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Fragmentation of a single jet

In the IF approach, it is assumed that the fragmentation of any system of partons can be described as an incoherent sum of independent fragmentation procedures for each parton separately. The process is to be carried out in the overall c.m. frame of the jet system, with each jet fragmentation axis given by the direction of motion of the corresponding parton in that frame.

Exactly as in string fragmentation, an iterative ansatz can be used to describe the successive production of one hadron after the next. Assume that a quark is kicked out by some hard interaction, carrying a well-defined amount of energy and momentum. This quark jet $\mathrm{q}$ is split into a hadron $\mathrm{q}\overline{\mathrm{q}}_1$ and a remainder-jet $\mathrm{q}_1$, essentially collinear with each other. New quark and hadron flavours are picked as already described. The sharing of energy and momentum is given by some probability distribution $f(z)$, where $z$ is the fraction taken by the hadron, leaving $1-z$ for the remainder-jet. The remainder-jet is assumed to be just a scaled-down version of the original jet, in an average sense. The process of splitting off a hadron can therefore be iterated, to yield a sequence of hadrons. In particular, the function $f(z)$ is assumed to be the same at each step, i.e. independent of remaining energy. If $z$ is interpreted as the fraction of the jet $E+p_{\mathrm{L}}$, i.e. energy plus longitudinal momentum with respect to the jet axis, this leads to a flat central rapidity plateau $dn/dy$ for a large initial energy.

Fragmentation functions can be chosen among those listed above for string fragmentation, but also here the default is the Lund symmetric fragmentation function.

The normal $z$ interpretation means that a choice of a $z$ value close to $0$ corresponds to a particle moving backwards, i.e. with $p_{\mathrm{L}} < 0$. It makes sense to allow only the production of particles with $p_{\mathrm{L}} > 0$, but to explicitly constrain $z$ accordingly would destroy longitudinal invariance. The most straightforward way out is to allow all $z$ values but discard hadrons with $p_{\mathrm{L}} < 0$. Flavour, transverse momentum and $E+p_{\mathrm{L}}$ carried by these hadrons are `lost' for the forward jet. The average energy of the final jet comes out roughly right this way, with a spread of 1-2 GeV around the mean. The jet longitudinal momentum is decreased, however, since the jet acquires an effective mass during the fragmentation procedure. For a 2-jet event this is as it should be, at least on average, because also the momentum of the compensating opposite-side parton is decreased.

Flavour is conserved locally in each $\mathrm{q}_i \overline{\mathrm{q}}_i$ splitting, but not in the jet as a whole. First of all, there is going to be a last meson $\mathrm{q}_{n-1}\overline{\mathrm{q}}_n$ generated in the jet, and that will leave behind an unpaired quark flavour $\mathrm{q}_n$. Independent fragmentation does not specify the fate of this quark. Secondly, also a meson in the middle of the flavour chain may be selected with such a small $z$ value that it obtains $p_{\mathrm{L}} < 0$ and is rejected. Thus a $\u $ quark jet of charge $2/3$ need not only gives jets of charge 0 or 1, but also of $-1$ or $+2$, or even higher. Like with the jet longitudinal momentum above, one could imagine this compensated by other jets in the event.

It is also assumed that transverse momentum is locally conserved, i.e. the net $p_{\perp}$ of the $\mathrm{q}_i \overline{\mathrm{q}}_i$ pair as a whole is assumed to be vanishing. The $p_{\perp}$ of the $\mathrm{q}_i$ is taken to be a Gaussian in the two transverse degrees of freedom separately, with the transverse momentum of a hadron obtained by the sum of constituent quark transverse momenta. The total $p_{\perp}$ of a jet can fluctuate for the same two reasons as discussed above for flavour. Furthermore, in some scenarios one may wish to have the same $p_{\perp}$ distribution for the first-rank hadron $\mathrm{q}\overline{\mathrm{q}}_1$ as for subsequent ones, in which case also the original $\mathrm{q}$ should be assigned an unpaired $p_{\perp}$ according to a Gaussian.

Within the IF framework, there is no unique recipe for how gluon jet fragmentation should be handled. One possibility is to treat it exactly like a quark jet, with the initial quark flavour chosen at random among $\u $, $\overline{\mathrm{u}}$, $\d $, $\overline{\mathrm{d}}$, $\mathrm{s}$ and $\overline{\mathrm{s}}$, including the ordinary $\mathrm{s}$ quark suppression factor. Since the gluon is supposed to fragment more softly than a quark jet, the fragmentation function may be chosen independently. Another common option is to split the $\mathrm{g}$ jet into a pair of parallel $\mathrm{q}$ and $\overline{\mathrm{q}}$ ones, sharing the energy, e.g. as in a perturbative branching $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$, i.e. $f(z) \propto z^2 + (1-z)^2$. The fragmentation function could still be chosen independently, if so desired. Further, in either case the fragmentation $p_{\perp}$ could be chosen to have a different mean.


next up previous contents
Next: Fragmentation of a jet Up: Independent Fragmentation Previous: Independent Fragmentation   Contents
Stephen Mrenna 2007-10-30