3 <title>Multiple Interactions</title>
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30 <h2>Multiple Interactions</h2>
32 The starting point for the multiple interactions physics scenario in
33 PYTHIA is provided by [<a href="Bibliography.php" target="page">Sjo87</a>]. Recent developments have
34 included a more careful study of flavour and colour correlations,
35 junction topologies and the relationship to beam remnants
36 [<a href="Bibliography.php" target="page">Sjo04</a>], interleaving with initial-state radiation
37 [<a href="Bibliography.php" target="page">Sjo05</a>], making use of transverse-momentum-ordered
38 initial- and final-state showers, with the extension to fully
39 interleaved evolution covered in [<a href="Bibliography.php" target="page">Cor10a</a>]. A framework to
40 handle rescattering is described in [<a href="Bibliography.php" target="page">Cor09</a>].
43 A big unsolved issue is how the colour of all these subsystems is
44 correlated. For sure there is a correlation coming from the colour
45 singlet nature of the incoming beams, but in addition final-state
46 colour rearrangements may change the picture. Indeed such extra
47 effects appear necessary to describe data, e.g. on
48 <i><pT>(n_ch)</i>. A simple implementation of colour
49 rearrangement is found as part of the
50 <?php $filepath = $_GET["filepath"];
51 echo "<a href='BeamRemnants.php?filepath=".$filepath."' target='page'>";?>beam remnants</a> description.
53 <h3>Main variables</h3>
55 <h4>Matching to hard process</h4>
57 The maximum <i>pT</i> to be allowed for multiple interactions is
58 related to the nature of the hard process itself. It involves a
59 delicate balance between not doublecounting and not leaving any
60 gaps in the coverage. The best procedure may depend on information
61 only the user has: how the events were generated and mixed (e.g. with
62 Les Houches Accord external input), and how they are intended to be
63 used. Therefore a few options are available, with a sensible default
65 <br/><br/><table><tr><td><strong>MultipleInteractions:pTmaxMatch </td><td> (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
66 Way in which the maximum scale for multiple interactions is set
67 to match the scale of the hard process itself.
69 <input type="radio" name="1" value="0" checked="checked"><strong>0 </strong>: <b>(i)</b> if the final state of the hard process (not counting subsequent resonance decays) contains only quarks (<ei>u, d, s, c ,b</ei>), gluons and photons then <ei>pT_max</ei> is chosen to be the factorization scale for internal processes and the <code>scale</code> value for Les Houches input; <b>(ii)</b> if not, interactions are allowed to go all the way up to the kinematical limit. The reasoning is that the former kind of processes are generated by the multiple-interactions machinery and so would doublecount hard processes if allowed to overlap the same <ei>pT</ei> range, while no such danger exists in the latter case. <br/>
70 <input type="radio" name="1" value="1"><strong>1 </strong>: always use the factorization scale for an internal process and the <code>scale</code> value for Les Houches input, i.e. the lower value. This should avoid doublecounting, but may leave out some interactions that ought to have been simulated. <br/>
71 <input type="radio" name="1" value="2"><strong>2 </strong>: always allow multiple interactions up to the kinematical limit. This will simulate all possible event topologies, but may lead to doublecounting. <br/>
73 <h4>Cross-section parameters</h4>
75 The rate of interactions is determined by
76 <br/><br/><table><tr><td><strong>MultipleInteractions:alphaSvalue </td><td></td><td> <input type="text" name="2" value="0.127" size="20"/> (<code>default = <strong>0.127</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)</td></tr></table>
77 The value of <i>alpha_strong</i> at <i>m_Z</i>. Default value is
78 picked equal to the one used in CTEQ 5L.
