%Title: Summing next-to-next-to-leading logarithms in b->c transitions at zero recoil
%Author: Andrey Grozin
%hep-ph/0509328
\documentclass{elsart}
\newcommand{\eo}[2]{\left\{\begin{array}{c}#1\\[-2mm]#2\end{array}\right\}}

\begin{document}
\begin{frontmatter}
\title{Summing next-to-next-to-leading logarithms\\
in $b \to c$ transitions at zero recoil}
\author{A.G.~Grozin}
\ead{grozin@particle.uni-karlsruhe.de}
\address{Institut f\"ur Theoretische Teilchenphysik,
Universit\"at Karlsruhe}
\begin{abstract}
The HQET-1 $\to$ HQET-2 matching coefficient for the $b\to c$ currents
at zero recoil has been found with two-loop accuracy.
The next-to-next-to-leading logarithms has been summed
in the leading order in $m_c/m_b$.
Higher-order corrections are estimated in the large-$\beta_0$ limit.
\end{abstract}
\begin{keyword}
\PACS 12.38.Bx \sep 12.39.Hg
\end{keyword}
\end{frontmatter}

\section{Introduction}
\label{S:Intro}

Weak $b\to c$ transitions are described
by QCD currents $\bar{c}\Gamma b$
with $\Gamma=\gamma^\mu$ and $\gamma_5 \gamma^\mu$.
If we are only interested in their matrix elements
for $p_b=m_b v+k$, $p_c=m_c v'+k'$,
where the residual momenta $k$, $k'$
are small as compared to the heavy-quark masses $m_{c,b}$,
then we may expand these QCD currents in HQET operators
of the form $\bar{c}_{v'}\cdots b_v$
with the appropriate quantum numbers
divided by powers of $m_{c,b}$.
The coefficients are calculated by matching.
This is descussed, e.g., in the review~\cite{Ne:94}
or in the textbook~\cite{G:04}.
In this paper we consider the zero-recoil case $v'=v$.
Then, in the $v$ rest frame,
\begin{eqnarray}
\bar{c} \gamma_0 b &=& \eta_V\,\bar{c}_v b_v
+ \mathcal{O}(1/m_{c,b}^2)\,,
\nonumber\\
\bar{c} \gamma_5 \vec{\gamma} b &=& \eta_A\,
\bar{c}_v \gamma_5 \vec{\gamma} b_v
+ \mathcal{O}(1/m_{c,b}^2)\,.
\label{Intro:def}
\end{eqnarray}
We use $\overline{\mathrm{MS}}$ renormalization
with anticommuting $\gamma_5$;
then the vector and axial currents in QCD do not depend
on the normalization scale $\mu$.
The HQET currents with $v'=v$ are also $\mu$-independent,
and hence the matching coefficients $\eta_{V,A}$
do not depend on any normalization scale.
We shall not discuss $1/m_{c,b}^2$ power corrections here.

The coefficients can be obtained by matching QCD matrix elements
with those in HQET:
\begin{equation}
\eta_\Gamma = 1 + \sum_n \eta_\Gamma^{(n)}
\left(\frac{\alpha_s^{(5)}(\mu_0)}{4\pi}\right)^n\,,\quad
\mu_0 = \sqrt{m_b m_c}\,.
\label{Intro:1step}
\end{equation}
Here $m_{b,c}$ are the on-shell heavy-quark masses.
In this way, we know the exact dependence of $\eta_\Gamma^{(n)}$
on $r=m_c/m_b$;
however, we cannot sum terms with leading powers of $L=\log(m_b/m_c)$
in all orders of perturbation theory.
The two-loop correction $\eta_\Gamma^{(2)}$
has been calculated in~\cite{CM:97}
and confirmed by an independent re-calculation~\cite{FT:98}.

