%Title: Recent Developments in the Treatment of Heavy Quarks
%Author: Matthias Steinhauser
%Published: <EM> AIP Conf.Proc </EM> <B> 756 </B> (2005) 357-359. 

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\begin{document}

\title{
\mbox{}\vspace*{-2.5em}\\
\mbox{}\hspace*{14em}{\small TTP04-22, SFB/CPP-04-65}\\
Recent Developments in the Treatment of Heavy Quarks}

\classification{12.38.Bx, 13.25.Gv}
\keywords      {Quarkonium, non-relativisic QCD, renormalization group}

\author{Matthias Steinhauser}{
  address={Institut f\"ur Theoretische Teilchenphysik, 
    Universit\"at Karlsruhe, D-76128 Karlsruhe}
}

\begin{abstract}
  We discuss new results obtained for the description of a system of
  two heavy
  quarks close to the production threshold. Special emphasis is put
  on the resummation of logarithms in the quark velocity.
  As physical applications we discuss the determination of the
  mass of the $\eta_b$ meson and the ratio of the photon mediated
  production or annihilation rates of spin triplet and spin singlet
  heavy quarkonium states.
\end{abstract}

\maketitle

\vspace*{-0.7em}

\section{Introduction}

\vspace*{-0.5em}

One of the most important tasks of a future linear collider is the
precise measurement of the cross section for the production of top quark pairs
close to their production threshold. Next to the mass and the width of the top
quark also the strong coupling and --- in case the Higgs boson is not too
heavy --- also the top quark Yukawa coupling can be determined with a quite
high accuracy. In order to match the expected experimental
precision~\cite{Martinez:2002st}  
it is important to compute higher-order quantum corrections to this process.
In this context we refer to the Refs.~\cite{Hoa_etal,PenSte,KPSS2,Hoa}.

In these proceedings we want to concentrate on the system of two
bottom quarks. In particular we report about results based on the
renormalization group equation within the framework of
non-relativistic QCD~\cite{PinSot1,Luk,Pineda02}. 
In particular, in the next Section we discuss the
determination of the mass of the pseudoscalar $\eta_b$ meson,
$M(\eta_b)$, and the last Section deals with the
ratio of the photon mediated
production or annihilation rates of spin triplet and spin singlet
heavy quarkonium states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{-0.7em}

\section{Prediction of $M(\eta_b)$}

\vspace*{-0.5em}

In order to determine $M(\eta_b)$ we exploit the relation
$M(\eta_b) = M(\Upsilon(1S)) - E_{\rm hfs}$
where $M(\Upsilon(1S)) = 9.46030(26)$~GeV and $E_{\rm hfs}$ is the hyperfine
splitting. The latter can be determined from the spin-dependent part
of the effective Lagrangian. The next-to-leading order (NLO) approximation of 
$E_{\rm hfs}$ is easily obtained from the N$^3$LO corrections to the
energy level~\cite{PenSte}. In order to improve the approximation 
one has to resum the logarithms in the velocity of the heavy quarks 
contained in the corresponding matching coefficient. This leads to
$E_{\rm hfs}$ in next-to-leading logarithmic (NLL) accuracy.
Its computation can be divided into
three steps: First, the renormalization group (RG) equations have to be
established within potential non-relativistic QCD (pNRQCD)~\cite{PinSot1}. 
One has to make sure to
include the running of all relevant operators and to consider the
soft, potential and ultra-soft regions.
In a second step the RG equations have to be solved. In the case at
hand this could be done analytically. As a result one obtains the
matching coefficient of the spin-dependent operator, $D_{S^2,s}^{(2)}$,
to the NLL accuracy. This expression is used in the third step to
evaluate within perturbation theory the corrections to the energy level. 
These steps have been performed in Refs.~\cite{KPPSS,PPSS}.

