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\begin{document}

\markboth{Johann H. K\"uhn}
{Implications of an R-Scan for Charm Physics}

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\title{Implications of an R-Scan for Charm Physics
\footnote{Presented at 
PHIPSI13 - International Workshop on e+e- collisions  from Phi to Psi 2013,
Rome, Italy, 9-12 September 2013.
}}

\author{Johann H. K\"uhn}

\address{Institut f\"ur Theoretische Teilchenphysik, Karlsruher Institut f\"ur
Technologie (KIT)\\
76128 Karlsruhe, Germany\\
Johann.Kuehn@KIT.edu}

\maketitle

%\begin{history}
%\received{Day Month Year}
%\revised{Day Month Year}
%\end{history}

\begin{abstract}
The impact of improved measurements of the cross section for 
electron-positron annihilation into hadrons in the charm threshold
region is discussed. Two aspects are studied in detail:
i.)~A significant reduction of the experimental error of the electronic
width of the narrow resonances $J/\psi$ and $\psi'$ and of the
continuum cross section from the open charm threshold up to 4.6~GeV
will lead to a correspondingly improved determination of the charmed quark
mass. ii)~A high luminosity measurement with 24~pb${}^{-1}$ at three
points and with a spacing of 2~MeV around $\sqrt{s}=3511\,{\rm MeV}$ may
allow to observe the direct, resonant production of $\chi_{c1}$.


\keywords{Perturbative quantum chromodynamics, charmed quarks, charmonium}
\end{abstract}

\ccode{PACS numbers:12.38.Bx, 14.65.Dw, 14.65.Fy}

\section{Introduction}
Significant improvements in luminosity are anticipated for electron positron collisions 
in the low energy region, with the BES III experiment at the BEPC II collider
in Beijing as characterisic example. BEPC II will cover the energy region between 
2 and 4.6~GeV and is therefore specifically suited for the exploration of the 
charm system. Indeed, recently a lot of emphasis has been put on the study of the 
newly discovered charmonium resonances denoted X,Y and Z.
(For discussions of recent theoretical and experimental developments see e.g.\
Refs.~\refcite{LiangYan,ThomasMehen}). In this brief contribution two different aspects of 
charm physics will be investigated which could profit from a high 
luminosity scan in the charm threshold region:

i) When combined with increasingly precise measurements of the electronic widths of 
$J/\psi$ and $\psi'$ an R-scan with larger luminosity and reduced systematical
error could lead to an improved determination of the charm quark mass.

ii.) Investing 24 pb${}^{-1}$ of integrated luminosity into the R-measurement
at three points each, with a spacing of 2 MeV around 3.511~GeV, the direct 
resonant production of $\chi_{c1}$
(and perhaps even $\chi_{c2}$), which can proceed through two virtual photons, 
seems within reach.

