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Qi-Li Liao

Publications and source records attributed to Qi-Li Liao.

15 recordsLinked to original sources
arXiv2023

Production of double heavy quarkonium at super $Z$ factory

Within the color singlet model, we calculate the exclusive production of double charmonia, double bottomonia, and double $B_c$ mesons at future super $Z$ factory. The two heavy quarkonia or $B_c$'s are either two S-wave Fock states ($^1S_0, ~^3S_1$), or one S-wave and one P-wave states ($^1P_1,~^3P_J~(J=0,1,2)$). The top three $Z^0$ propagated channels in cross sections for double charmonia are $J/ψ+h_c,η_c+χ_{c2}$, and $η_c+χ_{c0}$. For double bottomonia, they are $η_b+Υ, Υ+χ_{b2}$, and $Υ+Υ$. For double $B_c$ mesons, they are $B_c^{*+}+B_c^{*-}$,$B_c^{*+}+χ_{bc2}^-$, and $η_{bc}^++B_c^{*-}$. The cross sections of double $B_c$ mesons are roughly one order of magnitude larger than those of the double bottomonia, and two orders of magnitude larger than those of the double charmonia. To make it helpful for experimental study, we present the total cross sections $σ$ as functions of CM energy $\sqrt{s}$, $σ$ as functions of the renormalization scale $μ$, the angle distributions $dσ/dcosθ$, and the $p_T$ distributions $dσ/dp_{t}$. The uncertainties come from the varying masses of constituent heavy quarks bring up to 20% corrections. We also find that the initial state radiation can bring about 30%$\sim$40% suppresions when 1\%$m_Z$ energy is losing, and cross sections can increase by about $2\sim3$ times or decrease by an order of magnitude when adopting different potential models which becomes the major source of uncertainty. The numerical results show that it might be not optimistic for the experimental observation, but it is still far from excluded at the FCC-ee and also the CEPC running in the $Z$ factory mode.

hep-ph↗
arXiv2022

Production double heavy quarkonium via $e^+e^-\to γ^*/Z^0 \to|(Q\bar{Q'})[n]\rangle +|(Q'\bar{Q})[n']\rangle$ at Z mass pole

The exclusive production of double excited quarkonium is comprehensive studied, i.e., the production of double excitedcharmonium, double excited bottomonium, and double excited $Bc$-mesons via $e^+e^-\to γ^*/Z^0 \to|(Q\bar{Q'})[n]\rangle +|(Q'\bar{Q})[n']\rangle$ ($Q/Q'=c$- or $b$-quarks) at a future $Z$ factory under the nonrelativistic quantum chromodynamics framework, where the $[n]$~/$[n']$ represents the color-singlet heavy quarkonium states $[^1S_0]\rangle, ~[^3S_1]\rangle, ~[^1P_1]\rangle$, and $[^3P_J]\rangle$ ($J=0,1,2$). The "improved trace technology" is adopted for calculating the complicated $P$-wave channels for derive the analytic expressions at the amplitude level. According to our study, the production rates of double heavy quarkonium are considerable at the future $Z$ factory. We obtain the cross sections for the production of double excited charmonium for $σ{(|(c\bar{c})[n]\rangle+|(c\bar{c})[n']\rangle)_{total}}=1.167^{+0.113}_{-0.164}\times 10^{-2}~fb$, the cross sections of double excited bottomonium for $σ{(|(b\bar{b})[n]\rangle+|(b\bar{b})[n']\rangle)_{total}}=0.1132^{+0.0139}_{-0.0130}~fb$, and the cross sections of double excited $Bc$-mesons for $σ{(|(c\bar{b})[n]\rangle+|(b\bar{c})[n']\rangle)_{total}}=3.692^{+0.124}_{-0.097}~fb$. The main uncertainties come from the mass of the heavy quarkonium and the radial wave functions at the origin and their derivatives at the origin under different potential models. The numerical results show that such the super $Z$ factory should be a good platform to study the properties of the double excited charmonium, the double excited bottomonium , especially the double excited $Bc$-mesons.

