%Title: Strange Quark Mass from Pseudoscalar Sum Rule with O(alpha_s^4) Accuracy
%Author: K.G. Chetyrkin and A. Khodjamirian
%Published: Eur. Phys. J. C46 (2006) 721728.
%hepph/0512295
\documentclass[12pt]{article}
\usepackage{a4wide}
%\usepackage{axodraw}
%\usepackage{floatflt}
\usepackage{citesort}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage[center]{subfigure}
\newcommand{\dsp}{\displaystyle}
\newcommand{\ba}{\begin{eqnarray}}
\newcommand{\ea}{\end{eqnarray}}
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\as}{a_s}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\nnb{\nonumber}
\def\ga{\left(}
\def\dr{\right)}
\def\aga{\left\{}
\def\adr{\right\}}
\def\lb{\lbrack}
\def\rb{\rbrack}
\def\rar{\rightarrow}
\def\lrar{\Longrightarrow}
\def\nnb{\nonumber}
\def\la{\langle}
\def\ra{\rangle}
\def\nin{\noindent}
\def\ba{\begin{array}}
\def\ea{\end{array}}
\newcommand{\BreakO}{ \nonumber \\ &{}& }
\newcommand{\Break}{ \right. \nonumber \\ &{}& \left. }
\newcommand{\BreakI}{ \right. \nonumber \\ &{}& \left. }
\newcommand{\BreakII}{ \right. \right. \nonumber \\ &{}& \left.\left. }
\newcommand{\BreakIII}{ \right. \right. \right. \nonumber \\ &{}& \left.\left.\left. }
\newcommand{\BreakIV}{
\right. \right. \right.\right. \nonumber \\ &{}& \left. \left.\left.\left.
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\begin{titlepage}
\begin{flushright}
TTP0529\\
SIHEP200515\\
SFB/CPP0584\\
hepph/05122950
\end{flushright}
\vfill
\begin{center}
{\Large\bf
Strange Quark Mass from Pseudoscalar Sum Rule
with $O(\alpha_s^4)$ Accuracy }\\[2cm]
{\large K.G. Chetyrkin}\,$^{a,\!}$\footnote{{\small Permanent address:
Institute for Nuclear Research, Russian Academy of Sciences,
Moscow 117312, Russia}}
{\large and A.~Khodjamirian}\,$^b$ \\[3mm]
{\it $^{a}$ Institut f\"ur Theoretische Teilchenphysik, Universit\"at
Karlsruhe,\\ D76128 Karlsruhe, Germany }\\
{\it $^{b}$ Theoretische Physik 1,
Fachbereich Physik, Universit\"at Siegen,\\
D57068 Siegen, Germany }\\
\end{center}
\vfill
\begin{abstract}
We include the new, fiveloop, $O(\alpha_s^4)$ correction
into the QCD sum rule
used for the $s$quark mass determination.
The pseudoscalar Borel sum rule is taken as a study case.
The OPE for the correlation function with
$N^4LO$, $O(\alpha_s^4)$ accuracy in the perturbative part,
and with dimension $d\leq 6$ operators reveals
a good convergence.
We observe a significant improvement
of stability of the sum rule with respect to
the variation of the renormalization scale
after including the $O(\alpha_s^4)$ correction.
We obtain the interval
$\overline{m}_s(2 ~\mbox{GeV})=105\pm 6 \pm 7 $ MeV,
which exhibits about $ 2$ MeV increase of the central
value, if the $O(\alpha_s^4)$ terms are removed.
\end{abstract}
\vfill
\end{titlepage}
\section{Introduction}
A precise determination of the strange quark mass $m_s$
is extremely important for various tests of Standard Model.
A reach variety of approaches
is used to evaluate this fundamental parameter
in QCD. Recently,
the first unquenched
lattice QCD determinations became available
\cite{lat_Aubin04,lat_DellaMorte05,lat_Ishikawa05,lat_Gockeler05,lat_Becirevic05,lat_Mason05}.
In addition, ChPT provides rather accurate ratios of
strange and nonstrange quark masses \cite{Leutw,chpt}.
Furthermore, one evaluates $m_s$,
combining the operator product expansion (OPE)
of various correlation functions for strangenesschanging
quark currents with dispersion relations. These methods include modelindependent bounds \cite{bounds},
QCD analyses of hadronic $\tau$ decays (see
\cite{GJPPS04,BCKtau05,Maltman:2004sk,Gorbunov:2004wy,Narison:2005zg}
for the latest results), as well as different versions of
QCD sum rules \cite{SVZ}
and related finiteenergy sum rules (FESR) \cite{Chetyrkin:1978ta}.