Alternatively, at $r\ll1$ we can perform the matching in two steps.
First, we match QCD with HQET-1,
where $b$ is static and $c$ is dynamic, at $\mu\sim m_b$.
Second, we match HQET-1 with HQET-2,
where both $b$ and $c$ are static, at $\mu\sim m_c$.
In this way, we obtain $\eta_{V,A}$ as series in $r$:
\begin{equation}
\eta_\Gamma = C_\Gamma(m_b) G(m_b,m_c) E(m_c) + \mathcal{O}(r)\,.
\label{Intro:2step}
\end{equation}
In practice, few terms of this expansion can be found,
because the number of relevant operators in HQET-1
grows fast with the dimensionality.
However, in each term of the expansion in $r$,
we can sum leading, next-to-leading, etc., powers of $L$.
The leading term in the $r$-expansion is known
at the next-to-leading logarithmic order~\cite{Ne:92}.
The $\mathcal{O}(r)$ term is known
at the leading logarithmic order~\cite{FG:90}.
The $\mathcal{O}(r^2)$ term is discussed in~\cite{BO:97}.
The two approaches can be combined:
we subtract several first terms of expansion of~(\ref{Intro:1step})
in $r$ and add the RG-improved results instead.

The QCD $\to$ HQET-1 matching coefficients
\begin{equation}
C_\Gamma(m_b) = 1 + \sum_n c_\Gamma^{(n)}
\left(\frac{\alpha_s^{(5)}(m_b)}{4\pi}\right)^n
\label{Intro:C}
\end{equation}
are known at the next-to-next-to-leading (NNL) order~\cite{BG:95,G:97}.
The running factor $G(m_b,m_c)$
is determined by the anomalous dimension $\gamma$
of the heavy-light current in HQET-1.
Until recently, it was only known at two loops~\cite{JM:91,BG:91}.
Now it is has been calculated at three loops~\cite{CG:03},
i.e., also at the NNL order,
using the methods of~\cite{G:00}.
Finally, the HQET-1 $\to$ HQET-2 matching coefficient
\begin{equation}
E(m_c) = 1 + \sum_n e_n
\left(\frac{\alpha_s^{(4)}(m_c)}{4\pi}\right)^n
\label{Intro:E}
\end{equation}
is currently known at one loop~\cite{Ne:92} only.
In the present paper,
we shall find the two-loop coefficient $e_2$,
thus completing the NNL order calculation
of $\eta_{V,A}$ at $\mathcal{O}(r^0)$.

Quite a different kind of summation was performed in~\cite{Ne:95}:
terms with the highest powers of $\beta_0$ in the perturbative series
for $\eta_{V,A}$ were summed to all orders.

\section{Summing NNL logarithms}
\label{S:NNL}

By requiring that the two-step matching result~(\ref{Intro:2step})
reproduces the $\mathcal{O}(r^0)$ term in $\eta_{V,A}^{(2)}$~\cite{CM:97},
we have two independent derivations of the matching coefficient $E(m_c)$:
\begin{eqnarray}
E(m_c) &=& 1 - 4 C_F \frac{\alpha_s(m_c)}{4\pi}
\nonumber\\
&&{} + C_F \Biggl[ C_F \left( - 20 \pi^2 \log 2 + 14 \zeta_3 + 73 \zeta_2
+ \frac{81}{16} \right)
\nonumber\\
&&\hphantom{{}+C_F\Biggl[\Biggr.}
+ C_A \left( 10 \pi^2 \log 2 - 41 \zeta_3 - 6 \zeta_2 + \frac{503}{48}
\right)
\nonumber\\
&&\hphantom{{}+C_F\Biggl[\Biggr.}
+ T_F \left( - 17 \zeta_2 + \frac{233}{12} \right)
+ T_F n_l \left( - 4 \zeta_2 - \frac{31}{12} \right) \Biggr]
\left(\frac{\alpha_s(m_c)}{4\pi}\right)^2
\nonumber\\
&\approx& 1 - \frac{4}{3} \frac{\alpha_s(m_c)}{\pi}
+ 4.00 \left(\frac{\alpha_s(m_c)}{\pi}\right)^2\,.
\label{NNL:E}
\end{eqnarray}
Here $n_l=3$ is the number of flavours lighter than $c$;
$\alpha_s^{(4)}(\mu)$ is written simply as $\alpha_s(\mu)$.
This result can also be obtained by a direct calculation;
this will be discussed in a separate publication.

