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 | |
---|