The most recent sum rule determinations of $m_s$
in the channels of scalar, pseudoscalar and vector currents are presented in
\cite{JOP02,KM01,EJS03}, respectively, references to earlier analyses can
be found in reviews \cite{CK,Paver} (see also \cite{Narison05}).
The estimated accuracy of these results is 15 30\%.
To achieve a better
precision, one has to calculate higher orders in OPE
of the correlation functions and gain
a better control over potentially important
nonperturbative corrections beyond OPE (the so called
''direct instantons''). Furthermore, more
accurate data for the inputs in hadronic spectral
functions and a better assessment of the quarkhadron
duality are needed.
In this paper we concentrate on the QCD sum rules used
to evaluate the strange quark mass and make one further
step to improve the accuracy of this determination by including
the N$^4$LO perturbative QCD corrections of $O(\alpha_s^4)$
into the sum rule. The $O(\alpha_s^4)$, fiveloop
contribution has recently been calculated for the correlator
of the scalar quark currents in \cite{as4} and can be equally
well used for both scalar and pseudoscalar sum rules.
As a study case, we choose the pseudoscalar version
of the standard Borel sum rule. For the
hadronic spectral function we employ the threeresonance ansatz worked out in \cite{KM01}.
We find that both OPE for the correlation function
and the resulting sum rule
reveal a good numerical convergence in powers of $\alpha_s$.
The new $O(\alpha_s^4)$ correction to the sum rule
has a naturally small influence,
resulting in about 2 MeV decrease of the
squark mass $\overline{m}_s$ (in $\overline{MS}$ scheme)
determined with $O(\alpha_s^3)$ accuracy.
Importantly, after including the $O(\alpha_s^4)$
correction, we observe a significant improvement in the
stability of the extracted
value of $\overline{m}_s$
with respect to the renormalization scale variation
in the sum rule.
With $O(\alpha_s^4)$ accuracy we obtain the interval
\beq
\overline{m}_s(\mbox{2 GeV})=
\left(105\pm 6\Big_{param} \pm 7 \Big_{hadr} \right)\mbox{MeV}\,,
\label{interval}
\eeq
where the estimated uncertainties from the
sum rule parameters and hadronic inputs are shown separately and will be
explained below.
In what follows,
after a brief recapitulation of the pseudoscalar Borel sum rule
in Sect.~2, we present in Sect.~3 the QCD OPE
expressions for the underlying correlation function,
including the new $O(\alpha_s^4)$ terms in the
perturbative part. The Borel transform and the
imaginary part of the correlation function are
also given. In Sect.~4 we turn to the numerical analysis
of the sum rule and obtain the interval (\ref{interval}).
Sect.~5 contains the concluding discussion.
\section{ Pseudoscalar Sum Rule }
We consider the correlation function:
\beq
\Pi^{(5)}(q^2)=i\int d^4x ~e^{i q\cdot x}
\langle 0T\left\{j_5^{(s)}(x) j^{(s)\dagger}_5(0)\right \}0\rangle
\label{eqcorr}
\eeq
of the two pseudoscalar strangenesschanging quark
currents, defined as the divergences of
the corresponding axialvector currents:
\beq
j_5^{(s)}=\partial^\mu (\bar{s}\gamma_\mu \gamma_5 q) =
(m_s+m_q)\bar{s}i\gamma_5 q.
\label{eqcurrent}
\eeq
For definiteness, the light quark $q=u$ is taken.
The Borel sum rule is obtained following the
standard SVZ method \cite{SVZ} and is based on the (doublesubtracted)
dispersion relation for $\Pi^{(5)}(q^2)$.
This relation is more conveniently written in a form of the
second derivative:
\beq
\Pi^{(5)''}(q^2)\equiv \frac{d^2}{d(q^2)^2}\Pi^{(5)}(q^2)=
\frac{2}{\pi}\int\limits_0^{\infty}ds \frac{\mbox{Im}\,\Pi^{(5)}(s)}{(sq^2)^3}\,.
\label{eqdisp}
\eeq
After Borel transformation one obtains: \footnote{Here we use
the following normalization convention:
${\cal B}_{M^2}[1/(aq^2)]= e^{a/M^2}$.}
\beq
\Pi^{(5)''}(M^2)\equiv {\cal B}_{M^2}\left [\Pi^{(5)''}(q^2)\right ]=\frac{1}{\pi M^4}\int
\limits_0^{\infty}ds\, e^{s/M^2}\mbox{Im}\,\Pi^{(5)}(s)\,.
