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2<head>
3<title>Bose-Einstein Effects</title>
4<link rel="stylesheet" type="text/css" href="pythia.css"/>
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7<body>
8
9<h2>Bose-Einstein Effects</h2>
10
11The <code>BoseEinstein</code> class performs shifts of momenta
12of identical particles to provide a crude estimate of
13Bose-Einstein effects. The algorithm is the BE_32 one described in
14[<a href="Bibliography.html" target="page">Lon95</a>], with a Gaussian parametrization of the enhancement.
15We emphasize that this approach is not based on any first-principles
16quantum mechanical description of interference phenomena; such
17approaches anyway have many problems to contend with. Instead a cruder
18but more robust approach is adopted, wherein BE effects are introduced
19after the event has already been generated, with the exception of the
20decays of long-lived particles. The trick is that momenta of identical
21particles are shifted relative to each other so as to provide an
22enhancement of pairs closely separated, which is compensated by a
23depletion of pairs in an intermediate region of separation.
24
25<p/>
26More precisely, the intended target form of the BE corrrelations in
27BE_32 is
28<br/><i>
29f_2(Q) = (1 + lambda * exp(-Q^2 R^2))
30       * (1 + alpha * lambda * exp(-Q^2 R^2/9) * (1 - exp(-Q^2 R^2/4)))
31</i><br/>
32where <i>Q^2 = (p_1 + p_2)^2 - (m_1 + m_2)^2</i>.
33Here the strength <i>lambda</i> and effective radius <i>R</i>
34are the two main parameters. The first factor of the
35equation is implemented by pulling pairs of identical hadrons closer
36to each other. This is done in such a way that three-monentum is
37conserved, but at the price of a small but non-negligible negative
38shift in the energy of the event. The second factor compensates this
39by pushing particles apart. The negative <i>alpha</i> parameter is
40determined iteratively, separately for each event, so as to restore
41energy conservation. The effective radius parameter is here <i>R/3</i>,
42i.e. effects extend further out in <i>Q</i>. Without the dampening
43<i>(1 - exp(-Q^2 R^2/4))</i> in the second factor the value at the
44origin would become <i>f_2(0) = (1 + lambda) * (1 + alpha * lambda)</i>,
45with it the desired value <i>f_2(0) = (1 + lambda)</i> is restored.
46The end result can be viewed as a poor man's rendering of a rapidly
47dampened oscillatory behaviour in <i>Q</i>.
48
49<p/>
50Further details can be found in [<a href="Bibliography.html" target="page">Lon95</a>]. For instance, the
51target is implemented under the assumption that the initial distribution
52in <i>Q</i> can be well approximated by pure phase space at small
53values, and implicitly generates higher-order effects by the way
54the algorithm is implemented. The algorithm is applied after the decay
55of short-lived resonances such as the <i>rho</i>, but before the decay
56of longer-lived particles.
57
58<p/>
59This algorithm is known to do a reasonable job of describing BE
60phenomena at LEP. It has not been tested against data for hadron
61colliders, to the best of our knowledge, so one should exercise some
62judgement before using it. Therefore by default the master switch
63<a href="MasterSwitches.html" target="page">HadronLevel:BoseEinstein</a> is off.
64Furthermore, the implementation found here is not (yet) as
65sophisticated as the one used at LEP2, in that no provision is made
66for particles from separate colour singlet systems, such as
67<i>W</i>'s and <i>Z</i>'s, interfering only at a reduced rate.
68
69<p/>
70<b>Warning:</b> The algorithm will create a new copy of each particle
71with shifted momentum by BE effects, with status code 99, while the
72original particle with the original momentum at the same time will be
73marked as decayed. This means that if you e.g. search for all
74<i>pi+-</i> in an event you will often obtain the same particle twice.
75One way to protect yourself from unwanted doublecounting is to
76use only particles with a positive status code, i.e. ones for which
77<code>event[i].isFinal()</code> is <code>true</code>.
78 
79
80<h3>Main parameters</h3>
81
82<p/><code>flag&nbsp; </code><strong> BoseEinstein:Pion &nbsp;</strong> 
83 (<code>default = <strong>on</strong></code>)<br/>
84Include effects or not for identical <i>pi^+</i>, <i>pi^-</i>
85and <i>pi^0</i>.
86 
87
88<p/><code>flag&nbsp; </code><strong> BoseEinstein:Kaon &nbsp;</strong> 
89 (<code>default = <strong>on</strong></code>)<br/>
90Include effects or not for identical <i>K^+</i>, <i>K^-</i>,
91<i>K_S^0</i> and <i>K_L^0</i>.
92 
93
94<p/><code>flag&nbsp; </code><strong> BoseEinstein:Eta &nbsp;</strong> 
95 (<code>default = <strong>on</strong></code>)<br/>
96Include effects or not for identical <i>eta</i> and <i>eta'</i>.
97 
98
99<p/><code>parm&nbsp; </code><strong> BoseEinstein:lambda &nbsp;</strong> 
100 (<code>default = <strong>1.</strong></code>; <code>minimum = 0.</code>; <code>maximum = 2.</code>)<br/>
101The strength parameter for Bose-Einstein effects. On physical grounds
102it should not be above unity, but imperfections in the formalism
103used may require that nevertheless.
104 
105
106<p/><code>parm&nbsp; </code><strong> BoseEinstein:QRef &nbsp;</strong> 
107 (<code>default = <strong>0.2</strong></code>; <code>minimum = 0.05</code>; <code>maximum = 1.</code>)<br/>
108The size parameter of the region in <i>Q</i> space over which
109Bose-Einstein effects are significant.  Can be thought of as
110the inverse of an effective distance in normal space,
111<i>R = hbar / QRef</i>, with <i>R</i> as used in the above equation.
112That is, <i>f_2(Q) = (1 + lambda * exp(-(Q/QRef)^2)) * (...)</i>.
113 
114
115<p/><code>parm&nbsp; </code><strong> BoseEinstein:widthSep &nbsp;</strong> 
116 (<code>default = <strong>0.02</strong></code>; <code>minimum = 0.001</code>; <code>maximum = 1.</code>)<br/>
117Particle species with a width above this value (in GeV) are assumed
118to be so short-lived that they decay before Bose-Einstein effects
119are considered, while otherwise they do not. In the former case the
120decay products thus can obtain shifted momenta, in the latter not.
121The default has been picked such that both <i>rho</i> and
122<i>K^*</i> decay products would be modified.
123 
124
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