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2<head>
3<title>Phase Space Cuts</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
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7<body>
8
9<h2>Phase Space Cuts</h2>
10
11<code>PhaseSpace</code> is base class for all hard-process phase-space
12generators, either generic <i>2 -> 1</i> or <i>2 -> 2</i> ones,
13or specialized ones like for elastic and diffractive scattering.
14
15<p/>
16In it, it is possible to constrain the kinematics of most processes.
17(Exceptions are "soft physics", i.e. minimum bias, elastic and
18diffractive processes. The Coulomb singularity for elastic scatterings,
19if simulated, is <a href="TotalCrossSections.html" target="page">handled separately</a>.)
20These constraints apply in the rest frame of the hard subprocess, and
21topologies normally would be changed e.g. by subsequent showering
22activity. The cross section of a process is adjusted to only
23correspond to the allowed phase space.
24
25<p/>
26The more particles in the final state, the more cuts could be applied.
27Here we have tried to remain with the useful minimum, however. More
28generic possibilities could be handled by the
29<a href="UserHooks.html" target="page">user hooks</a> facility.
30
31<h3>Cuts in all processes</h3>
32
33<p/><code>parm&nbsp; </code><strong> PhaseSpace:mHatMin &nbsp;</strong> 
34 (<code>default = <strong>4.</strong></code>; <code>minimum = 0.</code>)<br/>
35The minimum invariant mass.
36 
37
38<p/><code>parm&nbsp; </code><strong> PhaseSpace:mHatMax &nbsp;</strong> 
39 (<code>default = <strong>-1.</strong></code>)<br/>
40The maximum invariant mass.
41A value below <code>mHatMin</code> means there is no upper limit.
42 
43
44<h3>Cuts in <i>2 -> 1</i> processes</h3>
45
46When a resonance <code>id</code> is produced, the
47<code><a href="ParticleDataScheme.html" target="page">mMin(id)</a></code> and
48<code><a href="ParticleDataScheme.html" target="page">mMax(id)</a></code> 
49methods restrict the allowed mass range
50of this resonance. Therefore the allowed range is chosen to be the
51overlap of this range and the <code>mHatMin</code> to
52<code>mHatMax</code> range above. Most resonances by default have no
53upper mass limit, so effects mainly concern the lower limit.
54Should there be no overlap between the two ranges then the process
55will be switched off.
56
57<h3>Cuts in <i>2 -> 2</i> processes</h3>
58
59<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHatMin &nbsp;</strong> 
60 (<code>default = <strong>0.</strong></code>; <code>minimum = 0.</code>)<br/>
61The minimum invariant <i>pT</i>.
62 
63
64<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHatMax &nbsp;</strong> 
65 (<code>default = <strong>-1.</strong></code>)<br/>
66The maximum invariant <i>pT</i>.
67A value below <code>pTHatMin</code> means there is no upper limit.
68 
69
70<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHatMinDiverge &nbsp;</strong> 
71 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.5</code>)<br/>
72Extra <i>pT</i> cut to avoid the divergences of some processes
73in the limit <i>pT -> 0</i>. Specifically, if either or both
74produced particles have a mass below <code>pTHatMinDiverge</code> 
75then <i>pT</i> is limited from below by the larger of
76<code>pTHatMin</code> and <code>pTHatMinDiverge</code>.
77 
78
79<p/><code>flag&nbsp; </code><strong> PhaseSpace:useBreitWigners &nbsp;</strong> 
80 (<code>default = <strong>on</strong></code>)<br/>
81Allows masses to be selected according to Breit-Wigner shapes in
82<i>2 -> 2</i> processes, whenever particles have been declared
83with a nonvanishing width above the threshold below. In those cases
84also the limits below will be used for the mass selection. For
85<i>2 -> 1</i> processes the Breit-Wigner shape is part of the
86cross section itself, and therefore always included.
87 
88
89<p/><code>parm&nbsp; </code><strong> PhaseSpace:minWidthBreitWigners &nbsp;</strong> 
90 (<code>default = <strong>0.01</strong></code>; <code>minimum = 1e-6</code>)<br/>
91The minimum width a resonance must have for the mass to be dynamically
92selected according to a Breit-Wigner shape, within the limits set below.
93Only applies when <code>useBreitWigners</code> is on; else the nominal
94mass value is always used.
95 
96
97<p/>
98For a particle with a Breit-Wigner shape selected, according to the
99rules above and to the rules of the particle species itself, the
100<code><a href="ParticleDataScheme.html" target="page">mMin(id)</a></code> and
101<code><a href="ParticleDataScheme.html" target="page">mMax(id)</a></code> 
102methods restrict the allowed mass range of the particle, just like for
103the <i>2 -> 1 </i> processes.   
