Phase Space Cuts
 
  - Cuts in all processes
- Cuts in 2 → 1 processes
- Cuts in 2 → 2 processes
- Cuts in 2 → 3 processes
- Generation strategy and documentation
- Reweighting of 2 → 2 processes
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
 
 
parm   PhaseSpace:mHatMin   
 (default = 4.; minimum = 0.)
The minimum invariant mass. 
   
 
parm   PhaseSpace:mHatMax   
 (default = -1.)
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
 
 
parm   PhaseSpace:pTHatMin   
 (default = 0.; minimum = 0.)
The minimum invariant pT. 
   
 
parm   PhaseSpace:pTHatMax   
 (default = -1.)
The maximum invariant pT. 
A value below pTHatMin means there is no upper limit. 
   
 
parm   PhaseSpace:pTHatMinDiverge   
 (default = 1.; minimum = 0.5)
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. 
   
 
flag   PhaseSpace:useBreitWigners   
 (default = on)
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. 
   
 
parm   PhaseSpace:minWidthBreitWigners   
 (default = 0.01; minimum = 1e-6)
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. 
The Breit-Wigner shape is deformed by the variation of the cross section 
across the peak. Only applies when useBreitWigners is on; 
else the nominal mass value is always used. 
   
 
parm   PhaseSpace:minWidthNarrowBW   
 (default = 1e-6; minimum = 1e-10)
A particle that is not wide enough to qualify for the dynamical mass 
generation above can still qualify for a simplified treatment if it has 
a width above this value. Then the mass is selected according to a simple 
symmetric Breit-Wigner that will not be (significantly) distorted by 
the cross section variation. Only applies when useBreitWigners 
is on; else the nominal mass value is always used. Note that this parameter 
is also used for mass selection in the MPI machinery. 
   
 
 
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. 
 
parm   PhaseSpace:Q2Min   
 (default = 0.0; minimum = 0.0)
The minimum value for the DIS variable Q^2 = - tHat. 
Can only meaningfully be used for scattering processes between two 
non-identical particles, i.e. where tHat and uHat 
are experimentally distinguishable. No cut will be applied for 
Q2Min < pTHatMinDiverge^2. 
   
 
 
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 labeling 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. 
 
parm   PhaseSpace:pTHat3Min   
 (default = 10.; minimum = 0.)
The minimum invariant pT of the highest-pT parton in 
QCD 2 → 3 processes. 
   
 
parm   PhaseSpace:pTHat3Max   
 (default = -1.)
The maximum invariant pT of the highest-pT parton in 
QCD 2 → 3 processes 
A value below pTHat3Min means there is no upper limit. 
   
 
parm   PhaseSpace:pTHat5Min   
 (default = 10.; minimum = 0.)
The minimum invariant pT of the lowest-pT parton in 
QCD 2 → 3 processes. 
   
 
parm   PhaseSpace:pTHat5Max   
 (default = -1.)
The maximum invariant pT of the lowest-pT parton in 
QCD 2 → 3 processes 
A value below pTHat5Min means there is no upper limit. 
   
 
parm   PhaseSpace:RsepMin   
 (default = 1.)
The minimum separation R in (eta, phi) space between 
any two outgoing partons in QCD 2 → 3 processes. 
   
 
 
 
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 efficiency 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. 
 
flag   PhaseSpace:showSearch   
 (default = off)
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. 
   
 
flag   PhaseSpace:showViolation   
 (default = off)
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. 
   
 
flag   PhaseSpace:increaseMaximum   
 (default = off)
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::stat() 
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. main322.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. 
 
flag   PhaseSpace:bias2Selection   
 (default = off)
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. 
   
 
parm   PhaseSpace:bias2SelectionPow   
 (default = 4.; minimum = 0.; maximum = 10.)
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. 
   
 
parm   PhaseSpace:bias2SelectionRef   
 (default = 10.; minimum = 1.)
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.