String Fragmentation

The original JETSET program is intimately connected with string fragmentation, in the form of the time-honoured `Lund model'. This is the default for all PYTHIA applications, but independent fragmentation options also exist, for applications where one wishes to study the importance of string effects.

All current models are of a probabilistic and iterative nature. This means that the fragmentation process as a whole is described in terms of one or a few simple underlying branchings, of the type jet $\to$ hadron + remainder-jet, string $\to$ hadron + remainder-string, and so on. At each branching, probabilistic rules are given for the production of new flavours, and for the sharing of energy and momentum between the products.

To understand fragmentation models, it is useful to start with the simplest possible system, a colour-singlet $\mathrm{q}\overline{\mathrm{q}}$ 2-jet event, as produced in $\mathrm{e}^+\mathrm{e}^-$ annihilation. Here lattice QCD studies lend support to a linear confinement picture (in the absence of dynamical quarks), i.e. the energy stored in the colour dipole field between a charge and an anticharge increases linearly with the separation between the charges, if the short-distance Coulomb term is neglected. This is quite different from the behaviour in QED, and is related to the presence of a triple-gluon vertex in QCD. The details are not yet well understood, however.

The assumption of linear confinement provides the starting point for the string model. As the $\mathrm{q}$ and $\overline{\mathrm{q}}$ partons move apart from their common production vertex, the physical picture is that of a colour flux tube (or maybe a colour vortex line) being stretched between the $\mathrm{q}$ and the $\overline{\mathrm{q}}$. The transverse dimensions of the tube are of typical hadronic sizes, roughly 1 fm. If the tube is assumed to be uniform along its length, this automatically leads to a confinement picture with a linearly rising potential. In order to obtain a Lorentz covariant and causal description of the energy flow due to this linear confinement, the most straightforward way is to use the dynamics of the massless relativistic string with no transverse degrees of freedom. The mathematical, one-dimensional string can be thought of as parameterizing the position of the axis of a cylindrically symmetric flux tube. From hadron spectroscopy, the string constant, i.e. the amount of energy per unit length, is deduced to be $\kappa \approx 1$ GeV/fm. The expression `massless' relativistic string is somewhat of a misnomer: $\kappa$ effectively corresponds to a `mass density' along the string.

Let us now turn to the fragmentation process. As the $\mathrm{q}$ and $\overline{\mathrm{q}}$ move apart, the potential energy stored in the string increases, and the string may break by the production of a new $\mathrm{q}' \overline{\mathrm{q}}'$ pair, so that the system splits into two colour-singlet systems $\mathrm{q}\overline{\mathrm{q}}'$ and $\mathrm{q}' \overline{\mathrm{q}}$. If the invariant mass of either of these string pieces is large enough, further breaks may occur. In the Lund string model, the string break-up process is assumed to proceed until only on-mass-shell hadrons remain, each hadron corresponding to a small piece of string with a quark in one end and an antiquark in the other.

In order to generate the quark-antiquark pairs $\mathrm{q}' \overline{\mathrm{q}}'$ which lead to string break-ups, the Lund model invokes the idea of quantum mechanical tunnelling. This leads to a flavour-independent Gaussian spectrum for the $p_{\perp}$ of $\mathrm{q}' \overline{\mathrm{q}}'$ pairs. Since the string is assumed to have no transverse excitations, this $p_{\perp}$ is locally compensated between the quark and the antiquark of the pair. The total $p_{\perp}$ of a hadron is made up out of the $p_{\perp}$ contributions from the quark and antiquark that together form the hadron. Some contribution of very soft perturbative gluon emission may also effectively be included in this description.

The tunnelling picture also implies a suppression of heavy-quark production, $\u : \d : \mathrm{s}: \c\approx 1 : 1 : 0.3 : 10^{-11}$. Charm and heavier quarks hence are not expected to be produced in the soft fragmentation, but only in perturbative parton-shower branchings $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$.

When the quark and antiquark from two adjacent string breaks are combined to form a meson, it is necessary to invoke an algorithm to choose between the different allowed possibilities, notably between pseudoscalar and vector mesons. Here the string model is not particularly predictive. Qualitatively one expects a $1 : 3$ ratio, from counting the number of spin states, multiplied by some wave-function normalization factor, which should disfavour heavier states.

A tunnelling mechanism can also be used to explain the production of baryons. This is still a poorly understood area. In the simplest possible approach, a diquark in a colour antitriplet state is just treated like an ordinary antiquark, such that a string can break either by quark-antiquark or antidiquark-diquark pair production. A more complex scenario is the `popcorn' one, where diquarks as such do not exist, but rather quark-antiquark pairs are produced one after the other. This latter picture gives a less strong correlation in flavour and momentum space between the baryon and the antibaryon of a pair.

