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Interconnection Effects

The widths of the $\mathrm{W}$, $\mathrm{Z}$ and $\t $ are all of the order of 2 GeV. A Standard Model Higgs with a mass above 200 GeV, as well as many supersymmetric and other beyond-the-Standard Model particles would also have widths in the multi-GeV range. Not far from threshold, the typical decay times $\tau = 1/\Gamma \approx 0.1 \, {\mathrm{fm}} \ll
\tau_{\mathrm{had}} \approx 1 \, \mathrm{fm}$. Thus hadronic decay systems overlap, between a resonance and the underlying event, or between pairs of resonances, so that the final state may not contain independent resonance decays.

So far, studies have mainly been performed in the context of $\mathrm{W}$ pair production at LEP2. Pragmatically, one may here distinguish three main eras for such interconnection:

112.
Perturbative: this is suppressed for gluon energies $\omega > \Gamma$ by propagator/timescale effects; thus only soft gluons may contribute appreciably.
113.
Nonperturbative in the hadroformation process: normally modelled by a colour rearrangement between the partons produced in the two resonance decays and in the subsequent parton showers.
114.
Nonperturbative in the purely hadronic phase: best exemplified by Bose-Einstein effects; see next section.
The above topics are deeply related to the unsolved problems of strong interactions: confinement dynamics, $1/N^2_{\mathrm{C}}$ effects, quantum mechanical interferences, etc. Thus they offer an opportunity to study the dynamics of unstable particles, and new ways to probe confinement dynamics in space and time [Gus88a,Sjö94], but they also threaten to limit or even spoil precision measurements.

The reconnection scenarios outlined in [Sjö94a] are now available, plus also an option along the lines suggested in [Gus94]. (A toy model presented in [San05] and intended mostly for hadron collisions is also available, as described in section [*].) Currently they can only be invoked in process 25, $\mathrm{e}^+ \mathrm{e}^- \to \mathrm{W}^+ \mathrm{W}^- \to \mathrm{q}_1 \overline{\mathrm{q}}_2 \mathrm{q}_3 \overline{\mathrm{q}}_4$, which is the most interesting one for the foreseeable future. (Process 22, $\mathrm{e}^+\mathrm{e}^- \to \gamma^* / \mathrm{Z}^0\; \gamma^* / \mathrm{Z}^0\to \mathrm{q}_1\overline{\mathrm{q}}_1\mathrm{q}_2\overline{\mathrm{q}}_2$ can also be used, but the travel distance is calculated based only on the $\mathrm{Z}^0$ propagator part. Thus, the description in scenarios I, II and II$'$ below would not be sensible e.g. for a light-mass $\gamma^*\gamma^*$ pair.) If normally the event is considered as consisting of two separate colour singlets, $\mathrm{q}_1 \overline{\mathrm{q}}_2$ from the $\mathrm{W}^+$ and $\mathrm{q}_3 \overline{\mathrm{q}}_4$ from the $\mathrm{W}^-$, a colour rearrangement can give two new colour singlets $\mathrm{q}_1 \overline{\mathrm{q}}_4$ and $\mathrm{q}_3 \overline{\mathrm{q}}_2$. It therefore leads to a different hadronic final state, although differences usually turn out to be subtle and difficult to isolate [Nor97]. When also gluon emission is considered, the number of potential reconnection topologies increases.

Since hadronization is not understood from first principles, it is important to remember that we deal with model building rather than exact calculations. We will use the standard Lund string fragmentation model [And83] as a starting point, but have to extend it considerably. The string is here to be viewed as a Lorentz covariant representation of a linear confinement field.

The string description is entirely probabilistic, i.e. any negative-sign interference effects are absent. This means that the original colour singlets $\mathrm{q}_1 \overline{\mathrm{q}}_2$ and $\mathrm{q}_3 \overline{\mathrm{q}}_4$ may transmute to new singlets $\mathrm{q}_1 \overline{\mathrm{q}}_4$ and $\mathrm{q}_3 \overline{\mathrm{q}}_2$, but that any effects e.g. of $\mathrm{q}_1\mathrm{q}_3$ or $\overline{\mathrm{q}}_2\overline{\mathrm{q}}_4$ dipoles are absent. In this respect, the nonperturbative discussion is more limited in outlook than a corresponding perturbative one. However, note that dipoles such as $\mathrm{q}_1\mathrm{q}_3$ do not correspond to colour singlets, and can therefore not survive in the long-distance limit of the theory, i.e. they have to disappear in the hadronization phase.

The imagined time sequence is the following. The $\mathrm{W}^+$ and $\mathrm{W}^-$ fly apart from their common production vertex and decay at some distance. Around each of these decay vertices, a perturbative parton shower evolves from an original $\mathrm{q}\overline{\mathrm{q}}$ pair. The typical distance that a virtual parton (of mass $m \sim 10$ GeV, say, so that it can produce separate jets in the hadronic final state) travels before branching is comparable with the average $\mathrm{W}^+ \mathrm{W}^-$ separation, but shorter than the fragmentation time. Each $\mathrm{W}$ can therefore effectively be viewed as instantaneously decaying into a string spanned between the partons, from a quark end via a number of intermediate gluons to the antiquark end. The strings expand, both transversely and longitudinally, at a speed limited by that of light. They eventually fragment into hadrons and disappear. Before that time, however, the string from the $\mathrm{W}^+$ and the one from the $\mathrm{W}^-$ may overlap. If so, there is some probability for a colour reconnection to occur in the overlap region. The fragmentation process is then modified.

