$ and $$ represent the mean, flavor-inclusive, forward-backward charge flows in left- and right-handed events. These quantities are measured using the momentum-weighted charge technique described below. %Since $\absp$ and $A_e$ factor out of the sums, the ratio of these charge The ratio of these charge asymmetries has the simple form %asymmetries (assuming the sum is not identically zero) has the simple form: \begin{eqnarray} %\qlrraw & \equiv & {<\tqr_{FB}> \over} \qlrraw & \equiv & {<\qr_{FB}> \over <\tqr_{FB}>} %\nonumber \\ % ~=~ {<\qr_{FB,L}> f_L\ +\ <\qr_{FB,R}> f_R \over % <\qr_{FB,L}> f_L\ -\ <\qr_{FB,R}> f_R} %\nonumber \\ % & = & {\absp \over A_e}. ~=~ {A_e \over \absp}. \label{QLR} \end{eqnarray} The expression for $\qlrraw$ shows that uncertainties in the detector %and jet-charge measurement cancel out, thus effectively eliminating the acceptance, charge measurement, and the dilution factors cancel out, thus effectively eliminating the dependence on Monte Carlo simulation for such corrections. Many systematic instrumental effects were investigated and are discussed below. %Many of the remaining errors due to instrumental effects are also %reduced in the $<\tilde{Q}_{FB}>$ measurement, but not in $ $, %\footnote{This is because in $<\tilde{Q}_{FB}>$ we measure the difference of %the charge flow for left and right handed electrons while in %$ $ we measure the sum} and %must be considered for corrections and systematic errors to $\qlrraw$. %In summary, comparing $A_e$ obtained from $A^{obs}_Q$ ~to that obtained from %$A^{obs}_{LR}$ %provides an inclusive test of the Standard Model description of hadronic %\z0 decays. Furthermore, By measuring the quantity $\qlrraw\absp$, ~$A_e$ can be obtained in a manner largely independent of the $A^{obs}_{LR}$ measurement \cite{9}. % Since $\qlrraw$ is formed from the ratio of $A_e$ %and $\absp$, whereas $\alrraw$ is formed from their product Furthermore, the two measurements can be combined to yield $A_e$ without a measurement of the electron polarization, using the expression \begin{equation} A_e\ =\ \sqrt{\alrraw \times \qlrraw}. \label{AeNOP} \end{equation} %This method takes advantage of both measurements of $A_e$ and eliminates the %systematic error due to the polarization measurement. This determination of A$_e$ is not independent of the more precise measurement using $A^{obs}_{LR}$ and the longitudinal polarization that has been published elsewhere \cite{2,3}. %Alternatively, %~$\qlrraw$~ and ~$\alrraw$~ may be combined to obtain a check on %the measured polarization value:\footnote{In principle, this would be an %effective polarization sensitive to both electron-beam and positron-beam %polarizations if present. However, we interpret it here as the electron-beam %polarization, since the positron-beam is unpolarized \cite{SLDALR2}.} % However, we estimate that the positron-beam %polarization is less than $1.5\times 10^{-5} \cite{SLDALR2}, which is %negligible compared to the electron-beam polarization of the data sample %on which this article is based.} %\begin{equation} %\absp \ =\ \sqrt{\alrraw ~\qlrraw}. %\label{P} %\end{equation} In this paper, we present the first measurement of $A_e$ from $\qlrraw$ and the electron-beam polarization. We also present an alternative measurement of $A_e$ from $\alrraw$ and $\qlrraw$ that does not require knowledge of the polarization magnitude. %\section{The SLD and Event Selection} Details of the slac Linear Collider (SLC), the polarized electron source, the measurement of the electron-beam polarization with the Compton polarimeter, and the SLD have been given elsewhere \cite{1,2,10}. The results presented in this article are based upon a sample of data corresponding to an integrated luminosity of $5.1~pb^{-1}$. The data