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29
30<h2>Multiparton Interactions</h2>
31
32The starting point for the multiparton interactions physics scenario in
33PYTHIA is provided by [<a href="Bibliography.php" target="page">Sjo87</a>]. Recent developments have
34included a more careful study of flavour and colour correlations,
35junction 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
38initial- and final-state showers, with the extension to fully
39interleaved evolution covered in [<a href="Bibliography.php" target="page">Cor10a</a>]. A framework to
40handle rescattering is described in [<a href="Bibliography.php" target="page">Cor09</a>].
41
42<p/>
43A big unsolved issue is how the colour of all these subsystems is
44correlated. For sure there is a correlation coming from the colour
45singlet nature of the incoming beams, but in addition final-state
46colour rearrangements may change the picture. Indeed such extra
47effects appear necessary to describe data, e.g. on
48<i>&lt;pT&gt;(n_ch)</i>. A simple implementation of colour
49rearrangement is found as part of the
50<?php $filepath = $_GET["filepath"];
51echo "<a href='BeamRemnants.php?filepath=".$filepath."' target='page'>";?>beam remnants</a> description.
52
53<h3>Main variables</h3>
54
55<h4>Matching to hard process</h4>
56
57The maximum <i>pT</i> to be allowed for multiparton interactions is
58related to the nature of the hard process itself. It involves a
59delicate balance between not doublecounting and not leaving any
60gaps in the coverage. The best procedure may depend on information
61only the user has: how the events were generated and mixed (e.g. with
62Les Houches Accord external input), and how they are intended to be
63used. Therefore a few options are available, with a sensible default
64behaviour.
65<br/><br/><table><tr><td><strong>MultipartonInteractions:pTmaxMatch </td><td>  &nbsp;&nbsp;(<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
66Way in which the maximum scale for multiparton interactions is set
67to match the scale of the hard process itself.
68<br/>
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 multiparton-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 multiparton interactions up to the  kinematical limit. This will simulate all possible event topologies,  but may lead to doublecounting. <br/>
72
73<h4>Cross-section parameters</h4>
74
75The rate of interactions is determined by
76<br/><br/><table><tr><td><strong>MultipartonInteractions:alphaSvalue </td><td></td><td> <input type="text" name="2" value="0.127" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.127</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)</td></tr></table>
77The value of <i>alpha_strong</i> at <i>m_Z</i>. Default value is
78picked equal to the one used in CTEQ 5L.
79   
80
81<p/>
82The 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>MultipartonInteractions:alphaSorder </td><td>  &nbsp;&nbsp;(<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
85The order at which <ei>alpha_strong</ei> runs at scales away from
86<ei>m_Z</ei>.
87<br/>
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/>
91
92<p/>
93QED interactions are regulated by the <i>alpha_electromagnetic</i>
94value at the <i>pT^2</i> scale of an interaction.
95 
96<br/><br/><table><tr><td><strong>MultipartonInteractions:alphaEMorder </td><td>  &nbsp;&nbsp;(<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)</td></tr></table>
97The running of <ei>alpha_em</ei> used in hard processes.
98<br/>
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/>
102
103<p/>
104Note that the choices of <i>alpha_strong</i> and <i>alpha_em</i>
105made here override the ones implemented in the normal process machinery,
106but only for the interactions generated by the
107<code>MultipartonInteractions</code> class.
108
109<p/>
110In addition there is the possibility of a global rescaling of
111cross sections (which could not easily be accommodated by a
112changed <i>alpha_strong</i>, since <i>alpha_strong</i> runs)
113<br/><br/><table><tr><td><strong>MultipartonInteractions:Kfactor </td><td></td><td> <input type="text" name="5" value="1.0" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 4.0</code>)</td></tr></table>
114Multiply all cross sections by this fix factor.
115 
116
117<p/> 
118The processes used to generate multiparton interactions form a subset
119of the standard library of hard processes. The input is slightly
120different from the standard hard-process machinery, however,
121since incoming flavours, the <i>alpha_strong</i> value and most
122of the kinematics are aready fixed when the process is called.
123It is possible to regulate the set of processes that are included in the
124multiparton-interactions framework.
