Comparison of Monte Carlo Models

There are many Monte Carlo event generators available in high energy physics each with its strengths and weaknesses and a focus on a particular family of physics

processes. For example, phojet [102] was not explicitly developed for Standard Model physics analysis, although it is capable of making reasonable predictions of LHC data. qgsjet [103] is tuned to very high energy cosmic ray data in an attempt to predict lower energy data from the Tevetron and the LHC. pythia 6 andpythia 8[104, 105] employ a different phenomenological model fromphojet and qgsjet. The following sections explain some of the differences between the three official models used in CMS; pythia 6, pythia 8 and phojet. These models were used in conjunction with the full GEANT4 detector simulation in the cross-section analysis of chapter 7.

6.3.1 PYTHIA 6

The model behind pythia 6 aims to combine perturbative Quantum Chromo-Dynamics (pQCD) of hard scattering the phenomenological model of Regge theory for soft scattering to provide a complete description of pp collisions. The total cross section is made up of elastic and inelastic components (Refer to Eq.6.16).

The Regge theory model tries to predict the total cross section σ_tot whilst the diffractive cross sections, σ_SD and σ_DD are predicted from pQCD. The Centrally diffractive cross section σ_CD is not included in the pythia 6 model. The elastic cross sectionσ_{N D} is then calculated fromσ_tot−(σ_{N D}+σ_SD+σ_DD). The total cross section of Hadron-Hadron interactions is calculated using the parameterisation developed by Donnachie and Landshoff [93] which appears as a sum of a Pomeron term and a Reggeon, or meson term;

σ_tot^AB(s) =

where s is the centre-of-momentum energy squared (s=E_cm² ),X^AB and Y^AB are parameters which depend on the exchanged field and and η are the Pomeron and Reggeon intercepts at α(t = 0) respectively. The Pomeron trajectory has an intercept = (α_P(0)−1) ≈ 0.081 while the Reggeon trajectory intercept η is αK(0) ≈ 0.5. It can be seen from equation 6.17 that as s increases, the Pomeron term dominates and the Reggeon term becomes negligible, resulting in the rise in the total cross section.

pythia 6 includes multi-parton interactions where the number of interactions is the ratio of the hard scattering cross section, σ_hard, and the non-diffractive cross

Forward Physics 111 section.

N_int= σ_hard σN D

(6.18) σ_hard can become greater than the total cross section. N_intis characterised as two overlapping density distributions (the protons). These distributions are double Gaussian, the widths and relative fractions of which are tuneable parameters.

After generating the initial collision, the branching of outgoing partons is modeled by a parton shower approach in which a shower is a repetitive sequence of 1→2 branchings. For hadronic collisions, these might be q → qg, q → q¯q or g → gg, where q and g represent quarks and gluons respectively. The branching probabil-ities are given by the DGLAP equations. Energy and momentum are conserved at each branching step. The diffractive cross sections are described by a model by Schuler and Sjöstrand[106].

The elastic cross section σ_el is approximated by a simple exponential, as long as the momentum transfer |t|, is not too large. The optical theorem gives:

σel = σ_tot²

16πB_el (6.19)

with Bel = 2bA+ 2bB+ 4·s^0.0808−4.2 and bA=bB= 2.3for protons.[104].

Hadronisation, the mechanism by which quarks remain colour confined by the creation of, and binding with other quarks to produce colourless mesons and light hadrons, is done using the Lund model[107]. It describes the evolution of initial state scattered partons to final state hadrons by String Fragmentation in which new qq¯ pairs are created from the vacuum as a result of the increasing energy stored in the colour field between the original quarks as the distance between them increases.

Full details of the pythia 6 physics generator can be found in [104]

6.3.2 PYTHIA 8

pythia 8 is a C++ version of the Fortran based pythia 6 and as such, shares many common features. It is the preferred generator for LHC studies. pythia 8 has been developed with a focus on pp and p¯p collisions as well asl¯l annihilation.

It includes multi-parton interactions which are important for predicting the total cross section. Hadronisation is again based solely on theLundstring fragmentation model.

Diffractive hadron-hadron interactions are modeled as inpythia 6(Pomeron and Reggeon model) but with additional Pomeron flux parameterisations beyond the Schuler and Sjöstrand model, namelyBruni and Ingelman[108],Berger et. al.[109]

and Donnachie and Landshoff [85]. Some useful information on pythia 8 can be found in [110].

6.3.3 PHOJET

phojet uses the Dual Parton Model (DPM) which provides a full phenomeno-logical description of soft processes. Soft processes, in which the strong coupling constant α_s is large, cannot be analysed using perturbative methods. DPM is a non-perturbative approach to modeling strong interactions consisting of per-forming a topological expansion of the interaction (per-forming cyclinders, toroids, spheres etc) and applying unitarity considerations^¶ and Regge field theory. More complicated topologies become important at higher energies but in each case, the topology represents the partonic nature of the interaction from which the momen-tum distributions of valance and sea quarks can be inferred. Detailed information can be found in [111]. phojet is a two component model with smooth transition between soft and hard interactions. The results from the DPM methodology have been confirmed at the ISR p¯pexperiment, UA4 [44] and in heavy ion collisions at RHIC. The phenomenological model underpinning phojet suppresses the triple-Pomeron exchange at higher energies, which was the largest contributor to the differences in cross section and rapidity distributions seen between phojet and pythia 6. Through the inclusion of the additional Pomeron flux parameterisa-tions in pythia 8, these differences have be reduced, making pythia 8 results closer to those of phojet. However, phojet includes central diffractive (CD) events which pythia 8 does not. Although the cross section σCD is small, it is not negligible and the pythia authors plan to include the mechanism in subse-quent versions. Figure 6.20 shows the inelastic cross section evolutions with √

s predicted from each model and demostrates the increasing diversions of the models at higher energies. The measurement from CDF [112] and the measurements of E710 [113] and E811 were expected to help in determining the correct model of ppinteractions at higher energies. However, the σtotal measurements disagreed by

¶Unitarity states that the total probability of an interaction cannot exceed 1

Forward Physics 113 2.6σwith CDF preferring a ln²(s)fit, while E710 and E811 results preferred a ln(s) fit [97], leading to an increase in the uncertainty for higher energy extrapolations.

(s)

Figure 6.20: Predictions of σ_inel for increasing collision energy from several MC models including those used in theσ_inel measurement of chapter 7.

Figure 6.21 shows the η distribution and the number of particles per event from each Monte Carlo generator used in the cross-section analysis. For the η distribu-tion of non-diffractive events, all generators are in good agreement. This is to be expected as ND events are the most abundant and have been extensively studied in experiments. The models differ in their topology of SD events, with some models, such asphojet showing a tendancy towards the central regions (-3.η.3), while others such asepos lhcand qgsjetII 4produce stable outgoing particle trajec-tories towards the more forward regions, beyond |η| ≈5. epos lhc in particular, has been tuned to LHC data, resulting in a modification of the mass distribution parameter which, in turn, produces lower mass, higher momentum (more forward going) particles [114]. These features play a role in the extrapolation to σ_inel in the following chapter.