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c6b60c38 | 1 | <chapter name="A Second Hard Process"> |
2 | ||
3 | <h2>A Second Hard Process</h2> | |
4 | ||
5 | When you have selected a set of hard processes for hadron beams, the | |
6 | <aloc href="MultipartonInteractions">multiparton interactions</aloc> | |
7 | framework can add further interactions to build up a realistic | |
8 | underlying event. These further interactions can come from a wide | |
9 | variety of processes, and will occasionally be quite hard. They | |
10 | do represent a realistic random mix, however, which means one cannot | |
11 | predetermine what will happen. Occasionally there may be cases | |
12 | where one wants to specify also the second hard interaction rather | |
13 | precisely. The options on this page allow you to do precisely that. | |
14 | ||
15 | <flag name="SecondHard:generate" default="off"> | |
16 | Generate two hard scatterings in a collision between hadron beams. | |
17 | The hardest process can be any combination of internal processes, | |
18 | available in the normal <aloc href="ProcessSelection">process | |
19 | selection</aloc> machinery, or external input. Here you must further | |
20 | specify which set of processes to allow for the second hard one, see | |
21 | the following. | |
22 | </flag> | |
23 | ||
24 | <h3>Process Selection</h3> | |
25 | ||
26 | In principle the whole <aloc href="ProcessSelection">process | |
27 | selection</aloc> allowed for the first process could be repeated | |
28 | for the second one. However, this would probably be overkill. | |
29 | Therefore here a more limited set of prepackaged process collections | |
30 | are made available, that can then be further combined at will. | |
31 | Since the description is almost completely symmetric between the | |
32 | first and the second process, you always have the possibility | |
33 | to pick one of the two processes according to the complete list | |
34 | of possibilities. | |
35 | ||
36 | <p/> | |
37 | Here comes the list of allowed sets of processes, to combine at will: | |
38 | ||
39 | <flag name="SecondHard:TwoJets" default="off"> | |
40 | Standard QCD <ei>2 -> 2</ei> processes involving gluons and | |
41 | <ei>d, u, s, c, b</ei> quarks. | |
42 | </flag> | |
43 | ||
44 | <flag name="SecondHard:PhotonAndJet" default="off"> | |
45 | A prompt photon recoiling against a quark or gluon jet. | |
46 | ||
47 | <flag name="SecondHard:TwoPhotons" default="off"> | |
48 | Two prompt photons recoiling against each other. | |
49 | ||
50 | <flag name="SecondHard:Charmonium" default="off"> | |
51 | Production of charmonium via colour singlet and colour octet channels. | |
52 | ||
53 | <flag name="SecondHard:Bottomonium" default="off"> | |
54 | Production of bottomonium via colour singlet and colour octet channels. | |
55 | ||
56 | <flag name="SecondHard:SingleGmZ" default="off"> | |
57 | Scattering <ei>q qbar -> gamma^*/Z^0</ei>, with full interference | |
58 | between the <ei>gamma^*</ei> and <ei>Z^0</ei>. | |
59 | </flag> | |
60 | ||
61 | <flag name="SecondHard:SingleW" default="off"> | |
62 | Scattering <ei>q qbar' -> W^+-</ei>. | |
63 | </flag> | |
64 | ||
65 | <flag name="SecondHard:GmZAndJet" default="off"> | |
66 | Scattering <ei>q qbar -> gamma^*/Z^0 g</ei> and | |
67 | <ei>q g -> gamma^*/Z^0 q</ei>. | |
68 | </flag> | |
69 | ||
70 | <flag name="SecondHard:WAndJet" default="off"> | |
71 | Scattering <ei>q qbar' -> W^+- g</ei> and | |
72 | <ei>q g -> W^+- q'</ei>. | |
73 | </flag> | |
74 | ||
75 | <flag name="SecondHard:TopPair" default="off"> | |
76 | Production of a top pair, either via QCD processes or via an | |
77 | intermediate <ei>gamma^*/Z^0</ei> resonance. | |
78 | </flag> | |
79 | ||
80 | <flag name="SecondHard:SingleTop" default="off"> | |
81 | Production of a single top, either via a <ei>t-</ei> or | |
82 | an <ei>s-</ei>channel <ei>W^+-</ei> resonance. | |
83 | </flag> | |
84 | ||
85 | <p/> | |
86 | A further process collection comes with a warning flag: | |
87 | ||
88 | <flag name="SecondHard:TwoBJets" default="off"> | |
89 | The <ei>q qbar -> b bbar</ei> and <ei>g g -> b bbar</ei> processes. | |
90 | These are already included in the <code>TwoJets</code> sample above, | |
91 | so it would be double-counting to include both, but we assume there | |
92 | may be cases where the <ei>b</ei> subsample will be of special interest. | |
93 | This subsample does not include flavour-excitation or gluon-splitting | |
