%Title: NNNLO results on top-quark pair production near threshold
%Author: M. Beneke (RWTH Aachen), Y. Kiyo (Karlsruhe U.), K. Schuller (RWTH Aachen)
%Published: * PoS * ** RADCOR2007: ** (2007.) 051.
%arxiv/0801.3464
% Please use the skeleton file you have received in the
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\documentclass{PoS}
\title{NNNLO results on top-quark pair production near
threshold\thanks{%08xx.xxxx [hep-ph],
preprint PITHA~08/03, TTP/08-04, SFB/CPP-08-07}}
\ShortTitle{NNNLO results on top-quark pair production near
threshold}
\author{\speaker{Martin Beneke}\\
%\thanks{}\\
Institut f{\"u}r Theoretische Physik~E,
RWTH Aachen, D--52056 Aachen, Germany\\
E-mail: \email{mbenekeATphysik.rwth-aachen.de}}
\author{Yuichiro Kiyo\\
Institut f{\"u}r Theoretische Teilchenphysik,
Universit{\"a}t Karlsruhe, D--76128 Karlsruhe, Germany}
\author{Kurt Schuller\\
Institut f{\"u}r Theoretische Physik~E,
RWTH Aachen, D--52056 Aachen, Germany}
\abstract{We present new results on the NNNLO top-antitop
production cross section near threshold from potential and
ultrasoft gluon corrections. The new non-logarithmic
third-order terms are
in the 10\% range and lead to a significant reduction in
the theoretical error.}
\FullConference{8th International Symposium on Radiative Corrections \\
October 1-5, 2007\\
Florence, Italy}
\newcommand{\bfm}[1]{\mbox{\boldmath$#1$}}
\newcommand{\bff}[1]{\mbox{\boldmath ${#1}$}}
\begin{document}
\section{Introduction}
High-order perturbative calculations of non-relativistic heavy
quark-antiquark systems are required for precise quark mass
determinations from QCD sum rules or the lowest upsilon states
(bottom quark mass) or the energy dependence of the threshold top-quark pair
production cross section in $e^+ e^-$ collisions (top quark mass).
Concerning the $t\bar t$ quark cross section the status is
as follows: the NNLO calculations performed about ten years ago
revealed a large uncertainty, up to $\pm 25\%$, in the cross
section in the resonance peak
region \cite{Hoang:2000yr,Beneke:1999qg}. Subsequent
calculations that include a summation of logarithms of
$\alpha_s$ find a much reduced scale dependence
around $\pm (3-6)\%$ \cite{Hoang:2001mm,Hoang:2003xg}. The
main effect comes from the logarithms at the third order (NNNLO)
rather than the all-order series \cite{Pineda:2006ri}.
Since at NNNLO the ultrasoft scale $m_t\alpha_s^2\sim 2\,
\mbox{GeV}$ appears for the first time, a complete calculation
of the (non-logarithmic) NNNLO correction is needed. Similar
conclusions apply to bottomonium systems, with larger
uncertainties.
Recently the NNNLO correction to the $S$-quarkonium wave-functions
at the origin (corresponding to the residues of the poles of the
heavy quark current spectral functions) from potential insertions
and ultrasoft gluons have been
completed~\cite{Beneke:2007gj,Beneke:2007pj}. The talk presented
at this conference summarized these results together with
NNNLO results on the full energy-dependent spectral function \cite{inprep}
relevant to the $t\bar t$ cross section. Since the combined
result of~\cite{Beneke:2007gj,Beneke:2007pj} has already been
discussed in another proceedings article \cite{Beneke:2007uf},
we focus here on the full spectral function and the case of
the top quark. That is, we consider the two-point function
\begin{eqnarray}
\label{eq:TwoPointFunction}
&&
\left(q^\mu q^\nu - g^{\mu\nu} q^2 \right)\Pi(q^2)
= i \int d^d x\, e^{i q x}\,
\langle \Omega | T(j^\mu(x)j^\nu(0)) |\Omega\rangle
\end{eqnarray}
of the electromagnetic top-quark current $j^\mu=\bar{t}\gamma^\mu
t$, choosing $q^\mu=(2 m_t+E,\vec{0})$ with $m_t$ the
pole mass of the top quark and $E$ of order of a few GeV. The
width of the top quark is taken into account by simply letting
$E\to E+i\Gamma_t$ become complex \cite{FK87}. However, one should
note that a fully consistent treatment of the top-quark decay
beyond the NLO approximation
requires the inclusion of many other electroweak effects that
are not yet known.