125
126<br/><br/><table><tr><td><strong>MultipartonInteractions:processLevel </td><td>  &nbsp;&nbsp;(<code>default = <strong>3</strong></code>; <code>minimum = 0</code>; <code>maximum = 3</code>)</td></tr></table>
127Set of processes included in the machinery.
128<br/>
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/>
133
134<h4>Cross-section regularization</h4>
135
136There are two complementary ways of regularizing the small-<i>pT</i>
137divergence, a sharp cutoff and a smooth dampening. These can be
138combined as desired, but it makes sense to coordinate with how the
139same issue is handled in <?php $filepath = $_GET["filepath"];
140echo "<a href='SpacelikeShowers.php?filepath=".$filepath."' target='page'>";?>spacelike
141showers</a>. Actually, by default, the parameters defined here are
142used also for the spacelike showers, but this can be overridden.
143
144<p/>
145Regularization 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>,
147and by using an <i>alpha_s(pT0^2 + pT^2)</i>. An energy dependence
148of the <i>pT0</i> choice is introduced by two further parameters,
149so that <i>pT0Ref</i> is the <i>pT0</i> value for the reference
150CM energy, <i>pT0Ref = pT0(ecmRef)</i>.   
151<br/><b>Warning:</b> if a large <i>pT0</i> is picked for multiparton
152interactions, such that the integrated interaction cross section is
153below the nondiffractive inelastic one, this <i>pT0</i> will
154automatically be scaled down to cope.
155
156<p/>
157The actual <i>pT0</i> parameter used at a given CM energy scale,
158<i>ecmNow</i>, is obtained as
159<br/><i>
160     pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
161</i><br/>
162where <i>pT0Ref</i>, <i>ecmRef</i> and <i>ecmPow</i> are the
163three parameters below.
164
165<br/><br/><table><tr><td><strong>MultipartonInteractions:pT0Ref </td><td></td><td> <input type="text" name="7" value="2.15" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>2.15</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10.0</code>)</td></tr></table>
166The <i>pT0Ref</i> scale in the above formula.
167<br/><b>Note:</b> <i>pT0Ref</i> is one of the key parameters in a
168complete PYTHIA tune. Its value is intimately tied to a number of other
169choices, such as that of colour flow description, so unfortunately it is
170difficult to give an independent meaning to <i>pT0Ref</i>.
171 
172
173<br/><br/><table><tr><td><strong>MultipartonInteractions:ecmRef </td><td></td><td> <input type="text" name="8" value="1800.0" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1800.0</strong></code>; <code>minimum = 1.</code>)</td></tr></table>
174The <i>ecmRef</i> reference energy scale introduced above.
175 
176
177<br/><br/><table><tr><td><strong>MultipartonInteractions:ecmPow </td><td></td><td> <input type="text" name="9" value="0.24" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.24</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)</td></tr></table>
178The <i>ecmPow</i> energy rescaling pace introduced above.
179 
180
181<p/>
182Alternatively, or in combination, a sharp cut can be used.
183<br/><br/><table><tr><td><strong>MultipartonInteractions:pTmin </td><td></td><td> <input type="text" name="10" value="0.2" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.2</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.0</code>)</td></tr></table>
184Lower cutoff in <i>pT</i>, below which no further interactions
185are allowed. Normally <i>pT0</i> above would be used to provide
186the main regularization of the cross section for <i>pT -> 0</i>,
187in which case <i>pTmin</i> is used  mainly for technical reasons.
188It is possible, however, to set <i>pT0Ref = 0</i> and use
189<i>pTmin</i> to provide a step-function regularization, or to
190combine them in intermediate approaches. Currently <i>pTmin</i>
191is taken to be energy-independent. 
192 
193
194<p/>
195G&ouml;sta Gustafson has proposed (private communication, unpublished)
196that the amount of screening, as encapsulated in the <i>pT0</i>
197parameter, fluctuates from one event to the next. Specifically,
198high-activity event are more likely to lead to interactions at large
199<i>pT</i> scales, but the high activity simultaneously leads to a
200larger screening of interactions at smaller <i>pT</i>. Such a scenario
201can approximately be simulated by scaling up the <i>pT0</i> by a
202factor <i>sqrt(n)</i>, where <i>n</i> is the number of interactions
203considered so far, including the current one. That is, for the first
204interaction the dampening factor is <i>pT^4 / (pT0^2 + pT^2)^2</i>,
205for 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
207may 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
209scenario the <i>pT0</i> scale must be lower than in the normal case
210to begin with, since it later is increased back up. Also note that the
211idea with this scenario is to propose an alternative to colour
212reconnection to understand the rise of <i>&lt;pT&gt;(n_ch)</i>,
213so that the amount of colour reconnection should be reduced.
