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%% subsection 3.1 Calculation of the Mass Dependence for the Running Coupling [slac-pub-7157-0-0-3-1 in slac-pub-7157-0-0-3: ^slac-pub-7157-0-0-3 >slac-pub-7157-0-0-3-2]
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\subsection{\usemenu{slac-pub-7157::context::slac-pub-7157-0-0-3-1}{Calculation of the Mass Dependence for the Running Coupling}}\label{subsection::slac-pub-7157-0-0-3-1}
The coupling $\av(Q)$, which is derived from heavy quark scattering, is
closely related to the renormalization of the gluon propagator. In
physical gauges with $Z_1=Z_2$ the coupling renormalization is due
purely to self-energy insertions in the propagator.\footnote{Strictly
speaking, this is only true up to one-loop in QCD and two-loops in QED.
At higher orders new types of diagrams appear in the potential which
cannot be described as simple self-energy insertions in the propagator.
In QCD such a diagram is the so called ``H-graph" \cite{37}
and in QED the light-by-light scattering diagram has the same effect.
In the QED case, the light-by-light scattering graphs have an anomalous
dependence on the external charges and a cut structure corresponding to
particle production.
In addition, we
note that the non-analytic contributions to \av\ in higher orders in QCD arise
from corrections to the ``H-graph". Therefore it could be argued that these
types of diagrams should be excluded when defining the V-scheme in QCD and QED.}
For the purposes of this paper it will be sufficient to restrict our
analysis to one-loop order\footnote{We expect the main effects from
including the quark masses at the one-loop level as this is
the leading term in the $\psi$-function. However, at small scales
the higher order terms will become important, especially since
the relative importance of the $N_F$ term is larger for $\psi^{(1)}$
than for $\psi^{(0)}$. A study at the two-loop level requires the
massive two-loop diagrams which is work in progress \cite{38}.},
i.e. $\psi^{(0)}$.
The physical running coupling in the \av\ scheme, normalized at an
arbitrary momentum transfer scale $Q_0$, may be represented as
\begin{equation}
\label{running}
\av(Q) \equiv \frac{\alpha_V(Q_0)}{1 -
\tilde{\Pi}\left(Q,Q_0,\alpha_V(Q_0)\right)} .
\end{equation}
The vacuum polarization function $\tilde{\Pi}$ may be computed from the
perturbative expansion of the renormalized propagator between heavy
quarks. The coupling is then
\begin{equation}
\av(Q) =\alpha_V(Q_0) \left[ 1 + \tilde{\Pi} +
\tilde{\Pi} ^2 + \tilde{\Pi} ^3 + \ .\ .\ . \right] ,
\end{equation}
where we have used the shorthand $ \tilde{\Pi}\equiv
\tilde{\Pi}(Q,Q_0,\alpha_V(Q_0)) $ for the renormalized sum of all
one-particle irreducible 1PI diagrams for the gluon self-energy. Since
the coupling has the value $\alpha_V(Q_0) \equiv \alpha_0 $ at the
physical renormalization point $Q=Q_0$, the self-energy obeys the
boundary condition $ \tilde{\Pi}(Q_0,Q_0,\alpha_0) = 0$.
\vspace{.5cm}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsfbox{onepidiag.ps}
\end{center}
\caption[*]{Single insertion of massive quark-antiquark loop into
a gluon propagator, giving the quark part of the one-loop gluon vacuum
polarization.}
\label{fig:onepidiag}
\end{figure}
We begin by considering the integral representation of the quark part
of the one-loop gluon vacuum
polarization diagram (see Fig.~\docLink{slac-pub-7157-0-0-3.tcx}[fig:onepidiag]{1}):
\begin{eqnarray*}
\tilde{\Pi}^{(0)}_{\mbox{\scriptsize{q}}}(Q,Q_0,\alpha_0) =
T_F \sum_{i=1}^{n} \frac{\alpha_0}{3 \pi} \left(
\int_{0}^{1} 6 z (1-z) \ln{\left(1+z (1-z) \rho_i(Q)\right)} - \right.
\\
\left. \int_{0}^{1} 6 z (1-z)
\ln{\left(1+z (1-z) \rho_i(Q_0)\right)} \right)
\end{eqnarray*}
where $\rho = Q^2/m^2$, $T_F={1 \over 2}$,
the superscript (0) indicates the one-loop order,
the subscript q indicates the quark-part
and the sum runs over all quarks ($n$).