104
105<h3>Cuts in <i>2 -> 3</i> processes</h3>
106
107There are two main classes of <i>2 -> 3</i> processes. One is the
108processes such as <i>WW/ZZ</i>-fusion Higgs production, i.e.
109<i>q q -> q q H</i>, where there are no special singularities
110associated with two partons in the final state being collinear,
111or even for <i>pT -> 0</i>. For this class, no further cuts
112have been introduced than those already available for <i>2 -> 2</i> 
113processes. Specifically, for now all three are restricted exactly the
114same way by <code>pTHatMin</code> and <code>pTHatMax</code>. As above,
115Breit-Wigner mass ranges can be restricted.
116
117<p/>
118The other <i>2 -> 3</i> event class is QCD processes, such as
119<i>g g -> g g g</i>. Here the soft and collinear singularities
120play a major role, and the phase space generation and cuts have
121been adapted to this. For this class, an alternative set of cuts
122is used, as outlined in the following. First of all the three
123outgoing partons are ordered in falling <i>pT</i>, i.e.
124<i>pT_3 > pT_4 > pT_5</i> (where the labelling 3, 4, 5 of the outgoing
125partons is random, i.e. unrelated to the order specified in the
126process name). The allowed ranges of <i>pT_3</i> and <i>pT_5</i>
127can be specified, but obviously <i>pT_3max >= pT_5max</i> and
128<i>pT_3min >= pT_5min</i>. The <i>pT_4</i> is not constrained
129explicitly, but is constructed from the vector sum of <i>pT_3</i>
130and <i>pT_5</i>, subject to the constraint that it has to lie
131between the two in magnitude. While the <i>pT</i> cuts take care
132of singularities collinear with the incoming beams, it is also
133necessary to handle final-state singularities, when two outgoing
134partons become collinear. This is done by requiring a minimal
135separation in <i>R</i>, where
136<i>R^2 = (Delta eta)^2 + (Delta phi)^2</i>.
137Finally, a note about efficiency. The QCD <i>2 -> 3</i> phase space
138is not set up to explicitly include <i>mHat</i> as one of the basic
139variables. Such a cut is only done after a phase space point is already
140selected, which means that a narrow mass choice will slow down the
141program appreciably. Also narrow <i>pT_3</i> and <i>pT_5</i> bins
142are likely to give inefficient generation, if it gives rise to
143significant indirect restrictions on <i>pT_4</i>.
144
145<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHat3Min &nbsp;</strong> 
146 (<code>default = <strong>10.</strong></code>; <code>minimum = 0.</code>)<br/>
147The minimum invariant <i>pT</i> of the highest-<i>pT</i> parton in
148QCD <i>2 -> 3</i> processes.
149 
150
151<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHat3Max &nbsp;</strong> 
152 (<code>default = <strong>-1.</strong></code>)<br/>
153The maximum invariant <i>pT</i> of the highest-<i>pT</i> parton in
154QCD <i>2 -> 3</i> processes
155A value below <code>pTHat3Min</code> means there is no upper limit.
156 
157
158<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHat5Min &nbsp;</strong> 
159 (<code>default = <strong>10.</strong></code>; <code>minimum = 0.</code>)<br/>
160The minimum invariant <i>pT</i> of the lowest-<i>pT</i> parton in
161QCD <i>2 -> 3</i> processes.
162 
163
164<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHat5Max &nbsp;</strong> 
165 (<code>default = <strong>-1.</strong></code>)<br/>
166The maximum invariant <i>pT</i> of the lowest-<i>pT</i> parton in
167QCD <i>2 -> 3</i> processes
168A value below <code>pTHat5Min</code> means there is no upper limit.
169 
170
171<p/><code>parm&nbsp; </code><strong> PhaseSpace:RsepMin &nbsp;</strong> 
172 (<code>default = <strong>1.</strong></code>)<br/>
173The minimum separation <i>R</i> in <i>(eta, phi)</i> space between
174any two outgoing partons in QCD <i>2 -> 3</i> processes.
175 
176
177
178<h3>Cuts for a second hard process</h3>
179
180If you use the machinery that allows the generation of a specified
181<a href="ASecondHardProcess.html" target="page">second hard process</a> then,
182by default, the same phase space cuts will be used for it as listed
183above. Optionally, however, you may use a second set of cuts, as
184described here. In this context "first" and "second" is merely a
185technical distinction; you are welcome e.g. to pick <i>pT</i> ranges
186such that the second interaction always has a larger <i>pT</i> than
187the first.
