Mixing processes

In the cross-section formulae given so far, we have deliberately suppressed a summation over the allowed incoming flavours. For instance, the process $\mathrm{f}\overline{\mathrm{f}}\to \mathrm{Z}^0$ in a hadron collider receives contributions from $\u\overline{\mathrm{u}}\to \mathrm{Z}^0$, $\d\overline{\mathrm{d}}\to \mathrm{Z}^0$, $\mathrm{s}\overline{\mathrm{s}}\to \mathrm{Z}^0$, and so on. These contributions share the same basic form, but differ in the parton-distribution weights and (usually) in a few coupling constants in the hard matrix elements. It is therefore convenient to generate the terms together, as follows:

61.

A phase-space point is picked, and all common factors related to this choice are evaluated, i.e. the Jacobian and the common pieces of the matrix elements (e.g. for a $\mathrm{Z}^0$ the basic Breit-Wigner shape, excluding couplings to the initial flavour).

62.

The parton-distribution-function library is called to produce all the parton distributions, at the relevant $x$ and $Q^2$ values, for the two incoming beams.

63.

A loop is made over the two incoming flavours, one from each beam particle. For each allowed set of incoming flavours, the full matrix-element expression is constructed, using the common pieces and the flavour-dependent couplings. This is multiplied by the common factors and the parton-distribution weights to obtain a cross-section weight.

64.

Each allowed flavour combination is stored as a separate entry in a table, together with its weight. In addition, a summed weight is calculated.

65.

The phase-space point is kept or rejected, according to a comparison of the summed weight with the maximum weight obtained at initialization. Also the cross-section Monte Carlo integration is based on the summed weight.

66.

If the point is retained, one of the allowed flavour combinations is picked according to the relative weights stored in the full table.

Generally, the flavours of the final state are either completely specified by those of the initial state, e.g. as in $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{g}$, or completely decoupled from them, e.g. as in $\mathrm{f}\overline{\mathrm{f}}\to \mathrm{Z}^0 \to \mathrm{f}'\overline{\mathrm{f}}'$. In neither case need therefore the final-state flavours be specified in the cross-section calculation. It is only necessary, in the latter case, to include an overall weight factor, which takes into account the summed contribution of all final states that are to be simulated. For instance, if only the process $\mathrm{Z}^0 \to \mathrm{e}^+\mathrm{e}^-$ is studied, the relevant weight factor is simply $\Gamma_{\mathrm{e}\mathrm{e}} / \Gamma_{\mathrm{tot}}$. Once the kinematics and the incoming flavours have been selected, the outgoing flavours can be picked according to the appropriate relative probabilities.

In some processes, such as $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$, several different colour flows are allowed, each with its own kinematical dependence of the matrix-element weight, see section . Each colour flow is then given as a separate entry in the table mentioned above, i.e. in total an entry is characterized by the two incoming flavours, a colour-flow index, and the weight. For an accepted phase-space point, the colour flow is selected in the same way as the incoming flavours.

The program can also allow the mixed generation of two or more completely different processes, such as $\mathrm{f}\overline{\mathrm{f}}\to \mathrm{Z}^0$ and $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{g}\mathrm{g}$. In that case, each process is initialized separately, with its own set of coefficients $c_i$ and so on. The maxima obtained for the individual cross sections are all expressed in the same units, even when the dimensionality of the phase space is different. (This is because we always transform to a phase space of unit volume, $\int h_{\tau}(\tau) \, \d\tau \equiv 1$, etc.) The above generation scheme need therefore only be generalized as follows:

67.

One process is selected among the allowed ones, with a relative probability given by the maximum weight for this process.

68.

A phase-space point is found, using the distributions $h_{\tau}(\tau)$ and so on, optimized for this particular process.

69.

The total weight for the phase-space point is evaluated, again with Jacobians, matrix elements and allowed incoming flavour combinations that are specific to the process.

70.

The point is retained with a probability given by the ratio of the actual to the maximum weight of the process. If the point is rejected, one has to go back to step 1 and pick a new process.

71.

Once a phase-space point has been accepted, flavours may be selected, and the event generated in full.

It is clear why this works: although phase-space points are selected among the allowed processes according to relative probabilities given by the maximum weights, the probability that a point is accepted is proportional to the ratio of actual to maximum weight. In total, the probability for a given process to be retained is therefore only proportional to the average of the actual weights, and any dependence on the maximum weight is gone.

In $\gamma\mathrm{p}$ and $\gamma\gamma$ physics, the different components of the photon give different final states, see section . Technically, this introduces a further level of administration, since each event class contains a set of (partly overlapping) processes. From an ideological point of view, however, it just represents one more choice to be made, that of event class, before the selection of process in step 1 above. When a weighting fails, both class and process have to be picked anew.