Fragmentation of multiparton systems

The full machinery needed for a multiparton system is very complicated, and is described in detail in [Sjö84]. The following outline is far from complete, and is complicated nonetheless. The main message to be conveyed is that a Lorentz covariant algorithm exists for handling an arbitrary parton configuration, but that the necessary machinery is more complex than in either cluster or independent fragmentation.

Assume $n$ partons, with ordering along the string, and related four-momenta, given by $\mathrm{q}(p_1) \mathrm{g}(p_2) \mathrm{g}(p_3) \cdots \mathrm{g}(p_{n-1}) \overline{\mathrm{q}}(p_n)$. The initial string then contains $n-1$ separate pieces. The string piece between the quark and its neighbouring gluon is, in four-momentum space, spanned by one side with four-momentum $p_+^{(1)} = p_1$ and another with $p_-^{(1)} = p_2/2$. The factor of 1/2 in the second expression comes from the fact that the gluon shares its energy between two string pieces. The indices `$+$' and `$-$' denotes direction towards the $\mathrm{q}$ and $\overline{\mathrm{q}}$ end, respectively. The next string piece, counted from the quark end, is spanned by $p_+^{(2)} = p_2/2$ and $p_-^{(2)} = p_3/2$, and so on, with the last one being $p_+^{(n-1)} = p_{n-1}/2$ and $p_-^{(n-1)} = p_n$.

For the algorithm to work, it is important that all $p_{\pm}^{(i)}$ be light-cone-like, i.e. $p_{\pm}^{(i)2} = 0$. Since gluons are massless, it is only the two endpoint quarks which can cause problems. The procedure here is to create new $p_{\pm}$ vectors for each of the two endpoint regions, defined to be linear combinations of the old $p_{\pm}$ ones for the same region, with coefficients determined so that the new vectors are light-cone-like. De facto, this corresponds to replacing a massive quark at the end of a string piece with a massless quark at the end of a somewhat longer string piece. With the exception of the added fictitious piece, which anyway ends up entirely within the heavy hadron produced from the heavy quark, the string motion remains unchanged by this.

In the continued string motion, when new string regions appear as time goes by, the first such string regions that appear can be represented as being spanned by one $p_+^{(j)}$ and another $p_-^{(k)}$ four-vector, with $j$ and $k$ not necessarily adjacent. For instance, in the $\mathrm{q}\mathrm{g}\overline{\mathrm{q}}$ case, the `third' string region is spanned by $p_+^{(1)}$ and $p_-^{(3)}$. Later on in the string evolution history, it is also possible to have regions made up of two $p_+$ or two $p_-$ momenta. These appear when an endpoint quark has lost all its original momentum, has accreted the momentum of an gluon, and is now re-emitting this momentum. In practice, these regions may be neglected. Therefore only pieces made up by a $(p_+^{(j)},p_-^{(k)})$ pair of momenta are considered in the program.

The allowed string regions may be ordered in an abstract parameter plane, where the $(j,k)$ indices of the four-momentum pairs define the position of each region along the two (parameter plane) coordinate axes. In this plane the fragmentation procedure can be described as a sequence of steps, starting at the quark end, where $(j,k) = (1,1)$, and ending at the antiquark one, $(j,k) = (n-1,n-1)$. Each step is taken from an `old' $\mathrm{q}_{i-1} \overline{\mathrm{q}}_{i-1}$ pair production vertex, to the production vertex of a `new' $\mathrm{q}_i \overline{\mathrm{q}}_i$ pair, and the string piece between these two string breaks represent a hadron. Some steps may be taken within one and the same region, while others may have one vertex in one region and the other vertex in another region. Consistency requirements, like energy-momentum conservation, dictates that vertex $j$ and $k$ region values be ordered in a monotonic sequence, and that the vertex positions are monotonically ordered inside each region. The four-momentum of each hadron can be read off, for $p_+$ ($p_-$) momenta, by projecting the separation between the old and the new vertex on to the $j$ ($k$) axis. If the four-momentum fraction of $p_{\pm}^{(i)}$ taken by a hadron is denoted $x_{\pm}^{(i)}$, then the total hadron four-momentum is given by

$$p = \sum_{j=j_1}^{j_2} x_+^{(j)} p_+^{(j)} + \sum_{k=k_1}^{... ...p_{x2} \hat{e}_x^{(j_2 k_2)} + p_{y2} \hat{e}_y^{(j_2 k_2)} ~,$$ (241)

for a step from region $(j_1,k_1)$ to region $(j_2,k_2)$. By necessity, $x_+^{(j)}$ is unity for a $j_1 < j < j_2$, and correspondingly for $x_-^{(k)}$.