188
189<p/><code>flag&nbsp; </code><strong> PhaseSpace:sameForSecond &nbsp;</strong> 
190 (<code>default = <strong>on</strong></code>)<br/>
191By default use the same cuts for a second hard process as for the
192first. If <code>off</code> then instead use the mass and <i>pT</i>
193cuts below, where relevant. (The other cuts above still remain the same.) 
194 
195
196<p/><code>parm&nbsp; </code><strong> PhaseSpace:mHatMinSecond &nbsp;</strong> 
197 (<code>default = <strong>4.</strong></code>; <code>minimum = 0.</code>)<br/>
198The minimum invariant mass for a second interaction, if separate.
199 
200
201<p/><code>parm&nbsp; </code><strong> PhaseSpace:mHatMaxSecond &nbsp;</strong> 
202 (<code>default = <strong>-1.</strong></code>)<br/>
203The maximum invariant mass for a second interaction, if separate.
204A value below <code>mHatMin</code> means there is no upper limit.
205 
206
207<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHatMinSecond &nbsp;</strong> 
208 (<code>default = <strong>0.</strong></code>; <code>minimum = 0.</code>)<br/>
209The minimum invariant <i>pT</i> for a second interaction, if separate.
210 
211
212<p/><code>parm&nbsp; </code><strong> PhaseSpace:pTHatMaxSecond &nbsp;</strong> 
213 (<code>default = <strong>-1.</strong></code>)<br/>
214The maximum invariant <i>pT</i> for a second interaction, if separate.
215A value below <code>pTHatMin</code> means there is no upper limit.
216 
217
218<h3>Generation strategy and documentation</h3>
219
220During the initialization stage a simplified function is found,
221that is intended to be above the true cross-section behaviour
222over the whole of phase space. It is chosen to be easily integrable
223and invertible. That way a trial phase space point can be selected
224according this simple function, and then be accepted by the ratio of
225true to the simple function. For a good efficieny the ratio should be
226close to unity,  yet never above it. This constrains the absolute
227normalization of the simple function. The initial search may fail to
228find the phase space point where the true-to-simple ratio is maximal,
229however. This then can lead to subsequent maximum violations, where the
230ratio is above unity. Two alternative strategies are implemented to
231handle such situations, see below.
232
233<p/><code>flag&nbsp; </code><strong> PhaseSpace:showSearch &nbsp;</strong> 
234 (<code>default = <strong>off</strong></code>)<br/>
235Possibility to print information on the search for phase-space
236coefficients that (in a multichannel approach) provides an analytical
237upper envelope of the differential cross section, and the
238corresponding upper estimate of the cross section. Of interest
239for crosschecks by expert users only.
240 
241
242<p/><code>flag&nbsp; </code><strong> PhaseSpace:showViolation &nbsp;</strong> 
243 (<code>default = <strong>off</strong></code>)<br/>
244Possibility to print information whenever the assumed maximum
245differential cross section of a process is violated, i.e. when
246the initial maximization procedure did not find the true maximum.
247Also, should negative cross sections occur, print whenever a more
248negative value is encountered.
249 
250
251<p/><code>flag&nbsp; </code><strong> PhaseSpace:increaseMaximum &nbsp;</strong> 
252 (<code>default = <strong>off</strong></code>)<br/>
253Strategy for handling cases where a larger cross section is
254obtained during the event generation than was assumed at initialization,
255i.e. when a violation occurs.
256<br/><b>off:</b>each event comes with a weight, which normally is unity
257(as a consequence of the acceptance/rejection step), and is found in
258<code><a href="EventInformation.html" target="page">Info::weight()</a></code>.
259For events which exceed the maximum instead the true-to-simple ratio
260is stored as event weight, which then is above unity. If the user so
261wishes this weight can then be carried along when event properties are
262histogrammed. Since normally such violations should be rare and not
263too much above unity one could expect most users to ignore such issues
264be default. Should maximum violations turn out to be frequent (visible
265in the <code><a href="EventStatistics.html" target="page">Pythia::statistics()</a></code>
266output) the option exists to use the information.
267<br/><b>on:</b>the maximum is increased whenever it is exceeded. Thus
268events generated after this point will be "correctly" distributed,
269while ones generated previously obviously then have had too high a
270relative weight. If violations occur early on and/or are small this
271strategy should do a good job of correcting to the desired phase-space
272distribution. This strategy may be more convenient for the normal user,
273who would not wish to worry about event weights. It does have the
274disadvantage that the raised maximum introduces an extra amount of
275"history memory" to the generation sequence, so that it becomes less
276easy to save-and-restore the <a href="RandomNumbers.html" target="page">random-number
277state</a> for debugging purposes. 
