String motion and infrared stability

We have now discussed the SF scheme for the fragmentation of a simple $\mathrm{q}\overline{\mathrm{q}}$ jet system. In order to understand how these results generalize to arbitrary jet systems, it is first necessary to understand the string motion for the case when no fragmentation takes place. In the following we will assume that quarks as well as gluons are massless, but all arguments can be generalized to massive quarks without too much problem.

For a $\mathrm{q}\overline{\mathrm{q}}$ event viewed in the c.m. frame, with total energy $W$, the partons start moving out back-to-back, carrying half the energy each. As they move apart, energy and momentum is lost to the string. When the partons are a distance $W / \kappa$ apart, all the energy is stored in the string. The partons now turn around and come together again with the original momentum vectors reversed. This corresponds to half a period of the full string motion; the second half the process is repeated, mirror-imaged. For further generalizations to multiparton systems, a convenient description of the energy and momentum flow is given in terms of `genes' [Art83], infinitesimal packets of the four-momentum given up by the partons to the string. Genes with $p_z = E$, emitted from the $\mathrm{q}$ end in the initial stages of the string motion above, will move in the $\overline{\mathrm{q}}$ direction with the speed of light, whereas genes with $p_z = -E$ given up by the $\overline{\mathrm{q}}$ will move in the $\mathrm{q}$ direction. Thus, in this simple case, the direction of motion for a gene is just opposite to that of a free particle with the same four-momentum. This is due to the string tension. If the system is not viewed in the c.m. frame, the rules are that any parton gives up genes with four-momentum proportional to its own four-momentum, but the direction of motion of any gene is given by the momentum direction of the genes it meets, i.e. that were emitted by the parton at the other end of that particular string piece. When the $\mathrm{q}$ has lost all its energy, the $\overline{\mathrm{q}}$ genes, which before could not catch up with $\mathrm{q}$, start impinging on it, and the $\mathrm{q}$ is pulled back, accreting $\overline{\mathrm{q}}$ genes in the process. When the $\mathrm{q}$ and $\overline{\mathrm{q}}$ meet in the origin again, they have completely traded genes with respect to the initial situation.

A 3-jet $\mathrm{q}\overline{\mathrm{q}}\mathrm{g}$ event initially corresponds to having a string piece stretched between $\mathrm{q}$ and $\mathrm{g}$ and another between $\mathrm{g}$ and $\overline{\mathrm{q}}$. Gluon four-momentum genes are thus flowing towards the $\mathrm{q}$ and $\overline{\mathrm{q}}$. Correspondingly, $\mathrm{q}$ and $\overline{\mathrm{q}}$ genes are flowing towards the $\mathrm{g}$. When the gluon has lost all its energy, the $\mathrm{g}$ genes continue moving apart, and instead a third string region is formed in the `middle' of the total string, consisting of overlapping $\mathrm{q}$ and $\overline{\mathrm{q}}$ genes. The two `corners' on the string, separating the three string regions, are not of the gluon-kink type: they do not carry any momentum.

If this third region would only appear at a time later than the typical time scale for fragmentation, it could not affect the sharing of energy between different particles. This is true in the limit of high energy, well separated partons. For a small gluon energy, on the other hand, the third string region appears early, and the overall drawing of the string becomes fairly 2-jet-like, since the third string region consists of $\mathrm{q}$ and $\overline{\mathrm{q}}$ genes and therefore behaves exactly as a sting pulled out directly between the $\mathrm{q}$ and $\overline{\mathrm{q}}$. In the limit of vanishing gluon energy, the two initial string regions collapse to naught, and the ordinary 2-jet event is recovered [Sjö84]. Also for a collinear gluon, i.e. $\theta_{\mathrm{q}\mathrm{g}}$ (or $\theta_{\overline{\mathrm{q}}\mathrm{g}}$) small, the stretching becomes 2-jet-like. In particular, the $\mathrm{q}$ string endpoint first moves out a distance $\mathbf{p}_{\mathrm{q}} / \kappa$ losing genes to the string, and then a further distance $\mathbf{p}_{\mathrm{g}} / \kappa$, a first half accreting genes from the $\mathrm{g}$ and the second half re-emitting them. (This latter half actually includes yet another string piece; a corresponding piece appears at the $\overline{\mathrm{q}}$ end, such that half a period of the system involves five different string regions.) The end result is, approximately, that a string is drawn out as if there had only been a single parton with energy $\vert\mathbf{p}_{\mathrm{q}} + \mathbf{p}_{\mathrm{g}}\vert$, such that the simple 2-jet event again is recovered in the limit $\theta_{\mathrm{q}\mathrm{g}} \to 0$. These properties of the string motion are the reason why the string fragmentation scheme is `infrared safe' with respect to soft or collinear gluon emission.

The discussions for the 3-jet case can be generalized to the motion of a string with $\mathrm{q}$ and $\overline{\mathrm{q}}$ endpoints and an arbitrary number of intermediate gluons. For $n$ partons, whereof $n-2$ gluons, the original string contains $n-1$ pieces. Anytime one of the original gluons has lost its energy, a new string region is formed, delineated by a pair of `corners'. As the extra `corners' meet each other, old string regions vanish and new are created, so that half a period of the string contains $2n^2 - 6n + 5$ different string regions. Each of these regions can be understood simply as built up from the overlap of (opposite-moving) genes from two of the original partons, according to well specified rules.