1 <chapter name="Fragmentation">
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.
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.
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.
22 <h3>Fragmentation functions</h3>
24 The <code>StringZ</code> class handles the choice of longitudinal
25 lightcone fraction <ei>z</ei> according to one of two possible
29 The Lund symmetric fragmentation function <ref>And83</ref> is the
30 only alternative for light quarks. It is of the form
32 f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z)
34 with the two main free parameters <ei>a</ei> and <ei>b</ei> to be
35 tuned to data. They are stored in
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.
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.
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
50 f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z)
52 This is only implemented for diquarks relative to normal quarks:
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>.
60 Finally, the Bowler modification <ref>Bow81</ref> introduces an extra
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
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.
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.
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.
83 As an alternative, it is possible to switch over to the
84 Peterson/SLAC formula <ref>Pet83</ref>
86 f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 )
88 for charm, bottom and heavier (defined as above) by the three flags
90 <flag name="StringZ:usePetersonC" default="off">
91 use Peterson for <ei>c</ei> quarks.
94 <flag name="StringZ:usePetersonB" default="off">
95 use Peterson for <ei>b</ei> quarks.
98 <flag name="StringZ:usePetersonH" default="off">
99 use Peterson for hypothetical heavier quarks.
103 When switched on, the corresponding epsilon values are chosen to be
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.
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.
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.
121 <h3>Fragmentation <ei>pT</ei></h3>
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.
130 d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2).
132 Then the total squared width is
134 <pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2.
136 It is this latter number that is stored in
138 <parm name="StringPT:sigma" default="0.304" min="0.0" max="1.0">
139 the width <ei>sigma</ei> in the fragmentation process.
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>.
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
156 <parm name="StringPT:enhancedFraction" default="0.01" min="0.0" max="0.1">
157 <ei>enhancedFraction</ei>,the fraction of string breaks with enhanced
161 <parm name="StringPT:enhancedWidth" default="2.0" min="1.0" max="5.0">
162 <ei>enhancedWidth</ei>,the enhancement of the width in this fraction.
165 <h3>Jet joining procedure</h3>
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.
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.
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.
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>.
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.)
200 <h3>Simplifying systems</h3>
202 There are a few situations when it is meaningful to simplify the
203 original task, one way or another.
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.
213 <parm name="FragmentationSystems:mJoin" default="0.2" 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.)
223 <parm name="FragmentationSystems:mJoinJunction" default="1.0"min="0.5" max="2.">
224 When the invariant mass of two of the quarks in a three-quark junction
225 string system becomes too small, the system is simplified to a
226 quark-diquark simple string. The requirement for this simplification
227 is that the diquark mass, minus the two quark masses, falls below
228 <ei>mJoinJunction</ei>. Gluons on the string between the junction and
229 the respective quark, if any, are counted as part of the quark
230 four-momentum. Those on the two combined legs are clustered with the
231 diquark when it is formed.
236 The <code>MiniStringFragmentation</code> machinery is only used when a
237 string system has so small invariant mass that normal string fragmentation
238 is difficult/impossible. Instead one or two particles are produced,
239 in the former case shuffling energy-momentum relative to another
240 colour singlet system in the event, while preserving the invariant
241 mass of that system. With one exception parameters are the same as
242 defined for normal string fragmentation, to the extent that they are
243 at all applicable in this case.
245 A discussion of the relevant physics is found in <ref>Nor00</ref>.
246 The current implementation does not completely abide to the scheme
247 presented there, however, but has in part been simplified. (In part
248 for greater clarity, in part since the class is not quite finished yet.)
250 <modeopen name="MiniStringFragmentation:nTry" default="2" min="1" max="10">
251 Whenever the machinery is called, first this many attempts are made
252 to pick two hadrons that the system fragments to. If the hadrons are
253 too massive the attempt will fail, but a new subsequent try could
254 involve other flavour and hadrons and thus still succeed.
255 After <ei>nTry</ei> attempts, instead an attempt is made to produce a
256 single hadron from the system. Should also this fail, some further
257 attempts at obtaining two hadrons will be made before eventually
261 <h3>Junction treatment</h3>
263 A junction topology corresponds to an Y arrangement of strings
264 i.e. where three string pieces have to be joined up in a junction.
265 Such topologies can arise if several valence quarks are kicked out
266 from a proton beam, or in baryon-number-violating SUSY decays.
267 Special attention is necessary to handle the region just around
268 the junction, where the baryon number topologically is located.
269 The junction fragmentation scheme is described in <ref>Sjo03</ref>.
270 The parameters in this section should not be touched except by experts.
272 <parm name="StringFragmentation:eNormJunction" default="2.0" min="0.5" max="10">
273 Used to find the effective rest frame of the junction, which is
274 complicated when the three string legs may contain additional
275 gluons between the junction and the endpoint. To this end,
276 a pull is defined as a weighed sum of the momenta on each leg,
277 where the weight is <ei>exp(- eSum / eNormJunction)</ei>, with
278 <ei>eSum</ei> the summed energy of all partons closer to the junction
279 than the currently considered one (in the junction rest frame).
280 Should in principle be (close to) <ei>sqrt((1 + a) / b)</ei>, with
281 <ei>a</ei> and <ei>b</ei> the parameters of the Lund symmetric
282 fragmentation function.
285 <parm name="StringFragmentation:eBothLeftJunction" default="1.0" min="0.5">
286 Retry (up to 10 times) when the first two considered strings in to a
287 junction both have a remaining energy (in the junction rest frame)
291 <parm name="StringFragmentation:eMaxLeftJunction" default="10.0" min="0.">
292 Retry (up to 10 times) when the first two considered strings in to a
293 junction has a highest remaining energy (in the junction rest frame)
294 above a random energy evenly distributed between
295 <ei>eBothLeftJunction</ei> and
296 <ei>eBothLeftJunction + eMaxLeftJunction</ei>
297 (drawn anew for each test).
300 <parm name="StringFragmentation:eMinLeftJunction" default="0.2" min="0.">
301 Retry (up to 10 times) when the invariant mass-squared of the final leg
302 and the leftover momentum of the first two treated legs falls below
303 <ei>eMinLeftJunction</ei> times the energy of the final leg (in the
304 junction rest frame).
309 <!-- Copyright (C) 2010 Torbjorn Sjostrand -->