| 1 | <html> |
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| 2 | <head> |
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| 3 | <title>Fragmentation</title> |
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| 4 | <link rel="stylesheet" type="text/css" href="pythia.css"/> |
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| 5 | <link rel="shortcut icon" href="pythia32.gif"/> |
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| 6 | </head> |
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| 7 | <body> |
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| 8 | |
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| 9 | <h2>Fragmentation</h2> |
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| 10 | |
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| 11 | Fragmentation in PYTHIA is based on the Lund string model |
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| 12 | [<a href="Bibliography.html" target="page">And83, Sjo84</a>]. Several different aspects are involved in |
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| 13 | the physics description, which here therefore is split accordingly. |
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| 14 | This also, at least partly, reflect the set of classes involved in |
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| 15 | the fragmentation machinery. |
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| 16 | |
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| 17 | <p/> |
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| 18 | The variables collected here have a very wide span of usefulness. |
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| 19 | Some would be central in any hadronization tuning exercise, others |
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| 20 | should not be touched except by experts. |
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| 21 | |
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| 22 | <p/> |
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| 23 | The fragmentation flavour-choice machinery is also used in a few |
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| 24 | other places of the program, notably particle decays, and is thus |
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| 25 | described on the separate <a href="FlavourSelection.html" target="page">Flavour |
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| 26 | Selection</a> page. |
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| 27 | |
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| 28 | <h3>Fragmentation functions</h3> |
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| 29 | |
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| 30 | The <code>StringZ</code> class handles the choice of longitudinal |
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| 31 | lightcone fraction <i>z</i> according to one of two possible |
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| 32 | shape sets. |
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| 33 | |
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| 34 | <p/> |
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| 35 | The Lund symmetric fragmentation function [<a href="Bibliography.html" target="page">And83</a>] is the |
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| 36 | only alternative for light quarks. It is of the form |
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| 37 | <br/><i> |
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| 38 | f(z) = (1/z) * (1-z)^a * exp(-b m_T^2 / z) |
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| 39 | </i><br/> |
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| 40 | with the two main free parameters <i>a</i> and <i>b</i> to be |
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| 41 | tuned to data. They are stored in |
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| 42 | |
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| 43 | <p/><code>parm </code><strong> StringZ:aLund </strong> |
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| 44 | (<code>default = <strong>0.3</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 45 | The <i>a</i> parameter of the Lund symmetric fragmentation function. |
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| 46 | |
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| 47 | |
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| 48 | <p/><code>parm </code><strong> StringZ:bLund </strong> |
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| 49 | (<code>default = <strong>0.8</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 2.0</code>)<br/> |
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| 50 | The <i>b</i> parameter of the Lund symmetric fragmentation function. |
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| 51 | |
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| 52 | |
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| 53 | <p/> |
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| 54 | In principle, each flavour can have a different <i>a</i>. Then, |
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| 55 | for going from an old flavour <i>i</i> to a new <i>j</i> one |
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| 56 | the shape is |
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| 57 | <br/><i> |
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| 58 | f(z) = (1/z) * z^{a_i} * ((1-z)/z)^{a_j} * exp(-b * m_T^2 / z) |
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| 59 | </i><br/> |
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| 60 | This is only implemented for diquarks relative to normal quarks: |
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| 61 | |
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| 62 | <p/><code>parm </code><strong> StringZ:aExtraDiquark </strong> |
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| 63 | (<code>default = <strong>0.5</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 64 | allows a larger <i>a</i> for diquarks, with total |
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| 65 | <i>a = aLund + aExtraDiquark</i>. |
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| 66 | |
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| 67 | |
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| 68 | <p/> |
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| 69 | Finally, the Bowler modification [<a href="Bibliography.html" target="page">Bow81</a>] introduces an extra |
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| 70 | factor |
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| 71 | <br/><i> |
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| 72 | 1/z^{r_Q * b * m_Q^2} |
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| 73 | </i><br/> |
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| 74 | for heavy quarks. To keep some flexibility, a multiplicative factor |
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| 75 | <i>r_Q</i> is introduced, which ought to be unity (provided that |
