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\section{\usemenu{slacpub7063::context::slacpub7063002}{ THE BLM SCALEFIXING PROCEDURE}}\label{section::slacpub7063002}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The crucial idea of BLM procedure \cite{4,5} is that, in the
$\alpha_V$ scheme [$\alpha_V$ is defined through the heavy quark
potential $V(Q^2)$ by $V(Q^2) = 4\pi C_F\alpha_V(Q)/Q^2$], the scale
must be chosen to absorb all the vacuum polarization (nonzero beta
function) contributions into the running coupling constant. Thus one
has to choose the argument of the coupling constant in each order of
perturbation theory so that there is no $n_f$ dependence in the
coefficients of the coupling constants (those lightbylight
contributions that are not associated with renormalization are not
resummed). This is a good choice of scale since all the
vacuumpolarization contributions are then automatically summed to all
orders for any finiteorder prediction. For a nexttoleading order
(NLO) perturbative prediction in {\em any} renormalization scheme, the
procedure simply translates to choosing the scale such that there is
no $n_f$ dependence in the NLO coefficient of the coupling constant,
by the transitivity of the procedure to this order \cite{5}.
To show explicitly how to set the scale for an NLO perturbative QCD
prediction, consider a prediction in any scheme of the following form,
\begin{equation}
\rho = \rho_0 \alpha(\mu) \left\{ 1 +
\left[A(\mu)n_f + B(\mu)\right] {\alpha(\mu) \over \pi}
\right\}
+O(\alpha^3),
\label{e1}
\end{equation}
where $\mu$ is the renormalization scale. The parameters $\rho$,
$\rho_0$, $A$ and $B$ are all in general dependent on one or more
physical scales such as centerofmass energy, momentum transfer and
Bjorken parameter $x$. The $\mu$ dependence of the coefficients $A(\mu)$
and $B(\mu)$, fixed by the renormalization group
\begin{equation}
\frac{d\rho}{d\mu} = 0 + O(\alpha^3)
\label{e2}
\end{equation}
are of the form
\begin{eqnarray}
\nonumber
A(\mu) &=& A'  \frac{1}{3}\ln(\mu) \label{Amu}\\
B(\mu) &=& B' + \frac{11}{6}C_A\ln(\mu) \label{Bmu}\ ,
\end{eqnarray}
where $A'$ and $B'$ are $\mu$ independent.
We apply the BLM procedure to eliminate the $n_f$ and $\beta_0$ dependence in
Eq. (\docLink{slacpub7063002.tcx}[e1]{1}). We choose the renormalization scale $\mu$ to be
\begin{equation}
\label{blmscale}
Q^* = \mu\exp(3A(\mu)),
\end{equation}
so that
\begin{equation}
A(Q^*) = 0
\end{equation}
[note that the righthand side of Eq. (\docLink{slacpub7063002.tcx}[blmscale]{4}) is $\mu$ independent by
Eq. (\docLink{slacpub7063002.tcx}[Amu]{3})].
Using the BLM scale, the perturbative QCD prediction becomes
\begin{equation}
\rho = \rho_0 \alpha(Q^*)
\left[ 1 + B(Q^*) {\alpha (Q^*) \over \pi}
\right]
+O(\alpha^3).
\end{equation}
Rewrite it in terms of the original coefficients $A(\mu)$ and $B(\mu)$ using
Eq. (\docLink{slacpub7063002.tcx}[Bmu]{3}),
\begin{equation}
\rho = \rho_0 \alpha(Q^*)
\left\{1 + \left[ \frac{11}{2}C_AA(\mu)+B(\mu)\right] {\alpha (Q^*) \over \pi}
\right\}
+O(\alpha^3).
\label{scalefixed}
\end{equation}
This is the scalefixed perturbative QCD prediction.
With the chosen scale $Q^*$, all vacuum polarization is resummed into
the running coupling constant, with the coefficient of $\alpha^2$
independent of $n_f$.
In short, by comparing Eq. (\docLink{slacpub7063002.tcx}[e1]{1}) and Eq. (\docLink{slacpub7063002.tcx}[scalefixed]{7}),
setting the scale for an observable is simply equivalent to replacing $n_f$ by
$\frac{11}{2}C_A$, and using $Q^*$ as the argument of the coupling
constant ($\mu$ dependences in $A(\mu)$ and $B(\mu)$ cancel).
For a prediction with $\mu$ preset to any energy scale $Q$, as is
usually given in the literature, the procedure for resetting
the scale appropriately is identical to the one presented above except with
$Q$ replacing $\mu$.
As we have seen, to obtain the renormalization scale only the $n_f$
term of the NLO result is needed. Therefore, one can improve a
leadingorder result by setting an appropriate scale without
calculating the full NLO prediction. Extensions of the method to
higher orders are given in \cite{5}, \cite{6}, and \cite{7}.
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