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%%%%%% author definitions added by nc_fix
\newcommand{\ie}{{\it i.e.}}
\newcommand{\ms}{$\overline{\mbox{MS}}$}
\newcommand{\ams}{\mbox{$\alpha_{\overline{\mbox{\tiny MS}}}$}}
\newcommand{\amst}{\mbox{${\widetilde{\alpha}_{\overline{\mbox{\tiny MS}}}}$}}%$
\newcommand{\av}{\mbox{$\alpha_{V}$}}
\newcommand{\ar}{\mbox{$\alpha_{R}$}}
\newcommand{\nfz}{\mbox{$N_{F,V}^{ (0) }$}}
\newcommand{\nfo}{\mbox{$N_{F,V}^{ (1) }$}}
\newcommand{\nfdt}{\mbox{$N_{F,DT}$}}
\newcommand{\nfct}{\mbox{$\widetilde{N}_{F,\overline{\mbox{\tinyMS}}}^{(0)}(Q)$}}
\newcommand{\nf}{\mbox{$N_F$}}
\newcommand{\mz}{\mbox{$M_Z$}}
\newcommand{\mb}{\mbox{$m_b$}}
\newcommand{\mc}{\mbox{$m_c$}}
\newcommand{\betz}{\mbox{$\beta^(0)$}}
\newcommand{\adt}{\mbox{$\alpha_{DT}$}}
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\section{\usemenu{slacpub7157::context::slacpub7157001}{Motivation}}\label{section::slacpub7157001}
The running coupling in quantum chromodynamics (QCD)
in the modified minimal subtraction ($\overline{\mbox{MS}}$)
scheme \cite{1} and other dimensional regularization schemes
is traditionally constructed by solving the renormalization group
equations using perturbative approximants to the $\beta$ function which
change discontinuously at the quark mass thresholds
\cite{2,3,4}.
This is equivalent to using effective
Lagrangians with a fixed number of massless fermions in each energy range
between the quark mass thresholds.
Thus in the $\overline{\mbox{MS}}$ scheme, the
$\beta(\mu)$ function depends on the number of ``massless'' quarks
$N_{F}(\mu)$ which is taken as a step function of the renormalization
scale $\mu$. Matching conditions at threshold require the equivalence
of one effective theory with $n$ massless flavors to another effective
theory with one massive and ($n1$) massless quarks.
It should be noted that this does not prevent one from including quark
masses in the \ms\ scheme. However, the quark masses do not enter into
the $\beta$ function since the running of the coupling is mass
independent.
The oneloop matching conditions \cite{5,6,7} in the
$\overline{\mbox{MS}}$ scheme require the coupling to be continuous if the
matching is done at the quark masses, although the derivative is discontinuous.
In twoloop matching
\cite{8,9,10}
the coupling itself becomes discontinuous if the matching is done at
the quark masses, but it can be rendered continuous by modifying the
$\overline{\mbox{MS}}$ scheme \cite{3}.
Recently the threeloop matching conditions have been computed
\cite{10}, which together with the fourloop
$\beta$function \cite{11}, give the
possibility to evolve the $\overline{\mbox{MS}}$ coupling
to four loops with massless quarks. This gives a reduced
dependence on the matching scale, as shown in
\cite{12}, but possibly a nonphysical threshold
dependence. However,
in such a treatment the derivatives of the coupling remain discontinuous.
The inevitable result of the matching in a dimensional regularization
scheme is that the running of the $\overline{\mbox{MS}}$ coupling in
the renormalization scale is nonanalytic  nondifferentiable or
even discontinuous  as the quark mass thresholds are crossed. Thus
there is an intrinsic difficulty in expressing physical, smooth
observables as an expansion in the $\overline {\mbox{MS}}$ coupling.
It is clearly necessary to restore the finite quark mass effects in
their entirety in order to restore analyticity.
Aesthetically, it is unnatural to characterize physical theories in
terms of an artificiallyconstructed renormalization scheme such as
$\overline{\mbox{MS}}$; it is more physical to use an effective charge
as determined from experiment to define the fundamental coupling
\cite{13}. For example, in analogy to quantum electrodynamics, one
could choose to define the QCD coupling as the coefficient
$\alpha_V(Q)$ in the static limit of the scattering potential between
two heavy quarkantiquark test charges:
\begin{equation}
\label{eq:avdef}
V(Q^2) =  4 \pi C_F {\alpha_V(Q) \over Q^2}
\end{equation}
at the momentum transfer $q^2 = t = Q^2$, where $C_F=
({N_C^21})/({2N_C}) ={4}/{3}$ is the Casimir operator for the
fundamental representation in $SU(N_C)$, (with $N_C=3$ for QCD). Such
an effective charge automatically incorporates the quark mass threshold
effects in the running, and thus it has an analytic
$\beta$ function. The \av\ scheme is particularly wellsuited to
summing the effects of gluon exchange at low momentum transfer, such
as in evaluating the finalstate interaction corrections to heavy
quark production \cite{14}, or in evaluating the hardscattering
matrix elements underlying exclusive processes \cite{15}. A
physical effective charge has the additional advantage that the
AppelquistCarazzone decoupling theorem \cite{16} is automatically
incorporated.
