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[u/mrichter/AliRoot.git] / PYTHIA8 / pythia8130 / xmldoc / ASecondHardProcess.xml
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5ad4eb21 1<chapter name="A Second Hard Process">
2
3<h2>A Second Hard Process</h2>
4
5When you have selected a set of hard processes for hadron beams, the
6<aloc href="MultipleInteractions">multiple interactions</aloc>
7framework can add further interactions to build up a realistic
8underlying event. These further interactions can come from a wide
9variety of processes, and will occasionally be quite hard. They
10do represent a realistic random mix, however, which means one cannot
11predetermine what will happen. Occasionally there may be cases
12where one wants to specify also the second hard interaction rather
13precisely. The options on this page allow you to do precisely that.
14
15<flag name="SecondHard:generate" default="off">
16Generate two hard scatterings in a collision between hadron beams.
17You must further specify which set of processes to allow for
18the second, see below.
19</flag>
20
21<p/>
22In principle the whole <aloc href="ProcessSelection">process
23selection</aloc> allowed for the first process could be repeated
24for the second one. However, this would probably be overkill.
25Therefore here a more limited set of prepackaged process collections
26are made available, that can then be further combined at will.
27Since the description is almost completely symmetric between the
28first and the second process, you always have the possibility
29to pick one of the two processes according to the complete list
30of possibilities.
31
32<p/>
33Here comes the list of allowed sets of processes, to combine at will:
34
35<flag name="SecondHard:TwoJets" default="off">
36Standard QCD <ei>2 -> 2</ei> processes involving gluons and
37<ei>d, u, s, c, b</ei> quarks.
38</flag>
39
40<flag name="SecondHard:PhotonAndJet" default="off">
41A prompt photon recoiling against a quark or gluon jet.
42
43<flag name="SecondHard:TwoPhotons" default="off">
44Two prompt photons recoiling against each other.
45
46
47<flag name="SecondHard:SingleGmZ" default="off">
48Scattering <ei>q qbar -> gamma^*/Z^0</ei>, with full interference
49between the <ei>gamma^*</ei> and <ei>Z^0</ei>.
50</flag>
51
52<flag name="SecondHard:SingleW" default="off">
53Scattering <ei>q qbar' -> W^+-</ei>.
54</flag>
55
56<flag name="SecondHard:GmZAndJet" default="off">
57Scattering <ei>q qbar -> gamma^*/Z^0 g</ei> and
58<ei>q g -> gamma^*/Z^0 q</ei>.
59</flag>
60
61<flag name="SecondHard:WAndJet" default="off">
62Scattering <ei>q qbar' -> W^+- g</ei> and
63<ei>q g -> W^+- q'</ei>.
64</flag>
65
66<p/>
67A further process collection comes with a warning flag:
68
69<flag name="SecondHard:TwoBJets" default="off">
70The <ei>q qbar -> b bbar</ei> and <ei>g g -> b bbar</ei> processes.
71These are already included in the <code>TwoJets</code> sample above,
72so it would be doublecounting to include both, but we assume there
73may be cases where the <ei>b</ei> subsample will be of special interest.
74This subsample does not include flavour-excitation or gluon-splitting
75contributions to the <ei>b</ei> rate, however, so, depending
76on the topology if interest, it may or may not be a good approximation.
77</flag>
78
79<p/>
80The second hard process obeys exactly the same selection rules for
81<aloc href="PhaseSpaceCuts">phase space cuts</aloc> and
82<aloc href="CouplingsAndScales">couplings and scales</aloc>
83as the first one does. Specifically, a <ei>pTmin</ei> cut for
84<ei>2 -> 2</ei> processes would apply to the first and the second hard
85process alike, and ballpark half of the time the second could be
86generated with a larger <ei>pT</ei> than the first. (Exact numbers
87depending on the relative shape of the two cross sections.) That is,
88first and second is only used as an administrative distinction between
89the two, not as a physics ordering one.
90
91<h3>Cross-section calculation</h3>
92
93As an introduction, a brief reminder of Poissonian statistics.
