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63ba5337 | 1 | <chapter name="Multiparton Interactions"> |
2 | ||
3 | <h2>Multiparton Interactions</h2> | |
4 | ||
5 | The starting point for the multiparton 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>, interleaving with initial-state radiation | |
10 | <ref>Sjo05</ref>, making use of transverse-momentum-ordered | |
11 | initial- and final-state showers, with the extension to fully | |
12 | interleaved evolution covered in <ref>Cor10a</ref>. A framework to | |
13 | handle rescattering is described in <ref>Cor09</ref>. | |
14 | ||
15 | <p/> | |
16 | A big unsolved issue is how the colour of all these subsystems is | |
17 | correlated. For sure there is a correlation coming from the colour | |
18 | singlet nature of the incoming beams, but in addition final-state | |
19 | colour rearrangements may change the picture. Indeed such extra | |
20 | effects appear necessary to describe data, e.g. on | |
21 | <ei><pT>(n_ch)</ei>. A simple implementation of colour | |
22 | rearrangement is found as part of the | |
23 | <aloc href="BeamRemnants">beam remnants</aloc> description. | |
24 | ||
25 | <h3>Main variables</h3> | |
26 | ||
27 | <h4>Matching to hard process</h4> | |
28 | ||
29 | The maximum <ei>pT</ei> to be allowed for multiparton interactions is | |
30 | related to the nature of the hard process itself. It involves a | |
31 | delicate balance between not doublecounting and not leaving any | |
32 | gaps in the coverage. The best procedure may depend on information | |
33 | only the user has: how the events were generated and mixed (e.g. with | |
34 | Les Houches Accord external input), and how they are intended to be | |
35 | used. Therefore a few options are available, with a sensible default | |
36 | behaviour. | |
37 | <modepick name="MultipartonInteractions:pTmaxMatch" default="0" min="0" | |
38 | max="2"> | |
39 | Way in which the maximum scale for multiparton interactions is set | |
40 | to match the scale of the hard process itself. | |
41 | <option value="0"><b>(i)</b> if the final state of the hard process | |
42 | (not counting subsequent resonance decays) contains only quarks | |
43 | (<ei>u, d, s, c ,b</ei>), gluons and photons then <ei>pT_max</ei> | |
44 | is chosen to be the factorization scale for internal processes | |
45 | and the <code>scale</code> value for Les Houches input; | |
46 | <b>(ii)</b> if not, interactions are allowed to go all the way up | |
47 | to the kinematical limit. | |
48 | The reasoning is that the former kind of processes are generated by | |
49 | the multiparton-interactions machinery and so would doublecount hard | |
50 | processes if allowed to overlap the same <ei>pT</ei> range, | |
51 | while no such danger exists in the latter case. | |
52 | </option> | |
53 | <option value="1">always use the factorization scale for an internal | |
54 | process and the <code>scale</code> value for Les Houches input, | |
55 | i.e. the lower value. This should avoid doublecounting, but | |
56 | may leave out some interactions that ought to have been simulated. | |
57 | </option> | |
58 | <option value="2">always allow multiparton interactions up to the | |
59 | kinematical limit. This will simulate all possible event topologies, | |
60 | but may lead to doublecounting. | |
61 | </option> | |
62 | </modepick> | |
63 | ||
64 | <h4>Cross-section parameters</h4> | |
65 | ||
66 | The rate of interactions is determined by | |
67 | <parm name="MultipartonInteractions:alphaSvalue" default="0.127" | |
68 | min="0.06" max="0.25"> | |
69 | The value of <ei>alpha_strong</ei> at <ei>m_Z</ei>. Default value is | |