\section{Quark mass determination in $\rm N^3LO$}

As discussed in the literature% 
\cite{Kuhn:2001dm,Kuhn:2007vp,Chetyrkin:2009fv,Chetyrkin:2010ic}, moments of the 
charm production cross section
$\sigma(e^+e^-\to c\bar c+X) \equiv R_c \sigma_0$
with
$\sigma_0 \equiv \frac{4\pi}{3}\frac{\alpha^2}{s}$,
allow for a fairly precise 
determination of the charm mass. The $n$-th moment of $R_c$, as defined through
%
\begin{equation}
  {\cal M}^{\rm exp}_n \equiv \int \frac{{\rm d}s}{s^{n+1}} R_Q(s)
  \,,
  \label{eq:Mexp}
\end{equation}
%
can be experimentally determined from analysis of $e^+e^-$ data. The quark mass 
(defined in the $\overline{MS}$ scheme at renormalization scale $\mu$) is then obtained from
%
\begin{equation}
  m_Q(\mu) = \frac{1}{2}
  \left( \frac{9 Q_Q^2\bar{C}_n}{4 {\cal M}_n^{\rm exp}}\right)^{1/(2n)} 
  \,.
  \label{eq:m_Q}
\end{equation}
%
  \begin{table}[t]
\tbl{Experimental moments
  in $(\mbox{GeV})^{-2n}$ as defined in
  Eq.~(\ref{eq:Mexp}), separated according to the contributions from
  the narrow resonances,
  the charm threshold region and the continuum region
  above $\sqrt{s}=4.8$~GeV. In the last column the NLO contribution from the
  gluon condensate is shown.} 
%    \centering
{
\begin{tabular}{l|lll|l||l}
%%%%%
\hline
$n$ & ${\cal M}_n^{\rm res}$
& ${\cal M}_n^{\rm thresh}$
& ${\cal M}_n^{\rm cont}$
& ${\cal M}_n^{\rm exp}$
& ${\cal M}_n^{\rm np}$(NLO)
\\
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
& $\times 10^{(n-1)}$
\\
\hline
$1$&$  0.1201(25)$ &$  0.0318(15)$ &$  0.0646(11)$ &$  0.2166(31)$
 &$-0.0002(5)$ \\
$2$&$  0.1176(25)$ &$  0.0178(8)$ &$  0.0144(3)$ &$  0.1497(27)$
 &$-0.0005(10)$ \\
$3$&$  0.1169(26)$ &$  0.0101(5)$ &$  0.0042(1)$ &$  0.1312(27)$
 &$-0.0008(16)$ \\
$4$&$  0.1177(27)$ &$  0.0058(3)$ &$  0.0014(0)$ &$  0.1249(27)$
 &$-0.0013(25)$ \\
\hline
\end{tabular}
    \label{tab:contglu}
}
  \end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \begin{table}[t]
\tbl{Results for $m_c(3\, {\rm GeV} )$ in GeV. The errors
      are from experiment, $\alpha_s$, variation of $\mu$ and the
     gluon condensate.}
%    \centering
%\scalefont{1}{
{
\begin{tabular}{l|l|llll||l}
%%%%%
\hline
$n$ & $m_c(3~\mbox{GeV}) $ & exp & $\alpha_s$ & $\mu$ 
& ${\rm np}_{\rm NLO}$ & total\\
 \hline
1& 0.986& 0.009&  0.009&  0.002& 0.001& 0.013\\
2& 0.975& 0.006&  0.014&  0.005& 0.002& 0.016\\
3& 0.975& 0.005&  0.015&  0.007& 0.003& 0.017\\
4& 0.999& 0.003&  0.009&  0.031& 0.003& 0.032\\
\hline
\end{tabular}
%}
    \label{tab:charmTab}}
  \end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The coefficients $\bar C_n$ which are required for this analysis depend on $\mu$ and 
the strong coupling constant and have been evaluated in 
three-\cite{Chetyrkin:1995ii,Chetyrkin:1996cf},
later in four-loop 
approximation\cite{Chetyrkin:2006xg,Boughezal:2006px,Maier:2008he,Maier:2009fz,Sturm:2008eb}.
The corresponding NNLO and 
N${}^3$LO charm- and bottom-quark-mass determinations have been performed in
Ref.~\refcite{Kuhn:2001dm} and Refs.~\refcite{Kuhn:2007vp,Chetyrkin:2009fv,Chetyrkin:2010ic}, 
respectively. The moments are conveniently split into three pieces, one from the 
narrow resonances $J/\psi$ and $\psi'$, one from the threshold for open charm 
production up to 4.8~GeV and one from the perturbative continuum above 4.8~GeV.
Their respective contributions and the corresponding errors are shown in Table~1,
the results for the quark mass $m_c(3~{\rm GeV})$ are given in Table~2.
A detailled discussion of the theory uncertainty can be found in 
Refs.~\refcite{Kuhn:2007vp,Chetyrkin:2010ic}.

