Search

In this paper, we compute the total and differential cross sections for $e^+e^- \to J/\psi+c+\bar{c}$ at the $B$ factories up to next-to-leading order (NLO) corrections within the framework of nonrelativistic QCD factorization theory. We then obtain an improved pQCD prediction for this process by using the Principle of Maximum Conformality (PMC). We show the first time that the PMC can be applied for any pQCD calculable observable at the total or differential levels via a self-consistent way in perturbation theory. By the recursive use of renormalization group equation, the resultant PMC series satisfies all the requirements of renormalization group, which is free of conventional renormalization scheme-and-scale ambiguities at any fixed-order level. We thus obtain a more precise prompt total cross section at the NLO level, e.g. $\sigma|_{\rm prompt}^{\rm PMC}= 0.565^{+0.144}_{-0.125}~\text{pb}$. Here the uncertainty is the squared average of those from the $\alpha_s$ fixed-point uncertainty $\Delta\alpha_s(M_Z)$, the uncertainty of charm quark mass $\Delta m_c$, and an estimated contribution of the uncalculated NNLO-terms as predicted by the Pad\'{e} approximation approach. The differential cross sections $d\sigma/dP_{J/\psi}$, $d\sigma/d|\cos \theta|$, and $d\sigma/dz$ for $e^+e^- \to J/\psi+c+\bar{c}$ are further examined. Those results show that by further considering the feed-down contributions, the PMC predictions show better agreement with the Belle measurements., Comment: 22 pages, 11 figures

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 Pad\'{e} 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 $\Delta \alpha_s(M_Z)$, $\Delta {\overline m}_b({\overline m}_b)$, and the estimated magnitude of the higher-order terms., Comment: 5 pages, 2 figures, matches published version

In this paper, we present an improved calculation for the decay rate of the rare $Z$-boson decay into $J/\psi + J/\psi$. This decay is dominated by the photon fragmentation mechanism, i.e., the transition $Z\to J/\psi + \gamma^{*}$ followed by the fragmentation $\gamma^{*}\to J/\psi$. In our calculation, the amplitude of $\gamma^{*}\to J/\psi$ is extracted from the measured value of $\Gamma(J/\psi \to e^+ e^-)$, and the amplitude of $Z\to J/\psi + \gamma^{*}$ is calculate through the light-cone approach. The higher-order QCD and relativistic corrections in the amplitude of $\gamma^{*}\to J/\psi$ and the large logarithms of $m_{_Z}^2/m_c^2$ that appear in the amplitude of $Z\to J/\psi + \gamma^{*}$ 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 $8.66 ^{+1.48} _{-0.69}\times 10^{-11}$., Comment: 10 pages, 2 figures

In this paper, we calculate the decay widths and branching fractions for the decays $Z \to H+ \gamma$ up to ${\cal O}(\alpha_{s} v^2)$ accuracy within the framework of nonrelativistic QCD, where $H$ stands for the $S$-wave quarkonium $\eta_c$, $J/\psi$, $\eta_b$ or $\Upsilon$, 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 ${\cal O}(\alpha_{s}v^0)$, the relativistic correction ${\cal O}(\alpha^{0}_{s}v^2)$ and their joint correction ${\cal O}(\alpha_{s} v^2)$ 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., Comment: 8 pages, 2 figures

The photoprodution of doubly heavy baryon ($\Xi_{cc},\Xi_{bb},\Xi_{bc}$) 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 $\gamma+\gamma \rightarrow \Xi_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$ ($Q^{(')}=c,b$), the resolved photoproduction channels are specifically considered, encompassing the sub-processes $\gamma + g \rightarrow \Xi_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$, $g + g \rightarrow \Xi_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$, and $q + \bar{q} \rightarrow \Xi_{QQ^{'}} +\bar{Q}+\bar{Q^{'}}$ 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.

The photoproduction of $P$-wave doubly charmed baryon ($\Xi_{cc}$) is investigated in the context of future high-energy and high-luminosity $e^+e^-$ colliders. The direct photoproduction via the sub-process $\gamma+\gamma \rightarrow \Xi_{cc} +\bar{c}+\bar{c}$ and the resolved channel $\gamma+g \rightarrow \Xi_{cc} +\bar{c}+\bar{c}$ 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 $\Xi_{cc}$ is approximately one order of magnitude lower than that of the $S$-wave.

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 $\alpha_s^3$ 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 $\alpha_s$. 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 $15ab^{-1}$., Comment: 12 pages, 9 figures

In the present paper, we study the photoproduction of various $S$-wave $B_c$ states, i.e. $B_c(1{}^1S_0)$, $B_c^*(1{}^1S_0)$, $B_c(2{}^1S_0)$ and $B^*_c(2{}^3S_1)$, at future $e^+e^-$ colliders within the framework of non-relativistic QCD. Two sources of the initial photons can be realized in the production, i.e., the LBS photon and the WWA photon. In addition to the direct photoproduction via the sub-process, $\gamma + \gamma \rightarrow B_c + b + \bar{c}$, we also calculate the resolved photoproduction via the sub-processes, $\gamma + g \rightarrow B_c + b + \bar{c}$ and $q + \bar{q} \rightarrow B_c + b + \bar{c}$ with $q=u,d,s,g$, respectively. Numerical results indicate that the contributions from the single resolved photoproduction are significant and even dominant at certain collision configuration. At the future high-energy and high-luminosity $e^+e^-$ colliders, the $B_c$ meson generated via the photoproduction mechanism is promisingly observable and can be well studied.