70 | picked equal to the one used in CTEQ 5L. | |
71 | </parm> | |
72 | ||
73 | <p/> | |
74 | The actual value is then regulated by the running to the scale | |
75 | <ei>pT^2</ei>, at which it is evaluated | |
76 | <modepick name="MultipartonInteractions:alphaSorder" default="1" | |
77 | min="0" max="2"> | |
78 | The order at which <ei>alpha_strong</ei> runs at scales away from | |
79 | <ei>m_Z</ei>. | |
80 | <option value="0">zeroth order, i.e. <ei>alpha_strong</ei> is kept | |
81 | fixed.</option> | |
82 | <option value="1">first order, which is the normal value.</option> | |
83 | <option value="2">second order. Since other parts of the code do | |
84 | not go to second order there is no strong reason to use this option, | |
85 | but there is also nothing wrong with it.</option> | |
86 | </modepick> | |
87 | ||
88 | <p/> | |
89 | QED interactions are regulated by the <ei>alpha_electromagnetic</ei> | |
90 | value at the <ei>pT^2</ei> scale of an interaction. | |
91 | ||
92 | <modepick name="MultipartonInteractions:alphaEMorder" default="1" | |
93 | min="-1" max="1"> | |
94 | The running of <ei>alpha_em</ei> used in hard processes. | |
95 | <option value="1">first-order running, constrained to agree with | |
96 | <code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass. | |
97 | </option> | |
98 | <option value="0">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
99 | fixed at its value at vanishing momentum transfer.</option> | |
100 | <option value="-1">zeroth order, i.e. <ei>alpha_em</ei> is kept | |
101 | fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value | |
102 | at the <ei>Z^0</ei> mass. | |
103 | </option> | |
104 | </modepick> | |
105 | ||
106 | <p/> | |
107 | Note that the choices of <ei>alpha_strong</ei> and <ei>alpha_em</ei> | |
108 | made here override the ones implemented in the normal process machinery, | |
109 | but only for the interactions generated by the | |
110 | <code>MultipartonInteractions</code> class. | |
111 | ||
112 | <p/> | |
113 | In addition there is the possibility of a global rescaling of | |
114 | cross sections (which could not easily be accommodated by a | |
115 | changed <ei>alpha_strong</ei>, since <ei>alpha_strong</ei> runs) | |
116 | <parm name="MultipartonInteractions:Kfactor" default="1.0" min="0.5" | |
117 | max="4.0"> | |
118 | Multiply all cross sections by this fix factor. | |
119 | </parm> | |
120 | ||
121 | <p/> | |
122 | The processes used to generate multiparton interactions form a subset | |
123 | of the standard library of hard processes. The input is slightly | |
124 | different from the standard hard-process machinery, however, | |
125 | since incoming flavours, the <ei>alpha_strong</ei> value and most | |
126 | of the kinematics are aready fixed when the process is called. | |
127 | It is possible to regulate the set of processes that are included in the | |
128 | multiparton-interactions framework. | |
129 | ||
130 | <modepick name="MultipartonInteractions:processLevel" default="3" | |
131 | min="0" max="3"> | |
132 | Set of processes included in the machinery. | |
133 | <option value="0">only the simplest <ei>2 -> 2</ei> QCD processes | |
134 | between quarks and gluons, giving no new flavours, i.e. dominated by | |
135 | <ei>t</ei>-channel gluon exchange.</option> | |
136 | <option value="1">also <ei>2 -> 2</ei> QCD processes giving new flavours | |
137 | (including charm and bottom), i.e. proceeding through <ei>s</ei>-channel | |
138 | gluon exchange.</option> | |
139 | <option value="2">also <ei>2 -> 2</ei> processes involving one or two | |
140 | photons in the final state, <ei>s</ei>-channel <ei>gamma</ei> | |
