text:
\section{Introduction}

The exciting discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) in 2012 has completed the Standard Model of particle physics (SM) \cite{Aad:2012tfa,Chatrchyan:2012xdj}. However, there remain several fundamentally important issues that cannot be accommodated within the SM framework, i.e., tiny neutrino masses, possible candidates for dark matter and the naturalness problem. No significant deviations from the SM have been found at the energy frontier explored by the LHC, indicating that the energy scale of new physics is probably lying above $\mu \sim 1~{\rm TeV}$.
Although the SM has to be extended to solve those fundamental issues, it can still stand as a successful low-energy effective theory of a complete theory at some high energy scale, e.g., the Grand Unification Theory (GUT) at typically $\Lambda^{}_{\rm GUT} \sim 2 \times 10^{16}~{\rm GeV}$. Any complete theories at high energy scales should be able to reproduce the low-energy observables that are in accordance with the SM predictions.

A convenient and efficient approach was suggested by Steven Weinberg a long time ago to connect the physical parameters in the high-energy full theory to those in the low-energy effective theory~\cite{Weinberg:1980wa,Hall:1980kf}. The basic idea is to integrate out heavy degrees of freedom from the full theory at the decoupling mass scale $\mu^{}_0$ and construct an effective theory at $\mu \ll \mu^{}_0$, where heavy particles just disappear but nonrenormalizable operators should be taken into account. There are several practical advantages of this approach to the multi-scale field theories. First, the renormalized physical parameters in both full and effective theories can be defined without any ambiguities in the modified minimal subtraction ($\overline{\rm MS}$) scheme. Second, the renormalization-group equations (RGEs) of the running parameters can be calculated in a much simpler way than in other mass-dependent renormalization schemes. Third, the connection between the full and effective theories is simply represented as matching conditions at the decoupling scale. Finally, such a procedure is perfectly applicable to a wide class of field theories, whenever two well separated mass scales in question can be identified. In the present work, we apply this approach to the SM and focus on the running fermion masses.

Within the SM, the physical parameters are usually measured at different energy scales. For instance, the electromagnetic fine-structure constant $\alpha$ is experimentally extracted from low-energy processes in which the momentum transfer of photons is vanishingly small, and the corresponding value will be denoted as $\alpha^{}_0$ hereafter. The Fermi constant $G^{}_{\rm F}$ is precisely determined at very low energy transfer from the muon decay data. The strong coupling constant $\alpha^{}_{s}$ has been measured at many different energy scales, but commonly provided at the pole mass of $Z$-boson $M^{}_Z = 91.1876~{\rm GeV}$ via the RGE running as in the review by Particle Data Group (PDG)~\cite{Zyla:2020}. These parameters are always entangled with each other in a complicated manner when running from one energy scale to another. A complete collection of running parameters, especially the running masses, at the concerned energy scale is demanding~\cite{Xing:2019vks}. Over the last two decades, there have been several systematic calculations of the running parameters at various energy scales in the SM~\cite{Fusaoka:1998vc,Xing:2007fb,Xing:2011aa,Antusch:2013jca,Deppisch:2018flu}, which have been proved to be very useful not only for the model building but also for the SM precision physics. In this paper, we make a comprehensive update on the running parameters. The primary motivation for such an update is three-fold. First, tremendous progress has been made in the determination of light quark masses from low-energy data, in particular the sophisticated calculations in the lattice Quantum Chromodynamics (QCD)~\cite{Aoki:2019cca}\footnote{The mass ratios of light quarks are usually extracted from the precise measurements of the $\pi$- and $K$-meson masses within the framework of chiral perturbation theory, whereas the absolute masses of light quarks can be determined either from the Lattice QCD simulations of hadronic mass spectra or from spectral function sum rules for hadronic correlation functions. See, e.g. Ref.~\cite{Zyla:2020}, for a recent review.}. Second, as more and more data have been accumulated, the electroweak observables are measured more and more precisely, e.g., the pole mass of the Higgs boson $M^{}_{h} = 125.10 (14)~{\rm GeV}$ and that of the top quark $M^{}_{t} = 172.4 (7)~{\rm GeV}$ have been found by PDG with an unprecedentedly high precision, where the last digits in parentheses are the standard deviations.
Third, the matching conditions between the full SM with the gauge symmetry ${\rm SU}(3)^{}_{\rm c} \times {\rm SU}(2)^{}_{\rm L} \times {\rm U}(1)^{}_{\rm Y}$ and the effective field theory (EFT) with the unbroken gauge symmetry ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ have been computed up to the two-loop order~\cite{Martin:2018yow}. The matching between the full SM with the low-energy ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ effective theory with massive fermions has been ignored in the previous works~\cite{Fusaoka:1998vc,Xing:2007fb,Xing:2011aa,Antusch:2013jca,Deppisch:2018flu}.

Furthermore, the overall uncertainties on the running parameters, which are also useful for the model building when theoretical predictions are confronted with low-energy observables, should be treated in a consistent way. On the one hand, the RGEs obtained with perturbative computations cannot be exact and will be always terminated at finite orders, leading to additional uncertainties in the evaluation of running parameters.
On the other hand, the uncertainties of different running parameters obtained at a given energy scale could partly originate from a common source of error, e.g., the input value of $\alpha^{}_{s}(\mu)$ at $\mu = M^{}_{Z}$; therefore the correlation can help in reducing the total degree of uncertainties. To evaluate the running parameters at various interesting energy scales, we adopt the recently released code \texttt{SMDR} for the numerical computations~\cite{Martin:2019lqd}.
Other codes for similar purposes, including \texttt{RunDec} \cite{Chetyrkin:2000yt,Herren:2017osy} and \texttt{mr} \cite{Kniehl:2016enc}, are also publicly available.

The rest of this work is structured as follows. In Sec.~2, we describe in detail the general calculational framework, including the inputs at low energies, the running and matching routines, and our numerical method to propagate the uncertainties of various types. Then, the final results of running parameters are presented in Sec.~3. We summarize our main conclusions in Sec.~4.

\section{General Strategy}

\subsection{Theoretical framework}

In the SM with the full gauge symmetry ${\rm SU}(3)^{}_{\rm c} \times {\rm SU}(2)^{}_{\rm L} \times {\rm U}(1)^{}_{\rm Y}$, there are totally 14 independent parameters, which are collectively denoted as follows
\begin{eqnarray} \label{eq:Ya}
\mathcal{Y}^{}_{\rm sm} = \left\{g^{}_{s},~g,~g^{\prime},~v,~\lambda,~y^{}_{t},~y^{}_{b}, ~y^{}_{c}, ~y^{}_{s}, ~y^{}_{d},
~y^{}_{u}, ~y^{}_{\tau}, ~y^{}_{\mu}, ~y^{}_{e}\right\}
\end{eqnarray}
at a given renormalization scale $\mu$ for $\mu > \Lambda^{}_{\rm EW}$ with $\Lambda^{}_{\rm EW}$ being the energy scale of spontaneous gauge symmetry breaking. Some explanations for these parameters are in order.
The SM gauge couplings $g^{}_{s}$, $g$ and $g^{\prime}$ correspond to the ${\rm SU}(3)^{}_{\rm c}$, ${\rm SU}(2)^{}_{\rm L}$ and ${\rm U}(1)^{}_{\rm Y}$ gauge symmetries, respectively. In the Higgs sector, the vacuum expectation value $v$ and the quartic coupling $\lambda$ are chosen to be independent, so the quadratic coupling $m^2$ in the scalar potential can be expressed in terms of $v$ and $\lambda$. In addition, the Yukawa coupling for each fermion $f$ in the SM is denoted as $y^{}_{f}$. After the spontaneous breaking of the electroweak gauge symmetry, i.e., ${\rm SU}(3)^{}_{\rm c} \times {\rm SU}(2)^{}_{\rm L} \times {\rm U}(1)^{}_{\rm Y} \to {\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$, there are 16 physical parameters at low energies that can be found in the global-fit analysis from PDG~\cite{Zyla:2020}:
\begin{eqnarray} \label{eq:mqini}
& & \alpha^{}_{ s}(M^{}_{Z}),
~\alpha^{-1}_{0},
~M^{}_{t},
~m^{}_{b}(m^{}_{b}),
~m^{}_{c}(m^{}_{c}) ,
~m^{}_{s}(2{\rm ~GeV}),
~m^{}_{d}(2{\rm ~GeV}) ,
~m^{}_{u}(2{\rm ~GeV}) ,
~M^{}_{\tau} ,
~M^{}_{\mu} ,
~M^{}_{e} , \notag\\
& & G^{}_{\rm F},
~M^{}_{Z},
~M^{}_{h} ,
~M^{}_{W},
~\sin^2{\theta^{}_{\rm W}} \; .
\end{eqnarray}
Some comments on the experimental measurements of these parameters are helpful. The strong coupling constant $\alpha^{}_{s}(M^{}_{Z})$ is determined by combining the experimental data collected at different energy scales. The fine-structure constant $\alpha^{}_{0}$ is precisely derived from the measurement of $e^{\pm}_{}$ anomalous magnetic moment \cite{Mohr:2015ccw}. The pole masses of heavy SM particles, i.e.,~$M^{}_{t}$, $M^{}_{Z}$, $M^{}_{h}$ and $M^{}_{W}$, are extracted from the data accumulated at high-energy lepton and hadron colliders. It should be noticed that the quoted pole masses $M^{}_{Z}$ and $M^{}_{W}$ from PDG are actually the Breit-Wigner masses, which are defined as $(M^2_{\rm pole} + \Gamma^2)^{1/2}$ with $M^{}_{\rm pole}$ being the true pole mass and $\Gamma$ being the total decay width. The pole mass of $\tau$-lepton is obtained from various lepton collider experiments, while those of light-flavor charged leptons are determined from atomic physics, i.e., the mass ratio of electron to the nucleus in the carbon ions and the mass ratio of muon to electron in the muonic atoms. In addition, the weak mixing angle $\sin^2{\theta^{}_{\rm W}}$ is pinned down from experiments at different energy scales, such as the collider experiments running at the $Z$ pole, the neutrino-nucleon scattering and the atomic parity violation. It is worthwhile to mention that the running of flavor mixing parameters in the quark sector has been ignored in our calculations, as its impact on the running fermion masses is insignificant either in the full SM  above or in the EFTs below the electroweak scale.

