54
2 Phenomenology of Jet Substructure
1 GeV, which results in an infrared cut-off scale t c . This leads to an upper bound on
z, given by z < 1 − t c /t max , where t max is the maximum parton shower scale, usually
identified by the scale of the primary hard scattering. The upper bound on z results
in a bound on the number of soft gluons produced, which would otherwise tend to
infinity due to the soft singularity. The computation of the total number of branchings
and the corresponding values of t at which these occur in individual events are the
primary tasks of parton shower algorithms. The ensemble of partonic final states in
many events then corresponds to the solution of an evolution equation for parton
showers, similar to the DGLAP evolution. The probability for parton a to branch at
a given value of t can be expressed by an integral over all allowed z values, I (t).
Conversely, the probability for no branching to occur between an initial value of t 0 and
t exponentiates, which is known as Sudakov form factor S a (t 0 , t). Thus, the actual
probability for parton a to branch at scale t, is given by I (t) · S a (t 0 , t), if parton a was
produced at scale t 0 . This means that parton a can only branch at t if it has not branched
already at an earlier stage t
< t. The Sudakov factor is the appropriate suppression
factor, ensuring the conservation of total probability. Formally, the Sudakov factors
account for any unresolved splittings and virtual corrections which are assumed to
precisely cancel the singular real corrections when integrated over phase space [332].
The product I (t) · S a (t 0 , t) can be cast into an evolution equation, which is solved
by any parton-shower event generator.
While this prescription is fairly general, there are a number of choices that have
to be made in the exact implementation of a parton shower and the quality of parton
shower predictions depends significantly on these choices. One important example
is the exact choice of the evolution variable, which is one of the main differences
between the most commonly used event generators. The virtuality t was used in
Pythia versions earlier than 6.4 and Sherpa versions earlier than 1.2, whereas both
generators switched to a k ⊥ -based evolution for later versions. This change expedites the merging of parton showers with higher-order, multi-leg matrix element
generators and the use of recent multiple parton interaction models. The squared
energy-weighted emission angle, E
2
θ
2 is used in early Herwig versions, with a generalisation implemented in recent versions. Another important choice is the splitting
variable z, where the two common choices are the light-cone (LC) momentum fraction or the energy fraction taken by parton b. Coherence effects can be accounted
for by employing a dipole formalism or an angular ordering (AO) of emissions. If
neither option is viable, an angular ordering of emissions can be enforced through a
veto. Recently, a new shower model has been developed, called Dire [333], which
is a hybrid between dipole and parton shower. The shower has been implemented in
Pythia and Sherpa, and is the only algorithm which can be used in two independent programs, therefore easing the comparison of results and identifying differences
in other parts of the programs. The choices made in common parton-shower event
generators are summarised in Table 2.4.
Additional complications in the implementation of parton showers arise because
of differences between final state showers with time-like virtualities and initial state
showers with space-like virtualities. In final-state showers, once the shower cut off is
reached, partons are put on the mass shell. Conversely, initial state showers have to be
2 Phenomenology of Jet Substructure
1 GeV, which results in an infrared cut-off scale t c . This leads to an upper bound on
z, given by z < 1 − t c /t max , where t max is the maximum parton shower scale, usually
identified by the scale of the primary hard scattering. The upper bound on z results
in a bound on the number of soft gluons produced, which would otherwise tend to
infinity due to the soft singularity. The computation of the total number of branchings
and the corresponding values of t at which these occur in individual events are the
primary tasks of parton shower algorithms. The ensemble of partonic final states in
many events then corresponds to the solution of an evolution equation for parton
showers, similar to the DGLAP evolution. The probability for parton a to branch at
a given value of t can be expressed by an integral over all allowed z values, I (t).
Conversely, the probability for no branching to occur between an initial value of t 0 and
t exponentiates, which is known as Sudakov form factor S a (t 0 , t). Thus, the actual
probability for parton a to branch at scale t, is given by I (t) · S a (t 0 , t), if parton a was
produced at scale t 0 . This means that parton a can only branch at t if it has not branched
already at an earlier stage t
< t. The Sudakov factor is the appropriate suppression
factor, ensuring the conservation of total probability. Formally, the Sudakov factors
account for any unresolved splittings and virtual corrections which are assumed to
precisely cancel the singular real corrections when integrated over phase space [332].
The product I (t) · S a (t 0 , t) can be cast into an evolution equation, which is solved
by any parton-shower event generator.
While this prescription is fairly general, there are a number of choices that have
to be made in the exact implementation of a parton shower and the quality of parton
shower predictions depends significantly on these choices. One important example
is the exact choice of the evolution variable, which is one of the main differences
between the most commonly used event generators. The virtuality t was used in
Pythia versions earlier than 6.4 and Sherpa versions earlier than 1.2, whereas both
generators switched to a k ⊥ -based evolution for later versions. This change expedites the merging of parton showers with higher-order, multi-leg matrix element
generators and the use of recent multiple parton interaction models. The squared
energy-weighted emission angle, E
2
θ
2 is used in early Herwig versions, with a generalisation implemented in recent versions. Another important choice is the splitting
variable z, where the two common choices are the light-cone (LC) momentum fraction or the energy fraction taken by parton b. Coherence effects can be accounted
for by employing a dipole formalism or an angular ordering (AO) of emissions. If
neither option is viable, an angular ordering of emissions can be enforced through a
veto. Recently, a new shower model has been developed, called Dire [333], which
is a hybrid between dipole and parton shower. The shower has been implemented in
Pythia and Sherpa, and is the only algorithm which can be used in two independent programs, therefore easing the comparison of results and identifying differences
in other parts of the programs. The choices made in common parton-shower event
generators are summarised in Table 2.4.
Additional complications in the implementation of parton showers arise because
of differences between final state showers with time-like virtualities and initial state
showers with space-like virtualities. In final-state showers, once the shower cut off is
reached, partons are put on the mass shell. Conversely, initial state showers have to be