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\begin{document}
\draft
\preprint{AMU-97-02; US-97-07}
\newcommand{\dfrac}[2]{\displaystyle{\frac{#1}{#2}}}
\def\thefootnote{\fnsymbol{footnote}}
\title{Updated Estimate of Running Quark Masses}
\author{Hideo Fusaoka\thanks{Email: fusaoka@amugw.aichi-med-u.ac.jp}}
\address{Department of Physics, Aichi Medical University \\
Nagakute, Aichi 480-11, Japan}
\author{Yoshio Koide\thanks{E-mail: koide@u-shizuoka-ken.ac.jp}}
\address{Department of Physics, University of Shizuoka \\
52-1 Yada, Shizuoka 422, Japan}
\date{\today}
\maketitle
\begin{abstract}
Stimulated by recent development of the calculation methods of
the running quark masses $m_q(\mu)$ and renewal of the input data,
for the purpose of making a standard table of $m_q(\mu)$
for convenience of particle physicists,
the values of $m_q(\mu)$ at various energy scales $\mu$
($\mu = 1$ GeV, $\mu = m_c$, $\mu=m_b$, $\mu=m_t$ and so on),
especially at $\mu = m_Z$,
are systematically evaluated by using the mass renormalization
equations and by taking into consideration a matching condition
at the quark threshold.
\end{abstract}
\pacs{ }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage
\narrowtext
\section{Introduction}
\label{sec:level1}
\vglue.1in
It is very important to know reliable values of quark masses $m_q$ not
only for hadron physicists who intend to evaluate observable quantities on
the basis of an effective theory, but also for quark-lepton physicists
who intend to build a model for quark and lepton unification.
For such a purpose, for example, a review article\cite{ref1} of 1982
by Gasser and Leutwyler has offered useful information on the running
quark masses $m_q(\mu)$ to us.
However, during the fifteen years after the Gasser and Leutwyler's
review article, there have been some developments in the input data
and calculation methods: the QCD parameter
$\Lambda^{(n)}_{\overline{MS}}$ has been revised\cite{ref2}; top-quark mass
$m_t$ has been observed\onlinecite{ref3,ref4,ref5}; the three-loop
diagrams have been
evaluated for the pole mass $M_q^{pole}$\cite{ref6} and for the running
quark mass $m_q(\mu)$\cite{ref7}; a new treatment of the matching condition
at the quark threshold has been proposed\cite{ref8}.
On the other hand, so far, there are few articles which review masses
of all quarks systematically, although there have been some
re-estimates
\onlinecite{ref9,ref10,ref11,ref12,ref13,ref14,ref15,ref16,ref17,ref18}
for specific quark masses.
For recent one of such few works in systematical study of all quark
masses, for example, see Ref.\cite{ref19} by Rodrigo.
We will give further systematical studies on the basis of recent data
and obtain a renewed table of
the running quark mass values.
The purpose of the present paper is to offer a useful table of the running
quark masses $m_q(\mu)$ to hadron physicists and quark-lepton physicists.
In Sec. \ref{sec:level4}, by using the mass renormalization equation (4.1),
we will
evaluate the value of $m_q(\mu)$ at various energy scales $\mu$, e.g.,
$\mu = 1$ GeV, $\mu = m_q \ (q = c, b, t)$, $\mu = M_q^{pole}$, $\mu = m_Z$,
$\mu = \Lambda_W$, and so on, where $M_q^{pole}$ is a ^^ ^^ pole" mass of the
quark $q$, and $\Lambda_W$ is the symmetry breaking energy scale of
the electroweak gauge symmetry SU(2)$_L$ $\times$ U(1)$_Y$:
\begin{equation}
\Lambda_W \equiv \langle \phi^0 \rangle = (\sqrt{2}G_F)^{-\frac{1}{2}}/
\sqrt{2} = 174.1\ {\rm GeV} \ \ .