141 | boson exchange and <ei>t</ei>-channel <ei>gamma/Z^0/W^+-</ei> | |
142 | boson exchange.</option> | |
143 | <option value="3">also charmonium and bottomonium production, via | |
144 | colour singlet and colour octet channels.</option> | |
145 | </modepick> | |
146 | ||
147 | <h4>Cross-section regularization</h4> | |
148 | ||
149 | There are two complementary ways of regularizing the small-<ei>pT</ei> | |
150 | divergence, a sharp cutoff and a smooth dampening. These can be | |
151 | combined as desired, but it makes sense to coordinate with how the | |
152 | same issue is handled in <aloc href="SpacelikeShowers">spacelike | |
153 | showers</aloc>. Actually, by default, the parameters defined here are | |
154 | used also for the spacelike showers, but this can be overridden. | |
155 | ||
156 | <p/> | |
157 | Regularization of the divergence of the QCD cross section for | |
158 | <ei>pT -> 0</ei> is obtained by a factor <ei>pT^4 / (pT0^2 + pT^2)^2</ei>, | |
159 | and by using an <ei>alpha_s(pT0^2 + pT^2)</ei>. An energy dependence | |
160 | of the <ei>pT0</ei> choice is introduced by two further parameters, | |
161 | so that <ei>pT0Ref</ei> is the <ei>pT0</ei> value for the reference | |
162 | CM energy, <ei>pT0Ref = pT0(ecmRef)</ei>. | |
163 | <note>Warning:</note> if a large <ei>pT0</ei> is picked for multiparton | |
164 | interactions, such that the integrated interaction cross section is | |
165 | below the nondiffractive inelastic one, this <ei>pT0</ei> will | |
166 | automatically be scaled down to cope. | |
167 | ||
168 | <p/> | |
169 | The actual <ei>pT0</ei> parameter used at a given CM energy scale, | |
170 | <ei>ecmNow</ei>, is obtained as | |
171 | <eq> | |
172 | pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow | |
173 | </eq> | |
174 | where <ei>pT0Ref</ei>, <ei>ecmRef</ei> and <ei>ecmPow</ei> are the | |
175 | three parameters below. | |
176 | ||
177 | <parm name="MultipartonInteractions:pT0Ref" default="2.15" min="0.5" | |
178 | max="10.0"> | |
179 | The <ei>pT0Ref</ei> scale in the above formula. | |
180 | <note>Note:</note> <ei>pT0Ref</ei> is one of the key parameters in a | |
181 | complete PYTHIA tune. Its value is intimately tied to a number of other | |
182 | choices, such as that of colour flow description, so unfortunately it is | |
183 | difficult to give an independent meaning to <ei>pT0Ref</ei>. | |
184 | </parm> | |
185 | ||
186 | <parm name="MultipartonInteractions:ecmRef" default="1800.0" min="1."> | |
187 | The <ei>ecmRef</ei> reference energy scale introduced above. | |
188 | </parm> | |
189 | ||
190 | <parm name="MultipartonInteractions:ecmPow" default="0.24" min="0.0" | |
191 | max="0.5"> | |
192 | The <ei>ecmPow</ei> energy rescaling pace introduced above. | |
193 | </parm> | |
194 | ||
195 | <p/> | |
196 | Alternatively, or in combination, a sharp cut can be used. | |
197 | <parm name="MultipartonInteractions:pTmin" default="0.2" min="0.1" | |
198 | max="10.0"> | |
199 | Lower cutoff in <ei>pT</ei>, below which no further interactions | |
200 | are allowed. Normally <ei>pT0</ei> above would be used to provide | |
201 | the main regularization of the cross section for <ei>pT -> 0</ei>, | |
202 | in which case <ei>pTmin</ei> is used mainly for technical reasons. | |
203 | It is possible, however, to set <ei>pT0Ref = 0</ei> and use | |
204 | <ei>pTmin</ei> to provide a step-function regularization, or to | |
205 | combine them in intermediate approaches. Currently <ei>pTmin</ei> | |
206 | is taken to be energy-independent. | |
207 | </parm> | |
208 | ||
209 | <p/> | |
210 | Gösta Gustafson has proposed (private communication, unpublished) | |
