%% slac-pub-7173: page file slac-pub-7173-0-0-2-2.tcx.
%% subsection 2.2 NRQCD and Velocity Scaling [slac-pub-7173-0-0-2-2 in slac-pub-7173-0-0-2: slac-pub-7173-0-0-2-3]
%%%% latex2techexplorer block:
%% latex2techexplorer page setup:
\newmenu{slac-pub-7173::context::slac-pub-7173-0-0-2-2}{
\docLink{slac-pub-7173.tcx}[::Top]{Top}%
\sectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2}{Above: 2. Theory of Quarkonium Production}%
\subsectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2-1}{Previous: 2.1. Factorization}%
\subsectionLink{slac-pub-7173-0-0-2}{slac-pub-7173-0-0-2-3}{Next: 2.3. Fragmentation}%
}
%%%% end of latex2techexplorer block.
\subsection{\usemenu{slac-pub-7173::context::slac-pub-7173-0-0-2-2}{NRQCD and Velocity Scaling}}\label{subsection::slac-pub-7173-0-0-2-2}
The matrix elements $\langle {\cal O}^\psi_n\rangle$ contain all
interactions of the nonrelativistic $Q\bar{Q}$ pair (in its rest frame)
with degrees of freedom with low momentum compared to $m_Q$. These
interactions are accurately described by an effective field theory
called nonrelativistic QCD (NRQCD). The matrix elements are defined
in NRQCD as \cite{1}
\be
\label{matrix}
\langle {\cal O}^H_n\rangle =
\sum_X\sum_\lambda\,\langle 0|\,\chi^\dagger {\cal \kappa}_n\psi\,
|H(\lambda)X
\rangle\langle H(\lambda)X|\,\psi^\dagger {\cal \kappa}^\prime_n\chi\,
|0\rangle,
\ee
where the sum is over all polarizations $\lambda$ and light hadrons $X$,
and $\psi$, $\chi$ are two-spinor fields. [Matrix elements
with additional gluon fields
are suppressed in $v^2$ and will not be considered.] Typically,
the `kernels'
${\cal \kappa}_n$, ${\cal \kappa}_n^\prime$ specify the color, spin and
orbital angular momentum state of the quark-antiquark pair.
Without an additional
organizing principle, there would be too many matrix elements
to make the theory predictive. The NRQCD Lagrangian informs us
about the coupling of soft gluons to the $Q\bar{Q}$ pair. In particular,
spin symmetry holds to leading order in $v^2$ and reduces the number
of independent matrix elements considerably. There is no flavor symmetry
as in Heavy Quark Effective Theory, because the kinetic energy term is
part of the leading order Lagrangian. Quarkonium spectroscopy tells us
that the kinetic energy is approximately constant in the range of
reduced masses that comprises $c\bar{c}$ and $b\bar{b}$ states. Thus,
$v^2\sim 1/m_Q$ in the range of interest, while for very large quark masses
$v^2\sim 1/\ln^2 m_Q$. Furthermore, the overlap of the final state
$H X$ with a $Q\bar{Q}$ state $n$ can be estimated from a multipole
expansion, which allows us to drop some operators that acquire an
additional suppression compared to the scaling in $v^2$ of the kernels
themselves. The resulting `velocity scaling rules' are summarized in
Refs.~\cite{1,7}. The double expansion of a quarkonium
production cross section in $\alpha_s$ and $v^2$ (neglecting higher
twist corrections) is now complete.
As a result two matrix elements, $\langle {\cal O}_1^{\chi_{c0}}({}^3P_0)
\rangle$ and $\langle {\cal O}_8^{\chi_{c0}}({}^3S_1)\rangle$ are needed
to describe the production of all three P-wave states at leading order
in $v^2$. The notation refers to the kernels in Eq.~\docLink{slac-pub-7173-0-0-2.tcx}[matrix]{2}:
The subscript denotes the color state and the angular momentum state is
written in spectroscopic notation. The first matrix element reduces, to
leading order in $v^2$, to the familiar derivative of the
wavefunction at the origin. The
second color-octet term absorbs the IR senstive regions that would
otherwise appear in the short-distance cross section \cite{3,6}.
At leading order in $v^2$, $J/\psi$ production is described by the single
parameter $\langle {\cal O}_1^{J/\psi}({}^3S_1)\rangle$. Because of charge
conjugation, the gluon-gluon fusion short-distance cross section which
multiplies this matrix element is suppressed by $\alpha_s$ and is
proportional to $\alpha_s^3$. Since $v^2\sim 0.25-0.3$ is not very small for
charmonium, higher order corrections in $v^2$ can be important, if they
arise at lower order in $\alpha_s$. According to the velocity scaling
rules three color-octet matrix elements --
$\langle {\cal O}_8^{J/\psi}({}^3S_1)\rangle$,
$\langle {\cal O}_8^{J/\psi}({}^1S_0)\rangle$,
$\langle {\cal O}_8^{J/\psi}({}^3P_0)\rangle$ -- contribute at order
$\alpha_s^2 v^4$ (powers of $v^2$ are counted relative to the leading
order contribution), so that $J/\psi$ production is described by four
nonperturbative parameters. Contributions of order $\alpha_s^3 v^2$
also exist and can be important in specific regions of phase space.
Because $\chi_{c1}$ production at low tranverse momentum is also
suppressed by $\alpha_s$ compared to $\chi_{c0}$ and $\chi_{c2}$,
higher order corrections in $v^2$ would also be important for
$\chi_{c1}$ production.