The shower structure

A fast hadron may be viewed as a cloud of quasi-real partons. Similarly a fast lepton may be viewed as surrounded by a cloud of photons and partons; in the program the two situations are on an equal footing, but here we choose the hadron as example. At each instant, an individual parton can initiate a virtual cascade, branching into a number of partons. This cascade can be described in terms of a tree-like structure, composed of many subsequent branchings $a \to bc$. Each branching involves some relative transverse momentum between the two daughters. In a language where four-momentum is conserved at each vertex, this implies that at least one of the $b$ and $c$ partons must have a space-like virtuality, $m^2 < 0$. Since the partons are not on the mass shell, the cascade only lives a finite time before reassembling, with those parts of the cascade that are most off the mass shell living the shortest time.

A hard scattering, e.g. in deeply inelastic leptoproduction, will probe the hadron at a given instant. The probe, i.e. the virtual photon in the leptoproduction case, is able to resolve fluctuations in the hadron up to the $Q^2$ scale of the hard scattering. Thus probes at different $Q^2$ values will seem to see different parton compositions in the hadron. The change in parton composition with $t = \ln(Q^2/\Lambda^2)$ is given by the evolution equations

$$\frac{\d f_b(x,t)}{\d t} = \sum_{a,c} \int \frac{\d x'}{x'} ... ...a_{abc}}{2 \pi} \, P_{a \to bc} \left( \frac{x}{x'} \right) ~.$$ (185)

Here the $f_i(x,t)$ are the parton-distribution functions, expressing the probability of finding a parton $i$ carrying a fraction $x$ of the total momentum if the hadron is probed at virtuality $Q^2$. The $P_{a \to bc}(z)$ are given in eq. (). As before, $\alpha_{abc}$ is $\alpha_{\mathrm{s}}$ for QCD shower and $\alpha_{\mathrm{em}}$ for QED ones.

Eq. () is closely related to eq. (): $\d {\cal P}_a$ describes the probability that a given parton $a$ will branch (into partons $b$ and $c$), $\d f_b$ the influx of partons $b$ from the branchings of partons $a$. (The expression $\d f_b$ in principle also should contain a loss term for partons $b$ that branch; this term is important for parton-distribution evolution, but does not appear explicitly in what we shall be using eq. () for.) The absolute form of hadron parton distributions cannot be predicted in perturbative QCD, but rather have to be parameterized at some $Q_0$ scale, with the $Q^2$ dependence thereafter given by eq. (). Available parameterizations are discussed in section . The lepton and photon parton distributions inside a lepton can be fully predicted, but here for simplicity are treated on equal footing with hadron parton distributions.

If a hard interaction scatters a parton out of the incoming hadron, the `coherence' [Gri83] of the cascade is broken: the partons can no longer reassemble completely back to the cascade-initiating parton. In this semiclassical picture, the partons on the `main chain' of consecutive branchings that lead directly from the initiating parton to the scattered parton can no longer reassemble, whereas fluctuations on the `side branches' to this chain may still disappear. A convenient description is obtained by assigning a space-like virtuality to the partons on the main chain, in such a way that the partons on the side branches may still be on the mass shell. Since the momentum transfer of the hard process can put the scattered parton on the mass shell (or even give it a time-like virtuality, so that it can initiate a final-state shower), one is then guaranteed that no partons have a space-like virtuality in the final state. (In real life, confinement effects obviously imply that partons need not be quite on the mass shell.) If no hard scattering had taken place, the virtuality of the space-like parton line would still force the complete cascade to reassemble. Since the virtuality of the cascade probed is carried by one single parton, it is possible to equate the space-like virtuality of this parton with the $Q^2$ scale of the cascade, to be used e.g. in the evolution equations. Coherence effects [Gri83,Bas83] guarantee that the $Q^2$ values of the partons along the main chain are strictly ordered, with the largest $Q^2$ values close to the hard scattering.

Further coherence effects have been studied [Cia87], with particular implications for the structure of parton showers at small $x$. None of these additional complications are implemented in the current algorithm, with the exception of a few rather primitive options that do not address the full complexity of the problem.

Instead of having a tree-like structure, where all legs are treated democratically, the cascade is reduced to a single sequence of branchings $a \to bc$, where the $a$ and $b$ partons are on the main chain of space-like virtuality, $m_{a,b}^2 < 0$, while the $c$ partons are on the mass shell and do not branch. (Later we will include the possibility that the $c$ partons may have positive virtualities, $m_c^2 > 0$, which leads to the appearance of time-like `final-state' parton showers on the side branches.) This truncation of the cascade is only possible when it is known which parton actually partakes in the hard scattering: of all the possible cascades that exist virtually in the incoming hadron, the hard scattering will select one.

To obtain the correct $Q^2$ evolution of parton distributions, e.g., it is essential that all branches of the cascade be treated democratically. In Monte Carlo simulation of space-like showers this is a major problem. If indeed the evolution of the complete cascade is to be followed from some small $Q_0^2$ up to the $Q^2$ scale of the hard scattering, it is not possible at the same time to handle kinematics exactly, since the virtuality of the various partons cannot be found until after the hard scattering has been selected. This kind of `forward evolution' scheme therefore requires a number of extra tricks to be made to work. Further, in this approach it is not known e.g. what the $\hat{s}$ of the hard scattering subsystem will be until the evolution has been carried out, which means that the initial-state evolution and the hard scattering have to be selected jointly, a not so trivial task.

Instead we use the `backwards evolution' approach [Sjö85], in which the hard scattering is first selected, and the parton shower that preceded it is subsequently reconstructed. This reconstruction is started at the hard interaction, at the $Q_{\mathrm{max}}^2$ scale, and thereafter step by step one moves `backwards' in `time', towards smaller $Q^2$, all the way back to the parton-shower initiator at the cut-off scale $Q_0^2$. This procedure is possible if evolved parton distributions are used to select the hard scattering, since the $f_i(x,Q^2)$ contain the inclusive summation of all initial-state parton-shower histories that can lead to the appearance of an interacting parton $i$ at the hard scale. What remains is thus to select an exclusive history from the set of inclusive ones.