82 The actual value is then regulated by the running to the scale
83 <i>pT^2</i>, at which it is evaluated
84 <br/><br/><table><tr><td><strong>MultipleInteractions:alphaSorder </td><td> (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
85 The order at which <ei>alpha_strong</ei> runs at scales away from
88 <input type="radio" name="3" value="0"><strong>0 </strong>: zeroth order, i.e. <ei>alpha_strong</ei> is kept fixed.<br/>
89 <input type="radio" name="3" value="1" checked="checked"><strong>1 </strong>: first order, which is the normal value.<br/>
90 <input type="radio" name="3" value="2"><strong>2 </strong>: second order. Since other parts of the code do not go to second order there is no strong reason to use this option, but there is also nothing wrong with it.<br/>
93 QED interactions are regulated by the <i>alpha_electromagnetic</i>
94 value at the <i>pT^2</i> scale of an interaction.
96 <br/><br/><table><tr><td><strong>MultipleInteractions:alphaEMorder </td><td> (<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)</td></tr></table>
97 The running of <ei>alpha_em</ei> used in hard processes.
99 <input type="radio" name="4" value="1" checked="checked"><strong>1 </strong>: first-order running, constrained to agree with <code>StandardModel:alphaEMmZ</code> at the <ei>Z^0</ei> mass. <br/>
100 <input type="radio" name="4" value="0"><strong>0 </strong>: zeroth order, i.e. <ei>alpha_em</ei> is kept fixed at its value at vanishing momentum transfer.<br/>
101 <input type="radio" name="4" value="-1"><strong>-1 </strong>: zeroth order, i.e. <ei>alpha_em</ei> is kept fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value at the <ei>Z^0</ei> mass. <br/>
104 Note that the choices of <i>alpha_strong</i> and <i>alpha_em</i>
105 made here override the ones implemented in the normal process machinery,
106 but only for the interactions generated by the
107 <code>MultipleInteractions</code> class.
110 In addition there is the possibility of a global rescaling of
111 cross sections (which could not easily be accommodated by a
112 changed <i>alpha_strong</i>, since <i>alpha_strong</i> runs)
113 <br/><br/><table><tr><td><strong>MultipleInteractions:Kfactor </td><td></td><td> <input type="text" name="5" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 4.0</code>)</td></tr></table>
114 Multiply all cross sections by this fix factor.
118 The processes used to generate multiple interactions form a subset
119 of the standard library of hard processes. The input is slightly
120 different from the standard hard-process machinery, however,
121 since incoming flavours, the <i>alpha_strong</i> value and most
122 of the kinematics are aready fixed when the process is called.
123 It is possible to regulate the set of processes that are included in the
124 multiple-interactions framework.
126 <br/><br/><table><tr><td><strong>MultipleInteractions:processLevel </td><td> (<code>default = <strong>3</strong></code>; <code>minimum = 0</code>; <code>maximum = 3</code>)</td></tr></table>
127 Set of processes included in the machinery.
129 <input type="radio" name="6" value="0"><strong>0 </strong>: only the simplest <ei>2 -> 2</ei> QCD processes between quarks and gluons, giving no new flavours, i.e. dominated by <ei>t</ei>-channel gluon exchange.<br/>
130 <input type="radio" name="6" value="1"><strong>1 </strong>: also <ei>2 -> 2</ei> QCD processes giving new flavours (including charm and bottom), i.e. proceeding through <ei>s</ei>-channel gluon exchange.<br/>
131 <input type="radio" name="6" value="2"><strong>2 </strong>: also <ei>2 -> 2</ei> processes involving one or two photons in the final state, <ei>s</ei>-channel <ei>gamma</ei> boson exchange and <ei>t</ei>-channel <ei>gamma/Z^0/W^+-</ei> boson exchange.<br/>
132 <input type="radio" name="6" value="3" checked="checked"><strong>3 </strong>: also charmonium and bottomonium production, via colour singlet and colour octet channels.<br/>
134 <h4>Cross-section regularization</h4>
136 There are two complementary ways of regularizing the small-<i>pT</i>
137 divergence, a sharp cutoff and a smooth dampening. These can be
138 combined as desired, but it makes sense to coordinate with how the
139 same issue is handled in <?php $filepath = $_GET["filepath"];
140 echo "<a href='SpacelikeShowers.php?filepath=".$filepath."' target='page'>";?>spacelike
141 showers</a>. Actually, by default, the parameters defined here are
142 used also for the spacelike showers, but this can be overridden.