94 | contributions to the <ei>b</ei> rate, however, so, depending | |
95 | on the topology if interest, it may or may not be a good approximation. | |
96 | </flag> | |
97 | ||
98 | <h3>Cuts and scales</h3> | |
99 | ||
100 | The second hard process obeys exactly the same selection rules for | |
101 | <aloc href="PhaseSpaceCuts">phase space cuts</aloc> and | |
102 | <aloc href="CouplingsAndScales">couplings and scales</aloc> | |
103 | as the first one does. Specifically, a <ei>pTmin</ei> cut for | |
104 | <ei>2 -> 2</ei> processes would apply to the first and the second hard | |
105 | process alike, and ballpark half of the time the second could be | |
106 | generated with a larger <ei>pT</ei> than the first. (Exact numbers | |
107 | depending on the relative shape of the two cross sections.) That is, | |
108 | first and second is only used as an administrative distinction between | |
109 | the two, not as a physics ordering one. | |
110 | ||
111 | <p/> | |
112 | Optionally it is possible to pick the mass and <ei>pT</ei> | |
113 | <aloc href="PhaseSpaceCuts">phase space cuts</aloc> separately for | |
114 | the second hard interaction. The main application presumably would | |
115 | be to allow a second process that is softer than the first, but still | |
116 | hard. But one is also free to make the second process harder than the | |
117 | first, if desired. So long as the two <ei>pT</ei> (or mass) ranges | |
118 | overlap the ordering will not be the same in all events, however. | |
119 | ||
120 | <h3>Cross-section calculation</h3> | |
121 | ||
122 | As an introduction, a brief reminder of Poissonian statistics. | |
123 | Assume a stochastic process in time, for now not necessarily a | |
124 | high-energy physics one, where the probability for an event to occur | |
125 | at any given time is independent of what happens at other times. | |
126 | Then the probability for <ei>n</ei> events to occur in a finite | |
127 | time interval is | |
128 | <eq> | |
129 | P_n = <n>^n exp(-<n>) / n! | |
130 | </eq> | |
131 | where <ei><n></ei> is the average number of events. If this | |
132 | number is small we can approximate <ei>exp(-<n>) = 1 </ei>, | |
133 | so that <ei>P_1 = <n></ei> and | |
134 | <ei>P_2 = <n>^2 / 2 = P_1^2 / 2</ei>. | |
135 | ||
136 | <p/> | |
137 | Now further assume that the events actually are of two different | |
138 | kinds <ei>a</ei> and <ei>b</ei>, occurring independently of each | |
139 | other, such that <ei><n> = <n_a> + <n_b></ei>. | |
140 | It then follows that the probability of having one event of type | |
141 | <ei>a</ei> (or <ei>b</ei>) and nothing else is | |
142 | <ei>P_1a = <n_a></ei> (or <ei>P_1b = <n_b></ei>). | |
143 | From | |
144 | <eq> | |
145 | P_2 = (<n_a> + <n_b>)^2 / 2 = (P_1a + P_1b)^2 / 2 = | |
146 | (P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2 | |
147 | </eq> | |
148 | it is easy to read off that the probability to have exactly two | |
149 | events of kind <ei>a</ei> and none of <ei>b</ei> is | |
150 | <ei>P_2aa = P_1a^2 / 2</ei> whereas that of having one <ei>a</ei> | |
151 | and one <ei>b</ei> is <ei>P_2ab = P_1a P_1b</ei>. Note that the | |
152 | former, with two identical events, contains a factor <ei>1/2</ei> | |
153 | while the latter, with two different ones, does not. If viewed | |
154 | in a time-ordered sense, the difference is that the latter can be | |
155 | obtained two ways, either first an <ei>a</ei> and then a <ei>b</ei> | |
156 | or else first a <ei>b</ei> and then an <ei>a</ei>. | |
157 | ||
158 | <p/> | |
159 | To translate this language into cross-sections for high-energy | |
160 | events, we assume that interactions can occur at different <ei>pT</ei> | |
161 | values independently of each other inside inelastic nondiffractive | |
162 | (= "minbias") events. Then the above probabilities translate into | |
163 | <ei>P_n = sigma_n / sigma_ND</ei> where <ei>sigma_ND</ei> is the | |
164 | total nondiffractive cross section. Again we want to assume that | |
165 | <ei>exp(-<n>)</ei> is close to unity, i.e. that the total | |
166 | hard cross section above <ei>pTmin</ei> is much smaller than | |
167 | <ei>sigma_ND</ei>. The hard cross section is dominated by QCD | |
168 | jet production, and a reasonable precaution is to require a | |
169 | <ei>pTmin</ei> of at least 20 GeV at LHC energies. | |
170 | (For <ei>2 -> 1</ei> processes such as | |