were recorded at a mean center-of-mass energy of $91.27\pm 0.02$ GeV during the 1993 and 1994-1995 runs of the SLC. The momenta of charged particles were measured in the central drift chamber (CDC). Accepted particles were required to have: (i) a minimum momentum transverse to the beam axis $>$ 0.15 GeV/c; (ii) a polar angle $\theta$ with respect to the beam axis satisfying $\vert\cos\theta\vert < 0.8$; and (iii) a point of closest approach to the beam axis within a cylinder of $5~cm$ radius and $10~cm$ half-length about the interaction point. If any remaining particle in an event had a total momentum $>$ 55 GeV/c the event was rejected. Each event was divided into two hemispheres by a plane transverse to the thrust axis \cite{11} which was determined using all accepted charged particles in the event. Hadronic events were selected by the following requirements: (i) the polar angle of the thrust axis satisfied $\vert\cos\theta_{T}\vert < 0.7$; (ii) there were at least three particles per hemisphere; (iii) the total energy of the particles in the event (assuming the particles to be pions) was greater than 20\% of the center-of-mass energy; (iv) the scalar sum per hemisphere of particle momentum components parallel to the thrust axis was greater than 10\% of the beam energy; and (v) the invariant mass of the particles in at least one hemisphere was greater than $2~GeV/c^2$. A total of 49,850 hadronic Z decays produced by left-handed electrons and 39,988 produced by right-handed electrons were obtained with an estimated non-Z background of less than .05\% \cite{12}. The effect of the residual $\tau^+\tau^-$ events on the value of A$_Q^{obs}$ was estimated to be (0.028 $\pm$ 0.012)\% which is negligible. This is relevant because final-state polarization effects in this channel complicate its contribution to this quantity. The luminosity-weighted polarization for this sample of events was $0.730\pm 0.008$, where the error is predominantly systematic \cite{13}. %\section{Method of Analysis and Results} The forward-backward charge asymmetries were determined in the following manner. A unit vector along the thrust axis,~$\bf{\hat T}$, ~was chosen such that ${\bf{\hat T}\cdot} {\bf p}_{e^-} > 0$, where ${\bf p}_{e^-}$ is the electron beam direction. Tracks with momentum vector {\bf p} were defined as forward if ${\bf p \cdot {\hat T}} > 0$, ~and backward otherwise. The weighted charge in the forward hemisphere was then calculated for each event from \begin{equation} Q_F = {\sum_{{\bf p}_i \cdot {\bf\hat T}>0} \vert {\bf p}_i \cdot {\bf\hat T}\vert q_i \over \sum_{{\bf p}_i \cdot {\bf\hat T}>0} \vert {\bf p}_i \cdot {\bf\hat T}\vert} \label{QF} \end{equation} where $q_i$ is the charge of particle $i$. The charge in the backward hemisphere, $Q_B$, was determined in a similar manner for tracks with ${\bf p \cdot {\hat T}} < 0$. The quantity $Q_{FB}=Q_F-Q_B$ was then found for each event. %The distribution for $Q_{FB}$ is shown separately for left and %right handed events in Fig.~\ref{qfblrfigs}. The distribution of $Q_{FB}$ was formed separately for left- and right-handed events. The distributions for $<\!\tqr_{FB}\!>$ and $<\!Q_{FB}\!>$ were obtained in accordance with \mbox{Eqs.~(\docLink{slac-pub-7069.tcx}[QFBPOL]{5})~and~(\docLink{slac-pub-7069.tcx}[QFB]{6})} and are shown in Fig.~\docLink{slac-pub-7069.tcx}[qfbfigs]{1}. The averages $<\!\tqr_{FB}\!>$ and $<\!Q_{FB}\!>$ were obtained from their corresponding distributions \cite {14}. Then $\qlrraw$ was determined using \mbox{Eq.