Phase Space Cuts
PhaseSpace is base class for all hard-process phase-space
generators, either generic 2 -> 1 or 2 -> 2 ones,
or specialized ones like for elastic and diffractive scattering.
In it, it is possible to constrain the kinematics of most processes.
(Exceptions are "soft physics", i.e. minimum bias, elastic and
diffractive processes. The Coulomb singularity for elastic scatterings,
if simulated, is handled separately.)
These constraints apply in the rest frame of the hard subprocess, and
topologies normally would be changed e.g. by subsequent showering
activity. The cross section of a process is adjusted to only
correspond to the allowed phase space.
The more particles in the final state, the more cuts could be applied.
Here we have tried to remain with the useful minimum, however. More
generic possibilities could be handled by the
user hooks facility.
Cuts in all processes
The minimum invariant mass.
The maximum invariant mass.
A value below mHatMin means there is no upper limit.
Cuts in 2 -> 1 processes
When a resonance id is produced, the
mMin(id) and
mMax(id)
methods restrict the allowed mass range
of this resonance. Therefore the allowed range is chosen to be the
overlap of this range and the mHatMin to
mHatMax range above. Most resonances by default have no
upper mass limit, so effects mainly concern the lower limit.
Should there be no overlap between the two ranges then the process
will be switched off.
Cuts in 2 -> 2 processes
The minimum invariant pT.
The maximum invariant pT.
A value below pTHatMin means there is no upper limit.
Extra pT cut to avoid the divergences of some processes
in the limit pT -> 0. Specifically, if either or both
produced particles have a mass below pTHatMinDiverge
then pT is limited from below by the larger of
pTHatMin and pTHatMinDiverge.
Allows masses to be selected according to Breit-Wigner shapes in
2 -> 2 processes, whenever particles have been declared
with a nonvanishing width above the threshold below. In those cases
also the limits below will be used for the mass selection. For
2 -> 1 processes the Breit-Wigner shape is part of the
cross section itself, and therefore always included.
The minimum width a resonance must have for the mass to be dynamically
selected according to a Breit-Wigner shape, within the limits set below.
Only applies when useBreitWigners is on; else the nominal
mass value is always used.
For a particle with a Breit-Wigner shape selected, according to the
rules above and to the rules of the particle species itself, the
mMin(id) and
mMax(id)
methods restrict the allowed mass range of the particle, just like for
the 2 -> 1 processes.
Cuts in 2 -> 3 processes
There are two main classes of 2 -> 3 processes. One is the
processes such as WW/ZZ-fusion Higgs production, i.e.
q q -> q q H, where there are no special singularities
associated with two partons in the final state being collinear,
or even for pT -> 0. For this class, no further cuts
have been introduced than those already available for 2 -> 2
processes. Specifically, for now all three are restricted exactly the
same way by pTHatMin and pTHatMax. As above,
Breit-Wigner mass ranges can be restricted.
The other 2 -> 3 event class is QCD processes, such as
g g -> g g g. Here the soft and collinear singularities
play a major role, and the phase space generation and cuts have
been adapted to this. For this class, an alternative set of cuts
is used, as outlined in the following. First of all the three
outgoing partons are ordered in falling pT, i.e.
pT_3 > pT_4 > pT_5 (where the labelling 3, 4, 5 of the outgoing
partons is random, i.e. unrelated to the order specified in the
process name). The allowed ranges of pT_3 and pT_5
can be specified, but obviously pT_3max >= pT_5max and
pT_3min >= pT_5min. The pT_4 is not constrained
explicitly, but is constructed from the vector sum of pT_3
and pT_5, subject to the constraint that it has to lie
between the two in magnitude. While the pT cuts take care
of singularities collinear with the incoming beams, it is also
necessary to handle final-state singularities, when two outgoing
partons become collinear. This is done by requiring a minimal
separation in R, where
R^2 = (Delta eta)^2 + (Delta phi)^2.
Finally, a note about efficiency. The QCD 2 -> 3 phase space
is not set up to explicitly include mHat as one of the basic
variables. Such a cut is only done after a phase space point is already
selected, which means that a narrow mass choice will slow down the
program appreciably. Also narrow pT_3 and pT_5 bins
are likely to give inefficient generation, if it gives rise to
significant indirect restrictions on pT_4.
The minimum invariant pT of the highest-pT parton in
QCD 2 -> 3 processes.
The maximum invariant pT of the highest-pT parton in
QCD 2 -> 3 processes
A value below pTHat3Min means there is no upper limit.
The minimum invariant pT of the lowest-pT parton in
QCD 2 -> 3 processes.
The maximum invariant pT of the lowest-pT parton in
QCD 2 -> 3 processes
A value below pTHat5Min means there is no upper limit.
The minimum separation R in (eta, phi) space between
any two outgoing partons in QCD 2 -> 3 processes.
Cuts for a second hard process
If you use the machinery that allows the generation of a specified
second hard process then,
by default, the same phase space cuts will be used for it as listed
above. Optionally, however, you may use a second set of cuts, as
described here. In this context "first" and "second" is merely a
technical distinction; you are welcome e.g. to pick pT ranges
such that the second interaction always has a larger pT than
the first.
By default use the same cuts for a second hard process as for the
first. If off then instead use the mass and pT
cuts below, where relevant. (The other cuts above still remain the same.)
The minimum invariant mass for a second interaction, if separate.
The maximum invariant mass for a second interaction, if separate.