211 | that the amount of screening, as encapsulated in the <ei>pT0</ei> | |
212 | parameter, fluctuates from one event to the next. Specifically, | |
213 | high-activity event are more likely to lead to interactions at large | |
214 | <ei>pT</ei> scales, but the high activity simultaneously leads to a | |
215 | larger screening of interactions at smaller <ei>pT</ei>. Such a scenario | |
216 | can approximately be simulated by scaling up the <ei>pT0</ei> by a | |
217 | factor <ei>sqrt(n)</ei>, where <ei>n</ei> is the number of interactions | |
218 | considered so far, including the current one. That is, for the first | |
219 | interaction the dampening factor is <ei>pT^4 / (pT0^2 + pT^2)^2</ei>, | |
220 | for the second <ei>pT^4 / (2 pT0^2 + pT^2)^2</ei>, for the third | |
221 | <ei>pT^4 / (3 pT0^2 + pT^2)^2</ei>, and so on. Optionally the scheme | |
222 | may also be applied to ISR emissions. For simplicity the same | |
223 | <ei>alpha_s(pT0^2 + pT^2)</ei> is used throughout. Note that, in this | |
224 | scenario the <ei>pT0</ei> scale must be lower than in the normal case | |
225 | to begin with, since it later is increased back up. Also note that the | |
226 | idea with this scenario is to propose an alternative to colour | |
227 | reconnection to understand the rise of <ei><pT>(n_ch)</ei>, | |
228 | so that the amount of colour reconnection should be reduced. | |
229 | <modepick name="MultipartonInteractions:enhanceScreening" default="0" | |
230 | min="0" max="2"> | |
231 | Choice to activate the above screening scenario, i.e. an increasing | |
232 | effective <ei>pT0</ei> for consecutive interactions. | |
233 | <option value="0">No activity-dependent screening, i.e. <ei>pT0</ei> | |
234 | is fixed.</option> | |
235 | <option value="1">The <ei>pT0</ei> scale is increased as a function | |
236 | of the number of MPI's, as explained above. ISR is not affected, | |
237 | but note that, if <code>SpaceShower:samePTasMPI</code> is on, | |
238 | then <code>MultipartonInteractions:pT0Ref</code> is used also for ISR, | |
239 | which may or may not be desirable. | |
240 | </option> | |
241 | <option value="2">Both MPI and ISR influence and are influenced by the | |
242 | screening. That is, the dampening is reduced based on the total number | |
243 | of MPI and ISR steps considered so far, including the current one. | |
244 | This dampening is implemented both for MPI and for ISR emissions, | |
245 | for the latter provided that <code>SpaceShower:samePTasMPI</code> is on | |
246 | (default). | |
247 | </option> | |
248 | </modepick> | |
249 | ||
250 | <h4>Impact-parameter dependence</h4> | |
251 | ||
252 | The choice of impact-parameter dependence is regulated by several | |
253 | parameters. The ones listed here refer to nondiffractive topologies | |
254 | only, while their equivalents for diffractive events are put in the | |
255 | <aloc href="Diffraction">Diffraction</aloc> description. Note that | |
256 | there is currently no <code>bProfile = 4</code> option for diffraction. | |
257 | Other parameters are assumed to agree between diffractive and | |
258 | nondiffractive topologies. | |
259 | ||
260 | <modepick name="MultipartonInteractions:bProfile" default="1" | |
261 | min="0" max="4"> | |
262 | Choice of impact parameter profile for the incoming hadron beams. | |
263 | <option value="0">no impact parameter dependence at all.</option> | |
264 | <option value="1">a simple Gaussian matter distribution; | |
265 | no free parameters.</option> | |
266 | <option value="2">a double Gaussian matter distribution, | |
267 | with the two free parameters <ei>coreRadius</ei> and | |
268 | <ei>coreFraction</ei>.</option> | |
