%Paper: 
%From: GTB@hep.anl.gov (Geoffrey Bodwin (708) 252-6229)
%Date: Mon, 16 Nov 1992 16:36:11 -0600 (CST)
%Date (revised): Mon, 16 Nov 1992 18:11:33 -0600 (CST)
%Date (revised): Tue, 17 Nov 1992 14:54:44 -0600 (CST)
%Date (revised): Wed, 25 Nov 1992 17:55:03 -0600 (CST)

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%%%%Miscellaneous definitions%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\m{m}
\def\asMQ{\alpha_s(M_Q)}
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\begin{document}
\nskip{-2.5truecm} \rightline{
\vbox{\elevenrm
\halign{&#\hfil\cr
&ANL-HEP-CP-92-109\cr
&November 1992\cr}
}}\nskip {1.95truecm}%

\begin{center}{{\tenbf RIGOROUS QCD PREDICTIONS\\
               \vglue 3pt
               FOR DECAYS OF P-WAVE QUARKONIA\footnotemark\\}
\vglue 1.0cm
{\tenrm GEOFFREY T. BODWIN \\}
\baselineskip=13pt
{\tenit High Energy Physics Division, Argonne National Laboratory,
%\\}
%\baselineskip=12pt
%{\tenit
Argonne, IL 60439, USA\\}
%\vglue 0.3cm
%{\tenrm and\\}
\vglue 0.3cm
{\tenrm ERIC BRAATEN\\}
{\tenit Department of Physics and Astronomy, Northwestern University,
%\\}
%\baselineskip=12pt
%{\tenit
Evanston, IL 60208, USA\\}
%\vglue 0.3cm
%{\tenrm and\\}
\vglue 0.3cm
{\tenrm G. PETER LEPAGE\\}
{\tenit Newman Laboratory of Nuclear Studies, Cornell University,
%\\}
%\baselineskip=12pt
%{\tenit
Ithaca, New York 14853, USA\\}
\vglue 0.8cm
{\tenrm ABSTRACT}}
\end{center}
\vglue 0.3cm
{\rightskip=3pc
 \leftskip=3pc
 \tenrm\baselineskip=12pt
 \noindent
\footnotetext{\ninerm
Talk presented by G.
Bodwin at DPF92, Fermilab, Batavia, Illinois, November 10--14, 1992}%
We present a new factorization theorem for the decay rates of P-wave
states of heavy quarkonia.  Infrared logarithms that had appeared in
previous perturbative calculations of P-wave decays are absorbed into
a quantity that is related to the amplitude for the heavy quark and
antiquark to be in a relative color-octet S-wave state. We predict
all of the light-hadronic and electromagnetic decays rates of the
$\chi_c$ and $h_c$ states in terms of two phenomenological parameters.
\vglue 0.6cm}
%{\elevenbf\noindent 1. $Q\overline Q$ Annihilation: A Short Distance
%Process?}
%\vglue 0.2cm
%{\elevenit\noindent 1.1. Typeset Scripts}
%\vglue 0.1cm
\baselineskip=14pt
%\vglue 0.4cm
\elevenrm
The annihilation of the heavy $Q\overline Q$ pair in quarkonium is a
short-distance process, occurring when the $Q$ and $\overline Q$ are
within $O(1/M_Q)$ of each other. If all of the interactions associated
with the annihilation were to occur at distance scales set by $M_Q$,
then one could, because of asymptotic freedom, calculate the decay rate
in a perturbation series in $\alpha_s(M_Q)$. Unfortunately, quark-gluon
interactions invalidate this simple scenario, since the associated
infrared (IR) and collinear singularities imply the presence of
long-range interactions.  Nevertheless, for the annihilation of S-wave
states a relatively simple picture emerges. If one neglects the
relative velocity $\vec v$ of the $Q$ and $\overline Q$,
keeping the leading term in an expansion in powers of $v/c$,
then the annihilation cross section for a meson $m$ with total spin $S$
factors into a short-distance piece times a long-distance piece:
$$
	{
\Gamma \left(\m(^{2S+1}\S) \rightarrow X \right) \; = \;
{G_1(\m)}\; {\Gamhat_1
\left( Q \Qbar (^{2S+1}\S) \rightarrow X \right)}.
}
\auto\label{factS}
$$
The quantity $\hat\Gamma_1$ is the short distance piece. It is the
(on-shell) parton-level annihilation cross section, and asymptotic
freedom allows its computation in perturbative QCD. The perturbation
series for the annihilation of an on-shell $Q\overline Q$ in an S~wave
is well behaved:  final-state IR and collinear divergences cancel
according to the KLN theorem, and initial-state IR divergences cancel
because the meson is a color singlet. $G_1$ is the long-distance piece,
which contains all of the nonperturbative effects. It is analogous to a
parton distribution. $G_1$ is proportional to the probability to find
the $Q$ and $\overline Q$ at the same point:
$$
G_1(\m)\approx (3/2\pi)|R_{\m S}(0)|^2/ M_Q^2,
\auto
$$
where $R_{\m S}(0)$ is the nonrelativistic radial wave function at the origin.
%\vglue 0.6cm
%{\elevenbf\noindent 2. P-Wave Decays}
%\vglue 0.4cm

