The final-state evolution is normally started from some initial parton pair , at a scale determined by deliberations already discussed. When the evolution of parton 1 is considered, it is assumed that parton 2 is on-shell, so that the parton 1 energy and momentum are simple functions of its mass (and of the c.m. energy of the pair, which is fixed), and hence also the allowed range for splittings is a function of this mass, eq. (). Correspondingly, parton 2 is evolved under the assumption that parton 1 is on-shell. After both partons have been assigned masses, their correct energies may be found, which are smaller than originally assumed. Therefore the allowed ranges have shrunk, and it may happen that a branching has been assigned a value outside this range. If so, the parton is evolved downwards in mass from the rejected mass value; if both values are rejected, the parton with largest mass is evolved further. It may also happen that the sum of and is larger than the c.m. energy, in which case the one with the larger mass is evolved downwards. The checking and evolution steps are iterated until an acceptable set of , , and has been found.

The procedure is an extension of the veto algorithm, where an initial overestimation of the allowed range is compensated by rejection of some branchings. One should note, however, that the veto algorithm is not strictly applicable for the coupled evolution in two variables ( and ), and that therefore some arbitrariness is involved. This is manifest in the choice of which parton will be evolved further if both values are unacceptable, or if the mass sum is too large.

For a pair of particles which comes from the decay of a resonance within the Standard Model or its MSSM supersymmetric extension, the first branchings are matched to the explicit first-order matrix elements for decays with one additional gluon in the final state, see subsection below. Here we begin by considering in detail how is matched to the matrix element for [Ben87a].

The matching is based on a mapping of the parton-shower variables
on to the 3-jet phase space. To produce a 3-jet event,
, in the shower language,
one will pass through an intermediate
state, where either the or the
is off the mass shell.
If the former is the case then

where . The emission case is obtained with . The parton-shower splitting expression in terms of and , eq. (), can therefore be translated into the following differential 3-jet rate:

where the first term inside the curly bracket comes from emission off the quark and the second term from emission off the antiquark. The corresponding expression in matrix-element language is

With the kinematics choice of PYTHIA, the matrix-element expression is always smaller than the parton-shower one. It is therefore possible to run the shower as usual, but to impose an extra weight factor , which is just the ratio of the expressions in curly brackets. If a branching is rejected, the evolution is continued from the rejected value onwards (the veto algorithm). The weighting procedure is applied to the first branching of both the and the , in each case with the (nominal) assumption that none of the other partons branch (neither the sister nor the daughters), so that the relations of eq. () are applicable.

If a photon is emitted instead of a gluon, the emission rate in
parton showers is given by

(175) |

and in matrix elements by [Gro81]

(176) |

Compared with the standard matrix-element treatment, a few differences remain. The shower one automatically contains the Sudakov form factor and an running as a function of the scale of the branching. The shower also allows all partons to evolve further, which means that the naïve kinematics assumed for a comparison with matrix elements is modified by subsequent branchings, e.g. that the energy of parton 1 is reduced when parton 2 is assigned a mass. All these effects are formally of higher order, and so do not affect a first-order comparison. This does not mean that the corrections need be small, but experimental results are encouraging: the approach outlined does as good as explicit second-order matrix elements for the description of 4-jet production, better in some respects (like overall rate) and worse in others (like some angular distributions).