The $(p_x,p_y)$ pairs are the transverse momenta produced at the two string breaks, and the $(\hat{e}_x,\hat{e}_y)$ pairs four-vectors transverse to the string directions in the regions of the respective string breaks:

$$\hat{e}_x^{(jk)2} = \hat{e}_y^{(jk)2} = -1 ~,$$

$$\hat{e}_x^{(jk)} \hat{e}_y^{(jk)} = \hat{e}_{x,y}^{(jk)} p_+^{(j)} = \hat{e}_{x,y}^{(jk)} p_-^{(k)} = 0 ~.$$ (242)

The fact that the hadron should be on mass shell, $p^2 = m^2$, puts one constraint on where a new breakup may be, given that the old one is already known, just as eq. () did in the simple 2-jet case. The remaining degree of freedom is, as before, to be given by the fragmentation function $f(z)$. The interpretation of the $z$ is only well-defined for a step entirely constrained to one of the initial string regions, however, which is not enough. In the 2-jet case, the $z$ values can be related to the proper times of string breaks, as follows. The variable $\Gamma = (\kappa \tau)^2$, with $\kappa$ the string tension and $\tau$ the proper time between the production vertex of the partons and the breakup point, obeys an iterative relation of the kind

$$\Gamma_0 = 0 ~,$$

$$\Gamma_i = (1-z_i) \left( \Gamma_{i-1} + \frac{m_{\perp i}^2}{z_i} \right) ~.$$ (243)

Here $\Gamma_0$ represents the value at the $\mathrm{q}$ and $\overline{\mathrm{q}}$ endpoints, and $\Gamma_{i-1}$ and $\Gamma_i$ the values at the old and new breakup vertices needed to produce a hadron with transverse mass $m_{\perp i}$, and with the $z_i$ of the step chosen according to $f(z_i)$. The proper time can be defined in an unambiguous way, also over boundaries between the different string regions, so for multijet events the $z$ variable may be interpreted just as an auxiliary variable needed to determine the next $\Gamma$ value. (In the Lund symmetric fragmentation function derivation, the $\Gamma$ variable actually does appear naturally, so the choice is not as arbitrary as it may seem here.) The mass and $\Gamma$ constraints together are sufficient to determine where the next string breakup is to be chosen, given the preceding one in the iteration scheme. Actually, several ambiguities remain, but are of no importance for the overall picture.

The algorithm for finding the next breakup then works something like follows. Pick a hadron, $p_{\perp}$, and $z$, and calculate the next $\Gamma$. If the old breakup is in the region $(j,k)$, and if the new breakup is also assumed to be in the same region, then the $m^2$ and $\Gamma$ constraints can be reformulated in terms of the fractions $x_+^{(j)}$ and $x_-^{(k)}$ the hadron must take of the total four-vectors $p_+^{(j)}$ and $p_-^{(k)}$:

$$m^2 = c_1 + c_2 x_+^{(j)} + c_3 x_-^{(k)} + c_4 x_+^{(j)} x_-^{(k)} ~,$$

$$\Gamma = d_1 + d_2 x_+^{(j)} + d_3 x_-^{(k)} + d_4 x_+^{(j)} x_-^{(k)} ~.$$ (244)

Here the coefficients $c_n$ are fairly simple expressions, obtainable by squaring eq. (), while $d_n$ are slightly more complicated in that they depend on the position of the old string break, but both the $c_n$ and the $d_n$ are explicitly calculable. What remains is an equation system with two unknowns, $x_+^{(j)}$ and $x_-^{(k)}$. The absence of any quadratic terms is due to the fact that all $p_{\pm}^{(i)2} = 0$, i.e. to the choice of a formulation based on light-cone-like longitudinal vectors. Of the two possible solutions to the equation system (elimination of one variable gives a second degree equation in the other), one is unphysical and can be discarded outright. The other solution is checked for whether the $x_{\pm}$ values are actually inside the physically allowed region, i.e. whether the $x_{\pm}$ values of the current step, plus whatever has already been used up in previous steps, are less than unity. If yes, a solution has been found. If no, it is because the breakup could not take place inside the region studied, i.e. because the equation system was solved for the wrong region. One therefore has to change either index $j$ or index $k$ above by one step, i.e. go to the next nearest string region. In this new region, a new equation system of the type in eq. () may be written down, with new coefficients. A new solution is found and tested, and so on until a physically acceptable solution is found. The hadron four-momentum is now given by an expression of the type (). The breakup found forms the starting point for the new step in the fragmentation chain, and so on. The final joining in the middle is done as in the 2-jet case, with minor extensions.