%Title: Strange Quark Mass from Tau Lepton Decays with O(alpha_s^3) Accuracy %Author: P.A. Baikov, K.G. Chetyrkin and J.H. K\"uhn %Published: Phys. Rev. Lett. 95 (2005) 012003-1 -- 012003-4. %hep-ph/0412350 %\documentclass[preprint,prl,aps,showpacs]{revtex4} \documentclass[twocolumn,prl,aps,showpacs,floatfix]{revtex4} % today 7.0.3.05 — Œ¡Þ…Œ¡¿ –œƒ«œ‘¡◊Ã…◊¡‘ÿ new version for PRL \usepackage{graphicx}% \usepackage{dcolumn} \usepackage{amsmath} \newcommand{\ice}{\relax} \newcommand{\as}{a_s} \newcommand{\api}{\frac{\alpha_s}{\pi}} \newcommand{\als}{\alpha_s} \newcommand{\beq}{\begin{equation}} \newcommand{\eeq}{\end{equation}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\g}{\gamma} \newcommand{\asthree}{{}^{2\times{\cal O}(\as^3)}_{{\rm no}\;{\cal O}(\as^3)}} \newcommand{\asfour}{{}^{2\times{\cal O}(\as^4)}_{{\rm no}\;{\cal O}(\as^4)}} \begin{document} \title{ %%%%%%%%%% { \vspace*{-14mm} \centerline{\normalsize\hfill SFB/CPP-04-72} \centerline{\normalsize\hfill TTP04-28} \centerline{\normalsize\hfill hep-ph/0412350} %\baselineskip 11pt {}} \vspace{3mm} %%%%%%%%%%% Strange Quark Mass from Tau Lepton Decays with ${\cal O}(\alpha_s^3)$ Accuracy \vskip.3cm } \author{P.~A.~Baikov} \affiliation{Institute of Nuclear Physics, Moscow State University, Moscow~119899, Russia } \author{K.~G.~Chetyrkin\thanks{{\small Permanent address: Institute for Nuclear Research, Russian Academy of Sciences, Moscow 117312, Russia}}} \author{J.~H.~K\"uhn} %\affiliation{ %Institut f\"ur Theoretische Teilchenphysik, % Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany %} \affiliation{Institut f\"ur Theoretische Teilchenphysik, Universit\"at Karlsruhe, D-76128 Karlsruhe, Germany} \begin{abstract} \vspace*{.7cm} \noindent The first complete calculation of the quadratic quark mass correction to the correlator of the two currents relevant for the strangeness-changing semihadronic tau-decay rate is presented {\em including its real} part at the four-loop level. This allows to perform the extraction of the strange quark mass $m_s$ from the decay width of the tau-lepton with {\em full} ${\cal O}(\alpha_s^3)$ accuracy. In agreement with previous estimates the newly computed $\alpha_s^3$ term proves to be rather large. This justifies inclusion of the similarly estimated $\als^4$ term in phenomenological analysis. Combined with an updated value of $V_{us}$ and an 'improved'' version of the renormalization group improvement of the perturbative series this leads to an increase of the central value of $m_s$ by about 20\% and a partial reduction of the theoretical uncertainty by about 50\%. \end{abstract} \pacs{12.38.-t 12.38.Bx 13.65+i 12.20.m } \maketitle \section{Introduction} The investigation of Cabbibo suppressed semileptonic $\tau$ decays has developed into one of the important themes of $\tau$-lepton physics. Combined theoretical and experimental studies provide an independent and fairly precise value for the strange quark mass $m_s$ and may in the future also lead to a competitive determination of the Cabbibo-Kabayashi-Maskawa matrix element $V_{us}$ \cite{Barate:1999hj,Abbiendi:2004xa,Chetyrkin:1993hi,Maltman:1998qz,Chetyrkin:1998ej,Pich:1998yn,% Korner:2000wd,Chen:2001qf,Maltman:2002wb}. The decay proceeds into vector and axial current induced final states, which can be further separated experimentally into the spin zero and spin one components \cite{Kuhn:1992nz}. In addition to the total rate also various