1 <chapter name="Beam Remnants">
7 The <code>BeamParticle</code> class contains information on all partons
8 extracted from a beam (so far). As each consecutive multiple interaction
9 defines its respective incoming parton to the hard scattering a
10 new slot is added to the list. This information is modified when
11 the backwards evolution of the spacelike shower defines a new
12 initiator parton. It is used, both for the multiple interactions
13 and the spacelike showers, to define rescaled parton densities based
14 on the <ei>x</ei> and flavours already extracted, and to distinguish
15 between valence, sea and companion quarks. Once the perturbative
16 evolution is finished, further beam remnants are added to obtain a
17 consistent set of flavours. The current physics framework is further
18 described in <ref>Sjo04</ref>.
21 Much of the above information is stored in a vector of
22 <code>ResolvedParton</code> objects, which each contains flavour and
23 momentum information, as well as valence/companion information and more.
24 The <code>BeamParticle</code> method <code>list()</code> shows the
25 contents of this vector, mainly for debug purposes.
28 The <code>BeamRemnants</code> class takes over for the final step
29 of adding primordial <ei>kT</ei> to the initiators and remnants,
30 assigning the relative longitudinal momentum sharing among the
31 remnants, and constructing the overall kinematics and colour flow.
32 This step couples the two sides of an event, and could therefore
33 not be covered in the <code>BeamParticle</code> class, which only
34 considers one beam at a time.
37 The methods of these classes are not intended for general use,
38 and so are not described here.
41 In addition to the parameters described on this page, note that the
42 choice of <aloc href="PDFSelection">parton densities</aloc> is made
43 in the <code>Pythia</code> class. Then pointers to the pdf's are handed
44 on to <code>BeamParticle</code> at initialization, for all subsequent
47 <h3>Primordial <ei>kT</ei></h3>
49 The primordial <ei>kT</ei> of initiators of hard-scattering subsystems
50 are selected according to Gaussian distributions in <ei>p_x</ei> and
51 <ei>p_y</ei> separately. The widths of these distributions are chosen
52 to be dependent on the hard scale of the central process and on the mass
53 of the whole subsystem defined by the two initiators:
55 sigma = (sigma_soft * Q_half + sigma_hard * Q) / (Q_half + Q)
58 Here <ei>Q</ei> is the hard-process renormalization scale for the
59 hardest process and the <ei>pT</ei> scale for subsequent multiple
60 interactions, <ei>m</ei> the mass of the system, and
61 <ei>sigma_soft</ei>, <ei>sigma_hard</ei>, <ei>Q_half</ei> and
62 <ei>m_half</ei> parameters defined below. Furthermore each separately
63 defined beam remnant has a distribution of width <ei>sigma_remn</ei>,
64 independently of kinematical variables.
66 <flag name="BeamRemnants:primordialKT" default="on">
67 Allow or not selection of primordial <ei>kT</ei> according to the
68 parameter values below.
71 <parm name="BeamRemnants:primordialKTsoft" default="0.4" min="0.">
72 The width <ei>sigma_soft</ei> in the above equation, assigned as a
73 primordial <ei>kT</ei> to initiators in the soft-interaction limit.
76 <parm name="BeamRemnants:primordialKThard" default="2.1" min="0.">
77 The width <ei>sigma_hard</ei> in the above equation, assigned as a
78 primordial <ei>kT</ei> to initiators in the hard-interaction limit.
81 <parm name="BeamRemnants:halfScaleForKT" default="7." min="0.">
82 The scale <ei>Q_half</ei> in the equation above, defining the
83 half-way point between hard and soft interactions.
86 <parm name="BeamRemnants:halfMassForKT" default="2." min="0.">
87 The scale <ei>m_half</ei> in the equation above, defining the
88 half-way point between low-mass and high-mass subsystems.
89 (Kinematics construction can easily fail if a system is assigned
90 a primordial <ei>kT</ei> value higher than its mass, so the
91 mass-dampening is intended to reduce some troubles later on.)
94 <parm name="BeamRemnants:primordialKTremnant" default="0.4" min="0.">
95 The width <ei>sigma_remn</ei>, assigned as a primordial <ei>kT</ei>
96 to beam-remnant partons.
100 A net <ei>kT</ei> imbalance is obtained from the vector sum of the
101 primordial <ei>kT</ei> values of all initiators and all beam remnants.
102 This quantity is compensated by a shift shared equally between
103 all partons, except that the dampening factor <ei>m / (m_half + m)</ei>
104 is again used to suppress the role of small-mass systems.
107 Note that the current <ei>sigma</ei> definition implies that
108 <ei><pT^2> = <p_x^2>+ <p_y^2> = 2 sigma^2</ei>.
109 It thus cannot be compared directly with the <ei>sigma</ei>
110 of nonperturbative hadronization, where each quark-antiquark
111 breakup corresponds to <ei><pT^2> = sigma^2</ei> and only
112 for hadrons it holds that <ei><pT^2> = 2 sigma^2</ei>.
