The simple model

In an event with several interactions, it is convenient to impose an ordering. The logical choice is to arrange the scatterings in falling sequence of $x_{\perp} = 2 p_{\perp}/ E_{\mathrm{cm}}$. The `first' scattering is thus the hardest one, with the `subsequent' (`second', `third', etc.) successively softer. It is important to remember that this terminology is in no way related to any picture in physical time; we do not know anything about the latter. In principle, all the scatterings that occur in an event must be correlated somehow, naïvely by momentum and flavour conservation for the partons from each incoming hadron, less naïvely by various quantum mechanical effects. When averaging over all configurations of soft partons, however, one should effectively obtain the standard QCD phenomenology for a hard scattering, e.g. in terms of parton distributions. Correlation effects, known or estimated, can be introduced in the choice of subsequent scatterings, given that the `preceding' (harder) ones are already known.

With a total cross section of hard interactions $\sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}})$ to be distributed among $\sigma_{\mathrm{nd}}(s)$ (non-diffractive, inelastic) events, the average number of interactions per event is just the ratio $\overline{n} = \sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}}) / \sigma_{\mathrm{nd}}(s)$. As a starting point we will assume that all hadron collisions are equivalent (no impact parameter dependence), and that the different parton-parton interactions take place completely independently of each other. The number of scatterings per event is then distributed according to a Poissonian with mean $\overline{n}$. A fit to S $\mathrm{p}\overline{\mathrm{p}}$S collider multiplicity data [UA584] gave $p_{\perp\mathrm{min}}\approx 1.6$ GeV, which corresponds to $\overline{n} \approx 1$. For Monte Carlo generation of these interactions it is useful to define

$$f(x_{\perp}) = \frac{1}{\sigma_{\mathrm{nd}}(s)} \, \frac{\d\sigma}{\d x_{\perp}} ~,$$ (204)

with $\d\sigma / \d x_{\perp}$ obtained by analogy with eq. (). Then $f(x_{\perp})$ is simply the probability to have a parton-parton interaction at $x_{\perp}$, given that the two hadrons undergo a non-diffractive, inelastic collision.

The probability that the hardest interaction, i.e. the one with highest $x_{\perp}$, is at $x_{\perp 1}$, is now given by

$$f(x_{\perp 1}) \exp \left\{ - \int_{x_{\perp 1}}^1 f(x'_{\perp}) \, \d x'_{\perp} \right\} ~,$$ (205)

i.e. the naïve probability to have a scattering at $x_{\perp 1}$ multiplied by the probability that there was no scattering with $x_{\perp}$ larger than $x_{\perp 1}$. This is the familiar exponential dampening in radioactive decays, encountered e.g. in parton showers in section . Using the same technique as in the proof of the veto algorithm, section , the probability to have an $i$:th scattering at an $x_{\perp i} < x_{\perp i-1} < \cdots < x_{\perp 1} < 1$ is found to be

$$f(x_{\perp i}) \, \frac{1}{(i-1)!} \left( \int_{x_{\perp i}}... ...int_{x_{\perp i}}^1 f(x'_{\perp}) \, \d x'_{\perp} \right\} ~.$$ (206)

The total probability to have a scattering at a given $x_{\perp}$, irrespectively of it being the first, the second or whatever, obviously adds up to give back $f(x_{\perp})$. The multiple interaction formalism thus retains the correct perturbative QCD expression for the scattering probability at any given $x_{\perp}$.

With the help of the integral

$$F(x_{\perp}) = \int_{x_{\perp}}^1 f(x'_{\perp}) \, \d x'_{\p... ...}^2/4}^{s/4} \frac{\d\sigma}{\d p_{\perp}^2} \, \d p_{\perp}^2$$ (207)

(where we assume $F(x_{\perp}) \to \infty$ for $x_{\perp} \to 0$) and its inverse $F^{-1}$, the iterative procedure to generate a chain of scatterings $1 > x_{\perp 1} > x_{\perp 2} > \cdots > x_{\perp i}$ is given by

$$x_{\perp i} = F^{-1}(F(x_{\perp i-1}) - \ln R_i) ~.$$ (208)

Here the $R_i$ are random numbers evenly distributed between 0 and 1. The iterative chain is started with a fictitious $x_{\perp 0} = 1$ and is terminated when $x_{\perp i}$ is smaller than $x_{\perp \mathrm{min}} = 2 p_{\perp\mathrm{min}}/ E_{\mathrm{cm}}$. Since $F$ and $F^{-1}$ are not known analytically, the standard veto algorithm is used to generate a much denser set of $x_{\perp}$ values, whereof only some are retained in the end. In addition to the $p_{\perp}^2$ of an interaction, it is also necessary to generate the other flavour and kinematics variables according to the relevant matrix elements.