moments of the distribution in $s$, the invariant mass of the hadronic state, can be considered and will be discussed below. The theory prediction for the rate and moments is based on perturbative QCD with small contributions from nonperturbative condensates \cite{Braaten:1991qm}. The calculation is greatly simplified by the smallness of the strange quark mass which justifies an expansion in powers of $m_s^2/M_{\tau}^2$. Up to now the $m_s$ dependence of the total rate \ice{and a specific combination of moments $(L+T)$ which is more stable against inclusion of higher orders, } was only known up to order $\als^2$. To push the precision further, estimates for the third order coefficient were used which were based on the principle of minimal sensitivity \cite{Stevenson:1981vj} (PMS) or fastest apparent convergence \cite{Grunberg:1984fw} (FAC). In the present paper the exact result for this coefficient is presented. It is deduced from the first complete calculation of the finite part of a specific correlator in four loop approximation. This calculation is based on the conceptual developments described in \cite{Baikov:tadpoles:96,BkvSthr:tadples:98,Baikov:criterion:00} and requires extensive use of computer algebra \cite{Vermaseren:2000nd}. The result confirms the PMS/FAC estimate and justifies the use the same approach for an estimate of the $\als^4$ coefficient. Inclusion of this next term leads to a decrease of the central value of $m_s$ by a few MeV and a partial reduction of the theoretical uncertainty by 50\%. The phenomenological analysis will be based on the, by now standard, approach of contour improved perturbation theory (CIPT) \cite{Pivovarov:1991bi,LeDiberder:1992te}. Two variants will be discussed. We will argue in favour of one of them which leads in total to more stable results. The newest developments for $V_{us}$ will be included in the analysis. The main outcome of the study of the total hadronic $\tau$ width (normalized to the known leptonic width) \begin{equation} R_\tau=\frac{\Gamma(\tau\rightarrow{\rm hadrons} +\nu_\tau)}{\Gamma(\tau\rightarrow l+\bar\nu_l+\nu_\tau)} %= R_{\tau,NS} + R_{\tau,S} \label{Rtau1} \end{equation} has been a remarkable confirmation of perturbative QCD. It was found that the value of the strong coupling constant $\alpha_s$ as obtained from $R_{\tau}$ is in good agreement with those obtained from completely different experiments such as the $Z$ boson decay into hadrons \cite{Barate:1998uf,Ackerstaff:1998yj,Chetyrkin:1996ia,EWWG:2003_v3}. Furthermore, the strangeness changing (Cabbibo-suppressed) part $R_S$ of the decomposition of the decay rate and the moments \cite{LeDiberder:1992te} \bea \label{Rtaukl} R_\tau^{kl} &\equiv& \!\int\limits_0^{M_\tau^2}\! ds\, \biggl( 1 - \frac{s}{M_\tau^2} \biggr)^{\!k}\!\biggl(\frac{s}{M_\tau^2}\biggr)^{\!l} \frac{d R_\tau}{ds} = R_{\tau,NS}^{kl} + R_{\tau,S}^{kl} \,, \nonumber \\ R_\tau^{00} &\equiv& R_\tau, \ \ R_{\tau,NS}^{00} \equiv R_{\tau,NS}, \ \ R_{\tau,S}^{00} \equiv R_{\tau,S} {} \eea into non-strange (NS) and strange (S) components can be used to determine the strange quark mass, one of the fundamental parameters of the Standard Model. The moments as introduced above can be experimentally determined from the measured distribution in the invariant mass of the final state hadrons. Hadronic physics is encoded in the quantities $r^{kl}_{ij}$ defined through \begin{equation} R_{\tau,NS} + R_{\tau,S}= 3\, S_{\rm