269 | <option value="3">an overlap function, i.e. the convolution of | |
270 | the matter distributions of the two incoming hadrons, of the form | |
271 | <ei>exp(- b^expPow)</ei>, where <ei>expPow</ei> is a free | |
272 | parameter.</option> | |
273 | <option value="4">a Gaussian matter distribution with a width | |
274 | that varies according to the selected <ei>x</ei> value of an interaction, | |
275 | <ei>1. + a1 log (1 / x)</ei>, where <ei>a1</ei> is a free parameter. | |
276 | Note that once <ei>b</ei> has been selected for the hard process, | |
277 | it remains fixed for the remainder of the evolution. | |
278 | </option> | |
279 | </modepick> | |
280 | ||
281 | <parm name="MultipartonInteractions:coreRadius" default="0.4" | |
282 | min="0.1" max="1."> | |
283 | When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, | |
284 | the inner core is assumed to have a radius that is a factor | |
285 | <ei>coreRadius</ei> smaller than the rest. | |
286 | </parm> | |
287 | ||
288 | <parm name="MultipartonInteractions:coreFraction" default="0.5" | |
289 | min="0." max="1."> | |
290 | When assuming a double Gaussian matter profile, <ei>bProfile = 2</ei>, | |
291 | the inner core is assumed to have a fraction <ei>coreFraction</ei> | |
292 | of the matter content of the hadron. | |
293 | </parm> | |
294 | ||
295 | <parm name="MultipartonInteractions:expPow" default="1." min="0.4" max="10."> | |
296 | When <ei>bProfile = 3</ei> it gives the power of the assumed overlap | |
297 | shape <ei>exp(- b^expPow)</ei>. Default corresponds to a simple | |
298 | exponential drop, which is not too dissimilar from the overlap | |
299 | obtained with the standard double Gaussian parameters. For | |
300 | <ei>expPow = 2</ei> we reduce to the simple Gaussian, <ei>bProfile = 1</ei>, | |
301 | and for <ei>expPow -> infinity</ei> to no impact parameter dependence | |
302 | at all, <ei>bProfile = 0</ei>. For small <ei>expPow</ei> the program | |
303 | becomes slow and unstable, so the min limit must be respected. | |
304 | </parm> | |
305 | ||
306 | <parm name="MultipartonInteractions:a1" default="0.15" min="0." max="2."> | |
307 | When <ei>bProfile = 4</ei>, this gives the <ei>a1</ei> constant in the | |
308 | Gaussian width. When <ei>a1 = 0.</ei>, this reduces back to the single | |
309 | Gaussian case. | |
310 | </parm> | |
311 | ||
312 | <h4>Rescattering</h4> | |
313 | ||
314 | It is possible that a parton may rescatter, i.e. undergo a further | |
315 | interaction subsequent to the first one. The machinery to model this | |
316 | kind of physics has only recently become fully operational | |
317 | <ref>Cor09</ref>, and is therefore not yet so well explored. | |
318 | ||
319 | <p/> | |
320 | The rescatting framework has ties with other parts of the program, | |
321 | notably with the <aloc href="BeamRemnants">beam remnants</aloc>. | |
322 | ||
323 | <flag name="MultipartonInteractions:allowRescatter" default="off"> | |
324 | Switch to allow rescattering of partons; on/off = true/false.<br/> | |
325 | <b>Note:</b> the rescattering framework has not yet been implemented | |
326 | for the <code>MultipartonInteractions:bProfile = 4</code> option, | |
327 | and can therefore not be switched on in that case. | |
328 | <b>Warning:</b> use with caution since machinery is still not | |
329 | so well tested. | |
330 | </flag> | |
331 | ||
332 | <flag name="MultipartonInteractions:allowDoubleRescatter" default="off"> | |
333 | Switch to allow rescattering of partons, where both incoming partons | |
334 | have already rescattered; on/off = true/false. Is only used if | |