hep-ph↗
arXiv2021

Production of excited heavy quarkonia in $e^+e^- \to γ^*/Z^0 \to |(Q\bar{Q})[n]\rangle +γ$ at super $Z$ factory

Within the nonrelativistic quantum chromodynamics framework, we make a comprehensive study on the exclusive production of excited charmonium and bottomonium in $e^+e^-\to γ^*/Z^0 \to|(Q\bar{Q})[n]\rangle +γ$ ($Q=c$ or $b$ quarks) at future $Z$ factory, where the $[n]$ represents the color-singlet $n^1S_0,~n^3S_1,~n^1P_0$ and $n^3P_J$ ($n=1,2,3,4; J=0,1,2$) Fock states. The "improved trace technology" is adopted to derive the analytic expressions at the amplitude level, which is useful for calculating the complicated $nP$-wave channels. Total cross sections, differential distributions, and uncertainties are discussed in system. According to our study, production rates of heavy quarkonia of high excited Fock states are considerable at future $Z$ factory. The cross sections of charmonium for $2S$, $3S$, $4S$, $1P$, $2P$, $3P$ and $4P$-wave states are about $53.5\%$, $30.4\%$, $23.7\%$, $13.7\%$, $6.8\%$, $9.2\%$, and $9.2\%$ of that of the $1S$ state, respectively. And cross sections of bottomonium for $2S$, $3S$, $4S$, $1P$, $2P$, $3P$ and $4P$-wave states are about $39.3\%$, $12.3\%$, $14.3\%$, $7.1\%$, $3.1\%$, $2.7\%$, and $3.1\%$ of that of the $1S$ state, respectively. The main uncertainties come from the radial wave functions at the origin and their derivatives at the origin under different potential models. Then, such super $Z$ factory should be a good platform to study the properties of the high excited charmonium and bottomonium states.

hep-ph↗
arXiv2019

Excited heavy quarkonium production in Higgs boson decays

The rare decay channels of Higgs boson to heavy quarkonium offer vital opportunities to explore the coupling of Higgs to heavy quarks. We study the semi-exclusive decay channels of Higgs boson to heavy quarkonia, i.e., $H^0\to |(Q\bar{Q^{\prime}})[n]\rangle+\bar{Q}Q^{\prime}$ ($Q^{(\prime)}=c~\text{or}~b$ quark) within the NRQCD framework. In addition to the lower-level Fock states $|(Q\bar{Q'})[1S]\rangle$ continent, contributions of high excited states $|(Q\bar{Q'})[2S]\rangle$, $|(Q\bar{Q'})[3S]\rangle$, $|(Q\bar{Q'})[4S]\rangle$, $|(Q\bar{Q'})[1P]\rangle$, $|(Q\bar{Q'})[2P]\rangle$, $|(Q\bar{Q'})[3P]\rangle$ and $|(Q\bar{Q'})[4P]\rangle$ are also studied. According to our study, the contributions of high excited Fock states should be considered seriously. Differential distributions of total decay width with respect to invariant-mass and angles, as well as uncertainties caused by non-perturbative hadronic non-perturbative matrix elements are discussed. If all excited heavy quarkonium states decay to the ground spin-singlet state through electromagnetic or hadronic interactions, we obtain the decay widths for $|(Q\bar{Q'})\rangle$ quarkonium production through $H^0$ semi-exclusive decays: $25.10^{+11.6\%}_{-51.6\%}$ keV for $|(b\bar{c})[n]\rangle$ meson, $3.23^{+0\%}_{-62.2\%}$ keV for $|(c\bar{c})[n]\rangle$ and $2.36^{+0\%}_{-57.1\%}$ keV for $|(b\bar{b})[n]\rangle$, where uncertainties are caused by adopting different non-perturbative potential models. At future high energy LHC ($\sqrt{s}=27$ TeV), numerical results show that sizable amounts of events for those high excited states can be produced, which implies that one could also consider exploring the coupling properties of Higgs to heavy quarks in these high excited states channels, especially for the charmonium and bottomonium.

