Impact parameter dependence

As mentioned in Section 7.2, the so-called “pedestal effect” (see also Figs. 20–22) is partly driven by impact parameter dependence; in periph- eral collisions, only a small fraction of events contain any high-p⊥ activity,

whereas central collisions are more likely to contain at least one hard scat- tering; a sample with a high-p_⊥ selection cut will therefore be biased towards

small impact parameters. The ability of a model to describe the shape of the pedestal (e.g. to describe both minimum bias data and underlying event dis- tributions simultaneously) is therefore related to its modelling of the impact parameter dependence. A related effect is that, also for a fixed selected p_⊥, events at comparatively higher impact parameters should exhibit relatively less underlying event and vice versa.

All the models discussed here contain an explicit treatment of impact parameter, but we note that there are still substantial simplifications made. Most importantly, the impact parameter dependence is so far still assumed to be factorized from the x dependence, f (x, b) = f (x)g(b), where b de- notes impact parameter, a simplifying assumption that by no means should be treated as inviolate, see e.g. [111–113]. Also, the hadron-hadron impact parameter only enters in an averaged global sense, not as a vector, and the individual MPI are not assigned individual “locations” in transverse space.

In order to quantify the concept of hadronic matter overlap, one may assume a spherically symmetric distribution of matter inside a hadron at rest, ρ(x) d3_{x = ρ(r) d}3_{x. The form of ρ is a matter of some uncertainty,}

with various more or less phenomenologically motivated choices available in models. The options range from simple parameteric forms in Pythia-based models, such as Gaussians, double Gaussians, exponentials [96], and forms interpolating between them [89], to a form based on the electromagnetic form factor in the Herwig-based ones [114]. A possibility for future model refinements thus lies in the input of more detailed information on the flavour- or x-dependence of the transverse structure of the proton, e.g. obtained from sum rules, from analytic fits beyond the EM form factor, or from lattice studies.

For a collision with impact parameter b, the time-integrated overlapO(b) between the matter distributions of the colliding hadrons is given by

O(b) ∝ Z

dt Z

d3x ρ(x, y, z) ρ(x + b, y, z + t) , (48)

where the necessity to use boosted ρ(x) distributions has been circumvented by a suitable scale transformation of the z and t coordinates, see [96]. The overlap function_{O(b) is identical to A(b) in “Jimmy notation” [102, 103]. It is} closely related to the Ω(b) of eikonal models (see, for example, [112, 115, 116]), but is somewhat simpler in spirit.

between partons in the two colliding hadrons. In fact, to first approximation, there should be a linear relationship

h˜n(b)i = kO(b) , (49)

where ˜n = 0, 1, 2, . . . counts the number of interactions when two hadrons pass each other with an impact parameter b and k is an undefined constant of proportionality, to be specified below.

For each impact parameter, b, the number of interactions ˜n can be as- sumed to be distributed according to a Poissonian, modulo momentum con- servation, with the mean value of the Poisson distribution depending on impact parameter, _{h˜n(b)i. If the matter distribution has a tail to infinity} (as, e.g., Gaussians do), one may nominally obtain events with arbitrarily large b values. In order to obtain finite total cross sections, it is therefore necessary to give a separate interpretation to the “zero bin” of the Poisson distribution, which corresponds to “no-interaction” events.

In the Jimmy [102] and Herwig++ [103] models the part of the pp cross section containing hard scatters is calculated from the area overlap function, the parton densities and the partonic cross section; the “no-interaction” pos- sibility is then accounted for as a reduction of this cross section with respect to its value without allowing for MPI. The Jimmy model stops here, consid- ering only hard events, and so it can only be applied to underlying event. As mentioned above, the Herwig++ model also permits the possibility of soft scatters (see also Section 13.6) and so can also be used to simulate soft- inclusive physics.

In the framework of [96], used by Pythia and Sherpa, the restriction to at least one perturbative scattering for soft inclusive scatters implies that the probability that two hadrons, passing each other with an impact parameter b, will produce a real event is given by

Pint(b) = ∞ X ˜ n=1 P˜n(b) = 1−P0(b) = 1−exp(−h˜n(b)i) = 1−exp(−kO(b)) , (50)

according to Poisson statistics. The average number of interactions per event at impact parameter b is now hn(b)i = h˜n(b)i/Pint(b), where the denomina-

tor comes from the removal of hadron pairs that pass without interaction, i.e. which do not produce any events. While the removal of ˜n = 0 from the potential event sample gives a narrower-than-Poisson interaction distri-

bution at each fixed b, the variation of _{hn(b)i with b gives a b-integrated} broader-than-Poisson interaction multiplicity distribution.

Averaged over all b the relationshiphni = σ2j/σnd should still hold. Here,

as before, σ2j is the integrated interaction cross section for a given regulariza-

tion prescription at small p_⊥, while the inelastic non-diffractive cross section σnd is taken from parameterization [100, 101, 117]. This relation can be used

to solve for the proportionality factor k in Eq. (49). Note that, since now each event has to have at least one interaction, hni > 1, one must ensure that σ2j > σnd. The p⊥0 parameter has to be chosen accordingly small.