~(\docLink{slac-pub-7069.tcx}[QLR]{7})}. A value for $A^{obs}_{LR}$ was also obtained using the number of accepted left- and right-handed events. These results are summarized in \mbox{Table~\docLink{slac-pub-7069.tcx}[results]{\uppercase {i}}}. %\section{Systematic Errors} %\label{sec:sys} We investigated a number of possible systematic errors due to biases in instrumentation, analysis misindentification, charge dependent nuclear interactions of low momentum hadrons, unphysical measured momenta, material asymmetries, and various backgrounds. We studied the possibility of a charge-dependent, forward-backward bias in the measured track sagitta, or momenta, by means of the dimuon and Bhabha events in the data sample. This can produce an artificial change in $ $, while affecting $<\tqr_{FB}>$ very little, thus biasing $A^{obs}_Q$ \cite{16}. %For each of the tracks, we %determined $D=q [1/p_\perp - 1/(E_{beam}\cos\lambda)]$ as a %function of $\tan\lambda$, where $p_\perp$ is the magnitude of a particle's %momentum transverse to the beam axis, $\lambda$ is the %dip angle from the plane transverse to the beam axis, and $q$ is the %measured charge sign. An approximately linear dependence of D with %$\tan\lambda$ is to be expected for small sense wire displacements that grow %in proportion to the $z$ position of a measurement. A linear fit to %$ $ vs. $\tan\lambda$ describes the data well. This fit is shown in Fig.~ %\ref{sagbiasfigs}. The slope of this fit $s=(-0.4\pm 1.7) \times 10^{-4} %GeV^{-1}$ was used to correct the data by applying %a factor $1 + q s p_z$ to each component of a particle's momentum. This study led to a $(-1.6\pm 6.5)\%$ change in $A^{obs}_Q$. This error was the largest of the systematic errors studied. The systematic errors on the value of A$_Q^{obs}$ resulting from these studies are presented in Table II. %,with a relative uncertainty of 6.6\% in the corrected value %We measured and or simulated other possible effects due to geometrical and %material asymmetries in the detector that could affect the value of %$A_Q^{obs}$ \cite {COLO}. We included their contribution as possible %systematic errors and are shown in Table II. %A charge-independent, forward-backward bias to the measured particle sagitta %could occur if the mean position of the interaction point or the midpoint of %the CDC were displaced along the beam axis with respect to the symmetry point %of the magnetic field. This would produce a change in $<\!Q_{FB}\!>$ %if the mean, net event charge for the sample of events is non-zero. %Charge-dependent nuclear interactions of low momentum pions in the material %preceding the CDC has been determined to give rise to a small, net total %weighted %charge per event of $<\!Q_{tot}\!> = <\!Q_F+Q_B\!> = (11.7\pm 1.8)10^{-3}$. %This bias and its effect on $A^{obs}_Q$ was determined directly as for the %charge-dependent, forward-backward bias, but ignoring the particle's charge %This bias and its effect on $A^{obs}_Q$ was determined in a similar manner as %the charge-dependent, forward-backward bias by using the actual hadronic data %but ignoring the particle's charge sign. %A bias value of $(1.9\pm 1.7)10^{-4}~GeV^{-1}$ was obtained, giving an %uncertainty of $\pm 0.5\%$ in $A^{obs}_Q$. %Another consequence of the small, net total weighted charge per event is %that it can change $<\!Q_{FB}\!>$ if there is also a forward-backward %difference in the distribution of scattering material, or a %forward-backward difference in track reconstruction efficiency or %acceptance. The uniformity of the SLD beamline material in the central %region was determined by reconstructing gamma conversions. The %forward-backward production asymmetry of the conversions was %$(.22\pm .81)\%$. Multiplying this by the net, total weighted charge %per event, and comparing to $ $, yields a 1.5\% relative error in %$\qlrraw$. We investigated differences in the reconstruction efficiency in the %forward and backward direction. The asymmetry of the total number of tracks %observed in the forward and backward direction was $(7.7\pm 7.8) \times %10^{-4}$. Combining this again with $ $ gives a $\pm 0.2\%$ error in %$A^{obs}_Q$. %The fraction of tracks failing the maximum $p_{tot}$ cut was 0.2\%. Monte %Carlo studies showed that these tracks were due to pattern recognition failures %in dense jets. %Whether these tracks and their events were included in the %The inclusion of events with such tracks altered $A^{obs}_Q$ %by 3.3\%, which we take as an additional systematic uncertainty. %The values of $<\!