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2<head>
3<title>Beam Remnants</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
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7<body>
8
9<h2>Beam Remnants</h2>
10
11<h3>Introduction</h3>
12
13The <code>BeamParticle</code> class contains information on all partons
14extracted from a beam (so far). As each consecutive multiparton interaction
15defines its respective incoming parton to the hard scattering a
16new slot is added to the list. This information is modified when
17the backwards evolution of the spacelike shower defines a new
18initiator parton. It is used, both for the multiparton interactions
19and the spacelike showers, to define rescaled parton densities based
20on the <i>x</i> and flavours already extracted, and to distinguish
21between valence, sea and companion quarks. Once the perturbative
22evolution is finished, further beam remnants are added to obtain a
23consistent set of flavours. The current physics framework is further
24described in [<a href="Bibliography.html" target="page">Sjo04</a>].
25
26<p/>
27The introduction of <a href="MultipartonInteractions.html" target="page">rescattering</a> 
28in the multiparton interactions framework further complicates the
29processing of events. Specifically, when combined with showers,
30the momentum of an individual parton is no longer uniquely associated
31with one single subcollision. Nevertheless the parton is classified
32with one system, owing to the technical and administrative complications
33of more complete classifications. Therefore the addition of primordial
34<i>kT</i> to the subsystem initiator partons does not automatically
35guarantee overall <i>pT</i> conservation. Various tricks are used to
36minimize the mismatch, with a brute force shift of all parton
37<i>pT</i>'s as a final step.
38
39<p/>
40Much of the above information is stored in a vector of
41<code>ResolvedParton</code> objects, which each contains flavour and
42momentum information, as well as valence/companion information and more.
43The <code>BeamParticle</code> method <code>list()</code> shows the
44contents of this vector, mainly for debug purposes.
45
46<p/>
47The <code>BeamRemnants</code> class takes over for the final step
48of adding primordial <i>kT</i> to the initiators and remnants,
49assigning the relative longitudinal momentum sharing among the
50remnants, and constructing the overall kinematics and colour flow.
51This step couples the two sides of an event, and could therefore
52not be covered in the <code>BeamParticle</code> class, which only
53considers one beam at a time.
54
55<p/>
56The methods of these classes are not intended for general use,
57and so are not described here.
58
59<p/>
60In addition to the parameters described on this page, note that the
61choice of <a href="PDFSelection.html" target="page">parton densities</a> is made
62in the <code>Pythia</code> class. Then pointers to the pdf's are handed
63on to <code>BeamParticle</code> at initialization, for all subsequent
64usage.
65
66<h3>Primordial <i>kT</i></h3>
67
68The primordial <i>kT</i> of initiators of hard-scattering subsystems
69are selected according to Gaussian distributions in <i>p_x</i> and
70<i>p_y</i> separately. The widths of these distributions are chosen
71to be dependent on the hard scale of the central process and on the mass
72of the whole subsystem defined by the two initiators:
73<br/><i>
74sigma = (sigma_soft * Q_half + sigma_hard * Q) / (Q_half + Q)
75  * m / (m_half + m) 
76</i><br/>
77Here <i>Q</i> is the hard-process renormalization scale for the
78hardest process and the <i>pT</i> scale for subsequent multiparton
79interactions, <i>m</i> the mass of the system, and
80<i>sigma_soft</i>, <i>sigma_hard</i>, <i>Q_half</i> and
81<i>m_half</i> parameters defined below. Furthermore each separately
82defined beam remnant has a distribution of width <i>sigma_remn</i>,
83independently of kinematical variables.
84
85<p/><code>flag&nbsp; </code><strong> BeamRemnants:primordialKT &nbsp;</strong> 
86 (<code>default = <strong>on</strong></code>)<br/>
87Allow or not selection of primordial <i>kT</i> according to the
88parameter values below.
89 
90
91<p/><code>parm&nbsp; </code><strong> BeamRemnants:primordialKTsoft &nbsp;</strong> 
92 (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.</code>)<br/>
93The width <i>sigma_soft</i> in the above equation, assigned as a
94primordial <i>kT</i> to initiators in the soft-interaction limit.