In this paper, we make a detailed study on the hadronic production of the $\Xi_{bc}$ 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 $\Xi_{bc}$ production, and the $(g+b)$ mechanism dominates the production. The results show that there are about $3.40\times10^5$ $\Xi_{bc}$ 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., Comment: 13 pages, 18 figures

Within the framework of NRQCD, we calculate the production of excited doubly heavy baryons $\Xi_{bQ}$ through the semi-inclusive production process $t\rightarrow \langle bQ\rangle[n]\rightarrow \Xi_{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 $\Xi_{bc}$ and $2.47 \times10^{1-3}$ events of excited $\Xi_{bb}$ can be produced per year at the LHC or HL-LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$., Comment: 11 pages, 5 figures, 7 tables. Some typos have been corrected and rewrite some sentences. arXiv admin note: text overlap with arXiv:1810.03834

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 $\beta$-function and the quark mass anomalous dimension $\gamma_m$-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 $\Delta \alpha_s(M_Z)$ and the approximate uncertainty stemming from the uncalculated five-loop terms predicted through the Pad\'{e} approximation approach., Comment: 15 pages, 2 figures, to be published in Chinese Physics C

The event generator BEEC [Z. Yang, X.G. Wu and X.Y. Wang, Comput. Phys. Commun. 184, 2848 (2013)] was devoted to the simulation of heavy quarkonium production at an unpolarized electron-positron collider. We upgraded it here by adding the generation of quarkonium with polarized electron and positron beams. In addition, the production of color-singlet 2S-wave states were included. Several future electron-positron colliders with high luminosity have been under discussion in the past decade. Especially, their possibility of producing polarized beams is important for providing more insights into the underlying physics. This upgraded version offers a useful tool for the feasibility study on quarkonium from the experimental side., Comment: 5 pages, 1 figure

Based on the non-relativistic quantum chromodynamics factorization framework, we study the inclusive $J/\psi$ photoproduction at the future high energy $e^+e^-$ collider, International Linear Collider(ILC), where the initial photons come from the back-scattering of laser and electron (positron). The intermediate states, $c\bar{c}$$({}^3\!S_1^{[\textbf{1}]}$,${}^1\!S_0^{[\textbf{8}]}$,${}^3\!S_1^{[\textbf{8}]}$,${}^3\!P_{0,1,2}^{[\textbf{8}]})$, and resolved photoproduction processes are considered. Numerical results show that the cross section of $J/\psi$ photoproduction at the ILC could be very large, and the single resolved process via $c\bar{c}({}^1\!S_0^{[\textbf{8}]})$ intermediate state dominates primarily the production. We also present various kinematic distributions for $J/\psi$ photoproduction. Combining the high luminosity of the collision, $J/\psi$ photoproduction at the ILC shall provide a good platform to test the NRQCD factorization., Comment: 13 pages,7 figures,3 tables

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 $\eta_{A}$. By applying the PMC, an overall effective $\alpha_s$ 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 $\eta_{A}=0.9225^{+0.0117}_{-0.0168}$, where the error is the squared average of those from $\Delta\alpha_{s}(M_Z)=\pm0.0010$ and the uncertainties caused by the uncalculated higher-order perturbative terms. By using the data of $B\to D^{\ast}\ell\bar{\nu}_{\ell}$, 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., Comment: 7 pages,2 figures

It has been found that at a high luminosity $e^+ e^-$ collider, sizable $\eta_c+c\bar{c}X$ and $\eta_b+b\bar{b}X$ events can be produced when it works around the $Z$ peak. In this paper, we calculate the decay widths of $Z \to \eta_c+c+\bar{c}+X$ and $Z \to \eta_b+b+\bar{b}+X$ up to next-to-leading order (NLO) accuracy. We find that the NLO corrections are significant in these two processes. After including the NLO corrections, the decay widths of $Z \to \eta_c+c+\bar{c}+X$ and $Z \to \eta_b+b+\bar{b}+X$ are enhanced by about $37\%$ and $28\%$ for the case of $\mu_R=2m_c$ and $\mu_R=2m_b$, respectively. The differential decay widths $d\Gamma/ds_1$ and $d\Gamma/dz$ for these two decay processes are also analyzed., Comment: 10 pages, 11 figures

