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5ad4eb21 | 1 | <chapter name="Multiple Interactions"> |
2 | ||
3 | <h2>Multiple Interactions</h2> | |
4 | ||
5 | The starting point for the multiple interactions physics scenario in | |
6 | PYTHIA is provided by <ref>Sjo87</ref>. Recent developments have | |
7 | included a more careful study of flavour and colour correlations, | |
8 | junction topologies and the relationship to beam remnants | |
9 | <ref>Sjo04</ref>, and interleaving with initial-state radiation | |
10 | <ref>Sjo05</ref>, making use of transverse-momentum-ordered | |
11 | initial- and final-state showers. | |
12 | ||
13 | <p/> | |
14 | A big unsolved issue is how the colour of all these subsystems is | |
15 | correlated. For sure there is a correlation coming from the colour | |
16 | singlet nature of the incoming beams, but in addition final-state | |
17 | colour rearrangements may change the picture. Indeed such extra | |
18 | effects appear necessary to describe data, e.g. on | |
19 | <ei><pT>(n_ch)</ei>. A simple implementation of colour | |
20 | rearrangement is found as part of the | |
21 | <aloc href="BeamRemnants">beam remnants</aloc> description. | |
22 | ||
23 | <h3>Main variables</h3> | |
24 | ||
25 | The maximum <ei>pT</ei> to be allowed for multiple interactions is | |
26 | related to the nature of the hard process itself. It involves a | |
27 | delicate balance between not doublecounting and not leaving any | |
28 | gaps in the coverage. The best procedure may depend on information | |
29 | only the user has: how the events were generated and mixed (e.g. with | |
30 | Les Houches Accord external input), and how they are intended to be | |
31 | used. Therefore a few options are available, with a sensible default | |
32 | behaviour. | |
33 | <modepick name="MultipleInteractions:pTmaxMatch" default="0" min="0" max="2"> | |
34 | Way in which the maximum scale for multiple interactions is set | |
35 | to match the scale of the hard process itself. | |
36 | <option value="0"><b>(i)</b> if the final state of the hard process | |
37 | (not counting subsequent resonance decays) contains only quarks | |
38 | (<ei>u, d, s, c ,b</ei>), gluons and photons then <ei>pT_max</ei> | |
39 | is chosen to be the factorization scale for internal processes | |
40 | and the <code>scale</code> value for Les Houches input; | |
41 | <b>(ii)</b> if not, interactions are allowed to go all the way up | |
42 | to the kinematical limit. | |
43 | The reasoning is that the former kind of processes are generated by | |
44 | the multiple-interactions machinery and so would doublecount hard | |
45 | processes if allowed to overlap the same <ei>pT</ei> range, | |
46 | while no such danger exists in the latter case. | |
47 | </option> | |
48 | <option value="1">always use the factorization scale for an internal | |
49 | process and the <code>scale</code> value for Les Houches input, | |
50 | i.e. the lower value. This should avoid doublecounting, but | |
51 | may leave out some interactions that ought to have been simulated. | |
52 | </option> | |
53 | <option value="2">always allow multiple interactions up to the | |
54 | kinematical limit. This will simulate all possible event topologies, | |
55 | but may lead to doublecounting. | |
56 | </option> | |
57 | </modepick> | |
58 | ||
59 | <p/> | |
60 | The rate of interactions is determined by | |
61 | <parm name="MultipleInteractions:alphaSvalue" default="0.127" min="0.06" max="0.25"> | |
62 | The value of <ei>alpha_strong</ei> at <ei>m_Z</ei>. Default value is | |
63 | picked equal to the one used in CTEQ 5L. | |
64 | </parm> | |
65 | ||
66 | <p/> | |
67 | The actual value is then regulated by the running to the scale | |
68 | <ei>pT^2</ei>, at which it is evaluated | |
69 | <modepick name="MultipleInteractions:alphaSorder" default="1" min="0" max="2"> | |
70 | The order at which <ei>alpha_strong</ei> runs at scales away from | |
71 | <ei>m_Z</ei>. | |
72 | <option value="0">zeroth order, i.e. <ei>alpha_strong</ei> is kept | |
73 | fixed.</option> | |
74 | <option value="1">first order, which is the normal value.</option> | |