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| 76 | quark masses were uniquely defined) but can be set in |
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| 77 | |
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| 78 | <p/><code>parm </code><strong> StringZ:rFactC </strong> |
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| 79 | (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 80 | <i>r_c</i>, i.e. the above parameter for <i>c</i> quarks. |
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| 81 | |
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| 82 | |
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| 83 | <p/><code>parm </code><strong> StringZ:rFactB </strong> |
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| 84 | (<code>default = <strong>0.67</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 85 | <i>r_b</i>, i.e. the above parameter for <i>b</i> quarks. |
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| 86 | |
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| 87 | |
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| 88 | <p/><code>parm </code><strong> StringZ:rFactH </strong> |
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| 89 | (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 90 | <i>r_h</i>, i.e. the above parameter for heavier hypothetical quarks, |
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| 91 | or in general any new coloured particle long-lived enough to hadronize. |
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| 92 | |
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| 93 | |
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| 94 | <p/> |
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| 95 | As an alternative, it is possible to switch over to the |
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| 96 | Peterson/SLAC formula [<a href="Bibliography.html" target="page">Pet83</a>] |
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| 97 | <br/><i> |
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| 98 | f(z) = 1 / ( z * (1 - 1/z - epsilon/(1-z))^2 ) |
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| 99 | </i><br/> |
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| 100 | for charm, bottom and heavier (defined as above) by the three flags |
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| 101 | |
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| 102 | <p/><code>flag </code><strong> StringZ:usePetersonC </strong> |
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| 103 | (<code>default = <strong>off</strong></code>)<br/> |
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| 104 | use Peterson for <i>c</i> quarks. |
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| 105 | |
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| 106 | |
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| 107 | <p/><code>flag </code><strong> StringZ:usePetersonB </strong> |
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| 108 | (<code>default = <strong>off</strong></code>)<br/> |
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| 109 | use Peterson for <i>b</i> quarks. |
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| 110 | |
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| 111 | |
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| 112 | <p/><code>flag </code><strong> StringZ:usePetersonH </strong> |
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| 113 | (<code>default = <strong>off</strong></code>)<br/> |
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| 114 | use Peterson for hypothetical heavier quarks. |
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| 115 | |
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| 116 | |
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| 117 | <p/> |
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| 118 | When switched on, the corresponding epsilon values are chosen to be |
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| 119 | |
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| 120 | <p/><code>parm </code><strong> StringZ:epsilonC </strong> |
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| 121 | (<code>default = <strong>0.05</strong></code>; <code>minimum = 0.01</code>; <code>maximum = 0.25</code>)<br/> |
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| 122 | <i>epsilon_c</i>, i.e. the above parameter for <i>c</i> quarks. |
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| 123 | |
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| 124 | |
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| 125 | <p/><code>parm </code><strong> StringZ:epsilonB </strong> |
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| 126 | (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 0.025</code>)<br/> |
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| 127 | <i>epsilon_b</i>, i.e. the above parameter for <i>b</i> quarks. |
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| 128 | |
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| 129 | |
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| 130 | <p/><code>parm </code><strong> StringZ:epsilonH </strong> |
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| 131 | (<code>default = <strong>0.005</strong></code>; <code>minimum = 0.0001</code>; <code>maximum = 0.25</code>)<br/> |
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| 132 | <i>epsilon_h</i>, i.e. the above parameter for hypothetical heavier |
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| 133 | quarks, normalized to the case where <i>m_h = m_b</i>. The actually |
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| 134 | used parameter is then <i>epsilon = epsilon_h * (m_b^2 / m_h^2)</i>. |
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| 135 | This allows a sensible scaling to a particle with an unknown higher |
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| 136 | mass without the need for a user intervention. |
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| 137 | |
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| 138 | |
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| 139 | <h3>Fragmentation <i>pT</i></h3> |
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| 140 | |
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| 141 | The <code>StringPT</code> class handles the choice of fragmentation |
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| 142 | <i>pT</i>. At each string breaking the quark and antiquark of the pair are |
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| 143 | supposed to receive opposite and compensating <i>pT</i> kicks according |
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| 144 | to a Gaussian distribution in <i>p_x</i> and <i>p_y</i> separately. |
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| 145 | Call <i>sigma_q</i> the width of the <i>p_x</i> and <i>p_y</i> |
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| 146 | distributions separately, i.e. |
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| 147 | <br/><i> |