145 Regularization of the divergence of the QCD cross section for
146 <i>pT -> 0</i> is obtained by a factor <i>pT^4 / (pT0^2 + pT^2)^2</i>,
147 and by using an <i>alpha_s(pT0^2 + pT^2)</i>. An energy dependence
148 of the <i>pT0</i> choice is introduced by two further parameters,
149 so that <i>pT0Ref</i> is the <i>pT0</i> value for the reference
150 CM energy, <i>pT0Ref = pT0(ecmRef)</i>.
151 <br/><b>Warning:</b> if a large <i>pT0</i> is picked for multiple
152 interactions, such that the integrated interaction cross section is
153 below the nondiffractive inelastic one, this <i>pT0</i> will
154 automatically be scaled down to cope.
157 The actual <i>pT0</i> parameter used at a given CM energy scale,
158 <i>ecmNow</i>, is obtained as
160 pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
162 where <i>pT0Ref</i>, <i>ecmRef</i> and <i>ecmPow</i> are the
163 three parameters below.
165 <br/><br/><table><tr><td><strong>MultipleInteractions:pT0Ref </td><td></td><td> <input type="text" name="7" value="2.15" size="20"/> (<code>default = <strong>2.15</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10.0</code>)</td></tr></table>
166 The <i>pT0Ref</i> scale in the above formula.
167 <br/><b>Note:</b> <i>pT0Ref</i> is one of the key parameters in a
168 complete PYTHIA tune. Its value is intimately tied to a number of other
169 choices, such as that of colour flow description, so unfortunately it is
170 difficult to give an independent meaning to <i>pT0Ref</i>.
173 <br/><br/><table><tr><td><strong>MultipleInteractions:ecmRef </td><td></td><td> <input type="text" name="8" value="1800.0" size="20"/> (<code>default = <strong>1800.0</strong></code>; <code>minimum = 1.</code>)</td></tr></table>
174 The <i>ecmRef</i> reference energy scale introduced above.
177 <br/><br/><table><tr><td><strong>MultipleInteractions:ecmPow </td><td></td><td> <input type="text" name="9" value="0.24" size="20"/> (<code>default = <strong>0.24</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)</td></tr></table>
178 The <i>ecmPow</i> energy rescaling pace introduced above.
182 Alternatively, or in combination, a sharp cut can be used.
183 <br/><br/><table><tr><td><strong>MultipleInteractions:pTmin </td><td></td><td> <input type="text" name="10" value="0.2" size="20"/> (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.0</code>)</td></tr></table>
184 Lower cutoff in <i>pT</i>, below which no further interactions
185 are allowed. Normally <i>pT0</i> above would be used to provide
186 the main regularization of the cross section for <i>pT -> 0</i>,
187 in which case <i>pTmin</i> is used mainly for technical reasons.
188 It is possible, however, to set <i>pT0Ref = 0</i> and use
189 <i>pTmin</i> to provide a step-function regularization, or to
190 combine them in intermediate approaches. Currently <i>pTmin</i>
191 is taken to be energy-independent.
195 Gösta Gustafson has proposed (private communication, unpublished)
196 that the amount of screening, as encapsulated in the <i>pT0</i>
197 parameter, fluctuates from one event to the next. Specifically,
198 high-activity event are more likely to lead to interactions at large
199 <i>pT</i> scales, but the high activity simultaneously leads to a
200 larger screening of interactions at smaller <i>pT</i>. Such a scenario
201 can approximately be simulated by scaling up the <i>pT0</i> by a
202 factor <i>sqrt(n)</i>, where <i>n</i> is the number of interactions
203 considered so far, including the current one. That is, for the first
204 interaction the dampening factor is <i>pT^4 / (pT0^2 + pT^2)^2</i>,
205 for the second <i>pT^4 / (2 pT0^2 + pT^2)^2</i>, for the third
206 <i>pT^4 / (3 pT0^2 + pT^2)^2</i>, and so on. Optionally the scheme
207 may also be applied to ISR emissions. For simplicity the same
208 <i>alpha_s(pT0^2 + pT^2)</i> is used throughout. Note that, in this
209 scenario the <i>pT0</i> scale must be lower than in the normal case
210 to begin with, since it later is increased back up. Also note that the
211 idea with this scenario is to propose an alternative to colour
212 reconnection to understand the rise of <i><pT>(n_ch)</i>,
213 so that the amount of colour reconnection should be reduced.
