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5ad4eb21 | 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="MultipleInteractions">multiple 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 | You must further specify which set of processes to allow for | |

18 | the second, see below. | |

19 | </flag> | |

20 | ||

21 | <p/> | |

22 | In principle the whole <aloc href="ProcessSelection">process | |

23 | selection</aloc> allowed for the first process could be repeated | |

24 | for the second one. However, this would probably be overkill. | |

25 | Therefore here a more limited set of prepackaged process collections | |

26 | are made available, that can then be further combined at will. | |

27 | Since the description is almost completely symmetric between the | |

28 | first and the second process, you always have the possibility | |

29 | to pick one of the two processes according to the complete list | |

30 | of possibilities. | |

31 | ||

32 | <p/> | |

33 | Here comes the list of allowed sets of processes, to combine at will: | |

34 | ||

35 | <flag name="SecondHard:TwoJets" default="off"> | |

36 | Standard 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"> | |

41 | A prompt photon recoiling against a quark or gluon jet. | |

42 | ||

43 | <flag name="SecondHard:TwoPhotons" default="off"> | |

44 | Two prompt photons recoiling against each other. | |

45 | ||

46 | ||

47 | <flag name="SecondHard:SingleGmZ" default="off"> | |

48 | Scattering <ei>q qbar -> gamma^*/Z^0</ei>, with full interference | |

49 | between the <ei>gamma^*</ei> and <ei>Z^0</ei>. | |

50 | </flag> | |

51 | ||

52 | <flag name="SecondHard:SingleW" default="off"> | |

53 | Scattering <ei>q qbar' -> W^+-</ei>. | |

54 | </flag> | |

55 | ||

56 | <flag name="SecondHard:GmZAndJet" default="off"> | |

57 | Scattering <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"> | |

62 | Scattering <ei>q qbar' -> W^+- g</ei> and | |

63 | <ei>q g -> W^+- q'</ei>. | |

64 | </flag> | |

65 | ||

66 | <p/> | |

67 | A further process collection comes with a warning flag: | |

68 | ||

69 | <flag name="SecondHard:TwoBJets" default="off"> | |

70 | The <ei>q qbar -> b bbar</ei> and <ei>g g -> b bbar</ei> processes. | |

71 | These are already included in the <code>TwoJets</code> sample above, | |

72 | so it would be doublecounting to include both, but we assume there | |

73 | may be cases where the <ei>b</ei> subsample will be of special interest. | |

74 | This subsample does not include flavour-excitation or gluon-splitting | |

75 | contributions to the <ei>b</ei> rate, however, so, depending | |

76 | on the topology if interest, it may or may not be a good approximation. | |

77 | </flag> | |

78 | ||

79 | <p/> | |

80 | The 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> | |

83 | as 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 | |

85 | process alike, and ballpark half of the time the second could be | |

86 | generated with a larger <ei>pT</ei> than the first. (Exact numbers | |

87 | depending on the relative shape of the two cross sections.) That is, | |

88 | first and second is only used as an administrative distinction between | |

89 | the two, not as a physics ordering one. | |

90 | ||

91 | <h3>Cross-section calculation</h3> | |

92 | ||

93 | As an introduction, a brief reminder of Poissonian statistics. | |

94 | Assume a stochastic process in time, for now not necessarily a | |

95 | high-energy physics one, where the probability for an event to occur | |

96 | at any given time is independent of what happens at other times. | |

97 | Then the probability for <ei>n</ei> events to occur in a finite | |

98 | time interval is | |

99 | <eq> | |

100 | P_n = <n>^n exp(-<n>) / n! | |

101 | </eq> | |

102 | where <ei><n></ei> is the average number of events. If this | |

103 | number is small we can approximate <ei>exp(-<n>) = 1 </ei>, | |

104 | so that <ei>P_1 = <n></ei> and | |

105 | <ei>P_2 = <n>^2 / 2 = P_1^2 / 2</ei>. | |

106 | ||

107 | <p/> | |

108 | Now further assume that the events actually are of two different | |

109 | kinds <ei>a</ei> and <ei>b</ei>, occuring independently of each | |

110 | other, such that <ei><n> = <n_a> + <n_b></ei>. | |

111 | It 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 = <n_a></ei> (or <ei>P_1b = <n_b></ei>). | |

