%Title: Quadratic Mass Corrections of Order O(\alpha_s^3 m_q^2/s) to the Decay Rate of Z- and W-Bosons %Author: K.G. Chetyrkin and J.H. Kuehn %Published: Phys.Lett. 406B (1997) 102-109. % version as of 10.03.1997 \documentstyle{elsart} \voffset -1cm \renewcommand{\arraystretch}{2} \setlength\topmargin{-0.5cm} \setlength\textheight{23.0cm} \setlength\textwidth{16cm} \protect\setlength\oddsidemargin{-0.25cm} \protect\setlength\evensidemargin{0.3cm} \headsep 30pt \newcommand{\lgm}{{\,\rm ln }} \newcommand{\Break}{ \right. \nonumber \\ &{}& \left. } \newcommand{\ice}{\relax} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\prd}{\partial} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\api}{\frac{\alpha_s}{\pi}} \newcommand{\apis}{\frac{\dsp\alpha_s(s)}{\dsp \pi}} \newcommand{\apimu}{\frac{\dsp\alpha_s(\mu)}{\dsp \pi}} \newcommand{\apim}{\frac{\dsp\alpha_s(m)}{\dsp \pi}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\ds}{\displaystyle} \newcommand{\as}{\alpha_s} \newcommand{\gm}{\gamma_m} \newcommand{\G}{\Gamma} \newcommand{\g}{\gamma} \newcommand{\gaam}{\gamma^{AA}_m} \newcommand{\gaaq}{\gamma^{AA}_q} \newcommand{\gvvm}{\gamma^{VV}_m} \newcommand{\gvvq}{\gamma^{VV}_q} \newcommand{\gssm}{\gamma^{SS}_m} \newcommand{\gssq}{\gamma^{SS}_q} \newcommand{\gppm}{\gamma^{PP}_m} \newcommand{\gppq}{\gamma^{PP}_q} \newcommand{\dmu}{\mu^2\frac{d}{d\mu^2}} \newcommand{\msbar}{\overline{\mbox{MS}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\dsp}{\displaystyle} \newcommand{\EQN}{\label} \newcommand{\ovl}{\overline} \begin{document} \noindent \begin{frontmatter} \title{ Quadratic Mass Corrections of Order ${\cal O}(\alpha_s^3 m_q^2/s)$ to the Decay Rate of $Z$- and $W$- Bosons } \author[Moscow,Munchen]{K.G.~Chetyrkin,} \author[Karlsruhe]{J. H. K\"uhn} \address[Moscow]{Institute for Nuclear Research, Russian Academy of Sciences, 60th October Anniversary Prospect 7a, Moscow 117312, Russia } \address[Munchen]{% Max-Planck-Institut f\"ur Physik, Werner-Heisenberg-Institut, F\"ohringer Ring 6, 80805 Munich, Germany} \address[Karlsruhe]{ Institut f\"ur Theoretische Teilchenphysik, Universit\"at Karlsruhe D-76128 Karlsruhe, Germany } \begin{abstract} We analytically compute quadratic mass corrections of order ${\cal O}(\alpha_s^3 m_q^2/s)$ to the absorptive part of the (non-diagonal) correlator of two axial vector currents. This allows us to find the correction of order ${\cal O}(\alpha_s^3 m_q^2/M^2_W)$ to $\G(W \to \mbox{hadrons})$ as well as similar corrections to $\G(Z \to \mbox{hadrons})$. \end{abstract} \end{frontmatter} \noindent \medskip \medskip \medskip \noindent PACS numbers: 12.38.-t; 12.38.Bx; 13.38.Be; 13.38.Dg \noindent Keywords: QCD; Z boson decay; W boson decay; \vfill \noindent Corresponding author: K.G.~Chetyrkin, address: Max-Planck-Institut f\"ur Physik, Werner-Heisenberg-Institut, F\"ohringer Ring 6, 80805 Munich, Germany \\ e-mail: chet@mppmu.mpg.de \vfill \newpage \section{Introduction} Precision measurements of the total and as well as partial Z decay rates have provided one of the the most important and, from the theoretical viewpoint, clean determination of the strong coupling constant $\alpha_s$ with a present value of $\alpha_s= 0.1202 \pm 0.0033$ \cite{Blondel}. Theoretical ingredients were the knowledge of QCD corrections to order $\alpha_s^3$ in the limit of massless quarks plus charm and bottom quarks effects (see, e.g. \cite{review} and references therein). These mass corrections which indeed are relevant at the