In this paper, decay widths of the doubly heavy baryons ($\Xi_{cc} ~\text{and} ~\Xi_{bc}$) production are theoritically calculated in the whole phase space through $W^+ \to \Xi_{cc}+ \bar{c}+\bar{s}$ and $W^+ \to \Xi_{bc}+ \bar{b}+\bar{s}$, within the framework of nonrelativistic QCD (NRQCD). Differential widths $d\Gamma/ds_{12}$, $d\Gamma/ds_{23}$, $d\Gamma/dcos\theta_{12}$, and $d\Gamma/dcos\theta_{13}$ are also given. In addition to the ordinary S wave contributions for the baryons, we specifically calculate P wave contributions as a comparison, namely the high excited states of the intermediate diquark, including $[^1P_1]$ and $[^3P_J]$ (with $J=0,1,2$) in both color anti-triplet state $\overline{\mathbf{3}}$ and color sextuplet state $\mathbf{6}$. It shows that the contribution from P wave is about one order lower than S wave. According to the results, we can expect plentiful events produced at the LHC, i.e., $3.69\times10^5$ $\Xi_{cc}$ events and $4.91\times10^4$ $\Xi_{bc}$ events per year., Comment: 11 pages, 5 figures

In the paper, we calculate the fragmentation functions for $g \to B_c$ and $g \to B_c^*$. The ultraviolet divergences in the calculation are removed through the renormalization of the operator definition of the fragmentation functions under the modified minimal subtraction scheme. We then obtain the fragmentation functions $D_{g \to B_c}(z,\mu_F)$ and $D_{g \to B_c^*}(z,\mu_F)$, which are presented as figures and fitting functions. The obtained fragmentation functions are complementary to the previous work on the next-to-leading order fragmentation functions for $\bar{b}\to B_c(B_c^*)$ and $c\to B_c(B_c^*)$., Comment: 22 pages, 4 figures

In this paper, we make a detailed study on the production of the $B_c$, $B_c^*$, $B_c(2\,^1S_0)$, and $B^*_c(2\,^3S_1)$ mesons via the three planned running modes ($Z$, $W$, and $H$) at the future Circular Electron-Positron Collider (CEPC). The fragmentation-function and full nonrelativistic QCD approaches are adopted to calculate the production cross sections. Considering the excited states shall decay to the ground $B_c$ state (e.g. $1^1S_0$-state) with almost $100\%$ probability, our numerical results show that up to next-to-leading order QCD corrections, there are about $1.4\times 10^8$ $B_c$ events to be accumulated via the $Z$ mode ($\sqrt{s}=m_{_Z}$) of the CEPC, but only about $1.6\times 10^4$ and $1.1\times 10^4$ $B_c$ events to be accumulated via the $W$ ($\sqrt{s}=160\,{\rm GeV}$) and $H$ ($\sqrt{s}=240\,{\rm GeV}$) modes, respectively. Since the $Z$ mode is the best mode among the three planned modes of the CEPC for studying the production of the $B_c$, $B_c^*$, $B_c(2\,^1S_0)$, and $B^*_c(2\,^3S_1)$ mesons and the differential distributions of these mesons may be measured precisely at this mode, we further present the differential cross sections $d\sigma/(dz\,d{\rm cos}\theta)$ and $d\sigma/dz$ via the $Z$ mode of the CEPC., Comment: 8 pages, 8 figures

In the paper, we calculate the fragmentation functions for $c \to \eta_c$ and $b \to \eta_b$ up to next-to-leading-order (NLO) QCD accuracy. The ultraviolet divergences in the real corrections are removed through operator renormalization under the modified minimal subtraction scheme. We then obtain the fragmentation functions $D_{c \to \eta_c}(z,\mu_F)$ and $D_{b \to \eta_b}(z,\mu_F)$ up to NLO QCD accuracy, which are presented as figures and fitting functions. The numerical results show that the NLO corrections are significant. The sensitives of the fragmentation functions to the renormalization scale and the factorization scale are analyzed explicitly., Comment: 18 pages, 7 figures

In this paper, we present a new analysis on the $P$-wave charmonium annihilation into two photons up to next-to-next-to-leading order (NNLO) QCD corrections by using the principle of maximum conformality (PMC). The conventional perturbative QCD prediction shows strong scale dependence and deviates largely from the BESIII measurements. After applying the PMC, we obtain a more precise scale-invariant pQCD prediction, which also agrees with the BESIII measurements within errors, i.e. $R={\Gamma_{\gamma\gamma}(\chi_{c2})} /{\Gamma_{\gamma\gamma}(\chi_{c0})}=0.246\pm0.013$, where the error is for $\Delta\alpha_s(M_\tau)=\pm0.016$. By further considering the color-octet contributions, even the central value can be in agreement with the data. This shows the importance of a correct scale-setting approach. We also give a prediction for the ratio involving $\chi_{b0, b2} \to\gamma\gamma$, which could be tested in future Belle II experiment.