214 <br/><br/><table><tr><td><strong>MultipleInteractions:enhanceScreening </td><td> (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
215 Choice to activate the above screening scenario, i.e. an increasing
216 effective <ei>pT0</ei> for consecutive interactions.
218 <input type="radio" name="11" value="0" checked="checked"><strong>0 </strong>: No activity-dependent screening, i.e. <ei>pT0</ei> is fixed.<br/>
219 <input type="radio" name="11" value="1"><strong>1 </strong>: The <ei>pT0</ei> scale is increased as a function of the number of MI's, as explained above. ISR is not affected, but note that, if <code>SpaceShower:samePTasMI</code> is on, then <code>MultipleInteractions:pT0Ref</code> is used also for ISR, which may or may not be desirable. <br/>
220 <input type="radio" name="11" value="2"><strong>2 </strong>: Both MI and ISR influence and are influenced by the screening. That is, the dampening is reduced based on the total number of MI and ISR steps considered so far, including the current one. This dampening is implemented both for MI and for ISR emissions, for the latter provided that <code>SpaceShower:samePTasMI</code> is on (default). <br/>
222 <h4>Impact-parameter dependence</h4>
224 The choice of impact-parameter dependence is regulated by several
227 <br/><br/><table><tr><td><strong>MultipleInteractions:bProfile </td><td> (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 3</code>)</td></tr></table>
228 Choice of impact parameter profile for the incoming hadron beams.
230 <input type="radio" name="12" value="0"><strong>0 </strong>: no impact parameter dependence at all.<br/>
231 <input type="radio" name="12" value="1" checked="checked"><strong>1 </strong>: a simple Gaussian matter distribution; no free parameters.<br/>
232 <input type="radio" name="12" value="2"><strong>2 </strong>: a double Gaussian matter distribution, with the two free parameters <ei>coreRadius</ei> and <ei>coreFraction</ei>.<br/>
233 <input type="radio" name="12" value="3"><strong>3 </strong>: an overlap function, i.e. the convolution of the matter distributions of the two incoming hadrons, of the form <ei>exp(- b^expPow)</ei>, where <ei>expPow</ei> is a free parameter.<br/>
235 <br/><br/><table><tr><td><strong>MultipleInteractions:coreRadius </td><td></td><td> <input type="text" name="13" value="0.4" size="20"/> (<code>default = <strong>0.4</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 1.</code>)</td></tr></table>
236 When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
237 the inner core is assumed to have a radius that is a factor
238 <i>coreRadius</i> smaller than the rest.
241 <br/><br/><table><tr><td><strong>MultipleInteractions:coreFraction </td><td></td><td> <input type="text" name="14" value="0.5" size="20"/> (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>; <code>maximum = 1.</code>)</td></tr></table>
242 When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
243 the inner core is assumed to have a fraction <i>coreFraction</i>
244 of the matter content of the hadron.
247 <br/><br/><table><tr><td><strong>MultipleInteractions:expPow </td><td></td><td> <input type="text" name="15" value="1." size="20"/> (<code>default = <strong>1.</strong></code>; <code>minimum = 0.4</code>; <code>maximum = 10.</code>)</td></tr></table>
248 When <i>bProfile = 3</i> it gives the power of the assumed overlap
249 shape <i>exp(- b^expPow)</i>. Default corresponds to a simple
250 exponential drop, which is not too dissimilar from the overlap
251 obtained with the standard double Gaussian parameters. For
252 <i>expPow = 2</i> we reduce to the simple Gaussian, <i>bProfile = 1</i>,
253 and for <i>expPow -> infinity</i> to no impact parameter dependence
254 at all, <i>bProfile = 0</i>. For small <i>expPow</i> the program
255 becomes slow and unstable, so the min limit must be respected.