113 The comparison is further complicated by the reduction of
114 primordial <ei>kT</ei> values by the overall compensation mechanism.
118 The colour flows in the separate subprocesses defined in the
119 multiple-interactions scenario are tied together via the assignment
120 of colour flow in the beam remnant. This is not an unambiguous
121 procedure, but currently no parameters are directly associated with it.
122 However, a simple "minimal" procedure of colour flow only via the beam
123 remnants does not result in a scenario in
124 agreement with data, notably not a sufficiently steep rise of
125 <ei><pT>(n_ch)</ei>. The true origin of this behaviour and the
126 correct mechanism to reproduce it remains one of the big unsolved issues
127 at the borderline between perturbative and nonperturbative QCD.
128 As a simple attempt, an additional step is introduced, wherein the gluons
129 of a lower-<ei>pT</ei> system are merged with the ones in a higher-pT one.
131 <flag name="BeamRemnants:reconnectColours" default="on">
132 Allow or not a system to be merged with another one.
135 <parm name="BeamRemnants:reconnectRange" default="2.5" min="0." max="10.">
136 A system with a hard scale <ei>pT</ei> can be merged with one of a
137 harder scale with a probability that is
138 <ei>pT0_Rec^2 / (pT0_Rec^2 + pT^2)</ei>, where
139 <ei>pT0_Rec</ei> is <code>reconnectRange</code> times <ei>pT0</ei>,
140 the latter being the same energy-dependent dampening parameter as
141 used for multiple interactions.
142 Thus it is easy to merge a low-<ei>pT</ei> system with any other,
143 but difficult to merge two high-<ei>pT</ei> ones with each other.
147 The procedure is used iteratively. Thus first the reconnection probability
148 <ei>P = pT0_Rec^2 / (pT0_Rec^2 + pT^2)</ei> of the lowest-<ei>pT</ei>
149 system is found, and gives the probability for merger with the
150 second-lowest one. If not merged, it is tested with the third-lowest one,
151 and so on. For the <ei>m</ei>'th higher system the reconnection
152 probability thus becomes <ei>(1 - P)^(m-1) P</ei>. That is, there is
153 no explicit dependence on the higher <ei>pT</ei> scale, but implicitly
154 there is via the survival probability of not already having been merged
155 with a lower-<ei>pT</ei> system. Also note that the total reconnection
156 probability for the lowest-<ei>pT</ei> system in an event with <ei>n</ei>
157 systems becomes <ei>1 - (1 - P)^(n-1)</ei>. Once the fate of the
158 lowest-<ei>pT</ei> system has been decided, the second-lowest is considered
159 with respect to the ones above it, then the third-lowest, and so on.
162 Once it has been decided which systems should be joined, the actual merging
163 is carried out in the opposite direction. That is, first the hardest
164 system is studied, and all colour dipoles in it are found (including to
165 the beam remnants, as defined by the holes of the incoming partons).
166 Next each softer system to be merged is studied in turn. Its gluons are,
167 in decreasing <ei>pT</ei> order, inserted on the colour dipole <ei>i,j</ei>
168 that gives the smallest <ei>(p_g p_i)(p_g p_j)/(p_i p_j)</ei>, i.e.
169 minimizes the "disturbance" on the existing dipole, in terms of
170 <ei>pT^2</ei> or <ei>Lambda</ei> measure (string length). The insertion
171 of the gluon means that the old dipole is replaced by two new ones.
172 Also the (rather few) quark-antiquark pairs that can be traced back to
173 a gluon splitting are treated in close analogy with the gluon case.
174 Quark lines that attach directly to the beam remnants cannot be merged
178 The joining procedure can be viewed as a more sophisticated variant of
179 the one introduced already in <ref>Sjo87</ref>. Clearly it is ad hoc.
180 It hopefully captures some elements of truth. The lower <ei>pT</ei> scale
181 a system has the larger its spatial extent and therefore the larger its
182 overlap with other systems. It could be argued that one should classify
183 individual initial-state partons by <ei>pT</ei> rather than the system
184 as a whole. However, for final-state radiation, a soft gluon radiated off
185 a hard parton is actually produced at late times and therefore probably
186 less likely to reconnect. In the balance, a classification by system
187 <ei>pT</ei> scale appears sensible as a first try.
190 Note that the reconnection is carried out before resonance decays are
191 considered. Colour inside a resonance therefore is not reconnected.
192 This is a deliberate choice, but certainly open to discussion and
193 extensions at a later stage, as is the rest of this procedure.
195 <h3>Further variables</h3>
197 <modeopen name="BeamRemnants:maxValQuark" default="3" min="0" max="5">
198 The maximum valence quark kind allowed in acceptable incoming beams,
199 for which multiple interactions are simulated. Default is that hadrons
200 may contain <ei>u</ei>, <ei>d</ei> and <ei>s</ei> quarks,
201 but not <ei>c</ei> and <ei>b</ei> ones, since sensible
202 kinematics has not really been worked out for the latter.