hep-ph↗
arXiv2018

Heavy $P$-wave quarkonium production via Higgs decays

The production of the heavy quarkonium, i.e., $|(c\bar{b})[n]\rangle$ (or $|(b\bar{c})[n]\rangle$), $|(c\bar{c})[n]\rangle$, and $|(b\bar{b})[n]\rangle$- quarkonium [$|(Q\bar{Q'})[n]\rangle$-quarkonium for short], through Higgs $H^{0}$ boson semiexclusive decays is evaluated within the NRQCD framework, where $[n]$ stands for the production of the two color-singlet $S$-wave states, $|(Q\bar{Q'})[^1S_0]_{\textbf{1}} \rangle$ and $|(Q\bar{Q'})[^3S_1]_{\textbf{1}} \rangle$, the production of the four color-singlet $P$-wave states, i.e., $|(Q\bar{Q'})[^1P_0]_{\textbf{1}}\rangle$, $|(Q\bar{Q'})[^3P_J]_{\textbf{1}}\rangle$ (with $J =[0, 1, 2]$). Moreover, according to the velocity scaling rule of the NRQCD, the production of the two color-octet components, $|(Q\bar{Q'})g[^1S_0]_{\textbf{8}} \rangle$ and $|(Q\bar{Q'})g[^3S_1]_{\textbf{8}} \rangle$, are also taken into account. The "improved trace technology" to derive the simplified analytic expressions at the amplitude level is adopted, which shall be useful for dealing with these decay channels. If all higher heavy quarkonium states decay completely to the ground states, it should be obtained $Γ{(H^0\to |(c\bar{b})[^1S_0]_{\textbf{1}}\rangle)}=15.14$ KeV, $Γ{(H^0\to |(c\bar{c})[^1S_0]_{\textbf{1}}\rangle)}=1.547$ KeV, and $Γ{(H^0\to |(b\bar{b})[^1S_0]_{\textbf{1}}\rangle)}=1.311$ KeV. The production of $5.6\times10^{5}$ Bc meson, $4.7\times10^{4}$ charmonium meson, and $4.9\times10^{4}$ bottomonium meson per year in Higgs decays at the HE/HL-LHC can be obtained.

hep-ph↗
arXiv2018

${\bar{B}^{0}_{s}}$ and its excited meson production via top quark decays at the LHC

In this work we evaluate the masses of the $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ quarkonium ($\bar{B}^{0}_{s}$ or ${B}^{0}_{s}$ meson) under the B.T. potential, and the values of the Schr${\rm \ddot{o}}$dinger radial wave function at the origin of the $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ quarkonium within the five potential models. Then we investigate a systematic study on the production of the $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ quarkonium via top quark or antitop quark decays in the color-singlet QCD factorization formula (CSQCDFF), i.e., the two $S$-wave states, $|(b\bar{s})[1^1S_0] \rangle$ (or $|(\bar{b}s)[1^1S_0] \rangle$) and $|(b\bar{s})[1^3S_1] \rangle$ (or $|(\bar{b}s)[1^3S_1] \rangle$), and its four $P$-wave excited states, $|(b\bar{s})[1^1P_1] \rangle$ (or $|(\bar{b}s)[1^1P_1] \rangle$) and $|(b\bar{s})[1^3P_J] \rangle$ (or $|(\bar{b}s)[1^3P_J] \rangle$) (with $J =[0, 1, 2]$). For deriving compact analytical results for complex processes, the "improved trace technology" is adopted to deal with the decay channels at the amplitudes. Moreover, various differential distributions and uncertainties of the concerned processes are analyzed carefully. By adding the uncertainties caused by the ${b}$ and ${s}$-quark masses in quadrature, we obtain $Γ{(t\to |(b\bar{s})[n]\rangle +W^{+}s)}=14.19^{+4.36}_{-3.20}$~MeV. At the LHC with the luminosity ${\cal L}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$ TeV, sizable $|(b\bar{s})[n]\rangle$ or $|(\bar{b}s)[n]\rangle$ meson events can be produced through ${t}$-quark or ${\bar{t}}$-quark decays; i.e., about $1.3~\times10^6$ ${\bar{B}^0_s}$ or ${B^0_s}$ events per year can be obtained.