In the paper, the inclusive production of heavy quarkonium $\eta_Q$ ($Q=b$ or $c$) via $Z$ boson decays within the framework of nonrelativistic QCD effective theory are studied. The contributions from the leading color-singlet and color-octet Fock states are considered. Total and differential decay widths for the inclusive decays $Z \to \eta_Q+X$ are presented. It is found that the decays $Z\to \eta_Q +X$ are dominated by the $^3S_1^{[8]}$ component, so the decays can be inversely adopted to determine the values of the long-distance matrix elements $\langle {\cal O}^{\eta_{c}}(^{3}S_{1}^{[8]})\rangle$ and $\langle {\cal O}^{\eta_{b}}(^{3}S_{1}^{[8]})\rangle$, respectively. Our numerical results show that at an $e^+e^-$ collider running at the $Z$ pole with a high luminosity around $10^{35}{\rm cm}^{-2}{\rm s}^{-1}$ (a super $Z$ factory), there are about $4.5\times 10^7$ $\eta_c$ meson events and $6.1\times 10^5$ $\eta_b$ meson events to be produced per operation year, and the inclusive decays may be used for clarifying some problems on the heavy quarkonium $\eta_Q$ and nonrelativistic QCD., Comment: 13 pages, 11 figures

In the paper, we analyze the properties of Gross-Llewellyn Smith (GLS) sum rule by using the $\mathcal{O}(\alpha_s^4)$-order QCD corrections with the help of principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent fixed-order pQCD contribution for GLS sum rule, e.g. $S^{\rm GLS}(Q_0^2=3{\rm GeV}^2)|_{\rm PMC}=2.559^{+0.023}_{-0.024}$, where the error is squared average of those from $\Delta\alpha_s(M_Z)$, the predicted $\mathcal{O}(\alpha_s^5)$-order terms predicted by using the Pad\'{e} approximation approach. After applying the PMC, a more convergent pQCD series has been obtained, and the contributions from the unknown higher-order terms are highly suppressed. In combination with the nonperturbative high-twist contribution, our final prediction of GLS sum rule agrees well with the experimental data given by the CCFR collaboration., Comment: 6 pages, 5 figures

In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to \eta_Q}(z,\mu_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $\eta_c$ are presented. As an application of those fragmentation functions, we study the processes $Z \to \eta_c+q\bar{q}g(q=u,d,s)$ and $Z \to \eta_c+b\bar{b}g$ under the fragmentation and the direct nonrelativistic QCD approaches., Comment: 10 pages, 8 figures

In the paper, we make a detailed study on the generalized Crewther Relation (GCR) between the Adler function ($D$) and the Gross-Llewellyn Smith sum rules coefficient ($C^{\rm GLS}$) by using the newly suggested single-scale approach of the principle of maximum conformality (PMC). The resultant GCR is scheme-independent, whose residual scale dependence due to unknown higher-order terms are highly suppressed. Thus a precise test of QCD theory without renormalization scheme and scale ambiguities can be achieved by comparing with the data. Moreover, a demonstration of the scheme independence of commensurate scale relation up to all orders has been presented. And as the first time, the Pade approximation approach has been adopted for estimating the unknown $5_{\rm th}$-loop contributions from the known four-loop perturbative series., Comment: 8 pages, 2 figures

In the paper, we study the properties of the $Z$-boson hadronic decay width by using the $\mathcal{O}(\alpha_s^4)$-order quantum chromodynamics (QCD) corrections with the help of the principle of maximum conformality (PMC). By using the PMC single-scale approach, we obtain an accurate renormalization scale-and-scheme independent perturbative QCD (pQCD) correction for the $Z$-boson hadronic decay width, which is independent to any choice of renormalization scale. After applying the PMC, a more convergent pQCD series has been obtained; and the contributions from the unknown $\mathcal{O}(\alpha_s^5)$-order terms are highly suppressed, e.g. conservatively, we have $\Delta \Gamma_{\rm Z}^{\rm had}|^{{\cal O}(\alpha_s^5)}_{\rm PMC}\simeq \pm 0.004$ MeV. In combination with the known electro-weak (EW) corrections, QED corrections, EW-QCD mixed corrections, and QED-QCD mixed corrections, our final prediction of the hadronic $Z$ decay width is $\Gamma_{\rm Z}^{\rm had}=1744.439^{+1.390}_{-1.433}$ MeV, which agrees with the PDG global fit of experimental measurements, $1744.4\pm 2.0$ MeV., Comment: 7 pages, 3 figures

In the paper, we study the properties of the top-quark $\overline{\rm MS}$ running mass computed from its on-shell mass by using both the four-loop $\overline{\rm MS}$-on-shell relation and the principle of maximum conformality (PMC) scale-setting approach. The PMC adopts the renormalization group equation to set the correct magnitude of the strong running coupling of the perturbative series, its prediction avoids the conventional renormalization scale ambiguity, and thus a more precise pQCD prediction can be achieved. After applying the PMC to the four-loop $\overline{\rm MS}$-on-shell relation and taking the top-quark on-shell mass $M_t=172.9\pm0.4$ GeV as an input, we obtain the renormalization scale-invariant $\overline{\rm MS}$ running mass at the scale $m_t$, e.g., $m_t(m_t)\simeq 162.6\pm 0.4$ GeV, in which the error is the squared average of those from $\Delta \alpha_s(M_Z)$, $\Delta M_t$, and the approximate error from the uncalculated five-loop terms predicted by using the Pad\'{e} approximation approach., Comment: 8 pages, 2 figures

