%% slac-pub-7129: page file slac-pub-7129-0-0-4-2.tcx.
%% subsection 4.2 Polarization in fixed target experiments [slac-pub-7129-0-0-4-2 in slac-pub-7129-0-0-4: 0.25$ and
$0.028\pm 0.004$ for $J/\psi$ measured at $\sqrt{s}=15.3\,$GeV
in the region $x_F>0$. In the CSM, the $J/\psi$'s are predicted to be
significantly transversely polarized \cite{2}, in conflict with
experiment.
The polarization yield of color octet processes can be calculated
along the lines of the previous subsection. We first concentrate on
$\psi'$ production and define $\xi$ as the fraction of longitudinally
polarized $\psi'$. It is related to $\alpha$ by
\begin{equation}
\alpha=\frac{1-3\xi}{1+\xi}\,.
\end{equation}
\noindent For the different intermediate quark-antiquark
states we find the following ratios of longitudinal to transverse
quarkonia:
\begin{equation}
\addtolength{\arraycolsep}{0.3cm}
\begin{array}{ccc}
{}^3 S_1^{(1)} & 1:3.35 & \xi=0.23\\
{}^1 S_0^{(8)} & 1:2 & \xi=1/3\\
{}^3 P_J^{(8)} & 1:6 & \xi=1/7\\
{}^3 S_1^{(8)} & 0:1 & \xi=0
\end{array}
\end{equation}
\noindent where the number for the singlet process (first line) has been
taken from \cite{2}\footnote{This number is $x_F$-dependent and we
have approximated it
by a constant at low $x_F$, where the bulk data is obtained from.
The polarization fractions for the octet $2\to 2$ parton processes
are $x_F$-independent.}. Let us add the following remarks:
(i) The ${}^3 S_1^{(8)}$-subprocess yields pure transverse
polarization. Its contribution to the total polarization is not
large, because gluon-gluon fusion dominates the total rate.
(ii) For the ${}^3 P_J^{(8)}$-subprocess $J$ is not specified,
because interference between intermediate states with different
$J$ could occur as discussed in the previous subsection. As it
turns out, interference does in fact not occur at leading order
in $\alpha_s$, because the only non-vanishing short-distance
amplitudes in the $J J_z$ basis are $00$, $22$ and
$2(-2)$, which do not
interfere.
(iii) The ${}^1 S_0^{(8)}$-subprocess yields unpolarized quarkonia.
This follows from the fact that the NRQCD matrix element is
\begin{equation}
\label{above}
\langle 0|\chi^\dagger T^A{a_{\psi'}^{(\lambda)}}^\dagger
a_{\psi'}^{(\lambda)}\,\psi^\dagger T^A\chi|0\rangle =\frac{1}{3}\,
\langle {\cal O}_8^{\psi'}({}^1 S_0)\rangle\,,
\end{equation}
\noindent independent of the helicity state $\lambda$. At this point,
we differ from \cite{13}, who assume that this channel results
in pure transverse polarization, because the gluon in the
chromomagnetic dipole transition ${}^1 S_0^{(8)}\to {}^3 S_1^{(8)}+g$
is assumed to be transverse. However, one should keep in mind that
the soft gluon is off-shell and interacts with other partons with unit
probability prior to hadronization. The NRQCD formalism
applies only to inclusive quarkonium production. Eq.~(\docLink{slac-pub-7129-0-0-4.tcx}[above]{30})
then follows from rotational invariance.
(iv) Since the ${}^3 P_J^{(8)}$ and ${}^1 S_0^{(8)}$-subprocesses
give different longitudinal polarization fractions, the $\psi'$
polarization depends on a combination of the matrix elements
$\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ which is different
from $\Delta_8(\psi')$.
To obtain the total polarization the various subprocesses have to be
weighted by their partial cross sections. We define
\begin{equation}
\delta_8(H)=\frac{\langle {\cal O}_8^{H} ({}^1 S_0)\rangle}
{\Delta_8(H)}
\end{equation}
\noindent and obtain
\begin{eqnarray}
\xi &=& 0.23\,\frac{\sigma_{\psi'}({}^3 S_1^{(1)})}{\sigma_{\psi'}} +
\left[\frac{1}{3}\delta_8(\psi')+\frac{1}{7} (1-\delta_8(\psi'))\right]
\frac{\sigma_{\psi'}({}^1 S_0^{(8)}+{}^3 P_J^{(8)})}{\sigma_{\psi'}}
\nonumber\\
&=& 0.16+0.11\,\delta_8(\psi')\,,
\end{eqnarray}
\noindent where the last line holds at $\sqrt{s}=21.8\,$GeV (The
energy dependence is mild and the above formula can be used with
little error even at $\sqrt{s}=40\,$GeV). Since $0<\delta_8(H)<1$, we
have $0.16<\xi<0.27$ and therefore
\begin{equation}
0.15 < \alpha < 0.44\,.
\end{equation}
\noindent In quoting this range we do not attempt an estimate of
$\delta_8(\psi')$. Note that taking the Tevatron and fixed target
extractions of certain (and different) combinations of
$\langle {\cal O}_8^{\psi'} ({}^1 S_0)\rangle$ and
$\langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle$ seriously
(see Sect.~5.1), a large value of $\delta_8(\psi')$
and therefore low $\alpha$ would be favored. Within large errors,
such a scenario could be considered consistent with the
measurement quoted earlier. From a theoretical point of view, however,
the numerical violation of velocity counting rules implied by
this scenario would be rather disturbing.
In contrast, the more accurate measurement of polarization for
$J/\psi$ leads to a clear discrepancy with theory. In this case, we
have to incorporate the polarization inherited from decays of
the higher charmonium states $\chi_{cJ}$ and $\psi'$. This task
is simplified by observing that the
contribution from $\chi_{c0}$ and $\chi_{c1}$
feed-down is (theoretically) small as is the octet contribution
to the $\chi_{c2}$ production cross section. On the other hand, the
gluon-gluon fusion process produces $\chi_{c2}$ states only in
a helicity $\pm 2$ level, so that the $J/\psi$ in the subsequent
radiative decay is completely transversely polarized.
Weighting all subprocesses by their partial cross section
and neglecting the small $\psi'$ feed-down, we arrive at
\begin{equation}
\xi = 0.10 + 0.11\,\delta_8(J/\psi)
\end{equation}
\noindent at $\sqrt{s}=15.3\,$GeV, again with mild energy dependence.
This translates into sizeable transverse polarization
\begin{equation}
0.31 < \alpha < 0.63\,.
\end{equation}
\noindent The discrepancy with data could be ameliorated if the observed
number of $\chi_{c1}$ from
feed-down were used instead of the theoretical value. However,
we do not know the polarization yield of whatever mechanism is
responsible for copious $\chi_{c1}$ production.
Thus, color octet mechanisms do not help to solve the
polarization problem and
one has to invoke a significant higher-twist contribution as
discussed in \cite{2}. To our knowledge, no specific mechanism
has yet been proposed that would yield predominantly longitudinally
polarized $\psi'$ and $J/\psi$ in the low $x_F$ region which dominates
the total production cross section. One might speculate that both
the low $\chi_{c1}/\chi_{c2}$ ratio and the large transverse polarization
follow from the assumption of transverse gluons in the gluon-gluon fusion
process, as inherent to the leading-twist approximation. If gluons
in the proton and pion have
large intrinsic transverse momentum, as suggested by the
$p_t$-spectrum in open charm
production, one would be naturally led to higher-twist effects that
obviate the helicity constraint on on-shell gluons.