114 | From | |

115 | <eq> | |

116 | P_2 = (<n_a> + <n_b>)^2 / 2 = (P_1a + P_1b)^2 / 2 = | |

117 | (P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2 | |

118 | </eq> | |

119 | it is easy to read off that the probability to have exactly two | |

120 | events 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> | |

122 | and one <ei>b</ei> is <ei>P_2ab = P_1a P_1b</ei>. Note that the | |

123 | former, with two identical events, contains a factor <ei>1/2</ei> | |

124 | while the latter, with two different ones, does not. If viewed | |

125 | in a time-ordered sense, the difference is that the latter can be | |

126 | obtained two ways, either first an <ei>a</ei> and then a <ei>b</ei> | |

127 | or else first a <ei>b</ei> and then an <ei>a</ei>. | |

128 | ||

129 | <p/> | |

130 | To translate this language into cross-sections for high-energy | |

131 | events, we assume that interactions can occur at different <ei>pT</ei> | |

132 | values 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 | |

135 | total nondiffractive cross section. Again we want to assume that | |

136 | <ei>exp(-<n>)</ei> is close to unity, i.e. that the total | |

137 | hard cross section above <ei>pTmin</ei> is much smaller than | |

138 | <ei>sigma_ND</ei>. The hard cross section is dominated by QCD | |

139 | jet 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 | |

143 | similar 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/> | |

149 | Again different processes <ei>a, b, c, ...</ei> contribute, | |

150 | and 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>, | |

153 | and so on. | |

154 | ||

155 | <p/> | |

156 | There is one important correction to this picture: all collisions | |

157 | do no occur under equal conditions. Some are more central in impact | |

158 | parameter, others more peripheral. This leads to a further element of | |

159 | variability: central collisions are likely to have more activity | |

160 | than the average, peripheral less. Integrated over impact | |

161 | parameter standard cross sections are recovered, but correlations | |

162 | are affected by a "trigger bias" effect: if you select for events | |

163 | with a hard process you favour events at small impact parameter | |

164 | which have above-average activity, and therefore also increased | |

165 | chance for further interactions. (In PYTHIA this is the origin | |

166 | of the "pedestal effect", i.e. that events with a hard interaction | |

167 | have more underlying activity than the level found in minimum-bias | |

168 | events.) When you specify a matter overlap profile in the | |

169 | multiple-interactions scenario, such an enhancement/depletion factor | |

170 | <ei>f_impact</ei> is chosen event-by-event and can be averaged | |

171 | during the course of the run. As an example, the double Gaussian | |

172 | form used in Tune A gives approximately | |

173 | <ei><f_impact> = 2.5</ei>. The above equations therefore | |

174 | have to be modified to | |

175 | <ei>sigma_2aa = <f_impact> sigma_1a^2 / (2 sigma_ND)</ei>, | |

176 | <ei>sigma_2ab = <f_impact> sigma_1a sigma_1b / sigma_ND</ei>. | |

177 | Experimentalists often instead use the notation | |

178 | <ei>sigma_2ab = sigma_1a sigma_1b / sigma_eff</ei>, | |

179 | from which we see that PYTHIA "predicts" | |

180 | <ei>sigma_eff = sigma_ND / <f_impact></ei>. | |

181 | When the generation of multiple interactions is switched off it is | |

182 | not possible to calculate <ei><f_impact></ei> and therefore | |

183 | it is set to unity. | |

184 | ||

185 | <p/> | |

186 | When this recipe is to be applied to calculate | |

187 | actual cross sections, it is useful to distinguish three cases, | |

188 | depending on which set of processes are selected to study for | |

189 | the 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. | |

194 | For instance, the first could be <code>TwoJets</code> and the | |

195 | second <code>TwoPhotons</code>. In that case, the two interactions | |

196 | can be selected independently, and cross sections tabulated | |

197 | for each separate subprocess in the two above classes. At the | |

198 | end of the run, the cross sections in <ei>a</ei> should be multiplied | |

199 | by <ei><f_impact> sigma_1b / sigma_ND</ei> to bring them to | |

200 | the correct overall level, and those in <ei>b</ei> by | |

201 | <ei><f_impact> sigma_1a / sigma_ND</ei>. | |

202 | ||

203 | <p/> | |

204 | (2) Exactly the same processes <ei>a</ei> are selected for the | |

205 | first and second interaction. In that case it works as above, | |

206 | with <ei>a = b</ei>, and it is only necessary to multiply by an | |

207 | additional factor <ei>1/2</ei>. A compensating factor of 2 | |

208 | is automatically obtained for picking two different subprocesses, | |

209 | e.g. if <code>TwoJets</code> is selected for both interactions, | |

210 | then the combination of the two subprocesses <ei>q qbar -> g g</ei> | |

211 | and <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. | |

215 | For instance, the first process is allowed to contain <ei>a</ei> | |

216 | or <ei>c</ei> and the second <ei>b</ei> or <ei>c</ei>, where | |

217 | there is no overlap between <ei>a</ei> and <ei>b</ei>. Then, | |

218 | when an independent selection for the first and second interaction | |

219 | both pick one of the subprocesses in <ei>c</ei>, half of those | |

220 | events have to be thrown, and the stored cross section reduced | |

221 | accordingly. Considering the four possible combinations of first | |

222 | and second process, this gives a | |

223 | <eq> | |

224 | sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) / | |

225 | (sigma_2b + sigma_2c) | |

226 | </eq> | |