\end{equation}
In the next section, we review the light quark masses
$m_u(\mu)$, $m_d(\mu)$ and $m_s(\mu)$ at $\mu=1$ GeV.
In Sec. \ref{sec:level3},
we review pole mass values of the heavy quark masses $M_c^{pole}$,
$M_b^{pole}$ and $M_t^{pole}$.
In Sec. \ref{sec:level4},
running quark masses $m_q(\mu)$ are evaluated for various energy
scales $\mu$ below $\mu= \Lambda_W = 174.1$ GeV.
In Sec. \ref{sec:level5}, we comment on the reliability of the perturbative
calculations of the running quark masses $m_q(\mu)$ ($\mu\leq\Lambda_W$).
In Sec. \ref{sec:level6},
we summarize our numerical results of the running quark
mass values $m_q(\mu)$, the charged lepton masses $m_\ell(\mu)$, the
Cabibbo-Kobayashi-Maskawa (CKM)\cite{ref20} matrix $V_{CKM}(\mu)$, and the
SU(3)$_c \times$SU(2)$_L \times$U(1)$_Y$ gauge coupling constants
$g_i(\mu)$ ($i = 1, 2, 3$) at $\mu=m_Z$.
In Sec. \ref{sec:level7},
for reference, the evolution of the Yukawa coupling constants
is estimated at energy scales higher than $\mu=\Lambda_W$ for the cases
of (A) the standard model with one Higgs boson and (B) the minimal
SUSY model.
Finally, Sec. \ref{sec:level8} is devoted to summary and discussions.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{.2in}
\section{Light quark masses at $\mu=1$ G\lowercase{e}V}
\label{sec:level2}
\vglue.1in
Gasser and Leutwyler\cite{ref1} have concluded in their review article of
1982 that the light quark masses $m_u(\mu)$, $m_d(\mu)$ and $m_s(\mu)$
at $\mu=1$ GeV
are
\begin{eqnarray}
m_u(1 \ {\rm GeV}) & = & 5.1 \pm 1.5\ {\rm MeV} \ , \nonumber \\
m_d(1 \ {\rm GeV}) & = & 8.9 \pm 2.6\ {\rm MeV} \ , \\
m_s(1 \ {\rm GeV}) & = & 175 \pm 55 \ {\rm MeV} \ , \nonumber
\end{eqnarray}
from QCD sum rules.
On 1987, Dominguez and Rafael\cite{ref9} have re-estimated those values
from QCD finite energy sum rules.
They have obtained the same ratios of the light quark masses
with those estimated by Gasser and Leutwyler,
but they have used a new value of ($m_u+m_d$) at $\mu=1$ GeV
\begin{eqnarray}
(m_u+m_d)_{\mu=1 \ {\rm GeV}} & = & (15.5 \pm 2.0) \ {\rm MeV} \ ,
\end{eqnarray}
instead of Gasser--Leutwyler's value $(m_u+m_d)_{\mu=1 \ {\rm GeV}}=(14\pm3)$
MeV.
Therefore, Dominguez and Rafael have concluded as
\begin{eqnarray}
m_u(1 \ {\rm GeV}) & = & 5.6 \pm 1.1\ {\rm MeV} \ , \nonumber \\
m_d(1 \ {\rm GeV}) & = & 9.9 \pm 1.1\ {\rm MeV} \ , \\
m_s(1 \ {\rm GeV}) & = & 199 \pm 33 \ {\rm MeV} \ . \nonumber
\end{eqnarray}
Recently, by simulating $\tau$-like inclusive processes for the
old Das-Mathur-Okubo sum rule relating the $e^+ e^-$ into $I = 0$
and $I = 1$ hadron total cross-section data,
Narison (1995)\cite{ref10} has obtained the following values:
\begin{eqnarray}
m_u(1 \ {\rm GeV}) & = & 4 \pm 1 \ {\rm MeV} \ , \nonumber \\
m_d(1 \ {\rm GeV}) & = & 10 \pm 1\ {\rm MeV} \ , \\
m_s(1 \ {\rm GeV}) & = & 197 \pm 29 \ {\rm MeV} \ , \nonumber
\end{eqnarray}
which are roughly in agreement with (2.3).