EW} \left(|V_{ud}|^2 \, r_{ud} +|V_{us}|^2 \, r_{us} \right) {}, \label{Rtau2} \end{equation} where $V_{ij}$ stands for the CKM matrix and $S_{EW}$ for the universal electroweak correction \cite{Marciano:1988vm,Braaten:1990ef}. The functions $r^{kl}_{ij}$ are directly related to the correlator of the charged current $j_\mu(x) =\bar{\psi_i} \g_\mu(1-\g_5)\psi_j$ \bea i \int \mathrm{d} x\, e^{iqx} \langle T\left[ j_\mu(x) j^{\dagger}_\nu(0) \right] \rangle &=& \nonumber \\ (-g_{\mu\nu} q^2 + q_{\mu}q_{\nu}) \Pi^{(q)}_{ij}(q^2) &+& g_{\mu\nu} q^2 \Pi^{(0)}_{ij} \label{corr:def} {}. \eea through \bea \lefteqn{r_{ij}^{k,l} = r_{ij}^{(q)k,l} + r_{ij}^{(0)k,l} =} \label{rij} \\ &{}& 2i\pi {\displaystyle \oint_{|s|=M_{\tau}^2} } \frac{ds}{M_{\tau}^2} \left[ w^{(q)k,l}(x) \Pi^{(q)}_{ij}(s) + w^{(0)k,l}(x) \Pi^{(0)}_{ij}(s)\right] {}. \nonumber \eea where $x=s/M^2_{\tau}$, and the weight functions $w^{(q)k,l} = (1+2x)(1-x)^{k+2}\,x^l$, $w^{(0)k,l} = -2x^{l+1}(1-x)^{k+2}$. The behaviour of the perturbative series for the part of the integral arising from $\Pi^{(q)}$ is more stable than the one from $\Pi^{(0)}$ \footnote{ The combinations $(q)$ and $(0)$ are equivalent to $L+T$ and $L$ in the notation of ref.~\cite{Pich:1998yn}.}. However, the latter can either be modelled theoretically and phenomenologically on the basis of scalar and pseudoscalar resonance physics \cite{Maltman:1998qz,Kambor:2000dj,Gamiz:2002nu,Gamiz:2004ar}, or, being solely determined by spin zero contribution in the $\tau$ decays, determined experimentally through the analysis of angular distribution of the $K\pi$ and $K\pi\pi$, thus separating spin zero and spin one contributions \cite{Kuhn:1992nz}. Restricting to scalar and pseudoscalar channels one finds \bea R^{kl(J=0)}_{\tau,S} &=& \int_0^{M_{\tau}^2} {d s} \left(1-\frac{s}{M_{\tau}^2}\right)^k \left(\frac{s}{M_{\tau}^2}\right)^l \frac{d R^{J=0}}{d s} \nonumber \\ &=& \frac{-3}{2}\, S_{EW}\,\,|V_{us}|^2\,\ r^{(0)k,l-1}_{us} \label{J=0:(L):connection} {} \eea A specific weighted integral over scalar spectral function thus directly determines the moments $r^{(0 )kl}_{ij}$. (A closely related discussion along these lines, which deals with $J=1$ and $J=0$ spectral functions and the resulting moments separately and the corresponding predictions can be found in \cite{Chetyrkin:1998ej}.) Since $m_u, m_d \ll m_s$ and $m_s \ll M_\tau$, the function $r_{us}^{k,l}$ is well described by setting $m_u=m_d=0$ and keeping only the leading and quadratic contributions to the correlator eq.~(\ref{corr:def}), viz. \bea \Pi^{(q)}_{ij}(q^2,m_s) &=& \frac{3}{16\pi^2} \left( \Pi^{(q)}_0(q^2) + \frac{m_s^2}{Q^2} \Pi^{(q)}_{2,ij}(q^2) \right) \label{Pi:q} \\ \Pi^{(0)}_{ij}(q^2,m_s) &=& \frac{3}{16\pi^2} \frac{m_s^2}{Q^2} \Pi^{(0)}_{2,ij}(q^2) \label{Pi:0} {}. \eea Here $Q^2 = -q^2$. In the following, we limit ourselves by the perturbative contributions (for a recent discussion of power-suppressed contributions, see \cite{Gorbunov:2004wy}). The small non-perturbative terms, as well as the $m_s^4$ terms will be effectively included in the phenomenological analysis at the end. As a result one has a convenient decomposition \beq r_{u d} = r_0 + \delta_{ud}, \ \ \ r_{us} = r_0 + \delta_{us} \ \ \ {}. \eeq Let us in addition define the difference \begin{equation} \label{delRtaukl} \delta R_\tau^{kl} \equiv \frac{R_{\tau,NS}^{kl}}{|V_{ud}|^2} - \frac{R_{\tau,S}^{kl}}{|V_{us}|^2} = 3\, S_{{\rm EW}} \, \delta