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%% subsection 2.1 Cross sections [slac-pub-7129-0-0-2-1 in slac-pub-7129-0-0-2: ^slac-pub-7129-0-0-2 >slac-pub-7129-0-0-2-2]
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\sectionLink{slac-pub-7129-0-0-2}{slac-pub-7129-0-0-2}{Above: 2. Quarkonium production cross sections at fixed
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\subsection{\usemenu{slac-pub-7129::context::slac-pub-7129-0-0-2-1}{Cross sections}}\label{subsection::slac-pub-7129-0-0-2-1}
We begin with the production cross section for $\psi'$ which does
not receive contributions from radiative decays of higher charmonium
states. The $2\to 2$ parton diagrams produce a quark-antiquark pair
in a color-octet state or $P$-wave singlet state (not relevant
to $\psi'$) and therefore contribute to $\psi'$
production at order $\alpha_s^2 v^7$. (For charmonium $v^2\approx
0.25 - 0.3$, for bottomonium $v^2\approx 0.08 - 0.1$.) The $2\to3$
parton processes contribute to the color singlet processes at order
$\alpha_s^3 v^3$. Using the notation in (\docLink{slac-pub-7129-0-0-1.tcx}[fact]{2}):
\begin{eqnarray}
\label{psiprimecross}
\hat{\sigma}(gg\to\psi') &=& \frac{5\pi^3\alpha_s^2}{12 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\left[\langle {\cal O}_8^{\psi'} ({}^1 S_0)
\rangle+\frac{3}{m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_0)\rangle
+\frac{4}{5 m_c^2} \langle {\cal O}_8^{\psi'} ({}^3 P_2)\rangle
\right]\nonumber\\[0.0cm]
&&\hspace*{-1.5cm}
+\,\frac{20\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\,\langle
{\cal O}_1^{\psi'} ({}^3 S_1)\rangle\,z^2\left[\frac{1-z^2+2 z
\ln z}{(1-z)^2}+\frac{1-z^2-2z \ln z}{(1+z)^3}\right]\\[0.2cm]
\hat{\sigma}(gq\to\psi') &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \psi') &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\psi'} ({}^3 S_1)
\rangle
\end{eqnarray}
\noindent Here $z\equiv (2 m_c)^2/(s x_1 x_2)$, $\sqrt{s}$ is the
center-of-mass energy and $\alpha_s$ is normalized at the scale $2 m_c$.
Corrections to these cross sections are suppressed
by either $\alpha_s/\pi$ or $v^2$. Note that the
relativistic corrections
to the color singlet cross section are substantial in specific
kinematic regions $z\to 0,1$ \cite{17}. For $\sqrt{s}>15\,$GeV
these corrections affect the total cross section by less than $50\%$
and decrease
as the energy is raised \cite{1}. Furthermore, notice that we
have expressed the short-distance coefficients in terms of the
charm quark mass, $M_{\psi'}\approx 2 m_c$, rather than the true
$\psi'$
mass.
Although the difference is formally of higher order in $v^2$, this
choice is conceptually favored since the short-distance coefficients
depend only on the physics prior to quarkonium formation. All
quarkonium specific properties which can affect the cross section,
such as quarkonium mass differences,
are hidden in the matrix elements.