Thus the quark component of the one-loop $\psi$-function is:
\begin{equation}
\psi_{V,\mbox{\scriptsize{q}}}^{(0)}(Q)
= -{\nfz \over 3}
= -\left[ \frac{\pi}{\av^2}
\frac{d \av}{d \ln{Q}}\right]^{(0)}_{\mbox{\scriptsize{q}}}
= -\frac{\pi}{\alpha_0}
\frac{d \, \tilde{\Pi}^{(0)}_{\mbox{\scriptsize{q}}}(Q,Q_0,\alpha_0)}
{d \ln{Q}}.
\end{equation}
This gives\footnote{This result was first obtained by Georgi and
Politzer \cite{39} in the MOM scheme and was applied to
general gauge theories by Ross\cite{40}. } the contribution to $N_F$
from quark flavor
$i$,
\begin{equation}
\nfz( \rho_i)= 6 \int_{0}^{1}
\frac{z^{2}(1-z)^{2}\rho_i dz}{1+z(1-z)\rho_i}=
1- \frac{6}{\rho_i} + \frac{24}{\rho_i^{3/2}\sqrt{4+\rho_i}}
\tanh^{-1}{\sqrt{\frac{\rho_i}{\rho_i+4}}},
\end{equation}
which is displayed in Fig.~\docLink{slac-pub-7157-0-0-3.tcx}[fig:pnfo]{2} as a function of
$\rho$. Thus, by keeping the explicit quark mass dependence,
$N_F$ becomes an analytic function of the scale $Q$.
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=nfrho.eps,width=10cm}}
\end{center}
\caption[*]{The curve shows the contribution to the continuous
$\nfz$ for just one quark as a function of $\rho=Q^2/m^2$ where $m$ is
the mass of the quark. $\nfz$ is found by using the massive quark part
of the one-loop gluon propagator instead of using the theta function
thresholds conventionally used in dimensional regularization schemes.}
\label{fig:pnfo}
\end{figure}
In fact, the approximate form:
\begin{equation}
\nfz(\rho_i) \cong \left(1 + {5 \over \rho_i} \right) ^{-1}
\label{eq:nfzappr}
\end{equation}
gives an accurate approximation to the exact form to within
a percent over the entire range of the momentum
transfer\footnote{This
approximate form can be obtained from using a rigorous
double asymptotic series approach, knowing the behavior of the function
at the low and high momentum transfer. \cite{41}}.
The one-loop analytic $N_{F,V}$ is shown in Fig.~\docLink{slac-pub-7157-0-0-3.tcx}[fig:nfsep]{3} for
various quark flavors (for reference,
the quark masses (in GeV) we used are:
$m_u=.004$; $m_d=.008$; $m_s=.200$; $m_c=1.5$; $m_b=4.5$; $m_t=175$).
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=nfsep.eps,width=10cm}}
\end{center}
\caption[*]{Continuous \nfz\ for various quarks, lightest to heaviest goes
top to bottom
(d, c, b, t as one proceed downwards; the u and s plots are virtually
identical at this scale to the d).
Q runs from 1 to $M_Z$ GeV (for reference, the quark masses (in GeV) used are:
$m_u=.004$; $m_d=.008$; $m_s=.200$; $m_c=1.5$; $m_b=4.5$; $m_t=175$.)}
\label{fig:nfsep}
\end{figure}
We may now substitute the $N_{F,V}$ into the one-loop QCD $\psi$ function
coefficient:
\begin{eqnarray*}
\psi_{V}^{(0)}(Q)= \frac{11}{2} - \frac{1}{3} \nfz(\rho_i)
\end{eqnarray*}
and thence into the QCD one-loop renormalization group equation
for the coupling constant:
\begin{equation}
{ {d \av} \over d \ln{Q} } =
- \psi_{V}^{(0)} { \av^2 \over \pi}
\end{equation}
We may then solve this renormalization group equation to yield an
expression for \av\ which is
analytic at mass thresholds. Note that the mass-dependence of the
$\psi$ function applies specifically to the \av\ scheme\footnote{Given
$\alpha_V(Q_0)$ one can obtain the coupling at other scales including
the mass dependence by numerical iteration such as the fourth-order
Runge-Kutta algorithm.}.