A compact analytical expression for the HFS to NLL,
$E_{\rm hfs}^{\rm NLL}$, which is a function of
$\alpha_s(\mu)/\alpha_s(m_b)$, is given in Eq.~(1) of Ref.~\cite{KPPSS}.
In Fig.~\ref{fig1}(a), the HFS for the bottomonium
ground state is plotted as a function of $\mu$ in the LO, NLO, LL, and NLL
approximations. As can be seen, the LL curve shows a weaker scale dependence
compared to the LO one. The scale dependence of the NLO and NLL expressions is
further reduced, and, moreover, the NLL approximation remains stable up to
smaller scales than the fixed-order calculation. At the scale $\mu'\approx
1.3$~GeV. The NLL correction
vanishes.  Furthermore, at $\mu''\approx 1.5$~GeV, the result becomes
independent of $\mu$; {\it i.e.}, the NLL curve shows a local maximum.  This
suggests a nice convergence of the logarithmic expansion despite the presence
of the ultrasoft contribution with $\alpha_s$ normalized at the rather low
scale $\bar\mu^2/m_b\sim 0.8$~GeV.
From these observations the following prediction of the mass of
the as-yet undiscovered $\eta_b$ meson has been obtained~\cite{KPPSS} 
\begin{equation}
M(\eta_b)=9421\pm 11\,{(\rm th)} \,{}^{+9}_{-8}\, 
(\delta\alpha_s)~{\rm MeV}\,,
\end{equation}
where the errors due to the high-order perturbative corrections and the
nonperturbative effects are added up in quadrature in ``th'', whereas
``$\delta\alpha_s$'' stands for the uncertainty in
$\alpha_s(M_Z)=0.118\pm0.003$.  If the experimental error in future
measurements of $M(\eta_b)$ will not exceed a few MeV, the bottomonium HFS
will become a competitive source of $\alpha_s(M_Z)$ with an estimated accuracy
of $\pm 0.003$, as can be seen from Fig.~\ref{fig1}(a).


\begin{figure}
  \begin{tabular}{cc}
      \leavevmode
      \epsfxsize=5cm
      \epsffile{etab1S.eps}
      &
      \leavevmode
      \epsfxsize=5cm
      \epsffile{gratio.bottom.mod2.eps}
\\
(a) & (b)
\end{tabular}
  \vspace*{-2em}
\caption{\label{fig1} (a) HFS of 1S bottomonium
as a function of the
renormalization scale $\mu$ in the LO (dotted line), NLO (dashed line), LL
(dot-dashed line), and NLL (solid line) approximations. 
(b) ${\cal R}_q$  as the function of the
renormalization scale $\nu$ in LO$\equiv$LL (dotted line), NLO (short-dashed
line), NNLO (long-dashed line), NLL (dot-dashed
line), and NNLL (solid line) approximation for the bottomonium
ground state.
The (yellow) band reflects the errors due to $\alpha_s(M_Z)=0.118\pm
0.003$. }
\end{figure}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vspace*{-0.7em}

\section{Spin dependence of heavy quarkonium production and
  annihilation}

\vspace*{-0.5em}

In Ref.~\cite{PPSS2} a further step was undertaken and the 
ratio of the photon mediated
production or annihilation rates of spin triplet and spin singlet
heavy quarkonium states has been considered. In particular, we define
\begin{eqnarray}
  {\cal R}_q &=&
  {\sigma(e^+e^-  \rightarrow {\cal Q}(n^3S_1) )\over 
    \sigma(\gamma\gamma \rightarrow {\cal Q}(n^1S_0))}=
  {\Gamma({\cal Q}(n^3S_1)\to e^+e^-)\over 
    \Gamma({\cal Q}(n^1S_0)\to  \gamma\gamma)}\,,
\end{eqnarray}
which can be written as
\begin{eqnarray}
  {\cal R}_q &=& {c_s^2(\nu)\over 3Q_q^2}
  {|\psi_n^{v}(0)|^2\over|\psi_n^{p}(0)|^2}+{\cal O}(\alpha_s v^2)\,.
  \label{Rdef}
\end{eqnarray}
$\psi_n^{(v,p)}(\vec{r}\,)$ are
the spin triplet (vector) and spin singlet (pseudoscalar) quarkonium
wave functions of the principal quantum number $n$. The expression for
$|\psi_n^{v}(0)|^2/|\psi_n^{p}(0)|^2$ to NNLL
can be found in Ref.~\cite{PPSS2}.
$c_s$ is the ratio of the corresponding matching coefficients.
With the help of the NLL approximation of $D_{S^2,s}^{(2)}$
it was possible to obtain the NNLL corrections of $c_s$
and thus for the quantity ${\cal R}_q$~\cite{PPSS2}, i.e. for the
first time for a physical quantity.