75 | <option value="2">second order. Since other parts of the code do | |
76 | not go to second order there is no strong reason to use this option, | |
77 | but there is also nothing wrong with it.</option> | |
78 | </modepick> | |
79 | ||
80 | <p/> | |
81 | QED interactions are regulated by the <ei>alpha_electromagnetic</ei> | |
82 | value at the <ei>pT^2</ei> scale of an interaction. | |
83 | ||
84 | <modepick name="MultipleInteractions:alphaEMorder" default="1" min="-1" max="1"> | |
85 | The running of <ei>alpha_em</ei> used in hard processes. | |
86 | <option value="1">first-order running, constrained to agree with | |
87 | <code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass. | |
88 | </option> | |
89 | <option value="0">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
90 | fixed at its value at vanishing momentum transfer.</option> | |
91 | <option value="-1">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
92 | fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value | |
93 | at the <ei>Z^0</ei> mass. | |
94 | </option> | |
95 | </modepick> | |
96 | ||
97 | <p/> | |
98 | Note that the choices of <ei>alpha_strong</ei> and <ei>alpha_em</ei> | |
99 | made here override the ones implemented in the normal process machinery, | |
100 | but only for the interactions generated by the | |
101 | <code>MultipleInteractions</code> class. | |
102 | ||
103 | <p/> | |
104 | In addition there is the possibility of a global rescaling of | |
105 | cross sections (which could not easily be accommodated by a | |
106 | changed <ei>alpha_strong</ei>, since <ei>alpha_strong</ei> runs) | |
107 | <parm name="MultipleInteractions:Kfactor" default="1.0" min="0.5" max="4.0"> | |
108 | Multiply all cross sections by this fix factor. | |
109 | </parm> | |
110 | ||
111 | <p/> | |
112 | There are two complementary ways of regularizing the small-<ei>pT</ei> | |
113 | divergence, a sharp cutoff and a smooth dampening. These can be | |
114 | combined as desired, but it makes sense to coordinate with how the | |
115 | same issue is handled in <aloc href="SpacelikeShowers">spacelike | |
116 | showers</aloc>. Actually, by default, the parameters defined here are | |
117 | used also for the spacelike showers, but this can be overridden. | |
118 | ||
119 | <p/> | |
120 | Regularization of the divergence of the QCD cross section for | |
121 | <ei>pT -> 0</ei> is obtained by a factor <ei>pT^4 / (pT0^2 + pT^2)^2</ei>, | |
122 | and by using an <ei>alpha_s(pT0^2 + pT^2)</ei>. An energy dependence | |
123 | of the <ei>pT0</ei> choice is introduced by two further parameters, | |
124 | so that <ei>pT0Ref</ei> is the <ei>pT0</ei> value for the reference | |
125 | cm energy, <ei>pT0Ref = pT0(ecmRef)</ei>. | |
126 | <note>Warning:</note> if a large <ei>pT0</ei> is picked for multiple | |
127 | interactions, such that the integrated interaction cross section is | |
128 | below the nondiffractive inelastic one, this <ei>pT0</ei> will | |
129 | automatically be scaled down to cope. | |
130 | ||
131 | <p/> | |
132 | The actual pT0 parameter used at a given cm energy scale, <ei>ecmNow</ei>, | |
133 | is obtained as | |
134 | <eq> | |
135 | pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow | |
136 | </eq> | |
137 | where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the | |
138 | three parameters below. | |
139 | ||
140 | <parm name="MultipleInteractions:pT0Ref" default="2.15" min="0.5" max="10.0"> | |
141 | The <ei>pT0Ref</ei> scale in the above formula. | |
142 | <note>Note:</note> <ei>pT0Ref</ei> is one of the key parameters in a | |
143 | complete PYTHIA tune. Its value is intimately tied to a number of other | |
144 | choices, such as that of colour flow description, so unfortunately it is | |
145 | difficult to give an independent meaning to <ei>pT0Ref</ei>. | |
146 | </parm> | |
147 | ||
148 | <parm name="MultipleInteractions:ecmRef" default="1800.0" min="1."> | |
149 | The <ei>ecmRef</ei> reference energy scale introduced above. | |
150 | </parm> | |
151 | ||
152 | <parm name="MultipleInteractions:ecmPow" default="0.16" min="0.0" max="0.5"> | |
153 | The <ei>ecmPow</ei> energy rescaling pace introduced above. | |
