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1<html>
2<head>
3<title>Multiparton Interactions</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
5<link rel="shortcut icon" href="pythia32.gif"/>
6</head>
7<body>
8
9<h2>Multiparton Interactions</h2>
10
11The starting point for the multiparton interactions physics scenario in
12PYTHIA is provided by [<a href="Bibliography.html" target="page">Sjo87</a>]. Recent developments have
13included a more careful study of flavour and colour correlations,
14junction topologies and the relationship to beam remnants
15[<a href="Bibliography.html" target="page">Sjo04</a>], interleaving with initial-state radiation
16[<a href="Bibliography.html" target="page">Sjo05</a>], making use of transverse-momentum-ordered
17initial- and final-state showers, with the extension to fully
18interleaved evolution covered in [<a href="Bibliography.html" target="page">Cor10a</a>]. A framework to
19handle rescattering is described in [<a href="Bibliography.html" target="page">Cor09</a>].
20
21<p/>
22A big unsolved issue is how the colour of all these subsystems is
23correlated. For sure there is a correlation coming from the colour
24singlet nature of the incoming beams, but in addition final-state
25colour rearrangements may change the picture. Indeed such extra
26effects appear necessary to describe data, e.g. on
27<i>&lt;pT&gt;(n_ch)</i>. A simple implementation of colour
28rearrangement is found as part of the
29<a href="BeamRemnants.html" target="page">beam remnants</a> description.
30
31<h3>Main variables</h3>
32
33<h4>Matching to hard process</h4>
34
35The maximum <i>pT</i> to be allowed for multiparton interactions is
36related to the nature of the hard process itself. It involves a
37delicate balance between not doublecounting and not leaving any
38gaps in the coverage. The best procedure may depend on information
39only the user has: how the events were generated and mixed (e.g. with
40Les Houches Accord external input), and how they are intended to be
41used. Therefore a few options are available, with a sensible default
42behaviour.
43<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:pTmaxMatch &nbsp;</strong> 
44 (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
45Way in which the maximum scale for multiparton interactions is set
46to match the scale of the hard process itself.
47<br/><code>option </code><strong> 0</strong> : <b>(i)</b> if the final state of the hard process
48(not counting subsequent resonance decays) contains only quarks
49(<i>u, d, s, c ,b</i>), gluons and photons then <i>pT_max</i> 
50is chosen to be the factorization scale for internal processes
51and the <code>scale</code> value for Les Houches input;
52<b>(ii)</b> if not, interactions are allowed to go all the way up
53to the kinematical limit.
54The reasoning is that the former kind of processes are generated by
55the multiparton-interactions machinery and so would doublecount hard
56processes if allowed to overlap the same <i>pT</i> range,
57while no such danger exists in the latter case.
58 
59<br/><code>option </code><strong> 1</strong> : always use the factorization scale for an internal
60process and the <code>scale</code> value for Les Houches input,
61i.e. the lower value. This should avoid doublecounting, but
62may leave out some interactions that ought to have been simulated.
63 
64<br/><code>option </code><strong> 2</strong> : always allow multiparton interactions up to the
65kinematical limit. This will simulate all possible event topologies,
66but may lead to doublecounting.
67 
68 
69
70<h4>Cross-section parameters</h4>
71
72The rate of interactions is determined by
73<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:alphaSvalue &nbsp;</strong> 
74 (<code>default = <strong>0.127</strong></code>; <code>minimum = 0.06</code>; <code>maximum = 0.25</code>)<br/>
75The value of <i>alpha_strong</i> at <i>m_Z</i>. Default value is
76picked equal to the one used in CTEQ 5L.
77   
78
79<p/>
80The actual value is then regulated by the running to the scale
81<i>pT^2</i>, at which it is evaluated
82<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:alphaSorder &nbsp;</strong> 
83 (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
84The order at which <i>alpha_strong</i> runs at scales away from
85<i>m_Z</i>.
86<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_strong</i> is kept
87fixed. 
88<br/><code>option </code><strong> 1</strong> : first order, which is the normal value. 