In this paper we shall construct an analytic extension of the \ams\
scheme, which we call \amst, by connecting the coupling directly to the
analytic and physicallydefined \av\ scheme. The necessity for
an analytic coupling has been emphasized by Shirkov and his
collaborators\cite{17}.
Our
definition allows one to use a scheme based on dimensional regularization,
but which also, in a simple way,
treats mass effects properly between the mass thresholds. Thus,
instead of having the number of effective flavors ($N_F$) change discontinuously
at (or nearby) the quark threshold, we obtain an analytic $N_F(\mu)$
which is a continuous function of the renormalization scale $\mu$ and
the quark masses $m_i$. Thus the analyticallyextended scheme inherits
the mass dependence of the physical scheme. In addition, the
renormalization scale $\mu$ that appears in the analyticallyextended
scheme \amst\ is directly related to the momentum transfer appearing in
the \av\ scheme and thus has a definite and simple physical
interpretation\footnote{A somewhat similar approach has been tried in
\cite{17}, but using the unphysical MOM renormalization scheme to
implement the mass thresholds.}.
The essential advantage of the modified scheme \amst\ is that it
provides an analytic interpolation of conventional dimensional
regularization expressions by utilizing the mass dependence of the
physical \av\ scheme. In effect, quark thresholds are treated
analytically to all orders in $m^2/Q^2$; \ie, the evolution of our analytically
extended coupling in the intermediate regions reflects the actual mass
dependence of a physical effective charge and the analytic properties of
particle production in a physical process.
Just as in Abelian QED, the mass dependence of the effective potential
and the analyticallyextended scheme \amst\ reflects the analyticity of the
physical thresholds for particle production in the
crossed channel. Furthermore, the definiteness of the dependence in the quark
masses automatically constrains the renormalization scale.
Alternatively, one could connect \ams\ to another
physical charge such as $\alpha_R$ defined from $e^+e^$ annihilation.
Our approach should be compared with the standard treatment of quark
mass threshold effects in the \ms\ scheme.
For fixed order in $\alpha_{\mbox{\scriptsize{s}}}$
the corrections due to finite quark
mass threshold effects which we are considering in this paper
have been calculated for the hadronic
width of the Zboson and the $\tau$ lepton semihadronic decay rate
\cite{18,19,20,9}. The calculations
have been made both exactly to order ${\alpha}_{\overline{\mbox{\tiny MS}}}^2$
and as expansions in terms of $m^2/Q^2$ and $Q^2/m^2$ for light and
heavy quarks respectively.
Note that in principle the determination of the finite
mass threshold effects for physical observables in
dimensional regularization schemes would require
a complete allorders analysis of
the highertwist mass corrections to the effective Lagrangian of the
theory.
There are a number of other reasons to construct an analytic extension
of the \ams\ scheme:
\begin{itemize}
\item The comparison of the values of the coupling
$\alpha_{\mbox{\scriptsize{s}}}$ as determined from different
experiments and at different momentum scales is an essential
test of QCD (for a recent review of existing measurements see
\cite{21}). One source of error is neglect of quark
masses in the determination of $\alpha_{\mbox{\scriptsize{s}}}$ and
in the subsequent running of the coupling from the scale where it
has been determined to the conventional reference scale, the
$Z$boson mass.
\item Lattice calculations for the J/$\Psi$ and $\Upsilon$
spectra now provide the most precise determination of
$\alpha_{\mbox{\scriptsize{s}}}$ at low momentum
scales\cite{22,23,24,25}. It is
important to know how finite quark mass effects enter into the
running of this value of $\alpha_{\mbox{\scriptsize{s}}}$ to
lower and higher energy scales with as small an error as
possible.
\item
Finite mass threshold effects in supersymmetric
grand
unified theories are important when analyzing the running
and unification of couplings over very large ranges. It
has been discussed, for example, in refs. \cite{26,27}.
However, the scale used in the running and for the threshold
effects has not been related to the physical scale which is
naturally obtained in our approach.
\item
It is natural to unify theories by matching physical couplings
and masses at the unification scale. This can be accomplished in
the \av\ scheme or equivalently \amst.
\end{itemize}
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