present level of accuracy have been calculated up to the order $\alpha_s^3 m_q^2/s$ for the vector and $\alpha_s^2 m_q^2/s$ for the axial current induced decay rate. In this short note the prediction is extended to include $\alpha_s^3 m_q^2/s$ terms for the (non-singlet part) of the axial current induced rate. At the same time results are obtained for the non-diagonal current correlator with two different masses -- a case of relevance e.g. for the W decay rate into charmed and bottom quarks. The same formulae can also be applied to a subclass of corrections which enter single top production in the Drell-Yan like reaction $q \ovl{q}\to t \ovl{b}$ far above threshold. The calculation is based on an approach introduced in Refs.~\cite{ChetKuhn90,CheKueKwi92}. Knowledge of the polarization function to order $\alpha_s^2$, the appropriate anomalous dimensions at order $\alpha_s^3$, combined with the renormalization group equation allows one to predict the corresponding logarithmic terms of order $\alpha_s^3$ and hence the constant terms of the imaginary part. The first of these ingredients has been available since some time \cite{GorKatLarSur90,Chetyrkin93,Karl94,levan94} while the anomalous dimension can been obtained from Ref.~\cite{gssq} in a straightforward way. In this short note only the theoretical framework and the analytical results are presented -- numerical studies will presented elsewhere. \section{Renormalization Group Analysis} In analogy to the vector case, we take as a starting point the generic vector/axial quark current correlator $\Pi^{V/A}_{\mu\nu}$ which is defined by \beq \ba{ll} \Pi^{V/A}_{\mu\nu}(q,m_u,m_d,m{},\mu,\as) & = \ds i \int dx e^{iqx} \langle T[\, j^{V/A}_{\mu}(x) (j^{V/A}_\nu)^{\dagger} (0)\, ] \rangle \\ & = \ds g_{\mu\nu} \Pi^{(1)}_{V/A}(Q^2) + q_{\mu}q_{\nu} \Pi^{(2)}_{V/A}(Q^2)) {}. \ea \label{correlator} \eeq with $Q^2=-q^2$, $m^2 = \sum_f m_f^2$ and $j^{V/A}_{\mu} = \bar{q}\gamma_{\mu}(\gamma_5) q'$. Here $q$ and $q'$ are two (generically different) quarks with masses $m_u$ and $m_d$ respectively. Note that the vector and axial correlators are related through \beq \Pi^{A}_{\mu\nu}(q,m_u,m_d,m{},\mu,\as) = \Pi^{V}_{\mu\nu}(q,m_u,-m_d,m{},\mu,\as) \label{VAidentity} \eeq The polarization function $\Pi^{(1)}_{V/A}$ and the spectral density $R^{V/A}(s)$ which in turn governs the $Z$ and $W$ decays rate obey the following dispersion relation \beq \dsp \Pi_{V/A}^{(1)} (Q^2) = \frac{-1}{12\pi^2} \int_{(m_u+m_d)^2}^{\infty}ds \frac{s R^{V/A}(s,m_u,m_d,m{},\mu,\as)}{s+Q^2} \ \ \ \;\;{\rm mod \;sub.} \label{dispersion.rel} \eeq Whereas $R^{V/A}$ as a physical quantity is invariant under renormalization group transformations, the function $\langle T[\, j^{V/A}_{\mu}(x) (j^{V/A}_\nu)^{\dagger}(0)\, ] \rangle$ contains some non-integrable singularities in the vicinity of the point $x=0$. These cannot be removed by standard quark mass and coupling constant renormalizations, but must be subtracted independently. As a result the relevant renormalization group equation assumes the form \cite{review} \beq \label{rgea} \dmu \Pi^{V/A}_{\mu\nu} = (q_{\mu}q_{\nu}-g_{\mu\nu}q^2) \g^\pm_q(\as) \frac{1}{16\pi^2} + (m_u \mp m_d)^2 g_{\mu\nu} \g^\pm_m (\as) \frac{1}{16\pi^2}, \eeq where \beq \label{rgdef} \mu^2\frac{d}{d\mu^2} = \mu^2\frac{\partial}{\partial\mu^2} + \pi\beta(\alpha_s) \frac{\partial}{\partial \alpha_s} +\gm(\alpha_s) \sum_f \bar{m_f} \frac{\partial}{\partial \bar{m_f}} {}. \eeq Here and below the upper and lower signs give the results for vector and axial vector correlators respectively. {}From the identity (\ref{VAidentity}) we infer that both anomalous dimensions $\g^\pm_q$ and $\g^\pm_m$, being not dependent on any masses, also do not depend on the sign. In what follows we will denote $\g^\pm_q = \gvvq \ \ \ \mbox{and} \ \ \ \g^\pm_m = \gvvm \ {}.