171 | <ei>q qbar -> gamma^*/Z^0 (-> f fbar)</ei> one can instead make a | |
172 | similar cut on mass.) Then the generic equation | |
173 | <ei>P_2 = P_1^2 / 2</ei> translates into | |
174 | <ei>sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2</ei> or | |
175 | <ei>sigma_2 = sigma_1^2 / (2 sigma_ND)</ei>. | |
176 | ||
177 | <p/> | |
178 | Again different processes <ei>a, b, c, ...</ei> contribute, | |
179 | and by the same reasoning we obtain | |
180 | <ei>sigma_2aa = sigma_1a^2 / (2 sigma_ND)</ei>, | |
181 | <ei>sigma_2ab = sigma_1a sigma_1b / sigma_ND</ei>, | |
182 | and so on. | |
183 | ||
184 | <p/> | |
185 | There is one important correction to this picture: all collisions | |
186 | do no occur under equal conditions. Some are more central in impact | |
187 | parameter, others more peripheral. This leads to a further element of | |
188 | variability: central collisions are likely to have more activity | |
189 | than the average, peripheral less. Integrated over impact | |
190 | parameter standard cross sections are recovered, but correlations | |
191 | are affected by a "trigger bias" effect: if you select for events | |
192 | with a hard process you favour events at small impact parameter | |
193 | which have above-average activity, and therefore also increased | |
194 | chance for further interactions. (In PYTHIA this is the origin | |
195 | of the "pedestal effect", i.e. that events with a hard interaction | |
196 | have more underlying activity than the level found in minimum-bias | |
197 | events.) When you specify a matter overlap profile in the | |
198 | multiparton-interactions scenario, such an enhancement/depletion factor | |
199 | <ei>f_impact</ei> is chosen event-by-event and can be averaged | |
200 | during the course of the run. As an example, the double Gaussian | |
201 | form used in Tune A gives approximately | |
202 | <ei><f_impact> = 2.5</ei>. The above equations therefore | |
203 | have to be modified to | |
204 | <ei>sigma_2aa = <f_impact> sigma_1a^2 / (2 sigma_ND)</ei>, | |
205 | <ei>sigma_2ab = <f_impact> sigma_1a sigma_1b / sigma_ND</ei>. | |
206 | Experimentalists often instead use the notation | |
207 | <ei>sigma_2ab = sigma_1a sigma_1b / sigma_eff</ei>, | |
208 | from which we see that PYTHIA "predicts" | |
209 | <ei>sigma_eff = sigma_ND / <f_impact></ei>. | |
210 | When the generation of multiparton interactions is switched off it is | |
211 | not possible to calculate <ei><f_impact></ei> and therefore | |
212 | it is set to unity. | |
213 | ||
214 | <p/> | |
215 | When this recipe is to be applied to calculate | |
216 | actual cross sections, it is useful to distinguish three cases, | |
217 | depending on which set of processes are selected to study for | |
218 | the first and second interaction. | |
219 | ||
220 | <p/> | |
221 | (1) The processes <ei>a</ei> for the first interaction and | |
222 | <ei>b</ei> for the second one have no overlap at all. | |
223 | For instance, the first could be <code>TwoJets</code> and the | |
224 | second <code>TwoPhotons</code>. In that case, the two interactions | |
225 | can be selected independently, and cross sections tabulated | |
226 | for each separate subprocess in the two above classes. At the | |
227 | end of the run, the cross sections in <ei>a</ei> should be multiplied | |
228 | by <ei><f_impact> sigma_1b / sigma_ND</ei> to bring them to | |
229 | the correct overall level, and those in <ei>b</ei> by | |
230 | <ei><f_impact> sigma_1a / sigma_ND</ei>. | |
231 | ||
232 | <p/> | |
233 | (2) Exactly the same processes <ei>a</ei> are selected for the | |
234 | first and second interaction. In that case it works as above, | |
235 | with <ei>a = b</ei>, and it is only necessary to multiply by an | |
236 | additional factor <ei>1/2</ei>. A compensating factor of 2 | |
237 | is automatically obtained for picking two different subprocesses, | |
238 | e.g. if <code>TwoJets</code> is selected for both interactions, | |
239 | then the combination of the two subprocesses <ei>q qbar -> g g</ei> | |
240 | and <ei>g g -> g g</ei> can trivially be obtained two ways. | |
241 | ||
242 | <p/> | |
243 | (3) The list of subprocesses partly but not completely overlap. | |
244 | For instance, the first process is allowed to contain <ei>a</ei> | |
245 | or <ei>c</ei> and the second <ei>b</ei> or <ei>c</ei>, where | |
246 | there is no overlap between <ei>a</ei> and <ei>b</ei>. Then, | |