89<br/><code>option </code><strong> 2</strong> : second order. Since other parts of the code do
90not go to second order there is no strong reason to use this option,
91but there is also nothing wrong with it. 
92 
93
94<p/>
95QED interactions are regulated by the <i>alpha_electromagnetic</i>
96value at the <i>pT^2</i> scale of an interaction.
97 
98<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:alphaEMorder &nbsp;</strong> 
99 (<code>default = <strong>1</strong></code>; <code>minimum = -1</code>; <code>maximum = 1</code>)<br/>
100The running of <i>alpha_em</i> used in hard processes.
101<br/><code>option </code><strong> 1</strong> : first-order running, constrained to agree with
102<code>StandardModel:alphaEMmZ</code> at the <i>Z^0</i> mass.
103 
104<br/><code>option </code><strong> 0</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
105fixed at its value at vanishing momentum transfer. 
106<br/><code>option </code><strong> -1</strong> : zeroth order, i.e. <i>alpha_em</i> is kept
107fixed, but at <code>StandardModel:alphaEMmZ</code>, i.e. its value
108at the <i>Z^0</i> mass.
109   
110 
111
112<p/>
113Note that the choices of <i>alpha_strong</i> and <i>alpha_em</i> 
114made here override the ones implemented in the normal process machinery,
115but only for the interactions generated by the
116<code>MultipartonInteractions</code> class.
117
118<p/>
119In addition there is the possibility of a global rescaling of
120cross sections (which could not easily be accommodated by a
121changed <i>alpha_strong</i>, since <i>alpha_strong</i> runs)
122<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:Kfactor &nbsp;</strong> 
123 (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 4.0</code>)<br/>
124Multiply all cross sections by this fix factor.
125 
126
127<p/> 
128The processes used to generate multiparton interactions form a subset
129of the standard library of hard processes. The input is slightly
130different from the standard hard-process machinery, however,
131since incoming flavours, the <i>alpha_strong</i> value and most
132of the kinematics are aready fixed when the process is called.
133It is possible to regulate the set of processes that are included in the
134multiparton-interactions framework.
135
136<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:processLevel &nbsp;</strong> 
137 (<code>default = <strong>3</strong></code>; <code>minimum = 0</code>; <code>maximum = 3</code>)<br/>
138Set of processes included in the machinery.
139<br/><code>option </code><strong> 0</strong> : only the simplest <i>2 -> 2</i> QCD processes
140between quarks and gluons, giving no new flavours, i.e. dominated by
141<i>t</i>-channel gluon exchange. 
142<br/><code>option </code><strong> 1</strong> : also <i>2 -> 2</i> QCD processes giving new flavours
143(including charm and bottom), i.e. proceeding through <i>s</i>-channel
144gluon exchange. 
145<br/><code>option </code><strong> 2</strong> : also <i>2 -> 2</i> processes involving one or two
146photons in the final state, <i>s</i>-channel <i>gamma</i>
147boson exchange and <i>t</i>-channel <i>gamma/Z^0/W^+-</i>
148boson exchange. 
149<br/><code>option </code><strong> 3</strong> : also charmonium and bottomonium production, via
150colour singlet and colour octet channels.   
151 
152
153<h4>Cross-section regularization</h4>
154
155There are two complementary ways of regularizing the small-<i>pT</i> 
156divergence, a sharp cutoff and a smooth dampening. These can be
157combined as desired, but it makes sense to coordinate with how the
158same issue is handled in <a href="SpacelikeShowers.html" target="page">spacelike
159showers</a>. Actually, by default, the parameters defined here are
160used also for the spacelike showers, but this can be overridden.
161
162<p/>
163Regularization of the divergence of the QCD cross section for
164<i>pT -> 0</i> is obtained by a factor <i>pT^4 / (pT0^2 + pT^2)^2</i>,
165and by using an <i>alpha_s(pT0^2 + pT^2)</i>. An energy dependence
166of the <i>pT0</i> choice is introduced by two further parameters,
167so that <i>pT0Ref</i> is the <i>pT0</i> value for the reference
168CM energy, <i>pT0Ref = pT0(ecmRef)</i>.   