The momentum space subtraction (MOM) scheme is one of the most frequently used renormalization schemes in perturbative QCD (pQCD) theory. In the paper, we make a detailed discussion on the gauge dependence of the pQCD prediction under the MOM scheme. Conventionally, there is renormalization scale ambiguity for the fixed-order pQCD predictions, which assigns an arbitrary range and an arbitrary error for the fixed-order pQCD prediction. The principle of maximum conformality (PMC) adopts the renormalization group equation to determine the magnitude of the coupling constant and hence determines an effective momentum flow of the process, which is independent to the choice of renormalization scale. There is thus no renormalization scale ambiguity in PMC predictions. To concentrate our attention on the MOM gauge dependence, we first apply the PMC to deal with the pQCD series. We adopt the Higgs boson decay width, $\Gamma(H\to gg)$, up to five-loop QCD contributions as an example to show how the gauge dependence behaves before and after applying the PMC. It is found that the Higgs decay width $\Gamma (H\to gg)$ depends very weakly on the choices of the MOM schemes, being consistent with the renormalization group invariance. It is found that the gauge dependence of $\Gamma(H\to gg)$ under the $\rm{MOMgg}$ scheme is less than $\pm1\%$, which is the smallest gauge dependence among all the mentioned MOM schemes., Comment: 21 pages, 13 figures. References updated

In this paper, we calculate the total decay widths for the $W^+$-boson decays, $W^+ \to B_c+b+\bar{s}+X$ and $W^+ \to B^*_c+b+\bar{s}+X$, up to next-to-leading order (NLO) accuracy within the framework of the nonrelativistic QCD theory. Both the fixed-order and the fragmentation approaches are adopted to do the calculation. Differential decay widths $d\Gamma/dz$ and $d\Gamma/ds_1$ are also given. We find that the NLO corrections are significant in those two $W^+$ decay channels. Our numerical results show that at the LHC, there are about $7.03\times 10^4$ $B_c$-meson events and $5.10\times 10^4$ $B^*_c$-meson events to be produced via the $W^+$-boson decays per operation year., Comment: 11 pages, 9 figures

In the paper, we derive the next-to-leading order (NLO) fragmentation function for a heavy quark, either charm or bottom, into a heavy quarkonium $J/\Psi$ or $\Upsilon$. The ultra-violet divergences in the real corrections are removed through the operator renormalization, which is performed under the modified minimal subtraction scheme. We then obtain the NLO fragmentation function at an initial factorization scale, e.g. $\mu_{F}=3 m_c$ for $c\to J/\Psi$ and $\mu_{F}=3m_b$ for $b\to \Upsilon$, which can be evolved to any scale via the use of Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation. As an initial application of those fragmentation functions, we study the $J/\Psi$ ($\Upsilon$) production at a high luminosity $e^+e^-$ collider which runs at the energy around the $Z$ pole and could be a suitable platform for testing the fragmentation function., Comment: 13 pages, 11 figures

The fragmentation functions for $c$ or $\bar{b}$ quark to $B_c$ or $B_c^*$ meson are derived up to QCD next-to-leading order (NLO). They are further computed numerically and presented in figures precisely. In order to reach a higher accuracy, we have also tried properly to apply them to estimating the $B_c$ and $B_c^*$ production at a $Z$-factory (an $e^+e^-$ collider running at the energy of the $Z$-boson pole)., Comment: 20 pages, 13 figures

A proposal on determination of the effective electro-weak mixing angle ${\rm sin^2\theta_{eff}^f\;(f=c,b)}$ at a super Z-factory (an $e^+e^-$ collider designed with the possible highest luminosity and running around the center mass energy $\sqrt s=m_Z$) is proposed. It is to determine how the effective mixing angles depend on the flavors, especially the flavors $c$, $b$ (with the determined effective ones, the electro-weak mixing angles ${\rm sin^2\theta_W^f\;(f=c,b)}$ of Standard Model can be derived out precisely). The key point of the proposal is that at such a super Z-factory, via measuring the forward-backward ($A_{FB}$), left-right ($A_{LR}$) and combined left-right forward-backward ($A^{FB}_{LR}$) asymmetries of the produced doubly heavy-flavored hadrons (such as $B_c$, $B_c^*$ and the baryons $H_{QQ'q}$: $\Xi_{cc}$, $\Xi_{bc}$ and $\Xi_{bb}$ etc) in the relevant production processes $e^+e^-\to B_c+\cdots$ and $e^+e^-\to H_{QQ'q}+\cdots$ one may complement the final determination of the mixing angle ${\rm sin^2\theta_W^f\;(f=c,b)}$ of Standard Model. The advantage of the proposed way to determine the effective electro-weak mixing angles is that the doubly heavy flavor(s) and the out-going direction of the produced doubly-heavy hadron can be experimental determined precisely, i.e. there are no such errors caused by missing identification of the heavy flavor(s) and by determining the so-called thrust axis of the produced jets, which can not be avoided in the early determination at LEP-I and SLC., Comment: 9 pages, 7 figures