214<br/><br/><table><tr><td><strong>MultipartonInteractions:enhanceScreening </td><td>  &nbsp;&nbsp;(<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)</td></tr></table>
215Choice to activate the above screening scenario, i.e. an increasing
216effective <ei>pT0</ei> for consecutive interactions.
217<br/>
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 MPI's, as explained above. ISR is not affected, but note that, if <code>SpaceShower:samePTasMPI</code> is on, then <code>MultipartonInteractions: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 MPI and ISR influence and are influenced by the screening. That is, the dampening is reduced based on the total number of MPI and ISR steps considered so far, including the current one. This dampening is implemented both for MPI and for ISR emissions, for the latter provided that <code>SpaceShower:samePTasMPI</code> is on  (default).   <br/>
221
222<h4>Impact-parameter dependence</h4>
223 
224The choice of impact-parameter dependence is regulated by several
225parameters. The ones listed here refer to nondiffractive topologies
226only, while their equivalents for diffractive events are put in the
227<?php $filepath = $_GET["filepath"];
228echo "<a href='Diffraction.php?filepath=".$filepath."' target='page'>";?>Diffraction</a> description. Note that
229there is currently no <code>bProfile = 4</code> option for diffraction.
230Other parameters are assumed to agree between diffractive and
231nondiffractive topologies.
232
233<br/><br/><table><tr><td><strong>MultipartonInteractions:bProfile </td><td>  &nbsp;&nbsp;(<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)</td></tr></table>
234Choice of impact parameter profile for the incoming hadron beams.
235<br/>
236<input type="radio" name="12" value="0"><strong>0 </strong>: no impact parameter dependence at all.<br/>
237<input type="radio" name="12" value="1" checked="checked"><strong>1 </strong>: a simple Gaussian matter distribution;  no free parameters.<br/>
238<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/>
239<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/>
240<input type="radio" name="12" value="4"><strong>4 </strong>: a Gaussian matter distribution with a width that varies according to the selected <ei>x</ei> value of an interaction, <ei>1. + a1 log (1 / x)</ei>, where <ei>a1</ei> is a free parameter. Note that once <ei>b</ei> has been selected for the hard process, it remains fixed for the remainder of the evolution. <br/>
241
242<br/><br/><table><tr><td><strong>MultipartonInteractions:coreRadius </td><td></td><td> <input type="text" name="13" value="0.4" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.4</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 1.</code>)</td></tr></table>
243When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
244the inner core is assumed to have a radius that is a factor
245<i>coreRadius</i> smaller than the rest.
246   
247
248<br/><br/><table><tr><td><strong>MultipartonInteractions:coreFraction </td><td></td><td> <input type="text" name="14" value="0.5" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>; <code>maximum = 1.</code>)</td></tr></table>
249When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
250the inner core is assumed to have a fraction <i>coreFraction</i>
251of the matter content of the hadron.
252   
253
254<br/><br/><table><tr><td><strong>MultipartonInteractions:expPow </td><td></td><td> <input type="text" name="15" value="1." size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1.</strong></code>; <code>minimum = 0.4</code>; <code>maximum = 10.</code>)</td></tr></table>
255When <i>bProfile = 3</i> it gives the power of the assumed overlap
256shape <i>exp(- b^expPow)</i>. Default corresponds to a simple
257exponential drop, which is not too dissimilar from the overlap
258obtained with the standard double Gaussian parameters. For
259<i>expPow = 2</i> we reduce to the simple Gaussian, <i>bProfile = 1</i>,
260and for <i>expPow -> infinity</i> to no impact parameter dependence
261at all, <i>bProfile = 0</i>. For small <i>expPow</i> the program
262becomes slow and unstable, so the min limit must be respected.
263   
264
265<br/><br/><table><tr><td><strong>MultipartonInteractions:a1 </td><td></td><td> <input type="text" name="16" value="0.15" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.15</strong></code>; <code>minimum = 0.</code>; <code>maximum = 2.</code>)</td></tr></table>
266When <i>bProfile = 4</i>, this gives the <i>a1</i> constant in the
267Gaussian width. When <i>a1 = 0.</i>, this reduces back to the single
268Gaussian case.
