A study of the reaction ${e^+ e^- \rightarrow \mu^+ \mu^- \gamma_{\sc ISR}}$ at LEP and search for new physics at annihilation energies near 80 GeV

A study of the channel $e^{+}e^{-}~\rightarrow~\mu^{+}\mu^{-}\gamma_{\ sc ISR}$, where $\gamma_{\sc ISR}$ is an initial state radiation photon, is presented using data collected by the DELPHI experiment from 1991 to 1994. The total cross-section at effective annihilation energies ($\sqrt{s'}$) below the $Z^0$ peak is obtained by using the events with relatively hard initial state radiative photon(s) ($E_{\gamma}~ ~$~1~GeV). The differential cross-section as a function of the muon polar production angle is also determined in order to extract t s alhe forward-backward asymmetries for the reaction $e^+ e^- \rightarrow \mu^+ \mu^-$ at energies $\sqrt{s'}$ between 20 and 87~GeV. The ratio of the helicity cross-sections $\frac{\sigma_{LL}+\sigma_{RR}}{\sigma_{RL}+\sigma_{LR}}$, where the two subscripts stand for the helicities of the incoming $e^-$ and outgoing $\mu^-$ respectively, is extracted from the differential cross-sections in order to test the Standard Model and to look for new physics near 80 GeV. No deviations from the Standard Model were found.