Initial-state radiation

Initial-state photon radiation has been included using the formalism of ref. [Ber82]. Here each event contains either no photon or one, i.e. it is a first-order non-exponentiated description. The main formula for the hard radiative photon cross section is

$$\frac{\d\sigma}{\d x_{\gamma}} = \frac{\alpha_{\mathrm{em}}}... ...rac{1 + (1-x_{\gamma})^2}{x_{\gamma}} \, \sigma_0 (\hat{s}) ~,$$ (39)

where $x_{\gamma}$ is the photon energy fraction of the beam energy, $\hat{s} = (1-x_{\gamma}) s$ is the squared reduced hadronic c.m. energy, and $\sigma_0$ is the ordinary annihilation cross section at the reduced energy. In particular, the selection of jet flavours should be done according to expectations at the reduced energy. The cross section is divergent both for $x_{\gamma} \to 1$ and $x_{\gamma} \to 0$. The former is related to the fact that $\sigma_0$ has a $1/\hat{s}$ singularity (the real photon pole) for $\hat{s} \to 0$. An upper cut on $x_{\gamma}$ can here be chosen to fit the experimental setup. The latter is a soft photon singularity, which is to be compensated in the no-radiation cross section. A requirement $x_{\gamma} > 0.01$ has therefore been chosen so that the hard-photon fraction is smaller than unity. In the total cross section, effects from photons with $x_{\gamma} < 0.01$ are taken into account, together with vertex and vacuum polarization corrections (hadronic vacuum polarizations using a simple parameterization of the more complicated formulae of ref. [Ber82]).

The hard photon spectrum can be integrated analytically, for the full $\gamma^* / \mathrm{Z}^0$ structure including interference terms, provided that no new flavour thresholds are crossed and that the $R_{\mathrm{QCD}}$ term in the cross section can be approximated by a constant over the range of allowed $\hat{s}$ values. In fact, threshold effects can be taken into account by standard rejection techniques, at the price of not obtaining the exact cross section analytically, but only by an effective Monte Carlo integration taking place in parallel with the ordinary event generation. In addition to $x_{\gamma}$, the polar angle $\theta_{\gamma}$ and azimuthal angle $\varphi_{\gamma}$ of the photons are also to be chosen. Further, for the orientation of the hadronic system, a choice has to be made whether the photon is to be considered as having been radiated from the $\mathrm{e}^+$ or from the $\mathrm{e}^-$.

Final-state photon radiation, as well as interference between initial- and final-state radiation, has been left out of this treatment. The formulae for $\mathrm{e}^+\mathrm{e}^-\to \mu^+ \mu^-$ cannot be simply taken over for the case of outgoing quarks, since the quarks as such only live for a short while before turning into hadrons. Another simplification in our treatment is that effects of incoming polarized $\mathrm{e}^{\pm}$ beams have been completely neglected, i.e. neither the effective shift in azimuthal distribution of photons nor the reduction in polarization is included. The polarization parameters of the program are to be thought of as the effective polarization surviving after initial-state radiation.