The tree-level relations between the fundamental SM parameters in Eq.~(\ref{eq:Ya}) and the low-energy observables in Eq.~(\ref{eq:mqini}) are straightforward. First, the fermion masses and their Yukawa couplings are simply related by $m^{}_{f} = y^{}_{f}v/\sqrt{2}$. Then, the strong coupling constant and the gauge coupling are linked by definition as $\alpha^{}_{s} = g^{2}_{s}/{4\pi}$. The Higgs mass $M^{}_{h} \simeq 125~{\rm GeV}$ can be used to fix the quartic coupling $\lambda \simeq 0.129$ via $M^{}_{h} = \sqrt{2\lambda} v$, given the vacuum expectation value $v \simeq 246~{\rm GeV}$. Finally, the remaining tree-level relations are as follows
\begin{eqnarray} \label{eq:mqini2}
\alpha^{}_{0} = \frac{g^2 g^{\prime 2}}{4\pi \left(g^2 + g^{\prime 2}\right)} \; , \quad G^{}_{\rm F} = \frac{1}{\sqrt{2}v^2} \;, \quad M^{}_{Z} = \frac{v}{2}\sqrt{g^2+g^{\prime 2}} \; , \quad M^{}_{W} = \frac{g v}{2} \; , \quad \sin^{2}\theta^{}_{\rm W} =  \frac{g^{\prime 2}_{}}{g^2+g^{\prime 2}} \; , \quad
\end{eqnarray}
where only three parameters $g$, $g^{\prime}$ and $v$ are involved in the above five observables. According to Table~10.2, Table~10.4 and Fig.~10.2 of Ref.~\cite{Zyla:2020}, the values of  $\sin^2{\theta^{}_{\rm W}}$ and $M^{}_{W}$  derived from $\alpha^{}_{0}$, $G^{}_{\rm F}$ and $M^{}_{Z}$ are much more precise than direct measurements of them. Therefore, we will discard $\sin^2{\theta^{}_{\rm W}}$ and $M^{}_{W}$, and choose $\alpha^{}_{0}$, $G^{}_{\rm F}$ and $M^{}_{Z}$ as basic numerical inputs to fix $g$, $g^{\prime}$ and $v$. The complete set of totally 15 input parameters to be used in our numerical calculations is
\small
\begin{eqnarray} \label{eq:mqini3}
\mathcal{I}^{}_{\rm sm}&  = & \{\alpha^{}_{ s}(M^{}_{Z}) \;,~
\alpha^{}_{0} \;,~
M^{}_{t} \;,~
m^{}_{b}(m^{}_{b}) \;,~
m^{}_{c}(m^{}_{c}) \; ,~
m^{}_{s}(2{\rm ~GeV}) \;, ~
m^{}_{d}(2{\rm ~GeV}) \; ,~
m^{}_{u}(2{\rm ~GeV}) \; ,~
M^{}_{\tau} \; ,~
M^{}_{\mu} \; ,~
 \quad \\ \notag
& &  M^{}_{e} \; , ~
G^{}_{\rm F} \;,~
M^{}_{Z} \;,~
M^{}_{h} \;,~
\Delta \alpha^{(5)}_{\rm had}(M^{}_{Z})
\} \;. \quad
\end{eqnarray}
\normalsize
Following Ref.~\cite{Martin:2019lqd}, we treat $\Delta \alpha^{(5)}_{\rm had}(M^{}_{Z})$, the non-perturbative hadronic radiative contributions to the fine-structure constant $\alpha$, as an input parameter due to its non-perturbative nature. Let us explicitly write down the input parameters with their best-fit values and $1\sigma$ errors ~\cite{Zyla:2020}:
\small
\begin{alignat}{4} \label{eq:inputs}
& \alpha^{}_{ s}(M^{}_{Z}) = 0.1179(10),\,\,\,
&& \alpha^{-1}_{0} = 137.035999084(21),\,\,\,
&&G^{}_{\rm F} = 1.1663787(6)\times 10^{-5}~{\rm GeV^{-2}},\, \notag \\
& M^{}_{Z} = 91.1876(21)~{\rm GeV},\,\,\,
&&M^{}_{h} = 125.10(14)~{\rm GeV} ,\,\,\,
&&M^{}_{t} = 172.4(7)~{\rm GeV},\, \notag \\
&m^{}_{b}(m^{}_{b}) = 4.18(2)~{\rm GeV},\,\,\,
&&m^{}_{c}(m^{}_{c}) = 1.27(2)~{\rm GeV},\,\,\,
&&m^{}_{s}(2{\rm ~GeV}) = 0.093(8)~{\rm GeV},\, \notag \\
&m^{}_{d}(2{\rm ~GeV}) = 4.67(32)~{\rm MeV},\,\,\,
&&m^{}_{u}(2{\rm ~GeV}) = 2.16(38)~{\rm MeV},\,\,\,
&&M^{}_{\tau} = 1.77686(12)~{\rm GeV},\,  \notag \\
& M^{}_{\mu} = 0.1056583745(24)~{\rm GeV},\,\,\,
&&M^{}_{e} = 0.5109989461(31)~{\rm MeV},\,\,\,
&&\Delta \alpha^{(5)}_{\rm had}(M^{}_{Z}) = 0.02764(7)\;.
\end{alignat}
\normalsize
Those quoted errors have been symmetrized and assumed to be Gaussian distributed. We can clearly observe that the uncertainties of inputs are mainly coming from the hadronic sector, i.e., the strong coupling constant $\alpha^{}_{s}$, the quark masses $m^{}_{q}$ and the magnitude of hadronic correction $\Delta \alpha^{(5)}_{\rm had}(M^{}_{Z})$.

Beyond the tree-level relations, the radiative corrections should be taken into account to link the parameters defined at different renormalization scales. Combining the low-energy inputs $\mathcal{I}^{}_{\rm sm}$ in Eq.~(\ref{eq:inputs}), the RGEs and the matching conditions, one can uniquely determine the SM parameters $\mathcal{Y}^{}_{\rm sm}$ in Eq.~(\ref{eq:Ya}). The theoretical details have been elaborated in the associated paper of the \texttt{SMDR} code \cite{Martin:2019lqd}. Below we briefly summarize the key points.
\begin{itemize}
\item\emph{The fine-structure constant}. The relation between the fine-structure constant at the vanishing momentum transfer $\alpha^{}_0$ and the running gauge couplings defined in the $\overline{\rm MS}$ scheme has been given in Refs.~\cite{Chetyrkin:1996cf,Degrassi:2003rw,Degrassi:2014sxa}, where the contributions up to the three-loop order have been included. To be more specific, the one-loop correction appears in Eq.~(A.2) of Ref.~\cite{Degrassi:2014sxa}, while the higher-order corrections at $\mu = M^{}_{Z}$ can be expressed as the interpolating result, i.e., Eq.~(3.7) in the published version of Ref.~\cite{Degrassi:2014sxa}. Note that $\alpha^{-1}_{0}$ is the most precisely measured parameter of the SM, with a relative precision as high as $10^{-10}$. However, as we will show later, its $\overline{\rm MS}$ value $\alpha(M^{}_{Z})$ at $\mu = M^{}_{Z}$ suffers considerable contamination of radiative corrections, and thus is determined with a relative precision of $\sim 2\times 10^{-4}$.

\item\emph{The Fermi constant}. The Fermi constant measured at low energies can be used to fix the vacuum expectation value $v$ of the Higgs field. Including the loop corrections, it is related to $v$ defined in the Landau gauge via
\begin{eqnarray}
G^{}_{\rm F} = \frac{1+\Delta \widetilde{r}}{\sqrt{2} v^2} \;,
\end{eqnarray}
where $\Delta \widetilde{r} = {\Delta \widetilde{r}^{(1)}}/\left(16\pi^2\right) + {\Delta \widetilde{r}^{(2)}}/\left(16\pi^2\right)^2$ has been written as a sum of one-loop~\cite{Degrassi:2012ry,Martin:2019lqd} and two-loop~\cite{Martin:2019lqd} contributions.
We will use the loop-corrected expectation value $v$ which is defined by the minimum of the Higgs effective potential in the tadpole-free Landau gauge~\cite{Martin:2019lqd},
while the results with the tree-level definition $v^{}_{\rm tree} \equiv \sqrt{-m^2/\lambda}$ can be found in Refs.~\cite{Kniehl:2014yia,Kniehl:2015nwa}.

\item\emph{The RGE running and matching.} In the symmetric phase of the SM, the state-of-the-art RGEs have been implemented in \texttt{SMDR}, including the one-loop, two-loop \cite{Cheng:1973nv,Machacek:1983fi,Machacek:1984zw,Machacek:1983tz,Arason:1991ic,Ford:1992mv,Luo:2002ey} and three-loop~\cite{Tarasov,Mihaila:2012fm,Chetyrkin:2012rz,Bednyakov:2012rb,Bednyakov:2012en,
Chetyrkin:2013wya,Bednyakov:2013eba,Bednyakov:2013cpa,Bednyakov:2014pia,Herren:2017uxn} complete beta functions of gauge couplings, Yukawa couplings and the Higgs quartic coupling and the anomalous dimension of $m^2$,
as well as the independent beta functions for gauge couplings at four-loop order~\cite{Davies:2019onf}.
Other higher-order beta functions are only available with some incomplete leading terms, see e.g. Refs~\cite{Martin:2015eia,Bednyakov:2015ooa,Zoller:2015tha,Chetyrkin:2016ruf}, thus will not be included in our calculation.
Below the electroweak scale, we assume the heavy degrees of freedom in the SM (i.e., the top quark $t$, the Higgs boson $h$, the weak gauge bosons $Z$ and $W^\pm$) to simultaneously decouple from the theory. After the decoupling of these heavy particles, a low-energy EFT with five quarks, three generations of leptons, photon and gluons can be constructed, where the QCD gauge symmetry ${\rm SU}(3)^{}_{\rm c}$ and the gauge symmetry ${\rm U}(1)^{}_{\rm EM}$ for Quantum Electrodynamics (QED) are preserved. To match the parameters in the full theory with those in the EFTs, we should utilize the matching conditions to include the threshold effects from the decoupling of heavy particles. In principle, the matching scale or the decoupling scale can be chosen arbitrarily if radiative corrections at all orders are known. However, to avoid large logarithms arising from the heavy particle masses, it is in practice convenient to set the matching scale to be around the heavy particle masses, $\mu \sim \left\{M^{}_{t}, M^{}_{h}, M^{}_{Z}, M^{}_{W}\right\} \in \left( 80\cdots 173\right)~{\rm GeV}$.
In this work, we fix the matching scale at the pole mass of $Z$, i.e., $M^{}_Z = 91.1876~{\rm GeV}$. The uncertainties associated with the choice of the matching scale can be consistently treated as the theoretical errors. After the electroweak matching procedure, the resultant low-energy EFT of QCD$\times$QED involves the following set of parameters
\begin{eqnarray} \label{eq:Yb}
\mathcal{Y}^{}_{\rm eff} = \left\{\alpha^{}_{s},~\alpha,~m^{}_{b},~m^{}_{c},~m^{}_{s},~ m^{}_{d},~
m^{}_{u},~ m^{}_{\tau},~ m^{}_{\mu}, ~m^{}_{e}\right\} \;.
\end{eqnarray}
The electroweak matching conditions between $\mathcal{Y}^{}_{\rm eff}$ in Eq.~(\ref{eq:Yb}) and $\mathcal{Y}^{}_{\rm sm}$  in Eq.~(\ref{eq:Ya}) have been presented in Ref.~\cite{Martin:2018yow}. The RGEs of the physical parameters for the low-energy EFT of QCD$\times$QED have been extensively studied and updated in the literature. The beta functions of $\alpha^{}_{s}$ have been calculated up to five loops, e.g., in Refs.~\cite{vanRitbergen:1997va,Czakon:2004bu,Poole:2019txl,Baikov:2016tgj,Luthe:2017ttc,Herzog:2017ohr}. These five-loop results have been incorporated into the latest version of the \texttt{RunDec} package~\cite{Herren:2017osy} to calculate the running strong coupling and quark masses below the electroweak scale. The anomalous dimensions of quark masses with pure QCD contributions have been updated in Refs.~\cite{Baikov:2014qja,Luthe:2016xec,Baikov:2017ujl,Chetyrkin:1997dh,Vermaseren:1997fq} up to the five-loop order. In Refs.~\cite{Kataev:1992dg,Surguladze:1996hx,Erler:1998sy,Mihaila:2014caa,Bednyakov:2016onn}, one can find the complete three-loop QCD$\times$QED contributions (including pure QCD, pure QED and their mixture) to the beta functions of $\alpha^{}_{s}$ and $\alpha$ and the anomalous dimensions of fermion masses. At low energies, when a fermion mass threshold (e.g., $b$, $c$ or $\tau$) is crossed, the matching of running parameters between two successive EFTs needs to be performed explicitly. The matching conditions for the gauge couplings and fermion masses are usually given in the form
$
\alpha^{(n^{}_{f}-1)}_{s}(\mu) = \zeta^2_{g^{}_{s}} \alpha^{(n^{}_{f})}_{s}(\mu)
$,
$
\alpha^{(n^{}_{f}-1)}(\mu) = \zeta^2_{e} \alpha^{(n^{}_{f})}(\mu)
$
and
$m^{(n^{}_{f}-1)}_{f} = \zeta^{}_{f} m^{(n^{}_{f})}_{f}
$,
where $\zeta^2_{g^{}_{s}}$, $\zeta^2_{e}$ and $\zeta^{}_{f}$ are the decoupling constants and $n^{}_f$ refers to the number of active fermions in the corresponding EFTs. The expressions of $\zeta^2_{g^{}_{s}}$ and $\zeta^{}_{q}$ (i.e., $\zeta^{}_f$ for quarks) are already known up to the order of four loops for the pure QCD contributions~\cite{Schroder:2005hy,Chetyrkin:2005ia,Liu:2015fxa}. However, the other contributions, e.g. the mixed QCD$\times$QED one, are obtained in Ref.~\cite{Martin:2018yow,Bednyakov:2014fua} only to the two-loop order.

