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%% section 5 THE EFFECTIVE CHARGE $\alpha_V(Q^2)$ AND \hfill\break LIGHTCONE
QUANTIZATION [slacpub7152005 in slacpub7152005: slacpub7152006]
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\section{\usemenu{slacpub7152::context::slacpub7152005}{THE EFFECTIVE CHARGE $\alpha_V(Q^2)$ AND \hfill\break LIGHTCONE
QUANTIZATION}}\label{section::slacpub7152005}
The heavy quark potential plays a central role in QCD, not only in
determining the spectrum and wavefunctions of heavy quarkonium, but
also in providing a physical definition of the running coupling for
QCD. The heavy quark potential $V(Q^2)$ is defined as the
twoparticle irreducible amplitude controlling the scattering of two
infinitely heavy test quarks $Q\overline Q$ in an overall
colorsinglet state. Here $Q^2=q^2={\vec q}^2$ is the momentum
transfer. The effective charge $\alpha_V(Q^2)$ is then defined
through the relation $V(Q^2) =  4 \pi C_F\alpha_V(Q^2)/Q^2$ where
$C_F=(N_c^21)/2 N_c = 4/3.$ The running coupling $\alpha_V(Q^2)$
satisfies the usual renormalization group equation, where the first
two terms $\beta_0$ and $\beta_1$ in the perturbation series are
universal coefficients independent of the renormalization scheme or
choice of effective charge. Thus $\alpha_V$ provides a physical
expansion parameter for perturbative expansions in PQCD.
By definition, all quark and gluon vacuum polarization contributions
are summed into $\alpha_V$; the scale $Q$ of $\alpha_V(Q^2)$ that
appears in perturbative expansions is thus fixed by the requirement
that no terms involving the QCD $\beta$function appear in the
coefficients. Thus expansions in $\alpha_V$ are identical to that
of conformally invariant QCD. This argument is the basis for BLM
scalefixing \cite{36} and commensurate scale relations
\cite{37}, which relate physical observables together without
renormalization scale, renormalization scheme, or other ambiguities
arising from theoretical conventions.
There has recently been remarkable progress \cite{38} in
determining the running coupling $\alpha_V(Q^2)$ from heavy quark
lattice gauge theory using as input a measured level splitting in
the $\Upsilon$ spectrum. The heavy quark potential can also be
determined in a direct way from experiment by measuring $e^+ e^ \to
c \bar c$ and $e^+ e^ \to b \bar b$ at threshold \cite{39}. The
cross section at threshold is strongly modified by the QCD
Sommerfeld rescattering of the heavy quarks through their Coulombic
gluon interactions. The amplitude near threshold is modified by a
factor $S(\beta,Q^2) = x/(1\exp(x))$, where
$x=C_F\alpha_V(Q^2)/\beta$ and $\beta=\sqrt{14 m_Q^2)/s}$ is the
relative velocity between the produced quark and heavy quark. The
scale $Q$ reflects the mean exchanged momentum transfer in the
Coulomb rescattering. For example, the angular distribution for
$e^+ e^ \to Q\overline Q$ has the form $1 + A(\beta)\cos^2
\theta_{\rm cm}.$ The anisotropy predicted in QCD for small $\beta$
is then $A={\widetilde A}/(1+{\widetilde A})$, where
\begin{equation}
{\widetilde A}=
\frac{\beta^2}{2}
\frac{S(\beta, 4 m_Q^2 \beta^2/e)}{S(\beta, 4 m_Q^2 \beta^2)}
\frac{1\frac{4}{\pi} \alpha_V(m_Q^2 \exp 7/6)}{
1\frac{16}{3 \pi}\alpha_V(m_Q^2 \exp 3/4)}.
\end{equation}
The last factor is due to hard virtual radiative corrections. The
anisotropy in $e^+ e^ \to Q \overline Q$ will be reflected in the
angular distribution of the heavy mesons produced in the
corresponding exclusive channels.
The renormalization scheme corresponding to the choice of $\alpha_V$
as the coupling is the natural one for analyzing QCD in the
lightcone formalism, since it automatically sums all vacuum
polarization contributions into the coupling. For example, once one
knows the form of $\alpha_V(Q^2),$ it can be used directly in the
lightcone formalism as a means to compute the wavefunctions and
spectrum of heavy quark systems. The effects of the light quarks
and higher Fock state gluons that renormalize the coupling are
already contained in $\alpha_V.$
The same coupling can also be used for computing the hard scattering
amplitudes that control large momentum transfer exclusive reactions
and heavy hadron weak decays. Thus when evaluating $T_{\rm quark}$
the scale appropriate for each appearance of the running coupling
$\alpha_V$ is the momentum transfer of the corresponding exchanged
gluon \cite{40}. This prescription agrees with the BLM
procedure. The connection between $\alpha_V$ and the usual
$\alpha_{\overline{MS}}$ scheme is described elsewhere \cite{37}.
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