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<title>Fragmentation</title>
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<h2>Fragmentation</h2>
32
Fragmentation in PYTHIA is based on the Lund string model
33
[<a href="Bibliography.php" target="page">And83, Sjo84</a>]. Several different aspects are involved in
34
the physics description, which here therefore is split accordingly.
35
This also, at least partly, reflect the set of classes involved in
36
the fragmentation machinery.
39
The variables collected here have a very wide span of usefulness.
40
Some would be central in any hadronization tuning exercise, others
41
should not be touched except by experts.
44
The fragmentation flavour-choice machinery is also used in a few
45
other places of the program, notably particle decays, and is thus
46
described on the separate <?php $filepath = $_GET["filepath"];
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echo "<a href='FlavourSelection.php?filepath=".$filepath."' target='page'>";?>Flavour
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<h3>Fragmentation functions</h3>
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The <code>StringZ</code> class handles the choice of longitudinal
53
lightcone fraction <i>z</i> according to one of two possible
57
The Lund symmetric fragmentation function [<a href="Bibliography.php" target="page">And83</a>] is the
58
only alternative for light quarks. It is of the form
60
f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z)
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with the two main free parameters <i>a</i> and <i>b</i> to be
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tuned to data. They are stored in
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<br/><br/><table><tr><td><strong>StringZ:aLund </td><td></td><td> <input type="text" name="1" value="0.3" size="20"/> (<code>default = <strong>0.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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The <i>a</i> parameter of the Lund symmetric fragmentation function.
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<br/><br/><table><tr><td><strong>StringZ:bLund </td><td></td><td> <input type="text" name="2" value="0.8" size="20"/> (<code>default = <strong>0.8</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 2.0</code>)</td></tr></table>
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The <i>b</i> parameter of the Lund symmetric fragmentation function.
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In principle, each flavour can have a different <i>a</i>. Then,
75
for going from an old flavour <i>i</i> to a new <i>j</i> one
78
f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z)
80
This is only implemented for diquarks relative to normal quarks:
82
<br/><br/><table><tr><td><strong>StringZ:aExtraDiquark </td><td></td><td> <input type="text" name="3" value="0.5" size="20"/> (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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allows a larger <i>a</i> for diquarks, with total
84
<i>a = aLund + aExtraDiquark</i>.
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Finally, the Bowler modification [<a href="Bibliography.php" target="page">Bow81</a>] introduces an extra
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for heavy quarks. To keep some flexibility, a multiplicative factor
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<i>r_Q</i> is introduced, which ought to be unity (provided that
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quark masses were uniquely defined) but can be set in
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<br/><br/><table><tr><td><strong>StringZ:rFactC </td><td></td><td> <input type="text" name="4" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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<i>r_c</i>, i.e. the above parameter for <i>c</i> quarks.
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<br/><br/><table><tr><td><strong>StringZ:rFactB </td><td></td><td> <input type="text" name="5" value="0.67" size="20"/> (<code>default = <strong>0.67</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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<i>r_b</i>, i.e. the above parameter for <i>b</i> quarks.
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<br/><br/><table><tr><td><strong>StringZ:rFactH </td><td></td><td> <input type="text" name="6" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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<i>r_h</i>, i.e. the above parameter for heavier hypothetical quarks,
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or in general any new coloured particle long-lived enough to hadronize.
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As an alternative, it is possible to switch over to the
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Peterson/SLAC formula [<a href="Bibliography.php" target="page">Pet83</a>]
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f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 )
116
for charm, bottom and heavier (defined as above) by the three flags
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<br/><br/><strong>StringZ:usePetersonC</strong> <input type="radio" name="7" value="on"><strong>On</strong>
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<input type="radio" name="7" value="off" checked="checked"><strong>Off</strong>
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(<code>default = <strong>off</strong></code>)<br/>
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use Peterson for <i>c</i> quarks.
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<br/><br/><strong>StringZ:usePetersonB</strong> <input type="radio" name="8" value="on"><strong>On</strong>
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<input type="radio" name="8" value="off" checked="checked"><strong>Off</strong>
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(<code>default = <strong>off</strong></code>)<br/>
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use Peterson for <i>b</i> quarks.