On the other hand, by combining various pieces of the information
on the quark mass ratios,
Leutwyler (1996)\cite{ref11} has recently re-estimated
the ratios
\begin{eqnarray}
m_u/m_d & = & 0.553 \pm 0.043 \ , \nonumber \\
m_s/m_d & = & 18.9 \pm 0.8 \ ,
\end{eqnarray}
and has obtained
\begin{eqnarray}
m_u(1 \ {\rm GeV}) & = & 5.1 \pm 0.9 \ {\rm MeV} \ , \nonumber \\
m_d(1 \ {\rm GeV}) & = & 9.3 \pm 1.4 \ {\rm MeV} \ , \\
m_s(1 \ {\rm GeV}) & = & 175 \pm 25 \ {\rm MeV} \ , \nonumber
\end{eqnarray}
The values (2.6) are in agreement with (2.1), (2.3) and (2.4).
There is not so large discrepancy among these estimates as far as
$m_u$ and $m_d$ are concerned,
except for estimates by Donoghue, Holstein and Wyler (1992)\cite{ref12},
who have obtained
\begin{eqnarray}
m_d/m_u= 3.49 , \ \ \ m_s/m_d= 20.7 \ ,
\end{eqnarray}
from the constraints of chiral symmetry treated to next-to-leading order.
Eletsky and Ioffe (1993)\cite{ref13}, and Adami, Drukarev and Ioffe
(1993)\cite{ref14} have obtained
\begin{equation}
(m_d-m_u)_{\mu = 0.5\ {\rm GeV}} = 3 \pm 1 \ {\rm MeV} \ ,
\end{equation}
from the QCD sum rules on the isospin-violating effects for $D$ and $D^*$
and for $N$, $\Sigma$ and $\Xi$, respectively.
The value (2.8) is consistent with (2.3) and (2.6).
The value
\begin{equation}
(m_u + m_d)_{\mu=1 \ {\rm GeV}} = (12 \pm 2.5) \ {\rm MeV} \ ,
\end{equation}
obtained from QCD finite energy sum rules and Laplace sum rules
by Bijnens, Prades and Rafael (1995)\cite{ref15} is consistent with
(2.2).
On the contrary, for the strange quark mass $m_s$, two different values,
$m_s \simeq 175$ MeV, [(2.1) and (2.6)], and $m_s \simeq 200$ MeV ,
[(2.3) and (2.4)],
have been reported.
Recently, Chetyrkin {\it et al}. (1997)\cite{ref16} have estimated
\begin{equation}
m_s (1 \ {\rm GeV}) = 205.5 \pm 19.1 \ {\rm MeV} \ \ ,
\end{equation}
by an order-$\alpha_s^3$ determination from the QCD sum rules.
The value (2.10) is consistent
with (2.3). (Of course,
if we take their errors into consideration, these values are
consistent.)
Hereafter, as the light quark masses at $\mu = 1$ GeV , we will use the
following values which are weighted averages of the values (2.3),
(2.4), (2.6) and (2.10).
\begin{eqnarray}
m_u (1 \ {\rm GeV}) & = & 4.88 \pm 0.57 \ {\rm MeV} \ \ , \nonumber \\
m_d (1 \ {\rm GeV}) & = & 9.81 \pm 0.65 \ {\rm MeV} \ \ , \\
m_s (1 \ {\rm GeV}) & = & 195.4 \pm 12.5 \ {\rm MeV} \ \ . \nonumber
\end{eqnarray}
\vglue.2in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{heavy quark masses}
\label{sec:level3}
\vglue.1in
\noindent {\bf A. Charm and bottom quark masses}
Gasser and Leutwyler (1982)\cite{ref1} have estimated charm and bottom
quark masses
$m_c$ and $m_b$ from the QCD sum rules as
\begin{eqnarray}
m_c(m_c) & = & 1.27\pm0.05 \ {\rm GeV} \ , \nonumber \\
m_b(m_b) & = & 4.25\pm0.10 \ {\rm GeV} \ .