\label{eqBorel}
\eeq
The l.h.s. of the above relation is calculated in QCD
at large $ M^ 2 \gg \Lambda_{QCD}^2$ in a form of OPE
(perturbative and condensate expansion), in
powers of $\alpha_s$ and $m_s/M$, and
up to a certain dimension of vacuum condensates. In r.h.s. the hadronic
spectral density $\rho^{(5)}_{hadr}(s)=(1/\pi)\mbox{Im}\,\Pi^{(5)}(s)$
is substituted. At $ss_0$ is approximated using quarkhadron duality,
$\rho^{(5)}_{hadr}(s)\simeq \rho^{(5)}_{OPE}(s)$,
with the spectral function calculated from OPE:
$ \rho^{(5)}_{OPE}(s)=(1/\pi) \mbox{Im}\,[\Pi^{(5)}(s)]_{OPE}$.
The final form of the sum rule is:
\beq
M^4[\Pi^{(5)''}(M^2)]_{OPE}=
\int \limits_0^{s_0}ds\, e^{s/M^2}\rho^{(5)}_{hadr}(s) +
\int\limits_{s_0}^{\infty}ds\, e^{s/M^2}\rho^{(5)}_{OPE}(s)\,.
\label{SR}
\eeq
In the following we discuss both parts of this equation
in detail.
\section{OPE results to $O(\alpha_s^4)$ }
In this section we present the expressions for $[\Pi^{(5)''}(q^2)]_{OPE}$
and, correspondingly, for $[\Pi^{(5)''}(M^2)]_{OPE}$ and $ \rho^{(5)}_{OPE}(s)$
determining the QCD input in the sum rule (\ref{SR}).
The OPE for $[\Pi^{(5)''}(q^2)]_{OPE}$ goes over powers of
$(1/q^2)^{d+2}$ ordered by the dimension $d=0,2,4,6$. The OPE terms
with $d>6$ are neglected, while already the $d=6$ contribution
is very small in the working region of the variables $Q^2$ and $M^2$.
The $d=0,2$ terms of OPE originate from the
perturbative part of the correlation function. The expansion
in quarkgluon coupling up to four loops, that is, up to $O(\alpha_s^3)$,
can be taken from \cite{Gorishnii,Chetyrkin:1996sr,CPS96}. The new $O(\alpha_s^4)$
terms are obtained in \cite{as4}. Putting them together, we obtain:
\beq
[\Pi^{(5)''}(Q^2)]_{OPE}^{(d=0,2)}
=\frac{3(m_s+m_u)^2}{8\pi^2Q^2}\Bigg\{1+
\sum_i\bar{d}_{0,i}\, \as^i
2\frac{m_s^2}{Q^2}
\left(1+\sum_i\bar{d}_{2,i} \, \as^i
\right)
\Bigg\}
\label{piOPE}
{},
\eeq
where $Q^2=q^2$, and the coefficients multiplying the
powers of the quarkgluon coupling
$a_s=\alpha_s(\mu)/\pi$ are
\begin{eqnarray}
{\bar{d}_{0,1} =
\frac{11}{3}
2 \,l_{Q}
%zero == 0
{},
}
\ \
\bar{d}_{0,2} =
\frac{5071}{144}
\frac{35}{2} \,\zeta_{3}
\frac{139}{6} \,l_{Q}
+\frac{17}{4} \,l_Q^2
{},
\label{d01_d02}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\bar{d}_{0,3} =
\frac{1995097}{5184}
\frac{1}{36} \pi^4
\frac{65869}{216} \,\zeta_{3}
+\frac{715}{12} \,\zeta_{5}
\frac{2720}{9} \,l_{Q}
+\frac{475}{4} \,\zeta_{3} \,l_{Q}
+\frac{695}{8} \,l_Q^2
\frac{221}{24} \,l_Q^3
%zero == 0
{},
\label{d03}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
{\bar{d}_{0,4} = }
&{}&
\frac{2361295759}{497664}
\frac{2915}{10368} \pi^4
\frac{25214831}{5184} \,\zeta_{3}
+\frac{192155}{216} \,\zeta_3^2
+\frac{59875}{108} \,\zeta_{5}
\frac{625}{48} \,\zeta_{6}
%zero == 0
\nonumber\\
&{}& \frac{52255}{256} \,\zeta_{7}
+l_{Q}\left[
\frac{43647875}{10368}
+\frac{1}{18} \pi^4
+\frac{864685}{288} \,\zeta_{3}
\frac{24025}{48} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& \,l_Q^2
\left[
\frac{1778273}{1152}
\frac{16785}{32} \,\zeta_{3}
%zero == 0
\right]
{+} \,l_Q^3
\left[
\frac{79333}{288}\right]
{+} \,l_Q^4
\left[
\frac{7735}{384}\right]
{},
\label{d04}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\bar{d}_{2,1} =
\frac{28}{3}
4 \,l_{Q}
{},
\ \
\bar{d}_{2,2} =
\frac{8557}{72}
\frac{77}{3} \,\zeta_{3}
%zero == 0
\frac{147}{2}
\,l_{Q}
%
+ \,\frac{25}{2}
\,l_Q^2
{}\,,
\label{d21+d22}
\end{eqnarray}
%zero == 0
%zero == 0
including the new result for $\bar{d}_{0,4}$.