One might guess that a factorization formula similar to
Eq.~(\ref{factS}) would hold for P-wave decays as well.  However, in
this case the situation is more complicated. Owing to the angular
dependence of the wave function, the leading term in the $v/c$ expansion
vanishes upon integration over the angular orientation of $\vec v$.
Consequently, the annihilation cross section for P~waves is suppressed
by $v^2/c^2$ relative to S-wave annihilation. The first subleading term
in the $v/c$ expansion yields, in perturbation theory, a contribution
that is proportional to the {\elevenit derivative} of the wave function
at the origin.

Now, the first subleading term in the $v/c$ expansion measures the color
currents of the $Q$ and $\overline Q$, rather than their color charges.
Thus, for the subleading term, infrared divergences need not cancel, even
though the quarkonium has no net color charge. In fact, the perturbation
series for $Q\overline Q$ annihilation on shell in a P~wave contains a
logarithmic IR divergence. This is a clear signal that long distance
effects are present and that the use of perturbative QCD is not valid.
Past (nonrigorous) treatments of P-wave decays have invoked the
confinement scale or the binding energy as an IR cutoff---but with no
fundamental justification.
%\vglue 0.6cm
%{\elevenbf\noindent 2. Interpretation of the IR Divergences}
%\vglue 0.4cm

The structure of these IR divergences has a simple physical
interpretation.  The divergences arise when the P-wave color-singlet
state converts to an S-wave color-octet state through the emission of
a soft gluon.  Then the $Q$ and $\overline Q$ annihilate from the
S-wave state. The soft-gluon transition costs a factor $v^2/c^2$.
However, S-wave annihilation is enhanced by a factor $c^2/v^2$ relative
to P-wave annihilation, so the color-octet process is competitive with
direct color-singlet P-wave annihilation.
%\vglue 0.6cm
%{\elevenbf\noindent 2. New factorization theorem}
%\vglue 0.4cm

