Transverse evolution

We have above seen that two parton lines may be defined, stretching back from the hard scattering to the initial incoming hadron wavefunctions at small . Specifically, all parton flavours , virtualities and energy fractions may be found. The exact kinematical interpretation of the variable is not unique, however. For partons with small virtualities and transverse momenta, essentially all definitions agree, but differences may appear for branchings close to the hard scattering.

In first-order QED [Ber85] and in some simple QCD toy models [Got86], one may show that the `correct' choice is the ` approach'. Here one requires that , both at the hard-scattering scale and at any lower scale, i.e. , where and are the values of the two resolved partons (one from each incoming beam particle) at the given scale. In practice this means that, at a branching with the splitting variable , the total has to be increased by a factor in the backwards evolution. It also means that branchings on the two incoming legs have to be interleaved in a single monotonic sequence of values of branchings. A problem with this interpretation is that it is not quite equivalent with an definition of parton densities [Col00], or any other standard definition. In practice, effects should not be large from this mismatch.

For a reconstruction of the complete kinematics in this approach,
one should start with the hard scattering, for which
has been chosen according to the hard-scattering matrix element.
By backwards evolution, the virtualities
and
of
the two interacting partons are reconstructed. Initially the two
partons are considered in their common c.m. frame, coming in along
the directions. Then the four-momentum vectors have the
non-vanishing components

(190) |

with .

If, say, , then the branching , which produced parton 1, is the one that took place closest to the hard scattering, and the one to be reconstructed first. With the four-momentum known, is automatically known, so there are four degrees of freedom. One corresponds to a trivial azimuthal angle around the axis. The splitting variable for the vertex is found as the same time as , and provides the constraint . The virtuality is given by backwards evolution of parton 3.

One degree of freedom remains to be specified, and this is related
to the possibility that parton 4 initiates a time-like parton shower,
i.e. may have a non-zero mass. The maximum allowed squared mass
is found for a collinear branching .
In terms of the combinations

(191) |

one obtains

(192) |

These constraints on are only the kinematical ones, in addition coherence phenomena could constrain the values further. Some options of this kind are available; the default one is to require additionally that , i.e. lesser than the space-like virtuality of the sister parton.

With the maximum virtuality given, the final-state showering machinery may be used to give the development of the subsequent cascade, including the actual mass , with . The evolution is performed in the c.m. frame of the two `resolved' partons, i.e. that of partons 1 and 2 for the branching , and parton 4 is assumed to have a nominal energy . (Slight modifications appear if parton 4 has a non-vanishing mass or .)

Using the relation
, the momentum of parton
3 may now be found as

(194) |

The requirement that (or for heavy flavours) imposes a constraint on allowed values. This constraint cannot be included in the choice of , where it logically belongs, since it also depends on and , which are unknown at this point. It is fairly rare (in the order of 10% of all events) that a disallowed value is generated, and when it happens it is almost always for one of the two branchings closest to the hard interaction: for eq. () may be solved to yield , which is a more severe cut for small and large. Therefore an essentially bias-free way of coping is to redo completely any initial-state cascade for which this problem appears.

This completes the reconstruction of the vertex. The subsystem made out of partons 3 and 2 may now be boosted to its rest frame and rotated to bring partons 3 and 2 along the directions. The partons 1 and 4 now have opposite and compensating transverse momenta with respect to the event axis. When the next vertex is considered, either the one that produces parton 3 or the one that produces parton 2, the 3-2 subsystem will fill the function the 1-2 system did above, e.g. the rôle of in the formulae above is now played by . The internal structure of the 3-2 system, i.e. the branching , appears nowhere in the continued description, but has become `unresolved'. It is only reflected in the successive rotations and boosts performed to bring back the new endpoints to their common rest frame. Thereby the hard-scattering subsystem 1-2 builds up a net transverse momentum and also an overall rotation of the hard-scattering subsystem.

After a number of steps, the two outermost partons have virtualities and then the shower is terminated and the endpoints assigned . Up to small corrections from primordial , discussed in section , a final boost will bring the partons from their c.m. frame to the overall c.m. frame, where the values of the outermost partons agree also with the light-cone definition. The combination of several rotations and boosts implies that the two colliding partons have a nontrivial orientation: when boosted back to their rest frame, they will not be oriented along the axis. This new orientation is then inherited by the final state of the collision, including resonance decay products.