Here $l_Q= \log\frac{Q^2}{\mu^2}$, and $\zeta_n\equiv \zeta(n)$ is the Riemann's Zetafunction. The coupling $a_s$
and the quark masses $m_s$ and $m_u$ are
all taken in $\overline{MS}$ scheme at the renormalization scale $\mu$.
We have neglected the lightquark mass $m_u$,
except in the overall factors. Note also that in the
subleading $d=2$, $O(m_s^4)$ terms of the above expansion, the currently
achieved $O(\alpha_s^2)$ accuracy is quite sufficient.
The contributions with $d=4,6$ in the correlation function
originate both from nonperturbative (condensate) terms and from $O(m_s^6)$
corrections, and we use the known expressions \cite{JM,CPS96}
\bea
&{}&[\Pi^{(5)''}(q^2)]_{OPE}^{(d=4,6)}=
\frac{(m_s+m_u)^2}{Q^6}\left\{2m_s\langle \bar uu\rangle
\left(1 + a_s(\frac{23}{3}2l_Q)\right)
\right.
\nonumber \\&{} &\left.
\frac19 I_G\left(1 + a_s(\frac{121}{18}  2l_Q)\right)
%\\&{} &\left.
+ I_s\left(1+a_s(\frac{64}{9}2l_Q)\right)
\nonumber \right. \\&{} &\left.
\frac{3}{7\pi^2}m_s^4\left(\frac{1}{a_s}+
\frac{155}{24}  \frac{15}{4}l_Q \right)
+\frac{I_6}{Q^2} \right\}\,,
\label{piOPEnp}
\eea
where
\be
I_s=m_s\langle \bar{s}s\rangle+\frac{3}{7\pi^2}
m_s^4\left(\frac{1}{a_s}\frac{53}{24}\right )
\ee
and
\be
I_G=\frac{9}{4}\langle \frac{\alpha_s}{\pi}
G^2\rangle\left (1+\frac{16}{9}a_s\right)+
4a_s\left(1+\frac{91}{24}a_s\right)m_s\langle \bar{s}s\rangle+\frac{3}{4\pi^2}\left(1+\frac{4}{3}a_s\right)
m_s^4
\ee
are the vacuum expectation values of two RGinvariant combinations of
dimension 4 containing quark and gluon condensate densities
(for details and explanation see
\cite{Spiridonov:1988md,Chetyrkin:1994qu,JM,CPS96}). Finally,
\be
I_6= 3m_s\langle\bar{u}uG\rangle
\frac{32}9\pi^2a_s\Big (\langle\bar{u}u\rangle^2
+\langle\bar{s}s\rangle^29\langle\bar{u}u\rangle
\langle\bar{s}s\rangle\Big)
\ee
is the combination of dimension6 contributions
of the quarkgluon and 4quark condensates
(the vacuum saturation is assumed for the latter).