335 | <code>MultipartonInteractions:allowRescatter</code> is switched on.<br/> | |
336 | <b>Warning:</b> currently there is no complete implementation that | |
337 | combines it with shower evolution, so you must use | |
338 | <code>PartonLevel:ISR = off</code> and <code>PartonLevel:FSR = off</code>. | |
339 | If not, a warning will be issued and double rescattering will not be | |
340 | simulated. The rate also comes out to be much lower than for single | |
341 | rescattering, so to first approximation it can be neglected. | |
342 | </flag> | |
343 | ||
344 | <modepick name="MultipartonInteractions:rescatterMode" default="0" | |
345 | min="0" max="4"> | |
346 | Selection of which partons rescatter against unscattered partons | |
347 | from the incoming beams A and B, based on their rapidity value | |
348 | <ei>y</ei> in the collision rest frame. Here <ei>ySep</ei> is | |
349 | shorthand for <code>MultipartonInteractions:ySepRescatter</code> and | |
350 | <ei>deltaY</ei> for <code>MultipartonInteractions:deltaYRescatter</code>, | |
351 | defined below. The description is symmetric between the two beams, | |
352 | so only one case is described below. | |
353 | <option value="0">only scattered partons with <ei>y > 0</ei> | |
354 | can collide with unscattered partons from beam B.</option> | |
355 | <option value="1">only scattered partons with <ei>y > ySep</ei> | |
356 | can collide with unscattered partons from beam B.</option> | |
357 | <option value="2">the probability for a scattered parton to be considered | |
358 | as a potential rescatterer against unscattered partons in beam B increases | |
359 | linearly from zero at <ei>y = ySep - deltaY</ei> to unity at | |
360 | <ei>y = ySep + deltaY</ei>.</option> | |
361 | <option value="3">the probability for a scattered parton to be considered | |
362 | as a potential rescatterer against unscattered partons in beam B increases | |
363 | with <ei>y</ei> according to | |
364 | <ei>(1/2) * (1 + tanh( (y - ySep) / deltaY))</ei>.</option> | |
365 | <option value="4">all partons are potential rescatterers against both | |
366 | beams.</option> | |
367 | </modepick> | |
368 | ||
369 | <parm name="MultipartonInteractions:ySepRescatter" default="0."> | |
370 | used for some of the <code>MultipartonInteractions:rescatterMode</code> | |
371 | options above, as the rapidity for which a scattered parton has a 50% | |
372 | probability to be considered as a potential rescatterer. | |
373 | A <ei>ySep > 0</ei> generally implies that some central partons cannot | |
374 | rescatter at all, while a <ei>ySep < 0</ei> instead allows central | |
375 | partons to scatter against either beam. | |
376 | </parm> | |
377 | ||
378 | <parm name="MultipartonInteractions:deltaYRescatter" default="1." min="0.1"> | |
379 | used for some of the <code>MultipartonInteractions:rescatterMode</code> | |
380 | options above, as the width of the rapidity transition region, where the | |
381 | probability rises from zero to unity that a scattered parton is considered | |
382 | as a potential rescatterer. | |
383 | </parm> | |
384 | ||
385 | ||
386 | <h3>Further variables</h3> | |
387 | ||
388 | These should normally not be touched. Their only function is for | |
389 | cross-checks. | |
390 | ||
391 | <modeopen name="MultipartonInteractions:nQuarkIn" default="5" min="0" | |
392 | max="5"> | |
393 | Number of allowed incoming quark flavours in the beams; a change | |
394 | to 4 would thus exclude <ei>b</ei> and <ei>bbar</ei> as incoming | |
395 | partons, etc. | |
396 | </modeopen> | |
397 | ||
398 | <modeopen name="MultipartonInteractions:nSample" default="1000" min="100"> | |
