Shower Evolution

In the leading-logarithmic picture, a shower may be viewed as a sequence of $1 \to 2$ branchings $a \to bc$. Here $a$ is called the mother and $b$ and $c$ the two daughters. Each daughter is free to branch in its turn, so that a tree-like structure can evolve. We will use the word `parton' for all the objects $a$, $b$ and $c$ involved in the branching process, i.e. not only for quarks and gluons but also for leptons and photons. The branchings included in the program are $\mathrm{q}\to \mathrm{q}\mathrm{g}$, $\mathrm{g}\to \mathrm{g}\mathrm{g}$, $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$, $\mathrm{q}\to \mathrm{q}\gamma$ and $\ell \to \ell \gamma$. Photon branchings, i.e. $\gamma \to \mathrm{q}\overline{\mathrm{q}}$ and $\gamma \to \ell \overline{\ell}$, have not been included so far, since they are reasonably rare and since no urgent need for them has been perceived.

A word on terminology may be in place. The algorithms described here are customarily referred to as leading-log showers. This is correct insofar as no explicit corrections from higher orders are included, i.e. there are no $\mathcal{O}(\alpha_{\mathrm{s}}^2)$ terms in the splitting kernels, neither by new $1\to 3$ processes nor by corrections to the $1 \to 2$ ones. However, it is grossly misleading if leading-log showers are equated with leading-log analytical calculations. In particular, the latter contain no constraints from energy-momentum conservation: the radiation off a quark is described in the approximation that the quark does not lose any energy when a gluon is radiated, so that the effects of multiple emissions factorize. Therefore energy-momentum conservation is classified as a next-to-leading-log correction. In a Monte Carlo shower, on the other hand, energy-momentum conservation is explicit branching by branching. By including coherence phenomena and optimized choices of $\alpha_{\mathrm{s}}$ scales, further information on higher orders is inserted. While the final product is still not certified fully to comply with a NLO/NLL standard, it is well above the level of an unsophisticated LO/LL analytic calculation.