269 
270
271<h4>Rescattering</h4>
272 
273It is possible that a parton may rescatter, i.e. undergo a further
274interaction subsequent to the first one. The machinery to model this
275kind of physics has only recently become fully operational
276[<a href="Bibliography.php" target="page">Cor09</a>], and is therefore not yet so well explored.
277
278<p/>
279The rescatting framework has ties with other parts of the program,
280notably with the <?php $filepath = $_GET["filepath"];
281echo "<a href='BeamRemnants.php?filepath=".$filepath."' target='page'>";?>beam remnants</a>.
282
283<br/><br/><strong>MultipartonInteractions:allowRescatter</strong>  <input type="radio" name="17" value="on"><strong>On</strong>
284<input type="radio" name="17" value="off" checked="checked"><strong>Off</strong>
285 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
286Switch to allow rescattering of partons; on/off = true/false.<br/>
287<b>Note:</b> the rescattering framework has not yet been implemented
288for the <code>MultipartonInteractions:bProfile = 4</code> option,
289and can therefore not be switched on in that case. 
290<b>Warning:</b> use with caution since machinery is still not
291so well tested.
292 
293
294<br/><br/><strong>MultipartonInteractions:allowDoubleRescatter</strong>  <input type="radio" name="18" value="on"><strong>On</strong>
295<input type="radio" name="18" value="off" checked="checked"><strong>Off</strong>
296 &nbsp;&nbsp;(<code>default = <strong>off</strong></code>)<br/>
297Switch to allow rescattering of partons, where both incoming partons
298have already rescattered; on/off = true/false. Is only used if
299<code>MultipartonInteractions:allowRescatter</code> is switched on.<br/>
300<b>Warning:</b> currently there is no complete implementation that
301combines it with shower evolution, so you must use
302<code>PartonLevel:ISR = off</code> and <code>PartonLevel:FSR = off</code>.
303If not, a warning will be issued and double rescattering will not be
304simulated. The rate also comes out to be much lower than for single
305rescattering, so to first approximation it can be neglected.
306 
307
308<br/><br/><table><tr><td><strong>MultipartonInteractions:rescatterMode </td><td>  &nbsp;&nbsp;(<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)</td></tr></table>
309Selection of which partons rescatter against unscattered partons
310from the incoming beams A and B, based on their rapidity value
311<ei>y</ei> in the collision rest frame. Here <ei>ySep</ei> is
312shorthand for <code>MultipartonInteractions:ySepRescatter</code> and
313<ei>deltaY</ei> for <code>MultipartonInteractions:deltaYRescatter</code>,
314defined below. The description is symmetric between the two beams,
315so only one case is described below.
316<br/>
317<input type="radio" name="19" value="0" checked="checked"><strong>0 </strong>: only scattered partons with <ei>y > 0</ei>  can collide with unscattered partons from beam B.<br/>
318<input type="radio" name="19" value="1"><strong>1 </strong>: only scattered partons with <ei>y > ySep</ei>  can collide with unscattered partons from beam B.<br/>
319<input type="radio" name="19" 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/>
320<input type="radio" name="19" 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/>
321<input type="radio" name="19" value="4"><strong>4 </strong>: all partons are potential rescatterers against both  beams.<br/>
322
323<br/><br/><table><tr><td><strong>MultipartonInteractions:ySepRescatter </td><td></td><td> <input type="text" name="20" value="0." size="20"/>  &nbsp;&nbsp;(<code>default = <strong>0.</strong></code>)</td></tr></table>
324used for some of the <code>MultipartonInteractions:rescatterMode</code>
325options above, as the rapidity for which a scattered parton has a 50%
326probability to be considered as a potential rescatterer.
327A <i>ySep > 0</i> generally implies that some central partons cannot
328rescatter at all, while a <i>ySep < 0</i>  instead allows central
329partons to scatter against either beam.
330   
331
332<br/><br/><table><tr><td><strong>MultipartonInteractions:deltaYRescatter </td><td></td><td> <input type="text" name="21" value="1." size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>)</td></tr></table>
333used for some of the <code>MultipartonInteractions:rescatterMode</code>
334options above, as the width of the rapidity transition region, where the
335probability rises from zero to unity that a scattered parton is considered
336as a potential rescatterer.