The path to further improvements is evident from Tables 1 and 2. The experimental input 
is dominated by the electronic width $\Gamma_e$ of the narrow resonances. Considering the 
enormous improvements in luminosity and experimental techniques during the past years,
a reduction of the error on $\Gamma_e$
by a factor two seems feasible\cite{EvgeniiBaldin}.
The planned scanning with drastically increased luminosity will, 
furthermore, allow for a similar reduction of the error in the threshold 
region\cite{AchimDenig}. At the same time a precise measurement at the highest 
accessible energies arount 4.6~GeV and the comparison of the 
result with the prediction of perturbative QCD will strengthen the confidence in the 
applicability of pQCD at higher energies. Performing a combined fit for the lowest 
three moments, thus exploiting the information of the data in an optimal way, may 
lead to a further reduction of the experimental error. The reduction by a factor 
two of the error on ${\cal M}^{\rm exp}_n$ is thus well within reach.

The choice of the strong coupling constant $\alpha_s$, its central value as well 
as its error, is a matter of disput-\cite{Bethke:2011tr}. In our analysis
$\alpha_s(M_Z)=0.1189\pm0.0020$ has been adopted. The 
dependence of our result on $\alpha_s$ can be directly deduced from the  
final result
%
\begin{equation} 
m_c(3~{\rm GeV})= (986 -\frac{\alpha_s -0.1189}{0.002}\,9 \pm 10)\, {\rm MeV}
\end{equation}
%
which had been based on the second moment. Evolving, for example, to the
scale of $m_H$, one obtains
%
\begin{equation} 
m_c(126~{\rm GeV})= (609 - \frac{\alpha_s-0.0189}{2}\,15    \pm 6)\, {\rm MeV}
\end{equation}
%
where the parametric $\alpha_s$-dependence affecting both the evaluation of 
$m_c(3~{\rm GeV})$ and its evolution from $\mu=3~{\rm GeV}$ to 
126~GeV are combined. This result is evidently a critical input for 
predictions of the Higgs boson decay rate into $c\bar c$ and implies 
an error of 5.3\% from this source. Note that this error is largely dominated by 
the error on the strong coupling. Adopting a more optimistic attitude\cite{Beringer:1900zz}
and assuming $\alpha_s=0.1184\pm0.0007$ 
evidently leads to a reduction down to 2.5\%.

Similar considerations apply to the analysis of the bottom quark mass but are 
outside the scope of this presentation.



\section{Direct Production of $\chi_{c1}$ and $\chi_{c2}$ at Electron-Positron colliders} 
%================================================================================
%================================================================================
%================================================================================

\subsection{Search for $\chi_{c1}$ in the scanning mode}
With the forementioned high luminosity operation of BEPC in the energy region 
between 2 and 4.6~GeV and plans to perform a dedicated energy scan, the possibility of 
direct resonant production of $\chi_{c1}$ and $\chi_{c2}$ through the reaction
\begin{equation}
e^+e^-\to \chi_{cJ}
\end{equation}
with $J=1$ or 2 is within reach. This reaction, originally suggested in 
Ref.~\refcite{Kuhn:1979bb}
long time ago, proceeds through two virtual photons.
Its strength is characterized by the respective electronic widths, 
$\Gamma(\chi_{cJ}\to e^+e^-)\equiv \Gamma_J$, which were investigated in 
Ref.~\refcite{Kuhn:1979bb}. 
(At that time this investigation was motivated by the idea\cite{Kaplan:1978wu} that 
direct resonant production of a 
$J^{PC}=1^{++}$ quarkonium state, composed of bottom or top quarks, could  
proceed through the axial part of the neutral current, a possibility which seems 
to be out of reach.) 
The peak cross section for a narrow resonance,
normalized to the point cross section
$\sigma_0 $, is given by 
%
\begin{equation}
R_{\rm peak}\equiv \frac{\sigma_{\rm peak}}{\sigma_0} =
(2J+1)\frac{3\pi}{2\alpha^2\sqrt{2\pi}}\frac{\Gamma_J}{\Delta M}C_{\rm rad}.
\end{equation}
%
For the present discussion we assume a Gaussian spread cms energy spread
of the center-of-mass energy with width
$\Delta M=1.5\,{\rm MeV}$. The factor $C_{\rm rad}\approx (2\Delta M/M)^t\approx 0.5$ with
$t=2 \frac{\alpha}{\pi}(\ln (s/m_e^2)-1)$ 
arises from radiative corrections due to initial-state radiation. In numerical form 
%
\begin{equation}
R_{\rm peak}=
(2J+1)0.0176 \frac{\Gamma_J[{\rm eV}]}{\Delta M[{\rm MeV}]}.
\end{equation}
%
Estimates for the electronic widths $\Gamma_J$ can be found in Table 2
of Ref.~\refcite{Kuhn:1979bb}.
A sizable model dependence has been observed with results for $J=1$ varying between 0.044~eV,
as nearly rigorous lower limit  and 0.46~eV as optimistic choice. 
The results for $J=2$ are even more uncertain and vary between 0.0023~eV and 0.027~eV.
Helicity conservation leads to $\Gamma_J=0$ for $J=0$ up to terms of order $(m_e/M_\chi)^2$.
Let us, for the moment, concentrate on $J=1$ and use $\Gamma_1=0.4~{\rm eV}$ for the 
analysis and 1.5~MeV for the energy spread. In this case we find 
%
\begin{equation}
R_{\rm peak}(J=1)\approx 0.014
\end{equation}
%
for the peak cross section on top of a continuum cross section of $R_{\rm cont}\approx 2$.
This would require the sensitivity to a variation of the cross section
to exceed a relative precision of 7 permille.