\end{eqnarray}
Titard and Yndur\'{a}in (1994)\cite{ref17} have re-estimated charm and
bottom quark masses by using the three-level QCD and the full one-loop
potential.
They have concluded that
\begin{eqnarray}
M_c^{pole} & = & 1.570 \pm 0.019 \mp 0.007 \ {\rm GeV} \ , \nonumber \\
M_b^{pole} & = & 4.906_{-0.051}^{+0.069} \mp 0.004_{-0.040}^{+0.011}
\ {\rm GeV}\ ,
\end{eqnarray}
\begin{eqnarray}
m_c(m_c) & = & 1.306_{-0.034}^{+0.021} \pm 0.006 \ {\rm GeV} \ , \nonumber \\
m_b(m_b) & = & 4.397_{-0.002+0.004-0.032}^{+0.007-0.003+0.016} \ {\rm GeV}
\ ,
\end{eqnarray}
where the first- and second-errors come from the use of the QCD parameter
$\Lambda_{\overline{MS}}^{(4)}=0.20_{-0.06}^{+0.08}$ GeV and
the gluon condensate value $\langle\alpha_s G^2\rangle=0.042\pm 0.020$
GeV$^4$, respectively, and the third error denotes a systematic error.
On the other hand, from the QCD spectral sum rules to two-loops for
$\psi$ and $\Upsilon$,
Narison (1995)\cite{ref18} has estimated the running quark masses
\begin{eqnarray}
m_c (M_c^{PT2}) & = & 1.23^{+0.02}_{-0.04}
\pm 0.03 \ {\rm GeV} \ , \nonumber \\
m_b (M_b^{PT2}) & = & 4.23^{+0.03}_{-0.04}
\pm 0.02 \ {\rm GeV} \ ,
\end{eqnarray}
corresponding to the short-distance perturbative
pole masses to two-loops
\begin{eqnarray}
M_c^{PT2} & = & 1.42 \pm 0.03 \ {\rm GeV} \ , \nonumber \\
M_b^{PT2} & = & 4.62 \pm 0.02 \ {\rm GeV} \ ,
\end{eqnarray}
and the three-loop values of the short-distance pole masses
\begin{eqnarray}
M_c^{PT3} & = & 1.64^{+0.10}_{-0.07} \pm 0.03 \ {\rm GeV} \ , \nonumber \\
M_b^{PT3} & = & 4.87 \pm 0.05 \pm 0.02 \ {\rm GeV} \ .
\end{eqnarray}
The values (3.6) are in agreement with the values (3.2)
estimated by Titard and Yndur\'{a}in while the values (3.5)
are not so.
Narison asserts that one should not use $M_q^{PT3}$ because the hadronic
correlators are only known to two-loop accuracy.
Although we must keep the Narison's statement in mind, since we use the
three-loop formula (4.5) for the running quark masses $m_q(\mu)$
for all quarks $q = u, d, \cdots, t,$ hereafter, we adopt
the following weighted averages of (3.2) and (3.6),
\begin{eqnarray}
M_c^{pole} & = & 1.59 \pm 0.02 \ {\rm GeV} \ , \nonumber \\
M_b^{pole} & = & 4.89 \pm 0.05 \ {\rm GeV} \ ,
\end{eqnarray}
as the pole mass values.
\vglue.1in
\noindent {\bf B. Top quark mass}
The explicit value of the top quark mass was first reported
by the CDF collaboration (1994)\cite{ref3} from the data of
$p \overline{p}$ collisions at $\sqrt{s} = 1.8$ {TeV}:
\begin{equation}
m_t = 174 \pm 10^{+13}_{-12} \ {\rm GeV} \ .