r^{kl} %\nonumber {}. \end{equation} which is a useful combination to probe the $SU(3)$ breaking effects as $\delta r^{kl} \equiv \Big(\delta_{ud}^{kl} -\delta_{us}^{kl}\Big)$ vanishes in the limit of exact $SU(3)$ flavour symmetry. In the spirit of the decomposition (\ref{rij}) one similarly defines related quantities $\delta R_\tau^{(q)kl}$, $\delta R_\tau^{(0)kl}$, $\delta r^{(q)kl}$ and $\delta r_\tau^{(0)kl}$ such that $\delta R_\tau^{kl} = \delta R_\tau^{(q)kl} + \delta R_\tau^{(0)kl} \nonumber$ , $\delta r^{kl} = \delta r^{(q)kl} + \delta r^{(0)kl} \nonumber {}.$ The currently known fixed order perturbative predictions for $r_0$, $\delta_{us}$ and $\delta_{ud}$ can be shortly summarized as follows \cite{Gorishnii:1991vf,Chetyrkin:1993hi,Maltman:1998qz}: \ice{ \footnote{displayed below are so called Fixed Order Perturbation Theory results. A procedure of improving'' of pQCD predictions by a partial resummation of higher order terms (CIPT) will be discussed later.}. } \bea %r_0^{\mathrm{FOPT}} &=& r_0 &=& \label{Rtau:FO} \left( 1 + \, a_s+ 5.202 \, a^2_s+ 26.37 \, a_s^3 \right) {}, \\ \delta_{us} &=& \frac{m_s^2}{M_{\tau}^2} \left( 1 + 5.33\, a_s+ 46.0\, a^2_s \right) {}, \label{delta_us:FO} \\ \delta_{ud} &=& -0.35\ a^2_s \ \frac{m_s^2}{M_{\tau}^2} \label{delta_ud:FO} {}, \eea where $a_s = \als(M_\tau)/\pi$. Now, with $a_s \approx .1$ one observes that the apparent convergency'' of the series is acceptable for $r_0$ but should be considered at best as marginal for $\delta_{us}$. Clearly, the next term in (\ref{delta_us:FO}) it is important for the interpretation of the measurements. Partial results for the $\as^4$ term in eq.~(\ref{Rtau:FO}) and the $\as^3$ term in eq.~(\ref{delta_us:FO}) have been published \cite{ChBK:tau:as4nf2}. We give the results of the calculation of the missing term as well as its phenomenological implications. \begin{table}[ht] \caption{\label{tab1} Contributions of successive orders of in $\alpha_s$ to $\delta \, r^{(q)kl}$ for the value of $\alpha_s(M_{\tau}) = .334$, and normalized to the value of $\delta \, r^{(q)kl}$ in the Born approximation. First five lines: fixed order PT, second five lines: contour improved version, with RG improvement for the Adler function $D^{(q)}$ ; last five lines: contour improved version with RG summation made for the polarization operator $\Pi^{(q)}$. } \begin{ruledtabular} \begin{tabular*}{\hsize}{l@{\extracolsep{0ptplus1fil}}c@{\extracolsep{0ptplus1fil}}r} $(kl)$ & Perturbative series \\ \colrule (0,0) & $1\ + \ 0.425\ + \ 0.283\ + \ 0.178\ + \ 0.0987 = \ 1.98$ \\ (1,0) & $1\ + \ 0.532\ + \ 0.458\ + \ 0.423\ + \ 0.437 = \ 2.85$ \\ (2,0) & $1\ + \ 0.606\ + \ 0.589\ + \ 0.623\ + \ 0.734 = \ 3.55$ \\ (3,0) & $1\ + \ 0.663\ + \ 0.695\ + \ 0.793\ + \ 1.00 = \ 4.15$ \\ (4,0) & $1\ + \ 0.708\ + \ 0.784\ + \ 0.943\ + \ \ 1.24 = \ 4.68$ \\ \colrule (0,0) & $0.753\ + \ 0.214\ + \ 0.065\ - \ 0.0611\ - \ 0.213 = \ 0.75$ \\ (1,0) & $0.912\ + \ 0.334\ + \ 0.192\ + \ 0.0675\ - \ 0.0969 = \ 1.41$ \\ (2,0) & $1.05\ + \ 0.451\ + \ 0.33\ + \ 0.228\ + \ 0.0802 = \ 2.14$ \\ (3,0) & $1.19\ + \ 0.571\ + \ 0.484\ + \ 0.425\ + \ 0.33 = \ 3.