278 
279
280<h3>Reweighting of <i>2 -> 2</i> processes</h3>
281
282Events normally come with unit weight, i.e. are distributed across
283the allowed phase space region according to the appropriate differential
284cross sections. Sometimes it may be convenient to have an uneven
285distribution of events. The classical example here is that many cross
286sections drop off with transverse momentum <i>pT</i>, such that few
287events are generated at large <i>pT</i> scales. If one wants to
288plot the <i>pT</i> cross section, and all that comes with it, the
289statistical error will then degrade with increasing <i>pT</i> 
290where fewer events end up.
291
292<p/>
293One solution is to split the full <i>pT</i> range into several
294separate subranges, where the events of each subsample obtains a
295different overall normalization. Specifically, if you generate a
296comparable number of events in each <i>pT</i> bin, such that
297larger <i>pT</i> bins are oversampled, these bins come with a
298correspondingly reduced overall weight, that needs to be taken into
299account when the bins are combined. The other is to have a continuously
300increasing oversampling of events at larger <i>pT</i> scales, which
301is compensated by a continuously decreasing weight for the event.
302
303<p/>
304Both of these solutions are supported. Specifically, for
305<i>2 -> 2</i> processes, the <i>pTHat</i> scale offers a
306convenient classification of the event. (Of course, two events
307starting out from the same <i>pTHat</i> scale will experience
308different parton shower evolutions, etc., and may therefore look
309quite different at the end.) The two cuts
310<code>PhaseSpace:pTHatMin</code> and <code>PhaseSpace:pTHatMax</code>
311therefore offers a way to slice a <i>pT</i> range into subranges,
312see e.g. <code>main08.cc</code>. Alternatively the
313<a href="UserHooks.html" target="page">User Hooks</a> machinery offers the
314possibility for you to define your own reweighting of phase space
315sampling, with a corresponding event weight, with
316<code>UserHooks::canBiasSelection</code> and related methods.
317
318<p/>
319As a simplified option, we here offer the possibility to bias the
320<i>2 -> 2</i> sampling by a power of <i>pTHat</i>, then with     
321events having a weight the inverse of this. This fast track will only
322work under a number of strict conditions, implemented to reduce the
323risk of abuse. (Whereas a <code>UserHooks</code> setup can be more
324flexible.) Specifically it will work if only high-<i>pT</i>
325<i>2 -> 2</i> processes already implemented in PYTHIA are requested,
326notably the <code>HardQCD</code> ones. That is, you cannot mix with
327<i>2 -> 1</i> or <i>2 -> 3</i> processes, nor with external
328processes (notably Les Houches input) or <code>SoftQCD</code> ones,
329and  you cannot use the option to define a
330<a href="ASecondHardProcess.html" target="page">second hard process</a> in
331the same event. Furthermore you have to be careful about the choice
332of <code>PhaseSpace:pTHatMin</code>, since a <i>pTHat = 0</i> 
333event would come with an infinite weight.
334
335<p/><code>flag&nbsp; </code><strong> PhaseSpace:bias2Selection &nbsp;</strong> 
336 (<code>default = <strong>off</strong></code>)<br/>
337Possibility to switch on a biased phase space sampling,
338with compensatingly weighted events, for <i>2 -> 2</i> processes.
339Can only be used under the specific conditions explained in
340the paragraph above; under other conditions the initialization
341will abort.
342 
343
344<p/><code>parm&nbsp; </code><strong> PhaseSpace:bias2SelectionPow &nbsp;</strong> 
345 (<code>default = <strong>4.</strong></code>; <code>minimum = 0.</code>; <code>maximum = 10.</code>)<br/>
346If the above flag is on, then a <i>2 -> 2</i> process at a scale
347<i>pTHat</i> will be oversampled in phase space by an amount
348<i>(pTHat/pTRef)^pow</i>, where you set the power <i>pow</i>
349here. Events are assigned a compensating
350<a href="EventInformation.html" target="page">weight</a> the inverse of this,
351i.e. <code>Info::weight()</code> will return <i>(pTRef/pTHat)^pow</i>.
352This weight should then be used in the histogramming of event properties.
353The final overall normalization also involves the
354<code>Info::weightSum()</code> value. 
355 
356
357<p/><code>parm&nbsp; </code><strong> PhaseSpace:bias2SelectionRef &nbsp;</strong> 
358 (<code>default = <strong>10.</strong></code>; <code>minimum = 1.</code>)<br/>
359The reference scale <i>pTRef</i> introduced above, such that events
360with this <i>pTHat</i> obtain unit weight in the reweighting procedure.
361The value of this parameter has no impact on the final result of the
362reweighting procedure, but is only there for convenience, i.e. to
363give "reasonably-sized" weights. 
364 
365
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