The Borel transform of Eqs.~(\ref{piOPE}) and (\ref{piOPEnp})
is given by
\beq
[\Pi^{(5)''}(M^2)]_{OPE}^{(d=0,2)}=
\frac{3(m_s+m_u)^2}{8\pi^2}\Bigg\{1+
\sum_i\bar{b}_{0,i}\, \as^i

2
\frac{m_s^2}{M^2}
\left(
1
+
\sum_i\bar{b}_{2,i} \, \as^i
\right)
\Bigg \}
\label{BpiOPE}
{},
\eeq
where $l_M = \log\frac{M^2}{\mu^2}$ and the coefficients are
\begin{eqnarray}
{\bar{b}_{0,1} = }
\frac{11}{3}
+2 \, \gamma_E
2 \,l_{M}
%zero == 0
{},
\label{b01}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
{\bar{b}_{0,2} = }
\frac{5071}{144}
+\frac{139}{6} \, \gamma_E
+\frac{17}{4} \,\gamma_E^2
\frac{17}{24} \pi^2
\frac{35}{2} \,\zeta_{3}
\frac{139}{6} \,l_{M}
\frac{17}{2} \, \gamma_E \,l_{M}
+\frac{17}{4} \,l_M^2
%zero == 0
{},
\label{b02}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\bar{b}_{0,3} &=&
\frac{1995097}{5184}
+\frac{2720}{9} \, \gamma_E
+\frac{695}{8} \,\gamma_E^2
+\frac{221}{24} \,\gamma_E^3
\frac{695}{48} \pi^2
\frac{221}{48} \, \gamma_E \pi^2
\nonumber
\\
&{}&
\phantom{+}
\frac{1}{36} \pi^4
\frac{61891}{216} \,\zeta_{3}
\frac{475}{4} \, \gamma_E \,\zeta_{3}
+\frac{715}{12} \,\zeta_{5}
%zero == 0
\nonumber\\
&{+}& \,l_{M}
\left[
\frac{2720}{9}
\frac{695}{4} \, \gamma_E
\frac{221}{8} \,\gamma_E^2
+\frac{221}{48} \pi^2
+\frac{475}{4} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}& \,l_M^2
\left[
\frac{695}{8}
+\frac{221}{8} \, \gamma_E
%zero == 0
\right]
\frac{221}{24} \,l_M^3
{},
\label{b03}
\end{eqnarray}
%zero == 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
b_{0,4} &=&
\left.
\frac{2361295759}{497664}
+\frac{43647875}{10368} \, \gamma_E
+\frac{1778273}{1152} \,\gamma_E^2
+\frac{79333}{288} \,\gamma_E^3
+\frac{7735}{384} \,\gamma_E^4
\frac{1778273}{6912} \pi^2
\BreakI
\phantom{+}
\frac{79333}{576} \, \gamma_E \pi^2
\frac{7735}{384} \,\gamma_E^2 \pi^2
+\frac{2263}{41472} \pi^4
\frac{1}{18} \, \gamma_E \pi^4
\frac{22358843}{5184} \,\zeta_{3}
\frac{818275}{288} \, \gamma_E \,\zeta_{3}
\BreakI
\phantom{+}
\frac{16785}{32} \,\gamma_E^2 \,\zeta_{3}
+\frac{5595}{64} \pi^2 \,\zeta_{3}
+\frac{192155}{216} \,\zeta_3^2
+\frac{59875}{108} \,\zeta_{5}
+\frac{24025}{48} \, \gamma_E \,\zeta_{5}
\frac{625}{48} \,\zeta_{6}
\frac{52255}{256} \,\zeta_{7}
%zero == 0
\right.
\nonumber\\
&{+}& \,l_{M}
\left[
\frac{43647875}{10368}
\frac{1778273}{576} \, \gamma_E
\frac{79333}{96} \,\gamma_E^2
\frac{7735}{96} \,\gamma_E^3
+\frac{79333}{576} \pi^2
+\frac{7735}{192} \, \gamma_E \pi^2
\BreakI
\phantom{+ \,l_{M}}
+\frac{1}{18} \pi^4
+\frac{818275}{288} \,\zeta_{3}
+\frac{16785}{16} \, \gamma_E \,\zeta_{3}
\frac{24025}{48} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& \,l_M^2
\left[
\frac{1778273}{1152}
+\frac{79333}{96} \, \gamma_E
+\frac{7735}{64} \,\gamma_E^2
\frac{7735}{384} \pi^2
\frac{16785}{32} \,\zeta_{3}
%zero == 0
\right]
\nonumber\\
&{+}& \,l_M^3
\left[
\frac{79333}{288}
\frac{7735}{96} \, \gamma_E
%zero == 0
\right]
+ \,l_M^4
\left[
\frac{7735}{384}
\right]
{},
\label{b04}
\end{eqnarray}
%zero == 0
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
{\bar{b}_{2,1} = }
\frac{16}{3}
+4 \, \gamma_E
4 \,l_{M}
%zero == 0
{},
\label{b21}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\bar{b}_{2,2} =
\frac{5065}{72}
+\frac{97}{2} \, \gamma_E
+\frac{25}{2} \,\gamma_E^2
\frac{25}{12} \pi^2
\frac{77}{3} \,\zeta_{3}
\frac{97}{2} \,l_{M}
25 \, \gamma_E \,l_{M}
+\frac{25}{2} \,l_M^2
%zero == 0
{},
\label{b22}
\end{eqnarray}
%zero == 0and
and, respectively,
\bea
[\Pi^{(5)''}(M^2)]_{OPE}^{(d=4,6)} &=&
\frac{(m_s+m_u)^2}{2M^4}\left\{2m_s\langle \bar uu\rangle
\left(1 + a_s(\frac{14}{3}+2\gamma_E2l_M)
\right)
\right.
