2.6 Event Generators
53
A large number of 2 → 1 and 2 → 2 processes, as well as some 2 → 3 processes
are implemented in the standard event generators Pythia and Herwig. The event
generator Sherpa allows for the simulation of tree-level 2 → n matrix elements. In
case of resonance production with 2 → 1 graphs and the resonance being either a W ,
Z , H boson or some BSM resonance, the decay is simulated through a Breit-Wigner
shape with energy-dependent width, resulting in 2 → 1 → 2 processes. A similar
procedure is applied when more than one resonance is produced (e.g. gg → tt),
where Breit-Wigner distributions with fixed widths are used to assign masses to the
resonances.
Modern event generators often go beyond the generation of LO 2 → 2 processes
at the matrix element level, which is accomplished through an interface to external
matrix element generators. These have to be matched to the parton shower simulation
to avoid double-counting of higher order contributions, as described below.
2.6.3 Parton Showers
In event generators the calculation of inclusive cross sections from (2.48) is augmented by additional particles originating from parton branchings of the form
a → bc. These 1 → 2 splittings are crucial for the description of local observables
(i.e. observables sensitive to emissions in localised parts of the phase space), in
particular all quantities related to jet substructure. The total inclusive cross section
is unchanged by the parton shower. The simulated parton branchings result in a
tree-like structure for individual events (a shower), and correspond to a resummation
of the leading logarithmic (LL) corrections when considering an ensemble of all
possible shower configurations. While this suggests a formal accuracy of event generators at the (N)LO+LL level, to equate a modern parton shower with LL accuracy
of analytic calculations is a gross underestimation. A modern parton shower respects
energy and momentum conservation through partonic recoils and includes coherence
effects, which enter only at NLO in analytic calculations. The final product are predictions on a similar footing as (N)LO+NLL calculations, even though O(α
2
S
) terms
are not included in the splitting kernels.
13
The basic building blocks of parton showers are the 1 → 2 DGLAP splitting
kernels P a→bc , describing the collinear splitting of parton a into partons b and
c.
14 For a given branching a → bc, parton b takes the fractional momentum z and
parton c a fraction 1 − z. The second variable specifying the kinematics is given
by t = −2 p a p b , which has dimension of a squared mass and is related to the mass
or transverse momentum scale k ⊥ of the branching. Confinement implies a natural
cut-off scale for the transverse momentum generated by the parton shower of about
13 These would result in 1 → 3 splittings and corrections to the 1 → 2 splitting functions.
14 In parton showers, the ‘± prescriptions’ and δ(1 − z) terms are absent. These ensure flavour and
energy conservation in analytical calculations, which are encoded at each step of the parton shower
where each step is traced in detail.
53
A large number of 2 → 1 and 2 → 2 processes, as well as some 2 → 3 processes
are implemented in the standard event generators Pythia and Herwig. The event
generator Sherpa allows for the simulation of tree-level 2 → n matrix elements. In
case of resonance production with 2 → 1 graphs and the resonance being either a W ,
Z , H boson or some BSM resonance, the decay is simulated through a Breit-Wigner
shape with energy-dependent width, resulting in 2 → 1 → 2 processes. A similar
procedure is applied when more than one resonance is produced (e.g. gg → tt),
where Breit-Wigner distributions with fixed widths are used to assign masses to the
resonances.
Modern event generators often go beyond the generation of LO 2 → 2 processes
at the matrix element level, which is accomplished through an interface to external
matrix element generators. These have to be matched to the parton shower simulation
to avoid double-counting of higher order contributions, as described below.
2.6.3 Parton Showers
In event generators the calculation of inclusive cross sections from (2.48) is augmented by additional particles originating from parton branchings of the form
a → bc. These 1 → 2 splittings are crucial for the description of local observables
(i.e. observables sensitive to emissions in localised parts of the phase space), in
particular all quantities related to jet substructure. The total inclusive cross section
is unchanged by the parton shower. The simulated parton branchings result in a
tree-like structure for individual events (a shower), and correspond to a resummation
of the leading logarithmic (LL) corrections when considering an ensemble of all
possible shower configurations. While this suggests a formal accuracy of event generators at the (N)LO+LL level, to equate a modern parton shower with LL accuracy
of analytic calculations is a gross underestimation. A modern parton shower respects
energy and momentum conservation through partonic recoils and includes coherence
effects, which enter only at NLO in analytic calculations. The final product are predictions on a similar footing as (N)LO+NLL calculations, even though O(α
2
S
) terms
are not included in the splitting kernels.
13
The basic building blocks of parton showers are the 1 → 2 DGLAP splitting
kernels P a→bc , describing the collinear splitting of parton a into partons b and
c.
14 For a given branching a → bc, parton b takes the fractional momentum z and
parton c a fraction 1 − z. The second variable specifying the kinematics is given
by t = −2 p a p b , which has dimension of a squared mass and is related to the mass
or transverse momentum scale k ⊥ of the branching. Confinement implies a natural
cut-off scale for the transverse momentum generated by the parton shower of about
13 These would result in 1 → 3 splittings and corrections to the 1 → 2 splitting functions.
14 In parton showers, the ‘± prescriptions’ and δ(1 − z) terms are absent. These ensure flavour and
energy conservation in analytical calculations, which are encoded at each step of the parton shower
where each step is traced in detail.