$ \\ (4,0) & $1.32\ + \ 0.697\ + \ 0.657\ + \ 0.665\ + \ 0.664 = \ 4.01$ \\ \colrule (0,0) & $0.857\ + \ 0.122\ - \ 0.0090\ - \ 0.158\ - \ 0.355 = \ 0.458$ \\ (1,0) & $1.11\ + \ 0.232\ + \ 0.11\ + \ -0.0417\ - \ 0.280 = \ 1.13$ \\ (2,0) & $1.35\ + \ 0.347\ + \ 0.251\ + \ 0.124\ - \ 0.121 = \ 1.95$ \\ (3,0) & $1.59\ + \ 0.471\ + \ 0.42\ + \ 0.35\ + \ 0.145 = \ 2.97$ \\ (4,0) & $1.83\ + \ 0.61\ + \ 0.623\ + \ 0.648\ + \ 0.544 = \ 4.26$ \\ \end{tabular*} \end{ruledtabular} \end{table} \section{Calculation and Results} To compute the real part of $\Pi^{(q)}_2(q^2)$ we proceed as follows. First, using the criterion of irreducibility of Feynman integrals \cite{Baikov:criterion:00}, the set of irreducible integrals involved in the problem was constructed. Second, the coefficients multiplying these integrals were calculated as series in the $1/D\rightarrow0$ expansion. Third, the exact answer, i.e. a rational function of $D$, was reconstructed from this expansion. Our results read: \bea \lefteqn{\Pi^{(q)}_{2,us} = -4 -\frac{28}{3}\, a_s + a_s^2 \left\{ -\frac{13981}{108} - \frac{646}{27} \, \zeta_{3} +\frac{2080}{27} \,\zeta_{5}\right\} \nonumber } \ \ \ \ \ \ \ \\ &{+}& a_s^3 \left\{ -\frac{2092745}{1296}-\frac{14713}{162} \,\zeta_{3} -122 \,\zeta_3^2 +10 \,\zeta_{4} %%%%%%%%%% \right. \nonumber \\ &{}& \left. %\left. %\hspace{-1.3cm} %%%%%%%%%%%%%%%% \ \ \ \ \ \ +\frac{41065}{27} \,\zeta_{5} -\frac{79835}{162}\, \zeta_7 \right\} \\ &{}& \hspace{-.6cm}= -4 \left( 1 +2.333\, a_s + 19.58\ a_s^2 + 202.309\ a_s^3 %\left\{(k^{{(q)}3}_{2,us} = 202.38)\right\} \right) {}, \nonumber \label{Pi2m2us:exact} \end{eqnarray} \bea \Pi^{(q)}_{2,ud} &=& a_s^2 \left\{ \frac{128}{9} - \frac{32}{3} \, \zeta_{3} \right\} \\ &+& a_s^3 \left\{ \frac{6392}{27} - \frac{4496}{27} \,\zeta_{3} - 16 \,\zeta_3^2 +\frac{320}{27}\,\zeta_{5} \right\} \nonumber \\ %\hspace{-.6cm} &=& -4 \left( -0.35\, a_s^2 -6.437\ a_s^3 \right) \nonumber \label{Pi2m2ud:exact} {}. \end{eqnarray} \section{Phenomenology} % % First of all, it is instructive to compare the exact result for the ${\cal O}(\als^3)$ contribution to (\ref{Pi2m2us:exact}) with the recently obtained predictions (\cite{ChBK:tau:as4nf2}) based on the optimization schemes as PMS and FAC: \beq k^{(q)3}_{2,us} = 202.309\,(\mbox{exact}), \ \ \ 201 \,(\mbox{PMS}), \ \ \ 199 \,(\mbox{FAC}) \label{Kq3_2:predcitions} {}. \eeq This astonishingly good agreement (a similar phenomenon has been observed for the prediction of the $O(\alpha_s^3)$ term for the correlator of the diagonal currents \cite{ChBK:LL04} can be considered as a strong argument to repeat the procedure and predict, starting from the now completely known $k_2^{(q)3}$, the corresponding result for one loop more, that is for $k_2^{(q)4}$. To be definite, we use the PMS predictions (again for $n_f=3$; the FAC result is very similar) \beq k^{(q)4}_{2,us} = 2276 \pm 200 \ \ \mbox{and} \ \ k^{(q)4}_{2,us} - k^{(q)4}_{2,ud} = 2378 \pm 200 {}. \eeq It is, of course, difficult to estimate uncertainty in the above predictions; however, the simple comparison with eq.~(\ref{Kq3_2:predcitions}) clearly demonstrates that an error of about 10\% should be considered quite conservative. Apart of evaluations using fixed order perturbation theory two formally equivalent versions of the contour improved procedure can be found in the literature. The first \cite{Chetyrkin:1998ej} is based directly on the integration of the polarization function $\Pi^{(q)}_2$, the second \cite{Pich:1998yn} is based on the integration of the Adler function $%$D^{(q)}_2 \equiv s\frac{d}{d s}\Pi^{(q)}_2 %$$ and is obtained from the first one by partial integration. %\input{tables} Partial integration and renormalization group improvement do not commute as long as finite orders are considered. Since the second procedure moves part of the lower order input to higher orders (contrary to the spirit of CIPT) and since, furthermore, the first procedure leads to a somewhat more stable perturbation series \footnote{At least this is true for small values of $\alpha_s(M_{\tau})$; for realistic higher value $\alpha_s(M_{\tau}) = .334$ both series start to oscillate.