\nonumber
\\
& &\left.
\hspace{2cm} \frac19 I_G\left(1 + a_s(
\frac{67}{18} + 2\gamma_E2l_M)\right)
+ I_s\left(1+ a_s(\frac{37}{9}+2\gamma_E
2l_M)\right)
\right.\nonumber\\
& &\left.
\hspace{2cm} \frac{3}{7\pi^2}m_s^4\left(\frac{1}{a_s}+
\frac{5}{6} + \frac{15}{4}\gamma_E\frac{15}{4}l_M
\right)+\frac{I_6}{3M^2}\right\}
\,.\label{BpiOPEnp}
\eea
In addition, we need the imaginary part of the
correlation function calculated
with the same $\alpha_s^4$ accuracy as Eqs.~(\ref{BpiOPE})
and (\ref{BpiOPEnp}):
\bea
\rho^{(5)}_{OPE}(s) &=&\frac{1}{\pi}\mbox{Im}\Pi^{(5)}(s)=
\frac{3(m_s+m_u)^2}{8\pi^2}\,s\,\Bigg\{1+
\sum_i\tilde{r}_{0,i}\, \as^i

2 \frac{m_s^2}{s}
\left(
1+\sum_i\tilde{r}_{2,i} \, \as^i
\right)
\Bigg \}
\nonumber\\
%\hspace*{1.5cm}
&+&
\frac{m_s^2(s)}{s}\left\{ \frac{45}{56\pi^2}m_s^4(s)
+
2a_s(s)m_s\langle \bar uu\rangle +
\frac{a_s(s)}{9}I_G  a_s(s)I_s\right\}
%\nonumber
\,,
\label{ImpiOPE}
\eea
where $l_s = \log\frac{s}{\mu^2}$ and
\begin{eqnarray}
\tilde{r}_{0,1} =
\frac{17}{3}
2 \, l_s,
\ \
{\tilde{r}_{0,2} = }
\frac{9631}{144}
\frac{17}{12} \pi^2
\frac{35}{2} \,\zeta_{3}
\frac{95}{3} \, l_s
+\frac{17}{4} \,l_s^2
%zero == 0
{},
\label{r02}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\tilde{r}_{0,3} =
&{}&
\frac{4748953}{5184}
\frac{229}{6} \pi^2
\frac{1}{36} \pi^4
\frac{91519}{216} \,\zeta_{3}
+\frac{715}{12} \,\zeta_{5}
\frac{4781}{9} \, l_s
+\frac{221}{24} \pi^2 \, l_s
\nonumber
\\
&{}&
\phantom{+}
+\frac{475}{4} \,\zeta_{3} \, l_s
+\frac{229}{2} \,l_s^2
\frac{221}{24} \,l_s^3
%zero == 0
{},
\label{r03}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\tilde{r}_{0,4} =
&{}&
\frac{7055935615}{497664}
\frac{3008729}{3456} \pi^2
+\frac{19139}{5184} \pi^4
\frac{46217501}{5184} \,\zeta_{3}
+\frac{5595}{32} \pi^2 \,\zeta_{3}
\nonumber
\\
&{}&
\phantom{+}
+\frac{192155}{216} \,\zeta_3^2
+\frac{455725}{432} \,\zeta_{5}
\frac{625}{48} \,\zeta_{6}
\frac{52255}{256} \,\zeta_{7}
%zero == 0
\nonumber\\
&{+}& \, l_s
\left[
\frac{97804997}{10368}
+\frac{51269}{144} \pi^2
+\frac{1}{18} \pi^4
+\frac{1166815}{288} \,\zeta_{3}
\frac{24025}{48} \,\zeta_{5}
%zero == 0
\right]
\nonumber\\
&{+}& \,l_s^2
\left[
\frac{3008729}{1152}
\frac{7735}{192} \pi^2
\frac{16785}{32} \,\zeta_{3}
%zero == 0
\right]
%+ \,l_s^3
\frac{51269}{144}\,l_s^3
{+}
% \,l_s^4
\frac{7735}{384}\,l_s^4
{},
\label{r04}
\end{eqnarray}
%zero == 0
\begin{eqnarray}
\tilde{r}_{2,1} =
\frac{16}{3}
4 \, l_s
{},
\ \
\tilde{r}_{2,2} =
\frac{5065}{72}
\frac{25}{6} \pi^2
\frac{77}{3} \,\zeta_{3}
\frac{97}{2} \, l_s
+\frac{25}{2} \,l_s^2
%zero == 0
{}.