hep-ph↗
arXiv2015

Excited Heavy Quarkonium Production via Z^0 Decays at a High Luminosity Collider

We present a systematic study of the production of the heavy quarkonium, i.e., $|(c\bar{c})[n] \rangle$ , $|(b\bar{c})[n] \rangle$ (or $|(c\bar{b})[n] \rangle$), and $|(b\bar{b})[n] \rangle$ quarkonium [$|(Q\bar{Q'})[n]\rangle$ quarkonium for short], through $Z^0$ boson semi-exclusive decays with new parameters \cite{lx} for the heavy quarkonium under the framework of the NRQCD, where $[n]$ stands for $n^1S_0$, $n^3S_1$, $n^1P_0$, $n^3P_J$ ($n=1, \cdots, 6$; $J=(0, 1, 2)$). "Improved trace technology" is adopted to derive the simplified analytic expressions at the amplitude level, which shall be useful for dealing with these decay channels. If all higher $|(Q\bar{Q'})[n]\rangle$ quarkonium states decay to the ground state $|(Q\bar{Q'})[1^1S_0]\rangle$ with $100\%$ efficiency via electromagnetic or hadronic interactions, we obtain $Γ{(Z^0\to |(c\bar{c})[1^1S_0]\rangle)}=1476$ KeV, $Γ{(Z^0\to |(b\bar{c})[1^1S_0]\rangle)}=1485$ KeV, $Γ{(Z^0\to |(b\bar{b})[1^1S_0]\rangle)}=127.5$ KeV. At the LHC and ILC with the luminosity ${\cal L}\propto 10^{34}cm^{-2}s^{-1}$, sizable heavy quarkonium events can be produced through $Z^0$ boson decays, i.e., about $5.9~\times10^{5}$ $(c\bar{c})$, $6.0~\times10^{5}$ $(b\bar{c})$ (or $(c\bar{b})$), $5.1~\times10^{4}$ $(b\bar{b})$ events per year can be obtained.

hep-ph↗
arXiv2014

Heavy quarkonium wave functions at the origin and excited heavy quarkonium production via top quark decays at the LHC

The value of quarkonium wave function at the origin is an important quantity while studying many physical problems concerning a heavy quarkonium. This is because that it is widely used to evaluate the production and decay amplitudes of the heavy quarkonium within the effective filed theory framework, e.g., the non-relativistic QCD (NRQCD). In this paper, the value of the Schr${\rm \ddot{o}}$dinger radial wave function or its first nonvanishing derivative at zero quark-antiquark separation, i.e., $|(|c\bar{c})[n]\rangle$-, $|(|b\bar{c})[n]\rangle$-, and $|(b\bar{b})[n]\rangle$-quarkonium, have been tabulated under five potential models with new parameters of the heavy quarkonium. Moreover, the production of the lower-level Fock states $|(b\bar{Q})[1S]\rangle$ and $|(b\bar{Q})[1P]\rangle$, together with the higher excited Fock states $|(b\bar{Q})[nS]\rangle$ and $|(b\bar{Q})[nP]\rangle$ ($Q$ stands for $c$- or $b$-quark; $n=2,\cdots,6 $) through top quark decays have been studied with the new values of heavy quarkonium wave functions at the origin under the framework of NRQCD. At the LHC with the luminosity ${\cal L}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$ TeV, sizable heavy quarkonium events can be produced through top quark decays, i.e., $4 \times10^5$ $B_c$ and $B^*_c$, and $2 \times10^4$ $η_b$ and $Υ$ events per year can be obtained according to our calculation.

hep-ph↗
arXiv2012

Doubly Heavy Baryon Production at A High Luminosity $e^+ e^-$ Collider

Within the framework of nonrelativistic QCD, we make a detailed discussion on the doubly heavy baryon production through the $e^+ e^-$ annihilation channel, $e^{+}e^{-}\rightarrowγ/Z^0 \rightarrow Ξ_{QQ^{\prime}} +\bar{Q} +\bar{Q^{\prime}}$, 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 $1.1\times10^{5 \sim 7}$ $Ξ_{cc}$ events, $2.6\times10^{5 \sim 7}$ $Ξ_{bc}$ events and $1.2\times 10^{4 \sim 6}$ $Ξ_{bb}$ 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.