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<br/><br/><strong>StringZ:usePetersonH</strong> <input type="radio" name="9" value="on"><strong>On</strong>
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<input type="radio" name="9" value="off" checked="checked"><strong>Off</strong>
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(<code>default = <strong>off</strong></code>)<br/>
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use Peterson for hypothetical heavier quarks.
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When switched on, the corresponding epsilon values are chosen to be
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<br/><br/><table><tr><td><strong>StringZ:epsilonC </td><td></td><td> <input type="text" name="10" value="0.05" size="20"/> (<code>default = <strong>0.05</strong></code>; <code>minimum = 0.01</code>; <code>maximum = 0.25</code>)</td></tr></table>
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<i>epsilon_c</i>, i.e. the above parameter for <i>c</i> quarks.
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<br/><br/><table><tr><td><strong>StringZ:epsilonB </td><td></td><td> <input type="text" name="11" value="0.005" size="20"/> (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 0.025</code>)</td></tr></table>
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<i>epsilon_b</i>, i.e. the above parameter for <i>b</i> quarks.
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<br/><br/><table><tr><td><strong>StringZ:epsilonH </td><td></td><td> <input type="text" name="12" value="0.005" size="20"/> (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.0001</code>; <code>maximum = 0.25</code>)</td></tr></table>
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<i>epsilon_h</i>, i.e. the above parameter for hypothetical heavier
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quarks, normalized to the case where <i>m_h = m_b</i>. The actually
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used parameter is then <i>epsilon = epsilon_h * (m_b^2 / m_h^2)</i>.
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This allows a sensible scaling to a particle with an unknown higher
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mass without the need for a user intervention.
155
<h3>Fragmentation <i>pT</i></h3>
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The <code>StringPT</code> class handles the choice of fragmentation
158
<i>pT</i>. At each string breaking the quark and antiquark of the pair are
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supposed to receive opposite and compensating <i>pT</i> kicks according
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to a Gaussian distribution in <i>p_x</i> and <i>p_y</i> separately.
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Call <i>sigma_q</i> the width of the <i>p_x</i> and <i>p_y</i>
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distributions separately, i.e.
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d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2).
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Then the total squared width is
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<pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2.
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It is this latter number that is stored in
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<br/><br/><table><tr><td><strong>StringPT:sigma </td><td></td><td> <input type="text" name="13" value="0.304" size="20"/> (<code>default = <strong>0.304</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.0</code>)</td></tr></table>
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the width <i>sigma</i> in the fragmentation process.
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Since a normal hadron receives <i>pT</i> contributions for two string
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breakings, it has a <i><p_x^2>_had = <p_y^2>_had = sigma^2</i>,
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and thus <i><pT^2>_had = 2 sigma^2</i>.
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Some studies on isolated particles at LEP has indicated the need for
183
a slightly enhanced rate in the high-<i>pT</i> tail of the above
184
distribution. This would have to be reviewed in the context of a
185
complete retune of parton showers and hadronization, but for the
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moment we stay with the current recipe, to boost the above <i>pT</i>
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by a factor <i>enhancedWidth</i> for a small fraction
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<i>enhancedFraction</i> of the breakups, where
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<br/><br/><table><tr><td><strong>StringPT:enhancedFraction </td><td></td><td> <input type="text" name="14" value="0.01" size="20"/> (<code>default = <strong>0.01</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.</code>)</td></tr></table>
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<i>enhancedFraction</i>,the fraction of string breaks with enhanced
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<br/><br/><table><tr><td><strong>StringPT:enhancedWidth </td><td></td><td> <input type="text" name="15" value="2.0" size="20"/> (<code>default = <strong>2.0</strong></code>; <code>minimum = 1.0</code>; <code>maximum = 10.0</code>)</td></tr></table>
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<i>enhancedWidth</i>,the enhancement of the width in this fraction.