Now we can sum the logarithms (up to NNL ones) to all orders:
\begin{equation}
\eta_\Gamma = x^{-\gamma_0/(2\beta_0)}
\left[ 1 + h_\Gamma^{(1)} \frac{\alpha_s(m_b)}{4\pi}
+ h_\Gamma^{(2)} \left(\frac{\alpha_s(m_b)}{4\pi}\right)^2
+ \cdots \right]\,.
\label{NNL:logs}
\end{equation}
Here
\begin{equation}
x = \frac{\alpha_s(m_c)}{\alpha_s(m_b)}\,,
\label{NNL:x}
\end{equation}
$\alpha_s(\mu)$ and $\beta_0$ are for 4 flavours.
The NL correction is
\begin{eqnarray}
\frac{h_{V,A}^{(1)}}{C_F} &=&
- \frac{1}{4} \left( 11 x + \eo{13}{21} \right)
+ \left[ \left( 8 \zeta_2 - \frac{23}{4} \right) C_F
- 2 ( \zeta_2 + 4 ) C_A \right]
\frac{x-1}{\beta_0}
\nonumber\\
&&{} + \frac{3 C_A (11 C_F + 7 C_A) (x-1)}{2 \beta_0^2}\,,
\label{NNL:NL}
\end{eqnarray}
where the upper number in braces is for $V$
and the lower one for $A$.
The NNL correction is
\begin{eqnarray}
&&\frac{h_{V,A}^{(2)}}{C_F} =
\left[ \left( 3 \zeta_2 + \frac{29}{24} \right) x^2
- 3 \zeta_2 - \frac{1}{24} \eo{53}{205} \right] \beta_0
\nonumber\\
&&{} + \left[ \left( - 20 \pi^2 \log 2 + 3 \zeta_3 + \frac{247}{3} \zeta_2
+ \frac{643}{96} \right) x^2
+ \eo{143}{231} \frac{x}{16} \right.
\nonumber\\
&&\quad\left.{} + \eo{4}{28/3} \pi^2 \log 2 - \eo{3}{11} \zeta_3
- \eo{73/3}{67} \zeta_2 + \frac{1}{96} \eo{1283}{2899} \right] C_F
\nonumber\\
&&{} + \left[ \left( 10 \pi^2 \log 2 - \frac{63}{2} \zeta_3
- \frac{58}{3} \zeta_2 - \frac{41}{96} \right) x^2
\vphantom{\eo{1}{1}} \right.
\nonumber\\
&&\quad\left.{} - \eo{2}{14/3} \pi^2 \log 2 - \eo{9}{1} \frac{\zeta_3}{2}
+ \eo{55}{103} \frac{\zeta_2}{3} - \frac{1}{96} \eo{343}{919} \right] C_A
\nonumber\\
&&{} + \left[ - (13 \zeta_2 - 22) x^2
- \eo{24}{40/3} \zeta_2 + \eo{727/18}{133/6} \right] T_F
\nonumber\\
&&{} + \left\{ \left[ \left( 20 \zeta_4 + 9 \zeta_3 - 50 \zeta_2
+ \frac{483}{16} \right) x
\vphantom{\eo{1}{1}} \right. \right.
\nonumber\\
&&\qquad\left.{} + 20 \zeta_4 + 9 \zeta_3 - \eo{54}{70} \zeta_2
+ \frac{1}{16} \eo{529}{713} \right] C_F^2
\nonumber\\
&&\quad{} + \left[ \left( 4 \zeta_4 + \frac{57}{2} \zeta_3
- \frac{77}{6} \zeta_2 + \frac{629}{24} \right) x
\vphantom{\eo{1}{1}} \right.
\nonumber\\
&&\qquad\left.{} + 4 \zeta_4 + \frac{57}{2} \zeta_3
- \eo{71}{47} \frac{\zeta_2}{6} + \frac{1}{24} \eo{725}{1109} \right] C_F C_A
\nonumber\\
&&\quad\left.{} + \left( 6 \zeta_4 - 33 \zeta_3 + \frac{19}{3} \zeta_2
+ \frac{103}{8} \right) (x+1) C_A^2
\vphantom{\eo{1}{1}} \right\} \frac{x-1}{\beta_0}
\nonumber\\
&&{} + \left\{ \frac{1}{32} (32 \zeta_2 - 23)^2 (x-1) C_F^3
- \left( 7 \zeta_2 + \frac{609}{32} \right) (x+1) C_A^3
\vphantom{\eo{1}{1}} \right.
\nonumber\\
&&\quad{} + \left[ - \left( 40 \zeta_4 + \frac{17}{2} \zeta_2
+ \frac{479}{8} \right) x + 40 \zeta_4 + \frac{193}{2} \zeta_2
- \frac{1}{8} \eo{1281}{1545} \right] C_F^2 C_A
\nonumber\\
&&\quad\left.{} + \left[ \left( 5 \zeta_4 + 33 \zeta_2
- \frac{395}{4} \right) x
- 5 \zeta_4 + \zeta_2 - \eo{168}{189} \right] C_F C_A^2
\right\} \frac{x-1}{\beta_0^2}
\nonumber\\
&&{} + 3 C_A (11 C_F + 7 C_A) \left\{
\left( 4 \zeta_2 - \frac{23}{8} \right) (x-1) C_F^2
+ \frac{7}{4} (x+1) C_A^2 \right.
\nonumber\\
&&\quad\left.{} + \left[ - \left( \zeta_2 + \frac{5}{4} \right) x
+ \zeta_2 + \frac{27}{4} \right] C_F C_A \right\}
\frac{x-1}{\beta_0^3}
\nonumber\\
&&{} + \frac{9 C_F C_A^2 (11 C_F + 7 C_A)^2 (x-1)^2}{8 \beta_0^4}\,.
\label{NNL:NNL}
\end{eqnarray}
Numerically,
\begin{eqnarray}
&&\eta_{V,A} = x^{6/25} \left[ 1
- \left( 1.10588 x + \eo{0.89412}{1.56079} \right)
\frac{\alpha_s(m_b)}{\pi} \right.
\nonumber\\
&&\left.{} + \left( 3.94156 x^2 + \eo{0.98879}{1.72604} x
- \eo{2.44780}{8.22155} \right)
\left(\frac{\alpha_s(m_b)}{\pi}\right)^2
+ \cdots \right]\,.
\label{NNL:num}
\end{eqnarray}