154 | </parm> | |
155 | ||
156 | <p/> | |
157 | Alternatively, or in combination, a sharp cut can be used. | |
158 | <parm name="MultipleInteractions:pTmin" default="0.2" min="0.1" max="10.0"> | |
159 | Lower cutoff in <ei>pT</ei>, below which no further interactions | |
160 | are allowed. Normally <ei>pT0</ei> above would be used to provide | |
161 | the main regularization of the cross section for <ei>pT -> 0</ei>, | |
162 | in which case <ei>pTmin</ei> is used mainly for technical reasons. | |
163 | It is possible, however, to set <ei>pT0Ref = 0</ei> and use | |
164 | <ei>pTmin</ei> to provide a step-function regularization, or to | |
165 | combine them in intermediate approaches. Currently <ei>pTmin</ei> | |
166 | is taken to be energy-independent. | |
167 | </parm> | |
168 | ||
169 | <p/> | |
170 | The choice of impact-parameter dependence is regulated by several | |
171 | parameters. | |
172 | ||
173 | <modepick name="MultipleInteractions:bProfile" default="2" | |
174 | min="0" max="3"> | |
175 | Choice of impact parameter profile for the incoming hadron beams. | |
176 | <option value="0">no impact parameter dependence at all.</option> | |
177 | <option value="1">a simple Gaussian matter distribution; | |
178 | no free parameters.</option> | |
179 | <option value="2">a double Gaussian matter distribution, | |
180 | with the two free parameters <ei>coreRadius</ei> and | |
181 | <ei>coreFraction</ei>.</option> | |
182 | <option value="3">an overlap function, i.e. the convolution of | |
183 | the matter distributions of the two incoming hadrons, of the form | |
184 | <ei>exp(- b^expPow)</ei>, where <ei>expPow</ei> is a free | |
185 | parameter.</option> | |
186 | </modepick> | |
187 | ||
188 | <parm name="MultipleInteractions:coreRadius" default="0.4" min="0.1" max="1."> | |
189 | When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, | |
190 | the inner core is assumed to have a radius that is a factor | |
191 | <ei>coreRadius</ei> smaller than the rest. | |
192 | </parm> | |
193 | ||
194 | <parm name="MultipleInteractions:coreFraction" default="0.5" min="0." max="1."> | |
195 | When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, | |
196 | the inner core is assumed to have a fraction <ei>coreFraction</ei> | |
197 | of the matter content of the hadron. | |
198 | </parm> | |
199 | ||
200 | <parm name="MultipleInteractions:expPow" default="1." min="0.4" max="10."> | |
201 | When <ei>bProfile = 3</ei> it gives the power of the assumed overlap | |
202 | shape <ei>exp(- b^expPow)</ei>. Default corresponds to a simple | |
203 | exponential drop, which is not too dissimilar from the overlap | |
204 | obtained with the standard double Gaussian parameters. For | |
205 | <ei>expPow = 2</ei> we reduce to the simple Gaussian, <ei>bProfile = 1</ei>, | |
206 | and for <ei>expPow -> infinity</ei> to no impact parameter dependence | |
207 | at all, <ei>bProfile = 0</ei>. For small <ei>expPow</ei> the program | |
208 | becomes slow and unstable, so the min limit must be respected. | |
209 | </parm> | |
210 | ||
211 | <p/> | |
212 | It is possible to regulate the set of processes that are included in the | |
213 | multiple-interactions framework. | |
214 | ||
215 | <modepick name="MultipleInteractions:processLevel" default="3" | |
216 | min="0" max="3"> | |
217 | Set of processes included in the machinery. | |
218 | <option value="0">only the simplest <ei>2 -> 2</ei> QCD processes | |
219 | between quarks and gluons, giving no new flavours, i.e. dominated by | |
220 | <ei>t</ei>-channel gluon exchange.</option> | |
221 | <option value="1">also <ei>2 -> 2</ei> QCD processes giving new flavours | |
222 | (including charm and bottom), i.e. proceeding through <ei>s</ei>-channel | |
223 | gluon exchange.</option> | |
224 | <option value="2">also <ei>2 -> 2</ei> processes involving one or two | |
225 | photons in the final state, <ei>s</ei>-channel <ei>gamma</ei> | |
226 | boson exchange and <ei>t</ei>-channel <ei>gamma/Z^0/W^+-</ei> | |
227 | boson exchange.</option> | |
228 | <option value="3">also charmonium and bottomonium production, via | |
229 | colour singlet and colour octet channels.</option> | |