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<h3>Jet joining procedure</h3>
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String fragmentation is carried out iteratively from both string ends
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inwards, which means that the two chains of hadrons have to be joined up
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somewhere in the middle of the event. This joining is described by
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parameters that in principle follows from the standard fragmentation
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parameters, but in a way too complicated to parametrize. The dependence
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is rather mild, however, so for a sensible range of variation the
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parameters in this section should not be touched.
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<br/><br/><table><tr><td><strong>StringFragmentation:stopMass </td><td></td><td> <input type="text" name="16" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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Is used to define a <i>W_min = m_q1 + m_q2 + stopMass</i>,
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where <i>m_q1</i> and <i>m_q2</i> are the masses of the two
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current endpoint quarks or diquarks.
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<br/><br/><table><tr><td><strong>StringFragmentation:stopNewFlav </td><td></td><td> <input type="text" name="17" value="2.0" size="20"/> (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)</td></tr></table>
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Add to <i>W_min</i> an amount <i>stopNewFlav * m_q_last</i>,
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where <i>q_last</i> is the last <i>q qbar</i> pair produced
218
between the final two hadrons.
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<br/><br/><table><tr><td><strong>StringFragmentation:stopSmear </td><td></td><td> <input type="text" name="18" value="0.2" size="20"/> (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)</td></tr></table>
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The <i>W_min</i> above is then smeared uniformly in the range
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<i>W_min_smeared = W_min * [ 1 - stopSmear, 1 + stopSmear ]</i>.
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This <i>W_min_smeared</i> is then compared with the current remaining
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<i>W_transverse</i> to determine if there is energy left for further
229
particle production. If not, i.e. if
230
<i>W_transverse < W_min_smeared</i>, the final two particles are
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produced from what is currently left, if possible. (If not, the
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fragmentation process is started over.)
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<h3>Simplifying systems</h3>
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There are a few situations when it is meaningful to simplify the
237
original task, one way or another.
239
<br/><br/><table><tr><td><strong>HadronLevel:mStringMin </td><td></td><td> <input type="text" name="19" value="1." size="20"/> (<code>default = <strong>1.</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 1.5</code>)</td></tr></table>
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Decides whether a partonic system should be considered as a normal
241
string or a ministring, the latter only producing one or two primary
242
hadrons. The system mass should be above <i>mStringMin</i> plus the
243
sum of quark/diquark constituent masses for a normal string description,
244
else the ministring scenario is used.
247
<br/><br/><table><tr><td><strong>FragmentationSystems:mJoin </td><td></td><td> <input type="text" name="20" value="0.2" size="20"/> (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 1.</code>)</td></tr></table>
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When two colour-connected partons are very nearby, with at least
249
one being a gluon, they can be joined into one, to avoid technical
250
problems of very small string regions. The requirement for joining is
251
that the invariant mass of the pair is below <i>mJoin</i>, where a
252
gluon only counts with half its momentum, i.e. with its contribution
253
to the string region under consideration. (Note that, for technical
254
reasons, the 0.2 GeV lower limit is de facto hardcoded.)
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<br/><br/><table><tr><td><strong>FragmentationSystems:mJoinJunction </td><td></td><td> <input type="text" name="21" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 2.</code>)</td></tr></table>
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When the invariant mass of two of the quarks in a three-quark junction
259
string system becomes too small, the system is simplified to a
260
quark-diquark simple string. The requirement for this simplification
261
is that the diquark mass, minus the two quark masses, falls below
262
<i>mJoinJunction</i>. Gluons on the string between the junction and
263
the respective quark, if any, are counted as part of the quark
264
four-momentum. Those on the two combined legs are clustered with the
265
diquark when it is formed.
270
The <code>MiniStringFragmentation</code> machinery is only used when a
271
string system has so small invariant mass that normal string fragmentation
272
is difficult/impossible. Instead one or two particles are produced,
273
in the former case shuffling energy-momentum relative to another
274
colour singlet system in the event, while preserving the invariant
275
mass of that system. With one exception parameters are the same as
276
defined for normal string fragmentation, to the extent that they are
277
at all applicable in this case.