$ The $\beta$-function and the quark mass anomalous dimension $\gm$ are defined in the usual way \beq \dmu \left( \frac{\as(\mu)}{\pi} \right) = \beta(\as) \equiv -\sum_{i\geq0}\beta_i\left(\api\right)^{i+2}, \eeq \beq \dmu \bar{m}(\mu) = \bar{m}(\mu)\gm(\as) \equiv -\bar{m}\sum_{i\geq0}\gm^i\left(\api\right)^{i+1}. \eeq Their expansion coefficients up to order ${\cal O}(\as^3)$ are well known \cite{beta,Larin:betaQCD,gamma,Larin:massQCD} and read ($n_f$ is the number of quark flavours) \beq \ba{c}\dsp \beta_0=\left(11-\frac{2}{3}n_f\right)/4, \ \ \beta_1=\left(102-\frac{38}{3}f\right)/16, \\ \dsp \beta_2=\left(\frac{2857}{2}-\frac{5033}{18}n_f+ \frac{325}{54}n_f^2\right)/64, \ea \label{beta3} \eeq \beq \ba{c}\dsp \g^0_m=1, \ \ \ \g^1_m=\left(\frac{202}{3}-\frac{20}{9}f\right)/16, \\ \dsp \g^2_m=\left(1249 - \left[\frac{2216}{27}+\frac{160}{3}\zeta(3)\right] n_f-\frac{140}{81}f^2\right)/64. \ea \label{anom.mass3} \eeq Another useful and closely related object is the correlator of the (pseudo)scalar quark currents \begin{equation} \label{SP} \Pi^{{\rm S/P}}(Q^2,m_u,m_d,m{},\mu,\as) = \int e^{iqx}\langle0|\; T\; [\,j^{{\rm S/P}}(x) (j^{{\rm S/P}})^\dagger (0)\,] \;|0\rangle {}\, . \end{equation} Scalar and pseudoscalar current correlators are also related in a simple manner: \begin{equation} \Pi^{{\rm S}}(Q^2,m_u,m_d,m{},\mu) = \Pi^{{\rm P}}(Q^2,m_u,-m_d,m{},\mu) {}. \label{SPidentity} \end{equation} For vanishing quark masses scalar and pseudoscalar correlators are, therefore, identical: $\Pi^{{\rm S}}= \Pi^{{\rm P}}$ and meet the following RG equation \beq \label{rg:sc} \left( \dmu + 2 \gamma_m(\alpha_s) \right) \Pi^{{\rm S/P}} = Q^2 \g^{{\rm SS}}_q(\as) \frac{1}{16\pi^2} {}. \eeq The (axial) vector and (pseudo)scalar correlators are connected through a Ward identity \cite{david75} \begin{equation} q_\mu q_\nu \Pi^{{\rm V/A}}_{\mu\nu} = (m_u \mp m_d)^2 \Pi^{\rm S/P} + (m_u \mp m_d) ( \langle \overline{\psi}_{{\rm q}} \psi_{{\rm q}} \rangle \mp \langle \overline{\psi}_{{\rm q'}} \psi_{{\rm q'}} \rangle ) {}\, , \label{axial-ward} \end{equation} where the vacuum expectation values on the r.h.s. are understood within the framework of perturbation theory and the minimal subtractions. Equation~(\ref{axial-ward}) leads to the following relation between the corresponding anomalous dimensions \cite{CheKueKwi92}: \begin{equation} \gamma_{{m}}^{\rm VV} \equiv - \gamma_{{q}}^{{\rm SS}} {}\, . \label{VV-SS} \end{equation} This relation was used in Ref.~\cite{CheKueKwi92} in order to find the anomalous dimension $\gamma_{{m}}^{\rm AA}$ at the $\alpha_s^2$ order starting from the results of Ref.~\cite{GorKatLarSur90}. In what follows we will be interested in quadratic mass corrections to the polarization operator $\Pi^{(1)}_{A}$ which is convenient to represent in the form (${\bf m} = \{ m_u, m_d, m{} \}$): \beq \Pi^{(1)}_{V/A}( Q^2,{\bf m},\mu,\alpha_s) = \frac{3}{16\pi^2}\Pi^{(1)}_{V/A,0}(Q^2,\mu, \alpha_s) + \frac{3}{16\pi^2}\Pi^{(1)}_{V/A,2}(\frac{\mu^2}{Q^2},{\bf m},\alpha_s) + {\cal O}({\bf m^4}) {}. \label{mass-exp} \eeq Here the first term on the rhs corresponds to the massless limit while the second term stands for quadratic mass corrections. Note that $\Pi^{(1)}_{V/A,2}$ is a second order polynomial in quark masses: a logarithmic dependence on quark masses may appear starting from $m^4$ terms only\footnote{Provided of course that one uses a mass independent renormalization scheme like the $\msbar$-scheme employed in this work.