In the paper, we present a detailed discussion on the semi-inclusive production of doubly heavy baryons ($\Xi_{bc}$ and $\Xi_{bb}$) through top-quark decay channel, $t\rightarrow \Xi_{bQ^{\prime}}+ \bar {Q^{\prime}} + W^+ $, within the framework of non-relativistic QCD (NRQCD). In our calculations, the contributions from the intermediate diquark states, $\langle bc\rangle[^{3}S_{1}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bc\rangle[^{1}S_{0}]_{\mathbf{\bar 3}/ \mathbf{6}}$, $\langle bb\rangle[^{1}S_{0}]_{\mathbf{6}}$ and $\langle bb\rangle[^{3}S_{1}]_{\mathbf{\bar 3}}$, have been taken into consideration. Main uncertainties from the heavy quark mass $(m_c,~m_b~\mbox{or}~m_t)$, the renormalization scale $\mu_r$, and the nonperturbative transition probability have been estimated. For a comparison, we also analyze the production of doubly heavy baryons under the approximate fragmentation function approach. Estimated at the LHC or a High Luminosity LHC with $\mathcal{L}$ =$10^{34-36}~\rm{cm}^{-2}~\rm{s}^{-1}$, there will be about $2.25\times10^{4-6}$ events of $\Xi_{bc}$ and $9.49\times10^{2-4}$ events of $\Xi_{bb}$ produced in one operation year through top-quark decays., Comment: 17 pages, 5 figures, 5 tables

Production of the $B_c$ and $B_c^*$ mesons at $Z$-factory (an $e^+e^-$ collider running at energies around $Z$ pole) is calculated up-to the next-to-leading order (NLO) QCD corrections. The results show that the dependence of the total cross sections on the renormalization scale $\mu$ is reduced by the corrections, and the NLO corrections enhance the production cross section by $52\%$($33\%$) when the renormalization scale is taken at $\mu=2m_b$. For experimental observations of the $B_c$ and $B_c^*$ mesons, the differential cross sections for the production are also analyzed up-to the NLO accuracy., Comment: 11 pages, 10 figures

Production of the doubly heavy-flavored hadrons ($B_c$ meson, doubly heavy baryons $\Xi_{cc}$, $\Xi_{bc}$, $\Xi_{bb}$, their excited states and antiparticles of them as well) at $e^+e^-$ colliders is investigated under two different approaches: $LO$ (leading order QCD complete calculation) and $LL$ (leading logarithm fragmentation calculation). The results for the production obtained by the approaches $LO$ and $LL$, including the angle distributions of the produced hadrons with unpolarized and polarized incoming beams, the behaviors on the energy fraction of the produced doubly heavy hadron and comparisons between the two approaches' results, are presented in terms of tables and figures. Thus characteristics of the production and uncertainties of the approaches are shown precisely, and it is concluded that only if the colliders run at the eneries around $Z$-pole (which may be called as $Z$-factories) and additionally the luminosity of the colliders is as high as possible then to study the doubly heavy hadrons thoroughly is accessible., Comment: 13 pages, 13 figures

The principle of maximum conformality (PMC) provides a way to eliminate the conventional renormalization scale ambiguity in a systematic way. By applying the PMC scale setting, all non-conformal terms in perturbative series are summed into the running coupling, and one obtains a unique, scale-fixed prediction at any finite order. In the paper, we make a detailed PMC analysis for the spin-singlet heavy quarkoniums decay (into light hadrons) at the next-to-leading order. After applying the PMC scale setting, the decay widths for all those cases are almost independent of the initial renormalization scales. The PMC scales for $\eta_c$ and $h_c$ decays are below $1$ GeV, in order to achieve a confidential pQCD estimation, we adopt several low-energy running coupling models to do the estimation. By taking the MPT model, we obtain: $\Gamma(\eta_{c} \to LH)=25.09^{+5.52}_{-4.28}$ MeV, $\Gamma(\eta_{b} \to LH)=14.34^{+0.92}_{-0.84}$ MeV, $\Gamma(h_{c} \to LH)=0.54^{+0.06}_{-0.04}$ MeV and $\Gamma(h_{b} \to LH)=39.89^{+0.28}_{-0.46}$ KeV, where the errors are calculated by taking $m_{c}\in[1.40\rm GeV,1.60\rm GeV]$ and $m_{b}\in[4.50\rm GeV,4.70\rm GeV]$. These decay widths agree with the principle of minimum sensitivity estimations, in which the decay widths of $\eta_{c,b}$ are also consistent with the measured ones., Comment: 5 pages, 1 figures. Typos corrected, to be published in Chin.Phys.Lett