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| 148 | d(Prob) = exp( -(p_x^2 + p_y^2) / 2 sigma_q^2). |
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| 149 | </i><br/> |
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| 150 | Then the total squared width is |
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| 151 | <br/><i> |
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| 152 | <pT^2> = <p_x^2> + <p_y^2> = 2 sigma_q^2 = sigma^2. |
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| 153 | </i><br/> |
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| 154 | It is this latter number that is stored in |
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| 155 | |
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| 156 | <p/><code>parm </code><strong> StringPT:sigma </strong> |
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| 157 | (<code>default = <strong>0.304</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.0</code>)<br/> |
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| 158 | the width <i>sigma</i> in the fragmentation process. |
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| 159 | |
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| 160 | |
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| 161 | <p/> |
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| 162 | Since a normal hadron receives <i>pT</i> contributions for two string |
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| 163 | breakings, it has a <i><p_x^2>_had = <p_y^2>_had = sigma^2</i>, |
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| 164 | and thus <i><pT^2>_had = 2 sigma^2</i>. |
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| 165 | |
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| 166 | <p/> |
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| 167 | Some studies on isolated particles at LEP has indicated the need for |
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| 168 | a slightly enhanced rate in the high-<i>pT</i> tail of the above |
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| 169 | distribution. This would have to be reviewed in the context of a |
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| 170 | complete retune of parton showers and hadronization, but for the |
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| 171 | moment we stay with the current recipe, to boost the above <i>pT</i> |
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| 172 | by a factor <i>enhancedWidth</i> for a small fraction |
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| 173 | <i>enhancedFraction</i> of the breakups, where |
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| 174 | |
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| 175 | <p/><code>parm </code><strong> StringPT:enhancedFraction </strong> |
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| 176 | (<code>default = <strong>0.01</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 1.</code>)<br/> |
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| 177 | <i>enhancedFraction</i>,the fraction of string breaks with enhanced |
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| 178 | width. |
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| 179 | |
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| 180 | |
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| 181 | <p/><code>parm </code><strong> StringPT:enhancedWidth </strong> |
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| 182 | (<code>default = <strong>2.0</strong></code>; <code>minimum = 1.0</code>; <code>maximum = 10.0</code>)<br/> |
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| 183 | <i>enhancedWidth</i>,the enhancement of the width in this fraction. |
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| 184 | |
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| 185 | |
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| 186 | <h3>Jet joining procedure</h3> |
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| 187 | |
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| 188 | String fragmentation is carried out iteratively from both string ends |
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| 189 | inwards, which means that the two chains of hadrons have to be joined up |
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| 190 | somewhere in the middle of the event. This joining is described by |
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| 191 | parameters that in principle follows from the standard fragmentation |
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| 192 | parameters, but in a way too complicated to parametrize. The dependence |
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| 193 | is rather mild, however, so for a sensible range of variation the |
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| 194 | parameters in this section should not be touched. |
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| 195 | |
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| 196 | <p/><code>parm </code><strong> StringFragmentation:stopMass </strong> |
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| 197 | (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 198 | Is used to define a <i>W_min = m_q1 + m_q2 + stopMass</i>, |
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| 199 | where <i>m_q1</i> and <i>m_q2</i> are the masses of the two |
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| 200 | current endpoint quarks or diquarks. |
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| 201 | |
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| 202 | |
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| 203 | <p/><code>parm </code><strong> StringFragmentation:stopNewFlav </strong> |
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| 204 | (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 2.0</code>)<br/> |
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| 205 | Add to <i>W_min</i> an amount <i>stopNewFlav * m_q_last</i>, |
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| 206 | where <i>q_last</i> is the last <i>q qbar</i> pair produced |
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| 207 | between the final two hadrons. |
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| 208 | |
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| 209 | |
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| 210 | <p/><code>parm </code><strong> StringFragmentation:stopSmear </strong> |
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| 211 | (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.0</code>; <code>maximum = 0.5</code>)<br/> |
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| 212 | The <i>W_min</i> above is then smeared uniformly in the range |
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| 213 | <i>W_min_smeared = W_min * [ 1 - stopSmear, 1 + stopSmear ]</i>. |
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| 214 | |
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| 215 | |
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| 216 | <p/> |