\label{r21_22}
\end{eqnarray}
%zero == 0
The OPE expressions are valid at sufficiently large
$Q^2\gg \Lambda_{QCD}^2$ or, correspondingly, at large
$M^2$. It is well known that in the spin zero (scalar and pseudoscalar)
channels the breakdown of OPE is expected to occur at
relatively large
$Q^2 \simeq $ 1 GeV$^2$, due to the presence of nonperturbative
vacuum effects which are beyond the local
condensate expansion \cite{inst,NSVZ}.
Models of the correlation function based on instanton ensembles,
such as the instanton liquid model (ILM) \cite{ILM,ILM1}
allow to penetrate to smaller $Q^2$. A remedy used in previous analyses of
pseudoscalar sum rules is to
add to the OPE series an instanton correction calculated in ILM.
As realized, e.g., in \cite{KM01}, at sufficiently large $M^2$,
practically already at $M^2>$ 2 GeV$^2$ the ILM correction is small,
hence we will avoid it by choosing 2 GeV$^2$ as a lower limit of
the Borel mass.
For the reader's convenience, the lengthy
coefficients appearing
in eqs.~(\ref{piOPE}), (\ref{BpiOPE}) and (\ref{ImpiOPE})
are made available (in computerreadable form) in
\cite{comp}.
\section{Hadronic spectral density and the sum rule}
The spectral function $\rho^{(5)}_{hadr}(s)$
in Eq.~(\ref{SR}) is a positive definite sum of all hadronic states
with strangeness and $J^P=0^$, located
below the threshold $\sqrt{s_0}$, above which
$\rho^{(5)}_{hadr}(s)$ is approximated by the OPE spectral density.
Clearly, the larger is $s_0$,
the smaller is the sensitivity of the sum rule
to this quarkhadron duality ansatz.
The lowest hadronic state is the kaon.
Using the standard definition of the kaon decay constant
\beq
\langle 0 \bar{s}\gamma_\mu\gamma_5 uK^+(q)\rangle
= i q_\mu f_K\,,
\label{eqfKax}
\eeq
one obtains the relevant hadronic matrix element of the pseudoscalar
current:
\beq
\langle 0 j_5^{(s)}K^+(q)\rangle = f_K m_K^2\,,
\label{eqfKdef}
\eeq
so that the kaon contribution to the hadronic
spectral density reads:
\beq
\rho^{(5)}_{K}(s)= f_K^2m_K^4\delta(m_K^2s)\,.
\eeq
The two heavier pseudoscalar resonances \cite{PDG}
are $K_1 = K(1460)$ and $K_2=K(1830)$ with the masses
$m_{K_1}= 1460$ MeV and $m_{K_2}= 1830$ MeV and
total widths $\Gamma_{K_1}= 260$ MeV
and $\Gamma_{K_2}= 250 $ MeV, respectively.
These resonances are not yet well established, in particular,
no experimental errors are attributed to their masses and widths.
In any case, it seems plausible
that the hadronic spectral density in the pseudoscalar
channel with strangeness is dominated by the kaon
and $K_{1,2}$ resonances,
making this channel less complicated than
the scalar channel where the strong $K\pi$
scattering in Swave ($J^P=0^+$)
demands a dedicated analysis (see e.g., \cite{CDNP97,JOP02}).