}, we consider the first of the two choices as preferable. %%%%%%%% insertion 3 starts %%%%%%%%%%%%% The three options for the perturbative series are displayed in Table~\ref{tab1}. As a consequence of the large value of $\alpha_s$ and the rapidly growing coefficients of the perturbative series it seems at first glance difficult to consider any of the theoretical predictions as truly preferable. Nevertheless, the results for moments $(2,0), (3,0)$ and $(4,0)$ are at least in plausible agreement among the three methods and exhibit acceptably decreasing subsequent terms. Since these moments are also relatively most precise, as far as experiment is concerned, they will be used in the subsequent analysis. %%%%%%%% insertion 3 ends %%%%%%%%%%%%% The phenomenological analysis will be based on the most recent evaluation \cite{Czarnecki:2004cw} of $|V_{us}|$. For the phenomenological description of the contribution due to $r^{(0)kl}$ we adopt the analysis presented in \cite{Gamiz:2004ar} for the second version of contour improvement. The results for $m_s$, derived from different moments and different ways of implementing the contour improvement procedure are shown in Table~\ref{tab3}. (Details about the error estimates and the corresponding analysis will be given elsewhere.) The main difference, compared to the previous analysis, is a downward shift of $m_s$ by about 5 MeV from the inclusion of the $\als^4$ terms and an upward shift by as much as 20 MeV from the new input for $|V_{us}|$. The ambiguity for the determined value of $m_s$, including the newly computed $\als^3$ term, and the estimate for the $\als^4$ term is shown in Table~\ref{tab3}. The renormalzation scale $\mu = \xi M_{\tau}$ is allowed to vary between $\xi = 1 - 1.5$. Values of $\mu$ lower than $M_{\tau}$ lead to a blow up of $\als$ and destabilize the result. By others'' we mean all uncertainties (added in quadrature) of the input parameters different from the ${\cal O}(\as^3)$ (or ${\cal O}(\as^4)$) terms in the perturbative contribution. They include experimental errors in the moments as reported in \cite{Abbiendi:2004xa}, in $|V_{us}| = 0.2259(23)$ from \cite{Czarnecki:2004cw} as well as uncertainties in the parameters related to construction of the subtracted longitudinal part. We have computed the uncertainties using numbers from \cite{Gamiz:2004ar}. \begin{table}[ht] \renewcommand{\arraystretch}{1.2} \begin{center} \begin{tabular}{crrrr} \hline Parameter & Value & \quad(2,0)\quad & \quad(3,0)\quad & \quad(4,0) \\ \hline $m_s({\cal O}(\as^3),\mbox{exact})$ & & 123. & 103. & 88. \\ \hline ${\cal O}(\as^3)$ & $\asthree$ & ${}^{-3.5}_{3.9}$ & ${}^{-5.8}_{7.}$ & ${}^{-6.8}_{8.9}$ \\ \hline $\xi$ & ${}^{1.5}_{1}$ & ${}^{-1.4}_{20.}$ & ${}^{4.5}_{9.1}$ & ${}^{7.5}_{2.7}$ \\ \hline $\alpha_s(M_{\tau})$ & $.334 \pm 0.022$ & ${}^{3.8}_{-1.4}$ & ${}^{0.44}_{1.2}$ & ${}^{-1.5}_{2.8}$ \\ \hline others \cite{Abbiendi:2004xa,Gamiz:2004ar,Czarnecki:2004cw}& &${}^{+22.3}_{-25.7}$ & ${}^{+17.}_{-19.1}$ & ${}^{+14.3}_{-15.8}$ \\ \hline Total & & ${}^{+23.