A value below mHatMin means there is no upper limit.
The minimum invariant pT for a second interaction, if separate.
The maximum invariant pT for a second interaction, if separate.
A value below pTHatMin means there is no upper limit.
Generation strategy and documentation
During the initialization stage a simplified function is found,
that is intended to be above the true cross-section behaviour
over the whole of phase space. It is chosen to be easily integrable
and invertible. That way a trial phase space point can be selected
according this simple function, and then be accepted by the ratio of
true to the simple function. For a good efficieny the ratio should be
close to unity, yet never above it. This constrains the absolute
normalization of the simple function. The initial search may fail to
find the phase space point where the true-to-simple ratio is maximal,
however. This then can lead to subsequent maximum violations, where the
ratio is above unity. Two alternative strategies are implemented to
handle such situations, see below.
Possibility to print information on the search for phase-space
coefficients that (in a multichannel approach) provides an analytical
upper envelope of the differential cross section, and the
corresponding upper estimate of the cross section. Of interest
for crosschecks by expert users only.
Possibility to print information whenever the assumed maximum
differential cross section of a process is violated, i.e. when
the initial maximization procedure did not find the true maximum.
Also, should negative cross sections occur, print whenever a more
negative value is encountered.
Strategy for handling cases where a larger cross section is
obtained during the event generation than was assumed at initialization,
i.e. when a violation occurs.
off:each event comes with a weight, which normally is unity
(as a consequence of the acceptance/rejection step), and is found in
Info::weight().
For events which exceed the maximum instead the true-to-simple ratio
is stored as event weight, which then is above unity. If the user so
wishes this weight can then be carried along when event properties are
histogrammed. Since normally such violations should be rare and not
too much above unity one could expect most users to ignore such issues
be default. Should maximum violations turn out to be frequent (visible
in the Pythia::statistics()
output) the option exists to use the information.
on:the maximum is increased whenever it is exceeded. Thus
events generated after this point will be "correctly" distributed,
while ones generated previously obviously then have had too high a
relative weight. If violations occur early on and/or are small this
strategy should do a good job of correcting to the desired phase-space
distribution. This strategy may be more convenient for the normal user,
who would not wish to worry about event weights. It does have the
disadvantage that the raised maximum introduces an extra amount of
"history memory" to the generation sequence, so that it becomes less
easy to save-and-restore the random-number
state for debugging purposes.
Reweighting of 2 -> 2 processes
Events normally come with unit weight, i.e. are distributed across
the allowed phase space region according to the appropriate differential
cross sections. Sometimes it may be convenient to have an uneven
distribution of events. The classical example here is that many cross
sections drop off with transverse momentum pT, such that few
events are generated at large pT scales. If one wants to
plot the pT cross section, and all that comes with it, the
statistical error will then degrade with increasing pT
where fewer events end up.
One solution is to split the full pT range into several
separate subranges, where the events of each subsample obtains a
different overall normalization. Specifically, if you generate a
comparable number of events in each pT bin, such that
larger pT bins are oversampled, these bins come with a
correspondingly reduced overall weight, that needs to be taken into
account when the bins are combined. The other is to have a continuously
increasing oversampling of events at larger pT scales, which
is compensated by a continuously decreasing weight for the event.
Both of these solutions are supported. Specifically, for
2 -> 2 processes, the pTHat scale offers a
convenient classification of the event. (Of course, two events
starting out from the same pTHat scale will experience
different parton shower evolutions, etc., and may therefore look
quite different at the end.) The two cuts
PhaseSpace:pTHatMin and PhaseSpace:pTHatMax
therefore offers a way to slice a pT range into subranges,
see e.g. main08.cc. Alternatively the
User Hooks machinery offers the
possibility for you to define your own reweighting of phase space
sampling, with a corresponding event weight, with
UserHooks::canBiasSelection and related methods.
As a simplified option, we here offer the possibility to bias the
2 -> 2 sampling by a power of pTHat, then with
events having a weight the inverse of this. This fast track will only
work under a number of strict conditions, implemented to reduce the
risk of abuse. (Whereas a UserHooks setup can be more
flexible.) Specifically it will work if only high-pT
2 -> 2 processes already implemented in PYTHIA are requested,
notably the HardQCD ones. That is, you cannot mix with
2 -> 1 or 2 -> 3 processes, nor with external
processes (notably Les Houches input) or SoftQCD ones,
and you cannot use the option to define a
second hard process in
the same event. Furthermore you have to be careful about the choice
of PhaseSpace:pTHatMin, since a pTHat = 0
event would come with an infinite weight.
Possibility to switch on a biased phase space sampling,
with compensatingly weighted events, for 2 -> 2 processes.
Can only be used under the specific conditions explained in
the paragraph above; under other conditions the initialization
will abort.
If the above flag is on, then a 2 -> 2 process at a scale
pTHat will be oversampled in phase space by an amount
(pTHat/pTRef)^pow, where you set the power pow
here. Events are assigned a compensating
weight the inverse of this,
i.e. Info::weight() will return (pTRef/pTHat)^pow.
This weight should then be used in the histogramming of event properties.
The final overall normalization also involves the
Info::weightSum() value.
The reference scale pTRef introduced above, such that events
with this pTHat obtain unit weight in the reweighting procedure.
The value of this parameter has no impact on the final result of the
reweighting procedure, but is only there for convenience, i.e. to
give "reasonably-sized" weights.