The principle of maximum conformality (PMC) provides a convenient way for setting the optimal renormalization scales for high-energy processes, which can eliminate the conventional renormalization scale error via an order-by-order manner. At present, we make a detailed PMC analysis on the Higgs decay $H\rightarrow \gamma\gamma$ up to three-loop QCD corrections. As an important point of deriving reliable PMC estimation, it is noted that only those $\{\beta_i\}$-terms that rightly determine the running behavior of coupling constant via the renormalization group equation should be absorbed into the coupling constant, and those $\{\beta_i\}$-terms that pertain to the quark mass renormalization and etc. should be kept as a separate. To avoid confusion of separating and absorbing different types of $\{\beta_i\}$-terms into the coupling constant, we first transform the decay width in terms of top quark $\overline{\rm MS}$ mass into that of on-shell mass and then apply the PMC scale setting. After applying PMC scale setting, the final estimation is conformal and is scheme-independent and scale-independent. Up to three-loop QCD corrections, we obtain a PMC scale $\mu^{\rm PMC}_{r}=242.3$ GeV $\sim 2M_H$, which is optimal and highly independent of any choice of initial scale. Thus, we obtain a more accurate scale-independent prediction by taking the Higgs mass as the same as that of ATLAS and CMS measurements, i.e., $\Gamma(H\rightarrow \gamma\gamma)|_{\rm ATLAS}=9.504^{+0.226}_{-0.252}$ keV and $\Gamma(H\rightarrow \gamma\gamma)|_{\rm CMS}=9.568^{+0.195}_{-0.191}$ keV, where the error is caused by the measured Higgs mass, i.e. the Higgs mass $M_{H}$ is taken as $125.5\pm0.2^{+0.5}_{-0.6}$ GeV for ATLAS and $125.7\pm0.3\pm0.3$ GeV for CMS, respectively., Comment: 16 pages, 2 figures. References updated and discussion improved, to be published in J.Phys.G

In our previous work [2], it has been found that sizable charmonium events via the channel $e^+e^- \to \gamma^*/Z^0 \to H(|c\bar{c}\rangle) +\gamma$ can be produced at the suggested super $Z$ factory, where $H(|c\bar{c}\rangle)$ represents the dominant color-singlet $S$-wave and $P$-wave charmonium states $J/\psi$, $\eta_c$, $h_c$ and $\chi_{cJ}$ ($J=0, 1, 2$), respectively. As an important step forward, in the present paper, we present a next-to-leading order (NLO) QCD analysis within the framework of nonrelativistic QCD. In different to the case of $B$ factory in which the single charmonium production is dominated by the channel via a virtual photon, at the super $Z$ factory, its cross-section is dominated by the channel via a $Z^0$ boson. Estimations up to NLO level are done at both the $B$ factory and the super $Z$ factory experimental conditions. We observe that the NLO distributions have the same shapes as those of LO distributions, but their differences are sizable. This indicates that a NLO calculation is necessary and important to achieve a more accurate estimation. Due to $Z^0$ boson resonance effect, at the super $Z$ factory with a high luminosity up to $10^{36}{\rm cm}^{-2}{\rm s}^{-1}$, when summing all the color-singlet states' contribution together, one may observe about $8.0\times10^{4}$ charmonium events via the channel $e^+e^-\to Z^0 \to H(|c\bar{c}\rangle)+\gamma$ in one operation year. Then, such super $Z$ factory could provide another useful platform to study the charmonium properties, even for the higher charmonium states., Comment: 12 pages, 6 figures. To match the published version, to be published in Phys.Rev.D

The principle of maximum conformality (PMC) has been suggested to eliminate the renormalization scheme and renormalization scale uncertainties, which are unavoidable for the conventional scale setting and are usually important errors for theoretical estimations. In this paper, by applying PMC scale setting, we analyze two important inclusive Standard Model Higgs decay channels, $H\rightarrow b\bar{b}$ and $H\rightarrow gg$, up to four-loop and three-loop levels accordingly. After PMC scale setting, it is found that the conventional scale uncertainty for these two channels can be eliminated to a high degree. There is small residual initial scale dependence for the Higgs decay widths due to unknown higher-order $\{\beta_i\}$-terms. Up to four-loop level, we obtain $\Gamma(H\rightarrow b\bar{b}) = 2.389\pm0.073 \pm0.041$ MeV and up to three-loop level, we obtain $\Gamma(H\rightarrow gg) = 0.373\pm0.030$ MeV, where the first error is caused by varying $M_H=126\pm4$ GeV and the second error for $H\to b\bar{b}$ is caused by varying the $\overline{\rm MS}$-running mass $m_b(m_b)=4.18\pm0.03$ GeV. Taking $H\to b\bar{b}$ as an example, we present a comparison of three BLM-based scale setting approaches, e.g. the PMC-I approach based on the PMC-BLM correspondence, the $R_\delta$-scheme and the seBLM approach, all of which are designed to provide effective ways to identify non-conformal $\{\beta_i\}$-series at each perturbative order. At four-loop level, all those approaches lead to good pQCD convergence, they have almost the same pQCD series, and their predictions are almost independent on the initial renormalization scale. In this sense, those approaches are equivalent to each other., Comment: 14 pages, 7 figures. References updated and discussions improved. To be published in Eur.Phys.J.C