247 | when an independent selection for the first and second interaction | |
248 | both pick one of the subprocesses in <ei>c</ei>, half of those | |
249 | events have to be thrown, and the stored cross section reduced | |
250 | accordingly. Considering the four possible combinations of first | |
251 | and second process, this gives a | |
252 | <eq> | |
253 | sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) / | |
254 | (sigma_2b + sigma_2c) | |
255 | </eq> | |
256 | with the factor <ei>1/2</ei> for the <ei>sigma_1c sigma_2c</ei> term. | |
257 | At the end of the day, this <ei>sigma'_1</ei> should be multiplied | |
258 | by the normalization factor | |
259 | <eq> | |
260 | f_1norm = <f_impact> (sigma_2b + sigma_2c) / sigma_ND | |
261 | </eq> | |
262 | here without a factor <ei>1/2</ei> (or else it would have been | |
263 | double-counted). This gives the correct | |
264 | <eq> | |
265 | (sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b | |
266 | + sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2 | |
267 | </eq> | |
268 | The second interaction can be handled in exact analogy. | |
269 | ||
270 | <p/> | |
271 | For the considerations above it is assumed that the phase space cuts | |
272 | are the same for the two processes. It is possible to set the mass and | |
273 | transverse momentum cuts differently, however. This changes nothing | |
274 | for processes that already are different. For two collisions of the | |
275 | same type it is partly a matter of interpretation what is intended. | |
276 | If we consider the case of the same process in two non-overlapping | |
277 | phase space regions, most likely we want to consider them as | |
278 | separate processes, in the sense that we expect a factor 2 relative | |
279 | to Poissonian statistics from either of the two hardest processes | |
280 | populating either of the two phase space regions. In total we are | |
281 | therefore lead to adopt the same strategy as in case (3) above: | |
282 | only in the overlapping part of the two allowed phase space regions | |
283 | could two processes be identical and thus appear with a 1/2 factor, | |
284 | elsewhere the two processes are never identical and do not | |
285 | include the 1/2 factor. We reiterate, however, that the case of | |
286 | partly but not completely overlapping phase space regions for one and | |
287 | the same process is tricky, and not to be used without prior | |
288 | deliberation. | |
289 | ||
290 | <p/> | |
291 | The listing obtained with the <code>pythia.statistics()</code> | |
292 | already contain these corrections factors, i.e. cross sections | |
293 | are for the occurrence of two interactions of the specified kinds. | |
294 | There is not a full tabulation of the matrix of all the possible | |
295 | combinations of a specific first process together with a specific | |
296 | second one (but the information is there for the user to do that, | |
297 | if desired). Instead <code>pythia.statistics()</code> shows this | |
298 | matrix projected onto the set of processes and associated cross | |
299 | sections for the first and the second interaction, respectively. | |
300 | Up to statistical fluctuations, these two sections of the | |
301 | <code>pythia.statistics()</code> listing both add up to the same | |
302 | total cross section for the event sample. | |
303 | ||
304 | <p/> | |
305 | There is a further special feature to be noted for this listing, | |
306 | and that is the difference between the number of "selected" events | |
307 | and the number of "accepted" ones. Here is how that comes about. | |
308 | Originally the first and second process are selected completely | |
309 | independently. The generation (in)efficiency is reflected in the | |
310 | different number of initially tried events for the first and second | |
311 | process, leading to the same number of selected events. While | |
312 | acceptable on their own, the combination of the two processes may | |
313 | be unacceptable, however. It may be that the two processes added | |
314 | together use more energy-momentum than kinematically allowed, or, | |
315 | even if not, are disfavoured when the PYTHIA approach to provide | |
316 | correlated parton densities is applied. Alternatively, referring | |
317 | to case (3) above, it may be because half of the events should | |
318 | be thrown for identical processes. Taken together, it is these | |
319 | effects that reduced the event number from "selected" to "accepted". | |