\section{Higher orders in the large-$\beta_0$ limit}
\label{S:beta}

It would be extremely difficult to sum the N$^3$L logarithms.
However, it is easy to find the $\beta_0^2$ term in $h_\Gamma^{(3)}$.
The highest powers of $\beta_0$ in all orders of perturbation theory
for $\gamma$ were found in~\cite{BG:95}:
\begin{eqnarray}
\gamma &=& - \frac{C_F}{3} \frac{\beta}{\beta_0}
\frac{3+2\beta}{B(2+\beta,2+\beta)\Gamma(1-\beta)\Gamma(3+\beta)}
\nonumber\\
&=& - 3 C_F \frac{\beta}{\beta_0} \left[ 1 + \frac{5}{6} \beta
- \frac{35}{36} \beta^2 - \left( 2 \zeta_3 - \frac{83}{72} \right) \beta^3
+ \cdots \right]\,.
\label{NNL:gamma}
\end{eqnarray}
where $\beta=\beta_0\alpha_s/(4\pi)$.
The matching coefficients $C_\Gamma(m_b)$
were also considered in~\cite{BG:95}:
\begin{eqnarray}
C_{V,A}(m_b) &=& 1 - C_F \frac{\beta}{\beta_0}
\left[ \eo{2}{4} + \left( 3 \zeta_2 + \eo{47/16}{445/48} \right) \beta
\right.
\nonumber\\
&&\left.{} + \left( 7 \zeta_3 + \eo{13}{21} \zeta_2
+ \frac{1}{432} \eo{-1751}{7993} \right) \beta^2
+ \cdots \right]\,.
\label{NNL:C}
\end{eqnarray}