399 | The allowed <ei>pT</ei> range is split (unevenly) into 100 bins, | |
400 | and in each of these the interaction cross section is evaluated in | |
401 | <ei>nSample</ei> random phase space points. The full integral is used | |
402 | at initialization, and the differential one during the run as a | |
403 | "Sudakov form factor" for the choice of the hardest interaction. | |
404 | A larger number implies increased accuracy of the calculations. | |
405 | </modeopen> | |
406 | ||
407 | <h3>Technical notes</h3> | |
408 | ||
409 | Relative to the articles mentioned above, not much has happened. | |
410 | The main news is a technical one, that the phase space of the | |
411 | <ei>2 -> 2</ei> (massless) QCD processes is now sampled in | |
412 | <ei>dy_3 dy_4 dpT^2</ei>, where <ei>y_3</ei> and <ei>y_4</ei> are | |
413 | the rapidities of the two produced partons. One can show that | |
414 | <eq> | |
415 | (dx_1 / x_1) * (dx_2 / x_2) * d(tHat) = dy_3 * dy_4 * dpT^2 | |
416 | </eq> | |
417 | Furthermore, since cross sections are dominated by the "Rutherford" | |
418 | one of <ei>t</ei>-channel gluon exchange, which is enhanced by a | |
419 | factor of 9/4 for each incoming gluon, effective structure functions | |
420 | are defined as | |
421 | <eq> | |
422 | F(x, pT2) = (9/4) * xg(x, pT2) + sum_i xq_i(x, pT2) | |
423 | </eq> | |
424 | With this technical shift of factors 9/4 from cross sections to parton | |
425 | densities, a common upper estimate of | |
426 | <eq> | |
427 | d(sigmaHat)/d(pT2) < pi * alpha_strong^2 / pT^4 | |
428 | </eq> | |
429 | is obtained. | |
430 | ||
431 | <p/> | |
432 | In fact this estimate can be reduced by a factor of 1/2 for the | |
433 | following reason: for any configuration <ei>(y_3, y_4, pT2)</ei> also | |
434 | one with <ei>(y_4, y_3, pT2)</ei> lies in the phase space. Not both | |
435 | of those can enjoy being enhanced by the <ei>tHat -> 0</ei> | |
436 | singularity of | |
437 | <eq> | |
438 | d(sigmaHat) propto 1/tHat^2. | |
439 | </eq> | |
440 | Or if they are, which is possible with identical partons like | |
441 | <ei>q q -> q q</ei> and <ei>g g -> g g</ei>, each singularity comes | |
442 | with half the strength. So, when integrating/averaging over the two | |
443 | configurations, the estimated <ei>d(sigmaHat)/d(pT2)</ei> drops. | |
444 | Actually, it drops even further, since the naive estimate above is | |
445 | based on | |
446 | <eq> | |
447 | (4 /9) * (1 + (uHat/sHat)^2) < 8/9 < 1 | |
448 | </eq> | |
449 | The 8/9 value would be approached for <ei>tHat -> 0</ei>, which | |
450 | implies <ei>sHat >> pT2</ei> and thus a heavy parton-distribution | |
451 | penalty, while parton distributions are largest for | |
452 | <ei>tHat = uHat = -sHat/2</ei>, where the above expression | |
453 | evaluates to 5/9. A fudge factor is therefore introduced to go the | |
454 | final step, so it can easily be modifed when further non-Rutherford | |
455 | processes are added, or should parton distributions change significantly. | |
456 | ||
457 | <p/> | |
458 | At initialization, it is assumed that | |
459 | <eq> | |
460 | d(sigma)/d(pT2) < d(sigmaHat)/d(pT2) * F(x_T, pT2) * F(x_T, pT2) | |
461 | * (2 y_max(pT))^2 | |
462 | </eq> | |
463 | where the first factor is the upper estimate as above, the second two | |
464 | the parton density sum evaluated at <ei>y_3 = y_ 4 = 0</ei> so that | |
465 | <ei>x_1 = x_2 = x_T = 2 pT / E_cm</ei>, where the product is expected | |
466 | to be maximal, and the final is the phase space for | |
467 | <ei>-y_max < y_{3,4} < y_max</ei>. | |
468 | The right-hand side expression is scanned logarithmically in <ei>y</ei>, | |
469 | and a <ei>N</ei> is determined such that it always is below | |