230 | </modepick> | |
231 | ||
232 | <h3>Further variables</h3> | |
233 | ||
234 | These should normally not be touched. Their only function is for | |
235 | cross-checks. | |
236 | ||
237 | <modeopen name="MultipleInteractions:nQuarkIn" default="5" min="0" max="5"> | |
238 | Number of allowed incoming quark flavours in the beams; a change | |
239 | to 4 would thus exclude <ei>b</ei> and <ei>bbar</ei> as incoming | |
240 | partons, etc. | |
241 | </modeopen> | |
242 | ||
243 | <modeopen name="MultipleInteractions:nSample" default="1000" min="100"> | |
244 | The allowed <ei>pT</ei> range is split (unevenly) into 100 bins, | |
245 | and in each of these the interaction cross section is evaluated in | |
246 | <ei>nSample</ei> random phase space points. The full integral is used | |
247 | at initialization, and the differential one during the run as a | |
248 | "Sudakov form factor" for the choice of the hardest interaction. | |
249 | A larger number implies increased accuracy of the calculations. | |
250 | </modeopen> | |
251 | ||
252 | <h3>The process library</h3> | |
253 | ||
254 | The processes used to generate multiple interactions form a subset | |
255 | of the standard library of hard processes. The input is slightly | |
256 | different from the standard hard-process machinery, however, | |
257 | since incoming flavours, the <ei>alpha_strong</ei> value and most | |
258 | of the kinematics are aready fixed when the process is called. | |
259 | ||
260 | <h3>Technical notes</h3> | |
261 | ||
262 | Relative to the articles mentioned above, not much has happened. | |
263 | The main news is a technical one, that the phase space of the | |
264 | <ei>2 -> 2</ei> (massless) QCD processes is now sampled in | |
265 | <ei>dy_3 dy_4 dpT^2</ei>, where <ei>y_3</ei> and <ei>y_4</ei> are | |
266 | the rapidities of the two produced partons. One can show that | |
267 | <eq> | |
268 | (dx_1 / x_1) * (dx_2 / x_2) * d(tHat) = dy_3 * dy_4 * dpT^2 | |
269 | </eq> | |
270 | Furthermore, since cross sections are dominated by the "Rutherford" | |
271 | one of <ei>t</ei>-channel gluon exchange, which is enhanced by a | |
272 | factor of 9/4 for each incoming gluon, effective structure functions | |
273 | are defined as | |
274 | <eq> | |
275 | F(x, pT2) = (9/4) * xg(x, pT2) + sum_i xq_i(x, pT2) | |
276 | </eq> | |
277 | With this technical shift of factors 9/4 from cross sections to parton | |
278 | densities, a common upper estimate of | |
279 | <eq> | |
280 | d(sigmaHat)/d(pT2) < pi * alpha_strong^2 / pT^4 | |
281 | </eq> | |
282 | is obtained. | |
283 | ||
284 | <p/> | |
285 | In fact this estimate can be reduced by a factor of 1/2 for the | |
286 | following reason: for any configuration <ei>(y_3, y_4, pT2)</ei> also | |
287 | one with <ei>(y_4, y_3, pT2)</ei> lies in the phase space. Not both | |
288 | of those can enjoy being enhanced by the <ei>tHat -> 0</ei> | |
289 | singularity of | |
290 | <eq> | |
291 | d(sigmaHat) propto 1/tHat^2. | |
292 | </eq> | |
293 | Or if they are, which is possible with identical partons like | |
294 | <ei>q q -> q q</ei> and <ei>g g -> g g</ei>, each singularity comes | |
295 | with half the strength. So, when integrating/averaging over the two | |
296 | configurations, the estimated <ei>d(sigmaHat)/d(pT2)</ei> drops. | |
297 | Actually, it drops even further, since the naive estimate above is | |
298 | based on | |
299 | <eq> | |
300 | (4 /9) * (1 + (uHat/sHat)^2) < 8/9 < 1 | |
301 | </eq> | |
302 | The 8/9 value would be approached for <ei>tHat -> 0</ei>, which | |
303 | implies <ei>sHat >> pT2</ei> and thus a heavy parton-distribution | |
304 | penalty, while parton distributions are largest for | |
305 | <ei>tHat = uHat = -sHat/2</ei>, where the above expression | |
306 | evaluates to 5/9. A fudge factor is therefore introduced to go the | |
307 | final step, so it can easily be modifed when further non-Rutherford | |
308 | processes are added, or should parton distributions change significantly. | |
309 | ||
310 | <p/> | |
311 | At initialization, it is assumed that | |
312 | <eq> | |
313 | d(sigma)/d(pT2) < d(sigmaHat)/d(pT2) * F(x_T, pT2) * F(x_T, pT2) | |