Here we use the same method for $E$.
The solution of the RG equation can be written as
\begin{eqnarray}
&&E(\mu) = \hat{E}
\left(\frac{\alpha_s(\mu)}{\alpha_s(m_c)}\right)^{\gamma_0/(2\beta_0)}
K(\alpha_s(\mu))\,,
\nonumber\\
&&K(\alpha_s) = \exp \int_0^{\alpha_s}
\left(\frac{\gamma(\alpha_s)}{2\beta(\alpha_s)}
-\frac{\gamma_0}{2\beta_0}\right)
\frac{d\alpha_s}{\alpha_s}\,.
\label{NNL:K}
\end{eqnarray}
The RG invariant $\hat{E}$ is,
at the first order in $1/\beta_0$,
\begin{equation}
\hat{E} = 1 + \frac{1}{\beta_0} \int_0^\infty
S(u) e^{-u/\beta} du
+ \mathcal{O}\left(\frac{1}{\beta_0^2}\right)\,,
\label{NNL:Ehat}
\end{equation}
where $\beta=\beta_0\alpha_s(m_c)/(4\pi)$.
We obtain
\begin{equation}
S(u) = 3 C_F \left[ e^{(5/3)u}
\frac{\Gamma(u)\Gamma(2-2u)}{\Gamma(3-u)}
\frac{1-u-u^2}{1+2u} - \frac{1}{2u} \right]\,.
\label{NNL:Su}
\end{equation}
Note that the leading infrared renormalon is at $u=1$.
Collecting all this together, we obtain
\begin{eqnarray}
E(m_c) &=& 1 + C_F \frac{\beta}{\beta_0} \left[ - 4
+ \left( 3 \zeta_2 + \frac{31}{16} \right) \beta \right.
\nonumber\\
&&\left.{}
+ \left( 7 \zeta_3 - 11 \zeta_2 - \frac{7655}{432} \right) \beta^2
+ \cdots \right]\,.
\label{NNL:E3}
\end{eqnarray}
This reproduces the $\beta_0$ term at two loops~(\ref{NNL:E}),
and gives us the $\beta_0^2$ term at three loops.

Using this result, we obtain
\begin{eqnarray}
\frac{h_{V,A}^{(3)}}{C_F} &=& \left[
\left( 6 \zeta_3 - 11 \zeta_2 - \frac{3703}{216} \right) x^3
- 6 \zeta_3 - \eo{13}{21} \zeta_2 + \frac{1}{216} \eo{751}{-4121} \right]
\beta_0^2
\nonumber\\
&&{} + \mathcal{O}(\beta_0)\,.
\label{NNL:NNNL}
\end{eqnarray}
Numerically, this means adding the term
\[
- \left( 40.5461 x^3 + \eo{36.3421}{88.0131} \right)
\left(\frac{\alpha_s(m_b)}{\pi}\right)^3
\]
inside the bracket in~(\ref{NNL:num}).
Of course, this is an estimate only,
because the terms $\mathcal{O}(\beta_0)$ and $\mathcal{O}(1)$
are not known.

\section{What is known about the three-loop coefficients?}
\label{S:3}

Re-expressing $\alpha_s(m_b)$, $\alpha_s(m_c)$ via $\alpha_s^{(5)}(\mu_0)$,
we can write $\eta_\Gamma$ in the form~(\ref{Intro:1step}).
From our NNL result, we obtain the $L^3$, $L^2$, and $L$ terms
in the three-loop coefficients $\eta_\Gamma^{(3)}$.
The $\beta_0^2$ part of the non-logarithmic term can also be found.
We can even make one more improvement,
and obtain the $\beta_0^2$ part of $\eta_\Gamma^{(3)}$
exactly as a function of $r$ (including power corrections).
The highest powers of $\beta_0$ in all orders of perturbation theory
for $\eta_\Gamma$ were found in~\cite{Ne:95}.
They have the form similar to~(\ref{NNL:Ehat}),
with $\beta=\beta_0\alpha_s(\mu_0)/(4\pi)$ and
\begin{eqnarray}
S_V(u) &=& 6 C_F e^{(5/3)u}
\frac{\Gamma(u)\Gamma(1-2u)}{\Gamma(3-u)}
\frac{1-u-u^2}{1+2u}
\left[ (1-2u) R_0 - R_1 \right]\,,
\nonumber\\
S_A(u) &=& 2 C_F e^{(5/3)u}
\frac{\Gamma(u)\Gamma(1-2u)}{\Gamma(3-u)}
\frac{1}{1+2u}
\nonumber\\
&&{}\times
\left[ 3 (1-2u) (1-u-u^2) R_0 - (3-u+u^2) R_1 \right]\,,
\label{3:Su}
\end{eqnarray}
where
\begin{equation}
R_0 = \cosh(Lu)\,,\quad
R_1 = \frac{\sinh\frac{(1-2u)L}{2}}{\sinh\frac{L}{2}}\,.
\label{3:R}
\end{equation}
This gives us the $\beta_0^2$ parts of $\eta_{V,A}^{(3)}$
as exact functions of $r$.