169<br/><b>Warning:</b> if a large <i>pT0</i> is picked for multiparton
170interactions, such that the integrated interaction cross section is
171below the nondiffractive inelastic one, this <i>pT0</i> will
172automatically be scaled down to cope.
173
174<p/>
175The actual <i>pT0</i> parameter used at a given CM energy scale,
176<i>ecmNow</i>, is obtained as
177<br/><i>
178     pT0 = pT0(ecmNow) = pT0Ref * (ecmNow / ecmRef)^ecmPow
179</i><br/>
180where <i>pT0Ref</i>, <i>ecmRef</i> and <i>ecmPow</i> are the
181three parameters below.
182
183<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:pT0Ref &nbsp;</strong> 
184 (<code>default = <strong>2.15</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10.0</code>)<br/>
185The <i>pT0Ref</i> scale in the above formula.
186<br/><b>Note:</b> <i>pT0Ref</i> is one of the key parameters in a
187complete PYTHIA tune. Its value is intimately tied to a number of other
188choices, such as that of colour flow description, so unfortunately it is
189difficult to give an independent meaning to <i>pT0Ref</i>.
190 
191
192<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:ecmRef &nbsp;</strong> 
193 (<code>default = <strong>1800.0</strong></code>; <code>minimum = 1.</code>)<br/>
194The <i>ecmRef</i> reference energy scale introduced above.
195 
196
197<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:ecmPow &nbsp;</strong> 
198 (<code>default = <strong>0.24</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)<br/>
199The <i>ecmPow</i> energy rescaling pace introduced above.
200 
201
202<p/>
203Alternatively, or in combination, a sharp cut can be used.
204<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:pTmin &nbsp;</strong> 
205 (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 10.0</code>)<br/>
206Lower cutoff in <i>pT</i>, below which no further interactions
207are allowed. Normally <i>pT0</i> above would be used to provide
208the main regularization of the cross section for <i>pT -> 0</i>,
209in which case <i>pTmin</i> is used  mainly for technical reasons.
210It is possible, however, to set <i>pT0Ref = 0</i> and use
211<i>pTmin</i> to provide a step-function regularization, or to
212combine them in intermediate approaches. Currently <i>pTmin</i> 
213is taken to be energy-independent. 
214 
215
216<p/>
217G&ouml;sta Gustafson has proposed (private communication, unpublished)
218that the amount of screening, as encapsulated in the <i>pT0</i> 
219parameter, fluctuates from one event to the next. Specifically,
220high-activity event are more likely to lead to interactions at large
221<i>pT</i> scales, but the high activity simultaneously leads to a
222larger screening of interactions at smaller <i>pT</i>. Such a scenario
223can approximately be simulated by scaling up the <i>pT0</i> by a
224factor <i>sqrt(n)</i>, where <i>n</i> is the number of interactions
225considered so far, including the current one. That is, for the first
226interaction the dampening factor is <i>pT^4 / (pT0^2 + pT^2)^2</i>,
227for the second <i>pT^4 / (2 pT0^2 + pT^2)^2</i>, for the third
228<i>pT^4 / (3 pT0^2 + pT^2)^2</i>, and so on. Optionally the scheme
229may also be applied to ISR emissions. For simplicity the same
230<i>alpha_s(pT0^2 + pT^2)</i> is used throughout. Note that, in this
231scenario the <i>pT0</i> scale must be lower than in the normal case
232to begin with, since it later is increased back up. Also note that the
233idea with this scenario is to propose an alternative to colour
234reconnection to understand the rise of <i>&lt;pT&gt;(n_ch)</i>,
235so that the amount of colour reconnection should be reduced.
236<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:enhanceScreening &nbsp;</strong> 
237 (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 2</code>)<br/>
238Choice to activate the above screening scenario, i.e. an increasing
239effective <i>pT0</i> for consecutive interactions.