314 | * (2 y_max(pT))^2 | |
315 | </eq> | |
316 | where the first factor is the upper estimate as above, the second two | |
317 | the parton density sum evaluated at <ei>y_3 = y_ 4 = 0</ei> so that | |
318 | <ei>x_1 = x_2 = x_T = 2 pT / E_cm</ei>, where the product is expected | |
319 | to be maximal, and the final is the phase space for | |
320 | <ei>-y_max < y_{3,4} < y_max</ei>. | |
321 | The right-hand side expression is scanned logarithmically in <ei>y</ei>, | |
322 | and a <ei>N</ei> is determined such that it always is below | |
323 | <ei>N/pT^4</ei>. | |
324 | ||
325 | <p/> | |
326 | To describe the dampening of the cross section at <ei>pT -> 0</ei> by | |
327 | colour screening, the actual cross section is multiplied by a | |
328 | regularization factor <ei>(pT^2 / (pT^2 + pT0^2))^2</ei>, and the | |
329 | <ei>alpha_s</ei> is evaluated at a scale <ei>pT^2 + pT0^2</ei>, | |
330 | where <ei>pT0</ei> is a free parameter of the order of 2 - 4 GeV. | |
331 | Since <ei>pT0</ei> can be energy-dependent, an ansatz | |
332 | <eq> | |
333 | pT0(ecm) = pT0Ref * (ecm/ecmRef)^ecmPow | |
334 | </eq> | |
335 | is used, where <ei>ecm</ei> is the current cm frame energy, | |
336 | <ei>ecmRef</ei> is an arbitrary reference energy where <ei>pT0Ref</ei> | |
337 | is defined, and <ei>ecmPow</ei> gives the energy rescaling pace. For | |
338 | technical reasons, also an absolute lower <ei>pT</ei> scale <ei>pTmin</ei>, | |
339 | by default 0.2 GeV, is introduced. In principle, it is possible to | |
340 | recover older scenarios with a sharp <ei>pT</ei> cutoff by setting | |
341 | <ei>pT0 = 0</ei> and letting <ei>pTmin</ei> be a larger number. | |
342 | ||
343 | <p/> | |
344 | The above scanning strategy is then slightly modified: instead of | |
345 | an upper estimate <ei>N/pT^4</ei> one of the form | |
346 | <ei>N/(pT^2 + r * pT0^2)^2</ei> is used. At first glance, <ei>r = 1</ei> | |
347 | would seem to be fixed by the form of the regularization procedure, | |
348 | but this does not take into account the nontrivial dependence on | |
349 | <ei>alpha_s</ei>, parton distributions and phase space. A better | |
350 | Monte Carlo efficiency is obtained for <ei>r</ei> somewhat below unity, | |
351 | and currently <ei>r = 0.25</ei> is hardcoded. | |
352 | ||
353 | In the generation a trial <ei>pT2</ei> is then selected according to | |
354 | <eq> | |
355 | d(Prob)/d(pT2) = (1/sigma_ND) * N/(pT^2 + r * pT0^2)^2 * ("Sudakov") | |
356 | </eq> | |
357 | For the trial <ei>pT2</ei>, a <ei>y_3</ei> and a <ei>y_4</ei> are then | |
358 | selected, and incoming flavours according to the respective | |
359 | <ei>F(x_i, pT2)</ei>, and then the cross section is evaluated for this | |
360 | flavour combination. The ratio of trial/upper estimate gives the | |
361 | probability of survival. | |
362 | ||
363 | <p/> | |
364 | Actually, to profit from the factor 1/2 mentioned above, the cross | |
365 | section for the combination with <ei>y_3</ei> and <ei>y_4</ei> | |
366 | interchanged is also tried, which corresponds to exchanging <ei>tHat</ei> | |
367 | and <ei>uHat</ei>, and the average formed, while the final kinematics | |
368 | is given by the relative importance of the two. | |
369 | ||
370 | <p/> | |
371 | Furthermore, since large <ei>y</ei> values are disfavoured by dropping | |
372 | PDF's, a factor | |
373 | <eq> | |
374 | WT_y = (1 - (y_3/y_max)^2) * (1 - (y_4/y_max)^2) | |
375 | </eq> | |
376 | is evaluated, and used as a survival probability before the more | |
377 | time-consuming PDF+ME evaluation, with surviving events given a | |
378 | compensating weight <ei>1/WT_y</ei>. | |
379 | ||
380 | <p/> | |
381 | An impact-parameter dependencs is also allowed. Based on the hard | |
382 | <ei>pT</ei> scale of the first interaction, and enhancement/depletion | |
383 | factor is picked, which multiplies the rate of subsequent interactions. | |
384 | ||
385 | <p/> | |
386 | Parton densities are rescaled and modified to take into account the | |
387 | energy-momentum and flavours kicked out by already-considered | |
388 | interactions. | |
389 | ||
390 | </chapter> | |
391 | ||
392 | <!-- Copyright (C) 2008 Torbjorn Sjostrand --> |