As an alternative one may concentrate on a selected final state with significantly 
improved signal-to-background ratio, which arises from the decay chain
\begin{equation}
e^+e^-\to \chi_J\to J/\psi\gamma
\end{equation}
With a branching ratio ${\rm Br}(\chi_{c1}\to J/\psi\gamma)=0.348\pm0.015$
this would lead to
%
\begin{equation}
R_{\rm peak}(e^+e^-\to\chi_{c1}\to J/\psi\gamma)\approx 0.0049.
\end{equation}
%
In this case the continuum background originates from the 
(non-resonant) radiative return to 
the $J/\psi$, which proceeds through initial state radiation.
This cross section is given by ($z\equiv M_{J/\psi}^2/s$)
%
\begin{eqnarray}
R_B&=&\frac{9}{\alpha}\frac{\Gamma(J/\psi\to e^+e^-)}{M_{J/\psi}}(1-z)
\Bigg[\frac{1 + z^2}{(1- z)^2} 
\ln \frac{1+\Delta}{1-\Delta} - \Delta\Bigg]\\
%
&\approx& \frac{9 \Gamma(J/\psi\to e^+e^-)}{\alpha (M_\chi - M_{J/\psi})} 
\ln \frac{1+\Delta}{1-\Delta}\\\\\
\end{eqnarray}
%
with $z\equiv M_{J/\psi}^2/s$ and
 $\Delta \equiv \cos\theta_{\rm max}$ defining the acceptance range for the radiated photon.
With $\theta_{\rm max} = 30^\circ$ one finds $R_B=0.045$, to be contrasted with a 
resonance peak value $R_{\rm peak}=0.0049$.

We thus arrive at 
a signal/background ratio of roughly 0.10 and a relatively clean final state,
namely a photon plus a lepton pair with an invariant mass of the $J/\psi$.
To finally estimate the required event rates we consider the $e^+e^-$ 
and $\mu^+\mu^-$ decay modes of $J/\psi$ with branching ratios of 6\% each. This affects signal 
and background in the same way  with a reduction factor of 0.12. The final $R$ value for the 
background reaction
$e^+e^-\to J/\psi\gamma$
including the the branching ratio of  $J/\psi$ into
$e^+e^-$  or $\mu^+\mu^-$ 
amounts to 0.0054 and corresponds to 
38 picobarn. To observe a one-sigma effect at one point would require 100 events, corresponding 
to 2.6~pb${}^{-1}$ per point.  More realistically, with 3 points and 24~pb${}^{-1}$ per point 
(corresponding to $3\sigma$) one would definitely be on the safe side. (Note that these estimates 
are based on the optimistic assumption of $\Gamma_e=0.4\,{\rm eV}$.) 