\item\emph{The pole mass and the $\overline{\rm MS}$ running mass.} To derive the $\overline{\rm MS}$ running mass from the pole mass, or vice versa, the relations between the masses in the on-shell scheme and those in the $\overline{\rm MS}$ scheme must be known. In our analysis, the conversions will be carried out from the pole masses of $t$~\cite{Jegerlehner:2003py, Marquard:2015qpa, Kataev:2015gvt, Marquard:2016dcn, Martin:2016xsp}, $\tau$, $\mu$ and $e$ \cite{Gray:1990yh,Bekavac:2007tk,Kataev:2018gle} to their $\overline{\rm MS}$ running masses, while the pole masses of $h$ \cite{Martin:2014cxa} and $Z$ \cite{Jegerlehner:2001fb, Jegerlehner:2002em, Martin:2015rea} have been used to determine the relevant $\overline{\rm MS}$ running couplings.
\end{itemize}
With all the above information, we can compute the $\overline{\rm MS}$ running parameters at a given energy scale, i.e., $\mathcal{Y}^{}_{\rm sm}(\mu)$ and $\mathcal{Y}^{}_{\rm eff}(\mu)$, based on the physical inputs of $\mathcal{I}^{}_{\rm sm}$. A sketch of our computational procedure has been shown in Fig.~\ref{fig:1}. The \texttt{SMDR} code is primarily intended for running the parameters in the full SM at high energies to those in the EFTs at low energies. To conversely obtain the $\overline{\rm MS}$ parameters in the full SM from the low-energy input parameters, a fitting routine has been adopted in \texttt{SMDR}, leading to the determination of $\overline{\rm MS}$ parameters with quite a high precision.

\begin{figure*}
	\begin{center}
		\scriptsize
		\begin{overpic}[width=0.95\textwidth]{SMDR.png}
			\put(1,8.5) {\normalsize$\mathcal{Y}^{}_{\rm sm}$}
			\put(5,9.5) {\textrm{RGEs of full SM}}
			\put(6,7.7) {\cite{Cheng:1973nv,Machacek:1983fi,Machacek:1984zw,Machacek:1983tz,Arason:1991ic,Ford:1992mv,Luo:2002ey,Tarasov,Mihaila:2012fm,Chetyrkin:2012rz,Bednyakov:2012rb,Bednyakov:2012en,
					Chetyrkin:2013wya,Bednyakov:2013eba,Bednyakov:2013cpa,Bednyakov:2014pia,Herren:2017uxn,Davies:2019onf}}
			\put(34,9.5) {\textrm{RGEs of QCD$\times$QED}}
			\put(37,7.8) {\cite{Baikov:2016tgj,Luthe:2017ttc,Herzog:2017ohr,Baikov:2014qja,Luthe:2016xec,Baikov:2017ujl,Kataev:1992dg,Surguladze:1996hx,Erler:1998sy,Mihaila:2014caa,Bednyakov:2016onn}}
			\put(23,10.5) {$t,~h,~Z,~W$}
			\put(22.8,9) {\textrm{decoupling}}
			\put(22,7.1) {\cite{Martin:2018yow,Schroder:2005hy,Chetyrkin:2005ia,Bednyakov:2014fua,Liu:2015fxa}
			}
			\put(55,10.5) {$b$-\textrm{quark}}
			\put(54.7,9) {\textrm{decoupling}}
			\put(53.8,7.2) {\cite{Schroder:2005hy,Chetyrkin:2005ia,Liu:2015fxa,Martin:2018yow}}
			\put(72,10.5) {$\tau$-\textrm{lepton}}
			\put(72,9) {\textrm{decoupling}}
			\put(73,7.2) {\cite{Martin:2018yow}
			}
			\put(87.3,10.5) {$c$-\textrm{quark}}
			\put(87,9) {\textrm{decoupling}}
			\put(86.7,7.2) {\cite{Schroder:2005hy,Chetyrkin:2005ia,Liu:2015fxa,Martin:2018yow}}
			\put(2.3,-1.5) {$M^{}_{t}$ \cite{Jegerlehner:2003py,Marquard:2015qpa,Kataev:2015gvt,Marquard:2016dcn,Martin:2016xsp}, $M^{}_{h}$ \cite{Martin:2014cxa}, $M^{}_{Z}$ \cite{Jegerlehner:2001fb,Jegerlehner:2002em,Martin:2015rea}}
			\put(2.3,-3.5) {$\alpha^{}_{0}$ \cite{Chetyrkin:1996cf,Degrassi:2003rw,Degrassi:2014sxa}, $G^{}_{\rm F}$ \cite{Degrassi:2012ry,Martin:2019lqd}}
			\put(30,-1.5) {{$\alpha^{}_{s}(M^{}_{Z})$}, {\normalsize$\mathcal{Y}^{}_{\rm eff}$}}
			\put(43,-1.5) {$m^{}_{b}(m^{}_{b})$}
			\put(56,-1.5) {$m^{}_{s},m^{}_{d},m^{}_{u},M^{}_{\tau}$ \cite{Gray:1990yh,Bekavac:2007tk,Kataev:2018gle}}
			\put(83,-1.5) {$m^{}_{c}(m^{}_{c})$}
			\put(93,-1.5) {$M^{}_{e},M^{}_{\mu}$}
			\put(93,-3.3) {\cite{Gray:1990yh,Bekavac:2007tk,Kataev:2018gle}
			}
			%
			\put(24,22) {$\underline{\mu=M^{}_{Z}}$}
			\put(54,22) {$\underline{\mu=m^{}_{b}(m^{}_{b})}$}
			\put(73,22) {$\underline{\mu=M^{}_{\tau}}$}
			\put(87,22) {$\underline{\mu=m^{}_{c}(m^{}_{c})}$}
			%
			\put(3,18) {$(n^{}_{q},n^{}_{l})=(6,3)$}
			\put(35,18) {$(n^{}_{q},n^{}_{l})=(5,3)$}
			\put(61,18) {$(n^{}_{q},n^{}_{l})=(4,3)$}
			\put(78,18) {$(n^{}_{q},n^{}_{l})=(4,2)$}
		\end{overpic}
	\end{center}
	\vspace{0.2cm}
	\caption{The flow chart for numerical calculations. First, the running parameters $\mathcal{Y}^{}_{\rm sm}$ in the full SM are specified as the initial conditions above the electroweak scale. The RGEs of the full SM with $n^{}_q=6$ active quarks and $n^{}_l=3$ active leptons are utilized to evolve those parameters to the electroweak scale $\mu = M^{}_{Z}$. After the decoupling of heavy particles $t$, $h$, $Z$ and $W$ at the electroweak scale, an EFT of QCD$\times$QED with $(n^{}_q, n^{}_l) = (5, 3)$ is obtained. Then, we evolve the running parameters based on the resultant RGEs of the EFT until the next threshold is encountered and similar procedures are carried out again. The intermediate decoupling scales have been explicitly shown as the colored frames and the RGEs in between as green arrows, along which the theoretical inputs together with the references are given.}
	\label{fig:1}
\end{figure*}
\subsection{Loop truncation and error propagation}

As mentioned before, we can estimate the uncertainties on the running quark and lepton masses with all the precision data available. The error from the experimental inputs is not the only source of uncertainties. Another source, which we shall investigate in the present work, is the theoretical error coming from the perturbation calculations truncated at finite-order loops. More explicitly, the RGEs, the matching conditions and the conversions from on-shell masses to $\overline{\rm MS}$ running masses have been given only at some finite order of perturbations. This type of theoretical error has not been considered in Refs.~\cite{Fusaoka:1998vc,Xing:2007fb,Xing:2011aa,Antusch:2013jca,Deppisch:2018flu}. To quantify this theoretical error, we take the difference between the result obtained by using the highest-order formulas partially from QCD and QED contributions and that by using the formulas completely given at the order lower by one.\footnote{For instance, in the full SM above the electroweak scale, to truncate the three-loop order we compute the running parameters twice, namely, once with the highest three-loop RGEs and once with the two-loop RGEs. The difference between the outcome of two calculations will then be estimated as the uncertainties caused by the finite-loop RGEs~\cite{Bednyakov:2015sca}.}
The loop orders of our truncation procedure for various sources have been summarized in Table~\ref{table:truncationList}.
Some comments and clarifications are helpful.
\begin{itemize}
\item For the RGEs in the EFT of QCD$\times$QED below the electroweak scale, the highest complete RGEs are of three-loop order, including the pure QCD part of $\mathcal{O}(\alpha^{3}_{s})$, the mixed QCD$\times$QED of the order $\mathcal{O}(\alpha^{2}_{s}\alpha) + \mathcal{O}(\alpha^{}_{s}\alpha^{2})$ and the pure QED contribution of the order $\mathcal{O}(\alpha^{3})$. The pure QCD contribution (i.e., without the mixed QCD$\times$QED terms) has actually been calculated up to the five-loop order. Since $\alpha^{}_{s} \gg \alpha$ holds below the electroweak scale, we adopt the five-loop results of the pure QCD contribution and the contributions other than the pure QCD part at the three-loop order.

For the RGEs above the electroweak scale, the perturbation calculations should rely on the gauge couplings $g^{2}_{s}$, $g^2$ and $g^{\prime 2}$, the Yukawa couplings $y^{2}_{f}$ and the quartic Higgs coupling $\lambda$. The complete results are of four-loop order for gauge couplings, and three-loop order for other couplings. Some higher-loop contributions are available but incomplete, so we implement the complete results and take the high-order contributions as the theoretical error.

\item The truncation needs to be performed as well for the decoupling of heavy particles. For $t$, $h$, $W^\pm$ and $Z$ to be decoupled at the electroweak threshold, the complete matching conditions are known up to the two-loop order. The pure QCD contribution to the matching condition for the decoupling of $t$ is known up to four-loop order, whereas the complete matching conditions for the decoupling of $b$, $c$ and $\tau$ in the EFT of QCD$\times$QED are given at the two-loop order. The pure QCD results for the $b$ and $c$ decoupling are also available at the four-loop order. Since the available QCD contribution to the matching conditions is more precise than the complete QCD$\times$QED ones, we will truncate the pure QCD contribution at the four-loop order and the others at the two-loop order, respectively.