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| 217 | This <i>W_min_smeared</i> is then compared with the current remaining |
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| 218 | <i>W_transverse</i> to determine if there is energy left for further |
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| 219 | particle production. If not, i.e. if |
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| 220 | <i>W_transverse < W_min_smeared</i>, the final two particles are |
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| 221 | produced from what is currently left, if possible. (If not, the |
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| 222 | fragmentation process is started over.) |
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| 223 | |
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| 224 | <h3>Simplifying systems</h3> |
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| 225 | |
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| 226 | There are a few situations when it is meaningful to simplify the |
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| 227 | original task, one way or another. |
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| 228 | |
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| 229 | <p/><code>parm </code><strong> HadronLevel:mStringMin </strong> |
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| 230 | (<code>default = <strong>1.</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 1.5</code>)<br/> |
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| 231 | Decides whether a partonic system should be considered as a normal |
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| 232 | string or a ministring, the latter only producing one or two primary |
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| 233 | hadrons. The system mass should be above <i>mStringMin</i> plus the |
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| 234 | sum of quark/diquark constituent masses for a normal string description, |
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| 235 | else the ministring scenario is used. |
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| 236 | |
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| 237 | |
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| 238 | <p/><code>parm </code><strong> FragmentationSystems:mJoin </strong> |
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| 239 | (<code>default = <strong>0.3</strong></code>; <code>minimum = 0.2</code>; <code>maximum = 1.</code>)<br/> |
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| 240 | When two colour-connected partons are very nearby, with at least |
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| 241 | one being a gluon, they can be joined into one, to avoid technical |
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| 242 | problems of very small string regions. The requirement for joining is |
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| 243 | that the invariant mass of the pair is below <i>mJoin</i>, where a |
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| 244 | gluon only counts with half its momentum, i.e. with its contribution |
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| 245 | to the string region under consideration. (Note that, for technical |
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| 246 | reasons, the 0.2 GeV lower limit is de facto hardcoded.) |
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| 247 | |
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| 248 | |
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| 249 | <p/><code>parm </code><strong> FragmentationSystems:mJoinJunction </strong> |
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| 250 | (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 2.</code>)<br/> |
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| 251 | When the invariant mass of two of the quarks in a three-quark junction |
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| 252 | string system becomes too small, the system is simplified to a |
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| 253 | quark-diquark simple string. The requirement for this simplification |
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| 254 | is that the diquark mass, minus the two quark masses, falls below |
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| 255 | <i>mJoinJunction</i>. Gluons on the string between the junction and |
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| 256 | the respective quark, if any, are counted as part of the quark |
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| 257 | four-momentum. Those on the two combined legs are clustered with the |
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| 258 | diquark when it is formed. |
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| 259 | |
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| 260 | |
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| 261 | <h3>Ministrings</h3> |
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| 262 | |
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| 263 | The <code>MiniStringFragmentation</code> machinery is only used when a |
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| 264 | string system has so small invariant mass that normal string fragmentation |
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| 265 | is difficult/impossible. Instead one or two particles are produced, |
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| 266 | in the former case shuffling energy-momentum relative to another |
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| 267 | colour singlet system in the event, while preserving the invariant |
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| 268 | mass of that system. With one exception parameters are the same as |
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| 269 | defined for normal string fragmentation, to the extent that they are |
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| 270 | at all applicable in this case. |
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| 271 | |
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| 272 | A discussion of the relevant physics is found in [<a href="Bibliography.html" target="page">Nor00</a>]. |
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| 273 | The current implementation does not completely abide to the scheme |
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| 274 | presented there, however, but has in part been simplified. (In part |
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| 275 | for greater clarity, in part since the class is not quite finished yet.) |
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| 276 | |
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| 277 | <p/><code>mode </code><strong> MiniStringFragmentation:nTry </strong> |
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| 278 | (<code>default = <strong>2</strong></code>; <code>minimum = 1</code>; <code>maximum = 10</code>)<br/> |
