The basic cross sections

The QCD cross section for hard $2 \to 2$ processes, as a function of the $p_{\perp}^2$ scale, is given by

$$\frac{\d\sigma}{\d p_{\perp}^2} = \sum_{i,j,k} \int \d x_1 \... ...\left( p_{\perp}^2 - \frac{\hat{t}\hat{u}}{\hat{s}} \right) ~,$$ (202)

cf. section . Implicitly, from now on we are assuming that the `hardness' of processes is given by the $p_{\perp}$ scale of the scattering. For an application of the formula above to small $p_{\perp}$ values, a number of caveats could be made. At low $p_{\perp}$, the integrals receive major contributions from the small-$x$ region, where parton distributions are poorly understood theoretically (Regge limit behaviour, dense packing problems etc. [Lev90]) and not yet measured. Different sets of parton distributions can therefore give numerically rather different results for the phenomenology of interest. One may also worry about higher-order corrections to the jet rates, $K$ factors, beyond what is given by parton-shower corrections -- one simple option we allow here is to evaluate $\alpha_{\mathrm{s}}$ of the hard scattering process at an optimized scale, such as $\alpha_{\mathrm{s}}(0.075p_{\perp}^2)$ [Ell86].

The hard scattering cross section above some given $p_{\perp\mathrm{min}}$ is given by

$$\sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}}) = \int_{p_{\p... ...^2}^{s/4} \frac{\d\sigma}{\d p_{\perp}^2} \, \d p_{\perp}^2 ~.$$ (203)

Since the differential cross section diverges roughly like $\d p_{\perp}^2 / p_{\perp}^4$, $\sigma_{\mathrm{hard}}$ is also divergent for $p_{\perp\mathrm{min}}\to 0$. We may compare this with the total inelastic, non-diffractive cross section $\sigma_{\mathrm{nd}}(s)$ -- elastic and diffractive events are not the topic of this section. At current collider energies $\sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}})$ becomes comparable with $\sigma_{\mathrm{nd}}$ for $p_{\perp\mathrm{min}}\approx$ 1.5-2 GeV. This need not lead to contradictions: $\sigma_{\mathrm{hard}}$ does not give the hadron-hadron cross section but the parton-parton one. Each of the incoming hadrons may be viewed as a beam of partons, with the possibility of having several parton-parton interactions when the hadrons pass through each other. In this language, $\sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}}) / \sigma_{\mathrm{nd}}(s)$ is simply the average number of parton-parton scatterings above $p_{\perp\mathrm{min}}$ in an event, and this number may well be larger than unity.

While the introduction of several interactions per event is the natural consequence of allowing small $p_{\perp\mathrm{min}}$ values and hence large $\sigma_{\mathrm{hard}}$ ones, it is not the solution of $\sigma_{\mathrm{hard}} (p_{\perp\mathrm{min}})$ being divergent for $p_{\perp\mathrm{min}}\to 0$: the average $\hat{s}$ of a scattering decreases slower with $p_{\perp\mathrm{min}}$ than the number of interactions increases, so naïvely the total amount of scattered partonic energy becomes infinite. One cut-off is therefore obtained via the need to introduce proper multi-parton correlated parton distributions inside a hadron. This is not a part of the standard perturbative QCD formalism and is therefore not built into eq. (). In practice, even correlated parton-distribution functions seems to provide too weak a cut, i.e. one is lead to a picture with too little of the incoming energy remaining in the small-angle beam jet region [Sjö87a].

A more credible reason for an effective cut-off is that the incoming hadrons are colour neutral objects. Therefore, when the $p_{\perp}$ of an exchanged gluon is made small and the transverse wavelength correspondingly large, the gluon can no longer resolve the individual colour charges, and the effective coupling is decreased. This mechanism is not in contradiction to perturbative QCD calculations, which are always performed assuming scattering of free partons (rather than partons inside hadrons), but neither does present knowledge of QCD provide an understanding of how such a decoupling mechanism would work in detail. In the simple model one makes use of a sharp cut-off at some scale $p_{\perp\mathrm{min}}$, while a more smooth dampening is assumed for the complex scenario.

One key question is the energy-dependence of $p_{\perp\mathrm{min}}$; this may be relevant e.g. for comparisons of jet rates at different Tevatron energies, and even more for any extrapolation to LHC energies. The problem actually is more pressing now than at the time of the original study [Sjö87a], since nowadays parton distributions are known to be rising more steeply at small $x$ than the flat $xf(x)$ behaviour normally assumed for small $Q^2$ before HERA. This translates into a more dramatic energy dependence of the multiple-interactions rate for a fixed $p_{\perp\mathrm{min}}$.

The larger number of partons should also increase the amount of screening, however, as confirmed by toy simulations [Dis01]. As a simple first approximation, $p_{\perp\mathrm{min}}$ is assumed to increase in the same way as the total cross section, i.e. with some power $\epsilon \approx 0.08$ [Don92] that, via reggeon phenomenology, should relate to the behaviour of parton distributions at small $x$ and $Q^2$. Thus the default in PYTHIA is

$$p_{\perp\mathrm{min}}(s) = (1.9~{\mathrm{GeV}}) \left( \frac{s}{1~\mathrm{TeV}^2} \right)^{0.08}$$ (204)

for the simple model, with the same ansatz for $p_{\perp 0}$ in the impact-parameter-dependent approach, except that then 1.9 GeV $\to$ 2.1 GeV. At any energy scale, the simplest criterion to fix $p_{\perp\mathrm{min}}$ is to require the average charged multiplicity to agree with the experimentally determined one. In general, there is quite a strong dependence of the multiplicity on $p_{\perp\mathrm{min}}$, with a lower $p_{\perp\mathrm{min}}$ corresponding to more multiple interactions and therefore a higher multiplicity. This is the way the 1.9 GeV and 2.1 GeV numbers are fixed, based on a comparison with UA5 data in the energy range 200-900 GeV [UA584]. The energy dependence inside this range is also consistent with the chosen ansatz. However, clearly, neither the experimental nor the theoretical precision is high enough to make too strong statements. It should also be remembered that the $p_{\perp\mathrm{min}}$ values are determined within the context of a given calculation of the QCD jet cross section, and given model parameters within the multiple interactions scenario. If anything of this is changed, e.g. the parton distributions used, then $p_{\perp\mathrm{min}}$ ought to be retuned accordingly.