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\section{\usemenu{slacpub7063::context::slacpub7063001}{INTRODUCTION}}\label{section::slacpub7063001}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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One of the most serious problems preventing precise empirical tests of
QCD is the ambiguity of the renormalization scale $\mu$ of
perturbative predictions. Formally, any physical quantity should be
$\mu$ independent. However, in practice, spurious $\mu$ dependence
appears as one can only have finiteorder predictions in a
perturbative theory. Unless one specifies the argument of the
coupling $\alpha_s(\mu)$ in the truncated predictions, the range of
theoretical uncertainty can be much larger than the
experimental error. Although the uncertainty of the prediction
contributed from the renormalization scale ambiguity is expected to
reduce as the order of the prediction gets higher, we still need a
rational way of choosing a value for $\mu$ such that the truncated
prediction will approximate the true prediction as close as possible.
A conventional way of setting the renormalization scale in
perturbation theory is setting $\mu$ to be the momentum transfer or
energy scale $Q$ of the system (provided there is only one energy
scale); this eliminates large logarithmic terms $\ln(Q/\mu)$ and so
gives a more convergent series. However, often an order1 variation
of $\mu/Q$ leads to a significant uncertainty in the prediction of
perturbative QCD.
In multiscale processes, the problem of scale setting becomes
compounded, since the choice for $\mu$ can depend on any combination
of the available physical scales. An example of this is the proper
scale for the running coupling constant that appears in the DGLAP
evolution equation for the deepinelastic structure function. In
addition to the momentum transfer $Q$ of the lepton, the physical
scale can also depend on the Bjorken ratio $x =Q^2/(2p \cdot q).$
Equivalently, the scale controlling the evolution of each moment
$M_n(Q)$ of the structure function can depend on both $Q$ and $n$.
Collins\cite {1}, Neubert \cite{2}, and Lepage and
Mackenzie \cite{3} have emphasized that the renormalization scale
should not be fixed by an {\it ad hoc} procedure but rather, should be
determined systematically as the mean virtuality of the underlying
physical subprocess. From this point of view, the choice of the
renormalization scale $\mu$ for a particular prediction to a certain
order thus depends on the specific experimental measure and the
truncation order.
In this paper, we discuss how to obtain the optimal scale for the
structure function evolution using the BLM (BrodskyLepageMackenzie)
\cite{4} scalefixing procedure. In this procedure, the vacuum
polarization diagrams that contribute to the nonzero QCD beta
function are resummed into the running coupling. More technically, we
absorb into the running coupling all factors of the number of flavors
$n_f$ that appear in the coefficients of the perturbative expansion.
This criterion automatically sets the scale to a value that reflects
the average gluon virtuality of the subprocess.
On the one hand, we can calculate the evolution of the moment by
fixing the scale as a function of $Q$ and $x$ before integrating over
$x$ to obtain the moments; on the other hand, we can set the scale
from the momentevolution equation for each $n$. We give the
appropriate scale for the momentevolution equation for each $n$ and
show that the two procedures give the same result to the order that we
consider.
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