Collecting all this together, we obtain
\begin{eqnarray}
&&\frac{\eta_{V,A}^{(3)}}{C_F} =
\left[ L \coth\frac{L}{2} \left( L^2 + 12 \zeta_2 + \eo{41/6}{39/2} \right)
\right.
\nonumber\\
&&\quad\left.{}
- \eo{6}{8} L^2 - \eo{24}{32} \zeta_2 - \eo{41/3}{326/9} \right] \beta_0^2
\nonumber\\
&&{} + \left( \frac{9}{2} C_F^2 + 12 C_F T_F + \frac{16}{3} T_F^2 \right) L^3
\nonumber\\
&&{} + \left[
\left( \eo{3}{15} \frac{C_F}{2} + \eo{4}{20} \frac{T_F}{3} \right) \beta_0
+ 3 \left( 16 \zeta_2 - \frac{1}{2} \eo{23}{29} \right) C_F^2
\right.
\nonumber\\
&&\quad{}
- 3 ( 4 \zeta_2 + 1 ) C_F C_A
+ \frac{16}{3} \left( 8 \zeta_2 - \eo{8}{11} \right) C_F T_F
\nonumber\\
&&\quad\left.{}
- \frac{4}{3} ( 8 \zeta_2 - 13 ) C_A T_F
- \eo{32/3}{128/9} T_F^2 \right] L^2
\nonumber\\
&&{} + \left[
\left( - \eo{48}{176/3} \pi^2 \log 2 + \eo{12}{28} \zeta_3
+ \eo{640}{896} \frac{\zeta_2}{3} - \eo{79/12}{213/4} \right) C_F \beta_0
\right.
\nonumber\\
&&\quad{}
+ \left( \eo{24}{88/3} \pi^2 \log 2 - \eo{54}{62} \zeta_3
- \eo{226}{322} \frac{\zeta_2}{3} + \frac{1}{3} \eo{56}{122} \right)
C_A \beta_0
\nonumber\\
&&\quad{}
+ \left( \eo{22}{2/3} \zeta_2 - \frac{1}{9} \eo{355}{179} \right) T_F \beta_0
\nonumber\\
&&\quad{}
+ \left( - \eo{48}{32} \pi^2 \log 2 + 80 \zeta_4 + \eo{36}{12} \zeta_3
+ \eo{14}{-146} \zeta_2 + \frac{1}{2} \eo{241}{385} \right) C_F^2
\nonumber\\
&&\quad{}
+ \left( \eo{24}{16} \pi^2 \log 2 + 16 \zeta_4 + \eo{6}{18} \zeta_3
+ \eo{47}{215} \frac{\zeta_2}{3} - \frac{1}{12} \eo{1313}{1769} \right)
C_F C_A
\nonumber\\
&&\quad{}
+ \left( 24 \zeta_4 - 132 \zeta_3 + \frac{16}{3} \zeta_2
+ \frac{1}{2} \eo{27}{-1} \right) C_A^2
\nonumber\\
&&\quad{}
+ \left( - \eo{128/3}{256/9} \pi^2 \log 2 - \eo{0}{64} \frac{\zeta_3}{3}
+ \eo{393}{-343} \frac{\zeta_2}{9} + \eo{481/2}{4273/18} \right) C_F T_F
\nonumber\\
&&\quad{}
+ \left( \eo{64/3}{128/9} \pi^2 \log 2 - \eo{96}{256/3} \zeta_3
+ \eo{-8/3}{40} \zeta_2 - \eo{152/3}{80} \right) C_A T_F
\nonumber\\
&&\quad\left.{}
- \frac{4}{9} \left( \eo{222}{158} \zeta_2 - \eo{1123/3}{265} \right) T_F^2
\right] L
\nonumber\\
&&{} + \mathcal{O}(\beta_0)
\label{3:eta}
\end{eqnarray}
Non-logarithmic terms $\mathcal{O}(\beta_0)$ and $\mathcal{O}(1)$
are not known.
Numerically, the best estimate of the three-loop contribution
to~(\ref{Intro:1step}) is
\begin{eqnarray}
&&\left[ \left( 1.80787 + 2.89352 \frac{r}{1-r} \right) L^3
- \eo{8.14599}{9.76174} L^2 \right.
\nonumber\\
&&{} + \left( \eo{27.63733}{45.56374} + \eo{76.88814}{113.53938} \frac{r}{1-r} \right) L
\nonumber\\
&&\left.{} - \eo{76.88814}{128.55919} + \mathcal{O}(\beta_0) \right]
\left(\frac{\alpha_s^{(5)}(\mu_0)}{\pi}\right)^3
\label{3:1step}
\end{eqnarray}
(we used $\coth(L/2)=(1+r)/(1-r)$).

