1 | <chapter name="Bose-Einstein Effects"> |
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2 | |
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3 | <h2>Bose-Einstein Effects</h2> |
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4 | |
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5 | The <code>BoseEinstein</code> class performs shifts of momenta |
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6 | of identical particles to provide a crude estimate of |
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7 | Bose-Einstein effects. The algorithm is the BE_32 one described in |
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8 | <ref>Lon95</ref>, with a Gaussian parametrization of the enhancement. |
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9 | We emphasize that this approach is not based on any first-principles |
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10 | quantum mechanical description of interference phenomena; such |
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11 | approaches anyway have many problems to contend with. Instead a cruder |
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12 | but more robust approach is adopted, wherein BE effects are introduced |
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13 | after the event has already been generated, with the exception of the |
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14 | decays of long-lived particles. The trick is that momenta of identical |
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15 | particles are shifted relative to each other so as to provide an |
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16 | enhancement of pairs closely separated, which is compensated by a |
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17 | depletion of pairs in an intermediate region of separation. |
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18 | |
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19 | <p/> |
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20 | More precisely, the intended target form of the BE corrrelations in |
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21 | BE_32 is |
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22 | <eq> |
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23 | f_2(Q) = (1 + lambda * exp(-Q^2 R^2)) |
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24 | * (1 + alpha * lambda * exp(-Q^2 R^2/9) * (1 - exp(-Q^2 R^2/4))) |
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25 | </eq> |
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26 | where <ei>Q^2 = (p_1 + p_2)^2 - (m_1 + m_2)^2</ei>. |
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27 | Here the strength <ei>lambda</ei> and effective radius <ei>R</ei> |
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28 | are the two main parameters. The first factor of the |
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29 | equation is implemented by pulling pairs of identical hadrons closer |
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30 | to each other. This is done in such a way that three-monentum is |
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31 | conserved, but at the price of a small but non-negligible negative |
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32 | shift in the energy of the event. The second factor compensates this |
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33 | by pushing particles apart. The negative <ei>alpha</ei> parameter is |
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34 | determined iteratively, separately for each event, so as to restore |
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35 | energy conservation. The effective radius parameter is here <ei>R/3</ei>, |
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36 | i.e. effects extend further out in <ei>Q</ei>. Without the dampening |
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37 | <ei>(1 - exp(-Q^2 R^2/4))</ei> in the second factor the value at the |
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38 | origin would become <ei>f_2(0) = (1 + lambda) * (1 + alpha * lambda)</ei>, |
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39 | with it the desired value <ei>f_2(0) = (1 + lambda)</ei> is restored. |
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40 | The end result can be viewed as a poor man's rendering of a rapidly |
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41 | dampened oscillatory behaviour in <ei>Q</ei>. |
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42 | |
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43 | <p/> |
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44 | Further details can be found in <ref>Lon95</ref>. For instance, the |
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45 | target is implemented under the assumption that the initial distribution |
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46 | in <ei>Q</ei> can be well approximated by pure phase space at small |
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47 | values, and implicitly generates higher-order effects by the way |
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48 | the algorithm is implemented. The algorithm is applied after the decay |
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49 | of short-lived resonances such as the <ei>rho</ei>, but before the decay |
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50 | of longer-lived particles. |
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51 | |
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52 | <p/> |
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53 | This algorithm is known to do a reasonable job of describing BE |
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54 | phenomena at LEP. It has not been tested against data for hadron |
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55 | colliders, to the best of our knowledge, so one should exercise some |
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56 | judgement before using it. Therefore by default the master switch |
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57 | <aloc href="MasterSwitches">HadronLevel:BoseEinstein</aloc> is off. |
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58 | Furthermore, the implementation found here is not (yet) as |
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59 | sophisticated as the one used at LEP2, in that no provision is made |
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60 | for particles from separate colour singlet systems, such as |
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61 | <ei>W</ei>'s and <ei>Z</ei>'s, interfering only at a reduced rate. |
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62 | |
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63 | <p/> |
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64 | <b>Warning:</b> The algorithm will create a new copy of each particle |
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65 | with shifted momentum by BE effects, with status code 99, while the |
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66 | original particle with the original momentum at the same time will be |
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67 | marked as decayed. This means that if you e.g. search for all |
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68 | <ei>pi+-</ei> in an event you will often obtain the same particle twice. |
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69 | One way to protect yourself from unwanted doublecounting is to |
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70 | use only particles with a positive status code, i.e. ones for which |
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71 | <code>event[i].isFinal()</code> is <code>true</code>. |
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72 | |
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73 | |
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74 | <h3>Main parameters</h3> |
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75 | |
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76 | <flag name="BoseEinstein:Pion" default="on"> |
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77 | Include effects or not for identical <ei>pi^+</ei>, <ei>pi^-</ei> |
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78 | and <ei>pi^0</ei>. |
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79 | </flag> |
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80 | |
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81 | <flag name="BoseEinstein:Kaon" default="on"> |
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82 | Include effects or not for identical <ei>K^+</ei>, <ei>K^-</ei>, |
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83 | <ei>K_S^0</ei> and <ei>K_L^0</ei>. |
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84 | </flag> |
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85 | |
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86 | <flag name="BoseEinstein:Eta" default="on"> |
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87 | Include effects or not for identical <ei>eta</ei> and <ei>eta'</ei>. |
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88 | </flag> |
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89 | |
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90 | <parm name="BoseEinstein:lambda" default="1." min="0." max="2."> |
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91 | The strength parameter for Bose-Einstein effects. On physical grounds |
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92 | it should not be above unity, but imperfections in the formalism |
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93 | used may require that nevertheless. |
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94 | </parm> |
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95 | |
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96 | <parm name="BoseEinstein:QRef" default="0.2" min="0.05" max="1."> |
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97 | The size parameter of the region in <ei>Q</ei> space over which |
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98 | Bose-Einstein effects are significant. Can be thought of as |
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99 | the inverse of an effective distance in normal space, |
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100 | <ei>R = hbar / QRef</ei>, with <ei>R</ei> as used in the above equation. |
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101 | That is, <ei>f_2(Q) = (1 + lambda * exp(-(Q/QRef)^2)) * (...)</ei>. |
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102 | </parm> |
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103 | |
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104 | <parm name="BoseEinstein:widthSep" default="0.02" min="0.001" max="1."> |
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105 | Particle species with a width above this value (in GeV) are assumed |
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106 | to be so short-lived that they decay before Bose-Einstein effects |
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107 | are considered, while otherwise they do not. In the former case the |
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108 | decay products thus can obtain shifted momenta, in the latter not. |
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109 | The default has been picked such that both <ei>rho</ei> and |
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110 | <ei>K^*</ei> decay products would be modified. |
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111 | </parm> |
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112 | |
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113 | </chapter> |
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114 | |
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115 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |
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116 | |
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