\end{equation}
They (1995)\cite{ref4} have also reported an updated value
\begin{equation}
m_t = 176 \pm 8 \pm 10 \ {\rm GeV} \ .
\end{equation}
On the other hand, the D0 collaboration\cite{ref5} has reported
the value
\begin{equation}
m_t = 199^{+19}_{-21} \pm 22 \ {\rm GeV} \ .
\end{equation}
The particle data group (PDG96)\cite{ref21} has quoted the value
\begin{equation}
m_t = 180 \pm 12 \ {\rm GeV} \ ,
\end{equation}
as the top quark mass from direct observations of top quark events.
Hereafter, we use the value (3.11) as the pole mass of the top
quark.
\vglue.1in
\noindent {\bf C. Mass values $m_q(\mu)$ at $\mu=M_q^{pole}$}
The relation between the pole mass $M_q^{pole}$ and the running quark mass
$m_q(M_q^{pole})$ at $\mu = M_q^{pole}$ has been calculated by Gray
{\it et al} \cite{ref6}:
\begin{equation}
m_q(M_q^{pole})=M_q^{pole} \left[1
+\frac{4}{3}\frac{\alpha_s(M_q^{pole})}{\pi}+
K_q \left(\frac{\alpha_s(M_q^{pole})}{\pi}\right)^2
+O(\alpha_s^3)\right]^{-1} \ ,
\end{equation}
where $K_c = 14.5$, $K_b = 12.9$ and $K_t = 11.0$.
The definition of $K_q$ and their estimates are given in Appendix A.
The values of $\alpha_s (\mu)$ at various values of $\mu$ and errors
are given in Table VII in Appendix B.
By using (3.12), from (3.7) and (3.11), we obtain
\begin{eqnarray}
m_c (M_c^{pole}) & = & 1.213
\pm 0.018^{-0.040}_{+0.034} \ {\rm GeV} \ \ , \nonumber \\
m_b (M_b^{pole}) & = & 4.248
\pm 0.046^{-0.040}_{+0.036} \ {\rm GeV} \ \ , \\
m_t (M_t^{pole}) & = & 170.1 \pm 11.4 \mp 0.3 \ {\rm GeV} \ \ , \nonumber
\end{eqnarray}
where the first and second errors come from $\pm \Delta M_q^{pole}$ and
$\pm \Delta \Lambda_{\overline{MS}}^{(5)}$, respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{.3in}
%\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Behaviors of ${\text{\lowercase{\it m}}}_{\text{\lowercase{\it q}}}
(\mu)$ at the quark thresholds}
\label{sec:level4}
\vglue.1in
The scale dependence of a running quark mass $m_q(\mu)$ is governed by the
equation \cite{ref7}
\begin{equation}
\mu\frac{d}{d\mu} m_q(\mu) = -\gamma(\alpha_s)m_q(\mu) \ \ ,
\end{equation}
where
\begin{equation}
\gamma(\alpha_s) = \gamma_0\frac{\alpha_s}{\pi} +
\gamma_1\left(\frac{\alpha_s}{\pi}\right)^2 +
\gamma_2\left(\frac{\alpha_s}{\pi}\right)^3 +
O(\alpha_s^4) \ \ .
\end{equation}
\[
\gamma_0 = 2 \ \ , \ \ \ \gamma_1 = \frac{101}{12} - \frac{5}{18}n_q \ \ ,
\]
\begin{equation}
\gamma_2 = \frac{1}{32}\left[1249 - \left(\frac{2216}{27} +
\frac{160}{3}\zeta(3)
\right)n_q - \frac{140}{81}n_q^2\right] \ \ .