\section{Results and conclusion}
\label{S:Conc}

The program RunDec~\cite{CKS:00} was used for numerical estimates:
\[
r\approx0.34\,,\quad
L\approx1.08\,,\quad
\alpha_s(m_b)\approx0.216\,,\quad
x\approx1.56\,,\quad
\alpha_s^{(5)}(\mu_0)\approx0.258\,.
\]
To combine the results of the two-loop calculation~\cite{CM:97}
with our resummation,
we subtract from~\cite{CM:97} its $\mathcal{O}(r^0)$ term
and add the resummed result~(\ref{NNL:num}) instead.
This results in the corrections
\begin{equation}
\Delta\eta_V \approx 0.021\,,\quad
\Delta\eta_A \approx 0.029\,,
\label{Conc:corr1}
\end{equation}
to be added to the results of~\cite{CM:97}.
If we also add the $\beta_0^2$ term~(\ref{NNL:NNNL})
of the N$^3$L logarithmic correction,
these numbers change significantly:
\begin{equation}
\Delta\eta_V \approx -0.049\,,\quad
\Delta\eta_A \approx -0.059\,.
\label{Conc:corr2}
\end{equation}
We don't know $\mathcal{O}(\beta_0)$ and $\mathcal{O}(1)$ terms
in this correction;
if we suppose that they (taken together) don't exceed the $\mathcal{O}(\beta_0^2)$ terms,
then the theoretical error in $\eta_{V,A}$ is about $0.08$ ---
not a very encouraging estimate.

We can also discuss the situation from another side.
The $\alpha_s^3$ terms~(\ref{3:1step}) in~(\ref{Intro:1step}) are
\begin{eqnarray}
&&\Delta\eta_V \approx 0.012 - 0.018 \approx -0.005 \pm 0.04\,,
\nonumber\\
&&\Delta\eta_A \approx 0.022 - 0.035 \approx -0.013 \pm 0.07\,,
\label{Conc:corr3}
\end{eqnarray}
where the first contribution is all terms with powers of $L$
and without $\mathcal{O}(r)$ correction;
the second contribution is the $\beta_0^2$ part of the non-logarithmic term
and of the power corrections;
the $\beta_0^2$ part of the non-logarithmic term is taken as a measure
of the theoretical error.
We see that summation of powers of the ``large'' logarithm $L\approx1$
does not make much sense:
the logarithmic terms at $\mathcal{O}(\alpha_s^3)$ are largely canceled
by the non-logarithmic $\mathcal{O}(\beta_0^2)$ terms,
leaving very small corrections.

A substantial improvement of the theoretical accuracy of $\eta_{V,A}$
can only be achieved by the full three-loop calculation
(which seems very difficult),
or at least by the calculation of the $\mathcal{O}(\beta_0)$ part
of the three-loop term
(this reduced to light-quark loop insertions into the two-loop diagrams
of~\cite{CM:97}, and is also quite difficult,
because this loop can be inserted not only into a gluon propagator,
but also into a three-gluon vertex).

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\end{document}