\section{Remarks on the calculation}
After integrating out the hard and soft momentum scales the problem
is reduced to the calculation of a non-relativistic correlation
function $G(E)$ to third order in non-relativistic perturbation
theory. The perturbations consist of potential insertions
(instantaneous, spatially non-local operators) and ultrasoft
gluon interactions with the top quarks. Since an infinite number of
potential (Coulomb) gluons can be exchanged between the heavy quarks
without parametric suppression,
the free heavy quark-anti-quark propagators are promoted to the
Green function of the Schr\"odinger operator $H_0=-\vec{\nabla}^2/m_t
-(\alpha_s C_F)/r$ with the colour
Coulomb potential ($C_F=4/3$). Computing
Feynman integrals with Coulomb Green functions while simultaneously
regulating all divergences dimensionally to be consistent with
fixed-order matching calculations is the main challenge of the
NNNLO calculation. The third-order correction to $G(E)$ is composed of
\begin{eqnarray}
\delta_3 G &=& - \langle 0|\hat{G}_0 \delta V_1 \hat{G}_0
\delta V_1\hat{G}_0 \delta V_1 \hat{G}_0|0\rangle +
2\, \langle 0|\hat{G}_0 \delta V_1 \hat{G}_0 \delta V_2
\hat{G}_0|0\rangle - \langle 0|\hat{G}_0\delta V_3 \hat{G}_0
|0\rangle + \delta G_{\rm us},\qquad
\label{git}
\end{eqnarray}
where $\hat{G}_0= (H_0 - E-i \epsilon)^{-1}$,
$|0\rangle$ denotes a relative position eigenstate with
eigenvalue ${\bf r}=0$, and $\delta V_i$ the $i$th order
perturbation potentials (see \cite{Beneke:2007gj}). The
contributions to $G(E)$ involving only higher-order corrections to
the Coulomb potential have already been computed in \cite{Beneke:2005hg}
and are included in the following numerical result.
The ultrasoft contribution is ($D = d-1$)
\begin{eqnarray}
\delta G_{\rm us} &=&
(-i) (i g_s)^2 C_F \int\frac{d^d k}{(2\pi)^d}\,
\frac{-i}{k^2}\left(\frac{k^i k^j}{k_0^2}-\delta^{ij}\right)
\int \prod_{n=1}^6 \frac{d^{D}p_n}{(2\pi)^{D}}\,\;
i \tilde G^{(1)}(p_1,p_2;E)
\nonumber\\
&& \hspace*{0cm}
\times \,i\left[\frac{2 p_3^i}{m_t}
(2\pi)^{D}\delta^{(D)}(p_3-p_2)+(i g_s)^2 \frac{C_A}{2}
\frac{2 (p_2-p_3)^i}{(p_2-p_3)^4}\right]
\,i \tilde G^{(8)}(p_3,p_4;E+k^0)
\nonumber\\
&& \hspace*{0cm}
\times \,i\left[-\frac{2 p_4^j}{m_t}
(2\pi)^{D}\delta^{(D)}(p_4-p_5)+(i g_s)^2 \frac{C_A}{2}
\frac{2 (p_4-p_5)^j}{(p_4-p_5)^4}\right]
\,i \tilde G^{(1)}(p_5,p_6;E)
\end{eqnarray}
with $\tilde G^{(1,8)}(p,p^\prime;E)$ the colour-singlet/octet
momentum-space Coulomb Green functions. In position space
this expression simplifies to three instead of seven loop
integrations and similar simplifications apply to the other
terms in (\ref{git}). However, the integrals are divergent
and the $1/\epsilon$ poles must be extracted in momentum
space. A further complication is that the Coulomb Green
functions are not known in $D$ dimensions. The strategy
therefore consists of identifying all divergent
subgraphs, and to calculate them in $d$-dimensional momentum
space. Then combine the result with the sub-divergence
counterterms related to the renormalization of potentials
and non-relativistic currents and perform the remaining
integrations in three dimensions. Due to the incomplete
treatment of finite width effects an over-all divergence
$\alpha_s^{1,2}/\epsilon\times \Gamma_t$ remains in the
$t\bar t$ cross section, which is minimally subtracted in the
result below.