258 <h4>Rescattering</h4>
260 It is possible that a parton may rescatter, i.e. undergo a further
261 interaction subsequent to the first one. The machinery to model this
262 kind of physics has only recently become fully operational
263 [<a href="Bibliography.php" target="page">Cor09</a>], and is therefore not yet so well explored.
266 The rescatting framework has ties with other parts of the program,
267 notably with the <?php $filepath = $_GET["filepath"];
268 echo "<a href='BeamRemnants.php?filepath=".$filepath."' target='page'>";?>beam remnants</a>.
270 <br/><br/><strong>MultipleInteractions:allowRescatter</strong> <input type="radio" name="16" value="on"><strong>On</strong>
271 <input type="radio" name="16" value="off" checked="checked"><strong>Off</strong>
272 (<code>default = <strong>off</strong></code>)<br/>
273 Switch to allow rescattering of partons; on/off = true/false.<br/>
274 <b>Warning:</b> use with caution since machinery is still not
278 <br/><br/><strong>MultipleInteractions:allowDoubleRescatter</strong> <input type="radio" name="17" value="on"><strong>On</strong>
279 <input type="radio" name="17" value="off" checked="checked"><strong>Off</strong>
280 (<code>default = <strong>off</strong></code>)<br/>
281 Switch to allow rescattering of partons, where both incoming partons
282 have already rescattered; on/off = true/false. Is only used if
283 <code>MultipleInteractions:allowRescatter</code> is switched on.<br/>
284 <b>Warning:</b> currently there is no complete implementation that
285 combines it with shower evolution, so you must use
286 <code>PartonLevel:ISR = off</code> and <code>PartonLevel:FSR = off</code>.
287 If not, a warning will be issued and double rescattering will not be
288 simulated. The rate also comes out to be much lower than for single
289 rescattering, so to first approximation it can be neglected.
292 <br/><br/><table><tr><td><strong>MultipleInteractions:rescatterMode </td><td> (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)</td></tr></table>
293 Selection of which partons rescatter against unscattered partons
294 from the incoming beams A and B, based on their rapidity value
295 <ei>y</ei> in the collision rest frame. Here <ei>ySep</ei> is
296 shorthand for <code>MultipleInteractions:ySepRescatter</code> and
297 <ei>deltaY</ei> for <code>MultipleInteractions:deltaYRescatter</code>,
298 defined below. The description is symmetric between the two beams,
299 so only one case is described below.
301 <input type="radio" name="18" value="0" checked="checked"><strong>0 </strong>: only scattered partons with <ei>y > 0</ei> can collide with unscattered partons from beam B.<br/>
302 <input type="radio" name="18" value="1"><strong>1 </strong>: only scattered partons with <ei>y > ySep</ei> can collide with unscattered partons from beam B.<br/>
303 <input type="radio" name="18" value="2"><strong>2 </strong>: the probability for a scattered parton to be considered as a potential rescatterer against unscattered partons in beam B increases linearly from zero at <ei>y = ySep - deltaY</ei> to unity at <ei>y = ySep + deltaY</ei>.<br/>
304 <input type="radio" name="18" value="3"><strong>3 </strong>: the probability for a scattered parton to be considered as a potential rescatterer against unscattered partons in beam B increases with <ei>y</ei> according to <ei>(1/2) * (1 + tanh( (y - ySep) / deltaY))</ei>.<br/>
305 <input type="radio" name="18" value="4"><strong>4 </strong>: all partons are potential rescatterers against both beams.<br/>
307 <br/><br/><table><tr><td><strong>MultipleInteractions:ySepRescatter </td><td></td><td> <input type="text" name="19" value="0." size="20"/> (<code>default = <strong>0.</strong></code>)</td></tr></table>
308 used for some of the <code>MultipleInteractions:rescatterMode</code>
309 options above, as the rapidity for which a scattered parton has a 50%
310 probability to be considered as a potential rescatterer.