By taking into account the fact that the IR divergences are associated
with the color-octet mechanism, one can write a new factorization
theorem for P-wave decays:\cite{bbl}
$${
\Gamma \left(\m(^{2S+1}\P) \rightarrow X \right) \; = \;
{H_1(\m)}\; {\Gamhat_1
\left( Q \Qbar (^{2S+1}\P) \rightarrow X \right)}\;
+ \;  {H_8(\m)}\; {\Gamhat_8
\left( Q \Qbar (^{2S+1}\S) \rightarrow X \right)}.
}
\auto\label{factP}
$$
The first term in Eq.~(\ref{factP}) corresponds to the naive
(color-singlet) factorization picture. The second term is new and gives
the contribution of the color-octet mechanism. The $\hat\Gamma$'s are
the parton-level cross sections for on-shell $Q\overline Q$
annihilation, except that the IR divergent part of $\hat\Gamma_1$ is
extracted and put into $H_8$. The precise way in which this is done is
the factorization prescription. $H_1$ is proportional to the derivative
of the P-wave color-singlet $Q\bar Q$~wave function at the origin:
$$
{
H_1(\m) \; \approx \; (9/2\pi) |R^\prime_{\m P}(0)|^2 /M_Q^4.
}
\auto\label{hsinglet}
$$
$H_8$ is related to the amplitude to find the $Q$ and $\overline Q$ in a
relative color-octet S-wave state.  Since $H_8$ contains information
about the $Q\overline Q g$ Fock state, it is not simply expressible in
terms of the nonrelativistic $Q\overline Q$ wave function. In
perturbation theory
$$
{
 H_8(\m) \; \approx \; {16 \over 27 \beta_0}
\ln  \left( {\alpha_s(\epsilon_\m) \over
\alpha_s(M_Q)} \right) \, H_1(\m) \;
\sim \; {16\over 27\pi}\alpha_s\ln \left({M_Q\over \epsilon_\m}\right)\,
H_1(\m),
}
\auto\label{hoctet}
$$
where $\epsilon_\m$ is the binding energy.
The logarithm signals that Eq.~(\ref{hoctet}) is not trustworthy as an
estimate of $H_8$.  However, by comparing it with the IR divergent
parts of previous calculations of P-wave decay\cite{barbieri-et-al},
one can extract the $\hat\Gamma$'s. Both $H_1$ and $H_8$ have precise
definitions in terms of operator matrix elements, so they can, in
principle, be measured in lattice simulations, or they can simply be
treated as phenomenological parameters.
%\vglue 0.6cm
%{\elevenbf\noindent 2. Phenomenology of P-wave Decays in Charmonium}
%\vglue 0.4cm

In general, $H_1$ and $H_8$ depend on the total angular momentum $J$ and
the total spin $S$ of the P-wave state.  However, if one describes the
heavy $Q\overline Q$ system in terms of a low-energy effective
Lagrangian, then the spin-dependent terms are suppressed by powers of
$1/M_Q$, that is, by powers of $v/c$.  Thus, to leading order in $v/c$,
we can take $H_1$ and $H_8$ to be independent of $S$ and $J$.
Corrections to this give terms of order $v^2/c^2$ in the decay rates,
where $v^2/c^2\approx 20\%$ for charmonium.