94Assume a stochastic process in time, for now not necessarily a
95high-energy physics one, where the probability for an event to occur
96at any given time is independent of what happens at other times.
97Then the probability for <ei>n</ei> events to occur in a finite
98time interval is
99<eq>
100P_n = &lt;n&gt;^n exp(-&lt;n&gt;) / n!
101</eq>
102where <ei>&lt;n&gt;</ei> is the average number of events. If this
103number is small we can approximate <ei>exp(-&lt;n&gt;) = 1 </ei>,
104so that <ei>P_1 = &lt;n&gt;</ei> and
105<ei>P_2 = &lt;n&gt;^2 / 2 = P_1^2 / 2</ei>.
106
107<p/>
108Now further assume that the events actually are of two different
109kinds <ei>a</ei> and <ei>b</ei>, occuring independently of each
110other, such that <ei>&lt;n&gt; = &lt;n_a&gt; + &lt;n_b&gt;</ei>.
111It then follows that the probability of having one event of type
112<ei>a</ei> (or <ei>b</ei>) and nothing else is
113<ei>P_1a = &lt;n_a&gt;</ei> (or <ei>P_1b = &lt;n_b&gt;</ei>).
114From
115<eq>
116P_2 = (&lt;n_a&gt; + &lt;n_b&gt)^2 / 2 = (P_1a + P_1b)^2 / 2 =
117(P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2
118</eq>
119it is easy to read off that the probability to have exactly two
120events of kind <ei>a</ei> and none of <ei>b</ei> is
121<ei>P_2aa = P_1a^2 / 2</ei> whereas that of having one <ei>a</ei>
122and one <ei>b</ei> is <ei>P_2ab = P_1a P_1b</ei>. Note that the
123former, with two identical events, contains a factor <ei>1/2</ei>
124while the latter, with two different ones, does not. If viewed
125in a time-ordered sense, the difference is that the latter can be
126obtained two ways, either first an <ei>a</ei> and then a <ei>b</ei>
127or else first a <ei>b</ei> and then an <ei>a</ei>.
128
129<p/>
130To translate this language into cross-sections for high-energy
131events, we assume that interactions can occur at different <ei>pT</ei>
132values independently of each other inside inelastic nondiffractive
133(= "minbias") events. Then the above probabilities translate into
134<ei>P_n = sigma_n / sigma_ND</ei> where <ei>sigma_ND</ei> is the
135total nondiffractive cross section. Again we want to assume that
136<ei>exp(-&lt;n&gt;)</ei> is close to unity, i.e. that the total
137hard cross section above <ei>pTmin</ei> is much smaller than
138<ei>sigma_ND</ei>. The hard cross section is dominated by QCD
139jet production, and a reasonable precaution is to require a
140<ei>pTmin</ei> of at least 20 GeV at LHC energies.
141(For <ei>2 -> 1</ei> processes such as
142<ei>q qbar -> gamma^*/Z^0 (-> f fbar)</ei> one can instead make a
143similar cut on mass.) Then the generic equation
144<ei>P_2 = P_1^2 / 2</ei> translates into
145<ei>sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2</ei> or
146<ei>sigma_2 = sigma_1^2 / (2 sigma_ND)</ei>.
147
148<p/>
149Again different processes <ei>a, b, c, ...</ei> contribute,
150and by the same reasoning we obtain
151<ei>sigma_2aa = sigma_1a^2 / (2 sigma_ND)</ei>,
152<ei>sigma_2ab = sigma_1a sigma_1b / sigma_ND</ei>,
153and so on.