We apply the principle of maximum conformality (PMC) to the Balitsky-Fadin-Kuraev-Lipatov (BFKL) Pomeron intercept at the next-to-leading logarithmic (NLL) accuracy. The PMC eliminates the conventional renormalization scale ambiguity by absorbing the non-conformal $\{\beta_i\}$-terms into the running coupling, and a more accurate pQCD estimation can be obtained. After PMC scale setting, the QCD perturbative convergence can be greatly improved due to the elimination of renormalon terms in pQCD series, and the BFKL Pomeron intercept has a weak dependence on the virtuality of the reggeized gluon. For example, by taking the Fried-Yennie gauge, we obtain $\omega_{\rm MOM}^{\rm PMC}(Q^{2},0)\in [0.149,0.176]$ for $Q^2\in[1,100]\;{\rm GeV}^2$. This is a good property to apply to the high-energy phenomenology. Further more, to compare with the data, it is found that the physical ${\rm MOM}$-scheme is more reliable than the $\overline{\rm MS}$-scheme. The ${\rm MOM}$-scheme is gauge dependent, which can also be greatly suppressed after PMC scale setting. We discuss the MOM-scheme gauge dependence for the Pomeron intercept by adopting three gauges, i.e. the Landau gauge, the Feynman gauge and the Fried-Yennie gauge, and we obtain $\omega_{\rm MOM}^{\rm PMC}(Q^{2}=15\;{\rm GeV}^2,0) = 0.166^{+0.010}_{-0.017}$; i.e. about $10\%$ gauge dependence is observed. We apply the BFKL Pomeron intercept to the photon-photon collision process, and compare the theoretical predictions with the data from the OPAL and L3 experiments., Comment: 18 pages, 6 figures. References updated and discussions improved

Under the conventional scale setting, the renormalization scale uncertainty usually constitutes a systematic error for a fixed-order perturbative QCD estimation. The recently suggested principle of maximum conformality (PMC) provides a principle to eliminate such scale ambiguity in a step-by-step way. Using the PMC, all non-conformal terms in perturbative expansion series are summed into the running coupling, and one obtains a unique, scale-fixed, scheme-independent prediction at any finite order. In the paper, we make a detailed PMC analysis for both the polarized and the unpolarized cross sections for the double charmonium production process, $e^{+} + e^{-} \to J/ \psi (\psi') + \chi_{cJ}$ with $(J=0,1,2)$. The running behavior for the coupling constant, governed by the PMC scales, are determined exactly for the specific processes. We compare our predictions with the measurements at the $B$ factories, BABAR and Belle, and the theoretical estimations obtained in the literature. Because the non-conformal terms are different for various polarized and unpolarized cross sections, the PMC scales of these cross sections are different in principle. It is found that all the PMC scales are almost independent of the initial choice of renormalization scale. Thus, the large renormalization scale uncertainty usually adopted in the literature up to $sim40%$ at the NLO level, obtained from the conventional scale setting, for both the polarized and the unpolarized cross sections are greatly suppressed. It is found that the charmonium production is dominated by J=0 channel. After PMC scale setting, we obtain $\sigma(J/\psi+\chi_{c0}) = 12.25^{+3.70}_{-3.13}$ fb and $\sigma(\psi'+\chi_{c0}) =5.23^{+1.56}_{-1.32}$ fb,......, Comment: 10 pages, 2 figures. Numerical results and references updated, and discussions improved. To be published in Nucl.Phys.B

Within the framework of nonrelativistic QCD, we make a detailed discussion on the doubly heavy baryon production through the $e^+ e^-$ annihilation channel, $e^{+}e^{-}\rightarrow\gamma/Z^0 \rightarrow \Xi_{QQ^{\prime}} +\bar{Q} +\bar{Q^{\prime}}$, at a high luminosity $e^{+}e^{-}$ collider. Here $Q^{(\prime)}$ stands for the heavy $b$ or $c$ quark. In addition to the channel through the usually considered diquark state $(QQ^{\prime})[^3S_1]_{\bf\bar{3}}$, contributions from the channels through other same important diquark states such as $(QQ^{\prime})[^1S_0]_{\bf 6}$ have also been discussed. Uncertainties for the total cross sections are predicted by taking $m_c=1.80\pm0.30$ GeV and $m_b=5.10\pm0.40$ GeV. At a super $Z$-factory running around the $Z^0$ mass and with a high luminosity up to ${\cal L} \propto 10^{34}\sim 10^{36}{\rm cm}^{-2} {\rm s}^{-1}$, we estimate that about $1.1\times10^{5 \sim 7}$ $\Xi_{cc}$ events, $2.6\times10^{5 \sim 7}$ $\Xi_{bc}$ events and $1.2\times 10^{4 \sim 6}$ $\Xi_{bb}$ events can be generated in one operation year. Such a $Z$-factory, thus, will provide a good platform for studying the doubly heavy baryons in comparable to the CERN large hadronic collider., Comment: 9 pages, 4 figures. To be published in Phys.Rev. D