311 A <i>ySep > 0</i> generally implies that some central partons cannot
312 rescatter at all, while a <i>ySep < 0</i> instead allows central
313 partons to scatter against either beam.
316 <br/><br/><table><tr><td><strong>MultipleInteractions:deltaYRescatter </td><td></td><td> <input type="text" name="20" value="1." size="20"/> (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>)</td></tr></table>
317 used for some of the <code>MultipleInteractions:rescatterMode</code>
318 options above, as the width of the rapidity transition region, where the
319 probability rises from zero to unity that a scattered parton is considered
320 as a potential rescatterer.
324 <h3>Further variables</h3>
326 These should normally not be touched. Their only function is for
329 <br/><br/><table><tr><td><strong>MultipleInteractions:nQuarkIn </td><td></td><td> <input type="text" name="21" value="5" size="20"/> (<code>default = <strong>5</strong></code>; <code>minimum = 0</code>; <code>maximum = 5</code>)</td></tr></table>
330 Number of allowed incoming quark flavours in the beams; a change
331 to 4 would thus exclude <i>b</i> and <i>bbar</i> as incoming
335 <br/><br/><table><tr><td><strong>MultipleInteractions:nSample </td><td></td><td> <input type="text" name="22" value="1000" size="20"/> (<code>default = <strong>1000</strong></code>; <code>minimum = 100</code>)</td></tr></table>
336 The allowed <i>pT</i> range is split (unevenly) into 100 bins,
337 and in each of these the interaction cross section is evaluated in
338 <i>nSample</i> random phase space points. The full integral is used
339 at initialization, and the differential one during the run as a
340 "Sudakov form factor" for the choice of the hardest interaction.
341 A larger number implies increased accuracy of the calculations.
344 <h3>Technical notes</h3>
346 Relative to the articles mentioned above, not much has happened.
347 The main news is a technical one, that the phase space of the
348 <i>2 -> 2</i> (massless) QCD processes is now sampled in
349 <i>dy_3 dy_4 dpT^2</i>, where <i>y_3</i> and <i>y_4</i> are
350 the rapidities of the two produced partons. One can show that
352 (dx_1 / x_1) * (dx_2 / x_2) * d(tHat) = dy_3 * dy_4 * dpT^2
354 Furthermore, since cross sections are dominated by the "Rutherford"
355 one of <i>t</i>-channel gluon exchange, which is enhanced by a
356 factor of 9/4 for each incoming gluon, effective structure functions
359 F(x, pT2) = (9/4) * xg(x, pT2) + sum_i xq_i(x, pT2)
361 With this technical shift of factors 9/4 from cross sections to parton
362 densities, a common upper estimate of
364 d(sigmaHat)/d(pT2) < pi * alpha_strong^2 / pT^4
369 In fact this estimate can be reduced by a factor of 1/2 for the
370 following reason: for any configuration <i>(y_3, y_4, pT2)</i> also
371 one with <i>(y_4, y_3, pT2)</i> lies in the phase space. Not both
372 of those can enjoy being enhanced by the <i>tHat -> 0</i>
375 d(sigmaHat) propto 1/tHat^2.
377 Or if they are, which is possible with identical partons like
378 <i>q q -> q q</i> and <i>g g -> g g</i>, each singularity comes
379 with half the strength. So, when integrating/averaging over the two
380 configurations, the estimated <i>d(sigmaHat)/d(pT2)</i> drops.
381 Actually, it drops even further, since the naive estimate above is
384 (4 /9) * (1 + (uHat/sHat)^2) < 8/9 < 1
386 The 8/9 value would be approached for <i>tHat -> 0</i>, which
387 implies <i>sHat >> pT2</i> and thus a heavy parton-distribution
388 penalty, while parton distributions are largest for
389 <i>tHat = uHat = -sHat/2</i>, where the above expression
390 evaluates to 5/9. A fudge factor is therefore introduced to go the
391 final step, so it can easily be modifed when further non-Rutherford
392 processes are added, or should parton distributions change significantly.