240<br/><code>option </code><strong> 0</strong> : No activity-dependent screening, i.e. <i>pT0</i>
241is fixed. 
242<br/><code>option </code><strong> 1</strong> : The <i>pT0</i> scale is increased as a function
243of the number of MPI's, as explained above. ISR is not affected,
244but note that, if <code>SpaceShower:samePTasMPI</code> is on,
245then <code>MultipartonInteractions:pT0Ref</code> is used also for ISR,
246which may or may not be desirable.
247   
248<br/><code>option </code><strong> 2</strong> : Both MPI and ISR influence and are influenced by the
249screening. That is, the dampening is reduced based on the total number
250of MPI and ISR steps considered so far, including the current one.
251This dampening is implemented both for MPI and for ISR emissions,
252for the latter provided that <code>SpaceShower:samePTasMPI</code> is on
253(default). 
254 
255 
256
257<h4>Impact-parameter dependence</h4>
258 
259The choice of impact-parameter dependence is regulated by several
260parameters. The ones listed here refer to nondiffractive topologies
261only, while their equivalents for diffractive events are put in the
262<a href="Diffraction.html" target="page">Diffraction</a> description. Note that
263there is currently no <code>bProfile = 4</code> option for diffraction.
264Other parameters are assumed to agree between diffractive and
265nondiffractive topologies.
266
267<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:bProfile &nbsp;</strong> 
268 (<code>default = <strong>1</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)<br/>
269Choice of impact parameter profile for the incoming hadron beams.
270<br/><code>option </code><strong> 0</strong> : no impact parameter dependence at all. 
271<br/><code>option </code><strong> 1</strong> : a simple Gaussian matter distribution;
272no free parameters. 
273<br/><code>option </code><strong> 2</strong> : a double Gaussian matter distribution,
274with the two free parameters <i>coreRadius</i> and
275<i>coreFraction</i>. 
276<br/><code>option </code><strong> 3</strong> : an overlap function, i.e. the convolution of
277the matter distributions of the two incoming hadrons, of the form
278<i>exp(- b^expPow)</i>, where <i>expPow</i> is a free
279parameter.   
280<br/><code>option </code><strong> 4</strong> : a Gaussian matter distribution with a width
281that varies according to the selected <i>x</i> value of an interaction,
282<i>1. + a1 log (1 / x)</i>, where <i>a1</i> is a free parameter.
283Note that once <i>b</i> has been selected for the hard process,
284it remains fixed for the remainder of the evolution.
285 
286 
287
288<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:coreRadius &nbsp;</strong> 
289 (<code>default = <strong>0.4</strong></code>; <code>minimum = 0.1</code>; <code>maximum = 1.</code>)<br/>
290When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
291the inner core is assumed to have a radius that is a factor
292<i>coreRadius</i> smaller than the rest.
293   
294
295<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:coreFraction &nbsp;</strong> 
296 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>; <code>maximum = 1.</code>)<br/>
297When assuming a double Gaussian matter profile, <i>bProfile = 2</i>,
298the inner core is assumed to have a fraction <i>coreFraction</i> 
299of the matter content of the hadron.
300   
301
302<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:expPow &nbsp;</strong> 
303 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.4</code>; <code>maximum = 10.</code>)<br/>
304When <i>bProfile = 3</i> it gives the power of the assumed overlap
305shape <i>exp(- b^expPow)</i>. Default corresponds to a simple
306exponential drop, which is not too dissimilar from the overlap
307obtained with the standard double Gaussian parameters. For
308<i>expPow = 2</i> we reduce to the simple Gaussian, <i>bProfile = 1</i>,
309and for <i>expPow -> infinity</i> to no impact parameter dependence
310at all, <i>bProfile = 0</i>. For small <i>expPow</i> the program
311becomes slow and unstable, so the min limit must be respected.
312   
313
314<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:a1 &nbsp;</strong> 
315 (<code>default = <strong>0.15</strong></code>; <code>minimum = 0.</code>; <code>maximum = 2.</code>)<br/>
316When <i>bProfile = 4</i>, this gives the <i>a1</i> constant in the
317Gaussian width. When <i>a1 = 0.</i>, this reduces back to the single
318Gaussian case.