279
A discussion of the relevant physics is found in [<a href="Bibliography.php" target="page">Nor00</a>].
280
The current implementation does not completely abide to the scheme
281
presented there, however, but has in part been simplified. (In part
282
for greater clarity, in part since the class is not quite finished yet.)
284
<br/><br/><table><tr><td><strong>MiniStringFragmentation:nTry </td><td></td><td> <input type="text" name="22" value="2" size="20"/> (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 10</code>)</td></tr></table>
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Whenever the machinery is called, first this many attempts are made
286
to pick two hadrons that the system fragments to. If the hadrons are
287
too massive the attempt will fail, but a new subsequent try could
288
involve other flavour and hadrons and thus still succeed.
289
After <i>nTry</i> attempts, instead an attempt is made to produce a
290
single hadron from the system. Should also this fail, some further
291
attempts at obtaining two hadrons will be made before eventually
295
<h3>Junction treatment</h3>
297
A junction topology corresponds to an Y arrangement of strings
298
i.e. where three string pieces have to be joined up in a junction.
299
Such topologies can arise if several valence quarks are kicked out
300
from a proton beam, or in baryon-number-violating SUSY decays.
301
Special attention is necessary to handle the region just around
302
the junction, where the baryon number topologically is located.
303
The junction fragmentation scheme is described in [<a href="Bibliography.php" target="page">Sjo03</a>].
304
The parameters in this section should not be touched except by experts.
306
<br/><br/><table><tr><td><strong>StringFragmentation:eNormJunction </td><td></td><td> <input type="text" name="23" value="2.0" size="20"/> (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10</code>)</td></tr></table>
307
Used to find the effective rest frame of the junction, which is
308
complicated when the three string legs may contain additional
309
gluons between the junction and the endpoint. To this end,
310
a pull is defined as a weighed sum of the momenta on each leg,
311
where the weight is <i>exp(- eSum / eNormJunction)</i>, with
312
<i>eSum</i> the summed energy of all partons closer to the junction
313
than the currently considered one (in the junction rest frame).
314
Should in principle be (close to) <i>sqrt((1 + a) / b)</i>, with
315
<i>a</i> and <i>b</i> the parameters of the Lund symmetric
316
fragmentation function.
319
<br/><br/><table><tr><td><strong>StringFragmentation:eBothLeftJunction </td><td></td><td> <input type="text" name="24" value="1.0" size="20"/> (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>)</td></tr></table>
320
Retry (up to 10 times) when the first two considered strings in to a
321
junction both have a remaining energy (in the junction rest frame)
325
<br/><br/><table><tr><td><strong>StringFragmentation:eMaxLeftJunction </td><td></td><td> <input type="text" name="25" value="10.0" size="20"/> (<code>default = <strong>10.0</strong></code>; <code>minimum = 0.</code>)</td></tr></table>
326
Retry (up to 10 times) when the first two considered strings in to a
327
junction has a highest remaining energy (in the junction rest frame)
328
above a random energy evenly distributed between
329
<i>eBothLeftJunction</i> and
330
<i>eBothLeftJunction + eMaxLeftJunction</i>
331
(drawn anew for each test).
334
<br/><br/><table><tr><td><strong>StringFragmentation:eMinLeftJunction </td><td></td><td> <input type="text" name="26" value="0.2" size="20"/> (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.</code>)</td></tr></table>
335
Retry (up to 10 times) when the invariant mass-squared of the final leg
336
and the leftover momentum of the first two treated legs falls below
337
<i>eMinLeftJunction</i> times the energy of the final leg (in the
338
junction rest frame).