337   
338
339
340<h3>Further variables</h3>
341
342These should normally not be touched. Their only function is for
343cross-checks.
344
345<br/><br/><table><tr><td><strong>MultipartonInteractions:nQuarkIn </td><td></td><td> <input type="text" name="22" value="5" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>5</strong></code>; <code>minimum = 0</code>; <code>maximum = 5</code>)</td></tr></table>
346Number of allowed incoming quark flavours in the beams; a change
347to 4 would thus exclude <i>b</i> and <i>bbar</i> as incoming
348partons, etc.
349 
350
351<br/><br/><table><tr><td><strong>MultipartonInteractions:nSample </td><td></td><td> <input type="text" name="23" value="1000" size="20"/>  &nbsp;&nbsp;(<code>default = <strong>1000</strong></code>; <code>minimum = 100</code>)</td></tr></table>
352The allowed <i>pT</i> range is split (unevenly) into 100 bins,
353and in each of these the interaction cross section is evaluated in
354<i>nSample</i> random phase space points. The full integral is used
355at initialization, and the differential one during the run as a
356"Sudakov form factor" for the choice of the hardest interaction.
357A larger number implies increased accuracy of the calculations.
358 
359
360<h3>Technical notes</h3>
361
362Relative to the articles mentioned above, not much has happened.
363The main news is a technical one, that the phase space of the
364<i>2 -> 2</i> (massless) QCD processes is now sampled in
365<i>dy_3 dy_4 dpT^2</i>, where <i>y_3</i> and <i>y_4</i> are
366the rapidities of the two produced partons. One can show that
367<br/><i>
368    (dx_1 / x_1) * (dx_2 / x_2) * d(tHat) = dy_3 * dy_4 * dpT^2
369</i><br/>
370Furthermore, since cross sections are dominated by the "Rutherford"
371one of <i>t</i>-channel gluon exchange, which is enhanced by a
372factor of 9/4 for each incoming gluon, effective structure functions
373are defined as
374<br/><i>
375    F(x, pT2) = (9/4) * xg(x, pT2) + sum_i xq_i(x, pT2)
376</i><br/>
377With this technical shift of factors 9/4 from cross sections to parton
378densities, a common upper estimate of
379<br/><i>
380    d(sigmaHat)/d(pT2) &lt; pi * alpha_strong^2 / pT^4   
381</i><br/>
382is obtained.
383
384<p/>
385In fact this estimate can be reduced by a factor of 1/2 for the
386following reason: for any configuration <i>(y_3, y_4, pT2)</i> also
387one with <i>(y_4, y_3, pT2)</i> lies in the phase space. Not both
388of those can enjoy being enhanced by the <i>tHat -> 0</i>
389singularity of
390<br/><i>
391    d(sigmaHat) propto 1/tHat^2.
392</i><br/>
393Or if they are, which is possible with identical partons like
394<i>q q -> q q</i> and <i>g g -> g g</i>, each singularity comes
395with half the strength. So, when integrating/averaging over the two
396configurations, the estimated <i>d(sigmaHat)/d(pT2)</i> drops.
397Actually, it drops even further, since the naive estimate above is
398based on
399<br/><i>
400    (4 /9) * (1 + (uHat/sHat)^2) &lt; 8/9 &lt; 1
401</i><br/>
402The 8/9 value would be approached for <i>tHat -> 0</i>, which
403implies <i>sHat >> pT2</i> and thus a heavy parton-distribution
404penalty, while parton distributions are largest for
405<i>tHat = uHat = -sHat/2</i>, where the above expression
406evaluates to 5/9. A fudge factor is therefore introduced to go the
407final step, so it can easily be modifed when further non-Rutherford
408processes are added, or should parton distributions change significantly.
409
410<p/>
411At initialization, it is assumed that 
412<br/><i>
413    d(sigma)/d(pT2) &lt; d(sigmaHat)/d(pT2) * F(x_T, pT2) * F(x_T, pT2)
414       * (2 y_max(pT))^2
415</i><br/>
416where the first factor is the upper estimate as above, the second two
417the parton density sum evaluated at <i>y_3 = y_ 4 = 0</i> so that
418<i>x_1 = x_2 = x_T = 2 pT / E_cm</i>, where the product is expected
419to be maximal, and the final is the phase space for
420<i>-y_max &lt; y_{3,4} &lt; y_max</i>.