319 
320
321<h4>Rescattering</h4>
322 
323It is possible that a parton may rescatter, i.e. undergo a further
324interaction subsequent to the first one. The machinery to model this
325kind of physics has only recently become fully operational
326[<a href="Bibliography.html" target="page">Cor09</a>], and is therefore not yet so well explored.
327
328<p/>
329The rescatting framework has ties with other parts of the program,
330notably with the <a href="BeamRemnants.html" target="page">beam remnants</a>.
331
332<p/><code>flag&nbsp; </code><strong> MultipartonInteractions:allowRescatter &nbsp;</strong> 
333 (<code>default = <strong>off</strong></code>)<br/>
334Switch to allow rescattering of partons; on/off = true/false.<br/>
335<b>Note:</b> the rescattering framework has not yet been implemented
336for the <code>MultipartonInteractions:bProfile = 4</code> option,
337and can therefore not be switched on in that case. 
338<b>Warning:</b> use with caution since machinery is still not
339so well tested.
340 
341
342<p/><code>flag&nbsp; </code><strong> MultipartonInteractions:allowDoubleRescatter &nbsp;</strong> 
343 (<code>default = <strong>off</strong></code>)<br/>
344Switch to allow rescattering of partons, where both incoming partons
345have already rescattered; on/off = true/false. Is only used if
346<code>MultipartonInteractions:allowRescatter</code> is switched on.<br/>
347<b>Warning:</b> currently there is no complete implementation that
348combines it with shower evolution, so you must use
349<code>PartonLevel:ISR = off</code> and <code>PartonLevel:FSR = off</code>.
350If not, a warning will be issued and double rescattering will not be
351simulated. The rate also comes out to be much lower than for single
352rescattering, so to first approximation it can be neglected.
353 
354
355<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:rescatterMode &nbsp;</strong> 
356 (<code>default = <strong>0</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)<br/>
357Selection of which partons rescatter against unscattered partons
358from the incoming beams A and B, based on their rapidity value
359<i>y</i> in the collision rest frame. Here <i>ySep</i> is
360shorthand for <code>MultipartonInteractions:ySepRescatter</code> and
361<i>deltaY</i> for <code>MultipartonInteractions:deltaYRescatter</code>,
362defined below. The description is symmetric between the two beams,
363so only one case is described below.
364<br/><code>option </code><strong> 0</strong> : only scattered partons with <i>y > 0</i> 
365can collide with unscattered partons from beam B.   
366<br/><code>option </code><strong> 1</strong> : only scattered partons with <i>y > ySep</i> 
367can collide with unscattered partons from beam B. 
368<br/><code>option </code><strong> 2</strong> : the probability for a scattered parton to be considered
369as a potential rescatterer against unscattered partons in beam B increases
370linearly from zero at <i>y = ySep - deltaY</i> to unity at
371<i>y = ySep + deltaY</i>. 
372<br/><code>option </code><strong> 3</strong> : the probability for a scattered parton to be considered
373as a potential rescatterer against unscattered partons in beam B increases
374with <i>y</i> according to
375<i>(1/2) * (1 + tanh( (y - ySep) / deltaY))</i>. 
376<br/><code>option </code><strong> 4</strong> : all partons are potential rescatterers against both
377beams. 
378 
379
380<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:ySepRescatter &nbsp;</strong> 
381 (<code>default = <strong>0.</strong></code>)<br/>
382used for some of the <code>MultipartonInteractions:rescatterMode</code>
383options above, as the rapidity for which a scattered parton has a 50%
384probability to be considered as a potential rescatterer.
385A <i>ySep > 0</i> generally implies that some central partons cannot
386rescatter at all, while a <i>ySep < 0</i>  instead allows central
387partons to scatter against either beam.