\section{Size of logarithmic and non-logarithmic
terms}
We consider the residue of the correlation function
at the lowest-energy bound-state pole at
$E_1=-m_t (\alpha_s C_F)^2/4+\ldots$ to compare the new NNNLO
non-logarithmic terms~\cite{Beneke:2007gj,Beneke:2007pj} to
the previously known logarithms \cite{KniPen2,ManSte,KPSS2,Hoa2},
since in this case a simple numerical result can be given.
$Z_1$, defined by
\begin{equation}
\Pi(q^2) \stackrel{E\rightarrow E_1}{=}
\frac{3}{2 m_t^2}\,\frac{Z_1}{E_1-E-i \epsilon},
\end{equation}
is related to the height of the cross section peak by
$R_{\rm peak} \approx 18\pi e_t^2 Z_1/(m_t^2\Gamma_t)$,
so we expect similar conclusions to hold for the entire
$t\bar t$ cross section. The NNNLO expression for $Z_1$
reads
\begin{eqnarray}
Z_1 &=& \frac{(m_t \alpha_s C_F)^3}{8\pi}
\times \bigg( 1+
\alpha_s \Big[-2.13+3.66\,L\Big]
\nonumber \\
&& +\,
\alpha_s^2 \Big[8.38-7.26 \,\ln\alpha_s -13.40 \, L
+ 8.93 \,L^2\Big]
\nonumber\\
&& +\, \alpha_s^3\Big[11.01 + [37.58]_{c_3,n_f} -9.79 \,\ln\alpha_s
-16.35 \,\ln^2\alpha_s
\nonumber\\
&& \hspace*{0.7cm} +\,(53.17- 44.27 \,\ln\alpha_s) \,L
- 48.18 \,L^2+ 18.17 \,L^3\Big]
\bigg)
\nonumber\\
&=& \frac{(m_t \alpha_s C_F)^3}{8\pi}
\times \bigg( 1-2.13 \alpha_s +22.64 \alpha_s^2 +
[-32.96+[37.58]_{c_3,n_f}]\alpha_s^3\bigg).
\label{Z1}
\end{eqnarray}
Since the scales $m_t$, $m_t\alpha_s$ and $m_t\alpha_s^2$ are all
relevant here, there is an ambiguity in the representation of the
logarithms. The above result uses $\alpha_s\equiv \alpha_s(\mu)$
and puts the explicit $\mu$-dependence into
$L\equiv\ln \mu/(m_t C_F\alpha_s)$. The NNNLO result is
not yet complete: the constant 11.01 includes an estimate
of the NNNLO correction to the Coulomb potential,
$a_3=3840$ \cite{Chishtie:2001mf}, and sets certain unknown
${\cal O}(\epsilon)$ potential terms to zero. This is expected
to have a minor effect \cite{Beneke:2007gj}. More important
is that only the $n_f$-parts of the third-order matching
coefficient $c_3$ of the non-relativistic current $\psi^\dagger
\sigma^i\chi$ are known \cite{Marquard:2006qi}, which turn out
to be very large ($[37.58]_{c_3,n_f}$). In the following numerical
results for the $t\bar t$ cross section we therefore consider
two options, one where the constant part of $c_3$ is set
to the known $n_f$ terms, the other where it is set to zero.
(The logarithms are all known and always included.)
We observe (third and fourth line of (\ref{Z1})) that the
typical size of non-logarithmic terms of individual third-order
corrections (ultrasoft, non-Coulomb potentials, Wilson coefficient)
is about $40\alpha_s^3 \approx 10\% \,\,(\,>100\%)$
for toponium (bottomonium). However, large cancellations
between individual terms and between logarithmic
and non-logarithmic terms occur. Thus, to obtain a reliable
third-order result the non-logarithmic terms are crucial
and a final assessment needs the missing $n_f$-independent
term in $c_3$. This is seen in the last line
of (\ref{Z1}), which shows the series for $\alpha_s=0.14$
where $L=0$. The large NNLO correction is evident. On the
other hand, the third-order correction is not anomalously
large, although the final coefficient will only be known
when the term $[37.58]_{c_3,n_f}$ is replaced by the full result
for $c_3$. The NNNLO result shows a strong reduction of
the scale dependence compared to NNLO as discussed
in \cite{Beneke:2007uf}, but the perturbative prediction becomes
unstable for $\mu<20\,\mbox{GeV}$. A study of this problem
for the Coulomb corrections, where a resummation of the perturbative
series can be done by means of a numerical solution, has shown
\cite{Beneke:2005hg} that the perturbative prediction for
$\mu>25\,\mbox{GeV}$ is close to the true result, hence
we do not consider scales $\mu<25\,$GeV.