}. Working only within the second order in quark masses, the vacuum expectation values on the right hand side of (\ref{axial-ward}) can be safely discarded. As a result, the latter can be written as follows: \beq -\Pi^{(1)}_{V/A,2} + Q^2\Pi^{(2)}_{V/A,2} = \frac{(m_u \mp m_d)^2}{Q^2}\Pi^{\rm S/P,0} {}, \label{axial-ward3} \eeq with $\Pi^{\rm S/P,0}$ being the massless scalar correlator. {}From the RG equation (\ref{rgea}) we arrive at the following equation for $\Pi^{(1)}_{V/A,2}$: \beq \dmu \Pi^{(1)}_{V/A,2} = \frac{1}{3}(m_u \mp m_d)^2 \gvvm(\as) \label{rgPi1} {} \eeq or, equivalently, ($L_q = \ln\frac{\mu^2}{Q^2}$) \beq \frac{\prd }{\prd L_q} \Pi^{(1)}_{V/A,2} = \frac{1}{3}(m_u \mp m_d)^2 \gvvm -\left( \beta \frac{\prd }{\prd \alpha_s} + 2\gamma_m \right) \Pi^{(1)}_{V/A,2} \label{rgPi2} {}. \eeq The last relation explicitly demonstrates that $R^{V/A}_2$ --- the absorptive part of $\Pi^{(1)}_{V/A,2}$ --- depends in order $\alpha_s^n$ on the very function $\Pi^{(1)}_{V/A,2}$ which is multiplied by at least one factor of $\alpha_s$. This means that one needs to know $\Pi^{(1)}_{V/A,2}$ up to order $\alpha_s^{n-1}$ only to unambiguously reconstruct all $Q$-dependent terms in $\Pi^{(1)}_{V/A,2}$ to $\alpha_s^n$, provided, of course, the anomalous dimensions $\gamma_m$ and $\gvvm$ are known to $\alpha_s^n$ and the beta function to $\alpha_s^{n+1}$. This observation was made first in \cite{ChetKuhn90} where it was used to find the absorptive part $R^{V}_2$ in order $\alpha_s^3$ for the case of the diagonal vector current (that is for the case of $m_u = m_d$). In the present paper we will use the results of a recent calculation of $\gssq$ \cite{gssq} to order $\alpha_s^3$ to determine the absorptive part $R^{V/A}_2$ to the same order in the general case of non-diagonal currents. \section{Calculation and results} % The result for the function $\Pi^{(2)}_{V/A,2}$ in the general non-diagonal case to order $\as^2$ was first published in Ref.~\cite{Chetyrkin93}. On the other hand, the identity (\ref{axial-ward3}) expresses the combination $-\Pi^{(1)}_{V/A,2}/Q^2 + \Pi^{(2)}_{V/A,2}$ in terms of the {\em massless} polarization operator $\Pi^S$ known from Refs.~\cite{GorKatLarSur90,Karl94}. A sum of these two functions leads us to the following result for $\Pi^{(1)}_{V,2}$ \begin{eqnarray} \lefteqn{\Pi^{(1)}_{V,2} = {}m_{-}^2 \left[ 2 +2 \lgm\frac{\mu^2}{Q^2} %zero = 0 \right] {+}m_{+}^2 \left[ -2\right ]} \nonumber\\ &{+}&m_{-}^2 \frac{\alpha_s}{\pi} \left[ \frac{107}{6} -8 \,\zeta(3) +\frac{22}{3} \lgm\frac{\mu^2}{Q^2} +2 \lgm^2\frac{\mu^2}{Q^2} %zero = 0 \right] {+}m_{+}^2 \frac{\alpha_s}{\pi} \left[ -\frac{16}{3} -4 \lgm\frac{\mu^2}{Q^2} %zero = 0 \right] \nonumber\\ &{+}&m_{-}^2\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{3241}{18} -129 \,\zeta(3) -\frac{1}{2} \,\zeta(4) +55 \,\zeta(5) -\frac{857}{108} \,n_f +\frac{32}{9} \,\zeta(3) \,n_f \Break \phantom{+m_{-}^2 \left(\frac{\alpha_s}{\pi}\right)^2} %+\frac{32}{9} \,\zeta(3) \,n_f +\frac{8221}{72} \lgm\frac{\mu^2}{Q^2} -39 \,\zeta(3) \lgm\frac{\mu^2}{Q^2} -\frac{151}{36} \,n_f \lgm\frac{\mu^2}{Q^2} +\frac{4}{3} \,\zeta(3) \,n_f \lgm\frac{\mu^2}{Q^2} \Break \phantom{+m_{-}^2 \left(\frac{\alpha_s}{\pi}\right)^2} +\frac{155}{6} \lgm^2\frac{\mu^2}{Q^2} -\frac{8}{9} \,n_f \lgm^2\frac{\mu^2}{Q^2} +\frac{19}{6} \lgm^3\frac{\mu^2}{Q^2} -\frac{1}{9} \,n_f \lgm^3\frac{\mu^2}{Q^2} %zero = 0 \right] \nonumber\\ &{+}&m_{+}^2 \left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{19691}{216} -\frac{124}{27} \,\zeta(3) +\frac{1045}{27} \,\zeta(5) +\frac{95}{36} \,n_f -\frac{253}{6} \lgm\frac{\mu^2}{Q^2} \Break \phantom{+m_{+}^2 \left(\frac{\alpha_s}{\pi}\right)^2} +\frac{13}{9} \,n_f \lgm\frac{\mu^2}{Q^2} -\frac{19}{2} \lgm^2\frac{\mu^2}{Q^2} +\frac{1}{3} \,n_f \lgm^2\frac{\mu^2}{Q^2} %zero = 0 \right] \nonumber\\ &{+}&m^2 \left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{128}{9} -\frac{32}{3} \,\zeta(3) %zero = 0 \right] {}. \label{Pi1V2} \end{eqnarray} Here $m_- = m_u - m_d$ and $m_+ = m_u + m_d$, $Q^2 = -q^2$, %$L_s = \lgm\frac{\mu^2}{Q^s}$ and %$L_Q = \lgm\frac{\mu^2}{Q^2}$; all masses as well as QCD coupling constant $\alpha_s$ are understood to be taken at a generic value of the t' Hooft~mass $\mu$. All correlators are renormalized within $\msbar$-scheme. We have also checked (\ref{Pi1V2}) by a direct calculation with the help of the program MINCER \cite{mincer2} written for the symbolic manipulation system FORM \cite{Ver91}. In a particular case of $m_u = m_d\,\,\,$ Eq.~(\ref{Pi1V2}) is in agreement with Refs.~\cite{levan94,GorKatLar86}. Now, as was shown in \cite{ChetKuhn90} the anomalous dimension $\gaam \equiv = -\gssq$, and, thus, from the results of \cite{gssq} we have: \begin{eqnarray} \lefteqn{\gvvm = -\g^{SS}_q = 6\left\{ 1 {+} \frac{5}{3}\frac{\alpha_s}{\pi} \right. {+}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{455}{72} -\frac{1}{2} \,\zeta(3) -\frac{1}{3} \,n_f %zero = 0 \right]} \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{157697}{5184} -\frac{1645}{216} \,\zeta(3) +\frac{15}{8} \,\zeta(4) +\frac{65}{12} \,\zeta(5) -\frac{14131}{7776} \,n_f \Break \left. \phantom{+\left(\frac{\alpha_s}{\pi}\right)^3} -\frac{13}{9} \,\zeta(3) \,n_f -\frac{11}{12} \,\zeta(4) \,n_f -\frac{1625}{11664} \, n_f^2 +\frac{1}{9} \,\zeta(3) \, n_f^2 %zero = 0 \right] \right\} {}. \label{gVVm} \end{eqnarray} %zero = 0 At last, integrating eq.~(\ref{rgPi2}) we find the spectral density $R^{V}_2$ in general case to order $\as^3$: \beq R^V_2(s,m_u,m_d,m{},\mu,\as) = 3\left\{ \frac{m_+^2}{s} r^V_{2,+} + \frac{m_-^2}{s}r^V_{2,-} + \frac{m^2}{s}r^V_{2,0} \right\} {}, \label{R2V} \eeq where the functions $r^V$ are \begin{eqnarray} \lefteqn{r_{2,+}^{V} = 3 \frac{\alpha_s}{\pi} {+} \left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{253}{8} -\frac{13}{12} \,n_f +\frac{57}{4} \lgm\frac{\mu^2}{s} -\frac{1}{2} \,n_f \lgm\frac{\mu^2}{s} %zero == 0 \right]} \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{1261}{2} -\frac{285}{16} \pi^2 +\frac{155}{6} \,\zeta(3) -\frac{5225}{24} \,\zeta(5) -\frac{2471}{54} \,n_f +\frac{17}{12} \pi^2 \,n_f \label{r2+ } \right. \\ &{}& \left. \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} %+\frac{17}{12} \pi^2 \,n_f -\frac{197}{54} \,\zeta(3) \,n_f +\frac{1045}{108} \,\zeta(5) \,n_f +\frac{125}{216} \, n_f^2 -\frac{1}{36} \pi^2 \, n_f^2 +\frac{4505}{16} \lgm\frac{\mu^2}{s} \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} %+\frac{4505}{16} \lgm\frac{\mu^2}{s} -\frac{175}{8} \,n_f \lgm\frac{\mu^2}{s} +\frac{13}{36} \, n_f^2 \lgm\frac{\mu^2}{s} +\frac{855}{16} \lgm^2\frac{\mu^2}{s} -\frac{17}{4} \,n_f \lgm^2\frac{\mu^2}{s} +\frac{1}{12} \, n_f^2 \lgm^2\frac{\mu^2}{s} %\Break %\phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} %+\frac{1}{12} \, n_f^2 \lgm^2\frac{\mu^2}{s} %zero == 0 \right] {}, \nonumber %\label{r2+ } \end{eqnarray} %zero == 0 \begin{eqnarray} \lefteqn{r_{2,-}^{V} = -\frac{3}{2} {+} \frac{\alpha_s}{\pi} \left[ -\frac{11}{2} -3 \lgm\frac{\mu^2}{s} %zero == 0 \right] } \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{8221}{96} +\frac{19}{8} \pi^2 +\frac{117}{4} \,\zeta(3) +\frac{151}{48} \,n_f -\frac{1}{12} \pi^2 \,n_f \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^2} - \,\zeta(3) \,n_f -\frac{155}{4} \lgm\frac{\mu^2}{s} +\frac{4}{3} \,n_f \lgm\frac{\mu^2}{s} -\frac{57}{8} \lgm^2\frac{\mu^2}{s} +\frac{1}{4} \,n_f \lgm^2\frac{\mu^2}{s} %zero == 0 \right] \nonumber\\ &{+}& \left(\frac{\alpha_s}{\pi}\right)^3 \left[ -\frac{4544045}{3456} +\frac{335}{6} \pi^2 +\frac{118915}{144} \,\zeta(3) -\frac{635}{2} \,\zeta(5) +\frac{71621}{648} \,n_f \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{209}{48} \pi^2 \,n_f -54 \,\zeta(3) \,n_f +\frac{5}{4} \,\zeta(4) \,n_f +\frac{55}{4} \,\zeta(5) \,n_f -\frac{13171}{7776} \, n_f^2 \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{2}{27} \pi^2 \, n_f^2 +\frac{13}{18} \,\zeta(3) \, n_f^2 -\frac{4693}{6} \lgm\frac{\mu^2}{s} +\frac{285}{16} \pi^2 \lgm\frac{\mu^2}{s} +\frac{1755}{8} \,\zeta(3) \lgm\frac{\mu^2}{s} \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{8909}{144} \,n_f \lgm\frac{\mu^2}{s} -\frac{17}{12} \pi^2 \,n_f \lgm\frac{\mu^2}{s} -\frac{59}{4} \,\zeta(3) \,n_f \lgm\frac{\mu^2}{s} -\frac{209}{216} \, n_f^2 \lgm\frac{\mu^2}{s} %+\frac{1}{36} \pi^2 \, n_f^2 \lgm\frac{\mu^2}{s} \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} +\frac{1}{36} \pi^2 \, n_f^2 \lgm\frac{\mu^2}{s} +\frac{1}{3} \,\zeta(3) \, n_f^2 \lgm\frac{\mu^2}{s} -\frac{335}{2} \lgm^2\frac{\mu^2}{s} +\frac{209}{16} \,n_f \lgm^2\frac{\mu^2}{s} \Break \phantom{+ \left(\frac{\alpha_s}{\pi}\right)^3} -\frac{2}{9} \, n_f^2 \lgm^2\frac{\mu^2}{s} -\frac{285}{16} \lgm^3\frac{\mu^2}{s} +\frac{17}{12} \,n_f \lgm^3\frac{\mu^2}{s} -\frac{1}{36} \, n_f^2 \lgm^3\frac{\mu^2}{s} %zero == 0 \right] {}, \nonumber\\ \label{r2-} \end{eqnarray} %zero == 0 \beq r_{2,0}^{V} = \left(\frac{\alpha_s}{\pi}\right)^3 \left[ -80 +60 \,\zeta(3) +\frac{32}{9} \,n_f -\frac{8}{3} \,\zeta(3) \,n_f \right] {}. \label{r20} \eeq %zero == 0 The expressions for the hadronic decay rates of the intermediate bosons read: % \begin{eqnarray} \G(Z \to \mbox{hadrons} ) = \G^Z_0 % &{}& \left[ \right. &{}& \sum_f ((g^f_V)^2 + (g^f_A)^2)\left(R_0(s) + R_2^V(s,0,0,\sqrt{m^2_b+m_c^2}\right) %\right. \nonumber \\ &+&\sum_{f=b,c} (g^f_V)^2 R_2^V(s,m_f,m_f,0) \nonumber \\ &+& \left. \sum_{f=b,c}(g^f_A)^2 R_2^A(s,m_f,m_f,0) \right] \label{Zff} {}, \end{eqnarray} % \begin{eqnarray} \G(W \to \mbox{hadrons}) = \G^W_0 % &{}& \bigg\{ %\right. &{}& 2 \left[ R_0(s) +R_2^{V} \left(s,0,0,\sqrt{m^2_b+m_c^2} \right) \right] \label{Wff} \\ + %&+& \frac{1}{2}&{}& \sum_{\dsp {i=u,c}\atop{\dsp j=d,s,b}} |V_{i,j}|^2( R_2^V(s,m_{i},m_{j},0) + R_2^A(s,m_{i},m_{j},0) ) %\left. \bigg\} \nonumber {}, \end{eqnarray} with $\G_0 = \frac{G_F M_{Z/W}^3}{6\pi \sqrt{2}}$, $g^{{f}}_V = I^{{f}}_3 - 2Q_{{f}}\sin^2 \theta_{{\rm w}}, \ \ g^{{f}}_A = I^{{f}}_3$ and $V_{ij}$ being the CKM matrix. %\newpage Here $R_0(s)$ is the (non-singlet part) of the ratio $R(s)$ in massless QCD; it was computed to $\as^3$ in \cite{GorKatLar91,SurSam91} and confirmed in \cite{gluino}; it reads: %\begin{equation}\label{apxc2} $\begin{array}{rl}\displaystyle