The full machinery needed for a multiparton system is very complicated, and is described in detail in [Sjö84]. The following outline is far from complete, and is complicated nonetheless. The main message to be conveyed is that a Lorentz covariant algorithm exists for handling an arbitrary parton configuration, but that the necessary machinery is more complex than in either cluster or independent fragmentation.
Assume partons, with ordering along the string, and related four-momenta, given by . The initial string then contains separate pieces. The string piece between the quark and its neighbouring gluon is, in four-momentum space, spanned by one side with four-momentum and another with . The factor of 1/2 in the second expression comes from the fact that the gluon shares its energy between two string pieces. The indices `' and `' denotes direction towards the and end, respectively. The next string piece, counted from the quark end, is spanned by and , and so on, with the last one being and .
For the algorithm to work, it is important that all be light-cone-like, i.e. . Since gluons are massless, it is only the two endpoint quarks which can cause problems. The procedure here is to create new vectors for each of the two endpoint regions, defined to be linear combinations of the old ones for the same region, with coefficients determined so that the new vectors are light-cone-like. De facto, this corresponds to replacing a massive quark at the end of a string piece with a massless quark at the end of a somewhat longer string piece. With the exception of the added fictitious piece, which anyway ends up entirely within the heavy hadron produced from the heavy quark, the string motion remains unchanged by this.
In the continued string motion, when new string regions appear as time goes by, the first such string regions that appear can be represented as being spanned by one and another four-vector, with and not necessarily adjacent. For instance, in the case, the `third' string region is spanned by and . Later on in the string evolution history, it is also possible to have regions made up of two or two momenta. These appear when an endpoint quark has lost all its original momentum, has accreted the momentum of a gluon, and is now re-emitting this momentum. In practice, these regions may be neglected. Therefore only pieces made up by a pair of momenta are considered in the program.
The allowed string regions may be ordered in an abstract parameter
plane, where the indices of the four-momentum pairs define
the position of each region along the two (parameter plane)
coordinate axes. In this plane the fragmentation procedure can be
described as a sequence of steps, starting at the quark end,
where , and ending at the antiquark one,
. Each step is taken from an `old'
pair production vertex, to the
production vertex of a `new'
pair, and the string piece between these two string
breaks represent a hadron. Some steps may be taken within one and
the same region, while others may have one vertex in one region
and the other vertex in another region. Consistency requirements,
like energy-momentum conservation, dictates that vertex and
region values be ordered in a monotonic sequence, and that
the vertex positions are monotonically ordered inside each region.
The four-momentum of each hadron can be read off, for ()
momenta, by projecting the separation between the old and the new
vertex on to the () axis. If the four-momentum fraction of
taken by a hadron is denoted ,
then the total hadron four-momentum is given by
The pairs are the transverse momenta produced at
the two string breaks, and the
four-vectors transverse to the string directions in the regions
of the respective string breaks:
The fact that the hadron should be on mass shell, , puts
one constraint on where a new breakup may be, given that the old
one is already known, just as eq. () did in the
simple 2-jet case. The remaining degree of freedom is, as before,
to be given by
the fragmentation function . The interpretation of the
is only well-defined for a step entirely constrained to one of the
initial string regions, however, which is not enough. In the
2-jet case, the values can be related to the proper times
of string breaks, as follows. The variable
, with the string tension and
the proper time between the production vertex of the
partons and the breakup point, obeys an iterative relation of the
The algorithm for finding the next breakup then works something
like follows. Pick a hadron, , and , and calculate the next
. If the old breakup is in the region , and if the
new breakup is also assumed to be in the same region, then the
and constraints can be reformulated in terms of
the fractions and the hadron must take of the
total four-vectors and :