On the other hand, 24~pb${}^{-1}$ correspond to about 0.3 million hadronic events, 
which would give a statistical precision of better than  $3.3\times10^{-3}$ in the total cross section,
to be compared to the variation of $7\times10^{-3}$. The total cross section measurement thus could
lead to an independent confirmation of the resonance.

A more refined analysis\cite{CzyzKuhn} will exploit the difference in the 
angular distribution of photon and leptons from signal and background, as well as the non-vanishing 
relative phase of signal versus background, which could give rise to an interesting 
interference pattern.

\subsection{Search for $\chi_{c1}$ through the radiative return}
As an alternative one might consider using the radiative return (including small angle 
photon emission) from one fixed energy point where a large amount of luminosity has been collected.
For a case study let us take $E_{\rm cm} = 3.770 {\rm GeV}$. The $R$-value for the production of 
a narrow resonance with mass $M$ and electronic width $\Gamma_e$ plus a photon emitted at 
arbitrary angle is given by
%
\begin{equation}
R=\frac{9}{\alpha}\frac{\Gamma_e}{M}
\frac{1 + z^2}{(1- z)}(L-1) 
\end{equation}
%
For $M=3.511~$GeV and $\Gamma_e=0.4$~eV as before one finds $R_{\chi\gamma}=7.8\times 10^{-6}$, 
corresponding to $\sigma_{\chi\gamma}=190~{\rm fb}^{-1}$. Assuming, furthermore, 
an integrated luminosity of $3~{\rm fb}^{-1}$, one arrives at 23 events in the 
$\gamma\gamma\mu^+\mu^-\,{\rm or}\,e^+e^-$ configuration where the invariant mass of 
the lepton pair plus one of the photons is given by the mass of the $\chi_1$ state.
It remains to be seen if the energy resolution is sufficient to distinguish this 
signal from the non-resonant background 
$e^+e^-\to \gamma\gamma J/\psi\to\gamma\gamma \mu^+\mu^-\,{\rm or}\,e^+e^-$.




\section*{Acknowledgments}
The author would like to thank H. Czy\.z and A. Denig for helpful discussions.


\begin{thebibliography}{99}

%=======================Teil 1 ============================================
\bibitem{LiangYan}
For an overview see: Liang Yan, these proceedings.

\bibitem{ThomasMehen}
for an overview see: Thomas Mehen, these proceedings.

