2.6 Event Generators
57
regulated. This is usually done by including a phenomenological cut off p T,0 such
that
α S ( p
2
T )
p
4
T
→
α S ( p T,0
2
+ p
2
T )
( p T,0
2 + p
2
T ) 2 ,
(2.49)
which smoothly regularises the divergence [378]. The cut-off parameter p T,0 can
have a dependence on
√
s. In the process of simulating MPIs, the structure functions
used for subsequent scatterings must depend on all preceding x values and flavours
chosen. To achieve this, the ordinary PDFs are still used for the hardest scattering,
but are split into a valence and a sea part for MPIs, where the x values of the spectator
partons are rescaled to ensure energy and momentum conservation [379].
By definition, MPIs have a scale smaller than the scale of the hard scattering,
μ f = p T,1 , and can be introduced as scatterings ordered in a sequence p T,1 > p T,2 >
. . . > p T,n . In order to model the competition between MPI and ISR for energy
in the incoming beam, these two effects, together with FSR, are interleaved [336,
380], where ISR and FSR are modelled identical for the primary hard scattering
and MPI. More complicated MPI schemes have been studied as well, like joined
interactions where two partons participating in a MPI have a common ancestor [336],
or rescattering, where a parton from one incoming hadron scatters against two or more
partons of the other incoming hadron [381, 382]. Another important aspect of MPIs
is the modelling of the dependence on an impact parameter, where the interaction
rate is proportional to the overlap of the two colliding hadrons [378]. A complicated
aspect of the dynamics of many parton interactions with varying scales is the way
how the colours of partons are connected with each. There are different schemes
implemented, also depending on the hadronisation model chosen. Generally, for
substructure applications the effect of changing these choices has been found to be
small, but should be tested in analyses. For a recent review on MPIs in Pythia, see
[383].
2.6.6 Hadronisation
The parton showers from the hard scattering and from multiple parton interactions
terminate at a scale t min , at which the value of α S becomes large and perturbative
methods are expected to fail. At this stage, the event is populated with a number
of final state partons which have to undergo a non-perturbative transition, called
hadronisation, to produce the actual hadronic final state. While no rigorous approach
exists to describe non-perturbative hadronic phenomena, two independent models,
namely the string and cluster models, have been very successful in describing a
wealth of data. These models are inspired by QCD and incorporate features based
on phenomenological observations.
The hadronisation in the Pythia event generator is based on the Lund string
model [384–386]. The model is based on the expectation from confinement that the
potential between two colour charges increases linearly with distance, V (r ) = κr ,
for distances larger than about 1 fm, with the string tension κ ≈ 1 GeV/ fm. This
57
regulated. This is usually done by including a phenomenological cut off p T,0 such
that
α S ( p
2
T )
p
4
T
→
α S ( p T,0
2
+ p
2
T )
( p T,0
2 + p
2
T ) 2 ,
(2.49)
which smoothly regularises the divergence [378]. The cut-off parameter p T,0 can
have a dependence on
√
s. In the process of simulating MPIs, the structure functions
used for subsequent scatterings must depend on all preceding x values and flavours
chosen. To achieve this, the ordinary PDFs are still used for the hardest scattering,
but are split into a valence and a sea part for MPIs, where the x values of the spectator
partons are rescaled to ensure energy and momentum conservation [379].
By definition, MPIs have a scale smaller than the scale of the hard scattering,
μ f = p T,1 , and can be introduced as scatterings ordered in a sequence p T,1 > p T,2 >
. . . > p T,n . In order to model the competition between MPI and ISR for energy
in the incoming beam, these two effects, together with FSR, are interleaved [336,
380], where ISR and FSR are modelled identical for the primary hard scattering
and MPI. More complicated MPI schemes have been studied as well, like joined
interactions where two partons participating in a MPI have a common ancestor [336],
or rescattering, where a parton from one incoming hadron scatters against two or more
partons of the other incoming hadron [381, 382]. Another important aspect of MPIs
is the modelling of the dependence on an impact parameter, where the interaction
rate is proportional to the overlap of the two colliding hadrons [378]. A complicated
aspect of the dynamics of many parton interactions with varying scales is the way
how the colours of partons are connected with each. There are different schemes
implemented, also depending on the hadronisation model chosen. Generally, for
substructure applications the effect of changing these choices has been found to be
small, but should be tested in analyses. For a recent review on MPIs in Pythia, see
[383].
2.6.6 Hadronisation
The parton showers from the hard scattering and from multiple parton interactions
terminate at a scale t min , at which the value of α S becomes large and perturbative
methods are expected to fail. At this stage, the event is populated with a number
of final state partons which have to undergo a non-perturbative transition, called
hadronisation, to produce the actual hadronic final state. While no rigorous approach
exists to describe non-perturbative hadronic phenomena, two independent models,
namely the string and cluster models, have been very successful in describing a
wealth of data. These models are inspired by QCD and incorporate features based
on phenomenological observations.
The hadronisation in the Pythia event generator is based on the Lund string
model [384–386]. The model is based on the expectation from confinement that the
potential between two colour charges increases linearly with distance, V (r ) = κr ,
for distances larger than about 1 fm, with the string tension κ ≈ 1 GeV/ fm. This