341
<input type="hidden" name="saved" value="1"/>
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echo "<input type='hidden' name='filepath' value='".$_GET["filepath"]."'/>"?>
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<table width="100%"><tr><td align="right"><input type="submit" value="Save Settings" /></td></tr></table>
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if($_POST["saved"] == 1)
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$filepath = $_POST["filepath"];
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$handle = fopen($filepath, 'a');
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if($_POST["1"] != "0.3")
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$data = "StringZ:aLund = ".$_POST["1"]."\n";
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fwrite($handle,$data);
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if($_POST["2"] != "0.8")
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$data = "StringZ:bLund = ".$_POST["2"]."\n";
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fwrite($handle,$data);
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if($_POST["3"] != "0.5")
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$data = "StringZ:aExtraDiquark = ".$_POST["3"]."\n";
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fwrite($handle,$data);
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if($_POST["4"] != "1.0")
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$data = "StringZ:rFactC = ".$_POST["4"]."\n";
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fwrite($handle,$data);
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if($_POST["5"] != "0.67")
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$data = "StringZ:rFactB = ".$_POST["5"]."\n";
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fwrite($handle,$data);
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if($_POST["6"] != "1.0")
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$data = "StringZ:rFactH = ".$_POST["6"]."\n";
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fwrite($handle,$data);
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if($_POST["7"] != "off")
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$data = "StringZ:usePetersonC = ".$_POST["7"]."\n";
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fwrite($handle,$data);
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if($_POST["8"] != "off")
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$data = "StringZ:usePetersonB = ".$_POST["8"]."\n";
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fwrite($handle,$data);
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if($_POST["9"] != "off")
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$data = "StringZ:usePetersonH = ".$_POST["9"]."\n";
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fwrite($handle,$data);
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if($_POST["10"] != "0.05")
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$data = "StringZ:epsilonC = ".$_POST["10"]."\n";
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fwrite($handle,$data);
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if($_POST["11"] != "0.005")
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$data = "StringZ:epsilonB = ".$_POST["11"]."\n";
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fwrite($handle,$data);
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if($_POST["12"] != "0.005")
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$data = "StringZ:epsilonH = ".$_POST["12"]."\n";
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fwrite($handle,$data);
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if($_POST["13"] != "0.304")
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$data = "StringPT:sigma = ".$_POST["13"]."\n";
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fwrite($handle,$data);
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if($_POST["14"] != "0.01")
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$data = "StringPT:enhancedFraction = ".$_POST["14"]."\n";
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fwrite($handle,$data);
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if($_POST["15"] != "2.0")
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$data = "StringPT:enhancedWidth = ".$_POST["15"]."\n";
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fwrite($handle,$data);
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if($_POST["16"] != "1.0")
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$data = "StringFragmentation:stopMass = ".$_POST["16"]."\n";
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fwrite($handle,$data);
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if($_POST["17"] != "2.0")
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$data = "StringFragmentation:stopNewFlav = ".$_POST["17"]."\n";
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fwrite($handle,$data);
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if($_POST["18"] != "0.2")
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$data = "StringFragmentation:stopSmear = ".$_POST["18"]."\n";
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fwrite($handle,$data);
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if($_POST["19"] != "1.")
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$data = "HadronLevel:mStringMin = ".$_POST["19"]."\n";
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fwrite($handle,$data);
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if($_POST["20"] != "0.2")
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$data = "FragmentationSystems:mJoin = ".$_POST["20"]."\n";
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fwrite($handle,$data);
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if($_POST["21"] != "1.0")
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$data = "FragmentationSystems:mJoinJunction = ".$_POST["21"]."\n";
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fwrite($handle,$data);
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if($_POST["22"] != "2")
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$data = "MiniStringFragmentation:nTry = ".$_POST["22"]."\n";
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fwrite($handle,$data);
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if($_POST["23"] != "2.0")
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$data = "StringFragmentation:eNormJunction = ".$_POST["23"]."\n";
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fwrite($handle,$data);
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if($_POST["24"] != "1.0")
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$data = "StringFragmentation:eBothLeftJunction = ".$_POST["24"]."\n";
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fwrite($handle,$data);
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if($_POST["25"] != "10.0")
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$data = "StringFragmentation:eMaxLeftJunction = ".$_POST["25"]."\n";
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fwrite($handle,$data);
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if($_POST["26"] != "0.2")
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$data = "StringFragmentation:eMinLeftJunction = ".$_POST["26"]."\n";
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fwrite($handle,$data);
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