Whereas the ordinary parton distributions should be used for the hardest scattering, in order to reproduce standard QCD phenomenology, the parton distributions to be used for subsequent scatterings must depend on all preceding $x$ values and flavours chosen. We do not know enough about the hadron wave function to write down such joint probability distributions. To take into account the energy `already' used in harder scatterings, a conservative approach is to evaluate the parton distributions, not at $x_i$ for the $i$:th scattered parton from hadron, but at the rescaled value

$$x'_i = \frac{x_i}{\sum_{j=1}^{i-1} x_j} ~.$$ (209)

This is our standard procedure in the program; we have tried a few alternatives without finding any significantly different behaviour in the final physics.

In a fraction $\exp(-F(x_{\perp \mathrm{min}}))$ of the events studied, there will be no hard scattering above $x_{\perp \mathrm{min}}$ when the iterative procedure in eq. () is applied. It is therefore also necessary to have a model for what happens in events with no (semi)hard interactions. The simplest possible way to produce an event is to have an exchange of a very soft gluon between the two colliding hadrons. Without (initially) affecting the momentum distribution of partons, the `hadrons' become colour octet objects rather than colour singlet ones. If only valence quarks are considered, the colour octet state of a baryon can be decomposed into a colour triplet quark and an antitriplet diquark. In a baryon-baryon collision, one would then obtain a two-string picture, with each string stretched from the quark of one baryon to the diquark of the other. A baryon-antibaryon collision would give one string between a quark and an antiquark and another one between a diquark and an antidiquark.

In a hard interaction, the number of possible string drawings are many more, and the overall situation can become quite complex when several hard scatterings are present in an event. Specifically, the string drawing now depends on the relative colour arrangement, in each hadron individually, of the partons that are about to scatter. This is a subject about which nothing is known. To make matters worse, the standard string fragmentation description would have to be extended, to handle events where two or more valence quarks have been kicked out of an incoming hadron by separate interactions. In particular, the position of the baryon number would be unclear. We therefore here assume that, following the hardest interaction, all subsequent interactions belong to one of three classes.

$\bullet$

Scatterings of the $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$ type, with the two gluons in a colour-singlet state, such that a double string is stretched directly between the two outgoing gluons, decoupled from the rest of the system.

$\bullet$

Scatterings $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$, but colour correlations assumed to be such that each of the gluons is connected to one of the strings `already' present. Among the different possibilities of connecting the colours of the gluons, the one which minimizes the total increase in string length is chosen. This is in contrast to the previous alternative, which roughly corresponds to a maximization (within reason) of the extra string length.

$\bullet$

Scatterings $\mathrm{g}\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$, with the final pair again in a colour-singlet state, such that a single string is stretched between the outgoing $\mathrm{q}$ and $\overline{\mathrm{q}}$.

By default, the three possibilities are assumed equally probable. Note that the total jet rate is maintained at its nominal value, i.e. scatterings such as $\mathrm{q}\mathrm{g}\to \mathrm{q}\mathrm{g}$ are included in the cross section, but are replaced by a mixture of $\mathrm{g}\mathrm{g}$ and $\mathrm{q}\overline{\mathrm{q}}$ events for string drawing issues. Only the hardest interaction is guaranteed to give strings coupled to the beam remnants. One should not take this approach to colour flow too seriously -- clearly it is a simplification -- but the overall picture does not tend to be very dependent on the particular choice you make.

Since a $\mathrm{g}\mathrm{g}\to \mathrm{g}\mathrm{g}$ or $\mathrm{q}\overline{\mathrm{q}}$ scattering need not remain of this character if initial- and final-state showers were to be included (e.g. it could turn into a $\mathrm{q}\mathrm{g}$-initiated process), radiation is only included for the hardest interaction. In practice, this is not a serious problem: except for the hardest interaction, which can be hard because of experimental trigger conditions, it is unlikely for a parton scattering to be so hard that radiation plays a significant rôle.

In events with multiple interactions, the beam remnant treatment is slightly modified. First the hard scattering is generated, with its associated initial- and final-state radiation, and next any additional multiple interactions. Only thereafter are beam remnants attached to the initiator partons of the hardest scattering, using the same machinery as before, except that the energy and momentum already taken away from the beam remnants also include that of the subsequent interactions.