\tqr_{FB}\!>$ and $<\!Q_{FB}\!>$ depend on the value of %$\kappa$ used in \mbox{Eq.~(\ref{QF})}. This dependence was %expected to cancel out in their ratio, $\qlrraw$. This was %$\kappa$ used in \mbox{Eq.~(\ref{QF})}. This dependence was %determined to cancel out in their ratio, $\qlrraw$. This was %done by varying the value of $\kappa$ between zero (no momentum weighting) %and infinity (using only the highest momentum track in each hemisphere). %No dependence was observed in either the data or in Monte Carlo simulations. %We only include any such systematic error as evaluated for the corrections %using ZFITTER as discussed below. %A forward-backward charge asymmetry in background tracks possibly produced by %the operation of the SLC was estimated by reconstructing a luminosity weighted %subsample of random triggers that were taken close in time to the accepted %hadronic events. The same track selection cuts were applied to these triggers %as for hadronic event triggers, but no event cuts were %made. This resulted in $1.9 \times 10^{-3}$ tracks/event with %$ _{SLC}=(-1.3 \pm 2.0) \times 10^{-4}$. Weighting this asymmetry with %the ratio of the background track multiplicity to hadronic event track %multiplicity gave a negligible uncertainty in $A^{obs}_Q$. %The background due to $e^+e^-$ final states misidentified as hadronic was %estimated from a Monte Carlo simulation to be $(1.2 \pm 0.9)\times 10^{-5}$. %Assuming this background %is forward with maximal $Q_{FB}$ gave an uncertainty in $A^{obs}_Q$ of $\pm %0.5\%$. The background due to two-photon processes was similarly estimated %to be $< 5.8 \times 10^{-5} @ 68\% C.L.$. Assuming this background to be %forward with half-maximal $Q_{FB}$ gave an uncertainty in %$A^{obs}_Q$ of $\pm 0.7\%$. %Accepted $\tau^+\tau^-$ final states could %be included in the analysis since they have the same $\qlr$ value as %$q_f\bar q_f$ final states, providing the correction for final state %polarization effects in $\tau$ decays is negligible for momentum-weighted %charges that are normalized separately for the two hemispheres, event by %event. In order to minimize the uncertainty in %the size and simulation of this effect, we have chosen to largely exclude %the $\tau^+\tau^-$ final states by virtue of the minimum invariant jet mass %cut. %In order to minimize the effect on our results due to $\tau^+\tau^-$ final %states (only as a result of final state polarization effects), we have chosen %to largely exclude this final state by virtue of the minimum invariant jet mass %cut. Monte Carlo studies, including the final state polarization effects, show %that the remaining $\tau^+\tau^-$ background of $(2.8\pm 1.2)10^{-4}$ does not %provide any significant systematic uncertainty in $A^{obs}_Q$. %The $\mu^+\mu^-$ final state background could also be included in the analysis, %but had negligible acceptance. %Final-state polarization effects complicate the contribution of $\tau^+\tau^-$ %outgoing states to the charge asymmetry. Hence we remove these events from our %sample. These events are suppresed by the six track and minimum jet %mass requirements. The residual $\tau$ background is estimated to be %$(0.028\pm 0.012)\%$ which is negligible compared with the other errors. %%and does not affect significantly the measured value of $A^{obs}_Q %\section{Summary} The value for the left-right charge asymmetry, before radiative corrections and including the systematic error from Table II, is \begin{equation} A^{obs}_Q = 0.225\pm 0.056 \; (stat.)