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\section{\usemenu{slacpub7063::context::slacpub7063005}{ SCALE SETTING FOR THE MOMENTEVOLUTION EQUATION}}\label{section::slacpub7063005}%%%%%%%%%%%%%%%%%%%%
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The BLM scale fixing procedure can also be applied to the
momentevolution equation {\em after} the moment integration has been
carried out, with evolution kernels given by Eq. (\docLink{slacpub7063003.tcx}[kernels]{10}).
When the resulting equation is written in the form of Eq. (\docLink{slacpub7063002.tcx}[e1]{1}),
the coefficient of $n_f$ is given by
\begin{equation}
A =\frac{ \int_0^1dxx^{n1}\left(P_{qq}^0A(x)\right)_+}{A_n},
\end{equation}
where $A(x)$ is given by Eq. (\docLink{slacpub7063003.tcx}[Ax]{13}).
This is related to our previously defined $B_n$ as
\begin{equation}
A = \frac{B_n}{3}.
\end{equation}
Again, using Eq. (\docLink{slacpub7063002.tcx}[blmscale]{4}), the BLM scale for the momentevolution
equation is
\begin{equation}
Q^*_n = Q \exp(B_n).
\end{equation}
Now the scale is $x$ independent but $n$ dependent.
The scale for the momentevolution equation can also be read from
Fig. \docLink{slacpub7063006.tcx}[f3]{2} and has a fit of
$Q^*_n/Q = 0.656/n^{1/2}$. The $n^{1/2}$ behavior for large $n$ was
also predicted in \cite{4}.
Keeping only the leadingorder term of $P_{qq}$, the momentevolution
equation can be written analogous to Eq. (\docLink{slacpub7063004.tcx}[e15]{24}) as
\begin{equation}
\frac{\partial\ln M_n(Q^2)}{\partial\ln Q^2}
= \frac{\alpha(Q^*_n)}{2\pi}A_n.
\end{equation}
Again, it can be solved by rewriting the coupling constant using
Eq. (\docLink{slacpub7063004.tcx}[coupling constant]{27}) to be
\begin{eqnarray}
M_n(Q)^{1/d_n} & = & C_n\ln({Q^*_n}^2/\Lambda^2) \nonumber \\
& = & C_n\left[\ln(Q^2/\Lambda^2) 2B_n\right],
\end{eqnarray}
which is simply Eq. (\docLink{slacpub7063004.tcx}[e20]{29}). Actually, the commutativity of
scale setting and moment integration works at any higher order,
because a choice of the renormalization scale can only affect the
result by an order higher than that of the calculation.
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