hep-ph↗
arXiv2012

Revisiting the Twist-3 Distribution Amplitudes of $K$ Meson within the QCD Background Field Approach

In the present paper, we investigate the kaon twist-3 distribution amplitudes (DAs) $ϕ_{p,σ}^K$ within the QCD background field approach. The $SU_f(3)$-breaking effects are studied in detail under a systematical way, especially the sum rules for the moments of $ϕ_{p,σ}^K$ are obtained by keeping all the mass terms in the $s$-quark propagator consistently. After adding all the uncertainties in quadrature, the first two Gegenbauler moments of $ϕ_{p,σ}^K$ are $a^1_{K,p}(1 {\rm GeV}) = -0.376^{+0.103}_{-0.148}$, $a^2_{K,p}(1 {\rm GeV}) = 0.701^{+0.481}_{-0.491}$, $a^1_{K,σ}(1 {\rm GeV}) = -0.160^{+0.051}_{-0.074}$ and $a^2_{K,σ}(1 {\rm GeV}) = 0.369^{+0.163}_{-0.149}$, respectively. Their normalization parameters $μ_K^p |_{1\rm GeV} = 1.188^{+0.039}_{-0.043}$ GeV and $μ_K^σ|_{1\rm GeV} = 1.021^{+0.036}_{-0.055}$ GeV. A detailed discussion on the properties of $ϕ^K_{p,σ}$ moments shows that the higher-order $s$-quark mass terms can indeed provide sizable contributions. Furthermore, based on the newly obtained moments, a model for the kaon twist-3 wavefunction $Ψ_{p,σ}^K(x,\mathbf{k}_\perp)$ with a better end-point behavior is constructed, which shall be useful for perturbative QCD calculations. As a byproduct, we make a discussion on the properties of the pion twist-3 DAs.

hep-ph↗
arXiv2012

Excited Heavy Quarkonium Production at the LHC through $W$-Boson Decays

Sizable amount of heavy-quarkonium events can be produced through $W$-boson decays at the LHC. Such channels will provide a suitable platform to study the heavy-quarkonium properties. The "improved trace technology", which disposes the amplitude ${\cal M}$ at the amplitude-level, is helpful for deriving compact analytical results for complex processes. As an important new application, in addition to the production of the lower-level Fock states $|(Q\bar{Q'})[1S]>$ and $|(Q\bar{Q'})[1P]>$, we make a further study on the production of higher-excited $|(Q\bar{Q'})>$-quarkonium Fock states $|(Q\bar{Q'})[2S]>$, $|(Q\bar{Q'})[3S]>$ and $|(Q\bar{Q'})[2P]>$. Here $|(Q\bar{Q'})>$ stands for the $|(c\bar{c})>$-charmonium, $|(c\bar{b})>$-quarkonium and $|(b\bar{b})>$-bottomonium respectively. We show that sizable amount of events for those higher-excited states can also be produced at the LHC. Therefore, we need to take them into consideration for a sound estimation.

hep-ph↗
arXiv2012

$B_c$ Meson Production around the $Z^0$ Peak at a High Luminosity $e^+ e^-$ Collider