The factorization formula Eq.~(\ref{factP}) combined with the
leading-order expressions for the $\hat\Gamma$'s gives
$$
H_1={45\over 16\pi}{\Gamma(\chi_{c2}\rightarrow
\hbox{LH})-\Gamma(\chi_{c1}\rightarrow \hbox{LH})\over
\alpha_s^2(M_c)}, \qquad
H_8={1\over \pi}{\Gamma(\chi_{c1}\rightarrow
\hbox{LH})\over
\alpha_s^2(M_c)},
\auto
$$
where ``LH'' denotes light hadrons. Using the Particle Data
Group\cite{pdg} values for the branching ratios and the recent E760
data\cite{e760} for the $\chi_{c1}$ and $\chi_{c2}$ total widths, we
find that $H_1=15.3\pm 6.6\; \hbox{MeV}\;$ and $H_8=3.2\pm 1.4\;
\hbox{MeV}\;$. The quoted error is the experimental error, which
includes the uncertainty in $\alpha_s$, combined in quadrature with our
estimate of the theoretical uncertainty, which includes $v^2/c^2$
corrections and higher-order perturbative QCD corrections. Given $H_1$
and $H_8$ and the leading-order expressions for the $\hat\Gamma$'s, we
can use Eq.~(\ref{factP}) to compute the partial widths for the decays
of the $\chi_{c0}$ and $h_c$ into light hadrons,
the decay of $h_c$ into $\gamma$ plus light hadrons, and the decays of
$\chi_{c0}$ and $\chi_{c2}$ into two $\gamma$'s. There is also a simple
relationship between the radiative decay rates of the
$P$-states\cite{mcclary-beyers}, which is correct to leading order in
$v/c$:
$$
\Gamma( ^1\P_1 \rightarrow \gamma \; ^1\S_0 )/E_\gamma^3(11)
\; \simeq \;
\Gamma( ^3\P_J \rightarrow \gamma \; ^3\S_1 )/E_\gamma^3(3J),
\;\; J = 0, 1, 2,
\auto\label{eqptos}
$$
where $E_\gamma (11)$ and $E_\gamma (3J)$ are the energies of the
$\gamma$'s in the singlet and triplet decays, respectively.
Eq.~(\ref{eqptos}) is well satisfied for the $\chi_{c1}$ and
$\chi_{c2}$. We use it to obtain predictions for the radiative decay
widths of the $\chi_{c0}$ and $h_c$. Our predictions for the $\chi_{c0}$
are
$$
\eqalign{
\omit $\Gamma(\chi_{c0})=(5\pm 2)\;\hbox{MeV},\qquad$ \hfill
&B(\chi_{c0}\rightarrow
\hbox{LH})=(98\pm 1)\%, \cr
\omit $B(\chi_{c0}\rightarrow \gamma+J/\psi)=(2\pm 1)\%,
\qquad$ \hfill&B(\chi_{c0}\rightarrow
\gamma\gamma)=(7\pm 4)\times 10^{-4}. \cr
}
\auto
$$
The predictions for the $\chi_{c0}$ total width and branching fraction
into $\gamma +J/\psi$ differ significantly from the accepted values of
$14\pm 5$~MeV and $(0.66\pm 0.18)\%$, respectively. More precise data on
the $\chi_{c0}$ would provide useful tests of the QCD predictions. Our
prediction for the branching fraction of the $\chi_{c2}$ into two
$\gamma$'s is
$$
B(\chi_{c2}\rightarrow \gamma\gamma)=(4\pm 2)\times 10^{-4}.
\auto
$$
This is somewhat smaller than the old Particle Data Group\cite{pdg}
value of $(11\pm 6)\times 10^{-4}$.  However, a new E760
measurement\cite{e760-2}
%of $0.34\pm0.11$~keV, when taken together with
%total width of $1.98\pm 0.18$~MeV,
yields a branching fraction into two $\gamma$'s of $(1.7\pm 0.6)\times
10^{-4}$. Our predictions for the $h_c$ are
$$
\eqalign{
\omit $\Gamma(h_c)=(1.0\pm 0.2)\;\hbox{MeV},\qquad$ \hfill
&B(h_c\rightarrow \hbox{LH})=(52\pm 11)\%, \cr
\omit $B(h_c\rightarrow \eta_c+\gamma)=(46\pm 11)\%, \qquad$ \hfill
&B(h_c\rightarrow \gamma+\hbox{LH})=(2\pm 1)\%. \cr
}
\auto
$$
The prediction for the $h_c$ total width is consistent with the upper
bound of 1.1~MeV obtained recently by the E760 collaboration.\cite{e760-3}
We predict a significant rate for $h_c$ into $\gamma$ plus
light hadrons.  The hard $\gamma$ recoiling against a jet
plus soft hadrons could be a distinctive signature for this decay.  A
large component of the error in all of these predictions arises from the
theoretical uncertainty.  It could be reduced considerably by making a
complete next-to-leading-order calculation of the $\hat\Gamma$'s.
%\vglue 0.6cm
%{\elevenbf\noindent 2. Application to Quarkonium Production}
%\vglue 0.4cm

One can treat quarkonium production processes using techniques that are
very similar to those that we have described for quarkonium decay
processes.  Possible applications include photoproduction,
leptoproduction, and hadron-hadron production.  The application to
production of $\chi_c$ states in $B$-meson decay can be found in
Ref.~\ref{b-decay}. The nonperturbative quantities $G_1$ and $H_1$,
which appeared in quarkonium decay, appear in the color-singlet
production processes as well. However, color-octet production involves a
new nonperturbative quantity $H'_8$.  Whereas $H_8$ is analogous to a
parton distribution function, $H'_8$ is analogous to a fragmentation
function.  The two quantities are related by crossing, but that
relationship is a simple one only in lowest-order perturbation theory.
Consequently, one must determine $H'_8$ from experiment or extract it
from a lattice calculation.

This work was supported in part by the U.S. Department of Energy,
Division of High Energy Physics, under Contract W-31-109-ENG-38 and
under Grant DE-FG02-91-ER40684, and by the National Science Foundation.
\vglue 0.5cm
{\elevenbf\noindent References \hfil}
\vglue 0.4cm
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\end{document}