It is interesting to consider the various approximations for ${\cal R}_q$
as a function of the renormalization scale, $\nu$. One observes a very good
convergence for the top quark system~\cite{PPSS2}.
In the case of the bottom quark the result is shown in Fig.~\ref{fig1}(b).
A nice convergence of the logarithmic expansion is observed
despite the presence of ultrasoft contributions with
$\alpha_s$ normalized at a rather low scale $\nu^2/m_b$. 
At the same time, the perturbative corrections are important
and reduce the leading order result by approximately $41\%$.  
More details can be found in Ref.~\cite{PPSS2}.

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\vspace*{-0.7em}

\begin{theacknowledgments}
I would like to thank B. Kniehl, A. Penin, A. Pineda and V. Smirnov for a
very pleasant and fruitful collaboration. This work was supported by
HGF Grant No. VH-NG-008.
\end{theacknowledgments}

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\begin{thebibliography}{9}

%\cite{Martinez:2002st}
\bibitem{Martinez:2002st}
M.~Martinez and R.~Miquel,
%``Multi-parameter fits to the t anti-t threshold observables at a future  e+ e-
%linear collider,''
Eur.\ Phys.\ J.\ C {\bf 27} (2003) 49.
%[arXiv:hep-ph/0207315].
%%CITATION = HEP-PH 0207315;%%
\bibitem{Hoa_etal} A.H. Hoang et al.,
Eur.\ Phys.\ J.direct C {\bf 3}, 1 (2000).

\bibitem{PenSte} 
B.A. Kniehl, A.A. Penin, 
V.A. Smirnov, and M. Steinhauser, Nucl.\ Phys.\ {\bf B635}, 357 (2002);\\
A.A. Penin and M. Steinhauser, 
Phys.\ Lett.\ B {538} (2002) 335.

\bibitem{KPSS2} B.A. Kniehl, A.A. Penin, V.A. Smirnov, and M. Steinhauser, 
Phys.\ Rev.\ Lett.\ {90}  (2003) 212001; 
Erratum {\it ibid.}\ {91} (2003) 139903.

\bibitem{Hoa} A.H. Hoang, Phys.\ Rev.\ D {69} (2004) 034009.

\bibitem{PinSot1} A. Pineda and J. Soto,
Nucl.\ Phys.\ Proc.\ Suppl. {64}  (1998) 428;\\
N. Brambilla, A. Pineda, J. Soto, and A. Vairo,
Nucl.\ Phys.\ {B 566} (2000) 275.

\bibitem{Luk}
M.E.~Luke, A.V.~Manohar, and I.Z.~Rothstein,
Phys.\ Rev.\ D {\bf 61} (2000) 074025.

\bibitem{Pineda02}
A. Pineda,
Phys.\ Rev.\ D {65} (2002) 074007; Phys.\ Rev.\ D {66} (2002) 054022.

\bibitem{KPPSS} B.A. Kniehl, A.A. Penin, A. Pineda,
V.A. Smirnov, and M. Steinhauser,  
Phys.\ Rev.\ Lett.\ {\bf 92} (2004) 242001. 

\bibitem{PPSS} A.A. Penin, A. Pineda, V.A. Smirnov, and M. Steinhauser,   
Phys.\ Lett.\ B {\bf 593} (2004) 124

\bibitem{PPSS2} A.A. Penin, A. Pineda, V.A. Smirnov, and M. Steinhauser,   
Nucl.\ Phys.\ B {\bf 699} (2004) 183.

\end{thebibliography}

\end{document}