\end{equation}
Then, $m_q(\mu)$ is given by
\begin{equation}
m_q(\mu) = R(\alpha_s(\mu)) \widehat{m}_q \ \ ,
\end{equation}
\[
R(\alpha_s) = \left(\frac{\beta_0}{2}\frac{\alpha_s}{\pi} \right)^
{2\gamma_0/\beta_0} \left\{1 + \left(2\frac{\gamma_1}{\beta_0} -
\frac{\beta_1\gamma_0}{\beta_0^2} \right)\frac{\alpha_s}{\pi} \right. \]
\begin{equation}
\left. + \frac{1}{2}\left[\left(2\frac{\gamma_1}{\beta_0} -
\frac{\beta_1 \gamma_0}
{\beta_0^2} \right)^2 + 2\frac{\gamma_2}{\beta_0} -
\frac{\beta_1\gamma_1}{\beta_0^2} - \frac{\beta_2\gamma_0}{16\beta_0^2} +
\frac{\beta_1^2\gamma_0}{2\beta_0^3} \right]
\left(\frac{\alpha_s}{\pi} \right)^2 + O(\alpha_s^3)\right\} \ \ ,
\end{equation}
where $\widehat{m}_q$ is the renormalization group invariant mass
which is independent of $\ln(\mu^2/\Lambda^2)$, $\alpha_s(\mu)$ is
given by (B4) in Appendix B and $\beta_i$ $(i= 0, 1, 2)$ are
also defined by (B3).
By using (4.5) and $\Lambda^{(n)}_{\overline{\rm MS}}$ obtained
in Appendix B, we can evaluate $R^{(n)}(\mu)$ for $ \mu < \mu_{n + 1}$,
where $\mu_n$ is the $n$th quark flavor threshold and we take
$\mu_n = m_{qn}(m_{qn})$.
Quite recently, the four-loop beta function and quark mass anomalous
dimension have been obtained by several authors [22].
In this paper, we evaluate the running quark masses by using the three-loop
results (4.1)-(4.5). The effects of the four-loop results to the
three-loop results will be discussed in the next section.
We can evaluate the values of $m_q(m_q)$ ($q=c,b,t$)
by using the values of $M_q^{pole}$
given in Sec. \ref{sec:level3} and the relation
\begin{equation}
m_{qn}(\mu) = \left[R^{(n)}(\mu)/R^{(n)}(M_{qn}^{pole})\right]
m_{qn}(M_{qn}^{pole}) \ \ \ (\mu < \mu_{n+1}) \ \ .
\end{equation}
Similarly, we evaluate the light quark masses $m_q(m_q)$ $(q = u, d, s)$
by using the relation
\begin{equation}
m_q(\mu) = \left[R^{(3)}(\mu)/R^{(3)}(1 \ {\rm GeV})\right]
m_q(1 \ {\rm GeV}) \ \ \ (\mu < \mu_4) \ \ ,
\end{equation}
and the values $m_q(1 \ {\rm GeV})$ given in (2.11).
The results are summarized in Table \ref{table1}.
The values of $m_u(m_u)$, $m_d(m_d)$ and $m_s(m_s)$ should not been
taken rigidly, because
the perturbative calculation is not reliable for such a region
in which $\alpha_s(\mu)$ takes a large value (see the next section).
Exactly speaking, the estimates of $\Lambda_{\overline{\rm MS}}^{(n)}$
in Table \ref{tableB1} in Appendix \ref{sec:levelB} are dependent on
the choices of the quark threshold
$\mu_n = m_{qn}(m_{qn})$.
The values in Table \ref{tableB1} and Table \ref{table1} have been obtained by
iterating the
evaluation of $\Lambda_{\overline{\rm MS}}^{(n)}$ and $m_q(m_q)$.
Running quark mass values $m_{qn}(\mu)$ at $\mu \geq \mu_{n+1}$
cannot be evaluated by the formula (4.4) straightforwardly,
because of the quark threshold effects.
As seen in Fig. \ref{fig1}, the behavior of $R(\mu)$ is discontinuous at
$\mu=\mu_n \equiv m_{qn}(m_{qn})$.
The behavior of the $n$th quark mass $m_{qn}^{(N)} \ (n