205 <modeopen name="BeamRemnants:companionPower" default="4" min="0" max="4">
206 When a sea quark has been found, a companion antisea quark ought to be
207 nearby in <ei>x</ei>. The shape of this distribution can be derived
208 from the gluon mother distribution convoluted with the
209 <ei>g -> q qbar</ei> splitting kernel. In practice, simple solutions
210 are only feasible if the gluon shape is assumed to be of the form
211 <ei>g(x) ~ (1 - x)^p / x</ei>, where <ei>p</ei> is an integer power,
212 the parameter above. Allowed values correspond to the cases programmed.
214 Since the whole framework is approximate anyway, this should be good
215 enough. Note that companions typically are found at small <ei>Q^2</ei>,
216 if at all, so the form is supposed to represent <ei>g(x)</ei> at small
217 <ei>Q^2</ei> scales, close to the lower cutoff for multiple interactions.
221 When assigning relative momentum fractions to beam-remnant partons,
222 valence quarks are chosen according to a distribution like
223 <ei>(1 - x)^power / sqrt(x)</ei>. This <ei>power</ei> is given below
224 for quarks in mesons, and separately for <ei>u</ei> and <ei>d</ei>
225 quarks in the proton, based on the approximate shape of low-<ei>Q^2</ei>
226 parton densities. The power for other baryons is derived from the
227 proton ones, by an appropriate mixing. The <ei>x</ei> of a diquark
228 is chosen as the sum of its two constituent <ei>x</ei> values, and can
229 thus be above unity. (A common rescaling of all remnant partons and
230 particles will fix that.) An additional enhancement of the diquark
231 momentum is obtained by its <ei>x</ei> value being rescaled by the
232 <code>valenceDiqEnhance</code> factor.
234 <parm name="BeamRemnants:valencePowerMeson" default="0.8" min="0.">
235 The abovementioned power for valence quarks in mesons.
238 <parm name="BeamRemnants:valencePowerUinP" default="3.5" min="0.">
239 The abovementioned power for valence <ei>u</ei> quarks in protons.
242 <parm name="BeamRemnants:valencePowerDinP" default="2.0" min="0.">
243 The abovementioned power for valence <ei>d</ei> quarks in protons.
246 <parm name="BeamRemnants:valenceDiqEnhance" default="2.0" min="0.5"
248 Enhancement factor for valence diqaurks in baryons, relative to the
249 simple sum of the two constituent quarks.
252 <flag name="BeamRemnants:allowJunction" default="on">
253 The <code>off</code> option is intended for debug purposes only, as
254 follows. When more than one valence quark is kicked out of a baryon
255 beam, as part of the multiple interactions scenario, the subsequent
256 hadronization is described in terms of a junction string topology.
257 This description involves a number of technical complications that
258 may make the program more unstable. As an alternative, by switching
259 this option off, junction configurations are rejected (which gives
260 an error message that the remnant flavour setup failed), and the
261 multiple interactions and showers are redone until a
262 junction-free topology is found.
265 <h3>Diffractive system</h3>
267 When an incoming hadron beam is diffractively excited, it is modeled
268 as if either a valence quark or a gluon is kicked out from the hadron.
269 In the former case this produces a simple string to the leftover
270 remnant, in the latter it gives a hairpin arrangement where a string
271 is stretched from one quark in the remnant, via the gluon, back to the
272 rest of the remnant. The latter ought to dominate at higher mass of
273 the diffractive system. Therefore an approximate behaviour like
279 <parm name="BeamRemnants:pickQuarkNorm" default="5.0" min="0.">
280 The abovementioned normalization <ei>N</ei> for the relative quark
281 rate in diffractive systems.
284 <parm name="BeamRemnants:pickQuarkPower" default="1.0" min="0.">
285 The abovementioned mass-dependence power <ei>p</ei> for the relative
286 quark rate in diffractive systems.
290 When a gluon is kicked out from the hadron, the longitudinal momentum
291 sharing between the the two remnant partons is determined by the
292 same parameters as above. It is plausible that the primordial
293 <ei>kT</ei> may be lower than in perturbative processes, however:
295 <parm name="BeamRemnants:diffPrimKTwidth" default="0.5" min="0.">
296 The width of Gaussian distributions in <ei>p_x</ei> and <ei>p_y</ei>
297 separately that is assigned as a primordial <ei>kT</ei> to the two
298 beam remnants when a gluon is kicked out of a diffractive system.
301 <parm name="BeamRemnants:diffLargeMassSuppress" default="2." min="0.">
302 The choice of longitudinal and transverse structure of a diffractive
303 beam remnant for a kicked-out gluon implies a remnant mass
304 <ei>m_rem</ei> distribution (i.e. quark plus diquark invariant mass
305 for a baryon beam) that knows no bounds. A suppression like
306 <ei>(1 - m_rem^2 / m_diff^2)^p</ei> is therefore introduced, where
307 <ei>p</ei> is the <code>diffLargeMassSuppress</code> parameter.
313 <!-- Copyright (C) 2008 Torbjorn Sjostrand -->