395 At initialization, it is assumed that
397 d(sigma)/d(pT2) < d(sigmaHat)/d(pT2) * F(x_T, pT2) * F(x_T, pT2)
400 where the first factor is the upper estimate as above, the second two
401 the parton density sum evaluated at <i>y_3 = y_ 4 = 0</i> so that
402 <i>x_1 = x_2 = x_T = 2 pT / E_cm</i>, where the product is expected
403 to be maximal, and the final is the phase space for
404 <i>-y_max < y_{3,4} < y_max</i>.
405 The right-hand side expression is scanned logarithmically in <i>y</i>,
406 and a <i>N</i> is determined such that it always is below
410 To describe the dampening of the cross section at <i>pT -> 0</i> by
411 colour screening, the actual cross section is multiplied by a
412 regularization factor <i>(pT^2 / (pT^2 + pT0^2))^2</i>, and the
413 <i>alpha_s</i> is evaluated at a scale <i>pT^2 + pT0^2</i>,
414 where <i>pT0</i> is a free parameter of the order of 2 - 4 GeV.
415 Since <i>pT0</i> can be energy-dependent, an ansatz
417 pT0(ecm) = pT0Ref * (ecm/ecmRef)^ecmPow
419 is used, where <i>ecm</i> is the current CM frame energy,
420 <i>ecmRef</i> is an arbitrary reference energy where <i>pT0Ref</i>
421 is defined, and <i>ecmPow</i> gives the energy rescaling pace. For
422 technical reasons, also an absolute lower <i>pT</i> scale <i>pTmin</i>,
423 by default 0.2 GeV, is introduced. In principle, it is possible to
424 recover older scenarios with a sharp <i>pT</i> cutoff by setting
425 <i>pT0 = 0</i> and letting <i>pTmin</i> be a larger number.
428 The above scanning strategy is then slightly modified: instead of
429 an upper estimate <i>N/pT^4</i> one of the form
430 <i>N/(pT^2 + r * pT0^2)^2</i> is used. At first glance, <i>r = 1</i>
431 would seem to be fixed by the form of the regularization procedure,
432 but this does not take into account the nontrivial dependence on
433 <i>alpha_s</i>, parton distributions and phase space. A better
434 Monte Carlo efficiency is obtained for <i>r</i> somewhat below unity,
435 and currently <i>r = 0.25</i> is hardcoded.
437 In the generation a trial <i>pT2</i> is then selected according to
439 d(Prob)/d(pT2) = (1/sigma_ND) * N/(pT^2 + r * pT0^2)^2 * ("Sudakov")
441 For the trial <i>pT2</i>, a <i>y_3</i> and a <i>y_4</i> are then
442 selected, and incoming flavours according to the respective
443 <i>F(x_i, pT2)</i>, and then the cross section is evaluated for this
444 flavour combination. The ratio of trial/upper estimate gives the
445 probability of survival.
448 Actually, to profit from the factor 1/2 mentioned above, the cross
449 section for the combination with <i>y_3</i> and <i>y_4</i>
450 interchanged is also tried, which corresponds to exchanging <i>tHat</i>
451 and <i>uHat</i>, and the average formed, while the final kinematics
452 is given by the relative importance of the two.
455 Furthermore, since large <i>y</i> values are disfavoured by dropping
458 WT_y = (1 - (y_3/y_max)^2) * (1 - (y_4/y_max)^2)
460 is evaluated, and used as a survival probability before the more
461 time-consuming PDF+ME evaluation, with surviving events given a
462 compensating weight <i>1/WT_y</i>.
465 An impact-parameter dependencs is also allowed. Based on the hard
466 <i>pT</i> scale of the first interaction, and enhancement/depletion
467 factor is picked, which multiplies the rate of subsequent interactions.