The Lund string model does not constrain the nature of the string fully. At one extreme, the string may be viewed as an elongated bag, i.e. as a flux tube without any pronounced internal structure. At the other extreme, the string contains a very thin core, a vortex line, which carries all the topological information, while the energy is distributed over a larger surrounding region. The latter alternative is the chromoelectric analogue to the magnetic flux lines in a type II superconductor, whereas the former one is more akin to the structure of a type I superconductor. We use them as starting points for two contrasting approaches, with nomenclature inspired by the superconductor analogy.

In scenario I, the reconnection probability is proportional to the space-time volume over which the $\mathrm{W}^+$ and $\mathrm{W}^-$ strings overlap, with saturation at unit probability. This probability is calculated as follows. In the rest frame of a string piece expanding along the $\pm z$ direction, the colour field strength is assumed to be given by

\begin{displaymath}
\Omega(\mbox{\bf x},t) =
\exp \left\{ - (x^2 + y^2)/2r_{\ma...
...
\exp \left\{ - (t^2 - z^2)/\tau_{\mathrm{frag}}^2 \right\} ~.
\end{displaymath} (258)

The first factor gives a Gaussian fall-off in the transverse directions, with a string width $r_{\mathrm{had}} \approx 0.5$ fm of typical hadronic dimensions. The time retardation factor $\theta(t - \vert\mbox{\bf x}\vert)$ ensures that information on the decay of the $\mathrm{W}$ spreads outwards with the speed of light. The last factor gives the probability that the string has not yet fragmented at a given proper time along the string axis, with $\tau_{\mathrm{frag}} \approx 1.5$ fm. For a string piece e.g. from the $\mathrm{W}^+$ decay, this field strength has to be appropriately rotated, boosted and displaced to the $\mathrm{W}^+$ decay vertex. In addition, since the $\mathrm{W}^+$ string can be made up of many pieces, the string field strength $\Omega_{\mathrm{max}}^+(\mbox{\bf x},t)$ is defined as the maximum of all the contributing $\Omega^+$'s in the given point. The probability for a reconnection to occur is now given by
\begin{displaymath}
{\cal P}_{\mathrm{recon}} = 1 - \exp \left( - k_{\mathrm{I}}...
...f x},t) \,
\Omega_{\mathrm{max}}^-(\mbox{\bf x},t) \right) ~,
\end{displaymath} (259)

where $k_{\mathrm{I}}$ is a free parameter. If a reconnection occurs, the space-time point for this reconnection is selected according to the differential probability $\Omega_{\mathrm{max}}^+(\mbox{\bf x},t) \,
\Omega_{\mathrm{max}}^-(\mbox{\bf x},t)$. This defines the string pieces involved and the new colour singlets.

In scenario II it is assumed that reconnections can only take place when the core regions of two string pieces cross each other. This means that the transverse extent of strings can be neglected, which leads to considerable simplifications compared with the previous scenario. The position of a string piece at time $t$ is described by a one-parameter set $\mbox{\bf x}(t,\alpha)$, where $0 \leq \alpha \leq 1$ is used to denote the position along the string. To find whether two string pieces $i$ and $j$ from the $\mathrm{W}^+$ and $\mathrm{W}^-$ decays cross, it is sufficient to solve the equation system $\mbox{\bf x}_i^+(t , \alpha^+) =
\mbox{\bf x}_j^-(t , \alpha^-)$ and to check that this (unique) solution is in the physically allowed domain. Further, it is required that neither string piece has had time to fragment, which gives two extra suppression factors of the form $\exp \{ - \tau^2/\tau_{\mathrm{frag}}^2 \}$, with $\tau$ the proper lifetime of each string piece at the point of crossing, i.e. as in scenario I. If there are several string crossings, only the one that occurs first is retained. The II$'$ scenario is a variant of scenario II, with the requirement that a reconnection is allowed only if it leads to a reduction of the string length.

Other models include a simplified implementation of the `GH' model [Gus94], where the reconnection is selected solely based on the criterion of a reduced string length. The `instantaneous' and `intermediate' scenarios are two toy models. In the former (which is equivalent to that in [Gus88a]) the two reconnected systems $\mathrm{q}_1 \overline{\mathrm{q}}_4$ and $\mathrm{q}_3 \overline{\mathrm{q}}_2$ are immediately formed and then subsequently shower and fragment independently of each other. In the latter, a reconnection occurs between the shower and fragmentation stages. One has to bear in mind that the last two `optimistic' (from the connectometry point of view) toy approaches are oversimplified extremes and are not supposed to correspond to the true nature. These scenarios may be useful for reference purposes, but are already excluded by data.

While interconnection effects are primarily viewed as hadronization physics, their implementation is tightly coupled to the event generation of a few specific processes, and not to the generic hadronization machinery. Therefore the relevant main switch MSTP(115) and parameters PARP(115) - PARP(120) are described in section [*].


next up previous contents
Next: Bose-Einstein effects Up: Other Fragmentation Aspects Previous: Small-mass systems   Contents
Stephen Mrenna 2007-10-30