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-<head>
-<title>Fragmentation</title>
-<link rel="stylesheet" type="text/css" href="pythia.css"/>
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-</head>
-<body>
-
-<h2>Fragmentation</h2>
-
-Fragmentation in PYTHIA is based on the Lund string model
-[<a href="Bibliography.html" target="page">And83, Sjo84</a>]. Several different aspects are involved in
-the physics description, which here therefore is split accordingly.
-This also, at least partly, reflect the set of classes involved in
-the fragmentation machinery.
-
-<p/>
-The variables collected here have a very wide span of usefulness.
-Some would be central in any hadronization tuning exercise, others
-should not be touched except by experts.
-
-<p/>
-The fragmentation flavour-choice machinery is also used in a few
-other places of the program, notably particle decays, and is thus
-described on the separate <a href="FlavourSelection.html" target="page">Flavour
-Selection</a> page.
-
-<h3>Fragmentation functions</h3>
-
-The <code>StringZ</code> class handles the choice of longitudinal
-lightcone fraction <i>z</i> according to one of two possible
-shape sets.
-
-<p/>
-The Lund symmetric fragmentation function [<a href="Bibliography.html" target="page">And83</a>] is the
-only alternative for light quarks. It is of the form
-<br/><i>
- f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z)
-</i><br/>
-with the two main free parameters <i>a</i> and <i>b</i> to be
-tuned to data. They are stored in
-
-<p/><code>parm </code><strong> StringZ:aLund </strong>
- (<code>default = <strong>0.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-The <i>a</i> parameter of the Lund symmetric fragmentation function.
-
-
-<p/><code>parm </code><strong> StringZ:bLund </strong>
- (<code>default = <strong>0.8</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 2.0</code>)<br/>
-The <i>b</i> parameter of the Lund symmetric fragmentation function.
-
-
-<p/>
-In principle, each flavour can have a different <i>a</i>. Then,
-for going from an old flavour <i>i</i> to a new <i>j</i> one
-the shape is
-<br/><i>
- f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z)
-</i><br/>
-This is only implemented for diquarks relative to normal quarks:
-
-<p/><code>parm </code><strong> StringZ:aExtraDiquark </strong>
- (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-allows a larger <i>a</i> for diquarks, with total
-<i>a = aLund + aExtraDiquark</i>.
-
-
-<p/>
-Finally, the Bowler modification [<a href="Bibliography.html" target="page">Bow81</a>] introduces an extra
-factor
-<br/><i>
- 1/z^{r_Q * b * m_Q^2}
-</i><br/>
-for heavy quarks. To keep some flexibility, a multiplicative factor
-<i>r_Q</i> is introduced, which ought to be unity (provided that
-quark masses were uniquely defined) but can be set in
-
-<p/><code>parm </code><strong> StringZ:rFactC </strong>
- (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-<i>r_c</i>, i.e. the above parameter for <i>c</i> quarks.
-
-
-<p/><code>parm </code><strong> StringZ:rFactB </strong>
- (<code>default = <strong>0.67</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-<i>r_b</i>, i.e. the above parameter for <i>b</i> quarks.
-
-
-<p/><code>parm </code><strong> StringZ:rFactH </strong>
- (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-<i>r_h</i>, i.e. the above parameter for heavier hypothetical quarks,
-or in general any new coloured particle long-lived enough to hadronize.
-
-
-<p/>
-As an alternative, it is possible to switch over to the
-Peterson/SLAC formula [<a href="Bibliography.html" target="page">Pet83</a>]
-<br/><i>
- f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 )
-</i><br/>
-for charm, bottom and heavier (defined as above) by the three flags
-
-<p/><code>flag </code><strong> StringZ:usePetersonC </strong>
- (<code>default = <strong>off</strong></code>)<br/>
-use Peterson for <i>c</i> quarks.
-
-
-<p/><code>flag </code><strong> StringZ:usePetersonB </strong>
- (<code>default = <strong>off</strong></code>)<br/>
-use Peterson for <i>b</i> quarks.
-
-
-<p/><code>flag </code><strong> StringZ:usePetersonH </strong>
- (<code>default = <strong>off</strong></code>)<br/>
-use Peterson for hypothetical heavier quarks.
-
-
-<p/>
-When switched on, the corresponding epsilon values are chosen to be
-
-<p/><code>parm </code><strong> StringZ:epsilonC </strong>
- (<code>default = <strong>0.05</strong></code>; <code>minimum = 0.01</code>; <code>maximum = 0.25</code>)<br/>
-<i>epsilon_c</i>, i.e. the above parameter for <i>c</i> quarks.
-
-
-<p/><code>parm </code><strong> StringZ:epsilonB </strong>
- (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 0.025</code>)<br/>
-<i>epsilon_b</i>, i.e. the above parameter for <i>b</i> quarks.