388   
389
390<p/><code>parm&nbsp; </code><strong> MultipartonInteractions:deltaYRescatter &nbsp;</strong> 
391 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.1</code>)<br/>
392used for some of the <code>MultipartonInteractions:rescatterMode</code>
393options above, as the width of the rapidity transition region, where the
394probability rises from zero to unity that a scattered parton is considered
395as a potential rescatterer.
396   
397
398
399<h3>Further variables</h3>
400
401These should normally not be touched. Their only function is for
402cross-checks.
403
404<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:nQuarkIn &nbsp;</strong> 
405 (<code>default = <strong>5</strong></code>; <code>minimum = 0</code>; <code>maximum = 5</code>)<br/>
406Number of allowed incoming quark flavours in the beams; a change
407to 4 would thus exclude <i>b</i> and <i>bbar</i> as incoming
408partons, etc.
409 
410
411<p/><code>mode&nbsp; </code><strong> MultipartonInteractions:nSample &nbsp;</strong> 
412 (<code>default = <strong>1000</strong></code>; <code>minimum = 100</code>)<br/>
413The allowed <i>pT</i> range is split (unevenly) into 100 bins,
414and in each of these the interaction cross section is evaluated in
415<i>nSample</i> random phase space points. The full integral is used
416at initialization, and the differential one during the run as a
417"Sudakov form factor" for the choice of the hardest interaction.
418A larger number implies increased accuracy of the calculations.
419 
420
421<h3>Technical notes</h3>
422
423Relative to the articles mentioned above, not much has happened.
424The main news is a technical one, that the phase space of the
425<i>2 -> 2</i> (massless) QCD processes is now sampled in
426<i>dy_3 dy_4 dpT^2</i>, where <i>y_3</i> and <i>y_4</i> are
427the rapidities of the two produced partons. One can show that
428<br/><i>
429    (dx_1 / x_1) * (dx_2 / x_2) * d(tHat) = dy_3 * dy_4 * dpT^2
430</i><br/>
431Furthermore, since cross sections are dominated by the "Rutherford"
432one of <i>t</i>-channel gluon exchange, which is enhanced by a
433factor of 9/4 for each incoming gluon, effective structure functions
434are defined as
435<br/><i>
436    F(x, pT2) = (9/4) * xg(x, pT2) + sum_i xq_i(x, pT2)
437</i><br/>
438With this technical shift of factors 9/4 from cross sections to parton
439densities, a common upper estimate of
440<br/><i>
441    d(sigmaHat)/d(pT2) &lt; pi * alpha_strong^2 / pT^4   
442</i><br/>
443is obtained.
444
445<p/>
446In fact this estimate can be reduced by a factor of 1/2 for the
447following reason: for any configuration <i>(y_3, y_4, pT2)</i> also
448one with <i>(y_4, y_3, pT2)</i> lies in the phase space. Not both
449of those can enjoy being enhanced by the <i>tHat -> 0</i> 
450singularity of
451<br/><i>
452    d(sigmaHat) propto 1/tHat^2.
453</i><br/>
454Or if they are, which is possible with identical partons like
455<i>q q -> q q</i> and <i>g g -> g g</i>, each singularity comes
456with half the strength. So, when integrating/averaging over the two
457configurations, the estimated <i>d(sigmaHat)/d(pT2)</i> drops.
458Actually, it drops even further, since the naive estimate above is
459based on
460<br/><i>
461    (4 /9) * (1 + (uHat/sHat)^2) &lt; 8/9 &lt; 1
462</i><br/>
463The 8/9 value would be approached for <i>tHat -> 0</i>, which
464implies <i>sHat >> pT2</i> and thus a heavy parton-distribution
465penalty, while parton distributions are largest for
466<i>tHat = uHat = -sHat/2</i>, where the above expression
467evaluates to 5/9. A fudge factor is therefore introduced to go the
468final step, so it can easily be modifed when further non-Rutherford
469processes are added, or should parton distributions change significantly.