}_{-26.}$ & ${}^{+19.}_{-20.}$ & ${}^{+18.6}_{-17.3}$ \\ \hline $m_s({\cal O}(\as^4),\mbox{PMS})$ & & 126. & 100. & 82.4 \\ \hline ${\cal O}(\as^4)$ & $\asfour$ & ${}^{3.9}_{-3.6}$ & ${}^{-2.3}_{2.4}$ & ${}^{-4.6}_{5.6}$ \\ \hline $\xi$ & ${}^{1.5}_{0}$ & ${}^{-8.9}_{0}$ & ${}^{-2.}_{0}$ & ${}^{4.3}_{0}$ \\ \hline $\alpha_s(M_{\tau})$ & $.334 \pm 0.022$ & ${}^{13.}_{-5.4}$ & ${}^{4.2}_{-0.65}$ & ${}^{0.24}_{2.1}$ \\ \hline others \cite{Abbiendi:2004xa,Gamiz:2004ar,Czarnecki:2004cw}& &${}^{+22.9}_{-26.4}$ & ${}^{+16.7}_{-18.8}$ & ${}^{+13.8}_{-15.3}$ \\ \hline Total & & ${}^{+26.4}_{-28.6}$ & ${}^{+17.4}_{-19.}$ & ${}^{+15.7}_{-16.}$ \\ \hline \end{tabular} \end{center} \caption{ Result for $m_s$, derived from different levels of approximation, based on contour improvement from \cite{ChBK:tau:as4nf2} and a list of different contributions to the associated error. \label{tab3}} \end{table} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% From Table \ref{tab1} we find that the inclusion of the $\als^4$ term leads to a better agreement between predictions based on two different methods of implementing of the contour improvement'' approach for the third and the fourth moments. As the former moment also shows smaller theoretical error involved we choose it to derive our final result for $m_s$ which will be given below. In total we find \beq m_s(M_{\tau}) = 100 + \left({}^{+5.}_{-3.} \right)_{\mbox{theo}} \ \ + \left( {}^{+17.}_{-19.} \right)_{\mbox{rest}} \ \mbox{MeV} \label{our:fin} \ {}. \eeq If one compares (\ref{our:fin}) to the ${\cal O}(\als^3)$ result of \cite{Gamiz:2004ar} (corrected for a different $|V_{us}|$) $m_s(M_{\tau}) = 106\ \mbox{MeV}$ (with larger theoretical and identical remaining errors) one sees an essential but still not too large sensitivity to the $\alpha_s^4$ contribution. In fact, for the third moment the shift in $m_s$ due inclusion of the $\alpha_s^4$ term (-2.5 MeV) is about a third of of the corresponding change (-7 MeV) due to the $\alpha_s^3$ contribution. Thus, the purely theoretical uncertainty from not yet computed higher orders could be estimated as about 3 MeV. Unfortunately, one can hardly hope that the error from not yet computed higher orders in $\alpha_s$ could be reduced further by means of a direct calculation in any foreseeable future. The authors are grateful to A.~Pivovarov and K.~Maltman for very useful discussions. This work was supported by the the Deutsche Forschungsgemeinschaft in the Sonderforschungsbereich/Transregio SFB/TR-9 Computational Particle Physics'', by INTAS (grant 03-51-4007), by RFBR (grant 03-02-17177) and by Volkswagen Foundation (grant I/77788). %\input{final.bbl} \begin{thebibliography}{36} \expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi \expandafter\ifx\csname bibnamefont\endcsname\relax \def\bibnamefont#1{#1}\fi \expandafter\ifx\csname bibfnamefont\endcsname\relax \def\bibfnamefont#1{#1}\fi \expandafter\ifx\csname citenamefont\endcsname\relax \def\citenamefont#1{#1}\fi \expandafter\ifx\csname url\endcsname\relax \def\url#1{\texttt{#1}}\fi \expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi \providecommand{\bibinfo}{#2} \providecommand{\eprint}[]{\url{#2}} \bibitem[{\citenamefont{Barate et~al.}(1999)}]{Barate:1999hj} \bibinfo{author}{\bibfnamefont{R.}~\bibnamefont{Barate}} \bibnamefont{et~al.} (\bibinfo{collaboration}{ALEPH}), \bibinfo{journal}{Eur. 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