%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) 721-728. %hep-ph/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} TTP05-29\\ SI-HEP-2005-15\\ SFB/CPP-05-84\\ hep-ph/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,\\ D-76128 Karlsruhe, Germany }\\ {\it $^{b}$ Theoretische Physik 1, Fachbereich Physik, Universit\"at Siegen,\\ D-57068 Siegen, Germany }\\ \end{center} \vfill \begin{abstract} We include the new, five-loop, $O(\alpha_s^4)$ 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 strangeness-changing quark currents with dispersion relations. These methods include model-independent 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 finite-energy 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 quark-hadron 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 $O(\alpha_s^4)$, five-loop 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 three-resonance 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 s-quark 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 0|T\left\{j_5^{(s)}(x) j^{(s)\dagger}_5(0)\right \}|0\rangle \label{eq-corr} \eeq of the two pseudoscalar strangeness-changing quark currents, defined as the divergences of the corresponding axial-vector 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{eq-current} \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 (double-subtracted) 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)}{(s-q^2)^3}\,. \label{eq-disp} \eeq After Borel transformation one obtains: \footnote{Here we use the following normalization convention: ${\cal B}_{M^2}[1/(a-q^2)]= e^{-a/M^2}$.} \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{eq-Borel} \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 quark-hadron 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 quark-gluon 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 quark-gluon 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 Zeta-function. 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 light-quark 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 RG-invariant 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^2-9\langle\bar{u}u\rangle \langle\bar{s}s\rangle\Big) \ee is the combination of dimension-6 contributions of the quark-gluon and 4-quark 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_E-2l_M) \right) \right. \nonumber \\ & &\left. \hspace{-2cm} -\frac19 I_G\left(1 + a_s( \frac{67}{18} + 2\gamma_E-2l_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 computer-readable 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 quark-hadron 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 u|K^+(q)\rangle = i q_\mu f_K\,, \label{eq-fKax} \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{eq-fKdef} \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^2-s)\,. \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 S-wave ($J^P=0^+$) 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 ground-state 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^2-s)+\sum\limits_{i=1,2}f_{K_i}^2 m_{K_i}^4 B_{K_i}(s)\,, \label{eq-hadr_spectr} \eeq where $B_{K_i}(s)$ are the finite-width (Breit-Wigner type) replacements of the $\delta$-function in the spectral density for $K_{1,2}$: \beq \delta(m_{K_{i}}^2-s)\to B_{K_i}(s)= \frac{1}{\pi}\left (\frac{\Gamma_{K_i}m_{K_i}} {(s-m_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 20-30 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 two-particle 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 $\langle 0| j_5^{(s)} |K \rangle \langle K| K^*\pi\rangle \langle K^*\pi|K_1\rangle \langle K_1|j_5^{(s)}|0\rangle$. The analysis of the light-quark 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{eq-hadr_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{eq-hadr_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 four-loop approximation and employ the numerical program RunDec described in \cite{RunDec}. The reference value for the quark-gluon 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 (non-lattice) 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