95 
96
97<p/><code>parm&nbsp; </code><strong> BeamRemnants:primordialKThard &nbsp;</strong> 
98 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.</code>)<br/>
99The width <i>sigma_hard</i> in the above equation, assigned as a
100primordial <i>kT</i> to initiators in the hard-interaction limit.
101 
102
103<p/><code>parm&nbsp; </code><strong> BeamRemnants:halfScaleForKT &nbsp;</strong> 
104 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.</code>)<br/>
105The scale <i>Q_half</i> in the equation above, defining the
106half-way point between hard and soft interactions.
107 
108
109<p/><code>parm&nbsp; </code><strong> BeamRemnants:halfMassForKT &nbsp;</strong> 
110 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.</code>)<br/>
111The scale <i>m_half</i> in the equation above, defining the
112half-way point between low-mass and high-mass subsystems.
113(Kinematics construction can easily fail if a system is assigned
114a primordial <i>kT</i> value higher than its mass, so the
115mass-dampening is intended to reduce some troubles later on.)
116 
117
118<p/><code>parm&nbsp; </code><strong> BeamRemnants:primordialKTremnant &nbsp;</strong> 
119 (<code>default = <strong>0.4</strong></code>; <code>minimum = 0.</code>)<br/>
120The width <i>sigma_remn</i>, assigned as a primordial <i>kT</i> 
121to beam-remnant partons.
122 
123
124<p/>
125A net <i>kT</i> imbalance is obtained from the vector sum of the
126primordial <i>kT</i> values of all initiators and all beam remnants.
127This quantity is compensated by a shift shared equally between
128all partons, except that the dampening factor <i>m / (m_half + m)</i> 
129is again used to suppress the role of small-mass systems.
130
131<p/>
132Note that the current <i>sigma</i> definition implies that
133<i>&lt;pT^2&gt; = &lt;p_x^2&gt;+ &lt;p_y^2&gt; = 2 sigma^2</i>.
134It thus cannot be compared directly with the <i>sigma</i>
135of nonperturbative hadronization, where each quark-antiquark
136breakup corresponds to <i>&lt;pT^2&gt; = sigma^2</i> and only
137for hadrons it holds that <i>&lt;pT^2&gt; = 2 sigma^2</i>.
138The comparison is further complicated by the reduction of
139primordial <i>kT</i> values by the overall compensation mechanism.
140
141<p/><code>flag&nbsp; </code><strong> BeamRemnants:rescatterRestoreY &nbsp;</strong> 
142 (<code>default = <strong>off</strong></code>)<br/>
143Is only relevant when <a href="MultipartonInteractions.html" target="page">rescattering</a> 
144is switched on in the multiparton interactions scenario. For a normal
145interaction the rapidity and mass of a system is preserved when
146primordial <i>kT</i> is introduced, by appropriate modification of the
147incoming parton momenta. Kinematics construction is more complicated for
148a rescattering, and two options are offered. Differences between these
149can be used to explore systematic uncertainties in the rescattering
150framework.<br/>
151The default behaviour is to keep the incoming rescattered parton as is,
152but to modify the unrescattered incoming parton so as to preserve the
153invariant mass of the system. Thereby the rapidity of the rescattering
154is modified.<br/> 
155The alternative is to retain the rapidity (and mass) of the rescattered
156system when primordial <i>kT</i> is introduced. This is made at the
157expense of a modified longitudinal momentum of the incoming rescattered
158parton, so that it does not agree with the momentum it ought to have had
159by the kinematics of the previous interaction.<br/> 
160For a double rescattering, when both incoming partons have already scattered,
161there is no obvious way to retain the invariant mass of the system in the
162first approach, so the second is always used.
163 
164
165<h3>Colour flow</h3>
166
167The colour flows in the separate subprocesses defined in the
168multiparton-interactions scenario are tied together via the assignment
169of colour flow in the beam remnant. This is not an unambiguous
170procedure, but currently no parameters are directly associated with it.
171However, a simple "minimal" procedure of colour flow only via the beam
172remnants does not result in a scenario in
173agreement with data, notably not a sufficiently steep rise of 
174<i>&lt;pT&gt;(n_ch)</i>. The true origin of this behaviour and the
175correct mechanism to reproduce it remains one of the big unsolved issues
176at the borderline between perturbative and nonperturbative QCD.
