Lepton beams

Lepton beams have to be handled slightly differently from what has been described so far. One also has to distinguish between a lepton for which parton distributions are included and one which is treated as an unresolved point-like particle. The necessary modifications are the same for $2 \to 2$ and $2 \to 1$ processes, however, since the $\hat{t}$ degree of freedom is unaffected.

If one incoming beam is an unresolved lepton, the corresponding parton-distribution piece collapses to a $\delta$ function. This function can be used to integrate out the $y$ variable: $\delta(x_{1,2} - 1) = \delta(y \pm (1/2) \ln \tau)$. It is therefore only necessary to select the $\tau$ and the $z$ variables according to the proper distributions, with compensating weight factors, and only one set of parton distributions has to be evaluated explicitly.

If both incoming beams are unresolved leptons, both the $\tau$ and the $y$ variables are trivially given: $\tau = 1$ and $y = 0$. Parton-distribution weights disappear completely. For a $2 \to 2$ process, only the $z$ selection remains to be performed, while a $2 \to 1$ process is completely specified, i.e. the cross section is a simple number that only depends on the c.m. energy.

For a resolved electron, the $f_{\mathrm{e}}^{\mathrm{e}}$ parton distribution is strongly peaked towards $x = 1$. This affects both the $\tau$ and the $y$ distributions, which are not well described by either of the pieces in $h_{\tau}(\tau)$ or $h_y(y)$ in processes with interacting $\mathrm{e}^{\pm}$. (Processes which involve e.g. the $\gamma$ content of the $\mathrm{e}$ are still well simulated, since $f_{\gamma}^{\mathrm{e}}$ is peaked at small $x$.)

If both parton distributions are peaked close to 1, the $h_{\tau}(\tau)$ expression in eq. () is therefore increased with one additional term of the form $h_{\tau}(\tau) \propto 1 / (1 - \tau)$, with coefficients $c_7$ and ${\cal I}_7$ determined as before. The divergence when $\tau \to 1$ is cut off by our regularization procedure for the $f_{\mathrm{e}}^{\mathrm{e}}$ parton distribution; therefore we only need consider $\tau < 1 - 2 \times 10^{-10}$.

Correspondingly, the $h_y(y)$ expression is expanded with a term $1/(1 - \exp(y-y_0))$ when incoming beam number 1 consists of a resolved $\mathrm{e}^{\pm}$, and with a term $1/(1 - \exp(-y-y_0))$ when incoming beam number 2 consists of a resolved $\mathrm{e}^{\pm}$. Both terms are present for an $\mathrm{e}^+\mathrm{e}^-$ collider, only one for an $\mathrm{e}\mathrm{p}$ one. The coefficient $y_0 = - (1/2) \ln \tau$ is the naïve kinematical limit of the $y$ range, $\vert y\vert < y_0$. From the definitions of $y$ and $y_0$ it is easy to see that the two terms above correspond to $1/(1-x_1)$ and $1/(1-x_2)$, respectively, and thus are again regularized by our parton-distribution function cut-off. Therefore the integration ranges are $y < y_0 -10^{-10}$ for the first term and $y > - y_0 + 10^{-10}$ for the second one.