-
-
-<p/><code>parm </code><strong> StringZ:epsilonH </strong>
- (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.0001</code>; <code>maximum = 0.25</code>)<br/>
-<i>epsilon_h</i>, i.e. the above parameter for hypothetical heavier
-quarks, normalized to the case where <i>m_h = m_b</i>. The actually
-used parameter is then <i>epsilon = epsilon_h * (m_b^2 / m_h^2)</i>.
-This allows a sensible scaling to a particle with an unknown higher
-mass without the need for a user intervention.
-
-
-<h3>Fragmentation <i>pT</i></h3>
-
-The <code>StringPT</code> class handles the choice of fragmentation
-<i>pT</i>. At each string breaking the quark and antiquark of the pair are
-supposed to receive opposite and compensating <i>pT</i> kicks according
-to a Gaussian distribution in <i>p_x</i> and <i>p_y</i> separately.
-Call <i>sigma_q</i> the width of the <i>p_x</i> and <i>p_y</i>
-distributions separately, i.e.
-<br/><i>
- d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2).
-</i><br/>
-Then the total squared width is
-<br/><i>
- <pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2.
-</i><br/>
-It is this latter number that is stored in
-
-<p/><code>parm </code><strong> StringPT:sigma </strong>
- (<code>default = <strong>0.304</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.0</code>)<br/>
-the width <i>sigma</i> in the fragmentation process.
-
-
-<p/>
-Since a normal hadron receives <i>pT</i> contributions for two string
-breakings, it has a <i><p_x^2>_had = <p_y^2>_had = sigma^2</i>,
-and thus <i><pT^2>_had = 2 sigma^2</i>.
-
-<p/>
-Some studies on isolated particles at LEP has indicated the need for
-a slightly enhanced rate in the high-<i>pT</i> tail of the above
-distribution. This would have to be reviewed in the context of a
-complete retune of parton showers and hadronization, but for the
-moment we stay with the current recipe, to boost the above <i>pT</i>
-by a factor <i>enhancedWidth</i> for a small fraction
-<i>enhancedFraction</i> of the breakups, where
-
-<p/><code>parm </code><strong> StringPT:enhancedFraction </strong>
- (<code>default = <strong>0.01</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.</code>)<br/>
-<i>enhancedFraction</i>,the fraction of string breaks with enhanced
-width.
-
-
-<p/><code>parm </code><strong> StringPT:enhancedWidth </strong>
- (<code>default = <strong>2.0</strong></code>; <code>minimum = 1.0</code>; <code>maximum = 10.0</code>)<br/>
-<i>enhancedWidth</i>,the enhancement of the width in this fraction.
-
-
-<h3>Jet joining procedure</h3>
-
-String fragmentation is carried out iteratively from both string ends
-inwards, which means that the two chains of hadrons have to be joined up
-somewhere in the middle of the event. This joining is described by
-parameters that in principle follows from the standard fragmentation
-parameters, but in a way too complicated to parametrize. The dependence
-is rather mild, however, so for a sensible range of variation the
-parameters in this section should not be touched.
-
-<p/><code>parm </code><strong> StringFragmentation:stopMass </strong>
- (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-Is used to define a <i>W_min = m_q1 + m_q2 + stopMass</i>,
-where <i>m_q1</i> and <i>m_q2</i> are the masses of the two
-current endpoint quarks or diquarks.
-
-
-<p/><code>parm </code><strong> StringFragmentation:stopNewFlav </strong>
- (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/>
-Add to <i>W_min</i> an amount <i>stopNewFlav * m_q_last</i>,
-where <i>q_last</i> is the last <i>q qbar</i> pair produced
-between the final two hadrons.
-
-
-<p/><code>parm </code><strong> StringFragmentation:stopSmear </strong>
- (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)<br/>
-The <i>W_min</i> above is then smeared uniformly in the range
-<i>W_min_smeared = W_min * [ 1 - stopSmear, 1 + stopSmear ]</i>.
-
-
-<p/>
-This <i>W_min_smeared</i> is then compared with the current remaining
-<i>W_transverse</i> to determine if there is energy left for further
-particle production. If not, i.e. if
-<i>W_transverse < W_min_smeared</i>, the final two particles are
-produced from what is currently left, if possible. (If not, the
-fragmentation process is started over.)
-
-<h3>Simplifying systems</h3>
-
-There are a few situations when it is meaningful to simplify the
-original task, one way or another.
-
-<p/><code>parm </code><strong> HadronLevel:mStringMin </strong>
- (<code>default = <strong>1.</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 1.5</code>)<br/>
-Decides whether a partonic system should be considered as a normal
-string or a ministring, the latter only producing one or two primary
-hadrons. The system mass should be above <i>mStringMin</i> plus the
-sum of quark/diquark constituent masses for a normal string description,
-else the ministring scenario is used.