177As a simple attempt, an additional step is introduced, wherein the gluons
178of a lower-<i>pT</i> system are merged with the ones in a higher-pT one.
179
180<p/><code>flag&nbsp; </code><strong> BeamRemnants:reconnectColours &nbsp;</strong> 
181 (<code>default = <strong>on</strong></code>)<br/>
182Allow or not a system to be merged with another one.
183 
184
185<p/><code>parm&nbsp; </code><strong> BeamRemnants:reconnectRange &nbsp;</strong> 
186 (<code>default = <strong>10.0</strong></code>; <code>minimum = 0.</code>; <code>maximum = 10.</code>)<br/>
187A system with a hard scale <i>pT</i> can be merged with one of a
188harder scale with a probability that is
189<i>pT0_Rec^2 / (pT0_Rec^2 + pT^2)</i>, where
190<i>pT0_Rec</i> is <code>reconnectRange</code> times <i>pT0</i>,
191the latter being the same energy-dependent dampening parameter as
192used for multiparton interactions.
193Thus it is easy to merge a low-<i>pT</i> system with any other,
194but difficult to merge two high-<i>pT</i> ones with each other.
195 
196
197<p/>
198The procedure is used iteratively. Thus first the reconnection probability
199<i>P = pT0_Rec^2 / (pT0_Rec^2 + pT^2)</i> of the lowest-<i>pT</i> 
200system is found, and gives the probability for merger with the
201second-lowest one. If not merged, it is tested with the third-lowest one,
202and so on. For the <i>m</i>'th higher system the reconnection
203probability thus becomes <i>(1 - P)^(m-1) P</i>. That is, there is
204no explicit dependence on the higher <i>pT</i> scale, but implicitly
205there is via the survival probability of not already having been merged
206with a lower-<i>pT</i> system. Also note that the total reconnection
207probability for the lowest-<i>pT</i> system in an event with <i>n</i> 
208systems becomes <i>1 - (1 - P)^(n-1)</i>. Once the fate of the
209lowest-<i>pT</i> system has been decided, the second-lowest is considered
210with respect to the ones above it, then the third-lowest, and so on. 
211
212<p/>
213Once it has been decided which systems should be joined, the actual merging
214is carried out in the opposite direction. That is, first the hardest
215system is studied, and all colour dipoles in it are found (including to
216the beam remnants, as defined by the holes of the incoming partons).
217Next each softer system to be merged is studied in turn. Its gluons are,
218in decreasing <i>pT</i> order, inserted on the colour dipole <i>i,j</i>
219that gives the smallest <i>(p_g p_i)(p_g p_j)/(p_i p_j)</i>, i.e.
220minimizes the "disturbance" on the existing dipole, in terms of
221<i>pT^2</i> or <i>Lambda</i> measure (string length). The insertion
222of the gluon means that the old dipole is replaced by two new ones.
223Also the (rather few) quark-antiquark pairs that can be traced back to
224a gluon splitting are treated in close analogy with the gluon case.
225Quark lines that attach directly to the beam remnants cannot be merged
226but are left behind.
227
228<p/>
229The joining procedure can be viewed as a more sophisticated variant of
230the one introduced already in [<a href="Bibliography.html" target="page">Sjo87</a>]. Clearly it is ad hoc.
231It hopefully captures some elements of truth. The lower <i>pT</i> scale
232a system has the larger its spatial extent and therefore the larger its
233overlap with other systems. It could be argued that one should classify
234individual initial-state partons by <i>pT</i> rather than the system
235as a whole. However, for final-state radiation, a soft gluon radiated off
236a hard parton is actually produced at late times and therefore probably
237less likely to reconnect. In the balance, a classification by system
238<i>pT</i> scale appears sensible as a first try.
239
240<p/>
241Note that the reconnection is carried out before resonance decays are
242considered. Colour inside a resonance therefore is not reconnected.
243This is a deliberate choice, but certainly open to discussion and
244extensions at a later stage, as is the rest of this procedure.
245
246<h3>Further variables</h3>
247
248<p/><code>mode&nbsp; </code><strong> BeamRemnants:maxValQuark &nbsp;</strong> 
249 (<code>default = <strong>3</strong></code>; <code>minimum = 0</code>; <code>maximum = 5</code>)<br/>
250The maximum valence quark kind allowed in acceptable incoming beams,
251for which multiparton interactions are simulated. Default is that hadrons
252may contain <i>u</i>, <i>d</i> and <i>s</i> quarks,
253but not <i>c</i> and <i>b</i> ones, since sensible
254kinematics has not really been worked out for the latter.
255 
256
257<p/><code>mode&nbsp; </code><strong> BeamRemnants:companionPower &nbsp;</strong> 
258 (<code>default = <strong>4</strong></code>; <code>minimum = 0</code>; <code>maximum = 4</code>)<br/>
259When a sea quark has been found, a companion antisea quark ought to be
260nearby in <i>x</i>. The shape of this distribution can be derived
261from the gluon mother distribution convoluted with the
262<i>g -> q qbar</i> splitting kernel. In practice, simple solutions
263are only feasible if the gluon shape is assumed to be of the form
264<i>g(x) ~ (1 - x)^p / x</i>, where <i>p</i> is an integer power,
265the parameter above. Allowed values correspond to the cases programmed.
266<br/> 
267Since the whole framework is approximate anyway, this should be good
268enough. Note that companions typically are found at small <i>Q^2</i>,
269if at all, so the form is supposed to represent <i>g(x)</i> at small
270<i>Q^2</i> scales, close to the lower cutoff for multiparton interactions.
271 
272
273<p/>
274When assigning relative momentum fractions to beam-remnant partons,
275valence quarks are chosen according to a distribution like
276<i>(1 - x)^power / sqrt(x)</i>. This <i>power</i> is given below
277for quarks in mesons, and separately for <i>u</i> and <i>d</i> 
278quarks in the proton, based on the approximate shape of low-<i>Q^2</i> 
279parton densities. The power for other baryons is derived from the
280proton ones, by an appropriate mixing. The <i>x</i> of a diquark
281is chosen as the sum of its two constituent <i>x</i> values, and can
282thus be above unity. (A common rescaling of all remnant partons and
283particles will fix that.) An additional enhancement of the diquark
284momentum is obtained by its <i>x</i> value being rescaled by the
285<code>valenceDiqEnhance</code> factor.
286
287<p/><code>parm&nbsp; </code><strong> BeamRemnants:valencePowerMeson &nbsp;</strong> 
288 (<code>default = <strong>0.8</strong></code>; <code>minimum = 0.</code>)<br/>
289The abovementioned power for valence quarks in mesons.
290 
291
292<p/><code>parm&nbsp; </code><strong> BeamRemnants:valencePowerUinP &nbsp;</strong> 
293 (<code>default = <strong>3.5</strong></code>; <code>minimum = 0.</code>)<br/>
294The abovementioned power for valence <i>u</i> quarks in protons.
295 
296
297<p/><code>parm&nbsp; </code><strong> BeamRemnants:valencePowerDinP &nbsp;</strong> 
298 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.</code>)<br/>
299The abovementioned power for valence <i>d</i> quarks in protons.
300 
301
302<p/><code>parm&nbsp; </code><strong> BeamRemnants:valenceDiqEnhance &nbsp;</strong> 
303 (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10.</code>)<br/>
304Enhancement factor for valence diqaurks in baryons, relative to the
305simple sum of the two constituent quarks.
306 
307
308<p/><code>flag&nbsp; </code><strong> BeamRemnants:allowJunction &nbsp;</strong> 
309 (<code>default = <strong>on</strong></code>)<br/>
310The <code>off</code> option is intended for debug purposes only, as
311follows. When more than one valence quark is kicked out of a baryon
312beam, as part of the multiparton interactions scenario, the subsequent
313hadronization is described in terms of a junction string topology.
314This description involves a number of technical complications that
315may make the program more unstable. As an alternative, by switching
316this option off, junction configurations are rejected (which gives
317an error message that the remnant flavour setup failed), and the
318multiparton interactions and showers are redone until a
319junction-free topology is found.
320   
321
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