\item The conversions of the parameters from the on-shell scheme to those in $\overline{\rm MS}$ scheme, including $M^{}_{\tau}$, $M^{}_{\mu}$, $M^{}_{e}$, $M^{}_{h}$, $M^{}_{t}$, $M^{}_{Z}$, $G^{}_{\rm F}$ and $\alpha^{}_{0}$, are treated with the highest-loop results available. More explicitly, for $M^{}_{h}$, the leading three-loop contribution is included. For $M^{}_{t}$, the pure QCD contribution has been calculated up to the four-loop order, while the other contributions are known at the two-loop order. Therefore, we will truncate the QCD part at the four-loop order and the others at the two-loop order. The conversion between the fine structure constant $\alpha^{}_{0}$ and $\alpha(\mu)$ will be performed at the two-loop order.
\end{itemize}
The errors from the experimental inputs and the theoretical errors from the finite-loop truncation will be treated independently. However, the uncertainties from $b$ and $c$ decoupling in pure QCD are based on the same theory, and likewise for the uncertainties from the $M^{}_{\mu}, M^{}_{e} \leftrightarrow m^{}_{\mu}, m^{}_{e}$ conversions, so they are taken to be $100\%$ correlated.

\begin{table}[!t]
	\small
	\centering
	\caption{Summary of the higher-order contributions implemented in the numerical calculations and the corresponding theoretical errors from the RGEs and the matching conditions truncated at finite-order loops. The listed contributions to the errors are assumed to be independent, except that the uncertainties from $b$ and $c$ decoupling in pure QCD and those from the $M^{}_{\mu}, M^{}_{e} \leftrightarrow m^{}_{\mu}, m^{}_{e}$ conversions are taken to be $100\%$ correlated.}
\vspace{0.5cm}
		\begin{tabular*}{\textwidth}{|l@{\extracolsep{\fill}} c l c|  }
    \hline
	EW matching, complete~\cite{Martin:2018yow} & 2-loop & $M^{}_{\tau} \leftrightarrow m^{}_{\tau}$, complete~\cite{Gray:1990yh,Bekavac:2007tk,Kataev:2018gle} & 3-loop \\
	$b$ and $c$ decoupling, pure QCD~\cite{Schroder:2005hy,Chetyrkin:2005ia,Liu:2015fxa} & 4-loop & 	$M^{}_{\mu},M^{}_{e} \leftrightarrow m^{}_{\mu},m^{}_{e}$, complete~\cite{Gray:1990yh,Bekavac:2007tk,Kataev:2018gle} & 2-loop \\
	$b$ and $c$ decoupling, others~\cite{Martin:2018yow} & 2-loop & 	 $M^{}_{h} \leftrightarrow \lambda$, leading~\cite{Martin:2014cxa} & 3-loop \\
    $\tau$ decoupling, pure QED~\cite{Martin:2018yow}& 2-loop & $M^{}_{t} \leftrightarrow y^{}_{t}$, pure QCD~\cite{Jegerlehner:2003py,Marquard:2015qpa,Kataev:2015gvt,Marquard:2016dcn} & 4-loop \\
	RGEs of QCD$\times$QED, pure QCD~\cite{Baikov:2016tgj,Luthe:2017ttc,Herzog:2017ohr,Baikov:2014qja,Luthe:2016xec,Baikov:2017ujl} & 5-loop &$M^{}_{t} \leftrightarrow y^{}_{t}$, others~\cite{Martin:2016xsp} & 2-loop \\
	RGEs of QCD$\times$QED, others~\cite{Surguladze:1996hx,Erler:1998sy,Mihaila:2014caa,Bednyakov:2016onn} & 3-loop  & 	$M^{}_{Z} \leftrightarrow g,g^{\prime}$, complete~\cite{Martin:2015rea} & 2-loop \\
	RGEs of SM, gauge couplings~\cite{Davies:2019onf} & 4-loop & $G^{}_{\rm F} \leftrightarrow v$, complete~\cite{Martin:2019lqd}  & 2-loop \\
	RGEs of SM, others~\cite{Tarasov,Mihaila:2012fm,Chetyrkin:2012rz,Bednyakov:2012rb,Bednyakov:2012en,
	Chetyrkin:2013wya,Bednyakov:2013eba,Bednyakov:2013cpa,Bednyakov:2014pia,Herren:2017uxn} & 3-loop & $\alpha^{}_{0} \leftrightarrow \alpha$~\cite{Degrassi:2014sxa} & 2-loop \\
	 \hline
	\end{tabular*}
	\label{table:truncationList}
	\end{table}
We adopt the approach of linear error propagation to translate the input errors in Eq.~(\ref{eq:inputs}) into those of the outputs $\mathcal{Y}^{}_{\rm sm}$ and $\mathcal{Y}^{}_{\rm eff}$. The $1\sigma$ experimental errors of the outputs can be obtained by varying the inputs within their $1\sigma$ ranges. By switching on one input error $\delta \mathcal{I}^{}_{k}$ at each time, one can figure out the shift in the output
$\delta \mathcal{Y}^{}_{ik} = ({\partial \mathcal{Y}^{}_{i}}/{\partial \mathcal{I}^{}_{k}}) \cdot \delta \mathcal{I}^{}_{k} $, for the $k$-th input and $i$-th output. In this way, one can find the error matrix
\begin{eqnarray}
S^{}_{ij} \equiv \sum^{}_{k} \left(\frac{\partial \mathcal{Y}^{}_{i}}{\partial \mathcal{I}^{}_{k}} \delta \mathcal{I}^{}_{k}\right)
\left(\frac{\partial \mathcal{Y}^{}_{j}}{\partial \mathcal{I}^{}_{k}} \delta \mathcal{I}^{}_{k} \right)\;,
\end{eqnarray}
where the $1\sigma$ error for each output can be identified as $\sigma^{}_{i} \equiv \sqrt{S^{}_{ii}}$, and the non-diagonal terms with $i\neq j$ quantify the correlations among the output errors of different parameters. The error matrix can be normalized to give the correlation matrix $\rho^{}_{ij}\equiv S^{}_{ij}/\sqrt{S^{}_{ii}S^{}_{jj}}$ such that its elements directly reflect the level of correlation among different parameters. We have demonstrated that the estimation via the linear error propagation turns out to be in perfect agreement with the calculation by the Monte-Carlo approach.

\section{Numerical Results}

Following the strategy outlined above, we update the $\overline{\rm MS}$ running parameters~\cite{Fusaoka:1998vc,Xing:2007fb,Xing:2011aa,Antusch:2013jca,Deppisch:2018flu} in the SM at several representative energy scales, including their best-fit values and the inferred $1\sigma$ uncertainties. Below the electroweak scale, in the EFT with five quarks and three leptons, all the heavy particles including $t$, $h$, $Z$ and $W^\pm$ are integrated out, and we calculate the running parameters at two relevant energy scales: (i) $\mu = M^{}_{W}$, $(n^{}_{q},n^{}_{l})= (5,3)$ without $t$, $h$, $Z$ and $W^\pm$;
(ii) $\mu = M^{}_{Z}$, $(n^{}_{q},n^{}_{l})= (5,3)$ without $t$, $h$, $Z$ and $W^\pm$. Above the electroweak scale, the full gauge symmetry of the SM with the number of active fermions $(n^{}_{q},n^{}_{l})= (6,3)$ is preserved and the typical energy scales are chosen to be (i) $\mu = M^{}_{Z}$;
(ii) $\mu = M^{}_{h}$; (iii) $\mu = M^{}_{t}$; (iv) $\mu = 100~{\rm TeV}$;
(v) $\mu = 10^8~{\rm GeV}$; (vi) $\mu = 10^{12}~{\rm GeV}$. To match the fermion masses below the electroweak scale, we introduce the effective running masses above the electroweak scale as $m^{}_{f} \equiv y^{}_{f} v^{}_{\rm F}/\sqrt{2}$ with $v^{}_{\rm F}  \equiv 246~{\rm GeV} \simeq  2^{-1/4}G^{-1/2}_{\rm F}$, with which one can simply obtain the running Yukawa couplings via $y^{}_{f} = \sqrt{2} m^{}_{f}/v^{}_{\rm F}$ from the running masses $m^{}_f$.

\begin{table}[t!]
	\small
	\centering
	\caption{Running quark masses at some representative energy scales, including $ M^{}_{W} = 80.379~{\rm GeV}$ and $M^{}_{Z}= 91.1876~{\rm GeV}$ in the EFT with the exact ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ gauge symmetry and the number of active fermions $(n^{}_{q},n^{}_{l}) = (5,3)$, as well as $M^{}_{Z}= 91.1876~{\rm GeV}$, $M^{}_{h} = 125.10~{\rm GeV}$, $M^{}_{t} = 173.1~{\rm GeV}$, $\mu = 10^5~{\rm GeV}$, $\mu = 10^8~{\rm GeV}$ and $\mu = 10^{12}~{\rm GeV}$ in the full SM. Above the electroweak scale, the effective running masses have been defined as $m^{}_{f} \equiv y^{}_{f} v^{}_{\rm F}/\sqrt{2}$ with $v^{}_{\rm F}  \equiv 246~{\rm GeV}$. For each parameter, we present its best-fit value $\mathcal{Y}$ and its uncertainty in the form of $\mathcal{Y} \pm \sqrt{\delta^{2}_{\rm exp}+\delta^{2}_{\rm trunc}}$, where the experimental error $\delta^{}_{\rm exp}$ is induced by the input uncertainty while the theoretical error $\delta^{}_{\rm trunc}$ by yet-unknown higher-loop corrections.}
\vspace{0.5cm}
\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}} c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c@{\extracolsep{\fill}} }
		\hline
		\hline
		$\mu\,/\,{\rm GeV}$ & ${\rm Theory}$ & $m^{}_{t}\,/\,{\rm GeV}$ & $m^{}_{b} \,/\,{\rm GeV}$ & $m^{}_{c} \,/\,{\rm GeV}$ & $m^{}_{s} \,/\,{\rm MeV}$ & $m^{}_{d} \,/\,{\rm MeV}$ & $m^{}_{u} \,/\,{\rm MeV}$ \\
		\hline
		\addlinespace[0.8mm]
		$M^{}_{W} $ & EFT& $\cdots$ & $2.897 \pm 0.026$ & $0.635 \pm 0.018$ & $54.40 \pm 4.71$ & $2.73 \pm 0.19$ & $1.26 \pm 0.22$    \\
		\addlinespace[0.8mm]
		$M^{}_{Z} $ & EFT & $\cdots$ & $2.866 \pm 0.026$ & $0.628 \pm 0.018$ & $53.80 \pm 4.66$ & $2.70 \pm 0.19$ & $1.24 \pm 0.22$   \\ \addlinespace[0.8mm]
		$M^{}_{Z} $ & Full SM & $168.26 \pm 0.75$ & $2.839 \pm 0.026$ & $0.620 \pm 0.017$ & $53.16 \pm 4.61$ & $2.67 \pm 0.19$ & $1.23 \pm 0.21$   \\ \addlinespace[0.8mm]
		$M^{}_{h} $ & Full SM & $165.05 \pm 0.75$ & $2.768 \pm 0.026$ & $0.607 \pm 0.017$ & $52.00 \pm 4.51$ & $2.61 \pm 0.18$ & $1.20 \pm 0.21$    \\ \addlinespace[0.8mm]
		$M^{}_{t} $ & Full SM & $161.98 \pm 0.75$ & $2.702 \pm 0.025$ & $0.594 \pm 0.017$ & $50.90 \pm 4.41$ & $2.56 \pm 0.18$ & $1.18 \pm 0.20$ \\ \addlinespace[0.8mm]
		$10^5$ & Full SM & $123.77 \pm 0.85$ & $1.908 \pm 0.021$ & $0.435 \pm 0.013$ & $37.47 \pm 3.26$ & $1.88 \pm 0.13$ & $0.86 \pm 0.15$
 \\ \addlinespace[0.8mm]
		$10^8$ & Full SM & $102.49 \pm 0.89$ & $1.502 \pm 0.018$ & $0.350 \pm 0.011$ & $30.34 \pm 2.65$ & $1.52 \pm 0.11$ & $0.69 \pm 0.12$   \\ \addlinespace[0.8mm]
		$10^{12}$ & Full SM & $85.07 \pm 0.89$ & $1.194 \pm 0.015$ & $0.283 \pm 0.009$ & $24.76 \pm 2.17$ & $1.24 \pm 0.09$ & $0.56 \pm 0.10$ \\ \addlinespace[0.8mm]
		\hline
		\hline
	\end{tabular*}
	\label{table:quarkMass}
\end{table}