421The right-hand side expression is scanned logarithmically in <i>y</i>,
422and a <i>N</i> is determined such that it always is below
423<i>N/pT^4</i>.
424
425<p/>
426To describe the dampening of the cross section at <i>pT -> 0</i> by
427colour screening, the actual cross section is multiplied by a
428regularization factor <i>(pT^2 / (pT^2 + pT0^2))^2</i>, and the
429<i>alpha_s</i> is evaluated at a scale <i>pT^2 + pT0^2</i>,
430where <i>pT0</i> is a free parameter of the order of 2 - 4 GeV.
431Since <i>pT0</i> can be energy-dependent,  an ansatz
432<br/><i>
433    pT0(ecm) = pT0Ref * (ecm/ecmRef)^ecmPow
434</i><br/>
435is used, where <i>ecm</i> is the current CM frame energy,
436<i>ecmRef</i> is an arbitrary reference energy where <i>pT0Ref</i>
437is defined, and <i>ecmPow</i> gives the energy rescaling pace. For
438technical reasons, also an absolute lower <i>pT</i> scale <i>pTmin</i>,
439by default 0.2 GeV, is introduced. In principle, it is possible to
440recover older scenarios with a sharp <i>pT</i> cutoff by setting
441<i>pT0 = 0</i> and letting <i>pTmin</i> be a larger number.
442
443<p/>
444The above scanning strategy is then slightly modified: instead of
445an upper estimate <i>N/pT^4</i> one of the form
446<i>N/(pT^2 + r * pT0^2)^2</i> is used. At first glance, <i>r = 1</i>
447would seem to be fixed by the form of the regularization procedure,
448but this does not take into account the nontrivial dependence on
449<i>alpha_s</i>, parton distributions and phase space. A better
450Monte Carlo efficiency is obtained for <i>r</i> somewhat below unity,
451and currently <i>r = 0.25</i> is hardcoded.
452
453In the generation a trial <i>pT2</i> is then selected according to
454<br/><i>
455    d(Prob)/d(pT2) = (1/sigma_ND) * N/(pT^2 + r * pT0^2)^2 * ("Sudakov")
456</i><br/>
457For the trial <i>pT2</i>, a <i>y_3</i> and a <i>y_4</i> are then
458selected, and incoming flavours according to the respective
459<i>F(x_i, pT2)</i>, and then the cross section is evaluated for this
460flavour combination. The ratio of trial/upper estimate gives the
461probability of survival.
462
463<p/>
464Actually, to profit from the factor 1/2 mentioned above, the cross
465section for the combination with <i>y_3</i> and <i>y_4</i>
466interchanged is also tried, which corresponds to exchanging <i>tHat</i>
467and <i>uHat</i>, and the average formed, while the final kinematics
468is given by the relative importance of the two.
469
470<p/>
471Furthermore, since large <i>y</i> values are disfavoured by dropping
472PDF's, a factor
473<br/><i>
474   WT_y = (1 - (y_3/y_max)^2) * (1 - (y_4/y_max)^2)
475</i><br/>
476is evaluated, and used as a survival probability before the more
477time-consuming PDF+ME evaluation, with surviving events given a
478compensating weight <i>1/WT_y</i>.
479
480<p/>
481An impact-parameter dependencs is also allowed. Based on the hard
482<i>pT</i> scale of the first interaction, and enhancement/depletion
483factor is picked, which multiplies the rate of subsequent interactions.
484
485<p/>
486Parton densities are rescaled and modified to take into account the
487energy-momentum and flavours kicked out by already-considered
488interactions.
