Measurements of cross section and asymmetry for<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow><mml:mo>→</mml:mo><mml:mi>b</mml:mi><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>¯</mml:mi></mml:mrow></mml:mover></mml:mrow></mml:mrow></mml:math>and heavy quark fragmentation at KEK TRISTAN

Using 773 muons found in hadronic events from 142 ${\mathrm{pb}}^{\ensuremath{-}1}$ of data at a c.m. energy of 57.8 GeV, we extract the cross section and forward-backward charge asymmetry for the ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}b\overline{b}$ process, and the heavy quark fragmentation function parameters for the Peterson model. For the analysis of the ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}b\overline{b}$ process, we use a method in which the behavior of the $c$ quark and lighter quarks is assumed, with only that of the $b$ quark left indeterminate. The cross section and asymmetry for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}b\overline{b}$ are found to be ${R}_{b} = 0.57 \ifmmode\pm\else\textpm\fi{} 0.06(\mathrm{stat}) \ifmmode\pm\else\textpm\fi{} 0.08(\mathrm{syst})$ and ${A}_{b} = \ensuremath{-}0.59 \ifmmode\pm\else\textpm\fi{} 0.09 \ifmmode\pm\else\textpm\fi{} 0.09$, respectively. They are consistent with the standard model predictions. For the study of the fragmentation function we use the variable $〈{x}_{E}〉$, the fraction of the beam energy carried by the heavy hadrons. We obtain ${〈{x}_{E}〉}_{c}={0.56}_{\ensuremath{-}0.05\ensuremath{-}0.03}^{+0.04+0.03}$ and ${〈{x}_{E}〉}_{b}={0.65}_{\ensuremath{-}0.04\ensuremath{-}0.06}^{+0.06+0.05}$, respectively. These are in good agreement with previously measured values.