470
471<p/>
472At initialization, it is assumed that 
473<br/><i>
474    d(sigma)/d(pT2) &lt; d(sigmaHat)/d(pT2) * F(x_T, pT2) * F(x_T, pT2)
475       * (2 y_max(pT))^2
476</i><br/>
477where the first factor is the upper estimate as above, the second two
478the parton density sum evaluated at <i>y_3 = y_ 4 = 0</i> so that
479<i>x_1 = x_2 = x_T = 2 pT / E_cm</i>, where the product is expected
480to be maximal, and the final is the phase space for
481<i>-y_max &lt; y_{3,4} &lt; y_max</i>.
482The right-hand side expression is scanned logarithmically in <i>y</i>,
483and a <i>N</i> is determined such that it always is below
484<i>N/pT^4</i>.
485
486<p/>
487To describe the dampening of the cross section at <i>pT -> 0</i> by
488colour screening, the actual cross section is multiplied by a
489regularization factor <i>(pT^2 / (pT^2 + pT0^2))^2</i>, and the
490<i>alpha_s</i> is evaluated at a scale <i>pT^2 + pT0^2</i>,
491where <i>pT0</i> is a free parameter of the order of 2 - 4 GeV.
492Since <i>pT0</i> can be energy-dependent,  an ansatz
493<br/><i>
494    pT0(ecm) = pT0Ref * (ecm/ecmRef)^ecmPow
495</i><br/>
496is used, where <i>ecm</i> is the current CM frame energy,
497<i>ecmRef</i> is an arbitrary reference energy where <i>pT0Ref</i> 
498is defined, and <i>ecmPow</i> gives the energy rescaling pace. For
499technical reasons, also an absolute lower <i>pT</i> scale <i>pTmin</i>,
500by default 0.2 GeV, is introduced. In principle, it is possible to
501recover older scenarios with a sharp <i>pT</i> cutoff by setting
502<i>pT0 = 0</i> and letting <i>pTmin</i> be a larger number.
503
504<p/>
505The above scanning strategy is then slightly modified: instead of
506an upper estimate <i>N/pT^4</i> one of the form
507<i>N/(pT^2 + r * pT0^2)^2</i> is used. At first glance, <i>r = 1</i> 
508would seem to be fixed by the form of the regularization procedure,
509but this does not take into account the nontrivial dependence on
510<i>alpha_s</i>, parton distributions and phase space. A better
511Monte Carlo efficiency is obtained for <i>r</i> somewhat below unity,
512and currently <i>r = 0.25</i> is hardcoded.
513
514In the generation a trial <i>pT2</i> is then selected according to
515<br/><i>
516    d(Prob)/d(pT2) = (1/sigma_ND) * N/(pT^2 + r * pT0^2)^2 * ("Sudakov")
517</i><br/>
518For the trial <i>pT2</i>, a <i>y_3</i> and a <i>y_4</i> are then
519selected, and incoming flavours according to the respective
520<i>F(x_i, pT2)</i>, and then the cross section is evaluated for this
521flavour combination. The ratio of trial/upper estimate gives the
522probability of survival.
523
524<p/>
525Actually, to profit from the factor 1/2 mentioned above, the cross
526section for the combination with <i>y_3</i> and <i>y_4</i> 
527interchanged is also tried, which corresponds to exchanging <i>tHat</i>
528and <i>uHat</i>, and the average formed, while the final kinematics
529is given by the relative importance of the two.
530
531<p/>
532Furthermore, since large <i>y</i> values are disfavoured by dropping
533PDF's, a factor
534<br/><i>
535   WT_y = (1 - (y_3/y_max)^2) * (1 - (y_4/y_max)^2)
536</i><br/>
537is evaluated, and used as a survival probability before the more
538time-consuming PDF+ME evaluation, with surviving events given a
539compensating weight <i>1/WT_y</i>.
540
541<p/>
542An impact-parameter dependencs is also allowed. Based on the hard
543<i>pT</i> scale of the first interaction, and enhancement/depletion
544factor is picked, which multiplies the rate of subsequent interactions.
545
546<p/>
547Parton densities are rescaled and modified to take into account the
548energy-momentum and flavours kicked out by already-considered
549interactions.
550
551</body>
552</html>
553
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