In general, the different string breaks are causally disconnected. This means that it is possible to describe the breaks in any convenient order, e.g. from the quark end inwards. One therefore is led to write down an iterative scheme for the fragmentation, as follows. Assume an initial quark $\mathrm{q}$ moving out along the $+z$ axis, with the antiquark going out in the opposite direction. By the production of a $\mathrm{q}_1 \overline{\mathrm{q}}_1$ pair, a meson with flavour content $\mathrm{q}\overline{\mathrm{q}}_1$ is produced, leaving behind an unpaired quark $\mathrm{q}_1$. A second pair $\mathrm{q}_2 \overline{\mathrm{q}}_2$ may now be produced, to give a new meson with flavours $\mathrm{q}_1 \overline{\mathrm{q}}_2$, etc. At each step the produced hadron takes some fraction of the available energy and momentum. This process may be iterated until all energy is used up, with some modifications close to the $\overline{\mathrm{q}}$ end of the string in order to make total energy and momentum come out right.

The choice of starting the fragmentation from the quark end is arbitrary, however. A fragmentation process described in terms of starting at the $\overline{\mathrm{q}}$ end of the system and fragmenting towards the $\mathrm{q}$ end should be equivalent. This `left-right' symmetry constrains the allowed shape of the fragmentation function $f(z)$, where $z$ is the fraction of the remaining light-cone momentum $E \pm p_z$ (+ for the $\mathrm{q}$ jet, $-$ for the $\overline{\mathrm{q}}$ one) taken by each new particle. The resulting `Lund symmetric fragmentation function' has two free parameters, which are determined from data.

If several partons are moving apart from a common origin, the details of the string drawing become more complicated. For a $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ event, a string is stretched from the $\mathrm{q}$ end via the $\mathrm{g}$ to the $\overline{\mathrm{q}}$ end, i.e. the gluon is a kink on the string, carrying energy and momentum. As a consequence, the gluon has two string pieces attached, and the ratio of gluon to quark string force is 2, a number which can be compared with the ratio of colour charge Casimir operators, $N_C/C_F = 2/(1-1/N_C^2) = 9/4$. In this, as in other respects, the string model can be viewed as a variant of QCD where the number of colours $N_C$ is not 3 but infinite. Note that the factor 2 above does not depend on the kinematical configuration: a smaller opening angle between two partons corresponds to a smaller string length drawn out per unit time, but also to an increased transverse velocity of the string piece, which gives an exactly compensating boost factor in the energy density per unit string length.

The $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ string will fragment along its length. To first approximation this means that there is one fragmenting string piece between $\mathrm{q}$ and $\mathrm{g}$ and a second one between $\mathrm{g}$ and $\overline{\mathrm{q}}$. One hadron is straddling both string pieces, i.e. sitting around the gluon corner. The rest of the particles are produced as in two simple $\mathrm{q}\overline{\mathrm{q}}$ strings, but strings boosted with respect to the overall c.m. frame. When considered in detail, the string motion and fragmentation is more complicated, with the appearance of additional string regions during the time evolution of the system. These corrections are especially important for soft and collinear gluons, since they provide a smooth transition between events where such radiation took place and events where it did not. Therefore the string fragmentation scheme is `infrared safe' with respect to soft or collinear gluon emission.

For events that involve many partons, there may be several possible topologies for their ordering along the string. An example would be a $\mathrm{q}\overline{\mathrm{q}}\g_1 \g_2$ (the gluon indices are here used to label two different gluon-momentum vectors), where the string can connect the partons in either of the sequences $\mathrm{q}- \g_1 - \g_2 - \overline{\mathrm{q}}$ and $\mathrm{q}- \g_2 - \g_1 - \overline{\mathrm{q}}$. The matrix elements that are calculable in perturbation theory contain interference terms between these two possibilities, which means that the colour flow is not always well-defined. Fortunately, the interference terms are down in magnitude by a factor $1/N_C^2$, where $N_C = 3$ is the number of colours, so approximate recipes can be found. In the leading log shower description, on the other hand, the rules for the colour flow are well-defined.

A final comment: in the argumentation for the importance of colour flows there is a tacit assumption that soft-gluon exchanges between partons will not normally mess up the original colour assignment. Colour rearrangement models provide toy scenarios wherein deviations from this rule could be studied. Of particular interest has been the process $\mathrm{e}^+ \mathrm{e}^- \to \mathrm{W}^+ \mathrm{W}^- \to \mathrm{q}_1 \overline{\mathrm{q}}_2 \mathrm{q}_3 \overline{\mathrm{q}}_4$, where the original singlets $\mathrm{q}_1 \overline{\mathrm{q}}_2$ and $\mathrm{q}_3 \overline{\mathrm{q}}_4$ could be rearranged to $\mathrm{q}_1 \overline{\mathrm{q}}_4$ and $\mathrm{q}_3 \overline{\mathrm{q}}_2$. So far, there are no experimental evidence for dramatic effects of this kind, but the more realistic models predict effects sufficiently small that these have not been ruled out. Another example of nontrivial effects is that of Bose-Einstein correlations between identical final-state particles, which reflect the true quantum nature of the hadronization process.