\section{Top-quark cross section}
We next discuss the $t\bar t$ production cross section in
$e^+ e^-$ annihilation near threshold. More precisely, we
consider the $R$-ratio
\begin{equation}
R = \sigma_{t\bar tX}/\sigma_0 = 12\pi e_t^2 \,\mbox{Im}\;\Pi(q^2)
\quad\qquad \left(\sigma_0=4\pi\alpha_{\rm
em}^2/(3 s)\right),
\label{R}
\end{equation}
neglecting the axial-vector contribution from $Z$-exchange for
the purpose of discussing the impact of the QCD NNNLO correction.
The top quark pole mass should be avoided as an input parameter.
Here we use the potential-subtracted mass \cite{Beneke:1998rk}
implemented as explained in \cite{Beneke:2005hg}. The parameters for
the cross section calculation are:
$m_{t,\rm PS}(20\,\mbox{GeV})=175\,\mbox{GeV}$,
$\Gamma_t=1.4\,$GeV, $\alpha_s(M_Z)=0.1189$.
\begin{figure}[t]
\begin{center}
% \vspace*{-0.9cm}
\includegraphics[width=0.57\textwidth]{fig_orders.eps}
\vskip0.5cm
\includegraphics[width=0.58\textwidth]{fig_scale.eps}
\caption{(top) Successive approximations to the $R$-ratio
at fixed $\mu=30\,\mbox{GeV}$. At NNNLO two implementations of
$c_3$ are shown as discussed in the text. (bottom)
Renormalization scale dependence at NNLO and NNNLO. Here
all known terms in $c_3$ are included.}
\label{fig1}
\end{center}
\end{figure}
The successive LO ... NNNLO approximations to $R$ are shown
in Figure~\ref{fig1} (top panel). At $\mu=30\,\mbox{GeV}$
the size of the third-order correction is up to 10\%
depending on the assumption for $c_3$. When all known
terms in $c_3$ are included the peak cross section is
about 10\% larger than in the renormalization-group improved
NNLO calculations \cite{Hoang:2001mm,Hoang:2003xg,Pineda:2006ri}
due to the sizeable constant term related to
$11.01 + [37.58]_{c_3,n_f}$ in (\ref{Z1}). Contrary to
the NNLO approximation the third-order result shows good
convergence of the perturbative expansion. The bottom panel of
Figure~\ref{fig1} consequently displays a strong reduction
of the scale dependence from NNLO to NNNLO.
The residual scale dependence is
similar at NNNLO and in the renormalization-group improved
calculations, which already captures correctly
the logarithms of $\mu$. It therefore appears that with a
complete NNNLO result and a summation of higher-order
logarithms at hand, the demands on an accurate theoretical
prediction of the cross section near threshold can be met
{\em as far as QCD corrections are concerned.} In particular,
the scale dependence of the peak position which is indicative
of the accuracy of the top mass measurement is now well
below 100 MeV.
\section{Summary}
The NNNLO QCD correction to the $t\bar t$ cross section
near threshold is now nearly complete. We presented for the
first time the result of the third-order potential and
ultrasoft correction to the non-relativistic heavy-quark correlation
function. We find that the third-order correction
behaves well (contrary to the anomalously large effect at NNLO)
and removes a large part of the theoretical
uncertainty. The new non-logarithmic terms are numerically
important and increase the cross section relative to the
renormalization-group improved NNLO result by about 10\%,
when all presently
known terms of the three-loop matching coefficient $c_3$
are included. Further work is necessary on a consistent
treatment of electroweak and finite-width effects
(see \cite{Beneke:2003xh,Hoang:2004tg,Beneke:2007zg}).
\section*{Acknowledgments}
This work was supported by the DFG Sonderforschungsbereich/Transregio
9 ``Computer-gest\"utzte Theoretische Teilchenphysik'' and DFG
Graduiertenkolleg ``Elementarteilchen\-physik an der TeV-Skala''.
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\end{document}