489
490<input type="hidden" name="saved" value="1"/>
491
492<?php
493echo "<input type='hidden' name='filepath' value='".$_GET["filepath"]."'/>"?>
494
495<table width="100%"><tr><td align="right"><input type="submit" value="Save Settings" /></td></tr></table>
496</form>
497
498<?php
499
500if($_POST["saved"] == 1)
501{
502$filepath = $_POST["filepath"];
503$handle = fopen($filepath, 'a');
504
505if($_POST["1"] != "0")
506{
507$data = "MultipartonInteractions:pTmaxMatch = ".$_POST["1"]."\n";
508fwrite($handle,$data);
509}
510if($_POST["2"] != "0.127")
511{
512$data = "MultipartonInteractions:alphaSvalue = ".$_POST["2"]."\n";
513fwrite($handle,$data);
514}
515if($_POST["3"] != "1")
516{
517$data = "MultipartonInteractions:alphaSorder = ".$_POST["3"]."\n";
518fwrite($handle,$data);
519}
520if($_POST["4"] != "1")
521{
522$data = "MultipartonInteractions:alphaEMorder = ".$_POST["4"]."\n";
523fwrite($handle,$data);
524}
525if($_POST["5"] != "1.0")
526{
527$data = "MultipartonInteractions:Kfactor = ".$_POST["5"]."\n";
528fwrite($handle,$data);
529}
530if($_POST["6"] != "3")
531{
532$data = "MultipartonInteractions:processLevel = ".$_POST["6"]."\n";
533fwrite($handle,$data);
534}
535if($_POST["7"] != "2.15")
536{
537$data = "MultipartonInteractions:pT0Ref = ".$_POST["7"]."\n";
538fwrite($handle,$data);
539}
540if($_POST["8"] != "1800.0")
541{
542$data = "MultipartonInteractions:ecmRef = ".$_POST["8"]."\n";
543fwrite($handle,$data);
544}
545if($_POST["9"] != "0.24")
546{
547$data = "MultipartonInteractions:ecmPow = ".$_POST["9"]."\n";
548fwrite($handle,$data);
549}
550if($_POST["10"] != "0.2")
551{
552$data = "MultipartonInteractions:pTmin = ".$_POST["10"]."\n";
553fwrite($handle,$data);
554}
555if($_POST["11"] != "0")
556{
557$data = "MultipartonInteractions:enhanceScreening = ".$_POST["11"]."\n";
558fwrite($handle,$data);
559}
560if($_POST["12"] != "1")
561{
562$data = "MultipartonInteractions:bProfile = ".$_POST["12"]."\n";
563fwrite($handle,$data);
564}
565if($_POST["13"] != "0.4")
566{
567$data = "MultipartonInteractions:coreRadius = ".$_POST["13"]."\n";
568fwrite($handle,$data);
569}
570if($_POST["14"] != "0.5")
571{
572$data = "MultipartonInteractions:coreFraction = ".$_POST["14"]."\n";
573fwrite($handle,$data);
574}
575if($_POST["15"] != "1.")
576{
577$data = "MultipartonInteractions:expPow = ".$_POST["15"]."\n";
578fwrite($handle,$data);
579}
580if($_POST["16"] != "0.15")
581{
582$data = "MultipartonInteractions:a1 = ".$_POST["16"]."\n";
583fwrite($handle,$data);
584}
585if($_POST["17"] != "off")
586{
587$data = "MultipartonInteractions:allowRescatter = ".$_POST["17"]."\n";
588fwrite($handle,$data);
589}
590if($_POST["18"] != "off")
591{
592$data = "MultipartonInteractions:allowDoubleRescatter = ".$_POST["18"]."\n";
593fwrite($handle,$data);
594}
595if($_POST["19"] != "0")
596{
597$data = "MultipartonInteractions:rescatterMode = ".$_POST["19"]."\n";
598fwrite($handle,$data);
599}
600if($_POST["20"] != "0.")
601{
602$data = "MultipartonInteractions:ySepRescatter = ".$_POST["20"]."\n";
603fwrite($handle,$data);
604}
605if($_POST["21"] != "1.")
606{
607$data = "MultipartonInteractions:deltaYRescatter = ".$_POST["21"]."\n";
608fwrite($handle,$data);
609}
610if($_POST["22"] != "5")
611{
612$data = "MultipartonInteractions:nQuarkIn = ".$_POST["22"]."\n";
613fwrite($handle,$data);
614}
615if($_POST["23"] != "1000")
616{
617$data = "MultipartonInteractions:nSample = ".$_POST["23"]."\n";
618fwrite($handle,$data);
619}
620fclose($handle);
621}
622
623?>
624</body>
625</html>
626
627<!-- Copyright (C) 2012 Torbjorn Sjostrand -->
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