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Junction topologies

When several valence quarks are kicked out of an incoming proton, or when baryon number is violated, another kind of string topology can be produced. In its simplest form, it can be illustrated by the decay $\tilde{\chi}_1^0 \to \u\d\d $ which can take place in baryon-number-violating supersymmetric theories. If we assume that the `colour triplet' string kind encountered above is the only basic building block, we are led to a Y-shaped string topology, with a quark at each end and a `junction' where the strings meet. This picture is similar to that adopted, for instance, in models of baryon wavefunctions [Art75]. The following gives a (very) brief overview of a detailed model of junction fragmentation developed in [Sjö03].

As the quarks move out, also the string junction would move, so as to minimize the total string energy. It is at rest in a frame where the opening angle between any pair of quarks is 120$^{\circ}$, so that the forces acting on the junction cancel. In such a frame, each of the three strings would fragment pretty much as ordinary strings, e.g. in a back-to-back $\mathrm{q}\overline{\mathrm{q}}$ pair of jets, at least as far as reasonably high-momentum particles are concerned. Thus an iterative procedure can be used, whereby the leading $\mathrm{q}$ is combined with a newly produced $\overline{\mathrm{q}}_1$, to form a meson and leave behind a remainder-jet $\mathrm{q}_1$. (As above, this has nothing to do with the ordering in physical time, where the fragmentation process again starts in the middle and spreads outwards.) Eventually, when little energy is left, the three remainders $\mathrm{q}_i \mathrm{q}_j \mathrm{q}_k$ form a single baryon, which thus has a reasonably small momentum in the rest frame of the junction. We see that the junction thereby implicitly comes to be the carrier of the net baryon number of the system. Further baryon production can well occur at higher momenta in each of the three jets, but then always in pairs of a baryon and an antibaryon. While the fragmentation principles as such are clear, the technical details of the joining of the jets become more complicated than in the $\mathrm{q}\overline{\mathrm{q}}$ case. Some approximations can be made that allow a reasonably compact and efficient algorithm, which gives sensible results [Sjö03]. Specifically, two of the strings, preferably the ones with lowest energy, can be fragmented until their remaining energy is below some cut-off value. In the actual implementation, one of the two is required to have rather little energy left, while the other could have somewhat more. At this point, the two remainder flavours are combined into one effective diquark, which is assigned all the remaining energy and momentum. The final string piece, between this diquark and the third quark, can now be considered as described for simple $\mathrm{q}\overline{\mathrm{q}}$ strings above.

Among the additional complications are that the diquark formed from the leftovers may have a larger momentum than energy and thereby nominally may be space-like. If only by a little, it normally would not matter, but in extreme cases the whole final string may come to have a negative squared mass. Such configurations are rejected on grounds of being unphysical and the whole fragmentation procedure is restarted from the beginning.

As above, the fragmentation procedure can be formulated in a Lorentz-frame-independent manner, given the four-vector that describes the motion of the junction. Therefore, while the fragmentation picture is simpler to visualize in the rest frame of the junction, one may prefer to work in the rest frame of the system or in the lab frame, as the case may be.

Each of the strings considered above normally would not go straight from the junction to an endpoint quark, rather it would wind its way via a number of intermediate gluons, in the neutralino case generated by bremsstrahlung in the decay. It is straightforward to use the same formalism as for other multiparton systems to extend the description above to such cases. The one complication is that the motion of the junction may become more complicated, especially when the emission of reasonably soft gluons is considered. This can be approximated by a typical mean motion during the hadronization era [Sjö03].

The most general string topology foreseen is one with two junctions, i.e. a $>$ $-$ $<$ topology. Here one junction would be associated with a baryon number and the other with an antibaryon one. There would be two quark ends, two antiquark ones, and five string pieces (including the one between the two junctions) that each could contain an arbitrary number of intermediate gluons. Such topologies can arise, for instance, in $\mathrm{p}\overline{\mathrm{p}}$ collisions where a gluon exchange ties together the two beam baryon junctions, or in prompt BNV decays of $\tilde{t}$ squarks in $\mathrm{e}^+\mathrm{e}^-$ collisions. A further complication here is that it should in principle be possible for the junction and antijunction to meet and annihilate, producing a system instead with two separate $\mathrm{q}\overline{\mathrm{q}}$ strings and no junctions. This aspect is also modeled in the implementation, by selecting the topology that minimises the total potential energy, as defined in [Sjö03]. Omitting details here, the resulting behaviour is that junctions with little effective relative motion tend to annihilate, whereas junctions at large velocities tend to remain intact, with corresponding consequences for the presence or absence of `junction baryons'.


next up previous contents
Next: Independent Fragmentation Up: String Fragmentation Previous: Fragmentation of multiparton systems   Contents
Stephen Mrenna 2007-10-30