiv2023

Photoproduction of $P$-wave doubly charmed baryon at future $e^+e^-$ collider

The photoproduction of $P$-wave doubly charmed baryon ($Ξ_{cc}$) is investigated in the context of future high-energy and high-luminosity $e^+e^-$ colliders. The direct photoproduction via the sub-process $γ+γ\rightarrow Ξ_{cc} +\bar{c}+\bar{c}$ and the resolved channel $γ+g \rightarrow Ξ_{cc} +\bar{c}+\bar{c}$ are considered. Within the framework of non-relativistic QCD, the calculation encompasses four $P$-wave $(cc)$-diquark configurations: $(cc)_{\bar{\textbf{3}}}[{}^1P_1]$, $(cc)_{\textbf{6}}[{}^3P_0]$, $(cc)_{\textbf{6}}[{}^3P_1]$ and $(cc)_{\textbf{6}}[{}^3P_2]$. The two $S$-wave states, $(cc)_{\bar{\textbf{3}}}[{}^3S_1]$ and $(cc)_{\textbf{6}}[{}^1S_0]$, are also included for comparison. The cross sections, as well as the differential distributions involving transverse momentum, rapidity, and angular variables, have been computed. Numerical results reveal that the resolved photoproduction process plays a significant role and can provide dominant contributions. The photoproduction rate of the $P$-wave $Ξ_{cc}$ is approximately one order of magnitude lower than that of the $S$-wave.

hep-ph↗

arXiv2023

Higgs boson decays to $B_c$ meson in the fragmentation-function approach

In the paper, we present a calculation of the decay widths for the Higgs boson decays to the $B_c$, $B_c^*$, $B_c(2^1S_0)$ and $B_c^*(2^3S_1)$ mesons using the fragmentation-function approach. In the calculation, the fragmentation functions up to order $α_s^3$ based on the nonrelativistic QCD factorization theory are used, and the decay widths for $H\to Q+X$ and $H \to g+X$ at the partonic level are calculated up to order $α_s$. The large logarithms of $m_H^2/m_{Bc}^2$ are resummed up to next-to-leading logarithmic accuracy by solving the evolution equations for the running quark masses and the fragmentation functions. Compared to the leading-order decay widths based on the nonrelativistic QCD approach, the decay widths based on the fragmentation-function approach that include the higher-order QCD corrections are reduced significantly. Our numerical results show that there are about $1.2\times 10^5$ $B_c$ events via the Higgs decays to be produced at the HL-LHC with $3ab^{-1}$, and about $1.6\times 10^6$ $B_c$ events via the Higgs decays to be produced at the HE-LHC with $15ab^{-1}$.

hep-ph↗

arXiv2023

New determination of $|V_{\rm cb}|$ using the three-loop QCD corrections for the $B\to D^{\ast}$ semi-leptonic decays

We present a new determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\rm cb}|$ by using the three-loop perturbative QCD corrections for the $B\to D^{\ast}$ semi-leptonic decay. The decay width of $B\to D^{\ast}$ semi-leptonic decay can be factorized as perturbatively calculable short-distance part and the non-perturbative but universal long-distance part. We adopt the principle of maximum conformality (PMC) single-scale setting approach to deal with the perturbative series so as to achieve a precise fixed-order prediction for the short-distance parameter $η_{A}$. By applying the PMC, an overall effective $α_s$ value is achieved by recursively using the renormalization group equation, which inversely results in a precise scale-invariant pQCD series. Such scale-invariant series also provides a reliable basis for predicting the contributions from uncalculated perturbative terms. We then obtain $η_{A}=0.9225^{+0.0117}_{-0.0168}$, where the error is the squared average of those from $Δα_{s}(M_Z)=\pm0.0010$ and the uncertainties caused by the uncalculated higher-order perturbative terms. By using the data of $B\to D^{\ast}\ell\barν_{\ell}$, we finally obtain $|V_{\rm cb}|_{\rm PMC} =(40.60^{+0.53}_{-0.57})\times10^{-3}$, which is consistent with the PDG value within errors.

hep-ph↗

arXiv2023

Hadronic production of $Ξ_{bc}$ with the intrinsic heavy-quark content at a fixed-target experiment at the LHC

In this paper, we make a detailed study on the hadronic production of the $Ξ_{bc}$ baryon at a fixed target experiment at the LHC (After@LHC). In estimating the production cross sections, the $(g+g)$, $(g+c)$ and $(g+b)$ production mechanisms are considered. For the initial heavy quarks, in addition to the extrinsic component, we also consider the intrinsic component. It is found that the $(g+c)$ and $(g+b)$ production mechanisms give sizable contributions to the $Ξ_{bc}$ production, and the $(g+b)$ mechanism dominates the production. The results show that there are about $3.40\times10^5$ $Ξ_{bc}$ events can be produced per year at After@LHC if the integrated luminosity of After@LHC can be up to $2\,{\rm fb}^{-1}$ per year. Moreover, the intrinsic heavy quarks can have significant impact on the production, which inversely makes the intrinsic component be possibly tested at the After@LHC.

hep-ph↗

arXiv2022

Excited doubly heavy baryons production via top-quark decays

Within the framework of NRQCD, we calculate the production of excited doubly heavy baryons $Ξ_{bQ}$ through the semi-inclusive production process $t\rightarrow \langle bQ\rangle[n]\rightarrow Ξ_{bQ}+ \bar {Q} + W^+ $, where $Q= b$ or $c$ quark. The intermediate diquark state $\langle bQ\rangle[n]$ is in the excited $P$-wave state, including $[^1P_1]$ and $[^3P_J]$ ($J=$0, 1 or 2) in both color antitriplet state $\mathbf{\mathbf{\overline 3}}$ and color sixtuplet state $\mathbf{6}$, that is, $\langle bc\rangle[^{1}P_{1}]_{\mathbf{\overline 3}/ \mathbf{6}}$, $\langle bc\rangle[^{3}P_{J}]_{\mathbf{\overline 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}P_{1}]_{\mathbf{\overline 3}}$, and $\langle bb\rangle[^{3}P_{J}]_{\mathbf{6}}$. We find that the contributions from the P-wave states are about one order lower than the S-wave contributions, and this conclusion is consistent with others. We also analyze the invariant mass and angle differential distributions, and the theoretical uncertainty from the mass parameters and the renormalization scale. Finally, we can expect that about $1.14 \times10^{3-5}$ events of excited $Ξ_{bc}$ and $2.47 \times10^{1-3}$ events of excited $Ξ_{bb}$ can be produced per year at the LHC or HL-LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$.

hep-ph↗