Junction topologies

When several valence quarks are kicked out of an incoming proton, or when baryon number is violated in Supersymmetry, another kind of string topology can be produced. In its simplest form, it can be illustrated by the decay $\tilde{\chi}_1^0 \to \u\d\d$. 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. As the quarks move out, also this junction would move, so as to minimize the total string energy. It is at rest in a frame where the opening angle between any pairs 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 [Sjö02]. 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 fact, 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 spacelike. 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. Therefore additional checks are required.

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, but 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 technology 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 [Sjö02].

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.