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%% subsection 3.2 Commensurate scale relation between \av\ and \ams\ [slac-pub-7157-0-0-3-2 in slac-pub-7157-0-0-3: slac-pub-7157-0-0-3-3]
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\subsection{\usemenu{slac-pub-7157::context::slac-pub-7157-0-0-3-2}{Commensurate scale relation between \av\ and \ams\ }}\label{subsection::slac-pub-7157-0-0-3-2}
We now relate the mass dependence of the \av\ scheme to the
$\overline{\mbox{MS}}$ scheme using the commensurate scale relation
\cite{13,36} between the two schemes. We use the NNLO results of
Peter\cite{33}. The first step is to invert
Eq.~(\docLink{slac-pub-7157-0-0-2.tcx}[eq:av]{2}) to obtain $\alpha_{\overline{\mbox{\tiny MS}}}$ as an
expansion in $\alpha_V$,
\begin{eqnarray}\label{eq:avinv}
\alpha_{\overline{\mbox{\tiny MS}}}(Q)
& = &
\alpha_V(M)
+ m_{1,V}\left({Q \over M}\right)
{\alpha_V^2(M) \over \pi}+
m_{2,V}\left({Q \over M}\right)
{\alpha_V^3(M) \over \pi^2}+ \cdots \; .
\end{eqnarray}
The needed commensurate scale relation is obtained by fixing the scales
$M$ in Eq.~(\docLink{slac-pub-7157-0-0-3.tcx}[eq:avinv]{11}) such that the $\psi^{(0)}$ and $\psi^{(1)}$
dependent parts of the coefficients $m_{1,V}$ and $m_{2,V}$ are
absorbed into the running of the coupling $\alpha_V(M)$. This insures
that all vacuum polarization dependence is summed into the heavy quark
potential. Application of this procedure in next-to-next-to leading
order (NNLO), using the multi-scale approach \cite{36}, gives the
following scale-fixed relation between $\alpha_V$ and the
conventional $\overline{\mbox{MS}}$,
\begin{eqnarray} \label{eq:csrmsofv}
\alpha_{\overline{\mbox{\tiny MS}}}(Q)
& = & \alpha_V(Q^{*})
+ \frac{2}{3}N_C{\alpha_V^2(Q^{**}) \over \pi}
\nonumber \\ &&
+\left\{
-\left(\frac{5}{144}+\frac{24\pi^2-\pi^4}{64}-
\frac{11}{4}\zeta_3\right)N_C^2
+\left(\frac{385}{192}-\frac{11}{4}\zeta_3\right)C_F N_C\right\}
{\alpha_V^3(Q^{***}) \over \pi^2}
\nonumber \\ & = &
\alpha_V(Q^{*})
+ 2{\alpha_V^2(Q^{**}) \over \pi}
+ 4.625 {\alpha_V^3(Q^{***}) \over \pi^2} ,
\end{eqnarray}
above or below the quark mass threshold\footnote{Note that the
NNLO results depend
crucially on whether or not the ``H-graph" is included in the definition of
the heavy quark potential since it is the unique source of the $\pi^4 N_C^2$
terms in the NNLO coefficient. We thank M. Peter for communications on this
point.}. The coefficients in the
perturbation expansion have their conformal values, i.e. the same
coefficients would occur even if the theory had been conformally
invariant with $\psi^{(0)}=0$ and thus do not contain the diverging
$(\psi^{(0)}\alpha_{\mbox{\scriptsize{s}}})^n n!$ growth
characteristic of an infrared renormalon \cite{42}. The next-to
leading order (NLO) coefficient $\frac{2}{3}N_C$ is a feature of the
non-Abelian couplings of QCD and is not present in QED. The
commensurate scales $Q^*$ and $Q^{**}$ are given by
\begin{eqnarray}
Q^* & = & Q\exp\left[\frac{5}{6}\right] = 2.300 Q
\\
Q^{**} & = & Q\exp\left[
\left(\frac{105}{128}-\frac{9}{8}\zeta_3\right)\frac{C_F}{N_C}
+\left(\frac{103}{192}+\frac{21}{16}\zeta_3\right)\right] = 6.539 Q
\end{eqnarray}
whereas to this order $Q^{***}$ is not constrained.
However, a first approximation is
obtained by setting $Q^{***}=Q^{**}$.
Also note that $Q^*$ is unchanged when going from NLO to NNLO.
The scale $Q^*$ arises because of the
convention used in defining the modified minimal subtraction
scheme. Comparing the scales $Q$ and $Q^*$ we find that
the scale in the $\overline {\mbox{MS}}$ scheme ($Q$) is a factor
$\sim 0.4$ smaller than the physical scale ($Q^*$).
Alternatively, one can write the relation between
$\alpha_{\overline{\mbox{\tiny MS}}}$ and $\alpha_V$ as a single-scale
commensurate scale relation \cite{42}. In this procedure $Q^* = Q^{**}$
where
\begin{eqnarray}
Q^* & = & Q\exp\left[\frac{5}{6} +
\left[\left(\frac{35}{32}-\frac{3}{2}\zeta_3\right)C_F -
\left(\frac {19}{48} -\frac{7}{4}\zeta_3\right)N_C\right]
\frac{\alpha_V}{\pi}
\right]
\end{eqnarray}
The conformal coefficients are the same in the two
procedures\footnote{Both the multiple and single-scale
setting methods generate a term
proportional to $C_F N_C$ in the NNLO conformal coefficient. The origin of
this term, which has the same color factor as an iteration of the potential,
is not clear and should be further investigated.}.
However, the single-scale form has the advantage that the non-Abelian
perturbation theory matches in a simple way the corresponding Abelian
perturbation theory in the limit
$N_C
\to 0$ with
$C_F\alpha_s$ and
$N_F/C_F$ fixed \cite{43}. For
$N_C= 3$ we have
$\ln(Q^*/Q) = 5/6 + 4.178 \alpha_V/\pi.$