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| 279 | Whenever the machinery is called, first this many attempts are made |
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| 280 | to pick two hadrons that the system fragments to. If the hadrons are |
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| 281 | too massive the attempt will fail, but a new subsequent try could |
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| 282 | involve other flavour and hadrons and thus still succeed. |
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| 283 | After <i>nTry</i> attempts, instead an attempt is made to produce a |
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| 284 | single hadron from the system. Should also this fail, some further |
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| 285 | attempts at obtaining two hadrons will be made before eventually |
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| 286 | giving up. |
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| 287 | |
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| 288 | |
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| 289 | <h3>Junction treatment</h3> |
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| 290 | |
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| 291 | A junction topology corresponds to an Y arrangement of strings |
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| 292 | i.e. where three string pieces have to be joined up in a junction. |
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| 293 | Such topologies can arise if several valence quarks are kicked out |
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| 294 | from a proton beam, or in baryon-number-violating SUSY decays. |
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| 295 | Special attention is necessary to handle the region just around |
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| 296 | the junction, where the baryon number topologically is located. |
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| 297 | The junction fragmentation scheme is described in [<a href="Bibliography.html" target="page">Sjo03</a>]. |
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| 298 | The parameters in this section should not be touched except by experts. |
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| 299 | |
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| 300 | <p/><code>parm </code><strong> StringFragmentation:eNormJunction </strong> |
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| 301 | (<code>default = <strong>2.0</strong></code>; <code>minimum = 0.5</code>; <code>maximum = 10</code>)<br/> |
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| 302 | Used to find the effective rest frame of the junction, which is |
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| 303 | complicated when the three string legs may contain additional |
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| 304 | gluons between the junction and the endpoint. To this end, |
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| 305 | a pull is defined as a weighed sum of the momenta on each leg, |
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| 306 | where the weight is <i>exp(- eSum / eNormJunction)</i>, with |
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| 307 | <i>eSum</i> the summed energy of all partons closer to the junction |
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| 308 | than the currently considered one (in the junction rest frame). |
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| 309 | Should in principle be (close to) <i>sqrt((1 + a) / b)</i>, with |
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| 310 | <i>a</i> and <i>b</i> the parameters of the Lund symmetric |
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| 311 | fragmentation function. |
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| 312 | |
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| 313 | |
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| 314 | <p/><code>parm </code><strong> StringFragmentation:eBothLeftJunction </strong> |
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| 315 | (<code>default = <strong>1.0</strong></code>; <code>minimum = 0.5</code>)<br/> |
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| 316 | Retry (up to 10 times) when the first two considered strings in to a |
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| 317 | junction both have a remaining energy (in the junction rest frame) |
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| 318 | above this number. |
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| 319 | |
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| 320 | |
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| 321 | <p/><code>parm </code><strong> StringFragmentation:eMaxLeftJunction </strong> |
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| 322 | (<code>default = <strong>10.0</strong></code>; <code>minimum = 0.</code>)<br/> |
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| 323 | Retry (up to 10 times) when the first two considered strings in to a |
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| 324 | junction has a highest remaining energy (in the junction rest frame) |
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| 325 | above a random energy evenly distributed between |
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| 326 | <i>eBothLeftJunction</i> and |
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| 327 | <i>eBothLeftJunction + eMaxLeftJunction</i> |
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| 328 | (drawn anew for each test). |
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| 329 | |
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| 330 | |
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| 331 | <p/><code>parm </code><strong> StringFragmentation:eMinLeftJunction </strong> |
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| 332 | (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.</code>)<br/> |
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| 333 | Retry (up to 10 times) when the invariant mass-squared of the final leg |
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| 334 | and the leftover momentum of the first two treated legs falls below |
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| 335 | <i>eMinLeftJunction</i> times the energy of the final leg (in the |
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| 336 | junction rest frame). |
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| 337 | |
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| 338 | |
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| 339 | </body> |
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| 340 | </html> |
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| 341 | |
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| 342 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |
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| 343 | |
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