R_0(s) & = \displaystyle 3 \Bigg\{ 1+\frac{\alpha_s}{\pi} \\ & \displaystyle +\left(\frac{\alpha_s}{\pi}\right)^2 \left[ \frac{365}{24}-11\zeta(3) +n_f\left( -\frac{11}{12}+\frac{2}{3}\zeta(3) \right) +\left( -\frac{11}{4}+\frac{1}{6}n_f \right)\ln\,\frac{s}{\mu^2} \right] \\ & \displaystyle +\left(\frac{\alpha_s}{\pi}\right)^3 \Bigg[ \frac{87029}{288}-\frac{121}{48}\pi^2 -\frac{1103}{4}\zeta(3) +\frac{275}{6}\zeta(5) \\ & \displaystyle \hphantom{\left(\frac{}{}\right)} +n_f\left( -\frac{7847}{216}+\frac{11}{36}\pi^2 +\frac{262}{9}\zeta(3)-\frac{25}{9}\zeta(5) \right) +n_f^2\left( \frac{151}{162}-\frac{1}{108}\pi^2 -\frac{19}{27}\zeta(3) \right) \\ & \displaystyle \hphantom{\left(\frac{}{}\right)} +\left( -\frac{4321}{48}+\frac{121}{2}\zeta(3) +n_f\left[ \frac{785}{72}-\frac{22}{3}\zeta(3) \right] +n_f^2\left[ -\frac{11}{36}+\frac{2}{9}\zeta(3) \right] \right)\ln\,\frac{s}{\mu^2} \\ & \displaystyle \hphantom{\left(\frac{}{}\right)} +\left( \frac{121}{16}-\frac{11}{12}n_f +\frac{1}{36}n_f^2 \right)\ln^2\frac{s}{\mu^2} \Bigg] \Bigg\} {}. \end{array} \nonumber$ In deriving Eqs.~(\ref{Zff}) and (\ref{Wff}) we have assumed that (i) the top quark is completely decoupled (the power suppressed corrections to this approximation start from the order $\frac{s}{m_t^2}\as^2$ and have been studied in Refs.~\cite{Kniehl90,me93,Soper94,HoaKueTeu95,HoaKueTeu95}); (ii) all other quarks except for the charmed and bottom ones are massless. Note that for the case of {\em diagonal} currents there exist also so-called singlet contributions to $R(s)$. We will ignore these contributions in what follows as they are absent for the case of non-diagonal currents relevant for the $W$-decay (a detailed discussion of the $Z$-decay rate including singlet contributions can be found in \cite{review}). Taking into account the peculiar structure of the general result (\ref{R2V}), the last formula can be written in a simpler form, viz. \begin{eqnarray} \G(W \to \mbox{hadrons}) = \G^W_0 % &{}& \left[ \right. &{}& 2 \left( R_0(s) +R_2^V(s,0,0,\sqrt{m^2_b+m_c^2} ) \right) \nonumber \\ &+& R_2^V(s,m_{eff},0,0) \left. \right] \label{Wff2} {}. \end{eqnarray} Here $m_{eff}^2 = |V_{c,b}|^2 (m_c^2 + m_b^2) + |V_{u,b}|^2 m_b^2$ and we have taken into account the fact that \bea R^V(s,m_i,m_j,0) + R^A(s,m_i,m_j,0) & = & 2 R^V(s,\sqrt{m^2_i+m^2_j},0,0) \nonumber \\ &=&2 R^A(s,\sqrt{m^2_i+m^2_j},0,0) \nonumber {}. \eea As a direct consequence of Eqs.~(2,\ref{R2V}) we obtain the following expressions for particular functions entering into (\ref{Zff},\ref{Wff}) \begin{eqnarray} R_2^{V}(s,m,m,0) &=& \frac{4m^2}{s} 3\, r^V_{2,+}, \label{part1} \\ R_2^{A}(s,m,m,0) &=& \frac{4m^2}{s} 3\, r^V_{2,-}, \\ \label{part2} R_2^{V}(s,m,0,0) &=& R_2^{A}(s,m,0,0)= \frac{m^2}{s} 3\, ( r^V_{2,+}+ r^V_{2,-} ), \\ R_2^{V}(s,0,0,m) &=& R_2^{A}(s,0,0,m) = \frac{m^2}{s}3\,r_{2,0} {}. \label{part3} \end{eqnarray} At last, with $n_f=5$ and $\mu^2= s$ the above formulas are simplified to % \begin{eqnarray} R_2^{V}(s,m,m,0) = &{}&\frac{m^2}{s}3\left\{ 12\frac{\alpha_s}{\pi} {+}\frac{629}{6} \left(\frac{\alpha_s}{\pi}\right)^2 \right. \label{R2Vmm0nf5} \\ &{+}& \left. \left(\frac{\alpha_s}{\pi}\right)^3 \left[ \frac{89893}{54} -\frac{1645}{36} \pi^2 +\frac{820}{27} \,\zeta(3) -\frac{36575}{54} \,\zeta(5) \right] \right\} \nonumber \end{eqnarray} %zero == 0 %\end{document} \begin{eqnarray} &{}&\!\!\!\!\!\!\!\!