The production of $P$-wave quarkonia differs from $S$-waves since
color singlet and color octet processes enter at the same order in $v^2$
as well as $\alpha_s$ in general. An exception is $\chi_{c1}$, which
can not be produced in $2\to 2$ parton reactions through
gluon-gluon fusion in a color singlet state. Since at order $\alpha_s^2$,
the $\chi_{c1}$ would be produced only in a $q\bar{q}$ collision, we
also include the gluon fusion diagrams at order $\alpha_s^3$, which
are enhanced by the gluon distribution. We have for $\chi_{c0}$,
\begin{eqnarray}
\label{chi0cross}
\hat{\sigma}(gg\to\chi_{c0}) &=& \frac{2\pi^3\alpha_s^2}{3 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c0}} ({}^3 P_0)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c0}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c0}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c0}} ({}^3 S_1)
\rangle\,,
\end{eqnarray}
\noindent for $\chi_{c1}$,
\begin{eqnarray}
\label{chi1cross}
\hat{\sigma}(gg\to\chi_{c1}) &=& \frac{2\pi^2\alpha_s^3}{9 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\nonumber\\
&&\hspace*{-1.5cm}
\times\Bigg[\frac{4 z^2\ln z \, (z^8+9 z^7+26 z^6+28 z^5+17 z^4+7 z^3-
40 z^2-4 z-4}{(1+z)^5 (1-z)^4}\nonumber\\
&&\hspace*{-1.5cm}
\,+\frac{z^9+39 z^8+145 z^7+251 z^6+119 z^5-153 z^4-17 z^3-147 z^2-8 z
+10}{3 (1-z)^3 (1+z)^4}\Bigg]
\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c1}) &=& \frac{8\pi^2\alpha_s^3}{81 (2 m_c)^5}\,
\Theta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c1}} ({}^3 P_1)
\rangle\left[-z^2\ln z + \frac{4 z^3-9 z+5}{3}\right]\nonumber\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c1}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c1}} ({}^3 S_1)
\rangle\,,
\end{eqnarray}
\noindent and for $\chi_{c2}$
\begin{eqnarray}
\label{chi2cross}
\hat{\sigma}(gg\to\chi_{c2}) &=& \frac{8\pi^3\alpha_s^2}{45 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\frac{1}{m_c^2}
\langle {\cal O}_1^{\chi_{c2}} ({}^3 P_2)
\rangle\\[0.2cm]
\hat{\sigma}(gq\to\chi_{c2}) &=& 0\\[0.2cm]
\hat{\sigma}(q\bar{q}\to \chi_{c2}) &=& \frac{16\pi^3\alpha_s^2}
{27 (2 m_c)^3 s}\,
\delta(x_1 x_2-4 m_c^2/s)\,\langle {\cal O}_8^{\chi_{c2}} ({}^3 S_1)
\rangle\,.
\end{eqnarray}
\noindent Note that in the NRQCD formalism
the infrared sensitive contributions to
the $q\bar{q}$-induced color-singlet process at order $\alpha_s^3$
are factorized into the color octet matrix elements
$\langle {\cal O}_8^{\chi_{cJ}} ({}^3 S_1)\rangle$, so that
the $q\bar{q}$ reactions at order $\alpha_s^3$ are truly suppressed
by $\alpha_s$. The production of $P$-wave states through
octet quark-antiquark
pairs in a state other than ${}^3 S_1$ is higher order in $v^2$.
Taking into account indirect production of $J/\psi$ from decays of
$\psi'$ and $\chi_{cJ}$ states, the $J/\psi$ cross section is given by
\begin{equation}
\label{jpsicross}
\sigma_{J/\psi} = \sigma(J/\psi)_{dir} + \sum_{J=0,1,2}
\mbox{Br}(\chi_{cJ}\to J/\psi X)\,\sigma_{\chi_{cJ}} +
\mbox{Br}(\psi'\to J/\psi X)\,\sigma_{\psi'}\,,
\end{equation}
\noindent where `Br' denotes the corresponding branching fraction and
the direct $J/\psi$ production cross section
$\sigma(J/\psi)_{dir}$ differs from $\sigma_{\psi'}$ (see
(\docLink{slac-pub-7129-0-0-2.tcx}[psiprimecross]{4})) only by the replacement of $\psi'$ matrix elements
with $J/\psi$ matrix elements. Finally, we note that charmonium production
through $B$ decays is comparatively negligible at fixed target energies.
The $2\to 2$ parton processes contribute only to quarkonium
production at zero transverse momentum with respect to the beam axis.
The transverse momentum distribution of $H$ in reaction (\docLink{slac-pub-7129-0-0-1.tcx}[proc]{1})
is not calculable
in the $p_t<\Lambda_{QCD}$ region, but the total cross section (which averages
over
all $p_t$) is predicted even if the underlying parton process
is strongly peaked at zero $p_t$.
The transcription of the above formulae to bottomonium production is
straightforward. Since more bottomonium states exist below the
open bottom threshold than for the charmonium system,
a larger chain of cascade decays in the
bottomonium system must be included. In particular, there is indirect
evidence from $\Upsilon(3 S)$ production both at the Tevatron
\cite{18} as well as in fixed target experiments (to be discussed
below) that there exist yet unobserved $\chi_b(3P)$ states below
threshold whose decay into lower bottomonium states should also be
included. Our numerical results do not include indirect contributions
from potential $D$-wave states below threshold.
All color singlet cross sections compiled in this section have been
taken from the review \cite{1}. We have checked that the color
octet short-distance coefficients agree with those given in
\cite{9}, but disagree with those that enter the numerical
analysis of fixed target data in \cite{14}.