A detailed analysis of the hadronic part in the
pseudoscalar sum rules (in both FESR and Borel
versions) is presented in \cite{KM01},
employing the hadronic spectral density where the
contributions of two resonances
$K_{1,2}$ with finite widths are simply added to the groundstate
term of the kaon.
Here we adopt the same ansatz for the
hadronic spectral density
\footnote{For a different hadronic ansatz
including $K^* \pi$ state explicitly, see \cite{DPS97}.} in the sum rule
(\ref{SR}):
\beq
\rho^{(5)}_{hadr}(s)=f_K^2m_K^4\delta(m_K^2s)+\sum\limits_{i=1,2}f_{K_i}^2 m_{K_i}^4 B_{K_i}(s)\,,
\label{eqhadr_spectr}
\eeq
where $B_{K_i}(s)$ are the
finitewidth (BreitWigner type)
replacements of the $\delta$function
in the spectral density for $K_{1,2}$:
\beq
\delta(m_{K_{i}}^2s)\to B_{K_i}(s)=
\frac{1}{\pi}\left (\frac{\Gamma_{K_i}m_{K_i}}
{(sm_K^2)^2+(\Gamma_{K_i}m_{K_i})^2}\right)\, .
\label{BW}
\eeq
In \cite{KM01} using FESR,
the decay constants $f_{K_1}$ and $f_{K_2}$ of $K_1$ and $K_2$ resonances
(defined similarly to $f_K$) were fitted. As anticipated from
ChPT, small values, in the ballpark of 2030 MeV
for both $f_{K_1}$ and $f_{K_2}$
were obtained. On the other hand, due to
the large mass multiplying these constants, the effects
of $K_{1}$ and $K_2$ are quite noticeable in the hadronic part of the sum rule,
hence, one has to avoid too large values of $M^2$.
We will use the estimates of $f_{K_1,K_2}$ from \cite{KM01}
as hadronic inputs in our numerical analysis of (\ref{SR}).
Further improvements of the hadronic ansatz are possible, but they
our beyond our scope here.
In particular, it seems important
to investigate the role of multiparticle states
in the hadronic spectral function,
starting from the twoparticle states $K^* \pi, K\rho$.
In \cite{KM01} it is assumed that multiparticle effects are
at least partially
taken into account in the finite widths of $K_{1,2}$.
One usually neglects the possible contributions
of the nondiagonal transitions to the hadronic
spectral function, e.g., intermediate states
of the type $\langle 0 j_5^{(s)} K \rangle \langle K K^*\pi\rangle
\langle K^*\piK_1\rangle \langle K_1j_5^{(s)}0\rangle $.
The analysis of the lightquark
vector channel ($J^{P}=1^$) without and with strangeness
(see e.g.,\cite{BKK}) indicates that the effects of mixing between
separate resonances via intermediate multiparticle states
could be noticeable. Here, adopting the ansatz
(\ref{eqhadr_spectr}) we tacitly assume that the
total widths of $K_{1,2}$ account for the dominant contributions
of multiparticle states. In order to estimate the influence
of this effect, we will also consider a version of the hadronic spectral density
(\ref{eqhadr_spectr}) with the total widths of $K_{1,2}$ set to zero,
interpreting the difference
of the result with and without the widths as a rough estimate
of the uncertainty due to multiparticle hadronic states.
\section{Inputs and numerical results}
For the running of the strong coupling $a_s$ and of
the quark masses in $\overline{MS}$
scheme we use the fourloop approximation and employ
the numerical program RunDec described in \cite{RunDec}.
The reference value for the quarkgluon coupling
is taken as $\alpha_s(m_Z)= 0.1187$ \cite{PDG}.
The alternative choice $\alpha_s(m_\tau)= 0.334$
\cite{PDG} produces a small difference
which we include into the overall counting
of uncertainties.
We do not attempt to fit the $u$ and $d$quark masses
from the analogous sum rules, and simply take
the current (nonlattice) intervals from \cite{PDG}:
$\overline{m}_u (2~\mbox{GeV})=(1.5 ~\mbox{}~ 5.0) ~\mbox{MeV}$, $\overline{m}_d (2~\mbox{GeV})=
(5.0 ~\mbox{}~ 9.0) ~\mbox{MeV}$.
The renormalization scale in our numerical calculation is taken as $\mu=M$,
reflecting the average virtuality
of perturbative quarks and gluons in the correlator.
In order to study the scale dependence we also vary the scale
within $M^2/2 <\mu^2 < 2M^2$. The window of Borel parameter
is taken as in \cite{KM01}, $2