\!R_2^{A}(s,m,m,0) = {}\frac{m^2}{s} 3\left\{ -6 -22 \frac{\alpha_s}{\pi} +\left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{2237}{8} +\frac{47}{6} \pi^2 +97 \,\zeta(3) %zero == 0 \right] \right. \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left. \left[ -\frac{25024465}{7776} +\frac{15515}{108} \pi^2 +\frac{27545}{12} \,\zeta(3) +25 \,\zeta(4) -995 \,\zeta(5) %zero == 0 \right] \right\} \label{R2Amm0nf5} \end{eqnarray} %zero == 0 \begin{eqnarray} &{}&R_2^{V}(s,m,0,0)= \frac{m^2}{s} 3\left\{ -\frac{3}{2} -\frac{5}{2} \frac{\alpha_s}{\pi} {+}\left(\frac{\alpha_s}{\pi}\right)^2 \left[ -\frac{4195}{96} +\frac{47}{24} \pi^2 +\frac{97}{4} \,\zeta(3) %zero == 0 \right] \right. \nonumber\\ &{+}&\left(\frac{\alpha_s}{\pi}\right)^3 \left. \left[ -\frac{12079873}{31104} +\frac{2645}{108} \pi^2 +\frac{251185}{432} \,\zeta(3) +\frac{25}{4} \,\zeta(4) -\frac{90305}{216} \,\zeta(5) %zero == 0 \right] \right\} \EQN{ } \end{eqnarray} %zero == 0 or, in the numerical form, \beq R_2^{V}(s,m,m,0) = \frac{m^2}{s} 3\left\{ 12 \frac{\alpha_s}{\pi} {+}104.833\left(\frac{\alpha_s}{\pi}\right)^2 {+}547.879\left(\frac{\alpha_s}{\pi}\right)^3 \right\} \EQN{R2Vmm0NN} {}, \eeq \beq R_2^{A}(s,m,m,0) = \frac{m_{}^2}{s} 3\left\{ -6 -22 \frac{\alpha_s}{\pi} -85.7136\left(\frac{\alpha_s}{\pi}\right)^2 -45.7886 \left(\frac{\alpha_s}{\pi}\right)^3 \right\} {}, \EQN{R2Amm0NN} \eeq \bea R_2^{V}(s,m,0,0) &=& R_2^{A}(s,m,0,0) \nonumber \\ &=& \frac{m_{}^2}{s} 3\left\{ -1.5 -2.5 \frac{\alpha_s}{\pi} + 4.7799\left(\frac{\alpha_s}{\pi}\right)^2 + 125.523\left(\frac{\alpha_s}{\pi}\right)^3 \right\} {}. \EQN{R2Vm00NN} \eea To conclude, we have extended the results of Ref.~\cite{ChetKuhn90} by evaluating analytically the terms of order ${\cal O}(\alpha_s^3m_q^2)$ for the general case of vector or axial-vector correlator of two non-diagonal quark currents. We have applied the obtained results to find the ${\cal O}(\alpha_s^3)$ quadratic quark mass corrections to the hadronic decay rates of W- and Z-bosons. We have shown that to the order in $\alpha_s$ we are dealing with the decay width $\G(W \to \mbox{hadrons})$ depends on the charmed and bottom quark masses only through a combination $|V_{c,b}|^2 (m_c^2 + m_b^2) + |V_{u,b}|^2 m_b^2$. Our results can also be applied to determination of the strange quark mass from $\tau$ decays. This topic will be treated in a subsequent paper. \noindent This work was supported by BMFT under Contract 057KA92P(0) and INTAS under Contract INTAS-93-0744. % \sloppy \raggedright \def\app#1#2#3{{\it Act. Phys. Pol. }{\bf B #1} (#2) #3} \def\apa#1#2#3{{\it Act. Phys. Austr.}{\bf #1} (#2) #3} \def\lhc{Proc. LHC Workshop, CERN 90-10} \def\npb#1#2#3{{\it Nucl. Phys. }{\bf B #1} (#2) #3} \def\plb#1#2#3{{\it Phys. Lett. }{\bf B #1} (#2) #3} \def\prd#1#2#3{{\it Phys. Rev. }{\bf D #1} (#2) #3} \def\pR#1#2#3{{\it Phys. Rev. }{\bf #1} (#2) #3} \def\prl#1#2#3{{\it Phys. Rev. Lett. }{\bf #1} (#2) #3} \def\prc#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3} \def\cpc#1#2#3{{\it Comp. Phys. Commun. }{\bf #1} (#2) #3} \def\nim#1#2#3{{\it Nucl. Inst. Meth. }{\bf #1} (#2) #3} \def\pr#1#2#3{{\it Phys. Reports }{\bf #1} (#2) #3} \def\sovnp#1#2#3{{\it Sov. J. Nucl. Phys. }{\bf #1} (#2) #3} \def\jl#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3} \def\jet#1#2#3{{\it JETP Lett. }{\bf #1} (#2) #3} \def\zpc#1#2#3{{\it Z. Phys. }{\bf C #1} (#2) #3} \def\ptp#1#2#3{{\it Prog.~Theor.~Phys.~}{\bf #1} (#2) #3} \def\nca#1#2#3{{\it Nouvo~Cim.~}{\bf #1A} (#2) #3} \def\mpl#1#2#3{{\it Mod. Phys. 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