154
155<p/>
156There is one important correction to this picture: all collisions
157do no occur under equal conditions. Some are more central in impact
158parameter, others more peripheral. This leads to a further element of
159variability: central collisions are likely to have more activity
160than the average, peripheral less. Integrated over impact
161parameter standard cross sections are recovered, but correlations
162are affected by a "trigger bias" effect: if you select for events
163with a hard process you favour events at small impact parameter
164which have above-average activity, and therefore also increased
165chance for further interactions. (In PYTHIA this is the origin
166of the "pedestal effect", i.e. that events with a hard interaction
167have more underlying activity than the level found in minimum-bias
168events.) When you specify a matter overlap profile in the
169multiple-interactions scenario, such an enhancement/depletion factor
170<ei>f_impact</ei> is chosen event-by-event and can be averaged
171during the course of the run. As an example, the double Gaussian
172form used in Tune A gives approximately
173<ei>&lt;f_impact&gt; = 2.5</ei>. The above equations therefore
174have to be modified to
175<ei>sigma_2aa = &lt;f_impact&gt; sigma_1a^2 / (2 sigma_ND)</ei>,
176<ei>sigma_2ab = &lt;f_impact&gt; sigma_1a sigma_1b / sigma_ND</ei>.
177Experimentalists often instead use the notation
178<ei>sigma_2ab = sigma_1a sigma_1b / sigma_eff</ei>,
179from which we see that PYTHIA "predicts"
180<ei>sigma_eff = sigma_ND / &lt;f_impact&gt;</ei>.
181When the generation of multiple interactions is switched off it is
182not possible to calculate <ei>&lt;f_impact&gt;</ei> and therefore
183it is set to unity.
184
185<p/>
186When this recipe is to be applied to calculate
187actual cross sections, it is useful to distinguish three cases,
188depending on which set of processes are selected to study for
189the first and second interaction.
190
191<p/>
192(1) The processes <ei>a</ei> for the first interaction and
193<ei>b</ei> for the second one have no overlap at all.
194For instance, the first could be <code>TwoJets</code> and the
195second <code>TwoPhotons</code>. In that case, the two interactions
196can be selected independently, and cross sections tabulated
197for each separate subprocess in the two above classes. At the
198end of the run, the cross sections in <ei>a</ei> should be multiplied
199by <ei>&lt;f_impact&gt; sigma_1b / sigma_ND</ei> to bring them to
200the correct overall level, and those in <ei>b</ei> by
201<ei>&lt;f_impact&gt; sigma_1a / sigma_ND</ei>.
202
203<p/>
204(2) Exactly the same processes <ei>a</ei> are selected for the
205first and second interaction. In that case it works as above,
206with <ei>a = b</ei>, and it is only necessary to multiply by an
207additional factor <ei>1/2</ei>. A compensating factor of 2
208is automatically obtained for picking two different subprocesses,
209e.g. if <code>TwoJets</code> is selected for both interactions,
210then the combination of the two subprocesses <ei>q qbar -> g g</ei>
211and <ei>g g -> g g</ei> can trivially be obtained two ways.
212
213<p/>
214(3) The list of subprocesses partly but not completely overlap.
215For instance, the first process is allowed to contain <ei>a</ei>
216or <ei>c</ei> and the second <ei>b</ei> or <ei>c</ei>, where
217there is no overlap between <ei>a</ei> and <ei>b</ei>. Then,
218when an independent selection for the first and second interaction
219both pick one of the subprocesses in <ei>c</ei>, half of those
220events have to be thrown, and the stored cross section reduced
221accordingly. Considering the four possible combinations of first
222and second process, this gives a
223<eq>
224sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) /
225(sigma_2b + sigma_2c)
226</eq>
227with the factor <ei>1/2</ei> for the <ei>sigma_1c sigma_2c</ei> term.
228At the end of the day, this <ei>sigma'_1</ei> should be multiplied
229by the normalization factor
230<eq>
231f_1norm = &lt;f_impact&gt; (sigma_2b + sigma_2c) / sigma_ND
232</eq>
233here without a factor <ei>1/2</ei> (or else it would have been
234doublecounted). This gives the correct
235<eq>
236(sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b
237+ sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2
238</eq>
239The second interaction can be handled in exact analogy.
240
241<p/>
242The listing obtained with the <code>pythia.statistics()</code>
243already contain these corrections factors, i.e. cross sections
244are for the occurence of two interactions of the specified kinds.
