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%% subsection 3.3 Definition of the Analytic \amst\ [slac-pub-7157-0-0-3-3 in slac-pub-7157-0-0-3: slac-pub-7157-0-0-3-4]
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\subsection{\usemenu{slac-pub-7157::context::slac-pub-7157-0-0-3-3}{Definition of the Analytic \amst\ }}\label{subsection::slac-pub-7157-0-0-3-3}
We now adopt the commensurate scale relation with the effective charge of the
effective potential as a definition of the extended scheme \amst:
\begin{equation}
\widetilde {\alpha}_{\overline{\mbox{\tiny MS}}}(Q)
= \alpha_V(Q^*) + \frac{2N_C}{3} {\alpha_V^2(Q^{**})\over\pi} +
\cdots ,
\label{alpmsbar2}
\end {equation}
for all scales $Q$. Eq.~(\docLink{slac-pub-7157-0-0-3.tcx}[alpmsbar2]{16}) not only provides an analytic
extension of dimensionally regulated schemes, but it also ties down the
renormalization scale to the physical masses of the quarks as they
enter into the vacuum polarization contributions to $\alpha_V$. There
is thus no scale ambiguity in perturbative expansions in \av\ or \amst.
Taking the logarithmic derivative of the commensurate scale relation
given by Eq.~(\docLink{slac-pub-7157-0-0-3.tcx}[alpmsbar2]{16})
with respect to $\ln Q$ we can define the $\psi$-function for the
\amst\ scheme as follows,
\begin{equation}
\widetilde {\psi}_{\overline{\mbox{\tiny MS}}}(Q)
\equiv \psi_V(Q^*) + 2 \frac{2N_C}{3} {\alpha_V(Q^{**})\over\pi}
\psi_V(Q^{**}).
\end{equation}
To lowest order this gives
$\widetilde {\psi}_{\overline{\mbox{\tiny MS}}}^{ (0) }(Q)
= \psi_V^{ (0) }(Q^*)$,
which in turn gives the following relation between
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}$
and $N_{F,V}^{ (0) }$,
\begin{equation}
\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)
=N_{F,V}^{ (0) }(Q^*),
\end{equation}
where to lowest order, $Q^* = \exp(5/6) Q$.
We can also use the approximate form given
by Eq.~(\docLink{slac-pub-7157-0-0-3.tcx}[eq:nfzappr]{9}) to write
\begin{equation}
\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(\rho_i)
\cong \left(1 + {5 \over \rho_i{\exp({5\over 3})}} \right)^{-1}
\cong \left( 1 + {1 \over {\rho_i}} \right)^{-1}.
\end{equation}
In other words the contribution from one flavor is $\simeq 0.5$ when
the scale $Q$ equals the quark mass $m_i$. Thus the standard procedure
of matching $\alpha_{\overline{\mbox{\tiny MS}}}(\mu)$ at the quark
masses is a zeroth order approximation to the continuous $N_F$.
\begin{figure}[htb]
\begin{center}
\mbox{\epsfig{figure=nfsum.eps,width=10cm}}
\end{center}
\caption[*]{The continuous
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}$ in the analytic
extension of the $\overline{\mbox{MS}}$ scheme as a
function of the physical scale $Q$. (For reference the
continuous $N_F$ is also compared with
the conventional procedure of taking $N_F$ to be a step-function at the
quark-mass thresholds.)}
\label{fig:nfsum}
\end{figure}
Adding all flavors together gives the total
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$
which is shown in Fig.~\docLink{slac-pub-7157-0-0-3.tcx}[fig:nfsum]{4}. For reference the
continuous $N_F$ is also compared with
the conventional procedure of taking $N_F$ to be a step-function at the
quark-mass thresholds.
The figure shows clearly that there are hardly any plateaus at all
for the continuous
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$ in
between the quark masses.
Thus there is really no scale below 1 TeV where
$\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$
can be approximated by a constant.
In other words, for all $Q$ below 1 TeV there is always one quark
with mass $m_i$ such that $m_i^2 \ll Q^2$ or $Q^2 \gg m_i^2$ is not
true.
We also note that if one would use any other scale than the
BLM-scale for $\widetilde {N}_{F,\overline{\mbox{\tiny MS}}}^{(0)}(Q)$,
the result would be to increase the difference between the analytic
$N_F$ and the standard procedure of using the step-function at the
quark-mass thresholds.