\pm 0.019 \; (syst.). \label{QLRINV} \end{equation} To obtain the relevant quantities A$_e$ and $\swein$ from A$_Q^{obs}$ we must correct Eqs. 7 for Z-$\gamma$ interference, $\gamma$ exchange and radiative corrections. These were made to the measured asymmetries using the ZFITTER program \cite{17}. The cancellation of the flavor sum in Eq.~\docLink{slac-pub-7069.tcx}[QLR]{7} is not preserved by these higher order processes, and Eqs.~\docLink{slac-pub-7069.tcx}[QFBPOL]{5} and~\docLink{slac-pub-7069.tcx}[QFB]{6} must be used with ZFITTER to obtain $A_e/|P_e|$. The charge dilution factors $d_f$ were varied by $\pm 20\%$ in a manner that maximizes the variation of the radiative correction to $\qlr$. This results in an uncertainty of $\pm 4\%$ in the corrected value of $\qlr$. After these corrections, the following values are obtained: \begin{eqnarray} A_e & = & 0.162 \pm 0.041 \; (stat.)\pm 0.014 \; (syst.) \nonumber\\ \swein & = & 0.2297 \pm 0.0052 \; (stat.)\pm 0.0018 \; (syst.). \end{eqnarray} These results are largely independent of those previously obtained by SLD from $\alr$, and are in good agreement with them. We can also obtain $A_e$ from $\qlrraw$ and $\alrraw$ using \mbox{Eq.~(\docLink{slac-pub-7069.tcx}[AeNOP]{8})}, without the use of the Compton-measured polarization. After radiative corrections to the measured results, we obtain: \begin{eqnarray} A_e & = & 0.1574 \pm 0.0197 \; (stat.) \pm 0.0067 \; (syst.) \nonumber\\ \swein & = & 0.2302 \pm 0.0025 \; (stat.) \pm 0.0009 \; (syst.),\nonumber \end{eqnarray} %with a fitted top mass of %\begin{equation} %m_t = 216\,^{+\,52}_{-\,68}\,(stat.)\,^{+\,19}_{-\,21}\,(syst.)\, %^{+\,17}_{-\,21}\,(\delta M_H)\,GeV/c^2. %\end{equation} %The last error on $m_t$ is due to a variation of the Higgs mass %$65~<~M_H~<~10^3$ GeV/c$^2$. The values $M_Z=91.187$ GeV/c$^2$, %$\alpha_s=.123$, and $M_H = 300$ GeV/c$^2$ were used in calculating the %radiative corrections. This result is not independent of those obtained from $A_{LR}$ and $A^{obs}_Q$ separately. Rather, it is an alternative measurement of $A_e$ and $\swein$ that does not use the measured polarization. This is a completely new technique in the determination of these quantities. %From \mbox{Eq.~(\ref{P})}, they are also %equivalent to %\begin{equation} %\absp = 0.70\,^{+\,0.11}_{-\,0.07}\,(stat.)\,\pm .03 \; (syst.), %\end{equation} %which agrees well with the average, Compton-measured electron-beam %polarization of the selected data sample. % These results can be compared with the latest value of $\swein$ = 0.23049 $\pm$ 0.00050, obtained directly from a measurement of $A_{LR}$ and the electron longitudinal polarization \cite{3}. %\begin{equation} %\swein ~ = ~ 0.23049~\pm~0.00050 %\end{equation} %\acknowledgements We thank the personnel of the slac accelerator department and the technical staffs of our collaborating institutions for their efforts which resulted in the successful operation of the SLC and the SLD. This work was supported by the Department of Energy,; The National Science Foundation; the Instituto Nazionale di Fisica Nucleare of Italy; the Japan-US Cooperative Research Project on High Energy Physics; and the Science and Engineering Research Council of the United Kingdom. % \begin{references} %\bibitem[*]{people} K. Abe, ... \bibitem{1} SLD Collab., K.Abe {\em et al.}, Phys. Rev. Lett. {\bf 70}, 2515 (1993). \bibitem{2} SLD Collab., K.Abe {\em et al.}, Phys. Rev. Lett. {\bf 73}, 25 (1994). \bibitem{3} SLD Collab., K.Abe {\em et al.