470 Parton densities are rescaled and modified to take into account the
471 energy-momentum and flavours kicked out by already-considered
474 <input type="hidden" name="saved" value="1"/>
477 echo "<input type='hidden' name='filepath' value='".$_GET["filepath"]."'/>"?>
479 <table width="100%"><tr><td align="right"><input type="submit" value="Save Settings" /></td></tr></table>
484 if($_POST["saved"] == 1)
486 $filepath = $_POST["filepath"];
487 $handle = fopen($filepath, 'a');
489 if($_POST["1"] != "0")
491 $data = "MultipleInteractions:pTmaxMatch = ".$_POST["1"]."\n";
492 fwrite($handle,$data);
494 if($_POST["2"] != "0.127")
496 $data = "MultipleInteractions:alphaSvalue = ".$_POST["2"]."\n";
497 fwrite($handle,$data);
499 if($_POST["3"] != "1")
501 $data = "MultipleInteractions:alphaSorder = ".$_POST["3"]."\n";
502 fwrite($handle,$data);
504 if($_POST["4"] != "1")
506 $data = "MultipleInteractions:alphaEMorder = ".$_POST["4"]."\n";
507 fwrite($handle,$data);
509 if($_POST["5"] != "1.0")
511 $data = "MultipleInteractions:Kfactor = ".$_POST["5"]."\n";
512 fwrite($handle,$data);
514 if($_POST["6"] != "3")
516 $data = "MultipleInteractions:processLevel = ".$_POST["6"]."\n";
517 fwrite($handle,$data);
519 if($_POST["7"] != "2.15")
521 $data = "MultipleInteractions:pT0Ref = ".$_POST["7"]."\n";
522 fwrite($handle,$data);
524 if($_POST["8"] != "1800.0")
526 $data = "MultipleInteractions:ecmRef = ".$_POST["8"]."\n";
527 fwrite($handle,$data);
529 if($_POST["9"] != "0.24")
531 $data = "MultipleInteractions:ecmPow = ".$_POST["9"]."\n";
532 fwrite($handle,$data);
534 if($_POST["10"] != "0.2")
536 $data = "MultipleInteractions:pTmin = ".$_POST["10"]."\n";
537 fwrite($handle,$data);
539 if($_POST["11"] != "0")
541 $data = "MultipleInteractions:enhanceScreening = ".$_POST["11"]."\n";
542 fwrite($handle,$data);
544 if($_POST["12"] != "1")
546 $data = "MultipleInteractions:bProfile = ".$_POST["12"]."\n";
547 fwrite($handle,$data);
549 if($_POST["13"] != "0.4")
551 $data = "MultipleInteractions:coreRadius = ".$_POST["13"]."\n";
552 fwrite($handle,$data);
554 if($_POST["14"] != "0.5")
556 $data = "MultipleInteractions:coreFraction = ".$_POST["14"]."\n";
557 fwrite($handle,$data);
559 if($_POST["15"] != "1.")
561 $data = "MultipleInteractions:expPow = ".$_POST["15"]."\n";
562 fwrite($handle,$data);
564 if($_POST["16"] != "off")
566 $data = "MultipleInteractions:allowRescatter = ".$_POST["16"]."\n";
567 fwrite($handle,$data);
569 if($_POST["17"] != "off")
571 $data = "MultipleInteractions:allowDoubleRescatter = ".$_POST["17"]."\n";
572 fwrite($handle,$data);
574 if($_POST["18"] != "0")
576 $data = "MultipleInteractions:rescatterMode = ".$_POST["18"]."\n";
577 fwrite($handle,$data);
579 if($_POST["19"] != "0.")
581 $data = "MultipleInteractions:ySepRescatter = ".$_POST["19"]."\n";
582 fwrite($handle,$data);
584 if($_POST["20"] != "1.")
586 $data = "MultipleInteractions:deltaYRescatter = ".$_POST["20"]."\n";
587 fwrite($handle,$data);
589 if($_POST["21"] != "5")
591 $data = "MultipleInteractions:nQuarkIn = ".$_POST["21"]."\n";
592 fwrite($handle,$data);
594 if($_POST["22"] != "1000")
596 $data = "MultipleInteractions:nSample = ".$_POST["22"]."\n";
597 fwrite($handle,$data);
606 <!-- Copyright (C) 2010 Torbjorn Sjostrand -->