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5ad4eb21 | 1 | <chapter name="Bose-Einstein Effects"> |
2 | ||
3 | <h2>Bose-Einstein Effects</h2> | |
4 | ||
5 | The <code>BoseEinstein</code> class performs shifts of momenta | |
6 | of identical particles to provide a crude estimate of | |
7 | Bose-Einstein effects. The algorithm is the BE_32 one described in | |
8 | <ref>Lon95</ref>, with a Gaussian parametrization of the enhancement. | |
9 | We emphasize that this approach is not based on any first-principles | |
10 | quantum mechanical description of interference phenomena; such | |
11 | approaches anyway have many problems to contend with. Instead a cruder | |
12 | but more robust approach is adopted, wherein BE effects are introduced | |
13 | after the event has already been generated, with the exception of the | |
14 | decays of long-lived particles. The trick is that momenta of identical | |
15 | particles are shifted relative to each other so as to provide an | |
16 | enhancement of pairs closely separated, which is compensated by a | |
17 | depletion of pairs in an intermediate region of separation. | |
18 | ||
19 | <p/> | |
20 | More precisely, the intended target form of the BE corrrelations in | |
21 | BE_32 is | |
22 | <eq> | |
23 | f_2(Q) = (1 + lambda * exp(-Q^2 R^2)) | |
24 | * (1 + alpha * lambda * exp(-Q^2 R^2/9) * (1 - exp(-Q^2 R^2/4))) | |
25 | </eq> | |
26 | where <ei>Q^2 = (p_1 + p_2)^2 - (m_1 + m_2)^2</ei>. | |
27 | Here the strength <ei>lambda</ei> and effective radius <ei>R</ei> | |
28 | are the two main parameters. The first factor of the | |
29 | equation is implemented by pulling pairs of identical hadrons closer | |
30 | to each other. This is done in such a way that three-monentum is | |
31 | conserved, but at the price of a small but non-negligible negative | |
32 | shift in the energy of the event. The second factor compensates this | |
33 | by pushing particles apart. The negative <ei>alpha</ei> parameter is | |
34 | determined iteratively, separately for each event, so as to restore | |
35 | energy conservation. The effective radius parameter is here <ei>R/3</ei>, | |
36 | i.e. effects extend further out in <ei>Q</ei>. Without the dampening | |
37 | <ei>(1 - exp(-Q^2 R^2/4))</ei> in the second factor the value at the | |
38 | origin would become <ei>f_2(0) = (1 + lambda) * (1 + alpha * lambda)</ei>, | |
39 | with it the desired value <ei>f_2(0) = (1 + lambda)</ei> is restored. | |
40 | The end result can be viewed as a poor man's rendering of a rapidly | |
41 | dampened oscillatory behaviour in <ei>Q</ei>. | |
42 | ||
43 | <p/> | |
44 | Further details can be found in <ref>Lon95</ref>. For instance, the | |
45 | target is implemented under the assumption that the initial distribution | |
46 | in <ei>Q</ei> can be well approximated by pure phase space at small | |
47 | values, and implicitly generates higher-order effects by the way | |
48 | the algorithm is implemented. The algorithm is applied after the decay | |
49 | of short-lived resonances such as the <ei>rho</ei>, but before the decay | |
50 | of longer-lived particles. | |
51 | ||
52 | <p/> | |
53 | This algorithm is known to do a reasonable job of describing BE | |
54 | phenomena at LEP. It has not been tested against data for hadron | |
55 | colliders, to the best of our knowledge, so one should exercise some | |
56 | judgement before using it. Therefore by default the master switch | |
57 | <aloc href="MasterSwitches">HadronLevel:BoseEinstein</aloc> is off. | |
58 | Furthermore, the implementation found here is not (yet) as | |
59 | sophisticated as the one used at LEP2, in that no provision is made | |
60 | for particles from separate colour singlet systems, such as | |
61 | <ei>W</ei>'s and <ei>Z</ei>'s, interfering only at a reduced rate. | |
62 | ||
63 | <p/> | |
64 | <b>Warning:</b> The algorithm will create a new copy of each particle | |
65 | with shifted momentum by BE effects, with status code 99, while the | |
66 | original particle with the original momentum at the same time will be | |
67 | marked as decayed. This means that if you e.g. search for all | |
68 | <ei>pi+-</ei> in an event you will often obtain the same particle twice. | |
69 | One way to protect yourself from unwanted doublecounting is to | |
70 | use only particles with a positive status code, i.e. ones for which | |
71 | <code>event[i].isFinal()</code> is <code>true</code>. | |
72 | ||
73 | ||
74 | <h3>Main parameters</h3> | |
75 | ||
76 | <flag name="BoseEinstein:Pion" default="on"> | |
77 | Include effects or not for identical <ei>pi^+</ei>, <ei>pi^-</ei> | |
78 | and <ei>pi^0</ei>. | |
79 | </flag> | |
80 | ||
81 | <flag name="BoseEinstein:Kaon" default="on"> | |
82 | Include effects or not for identical <ei>K^+</ei>, <ei>K^-</ei>, | |
83 | <ei>K_S^0</ei> and <ei>K_L^0</ei>. | |
84 | </flag> | |
85 | ||
86 | <flag name="BoseEinstein:Eta" default="on"> | |
87 | Include effects or not for identical <ei>eta</ei> and <ei>eta'</ei>. | |
88 | </flag> | |
89 | ||
90 | <parm name="BoseEinstein:lambda" default="1." min="0." max="2."> | |
91 | The strength parameter for Bose-Einstein effects. On physical grounds | |
92 | it should not be above unity, but imperfections in the formalism | |
93 | used may require that nevertheless. | |
94 | </parm> | |
95 | ||
96 | <parm name="BoseEinstein:QRef" default="0.2" min="0.05" max="1."> | |
97 | The size parameter of the region in <ei>Q</ei> space over which | |
98 | Bose-Einstein effects are significant. Can be thought of as | |
99 | the inverse of an effective distance in normal space, | |
100 | <ei>R = hbar / QRef</ei>, with <ei>R</ei> as used in the above equation. | |
101 | That is, <ei>f_2(Q) = (1 + lambda * exp(-(Q/QRef)^2)) * (...)</ei>. | |
102 | </parm> | |
103 | ||
104 | <parm name="BoseEinstein:widthSep" default="0.02" min="0.001" max="1."> | |
105 | Particle species with a width above this value (in GeV) are assumed | |
106 | to be so short-lived that they decay before Bose-Einstein effects | |
107 | are considered, while otherwise they do not. In the former case the | |
108 | decay products thus can obtain shifted momenta, in the latter not. | |
109 | The default has been picked such that both <ei>rho</ei> and | |
110 | <ei>K^*</ei> decay products would be modified. | |
111 | </parm> | |
112 | ||
113 | </chapter> | |
114 | ||
115 | <!-- Copyright (C) 2008 Torbjorn Sjostrand --> | |
116 |