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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 |