\begin{table}[h!]
	\small
	\centering
	\caption{Running charged-lepton masses at some representative energy scales, including $ M^{}_{W} = 80.379~{\rm GeV}$ and $M^{}_{Z}= 91.1876~{\rm GeV}$ in the EFT with the exact ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ gauge symmetry and the number of active fermions $(n^{}_{q},n^{}_{l}) = (5,3)$, as well as $M^{}_{Z}= 91.1876~{\rm GeV}$, $M^{}_{h} = 125.10~{\rm GeV}$, $M^{}_{t} = 173.1~{\rm GeV}$, $\mu = 10^5~{\rm GeV}$, $\mu = 10^8~{\rm GeV}$ and $\mu = 10^{12}~{\rm GeV}$ in the full SM with the number of active fermions $(n^{}_{q},n^{}_{l}) = (6,3)$.}
\vspace{0.5cm}
	\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}}c c c c}
		\hline
		\hline
		$\mu\,/\,{\rm GeV}$ & ${\rm Theory}$ & $m^{}_{\tau} \,/\,{\rm GeV}$ & $m^{}_{\mu} \,/\,{\rm GeV}$ & $m^{}_{e} \,/\,{\rm MeV}$   \\
		\hline
		\addlinespace[0.8mm]
		$M^{}_{W} $ & EFT & $1.74826 \pm 0.00012$ & $0.102925 \pm 0.000018$ & $0.48858 \pm 0.00045$   \\  \addlinespace[0.8mm]
		$M^{}_{Z} $ & EFT& $1.74743 \pm 0.00012$ & $0.102877 \pm 0.000018$ & $0.48835 \pm 0.00045$\\ \addlinespace[0.8mm]
		$M^{}_{Z} $ & Full SM & $1.72856 \pm 0.00028$ & $0.101766 \pm 0.000023$ & $0.48307 \pm 0.00045$  \\ \addlinespace[0.8mm]
		$M^{}_{h} $ & Full SM & $1.73369 \pm 0.00020$ & $0.102065 \pm 0.000020$ & $0.48449 \pm 0.00045$ \\ \addlinespace[0.8mm]
		$M^{}_{t} $ & Full SM& $1.73850 \pm 0.00014$ & $0.102347 \pm 0.000019$ & $0.48583 \pm 0.00045$  \\ \addlinespace[0.8mm]
		$10^5 $ & Full SM & $1.78412 \pm 0.00158$ & $0.105015 \pm 0.000095$ & $0.49850 \pm 0.00064$  \\ \addlinespace[0.8mm]
		$10^8 $ & Full SM & $1.77852 \pm 0.00308$ & $0.104681 \pm 0.000183$ & $0.49691 \pm 0.00098$ \\ \addlinespace[0.8mm]
		$10^{12} $ & Full SM & $1.73194 \pm 0.00466$ & $0.101936 \pm 0.000277$ & $0.48388 \pm 0.00139$ \\ \addlinespace[0.8mm]
		\hline
		\hline
	\end{tabular*}
	\label{table:leptons}
\end{table}

The running masses in the $\overline{\rm MS}$ scheme at different renormalization scales have been summarized in Table~\ref{table:quarkMass} for six quarks and Table~\ref{table:leptons} for three leptons.
The other SM parameters, including the gauge coupling constants and the Higgs-related parameters, have been given in Table~\ref{table:gaugeBelow}, Table~\ref{table:gaugeAbove} and Table~\ref{table:higgs}. The values of the parameters that are not well-defined in the specified theory will be denoted as dots. For each parameter we present its best-fit value $\mathcal{Y}$ and its uncertainty in the form of $\mathcal{Y} \pm \sqrt{\delta^{2}_{\rm exp}+\delta^{2}_{\rm trunc}}$, where the experimental error $\delta^{}_{\rm exp}$ is induced by the input uncertainty while the theoretical error $\delta^{}_{\rm trunc}$ by the loop truncation.
\begin{table}[t!]
	\small
	\centering
	\caption{Running gauge couplings at $\mu = M^{}_{W} = 80.379~{\rm GeV}$ and $\mu = M^{}_{Z}= 91.1876~{\rm GeV}$ in the EFT with the ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ gauge symmetry and the number of active fermions $(n^{}_{q},n^{}_{l}) = (5,3)$.}
\vspace{0.5cm}
	\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}} c c c  }
		\hline
		\hline
		$\mu$ & ${\rm Theory}$ & $\alpha^{}_{s}$ & $\alpha^{-1}$  \\
		\hline
		\addlinespace[0.8mm]
		$M^{}_{W} $ & EFT & $0.1199 \pm 0.0010$ & $127.937 \pm 0.026$   \\ \addlinespace[0.8mm]
		$M^{}_{Z} $ & EFT & $0.1176 \pm 0.0010$ & $127.754 \pm 0.026$  \\ \addlinespace[0.8mm]
		\hline
		\hline
	\end{tabular*}
	\label{table:gaugeBelow}
\end{table}

\begin{table}[t!]
	\small
	\centering
	\caption{Running gauge couplings at some representative energy scales, including $M^{}_{Z}= 91.1876~{\rm GeV}$, $M^{}_{h} = 125.10~{\rm GeV}$, $M^{}_{t} = 173.1~{\rm GeV}$, $\mu = 10^5~{\rm GeV}$, $\mu = 10^8~{\rm GeV}$ and $\mu = 10^{12}~{\rm GeV}$ in the full SM with the number of active fermions $(n^{}_{q},n^{}_{l}) = (6,3)$.}
\vspace{0.5cm}
	\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}} c c  c c }
		\hline
		\hline
		$\mu$ & ${\rm Theory}$ &  $g^{}_{s}$ & $g$ & $g^{\prime}$  \\
		\hline
		\addlinespace[0.8mm]
		$M^{}_{Z} $ & Full SM & $1.2104 \pm 0.0051$ & $0.65100 \pm 0.00028$ & $0.357254 \pm 0.000069$  \\ \addlinespace[0.8mm]
		$M^{}_{h} $ & Full SM & $1.1855 \pm 0.0048$ & $0.64934 \pm 0.00028$ & $0.357893 \pm 0.000070$  \\ \addlinespace[0.8mm]
		$M^{}_{t} $ & Full SM & $1.1618 \pm 0.0045$ & $0.64765 \pm 0.00028$ & $0.358545 \pm 0.000070$  \\ \addlinespace[0.8mm]
		$10^5~{\rm GeV}$ & Full SM & $0.8711 \pm 0.0019$ & $0.61644 \pm 0.00025$ & $0.372179 \pm 0.000079$  \\ \addlinespace[0.8mm]
		$10^8~{\rm GeV}$ & Full SM & $0.7182 \pm 0.0010$ & $0.58690 \pm 0.00022$ & $0.388806 \pm 0.000091$  \\ \addlinespace[0.8mm]
		$10^{12}~{\rm GeV}$ & Full SM & $0.6017 \pm 0.0006$ & $0.55325 \pm 0.00019$ & $0.414821 \pm 0.000111$ \\ \addlinespace[0.8mm]
		\hline
		\hline
	\end{tabular*}
	\label{table:gaugeAbove}
\end{table}

\begin{table}[!t]
	\small
	\centering
	\caption{Running Higgs parameters at some representative energy scales, including $M^{}_{Z}= 91.1876~{\rm GeV}$, $M^{}_{h} = 125.10~{\rm GeV}$, $M^{}_{t} = 173.1~{\rm GeV}$, $\mu = 10^5~{\rm GeV}$, $\mu = 10^8~{\rm GeV}$ and $\mu = 10^{12}~{\rm GeV}$ in the full SM with the number of active fermions $(n^{}_{q},n^{}_{l}) = (6,3)$.}
\vspace{0.5cm}
	\begin{tabular*}{\textwidth}{c@{\extracolsep{\fill}} c c c c}
		\hline
		\hline
		$\mu$ & ${\rm Theory}$ & $v \,/\,{\rm GeV}$ & $\lambda$ & $-m^2 \,/\,{\rm GeV}^2$   \\
		\hline
		\addlinespace[0.8mm]
		$M^{}_{Z} $ & Full SM & $248.404 \pm 0.036$ & $0.13947 \pm 0.00045$ & $8434 \pm 21$  \\ \addlinespace[0.8mm]
		$M^{}_{h} $ & Full SM & $247.482 \pm 0.023$ & $0.13259 \pm 0.00035$ & $8525 \pm 22$
		\\ \addlinespace[0.8mm]
		$M^{}_{t} $ & Full SM & $246.605 \pm 0.011$ & $0.12607 \pm 0.00030$ & $8612 \pm 23$
		 \\ \addlinespace[0.8mm]
		$10^5~{\rm GeV} $ & Full SM & $236.500 \pm 0.209$ & $0.04993 \pm 0.00239$ & $9740 \pm 35$
		\\ \addlinespace[0.8mm]
		$10^8~{\rm GeV} $ & Full SM  & $232.831 \pm 0.402$ & $0.01525 \pm 0.00408$ & $10213 \pm 38$
		\\ \addlinespace[0.8mm]
		$10^{12}~{\rm GeV} $ & Full SM & $232.510 \pm 0.624$ & $-0.00402 \pm 0.00522$ & $10274 \pm 32$
		\\ \addlinespace[0.8mm]
		\hline
		\hline
	\end{tabular*}
	\label{table:higgs}
\end{table}

As briefly mentioned before, the $\overline{\rm MS}$ running parameters may share the common sources of input uncertainties, so their overall uncertainties are actually correlated.
For further reference and completeness, we summarize in Table~\ref{table:corr2} and Table~\ref{table:corr3} the correlation matrix of those $\overline{\rm MS}$ parameters at $\mu = M^{}_{Z}$ in the EFT of QCD$\times$QED and the full SM, respectively. To make use of these numerical results in confronting the model predictions with observations, one may first reconstruct the error matrix $S^{}_{ij}$ based on the normalized correlation matrix $\rho^{}_{ij}$ via $ S^{}_{ij} = \rho^{}_{ij} \sigma^{}_{i} \sigma^{}_{j}$, and then quantify the statistical significance of deviations by $\chi^2 = (\mathcal{Y}^{}_{i} - \mathcal{Y}^{\rm bf}_{i} )
\left(S^{-1}\right)^{}_{ij}(\mathcal{Y}^{}_{j} - \mathcal{Y}^{\rm bf}_{j} )$ with $\mathcal{Y}^{}_{j}$ being the model prediction and $\mathcal{Y}^{\rm bf}_{j}$ being the given central value.

Now we demonstrate that it is reasonable to implement the linear error propagation by the Monte-Carlo simulations. With $2\times 10^{5}$ sampling points, the posterior distributions of the running parameters have been generated in the EFT of QCD$\times$QED at $\mu = M^{}_{Z}$, and they are compared in Fig.~\ref{fig:2} with the distributions obtained from the linear error propagation.
As one can observe from Fig.~\ref{fig:2}, a perfect agreement between these two methods is found.
Since the uncertainties on the input parameters are at the perturbative level, if their values are updated in the future with new best-fit values and even smaller errors, it is straightforward to recalculate the best-fit values and errors of running parameters by utilizing the error dependence matrix provided in Table~\ref{table:coe2}. For instance, if the best-fit values of input parameters are changed by the amount of $\Delta \mathcal{I}^{}_{k}$, the best-fit values of the running parameters will be accordingly shifted as
$\mathcal{Y}^{\rm new-bf}_{i}  = \mathcal{Y}^{\rm old-bf}_{i} +\sum^{}_{k} ({\partial \mathcal{Y}^{}_{i}}/{\partial \mathcal{I}^{}_{k}}) \cdot \Delta \mathcal{I}^{}_{k} $. Similarly, if the uncertainties on the input parameters are improved, one can determine the $1\sigma$ experimental errors of the running parameters via $\delta \mathcal{Y}^{}_{ik} = ({\partial \mathcal{Y}^{}_{i}}/{\partial \mathcal{I}^{}_{k}}) \cdot \delta \mathcal{I}^{}_{k} $. Such a treatment is valid as long as the uncertainties are small and the linear error propagation is justified, which should be the case for the future measurements with more data.
\begin{figure}[t!]
	\begin{center}
		\hspace{0.0cm}
		\includegraphics[width=0.95\textwidth]{LEP_MCMC.pdf}
	\end{center}
	\vspace{-0.4cm}
	\caption{The comparison between the uncertainties of running parameters evaluated by the linear error propagation method (gray curves) and those by the Monte-Carlo simulations (red dots) with $2\times 10^{5}$ sampling points, where the running parameters are given at $\mu = M^{}_{Z}$ in the EFT with the exact ${\rm SU}(3)^{}_{\rm c} \times {\rm U}(1)^{}_{\rm EM}$ gauge symmetry and the number of active fermions $(n^{}_{q},n^{}_{l}) = (5,3)$. For clarity, the labels for the horizontal and vertical axes are not explicitly shown. The gray curves are generated by using the Gaussian distribution with the central values and $1\sigma$ experimental uncertainties, while the red dots stand for the posteriors yielded by the Monte-Carlo simulations.}
	\label{fig:2}
\end{figure}

Apart from the above comments, some important observations from the numerical calculations can be made.
\begin{itemize}
\item In Table~\ref{table:errorSourcesAbove}, we present the fractions of the output uncertainty contributed from each input parameter, which are characterized by $\left({\delta\mathcal{Y}^{}_{ik}}/{\sigma^{}_{i}} \right)^2$, at $\mu = M^{}_{Z}$ in the full SM. For each output parameter the total fraction by definition amounts to $100\%$. For the running masses of five quarks $b$, $c$, $s$, $d$ and $u$ at high energy scales, we identify two major sources of input errors: (i) the strong coupling constant $\alpha^{}_{s}(M^{}_{Z})$; (ii) their input $\overline{\rm MS}$ masses at low energies. For the masses of three light quarks, i.e., $m^{}_{s}$, $m^{}_{d}$ and $m^{}_{u}$, the uncertainties from their input $\overline{\rm MS}$ masses are dominant. A better knowledge of $m^{}_{q}$ and $\alpha^{}_{s}(M^{}_{Z})$ at low energies in the future will thus greatly improve the accuracy of  their running values at high energy scales. For the top quark $t$, its running mass above the electroweak scale is mainly affected by the input of its pole mass.