320 | (A further reduction may occur if a | |
321 | <aloc href="UserHooks">user hook</aloc> rejects some events.) | |
322 | ||
323 | <p/> | |
324 | It is allowed to use external Les Houches Accord input for the | |
325 | hardest process, and then pick an internal one for the second hardest. | |
326 | In this case PYTHIA does not have access to your thinking concerning | |
327 | the external process, and cannot know whether it overlaps with the | |
328 | internal or not. (External events <ei>q qbar' -> e+ nu_e</ei> could | |
329 | agree with the internal <ei>W</ei> ones, or be a <ei>W'</ei> resonance | |
330 | in a BSM scenario, to give one example.) Therefore the combined cross | |
331 | section is always based on the scenario (1) above. Corrections for | |
332 | correlated parton densities are included also in this case, however. | |
333 | That is, an external event that takes a large fraction of the incoming | |
334 | beam momenta stands a fair chance of being rejected when it has to be | |
335 | combined with another hard process. For this reason the "selected" and | |
336 | "accepted" event numbers are likely to disagree. | |
337 | ||
338 | <p/> | |
339 | In the cross section calculation above, the <ei>sigma'_1</ei> | |
340 | cross sections are based on the number of accepted events, while | |
341 | the <ei>f_1norm</ei> factor is evaluated based on the cross sections | |
342 | for selected events. That way the suppression by correlations | |
343 | between the two processes does not get to be double-counted. | |
344 | ||
345 | <p/> | |
346 | The <code>pythia.statistics()</code> listing contains two final | |
347 | lines, indicating the summed cross sections <ei>sigma_1sum</ei> and | |
348 | <ei>sigma_2sum</ei> for the first and second set of processes, at | |
349 | the "selected" stage above, plus information on the <ei>sigma_ND</ei> | |
350 | and <ei><f_impact></ei> used. The total cross section | |
351 | generated is related to this by | |
352 | <eq> | |
353 | <f_impact> * (sigma_1sum * sigma_2sum / sigma_ND) * | |
354 | (n_accepted / n_selected) | |
355 | </eq> | |
356 | with an additional factor of <ei>1/2</ei> for case 2 above. | |
357 | ||
358 | <p/> | |
359 | The error quoted for the cross section of a process is a combination | |
360 | in quadrature of the error on this process alone with the error on | |
361 | the normalization factor, including the error on | |
362 | <ei><f_impact></ei>. As always it is a purely statistical one | |
363 | and of course hides considerably bigger systematic uncertainties. | |
364 | ||
365 | <h3>Event information</h3> | |
366 | ||
367 | Normally the <code>process</code> event record only contains the | |
368 | hardest interaction, but in this case also the second hardest | |
369 | is stored there. If both of them are <ei>2 -> 2</ei> ones, the | |
370 | first would be stored in lines 3 - 6 and the second in 7 - 10. | |
371 | For both, status codes 21 - 29 would be used, as for a hardest | |
372 | process. Any resonance decay chains would occur after the two | |
373 | main processes, to allow normal parsing. The beams in 1 and 2 | |
374 | only appear in one copy. This structure is echoed in the | |
375 | full <code>event</code> event record. | |
376 | ||
377 | <p/> | |
378 | Most of the properties accessible by the | |
379 | <code><aloc href="EventInformation">pythia.info</aloc></code> | |
380 | methods refer to the first process, whether that happens to be the | |
381 | hardest or not. The code and <ei>pT</ei> scale of the second process | |
382 | are accessible by the <code>info.codeMPI(1)</code> and | |
383 | <code>info.pTMPI(1)</code>, however. | |
384 | ||
385 | <p/> | |
386 | The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide | |
387 | the cross section and its error for the event sample as a whole, | |
388 | combining the information from the two hard processes as described | |
389 | above. In particular, the former should be used to give the | |
390 | weight of the generated event sample. The statistical error estimate | |
391 | is somewhat cruder and gives a larger value than the | |
392 | subprocess-by-subprocess one employed in | |
393 | <code>pythia.statistics()</code>, but this number is | |
394 | anyway less relevant, since systematical errors are likely to dominate. | |
395 | ||
396 | </chapter> | |
397 | ||
398 | <!-- Copyright (C) 2013 Torbjorn Sjostrand --> |