-
-
-<p/><code>parm </code><strong> FragmentationSystems:mJoin </strong>
- (<code>default = <strong>0.3</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 1.</code>)<br/>
-When two colour-connected partons are very nearby, with at least
-one being a gluon, they can be joined into one, to avoid technical
-problems of very small string regions. The requirement for joining is
-that the invariant mass of the pair is below <i>mJoin</i>, where a
-gluon only counts with half its momentum, i.e. with its contribution
-to the string region under consideration. (Note that, for technical
-reasons, the 0.2 GeV lower limit is de facto hardcoded.)
-
-
-<p/><code>parm </code><strong> FragmentationSystems:mJoinJunction </strong>
- (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 2.</code>)<br/>
-When the invariant mass of two of the quarks in a three-quark junction
-string system becomes too small, the system is simplified to a
-quark-diquark simple string. The requirement for this simplification
-is that the diquark mass, minus the two quark masses, falls below
-<i>mJoinJunction</i>. Gluons on the string between the junction and
-the respective quark, if any, are counted as part of the quark
-four-momentum. Those on the two combined legs are clustered with the
-diquark when it is formed.
-
-
-<h3>Ministrings</h3>
-
-The <code>MiniStringFragmentation</code> machinery is only used when a
-string system has so small invariant mass that normal string fragmentation
-is difficult/impossible. Instead one or two particles are produced,
-in the former case shuffling energy-momentum relative to another
-colour singlet system in the event, while preserving the invariant
-mass of that system. With one exception parameters are the same as
-defined for normal string fragmentation, to the extent that they are
-at all applicable in this case.
-
-A discussion of the relevant physics is found in [<a href="Bibliography.html" target="page">Nor00</a>].
-The current implementation does not completely abide to the scheme
-presented there, however, but has in part been simplified. (In part
-for greater clarity, in part since the class is not quite finished yet.)
-
-<p/><code>mode </code><strong> MiniStringFragmentation:nTry </strong>
- (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 10</code>)<br/>
-Whenever the machinery is called, first this many attempts are made
-to pick two hadrons that the system fragments to. If the hadrons are
-too massive the attempt will fail, but a new subsequent try could
-involve other flavour and hadrons and thus still succeed.
-After <i>nTry</i> attempts, instead an attempt is made to produce a
-single hadron from the system. Should also this fail, some further
-attempts at obtaining two hadrons will be made before eventually
-giving up.
-
-
-<h3>Junction treatment</h3>
-
-A junction topology corresponds to an Y arrangement of strings
-i.e. where three string pieces have to be joined up in a junction.
-Such topologies can arise if several valence quarks are kicked out
-from a proton beam, or in baryon-number-violating SUSY decays.
-Special attention is necessary to handle the region just around
-the junction, where the baryon number topologically is located.
-The junction fragmentation scheme is described in [<a href="Bibliography.html" target="page">Sjo03</a>].
-The parameters in this section should not be touched except by experts.
-
-<p/><code>parm </code><strong> StringFragmentation:eNormJunction </strong>
- (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10</code>)<br/>
-Used to find the effective rest frame of the junction, which is
-complicated when the three string legs may contain additional
-gluons between the junction and the endpoint. To this end,
-a pull is defined as a weighed sum of the momenta on each leg,
-where the weight is <i>exp(- eSum / eNormJunction)</i>, with
-<i>eSum</i> the summed energy of all partons closer to the junction
-than the currently considered one (in the junction rest frame).
-Should in principle be (close to) <i>sqrt((1 + a) / b)</i>, with
-<i>a</i> and <i>b</i> the parameters of the Lund symmetric
-fragmentation function.
-
-
-<p/><code>parm </code><strong> StringFragmentation:eBothLeftJunction </strong>
- (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>)<br/>
-Retry (up to 10 times) when the first two considered strings in to a
-junction both have a remaining energy (in the junction rest frame)
-above this number.
-
-
-<p/><code>parm </code><strong> StringFragmentation:eMaxLeftJunction </strong>
- (<code>default = <strong>10.0</strong></code>; <code>minimum = 0.</code>)<br/>
-Retry (up to 10 times) when the first two considered strings in to a
-junction has a highest remaining energy (in the junction rest frame)
-above a random energy evenly distributed between
-<i>eBothLeftJunction</i> and
-<i>eBothLeftJunction + eMaxLeftJunction</i>
-(drawn anew for each test).
-
-
-<p/><code>parm </code><strong> StringFragmentation:eMinLeftJunction </strong>
- (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.</code>)<br/>
-Retry (up to 10 times) when the invariant mass-squared of the final leg
-and the leftover momentum of the first two treated legs falls below
-<i>eMinLeftJunction</i> times the energy of the final leg (in the
-junction rest frame).
-
-
-</body>
-</html>
-
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