245There is not a full tabulation of the matrix of all the possible
246combinations of a specific first process together with a specific
247second one (but the information is there for the user to do that,
248if desired). Instead <code>pythia.statistics()</code> shows this
249matrix projected onto the set of processes and associated cross
250sections for the first and the second interaction, respectively.
251Up to statistical fluctuations, these two sections of the
252<code>pythia.statistics()</code> listing both add up to the same
253total cross section for the event sample.
254
255<p/>
256There is a further special feature to be noted for this listing,
257and that is the difference between the number of "selected" events
258and the number of "accepted" ones. Here is how that comes about.
259Originally the first and second process are selected completely
260independently. The generation (in)efficiency is reflected in the
261different number of intially tried events for the first and second
262process, leading to the same number of selected events. While
263acceptable on their own, the combination of the two processes may
264be unacceptable, however. It may be that the two processes added
265together use more energy-momentum than kinematically allowed, or,
266even if not, are disfavoured when the PYTHIA approach to provide
267correlated parton densities is applied. Alternatively, referring
268to case (3) above, it may be because half of the events should
269be thrown for identical processes. Taken together, it is these
270effects that reduced the event number from "selected" to "accepted".
271(A further reduction may occur if a
272<aloc href="UserHooks">user hook</aloc> rejects some events.)
273
274<p/>
275In the cross section calculation above, the <ei>sigma'_1</ei>
276cross sections are based on the number of accepted events, while
277the <ei>f_1norm</ei> factor is evaluated based on the cross sections
278for selected events. That way the suppression by correlations
279between the two processes does not get to be doublecounted.
280
281<p/>
282The <code>pythia.statistics()</code> listing contains two final
283lines, indicating the summed cross sections <ei>sigma_1sum</ei> and
284<ei>sigma_2sum</ei> for the first and second set of processes, at
285the "selected" stage above, plus information on the <ei>sigma_ND</ei>
286and <ei>&lt;f_impact&gt;</ei> used. The total cross section
287generated is related to this by
288<eq>
289&lt;f_impact&gt; * (sigma_1sum * sigma_2sum / sigma_ND) *
290(n_accepted / n_selected)
291</eq>
292 with an additional factor of <ei>1/2</ei> for case 2 above.
293
294<p/>
295The error quoted for the cross section of a process is a combination
296in quadrature of the error on this process alone with the error on
297the normalization factor, including the error on
298<ei>&lt;f_impact&gt;</ei>. As always it is a purely statistical one
299and of course hides considerably bigger systematic uncertainties.
300
301<h3>Event information</h3>
302
303Normally the <code>process</code> event record only contains the
304hardest interaction, but in this case also the second hardest
305is stored there. If both of them are <ei>2 -> 2</ei> ones, the
306first would be stored in lines 3 - 6 and the second in 7 - 10.
307For both, status codes 21 - 29 would be used, as for a hardest
308process. Any resonance decay chains would occur after the two
309main processes, to allow normal parsing. The beams in 1 and 2
310only appear in one copy. This structure is echoed in the
311full <code>event</code> event record.
312
313<p/>
314Most of the properties accessible by the
315<aloc href="EventInformation"><code>pythia.info</code></aloc>
316methods refer to the first process, whether that happens to be the
317hardest or not. The code and <ei>pT</ei> scale of the second process
318are accessible by the <code>info.codeMI(1)</code> and
319<code>info.pTMI(1)</code>, however.
320
321<p/>
322The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide
323the cross section and its error for the event sample as a whole,
324combining the information from the two hard processes as described
325above. In particular, the former should be used to give the
326weight of the generated event sample. The statitical error estimate
327is somewhat cruder and gives a larger value than the
328subprocess-by-subprocess one employed in
329<code>pythia.statistics()</code>, but this number is
330anyway less relevant, since systematical errors are likely to dominate.
331
332</chapter>
333
334<!-- Copyright (C) 2008 Torbjorn Sjostrand -->