\item The vacuum expectation value $v$ of the SM Higgs field does not change much at different renormalization scales. As is well known, however, the quartic coupling $\lambda$ has the risk to be negative at the energy scales above $\mu = 10^{8}~{\rm GeV}$, e.g., $\lambda<0$ is within the $4\sigma$ range at $\mu = 10^{8}~{\rm GeV}$ and $\lambda<0$ is the best fit for $\mu = 10^{12}~{\rm GeV}$. New physics beyond the SM has to come into play to save the vacuum from being unstable. For the latest discussions in this aspect, one should be referred to Refs.~\cite{Xing:2011aa,Degrassi:2012ry,Holthausen:2011aa,EliasMiro:2011aa,Bezrukov:2012sa,Buttazzo:2013uya,DiLuzio:2014bua,Bednyakov:2015sca,Andreassen:2014gha}.

\item It is worthwhile to notice that the theoretical uncertainties arising from the truncation of finite loops could be dominant for the electroweak parameters $m^{}_{\mu}$, $m^{}_{e}$, $g$ and $g^{\prime}$. For the other parameters, the theoretical errors are relatively small compared to the experimental ones. The electroweak matching procedure dominates the theoretical errors of the running parameters $g^{}_{s}$, $m^{}_{b}$, $m^{}_{s}$, $m^{}_{d}$ and $m^{}_{u}$ above the electroweak scale.
\end{itemize}

\section{Concluding Remarks}

Although the SM has been experimentally proved to be extremely successful, it must be extended to explain tiny neutrino masses, and to provide the dark matter candidate and the solution to the hierarchy problem. The energy scale for new physics has been pushed by the experiments at the LHC to be beyond TeV. In the present paper, we update the running fermion masses and other $\overline{\rm MS}$ parameters at several representative energy scales, e.g., $\mu = 10^5~{\rm GeV}$, $10^8~{\rm GeV}$ and $10^{12}~{\rm GeV}$, where new physics may come into play. In addition, the running parameters are also evaluated at $\mu = M^{}_Z$, $M^{}_h$ and $M^{}_t$, which will be useful for the precision calculations in the SM.

Compared with the previous similar works in Refs.~\cite{Fusaoka:1998vc,Xing:2007fb,Xing:2011aa,Antusch:2013jca,Deppisch:2018flu}, a number of significant improvements should be emphasized. First, tremendous progress has been made in the theoretical high-order calculations of the RGEs and the matching conditions in the SM and its related EFTs in the past few years. In particular, stimulated by the exciting discovery of the Higgs boson in 2012, the theoretical treatment of radiative corrections from the electroweak sector has been significantly advanced. Using the publicly available code \texttt{SMDR} with the state-of-the-art theoretical knowledge, we have incorporated all the latest results of relevant RGEs and matching conditions. Second, the experimental information has also been greatly changed, as indicated in the latest version of Review of Particle Physics from PDG, which is the source of all the input parameters used in our numerical calculations. Third, we deal with the uncertainties of running parameters in a consistent manner. Both the experimental input errors and the theoretical uncertainties due to the finite-loop calculations  are relevant error sources.

Our main results of the running quark and lepton masses, gauge couplings and scalar parameters are summarized in a number of tables, namely, from Table~\ref{table:quarkMass} to Table~\ref{table:higgs}, which can be directly used for further applications. The linear error propagation has been verified to be an efficient and convenient method in deriving the uncertainties on the running parameters. We find that the theoretical uncertainties due to loop truncations can be dominant for the running parameters $m^{}_{e}$, $m^{}_{\mu}$, $g$ and $g^{\prime}$, whereas for other running parameters the experimental input uncertainties are the major error sources. The results presented here could be easily improved with future update on the experimental and theoretical knowledge.

\section*{Acknowledgments}

The authors are indebted to Prof.~Zhi-zhong Xing for helpful discussions and valuable suggestions. This work was supported in part by the National Natural Science Foundation of China under Grants No.~11775232 and No.~11835013, by the CAS Center for Excellence in Particle Physics, and by the Alexander von Humboldt Foundation.

\bibliographystyle{utcaps_mod}

redpajama_set_name: RedPajamaArXiv


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Business & Economy 0
Vuèiæ: "Belgrade-Sabac in a maximum of 45 minutes" VIDEO / PHOTO
Today, in Macvanski Prnjavor, the launch of works on the construction of the fast road Sabac - Loznica was marked, in the presence of President Vuèiæ.
Source: Tanjug, B92.net Monday, February 14, 2022 | 13:48
Foto: TANJUG/ JADRANKA ILIC/ nr
The Sabac-Loznica expressway is part of the Ruma-Sabac-Loznica highway construction project.
Vuèiæ pointed out that the road Ruma - Šabac will be ready by the end of September next year, which means that in a year and a half, the journey from Belgrade to Šabac will take maximum 45 minutes, and from Belgrade to Loznica - an hour and 10 minutes.
"It will additionally raise the entire Podrinje region for tourism and everything else. It will significantly strengthen Macva," said Vuèiæ and announced that the company "Minth", which employs about 500 people in Loznica, will also come to Sabac.
He reminded that Adient, which employs more than 1.300 people, also works in Loznica. "We brought all those factories, but the salaries in Loznica and Sabac are rather small. We have to work on that, to bring another investor, and a higher rank, in order to raise salaries," said Vucic.
He was informed that 66 percent of the works were completed on the bridge on the Sava within the project of building the highway and the fast road Ruma-Sabac-Loznica, which is also being done by the Azerbaijani "AzVirt".
President stated that "AzVirt" works well, and everything ends on time.
"If we manage to finish everything in Loznica in September 2024, it is really unbelievable. Nobody dreamed of that. Then it remains to be seen what we will do for Zvornik, towards Ljubovija and Bajina Basta, to build a highway, to build that area, although it is difficult because there are not many places along the Drina", said Vucic.
He added that after the completion of works on the road Ruma - Sabac - Loznica, the road from Loznica to Novi Sad will take an hour, unlike the current two hours.
"If you finish this on time, that is a great thing, you deserve a medal", Vucic stated. Vuèiæ said that history is being built, as he said, kilometers of new hope and perspective, because this large infrastructure project will mean accelerated progress in that part of Serbia.
Quoting Milorad Paviæ, that it is more important what kind of road he chose, than the goal itself, Vuèiæ said that the construction of the road is hard work, and that unlike many who mostly promise destruction, he always promises construction.
"We are building and strengthening a stable state. And the promised progress is there, and this great infrastructure project gives Macva, Jadar, a better life, makes people live easier, we will attract investors more easily, which means higher salaries and pensions," Vucic said.
He added that the value of the complete project, Ruma-Sabac-Loznica, is 467.5 million euros.
"Those are kilometers of new hope and perspective. It is a path in the right direction, towards general, economic, cultural progress, development in all spheres, towards faster realization of free flow of goods, people, services and capital," said Vucic.
New infrastructure projects for people here mean a better life, so that they can stay where they were born, where their grandfather is, said Vuèiæ and thanked the citizens, because, as he says, without their dreams, support and patience, they could not have implemented difficult reforms, which have stabilized public finances.
"They will see here how this beauty grows and how Belgrade and Novi Sad are getting closer. Today we are building history here, this means the accelerated progress of western Serbia," Vucic concluded.
A post shared by Aleksandar Vuèiæ (@buducnostsrbijeav)
The celebration of the launch of the works was also attended by the Minister of Construction, Transport and Infrastructure Tomislav Momiroviæ, the Ambassador of Azerbaijan Kamil Khasiyev, the Mayor of Šabac Aleksandar Pajiæ, the Mayor of Loznica Vidoje Petroviæ, Acting Director of the "Corridor of Serbia" Aleksandar Antic, acting Director of "Roads of Serbia" Zoran Drobnjak.
The commercial contract on the design and execution of works on the highway Ruma - Sabac and the highway Sabac - Loznica, was concluded on November 19, 2019, with the Azerbaijani company "AzVirt".
The total length of the section is 77 kilometers, of which the highway Ruma - Sabac is 22 kilometers long, and the length of the section Sabac - Loznica, which will be built in the profile of the fast road, is 55 kilometers.
The construction of a total of 56 objects is planned, of which 54 bridges and two loops.
The construction of the Ruma-Sabac highway and the Sabac-Loznica expressway is one of the priority tasks in road transport, given that this is a project that is important for the development and connection of Srem, Macva and western Serbia and Vojvodina, but also this part of Serbia with the region, primarily from Bosnia-Herzegovina.
The modern road that will connect Ruma and Loznica, in the total length of 77 km, will include three spatial units: section Ruma - (Sabac) bridge on the Sava, in the length of 22 km will be a highway in full profile, the second unit is the bridge over the Sava River and the third section, the construction of which begins today Sabac-Loznica.
That section will be a fast road, the route will bypass all populated places, through which the existing state road IB row number 26 (former highway M-19) passes almost the entire length. The total agreed price of works from Ruma to Loznica is 467.5 million euros, and from Sabac to Loznica 247.5 million euros.
Between these two cities, Sabac and Loznica, after the construction of this section, they will travel for a maximum of half an hour instead of an hour.
"Friendly relations between Serbia and Azerbaijan are developing successfully"
The Ambassador of Azerbaijan to Serbia, Kamil Khasiyev, stated today that the friendly relations between Serbia and Azerbaijan are successfully developing and that they have reached the level of strategic partnership, based on the principle of integrity and territorial sovereignty.
He also pointed out that economic cooperation between the two countries is also developing, which, as he said, shows today's start of works on the construction of the Sabac-Loznica highway and the entire project from Ruma to Loznica, which is being done by the Azerbaijani company AzVirt.
Khasiyev also stated that the Serbian economy has been showing sustainable growth in the last year, despite the pandemic, and expressed satisfaction that Azerbaijani companies are contributing to that growth. The Ambassador sent congratulations to the President of Serbia, Aleksandar Vuèiæ, and the Serbian people on the occasion of the Statehood Day.
He expressed confidence that the strategic partnership between Serbia and Azerbaijan will continue to deepen.
Vuèiæ at the Clinical Center: "We can say that Serbia has changed" PHOTO / VIDEO
President of Serbia, Aleksandar Vuèiæ, attends the opening of the Clinical Center.
Business & Economy Sunday, February 27, 2022 11:55 Comments: 1
Vuèiæ visited Nestlé: "This space is ideal for you" VIDEO / PHOTO
President of Serbia, Aleksandar Vuèiæ, attended the ceremony on the occasion of completion of works on the construction of the new Nestle factory in Belgrade.
Business & Economy Friday, February 25, 2022 14:40 Comments: 5
Vuèiæ in Spain; Serbia is buying two planes
Today, President of Serbia, Aleksandar Vuèiæ, will pay a one-day working visit to the Kingdom of Spain.
Business & Economy Wednesday, February 23, 2022 10:27 Comments: 3
Vuèiæ: "We have connected Belgrade and Novi Sad with this railway line" VIDEO / PHOTO
Today, President Aleksandar Vuèiæ attended the promotional ride of the new high-speed train "Stadler KISS" on the Belgrade -Petrovaradin line.
Business & Economy Thursday, February 17, 2022 18:11 Comments: 9
Vuèiæ: "Additional 100 euros for young people"
President of Serbia, Aleksandar Vuèiæ, said that young people will receive additional 100 euros by Vidovdan.
Business & Economy Monday, February 7, 2022 22:45 Comments: 0
Signed: An important center opens in Belgrade VIDEO / PHOTO
The agreement on the opening of the Center for the Fourth Industrial Revolution in Belgrade was signed today in Geneva.
Brnabiæ: The end for Rio Tinto in Serbia, it's over
Prime Minister Ana Brnabiæ is addressing the public after today's session of the Government of Serbia.
Business & Economy Thursday, January 20, 2022 18:30 Comments: 1
The end for Rio Tinto in Serbia?
Prime Minister of Serbia, Ana Brnabiæ, convened a government session for tomorrow at 5:00 PM, Tanjug has learned from the Prime Minister's office.
Business & Economy Wednesday, January 19, 2022 18:48 Comments: 5
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Vuèiæ: This is a plan that is little talked about in the Serbian public VIDEO / PHOTO
Brnabiæ: Extremely difficult topic for us; The question of mandate isn't for me
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Ready Colorado, an education reform organization, recently published a first-of-its-kind report about Colorado's public school open enrollment system. Open Doors, Open Districts: School Choice in Colorado's Traditional Public Schools analyzes enrollment trends, barriers to choice, and makes policy recommendations to create a more family-friendly process.
Putting aside open enrollment into charter public schools and online public schools, the researchers found that approximately 145,000 Colorado public school students, or 16 percent, choice into other traditional public schools, within and outside of their district boundaries. The data shows that this is the largest form of school choice in Colorado. There are a number of reasons why parents enroll their children in schools outside their neighborhood, but the primary reason is based on a school's student performance. Parents want their children to attend the best schools.
Because each school district sets its own open enrollment process, it can be confusing for families who are school shopping. In fact, the procedures can be dramatically different among districts and therefore could cause a family much distress to learn they missed a deadline or that their already enrolled child in the school of their choice may not be guaranteed for enrollment the next year. The Ready Colorado report discusses several of the challenges families face.
There is a fine balance between policies that create a more standard open enrollment process but still provide school districts the flexibility they need to provide space for neighborhood children or staffing requirements. Unfortunately, the resistance to change can be the result of not wanting to bother with a modification of a traditional practice or an unwillingness to serve students who "don't look like" the rest of the neighborhood. Yes, that does happen in Colorado. But we also have districts that have helped to nurture a culture of parental public school choice.
Several years ago, school districts killed legislation that would have required family-friendly information about open enrollment be posted on district websites. The lobbyist claimed it was another unfunded mandate. Also, that it was an Independence Institute bill, therefore it should die. However, a resolution, based on the language of the bill, did pass and many school districts improved their Internet communication about open enrollment.
Most states don't allow the level of public school choice Colorado families enjoy. Parents need to be educated about the opportunity to choose a public school. Eleven years ago this month the Independence Institute launched the School Choice for Kids website in both English and Spanish as a helpful tool to select a school and to learn about how to maneuver through the school open enrollment process.
The publication of Open Doors, Open Districts should re-open the conversation about how school districts can better serve students with policies that encourage parents to choose the school that best meets the need of their children.