470 | <ei>N/pT^4</ei>. | |
471 | ||
472 | <p/> | |
473 | To describe the dampening of the cross section at <ei>pT -> 0</ei> by | |
474 | colour screening, the actual cross section is multiplied by a | |
475 | regularization factor <ei>(pT^2 / (pT^2 + pT0^2))^2</ei>, and the | |
476 | <ei>alpha_s</ei> is evaluated at a scale <ei>pT^2 + pT0^2</ei>, | |
477 | where <ei>pT0</ei> is a free parameter of the order of 2 - 4 GeV. | |
478 | Since <ei>pT0</ei> can be energy-dependent, an ansatz | |
479 | <eq> | |
480 | pT0(ecm) = pT0Ref * (ecm/ecmRef)^ecmPow | |
481 | </eq> | |
482 | is used, where <ei>ecm</ei> is the current CM frame energy, | |
483 | <ei>ecmRef</ei> is an arbitrary reference energy where <ei>pT0Ref</ei> | |
484 | is defined, and <ei>ecmPow</ei> gives the energy rescaling pace. For | |
485 | technical reasons, also an absolute lower <ei>pT</ei> scale <ei>pTmin</ei>, | |
486 | by default 0.2 GeV, is introduced. In principle, it is possible to | |
487 | recover older scenarios with a sharp <ei>pT</ei> cutoff by setting | |
488 | <ei>pT0 = 0</ei> and letting <ei>pTmin</ei> be a larger number. | |
489 | ||
490 | <p/> | |
491 | The above scanning strategy is then slightly modified: instead of | |
492 | an upper estimate <ei>N/pT^4</ei> one of the form | |
493 | <ei>N/(pT^2 + r * pT0^2)^2</ei> is used. At first glance, <ei>r = 1</ei> | |
494 | would seem to be fixed by the form of the regularization procedure, | |
495 | but this does not take into account the nontrivial dependence on | |
496 | <ei>alpha_s</ei>, parton distributions and phase space. A better | |
497 | Monte Carlo efficiency is obtained for <ei>r</ei> somewhat below unity, | |
498 | and currently <ei>r = 0.25</ei> is hardcoded. | |
499 | ||
500 | In the generation a trial <ei>pT2</ei> is then selected according to | |
501 | <eq> | |
502 | d(Prob)/d(pT2) = (1/sigma_ND) * N/(pT^2 + r * pT0^2)^2 * ("Sudakov") | |
503 | </eq> | |
504 | For the trial <ei>pT2</ei>, a <ei>y_3</ei> and a <ei>y_4</ei> are then | |
505 | selected, and incoming flavours according to the respective | |
506 | <ei>F(x_i, pT2)</ei>, and then the cross section is evaluated for this | |
507 | flavour combination. The ratio of trial/upper estimate gives the | |
508 | probability of survival. | |
509 | ||
510 | <p/> | |
511 | Actually, to profit from the factor 1/2 mentioned above, the cross | |
512 | section for the combination with <ei>y_3</ei> and <ei>y_4</ei> | |
513 | interchanged is also tried, which corresponds to exchanging <ei>tHat</ei> | |
514 | and <ei>uHat</ei>, and the average formed, while the final kinematics | |
515 | is given by the relative importance of the two. | |
516 | ||
517 | <p/> | |
518 | Furthermore, since large <ei>y</ei> values are disfavoured by dropping | |
519 | PDF's, a factor | |
520 | <eq> | |
521 | WT_y = (1 - (y_3/y_max)^2) * (1 - (y_4/y_max)^2) | |
522 | </eq> | |
523 | is evaluated, and used as a survival probability before the more | |
524 | time-consuming PDF+ME evaluation, with surviving events given a | |
525 | compensating weight <ei>1/WT_y</ei>. | |
526 | ||
527 | <p/> | |
528 | An impact-parameter dependencs is also allowed. Based on the hard | |
529 | <ei>pT</ei> scale of the first interaction, and enhancement/depletion | |
530 | factor is picked, which multiplies the rate of subsequent interactions. | |
531 | ||
532 | <p/> | |
533 | Parton densities are rescaled and modified to take into account the | |
534 | energy-momentum and flavours kicked out by already-considered | |
535 | interactions. | |
536 | ||
537 | </chapter> | |
538 | ||
539 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |