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63ba5337 | 1 | <chapter name="Fragmentation"> |
2 | ||
3 | <h2>Fragmentation</h2> | |
4 | ||
5 | Fragmentation in PYTHIA is based on the Lund string model | |
6 | <ref>And83, Sjo84</ref>. Several different aspects are involved in | |
7 | the physics description, which here therefore is split accordingly. | |
8 | This also, at least partly, reflect the set of classes involved in | |
9 | the fragmentation machinery. | |
10 | ||
11 | <p/> | |
12 | The variables collected here have a very wide span of usefulness. | |
13 | Some would be central in any hadronization tuning exercise, others | |
14 | should not be touched except by experts. | |
15 | ||
16 | <p/> | |
17 | The fragmentation flavour-choice machinery is also used in a few | |
18 | other places of the program, notably particle decays, and is thus | |
19 | described on the separate <aloc href="FlavourSelection">Flavour | |
20 | Selection</aloc> page. | |
21 | ||
22 | <h3>Fragmentation functions</h3> | |
23 | ||
24 | The <code>StringZ</code> class handles the choice of longitudinal | |
25 | lightcone fraction <ei>z</ei> according to one of two possible | |
26 | shape sets. | |
27 | ||
28 | <p/> | |
29 | The Lund symmetric fragmentation function <ref>And83</ref> is the | |
30 | only alternative for light quarks. It is of the form | |
31 | <eq> | |
32 | f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z) | |
33 | </eq> | |
34 | with the two main free parameters <ei>a</ei> and <ei>b</ei> to be | |
35 | tuned to data. They are stored in | |
36 | ||
37 | <parm name="StringZ:aLund" default="0.3" min="0.0" max="2.0"> | |
38 | The <ei>a</ei> parameter of the Lund symmetric fragmentation function. | |
39 | </parm> | |
40 | ||
41 | <parm name="StringZ:bLund" default="0.8" min="0.2" max="2.0"> | |
42 | The <ei>b</ei> parameter of the Lund symmetric fragmentation function. | |
43 | </parm> | |
44 | ||
45 | <p/> | |
46 | In principle, each flavour can have a different <ei>a</ei>. Then, | |
47 | for going from an old flavour <ei>i</ei> to a new <ei>j</ei> one | |
48 | the shape is | |
49 | <eq> | |
50 | f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z) | |
51 | </eq> | |
52 | This is only implemented for diquarks relative to normal quarks: | |
53 | ||
54 | <parm name="StringZ:aExtraDiquark" default="0.5" min="0.0" max="2.0"> | |
55 | allows a larger <ei>a</ei> for diquarks, with total | |
56 | <ei>a = aLund + aExtraDiquark</ei>. | |
57 | </parm> | |
58 | ||
59 | <p/> | |
60 | Finally, the Bowler modification <ref>Bow81</ref> introduces an extra | |
61 | factor | |
62 | <eq> | |
63 | 1/z^{r_Q * b * m_Q^2} | |
64 | </eq> | |
65 | for heavy quarks. To keep some flexibility, a multiplicative factor | |
66 | <ei>r_Q</ei> is introduced, which ought to be unity (provided that | |
67 | quark masses were uniquely defined) but can be set in | |
68 | ||
69 | <parm name="StringZ:rFactC" default="1.0" min="0.0" max="2.0"> | |
70 | <ei>r_c</ei>, i.e. the above parameter for <ei>c</ei> quarks. | |
71 | </parm> | |
72 | ||
73 | <parm name="StringZ:rFactB" default="0.67" min="0.0" max="2.0"> | |
74 | <ei>r_b</ei>, i.e. the above parameter for <ei>b</ei> quarks. | |
75 | </parm> | |
76 | ||
77 | <parm name="StringZ:rFactH" default="1.0" min="0.0" max="2.0"> | |
78 | <ei>r_h</ei>, i.e. the above parameter for heavier hypothetical quarks, | |
79 | or in general any new coloured particle long-lived enough to hadronize. | |
80 | </parm> | |
81 | ||
82 | <p/> | |
83 | As an alternative, it is possible to switch over to the | |
84 | Peterson/SLAC formula <ref>Pet83</ref> | |
85 | <eq> | |
86 | f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 ) | |
87 | </eq> | |
88 | for charm, bottom and heavier (defined as above) by the three flags | |
89 | ||
90 | <flag name="StringZ:usePetersonC" default="off"> | |
91 | use Peterson for <ei>c</ei> quarks. | |
92 | </flag> | |
93 | ||
94 | <flag name="StringZ:usePetersonB" default="off"> | |
95 | use Peterson for <ei>b</ei> quarks. | |
96 | </flag> | |
97 | ||
98 | <flag name="StringZ:usePetersonH" default="off"> | |
99 | use Peterson for hypothetical heavier quarks. | |
100 | </flag> | |
101 | ||
102 | <p/> | |
103 | When switched on, the corresponding epsilon values are chosen to be | |
104 | ||
105 | <parm name="StringZ:epsilonC" default="0.05" min="0.01" max="0.25"> | |
106 | <ei>epsilon_c</ei>, i.e. the above parameter for <ei>c</ei> quarks. | |
107 | </parm> | |
108 | ||
109 | <parm name="StringZ:epsilonB" default="0.005" min="0.001" max="0.025"> | |
110 | <ei>epsilon_b</ei>, i.e. the above parameter for <ei>b</ei> quarks. | |
111 | </parm> | |
112 | ||
113 | <parm name="StringZ:epsilonH" default="0.005" min="0.0001" max="0.25"> | |
114 | <ei>epsilon_h</ei>, i.e. the above parameter for hypothetical heavier | |
115 | quarks, normalized to the case where <ei>m_h = m_b</ei>. The actually | |
116 | used parameter is then <ei>epsilon = epsilon_h * (m_b^2 / m_h^2)</ei>. | |
117 | This allows a sensible scaling to a particle with an unknown higher | |
118 | mass without the need for a user intervention. | |
119 | </parm> | |
120 | ||
121 | <h3>Fragmentation <ei>pT</ei></h3> | |
122 | ||
123 | The <code>StringPT</code> class handles the choice of fragmentation | |
124 | <ei>pT</ei>. At each string breaking the quark and antiquark of the pair are | |
125 | supposed to receive opposite and compensating <ei>pT</ei> kicks according | |
126 | to a Gaussian distribution in <ei>p_x</ei> and <ei>p_y</ei> separately. | |
127 | Call <ei>sigma_q</ei> the width of the <ei>p_x</ei> and <ei>p_y</ei> | |
128 | distributions separately, i.e. | |
129 | <eq> | |
130 | d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2). | |
131 | </eq> | |
132 | Then the total squared width is | |
133 | <eq> | |
134 | <pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2. | |
135 | </eq> | |
136 | It is this latter number that is stored in | |
137 | ||
138 | <parm name="StringPT:sigma" default="0.304" min="0.0" max="1.0"> | |
139 | the width <ei>sigma</ei> in the fragmentation process. | |
140 | </parm> | |
141 | ||
142 | <p/> | |
143 | Since a normal hadron receives <ei>pT</ei> contributions for two string | |
144 | breakings, it has a <ei><p_x^2>_had = <p_y^2>_had = sigma^2</ei>, | |
145 | and thus <ei><pT^2>_had = 2 sigma^2</ei>. | |
146 | ||
147 | <p/> | |
148 | Some studies on isolated particles at LEP has indicated the need for | |
149 | a slightly enhanced rate in the high-<ei>pT</ei> tail of the above | |
150 | distribution. This would have to be reviewed in the context of a | |
151 | complete retune of parton showers and hadronization, but for the | |
152 | moment we stay with the current recipe, to boost the above <ei>pT</ei> | |
153 | by a factor <ei>enhancedWidth</ei> for a small fraction | |
154 | <ei>enhancedFraction</ei> of the breakups, where | |
155 | ||
156 | <parm name="StringPT:enhancedFraction" default="0.01" min="0.0" max="1."> | |
157 | <ei>enhancedFraction</ei>,the fraction of string breaks with enhanced | |
158 | width. | |
159 | </parm> | |
160 | ||
161 | <parm name="StringPT:enhancedWidth" default="2.0" min="1.0" max="10.0"> | |
162 | <ei>enhancedWidth</ei>,the enhancement of the width in this fraction. | |
163 | </parm> | |
164 | ||
165 | <h3>Jet joining procedure</h3> | |
166 | ||
167 | String fragmentation is carried out iteratively from both string ends | |
168 | inwards, which means that the two chains of hadrons have to be joined up | |
169 | somewhere in the middle of the event. This joining is described by | |
170 | parameters that in principle follows from the standard fragmentation | |
171 | parameters, but in a way too complicated to parametrize. The dependence | |
172 | is rather mild, however, so for a sensible range of variation the | |
173 | parameters in this section should not be touched. | |
174 | ||
175 | <parm name="StringFragmentation:stopMass" default="1.0" min="0.0" max="2.0"> | |
176 | Is used to define a <ei>W_min = m_q1 + m_q2 + stopMass</ei>, | |
177 | where <ei>m_q1</ei> and <ei>m_q2</ei> are the masses of the two | |
178 | current endpoint quarks or diquarks. | |
179 | </parm> | |
180 | ||
181 | <parm name="StringFragmentation:stopNewFlav" default="2.0" min="0.0" max="2.0"> | |
182 | Add to <ei>W_min</ei> an amount <ei>stopNewFlav * m_q_last</ei>, | |
183 | where <ei>q_last</ei> is the last <ei>q qbar</ei> pair produced | |
184 | between the final two hadrons. | |
185 | </parm> | |
186 | ||
187 | <parm name="StringFragmentation:stopSmear" default="0.2" min="0.0" max="0.5"> | |
188 | The <ei>W_min</ei> above is then smeared uniformly in the range | |
189 | <ei>W_min_smeared = W_min * [ 1 - stopSmear, 1 + stopSmear ]</ei>. | |
190 | </parm> | |
191 | ||
192 | <p/> | |
193 | This <ei>W_min_smeared</ei> is then compared with the current remaining | |
194 | <ei>W_transverse</ei> to determine if there is energy left for further | |
195 | particle production. If not, i.e. if | |
196 | <ei>W_transverse < W_min_smeared</ei>, the final two particles are | |
197 | produced from what is currently left, if possible. (If not, the | |
198 | fragmentation process is started over.) | |
199 | ||
200 | <h3>Simplifying systems</h3> | |
201 | ||
202 | There are a few situations when it is meaningful to simplify the | |
203 | original task, one way or another. | |
204 | ||
205 | <parm name="HadronLevel:mStringMin" default="1." min="0.5" max="1.5"> | |
206 | Decides whether a partonic system should be considered as a normal | |
207 | string or a ministring, the latter only producing one or two primary | |
208 | hadrons. The system mass should be above <ei>mStringMin</ei> plus the | |
209 | sum of quark/diquark constituent masses for a normal string description, | |
210 | else the ministring scenario is used. | |
211 | </parm> | |
212 | ||
213 | <parm name="FragmentationSystems:mJoin" default="0.3" min="0.2" max="1."> | |
214 | When two colour-connected partons are very nearby, with at least | |
215 | one being a gluon, they can be joined into one, to avoid technical | |
216 | problems of very small string regions. The requirement for joining is | |
217 | that the invariant mass of the pair is below <ei>mJoin</ei>, where a | |
218 | gluon only counts with half its momentum, i.e. with its contribution | |
219 | to the string region under consideration. (Note that, for technical | |
220 | reasons, the 0.2 GeV lower limit is de facto hardcoded.) | |
221 | </parm> | |
222 | ||
223 | <parm name="FragmentationSystems:mJoinJunction" default="1.0"min="0.5" | |
224 | max="2."> | |
225 | When the invariant mass of two of the quarks in a three-quark junction | |
226 | string system becomes too small, the system is simplified to a | |
227 | quark-diquark simple string. The requirement for this simplification | |
228 | is that the diquark mass, minus the two quark masses, falls below | |
229 | <ei>mJoinJunction</ei>. Gluons on the string between the junction and | |
230 | the respective quark, if any, are counted as part of the quark | |
231 | four-momentum. Those on the two combined legs are clustered with the | |
232 | diquark when it is formed. | |
233 | </parm> | |
234 | ||
235 | <h3>Ministrings</h3> | |
236 | ||
237 | The <code>MiniStringFragmentation</code> machinery is only used when a | |
238 | string system has so small invariant mass that normal string fragmentation | |
239 | is difficult/impossible. Instead one or two particles are produced, | |
240 | in the former case shuffling energy-momentum relative to another | |
241 | colour singlet system in the event, while preserving the invariant | |
242 | mass of that system. With one exception parameters are the same as | |
243 | defined for normal string fragmentation, to the extent that they are | |
244 | at all applicable in this case. | |
245 | ||
246 | A discussion of the relevant physics is found in <ref>Nor00</ref>. | |
247 | The current implementation does not completely abide to the scheme | |
248 | presented there, however, but has in part been simplified. (In part | |
249 | for greater clarity, in part since the class is not quite finished yet.) | |
250 | ||
251 | <modeopen name="MiniStringFragmentation:nTry" default="2" min="1" max="10"> | |
252 | Whenever the machinery is called, first this many attempts are made | |
253 | to pick two hadrons that the system fragments to. If the hadrons are | |
254 | too massive the attempt will fail, but a new subsequent try could | |
255 | involve other flavour and hadrons and thus still succeed. | |
256 | After <ei>nTry</ei> attempts, instead an attempt is made to produce a | |
257 | single hadron from the system. Should also this fail, some further | |
258 | attempts at obtaining two hadrons will be made before eventually | |
259 | giving up. | |
260 | </modeopen> | |
261 | ||
262 | <h3>Junction treatment</h3> | |
263 | ||
264 | A junction topology corresponds to an Y arrangement of strings | |
265 | i.e. where three string pieces have to be joined up in a junction. | |
266 | Such topologies can arise if several valence quarks are kicked out | |
267 | from a proton beam, or in baryon-number-violating SUSY decays. | |
268 | Special attention is necessary to handle the region just around | |
269 | the junction, where the baryon number topologically is located. | |
270 | The junction fragmentation scheme is described in <ref>Sjo03</ref>. | |
271 | The parameters in this section should not be touched except by experts. | |
272 | ||
273 | <parm name="StringFragmentation:eNormJunction" default="2.0" min="0.5" | |
274 | max="10"> | |
275 | Used to find the effective rest frame of the junction, which is | |
276 | complicated when the three string legs may contain additional | |
277 | gluons between the junction and the endpoint. To this end, | |
278 | a pull is defined as a weighed sum of the momenta on each leg, | |
279 | where the weight is <ei>exp(- eSum / eNormJunction)</ei>, with | |
280 | <ei>eSum</ei> the summed energy of all partons closer to the junction | |
281 | than the currently considered one (in the junction rest frame). | |
282 | Should in principle be (close to) <ei>sqrt((1 + a) / b)</ei>, with | |
283 | <ei>a</ei> and <ei>b</ei> the parameters of the Lund symmetric | |
284 | fragmentation function. | |
285 | </parm> | |
286 | ||
287 | <parm name="StringFragmentation:eBothLeftJunction" default="1.0" min="0.5"> | |
288 | Retry (up to 10 times) when the first two considered strings in to a | |
289 | junction both have a remaining energy (in the junction rest frame) | |
290 | above this number. | |
291 | </parm> | |
292 | ||
293 | <parm name="StringFragmentation:eMaxLeftJunction" default="10.0" min="0."> | |
294 | Retry (up to 10 times) when the first two considered strings in to a | |
295 | junction has a highest remaining energy (in the junction rest frame) | |
296 | above a random energy evenly distributed between | |
297 | <ei>eBothLeftJunction</ei> and | |
298 | <ei>eBothLeftJunction + eMaxLeftJunction</ei> | |
299 | (drawn anew for each test). | |
300 | </parm> | |
301 | ||
302 | <parm name="StringFragmentation:eMinLeftJunction" default="0.2" min="0."> | |
303 | Retry (up to 10 times) when the invariant mass-squared of the final leg | |
304 | and the leftover momentum of the first two treated legs falls below | |
305 | <ei>eMinLeftJunction</ei> times the energy of the final leg (in the | |
306 | junction rest frame). | |
307 | </parm> | |
308 | ||
309 | </chapter> | |
310 | ||
311 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> | |
312 |