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Town Return Dream Come True For Mildenhall
New Goalkeeping Coach Over The Moon To Be Back Home
Town's new Goalkeeping Coach Steve Mildenhall has described his return to his boyhood club as a dream come true.
The former stopper was announced as Dean Thornton's replacement on Monday and is a familiar sight to Town fans having began his career at SN1.
And, having carved out a lengthy playing career in the EFL, Mildenhall is now looking forward to embarking on the next chapter of his career with a club that is very close to his heart.
"I'm ecstatic," he told swindontownfc.co.uk.
"It is something that I always hoped might happen one day and when I left as a player I always hoped that I might come back.
"There was a chance many moons ago when Danny Wilson was here to come back but we didn't quite get it over the line for whatever reason and it always bugged me never getting the chance to come back and play for the club.
"Being a local boy I left pretty early in my career and it is my hometown team that I've supported for a long, long time.
"The main draw was coming back to my old club and getting the opportunity to do my first-ever job as a first-team Goalkeeping Coach at a club I made my debut with was an opportunity I couldn't turn down.
"I am so pleased to be here and I am so pleased to be given the opportunity.
"I must also say a massive, massive thank you to Darrell Clarke and Bristol Rovers because I've had six unbelievable years there and it will be a club that will always be close to me now.
"I will always be indebted to that football club but now is a new chapter for me and a new path for me.
"I was in the building on Monday, had a really good days' worth of training, got to know the three goalkeepers collectively and I'll be talking to them individually over the next couple of days.
"It was good to get out on that grass and get down to the nitty gritty side of it and it was good to get a feel for it.
"I missed that cutting edge where it really meant something and I'm looking forward to it."
And Mildenhall is also relishing the opportunity to work with Lawrence Vigouroux, Luke McCormick and Will Henry during the course of the season.
"I've known Luke McCormick for a young time and I knew him when he was a young lad playing and I've played against him a lot of times," he added.
"I've watched Lawrence Vigouroux and I've bumped into him around the town a few times and chatted to him.
"You've got an older experienced goalkeeper with a lot of games under his belt and a young goalkeeper with all the attributes in the world.
"And then we've got another young keeper in Will Henry who has played some games and he's got potential.
"The goalkeeper department was very attractive from the outside looking in and there is a lot to work with there.
"This is the last week of pre-season training as such so this week is about me coming in and seeing where they are in terms of fitness, sharpness and they all look good.
"I'll push them to their limits this week and then next week I'll start to drip-feed into them what they will face from Macclessfield.
"It is then constructing the sessions to make sure whoever the Gaffer picks is ready to go come August 4th."
Steve Mildenhall
Mildenhall Returns Home!
Swindon Town Football Club are delighted to confirm that we have appointed Steve Mildenhall as our new first-team Goalkeeping Coach.
BROWN: I'M BLESSED WITH QUALITY GOALKEEPERS
Phil Brown has a selection dilemma ahead of Swindon's League Two clash with Bury at the Energy Check County Ground on Saturday.
BROWN: IT'LL BE A TOUGH TEST
The Gaffer spoke to the media ahead of tomorrow's match at the County Ground against MK Dons.
MATTHEWS: I CAN'T WAIT TO CARRY ON LEARNING
Archie Matthews has spent three and half years with Swindon Town's academy and, after being called up to the first-team squad towards the end of the 2018/19 season, has been rewarded with his first...
DOUGHTY: I WANTED TO BE PART OF THE PROJECT
Being part of the project was the main pull for Michael Doughty as he signed a two-year contract extension, keeping him at the club until 2022.
WELLENS: DION HAS HUGE POTENTIAL
Richie Wellens has challenged Dion Conroy to push on this season after an impressive end to the 2018/19 campaign.
DION CONROY SIGNS NEW ONE-YEAR DEAL
Swindon Town are delighted to announce that Dion Conroy has accepted a new one-year deal.
WELLENS DELIGHTED TO GET YATES SIGNING OVER THE LINE
Richie Wellens was delighted to get the loan move for Jerry Yates over the line following over a month of negotiations.


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Five minutes with a Legend: Melissa Connors
Photo: Melissa Connors of This Farm needs a Farmer.
It was after turning her back on a corporate career, a fashion degree, and the big smoke that she found her place in the food and drink industry. A move to Kyneton meant she could share her passion for the outdoors with others, and in 2013 she launched This Farm Needs a Farmer — a community-building program that would partner novice tree changers with seasoned farmers, local businesses, services and advice providers.
Melissa Connors was inducted into the MFWF Legends Hall of Fame as a Local Hero in 2017, and Connors' commitment to community and sustainability is common knowledge in Kyneton. Here, she reflects on her considerable contribution to the industry.
The proudest moment in my career has been winning the Victorian Rural Women's award in 2018. It is a real honour to be recognised for the work that I do and to be included among such prestigious alumni in Victoria's rural community. It leapfrogged our project a good five years forward, but to be part of this group of women that I can call on at any time has been nothing but positive. I'm very proud to be part of it.
The mistake that taught me the most was my corporate career crashing and burning when I was 25. I was working for IT companies in service delivery. It was a job with an attractive salary package and it barely motivated me at all. I wasn't good at playing the corporate games. Once I called it quits on that part of my career, I really did jump around and figure out what I wanted to do. That career path not working out gave me the chance to try things that I really had a passion for and just go for them. My time in the corporate world was a bit of drawn-out mistake, but it was the best thing that could have ever happened to me.
The reason I got into this industry was initially, it was to support my study, but I ended up staying a few years past that. Still to this day, my job at Jooce nightclub is one of my favourites, and I have taken a lot of the relationship building and multitasking skills I learned there into my career today.
The reason I stayed was the fun and camaraderie – it was a hoot. Yes, I took a break, but that sense of camaraderie is what I get to experience now every day at This Farm Needs a Farmer, and it's really one of the perks of what I do. It was honestly those years spent forging relationships and building up confidence to just chat to people in hospitality that helped me bring people together in our community. To be able to understand what people need, how to work with them, and do it all without causing offence or annoyance is a very underrated skill, and one I hope stays with me forever. Who would have thought it would be thanks to Jooce nightclub.
My mentor was is one of the most incredible women I had the good fortune to be paired with when I won a scholarship with Inspiring Rare Birds in 2019, Angela Elliot. The timing was impeccable, and she is truly the most intelligent person I've ever met. Even with her background being in fast-moving consumer goods rather than farming or rural community, she's been a constant pillar of guidance and support, especially as we've pivoted This Farm Needs a Farmer throughout 2020 and 2021. It has been a strong working relationship that has developed into a very solid friendship. I have a lot to thank Angela for.
If there's one positive thing to come out of this lockdown experience, I think it's the new appreciation for the importance of our farmers' markets. They have been the one constant for us during COVID, and the highlight of people's weeks here is getting out, chatting with producers and knowing you're getting absolute quality. Castlemaine's is just incredible. The farmers I know that work at these markets are the busiest they've ever been in these last two years; there's been a real recognition of how hard our farmers work, and the quality of the produce they're putting out there. There's no reason why we should be importing anything, and I think the more people that go to these farmers' markets, the more that consensus will grow.
If I could return to any moment in the Victorian hospitality industry of the last 50 years, I'd choose being able to eat out with as many friends as I want in a busy restaurant, beyond a five-kilometre radius and staying out as long as I want to with no curfew. COVID has been a very tough time, especially for rural communities, as we're already quite removed. We used to host Friday drinks for everyone involved in This Farm Needs a Farmer at The Vic in Woodend. We'll never take that for granted again.
But the most exciting development in the Victorian hospitality industry in the last five years is the growth and support of our local, boutique primary producers supplying the hospitality industry with high-quality, low-food-mile product.
Five minutes with a Legend: Alister Purbrick
Five Minutes with a Legend: Hana Assafiri
Five Minutes with a Legend: Rinaldo Di Stasio


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Gift will fund LSU entrepreneurial competition
BATON ROUGE, La. (AP) " LSU announced Thursday that a gift to its College of Business will be used to support entrepreneurship by funding its annual student competition for capital backing new business ideas.
The gift to the E.J. Ourso College of Business was announced in a news release. LSU said the gift was made in honor of a 1964 LSU graduate " J. Terrell Brown Sr. " by his children in recognition of Brown's business career, 'his love for LSU and his passion for entrepreneurship.'
The endowment ensures a steady stream of funding for the competition's $25,000 prize pool.
Brown received a bachelor's degree in finance and insurance. He has served on the board of the LSU Foundation and the E. J. Ourso College of Business Dean's Advisory Council. He served for over two decades on the board of LSU's Pennington Biomedical Research Foundation, including three years as chair.
The annual competition is entering its 10th year, according to the university. It is managed by the Stephenson Entrepreneurship Institute, part of the Ourso College of Business.
'By challenging students to create business plans and pitches, this competition is an important element for our strategy to provide entrepreneurship education and co-curricular opportunities across the LSU system," said Dean Richard D. White Jr.


Be the first to hear about their new releases and deals. Download the Booksprout app and search for Jessiqua Wylder today!
I like to create characters that you think about long after you've put the book down.


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Q: GCE Node Api use custom network with no external IP I struggle with creating a new vm via the node google compute engine API. The cloud console gives me the REST code when no external IP is selected:
 "networkInterfaces": [
    {
      "subnetwork": "projects/optum-80593/regions/us-central1/subnetworks/subnet-us-central1"
    }
  ]

But when I use this in my config for the node API, the external IP is still set to that VM:
 const Compute = require('@google-cloud/compute');
 const compute = new Compute();
 const zone = compute.zone('us-central1-a');

 const config = {
        machineType: machinetype,
        maintenancePolicy: 'MIGRATE',
        http: true,
        disks: [
          ...
        ],
        canIpForward: false,
        networkInterfaces: [
            {
                subnetwork: "projects/optum-80593/regions/us-central1/subnetworks/subnet-us-central1"
            }
        ],
   ...
}

await zone.createVM(vmName, config).then(async (data) => {

A: It works, when I copy the whole REST code to my node API.

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Q: Why do we need continuity at the end points of the interval for Rolle's theorem? For a function ($f:[a,b] ->R$) to satisfy Rolle's theorem it must be continuous in the interval $[a,b]$ , differentiable in $(a,b)$ and $f(a)=f(b)$. I don't really understand what's the need of function being continuous at $x=a$ and at $x=b$.
Because even if the functions are not continuous at $ a,b$ then also I can get a 'c' in $(a,b)$ such that $f'(c)=0$ by just replacing the third condition written above with
$$lim_{x->a}f(x)=lim_{x->b}f(x)$$
So is there any specific reason to define continuity at the end points ?


I use a Custom ArcMap Extension. I also use Stereo Analyst for ArcGIS Extension.
When running ArcMap 10 (with the two extensions), I get random, non-reproducible crashes. With ArcMap 9.3.1 these crashes didn't occur.
Your assumptions are not necessarily correct.
The first thing that you want to look at is the call stack when the crash occurs. Hopefully, the code which is the root cause of the problem is in the stack. When the problem occurs you should be able to attach the Visual Studio debugger to the process and view the call stack (make sure that you attach to debug both managed and unmanged code). I definitely recommend that you use the ESRI public symbols to assist.
If you (or ESRI) have corrupted memory then the crash can occur at some interdeterminate time later. These type of problems are very hard to debug.
What language are you using to program?
What type of extension(s) did you develop?
If you have random non-reproducible crashes then ESRI support will not file an issue for you. Is the problem really random? Perhaps it is data specific.
thank you for your information! As the crashes are random, it will probably take some time until the next crash occurs and I can look at the call stack.
Search your disk drive for ArcMap dump files (*.dmp). I think that something like the last 10 are kept. You can open them in Visual Studio and view the call stack.
If you are programming in .NET, not doing P/Invokes, and not doing any multi-threading then the problem is likely not yours.
attached, you can see the call stack. LFI.dll is the dll containing my application extension and my custom commands.
I have no clue what the calls stacks tells... (except where the crash in my program occurs, which I already know). Do you understand more?
Hopefully, the resulting call stack will have more information.
It appears that code is crashing due to a null pointer dereference. The obvious question is is what is dvodisplay?
I had already set up the debug symbols according to http://resources.esri.com/help/9.3/A...c81f9c0347.htm !
Ok, for me, the case is clear: dvodisplay.dll is an ERDAS Stereo Analyst dll. This is also what I suspected from the beginning: The problem is in Stereo Analyst.
Anyway, thanks a lot for your help, Richard, it was very useful!


text:
The zebrafish annexin gene family
Steven A. Farber, Robert A. De Rose, Eric S. Olson, Marnie E. Halpern
The Annexins (ANXs) are a family of calcium- and phospholipid-binding proteins that have been implicated in many cellular processes, including channel formation, membrane fusion, vesicle transport, and regulation of phospholipase A2 activity. As a first step toward understanding in vivo function, we have cloned 11 zebrafish anx genes. Four genes (anx1a, anx2a, anx5, and anx11a) were identified by screening a zebrafish cDNA library with a Xenopus anx2 fragment. For these genes, full-length cDNA sequences were used to cluster 212 EST sequences generated by the Zebrafish Genome Resources Project. The EST analysis revealed seven additional anx genes that were subsequently cloned. The genetic map positions of all 11 genes were determined by using a zebrafish radiation hybrid panel. Sequence and syntenic relationships between zebrafish and human genes indicate that the 11 genes represent orthologs of human anx1,2,4,5,6,11,13, and suggest that several zebrafish anx genes resulted from duplications that arose after divergence of the zebrafish and mammalian genomes. Zebrafish anx genes are expressed in a wide range of tissues during embryonic and larval stages. Analysis of the expression patterns of duplicated genes revealed both redundancy and divergence, with the most similar genes having almost identical tissue-specific patterns of expression and with less similar duplicates showing no overlap. The differences in gene expression of recently duplicated anx genes could explain why highly related paralogs were maintained in the genome and did not rapidly become pseudogenes.
https://doi.org/10.1101/gr.479603
10.1101/gr.479603
Dive into the research topics of 'The zebrafish annexin gene family'. Together they form a unique fingerprint.
Annexins Medicine & Life Sciences 100%
Zebrafish Medicine & Life Sciences 76%
Expressed Sequence Tags Medicine & Life Sciences 18%
Radiation Hybrid Mapping Medicine & Life Sciences 11%
Transport Vesicles Medicine & Life Sciences 9%
Pseudogenes Medicine & Life Sciences 9%
Farber, S. A., De Rose, R. A., Olson, E. S., & Halpern, M. E. (2003). The zebrafish annexin gene family. Genome research, 13(6 A), 1082-1096. https://doi.org/10.1101/gr.479603
The zebrafish annexin gene family. / Farber, Steven A.; De Rose, Robert A.; Olson, Eric S.; Halpern, Marnie E.
In: Genome research, Vol. 13, No. 6 A, 01.06.2003, p. 1082-1096.
Farber, SA, De Rose, RA, Olson, ES & Halpern, ME 2003, 'The zebrafish annexin gene family', Genome research, vol. 13, no. 6 A, pp. 1082-1096. https://doi.org/10.1101/gr.479603
Farber SA, De Rose RA, Olson ES, Halpern ME. The zebrafish annexin gene family. Genome research. 2003 Jun 1;13(6 A):1082-1096. https://doi.org/10.1101/gr.479603
Farber, Steven A. ; De Rose, Robert A. ; Olson, Eric S. ; Halpern, Marnie E. / The zebrafish annexin gene family. In: Genome research. 2003 ; Vol. 13, No. 6 A. pp. 1082-1096.
@article{4059880a11234835aa39bd60a249194f,
title = "The zebrafish annexin gene family",
abstract = "The Annexins (ANXs) are a family of calcium- and phospholipid-binding proteins that have been implicated in many cellular processes, including channel formation, membrane fusion, vesicle transport, and regulation of phospholipase A2 activity. As a first step toward understanding in vivo function, we have cloned 11 zebrafish anx genes. Four genes (anx1a, anx2a, anx5, and anx11a) were identified by screening a zebrafish cDNA library with a Xenopus anx2 fragment. For these genes, full-length cDNA sequences were used to cluster 212 EST sequences generated by the Zebrafish Genome Resources Project. The EST analysis revealed seven additional anx genes that were subsequently cloned. The genetic map positions of all 11 genes were determined by using a zebrafish radiation hybrid panel. Sequence and syntenic relationships between zebrafish and human genes indicate that the 11 genes represent orthologs of human anx1,2,4,5,6,11,13, and suggest that several zebrafish anx genes resulted from duplications that arose after divergence of the zebrafish and mammalian genomes. Zebrafish anx genes are expressed in a wide range of tissues during embryonic and larval stages. Analysis of the expression patterns of duplicated genes revealed both redundancy and divergence, with the most similar genes having almost identical tissue-specific patterns of expression and with less similar duplicates showing no overlap. The differences in gene expression of recently duplicated anx genes could explain why highly related paralogs were maintained in the genome and did not rapidly become pseudogenes.",
author = "Farber, {Steven A.} and {De Rose}, {Robert A.} and Olson, {Eric S.} and Halpern, {Marnie E.}",
doi = "10.1101/gr.479603",
journal = "Genome Research",
publisher = "Cold Spring Harbor Laboratory Press",
number = "6 A",
T1 - The zebrafish annexin gene family
AU - Farber, Steven A.
AU - De Rose, Robert A.
AU - Olson, Eric S.
AU - Halpern, Marnie E.
N2 - The Annexins (ANXs) are a family of calcium- and phospholipid-binding proteins that have been implicated in many cellular processes, including channel formation, membrane fusion, vesicle transport, and regulation of phospholipase A2 activity. As a first step toward understanding in vivo function, we have cloned 11 zebrafish anx genes. Four genes (anx1a, anx2a, anx5, and anx11a) were identified by screening a zebrafish cDNA library with a Xenopus anx2 fragment. For these genes, full-length cDNA sequences were used to cluster 212 EST sequences generated by the Zebrafish Genome Resources Project. The EST analysis revealed seven additional anx genes that were subsequently cloned. The genetic map positions of all 11 genes were determined by using a zebrafish radiation hybrid panel. Sequence and syntenic relationships between zebrafish and human genes indicate that the 11 genes represent orthologs of human anx1,2,4,5,6,11,13, and suggest that several zebrafish anx genes resulted from duplications that arose after divergence of the zebrafish and mammalian genomes. Zebrafish anx genes are expressed in a wide range of tissues during embryonic and larval stages. Analysis of the expression patterns of duplicated genes revealed both redundancy and divergence, with the most similar genes having almost identical tissue-specific patterns of expression and with less similar duplicates showing no overlap. The differences in gene expression of recently duplicated anx genes could explain why highly related paralogs were maintained in the genome and did not rapidly become pseudogenes.
AB - The Annexins (ANXs) are a family of calcium- and phospholipid-binding proteins that have been implicated in many cellular processes, including channel formation, membrane fusion, vesicle transport, and regulation of phospholipase A2 activity. As a first step toward understanding in vivo function, we have cloned 11 zebrafish anx genes. Four genes (anx1a, anx2a, anx5, and anx11a) were identified by screening a zebrafish cDNA library with a Xenopus anx2 fragment. For these genes, full-length cDNA sequences were used to cluster 212 EST sequences generated by the Zebrafish Genome Resources Project. The EST analysis revealed seven additional anx genes that were subsequently cloned. The genetic map positions of all 11 genes were determined by using a zebrafish radiation hybrid panel. Sequence and syntenic relationships between zebrafish and human genes indicate that the 11 genes represent orthologs of human anx1,2,4,5,6,11,13, and suggest that several zebrafish anx genes resulted from duplications that arose after divergence of the zebrafish and mammalian genomes. Zebrafish anx genes are expressed in a wide range of tissues during embryonic and larval stages. Analysis of the expression patterns of duplicated genes revealed both redundancy and divergence, with the most similar genes having almost identical tissue-specific patterns of expression and with less similar duplicates showing no overlap. The differences in gene expression of recently duplicated anx genes could explain why highly related paralogs were maintained in the genome and did not rapidly become pseudogenes.
U2 - 10.1101/gr.479603
DO - 10.1101/gr.479603
JO - Genome Research
JF - Genome Research
IS - 6 A


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