Cross sections of muon and charged pion pair production in electron-positron annihilation near the threshold

The processes of muon (tau) and charged pion pair production at electron-positron annihilation with $\mathcal{O}(\alpha )$ radiative corrections are considered. The calculation results are presented assuming the energies of final particles (center of mass system implied) to be not far significantly from threshold production. The invariant mass distributions for the muon (tau) and pion pairs are obtained both for the initial and final state radiation. Some analytical calculations are illustrated numerically. The pions were assumed to be pointlike objects and scalar QED was applied for calculation. The QED radiative corrections related to the final state radiation, in addition to the well-known Coulomb factor, are treated near threshold region exactly.

Figure 1

Final state radiation corrections to $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ process.

Figure 2

Distribution on $\nu$ for FSR of muons, i.e. the value $z=(2{\alpha }^3/3s{)}^{-1}(d{\sigma }_{\mathrm{FSR}}^h/d\nu )$ [see (5)] is shown.

Figure 3

The dependence of quantity ${\Delta }_{\mathrm{FSR}}^{{\mu }^{+}{\mu }^{-}}(\beta )$ as a function of $\beta$ for FSR of muons. See formula (16) and its asymptotic behavior.

Figure 4

Initial state radiation corrections to $e^{+}{e}^{-}\to {\mu }^{+}{\mu }^{-}$ process.

Figure 5

Distribution of muon pairs as a function on $\nu$ for ISR. Vertical axis represents the quantity $y=[(4{\alpha }^3/3s)(l_e-1){]}^{-1}(d{\sigma }_{\mathrm{ISR}}^h/d\nu )$ [see (21)] and horizontal axis represents the fraction of radiated photon energy $\nu$.

Figure 6

Distribution of muon pairs on $\beta$ for ISR. See formula (25) for the quantity ${\Delta }_{\mathrm{ISR}}^{{\mu }^{+}{\mu }^{-}}(\beta )$.

Figure 7

The pion invariant mass distribution on $\nu$ for FSR. The vertical and horizontal axes represent the value $z=[(2{\alpha }^3/3s){]}^{-1}(d{\sigma }_{\mathrm{FSR}}^h/d\nu )$ [see (31)] and fraction of photon energy, respectively.

Figure 8

The dependence of quantity ${\Delta }_{\mathrm{FSR}}^{{\pi }^{+}{\pi }^{-}}(\beta )$ on $\beta$ for FSR of pions. See formula (33) and its asymptotic behavior.

Figure 9

The ratio of the cross sections with $\mathrm{ISR}+\mathrm{FSR}$ divided on the cross section with ISR as a function of energy in c.m. frame. The different smooth curves represent this ratio vs threshold photon energy (in MeV) to be detected.

Figure 10

The distribution pion pairs as a function on $\nu$ for ISR. The vertical axis represents the quantity $y=({\alpha }^3/3s(l_e-1){)}^{-1}d{\sigma }_{\mathrm{ISR}}^{e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}\gamma }/d\nu$ [see (35)] and the horizontal axis represents the fraction of radiated photon energy.

Figure 11

The dependence of the quantity ${\Delta }_{\mathrm{ISR}}^{{\pi }^{+}{\pi }^{-}}(\beta )$ [see (39)] on $\beta$ for ISR.