Considering the possibility to build an $e^+ e^-$ collider at the energies around the $Z^0$-boson resonance with a planned luminosity so high as ${\cal L}\propto 10^{34}\sim 10^{36}cm^{-2}s^{-1}$ (super $Z$-factory), we make a detailed discussion on the $(c\bar{b})$-quarkonium production through $e^{+}+e^{-}\rightarrow (c\bar{b})[n]+b+\bar{c}$ within the framework of non-relativistic QCD. To simplify the hard-scattering amplitude as much as possible and to derive analytic expressions for the purpose of future events simulation, we adopt the "improved trace technology" to do our calculation, which deals with the hard scattering amplitude directly at the amplitude level other than the conventional way at the squared-amplitude level. Total cross-section uncertainties caused by the quark masses are predicted by taking $m_c=1.50\pm0.30$ GeV and $m_b=4.90\pm0.40$ GeV. If all higher $(c\bar{b})$-quarkonium states decay to the ground state $B_c$ ($|(c\bar{b})_{\bf 1}[^1S_0]>$) with 100% efficiency, we obtain $σ_{e^{+}+e^{-}\rightarrow B_{c}+b+\bar{c}} =5.190^{+6.222}_{-2.419}$ pb, which shows that about $10^5 \sim 10^7$ $B_c$ events per operation year can be accumulated in the super $Z$-factory. If taking the collider energy runs slightly off the $Z^0$-peak, i.e. $\sqrt{S}=(1.00\pm0.05) m_Z$, the total cross-section shall be lowered by about one-order from its peak value. Such a super $Z$-factory shall provide another useful platform to study the properties of $B_c$ meson, or even the properties of its excited $P$-wave states, in addition to its production at the hadronic colliders Tevatron and LHC.

hep-ph↗
arXiv2012

Heavy Quarkonium Production at LHC through $W$ Boson Decays

The production of the heavy $(c\bar{c})$-quarkonium, $(c\bar{b})$-quarkonium and $(b\bar{b})$-quarkonium states ($(Q\bar{Q'})$-quarkonium for short), via the $W^+$ semi-inclusive decays, has been systematically studied within the framework of the non-relativistic QCD. In addition to the two color-singlet $S$-wave states, we also discuss the production of the four color-singlet $P$-wave states $|(Q\bar{Q'})(^1P_1)_{\bf 1}>$ and $(Q\bar{Q'})(^3P_J)_{\bf 1}>$ (with $J=(1,2,3)$) together with the two color-octet components $|(Q\bar{Q'})(^1S_0)_{\bf 8}>$ and $|(Q\bar{Q'})(^3S_1)_{\bf 8}>$. Improved trace technology is adopted to derive the simplified analytic expressions at the amplitude level, which shall be useful for dealing with the following cascade decay channels. At the LHC with the luminosity ${\cal L}\propto 10^{34}cm^{-2}s^{-1}$ and the center-of-mass energy $\sqrt{S}=14$ TeV, sizable heavy-quarkonium events can be produced through the $W^+$ boson decays, i.e. $2.57\times10^6$ $η_c$, $2.65\times10^6$ $J/Ψ$ and $2.40\times10^6$ $P$-wave charmonium events per year can be obtained; and $1.01\times10^5$ $B_c$, $9.11\times10^4$ $B^*_c$ and $3.16\times10^4$ $P$-wave $(c\bar{b})$-quarkonium events per year can be obtained. Main theoretical uncertainties have also been discussed. By adding the uncertainties caused by the quark masses in quadrature, we obtain $Γ_{W^+\to (c\bar{c})+c\bar{s}} =524.8^{+396.3}_{-258.4}$ KeV, $Γ_{W^+\to (c\bar{b})+b\bar{s}} =13.5^{+4.73}_{-3.29}$ KeV, $Γ_{W^+\to (c\bar{b})+c\bar{c}}= 1.74^{+1.98}_{-0.73}$ KeV and $Γ_{W^+\to (b\bar{b})+c\bar{b}}= 38.6^{+13.4}_{-9.69}$ eV.

hep-ph↗
arXiv2010

$Z_0$ Boson Decays to $B^{(*)}_c$ Meson and Its Uncertainties

The programming new $e^{+}e^-$ collider with high luminosity shall provide another useful platform to study the properties of the doubly heavy $B_c$ meson in addition to the hadronic colliders as LHC and TEVATRON. Under the `New Trace Amplitude Approach', we calculate the production of the spin-singlet $B_c$ and the spin-triplet $B^*_c$ mesons through the $Z^0$ boson decays, where uncertainties for the production are also discussed. Our results show $Γ_{(^1S_0)}= 81.4^{+102.1}_{-40.5}$ KeV and $Γ_{(^3S_1)}=116.4^{+163.9}_{-62.8}$ KeV, where the errors are caused by varying $m_b$ and $m_c$ within their reasonable regions.

hep-ph↗