%\cite{Kuhn:2001dm}
\bibitem{Kuhn:2001dm} 
  J.~H.~K\"uhn and M.~Steinhauser,
  %``Determination of alpha(s) and heavy quark masses from recent measurements of R(s),''
  Nucl.\ Phys.\ B {\bf 619}, 588 (2001)
  [Erratum-ibid.\ B {\bf 640}, 415 (2002)]
  [hep-ph/0109084].
  %%CITATION = HEP-PH/0109084;%%
%\cite{Kuhn:2007vp}
\bibitem{Kuhn:2007vp} 
  J.~H.~K\"uhn, M.~Steinhauser and C.~Sturm,
  %``Heavy Quark Masses from Sum Rules in Four-Loop Approximation,''
  Nucl.\ Phys.\ B {\bf 778}, 192 (2007)
  [hep-ph/0702103 [HEP-PH]].
  %%CITATION = HEP-PH/0702103;%%
 %\cite{Chetyrkin:2009fv}%\cite{Chetyrkin:2010ic}
\bibitem{Chetyrkin:2009fv} 
  K.~G.~Chetyrkin, J.~H.~K\"uhn, A.~Maier, P.~Maierhofer, P.~Marquard, M.~Steinhauser and C.~Sturm,
  %``Charm and Bottom Quark Masses: An Update,''
  Phys.\ Rev.\ D {\bf 80}, 074010 (2009)
  [arXiv:0907.2110 [hep-ph]].
  %%CITATION = ARXIV:0907.2110;%%
 %\cite{Chetyrkin:2010ic}
\bibitem{Chetyrkin:2010ic} 
  K.~Chetyrkin, J.~H.~K\"uhn, A.~Maier, P.~Maierhofer, P.~Marquard, M.~Steinhauser and C.~Sturm,
  %``Precise Charm- and Bottom-Quark Masses: Theoretical and Experimental Uncertainties,''
  Theor.\ Math.\ Phys.\  {\bf 170}, 217 (2012)
  [arXiv:1010.6157 [hep-ph]].
  %%CITATION = ARXIV:1010.6157;%%
 %\cite{Chetyrkin:1995ii}{Chetyrkin:1996cf}{Chetyrkin:1997mb}
\bibitem{Chetyrkin:1995ii} 
  K.~G.~Chetyrkin, J.~H.~K\"uhn and M.~Steinhauser,
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  [hep-ph/9511430].
  %%CITATION = HEP-PH/9511430;%%
 %\cite{Chetyrkin:1996cf}
\bibitem{Chetyrkin:1996cf} 
  K.~G.~Chetyrkin, J.~H.~K\"uhn and M.~Steinhauser,
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  [hep-ph/9606230].
  %%CITATION = HEP-PH/9606230;%%
  %\cite{Chetyrkin:2006xg}
\bibitem{Chetyrkin:2006xg} 
  K.~G.~Chetyrkin, J.~H.~K\"uhn and C.~Sturm,
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  Eur.\ Phys.\ J.\ C {\bf 48}, 107 (2006)
  [hep-ph/0604234].
  %%CITATION = HEP-PH/0604234;%%
 %\cite{Boughezal:2006px}
\bibitem{Boughezal:2006px} 
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  [hep-ph/0605023].
  %%CITATION = HEP-PH/0605023;%%
 %\cite{Maier:2008he}
\bibitem{Maier:2008he} 
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  [arXiv:0806.3405 [hep-ph]].
  %%CITATION = ARXIV:0806.3405;%%
  %33 citations counted in INSPIRE as of 30 Apr 2013
%\cite{Maier:2009fz}
\bibitem{Maier:2009fz} 
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  %%CITATION = ARXIV:0907.2117;%%
 %\cite{Sturm:2008eb}
\bibitem{Sturm:2008eb} 
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  %%CITATION = ARXIV:0805.3358;%%
 \bibitem{EvgeniiBaldin}
Evgenii Baldin, private communication.
\bibitem{AchimDenig}
Achim Denig, private communication.
 %\cite{Bethke:2011tr}
\bibitem{Bethke:2011tr}
  S.~Bethke, A.~H.~Hoang, S.~Kluth, J.~Schieck, I.~W.~Stewart, S.~Aoki, M.~Beneke and S.~Bethke {\it et al.},
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  arXiv:1110.0016 [hep-ph].
  %%CITATION = ARXIV:1110.0016;%%
%\cite{Beringer:1900zz}
\bibitem{Beringer:1900zz}
  J.~Beringer {\it et al.}  [Particle Data Group Collaboration],
  %``Review of Particle Physics (RPP),''
  Phys.\ Rev.\ D {\bf 86} (2012) 010001.
  %%CITATION = PHRVA,D86,010001;%%
 %=======================Teil 2 =============================================================
%\cite{Kuhn:1979bb}
\bibitem{Kuhn:1979bb}
  J.~H.~K\"uhn, J.~Kaplan and E.~G.~O.~Safiani,
  ``Electromagnetic Annihilation of e+ e- Into Quarkonium States with Even Charge Conjugation,''
  Nucl.\ Phys.\ B {\bf 157} (1979) 125.
  %%CITATION = NUPHA,B157,125;%%
%\cite{Kaplan:1978wu}
\bibitem{Kaplan:1978wu}
  J.~Kaplan and J.~H.~K\"uhn,
``Direct Production of $^{++}$ States in $e^+e^-$ Annihilation,'',
  Phys.\ Lett.\ B {\bf 78} (1978) 252.
  %%CITATION = PHLTA,B78,252;%%
\bibitem{CzyzKuhn}
H.~Czy\.z, J.~H.~K\"uhn, in preparation.



\end{thebibliography}


\end{document}