227 | with the factor <ei>1/2</ei> for the <ei>sigma_1c sigma_2c</ei> term. | |

228 | At the end of the day, this <ei>sigma'_1</ei> should be multiplied | |

229 | by the normalization factor | |

230 | <eq> | |

231 | f_1norm = <f_impact> (sigma_2b + sigma_2c) / sigma_ND | |

232 | </eq> | |

233 | here without a factor <ei>1/2</ei> (or else it would have been | |

234 | doublecounted). 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> | |

239 | The second interaction can be handled in exact analogy. | |

240 | ||

241 | <p/> | |

242 | The listing obtained with the <code>pythia.statistics()</code> | |

243 | already contain these corrections factors, i.e. cross sections | |

244 | are for the occurence of two interactions of the specified kinds. | |

245 | There is not a full tabulation of the matrix of all the possible | |

246 | combinations of a specific first process together with a specific | |

247 | second one (but the information is there for the user to do that, | |

248 | if desired). Instead <code>pythia.statistics()</code> shows this | |

249 | matrix projected onto the set of processes and associated cross | |

250 | sections for the first and the second interaction, respectively. | |

251 | Up to statistical fluctuations, these two sections of the | |

252 | <code>pythia.statistics()</code> listing both add up to the same | |

253 | total cross section for the event sample. | |

254 | ||

255 | <p/> | |

256 | There is a further special feature to be noted for this listing, | |

257 | and that is the difference between the number of "selected" events | |

258 | and the number of "accepted" ones. Here is how that comes about. | |

259 | Originally the first and second process are selected completely | |

260 | independently. The generation (in)efficiency is reflected in the | |

261 | different number of intially tried events for the first and second | |

262 | process, leading to the same number of selected events. While | |

263 | acceptable on their own, the combination of the two processes may | |

264 | be unacceptable, however. It may be that the two processes added | |

265 | together use more energy-momentum than kinematically allowed, or, | |

266 | even if not, are disfavoured when the PYTHIA approach to provide | |

267 | correlated parton densities is applied. Alternatively, referring | |

268 | to case (3) above, it may be because half of the events should | |

269 | be thrown for identical processes. Taken together, it is these | |

270 | effects 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/> | |

275 | In the cross section calculation above, the <ei>sigma'_1</ei> | |

276 | cross sections are based on the number of accepted events, while | |

277 | the <ei>f_1norm</ei> factor is evaluated based on the cross sections | |

278 | for selected events. That way the suppression by correlations | |

279 | between the two processes does not get to be doublecounted. | |

280 | ||

281 | <p/> | |

282 | The <code>pythia.statistics()</code> listing contains two final | |

283 | lines, indicating the summed cross sections <ei>sigma_1sum</ei> and | |

284 | <ei>sigma_2sum</ei> for the first and second set of processes, at | |

285 | the "selected" stage above, plus information on the <ei>sigma_ND</ei> | |

286 | and <ei><f_impact></ei> used. The total cross section | |

287 | generated is related to this by | |

288 | <eq> | |

289 | <f_impact> * (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/> | |

295 | The error quoted for the cross section of a process is a combination | |

296 | in quadrature of the error on this process alone with the error on | |

297 | the normalization factor, including the error on | |

298 | <ei><f_impact></ei>. As always it is a purely statistical one | |

299 | and of course hides considerably bigger systematic uncertainties. | |

300 | ||

301 | <h3>Event information</h3> | |

302 | ||

303 | Normally the <code>process</code> event record only contains the | |

304 | hardest interaction, but in this case also the second hardest | |

305 | is stored there. If both of them are <ei>2 -> 2</ei> ones, the | |

306 | first would be stored in lines 3 - 6 and the second in 7 - 10. | |

307 | For both, status codes 21 - 29 would be used, as for a hardest | |

308 | process. Any resonance decay chains would occur after the two | |

309 | main processes, to allow normal parsing. The beams in 1 and 2 | |

310 | only appear in one copy. This structure is echoed in the | |

311 | full <code>event</code> event record. | |

312 | ||

313 | <p/> | |

314 | Most of the properties accessible by the | |

315 | <aloc href="EventInformation"><code>pythia.info</code></aloc> | |

316 | methods refer to the first process, whether that happens to be the | |

317 | hardest or not. The code and <ei>pT</ei> scale of the second process | |

318 | are accessible by the <code>info.codeMI(1)</code> and | |

319 | <code>info.pTMI(1)</code>, however. | |

320 | ||

321 | <p/> | |

322 | The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide | |

323 | the cross section and its error for the event sample as a whole, | |

324 | combining the information from the two hard processes as described | |

325 | above. In particular, the former should be used to give the | |

326 | weight of the generated event sample. The statitical error estimate | |

327 | is somewhat cruder and gives a larger value than the | |

328 | subprocess-by-subprocess one employed in | |

329 | <code>pythia.statistics()</code>, but this number is | |

330 | anyway less relevant, since systematical errors are likely to dominate. | |

331 | ||

332 | </chapter> | |

333 | ||

334 | <!-- Copyright (C) 2008 Torbjorn Sjostrand --> |