}, slac pub {\bf 96-7291}, (1995). \bibitem{4} In Ref. 1, 2 we use A$_m$ instead of A$_{LR}^{obs}$. \bibitem{5} ALEPH Collab., D.Decamp {\em et al.}, Phys. Lett. B {\bf 259}, 377 (1991). \bibitem{6} DELPHI Collab., P.Abreu {\em et al.}, Phys. Lett. B {\bf 277}, 371 (1992). \bibitem{7} OPAL Collab., P.D.Acton {\em et al.}, Phys. Lett. B {\bf 294}, 436 (1992). \bibitem{8} It is assumed that the difference in the $d_f$ factors for processes initiated by left handed and right handed electrons is negligible. \bibitem{9} The correlation between $\alrraw$ and $\qlrraw$ is approximately -6\%. Likewise, the correlation coefficient between $A_{LR}$ and $Q_{LR}$ is similar in value for the measurement presented in this letter, since the error in the measured polarization is small compared to the statistical errors in $\alrraw$ and $\qlrraw$. \bibitem{10} R. Alley {\em et al.}, NIM A365, 1 (1995). M. Woods, Proceedings of the 11th International Symposium on High Energy Spin Physics, AIP Conference Proceedings 343, 230 (1994). \bibitem{11} E. Farhi, Phys. Rev. Lett. 39, 1587 (1977). \bibitem{12} Our background is less than in previous studies of the SLD collaboration due to the more stringent cuts in our hadronic sample \cite{15}. \bibitem{13} This is the luminosity-weighted average polarization from two runs with polarization of 0.63 and 0.78. \cite{2,10}. \bibitem{14} In computing the charge asymmetries for each event, a correction for a charge-dependent, forward-backward bias to the track sagitta was applied to each track's momentum as determined by our systematic error studies. \bibitem{15} SLD Collab., K.Abe {\em et al.}, Phys. Rev. D {\bf 51}, 962 (1995). \bibitem{16}This is because in $<\tilde{Q}_{FB}>$ we measure the difference of the charge flow for left- and right-handed electrons, canceling these types of biases, while in $ $ we measure the sum. \bibitem{17} We use the software ZFITTER 4.9 from D.Bardin {\em et al.}, CERN-TH. 6443/92, May 1992 (unpublished). \end{references} % %\begin{figure} %\caption{Distributions of the forward-backward charge flows in left (a) %and right (b) events.} %\label{qfblrfigs} %\end{figure} % \begin{figure} \centerline{\epsfig{file=8109A01.eps,width=3.35 in}} \caption{Distributions of the polarized (a) and unpolarized (b), forward-backward charge flows.} \label{qfbfigs} \end{figure} % %\begin{figure} %\centerline{\epsfig{file=8109A02.eps,width=6.35 in}} %\caption{Study of the bias in the sagitta measurement of $\mu$ pair events.} %\label{sagbiasfigs} %\end{figure} % \begin{table} \caption{Summary of Results} \label{results} \begin{tabular}{lccc} Quantity & Value\\ \hline $f_L, f_R$ & 0.5549, 0.4451\\ %$<\!\tilde{Q}_{FB,L}^{obs}\!>$ & $ -0.0408\pm 0.0027$\\ $<\!{Q}^L_{FB}\!>$ & $ -0.0408\pm 0.0027$\\ %$<\!\tilde{Q}_{FB,R}^{obs}\!>$ & $\;\;\: 0.0322\pm 0.0031$\\ $<\!{Q}^R_{FB}\!>$ & $\;\;\: 0.0322\pm 0.0031$\\ $<\!\tilde{Q}_{FB}\!>$ & $ -0.03697\pm 0.00204$\\ $<\!Q_{FB}\!>$ & $ -0.00831\pm 0.00204$\\ $A_Q^{obs} $ & $\;\;\: 0.2247\pm 0.0556$\\ $A_{LR}^{obs}$ & $\;\;\: 0.1098\pm 0.0033$\\ \end{tabular} \end{table} % \begin{table} \caption{Summary of Systematic Errors} \label{sys} \begin{tabular}{lc} & $\delta A^{obs}_Q/A^{obs}_Q$ \\ Source of uncertainty & (\%) \\ \hline q dependent, F-B sagitta bias & 6.5 \\ q independent, F-B sagitta bias & 0.5 \\ q independent, F-B track efficiency biases & 0.2 \\ unphysical $p_{tot}$ tracks & 3.3 \\ F-B asymmetry of SLD central material & 1.5 \\ $e^+e^-$ final state backgrounds & 0.5 \\ two photon backgrounds & 0.7 \\ radiative corrections & 4.0 \\ polarization measurement (for result~(\docLink{slac-pub-7069.tcx}[QLRINV]{10}) only) & 1.1 \\ Residual $\tau^+\tau^-$ effect & 0.03 \\ SLC track backgrounds & 0.02 \\ \hline Total & 8.7 \\ \end{tabular} \end{table} %