1 | <chapter name="A Second Hard Process"> |
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2 | |
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3 | <h2>A Second Hard Process</h2> |
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4 | |
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5 | When you have selected a set of hard processes for hadron beams, the |
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6 | <aloc href="MultipartonInteractions">multiparton interactions</aloc> |
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7 | framework can add further interactions to build up a realistic |
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8 | underlying event. These further interactions can come from a wide |
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9 | variety of processes, and will occasionally be quite hard. They |
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10 | do represent a realistic random mix, however, which means one cannot |
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11 | predetermine what will happen. Occasionally there may be cases |
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12 | where one wants to specify also the second hard interaction rather |
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13 | precisely. The options on this page allow you to do precisely that. |
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14 | |
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15 | <flag name="SecondHard:generate" default="off"> |
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16 | Generate two hard scatterings in a collision between hadron beams. |
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17 | The hardest process can be any combination of internal processes, |
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18 | available in the normal <aloc href="ProcessSelection">process |
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19 | selection</aloc> machinery, or external input. Here you must further |
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20 | specify which set of processes to allow for the second hard one, see |
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21 | the following. |
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22 | </flag> |
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23 | |
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24 | <h3>Process Selection</h3> |
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25 | |
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26 | In principle the whole <aloc href="ProcessSelection">process |
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27 | selection</aloc> allowed for the first process could be repeated |
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28 | for the second one. However, this would probably be overkill. |
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29 | Therefore here a more limited set of prepackaged process collections |
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30 | are made available, that can then be further combined at will. |
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31 | Since the description is almost completely symmetric between the |
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32 | first and the second process, you always have the possibility |
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33 | to pick one of the two processes according to the complete list |
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34 | of possibilities. |
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35 | |
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36 | <p/> |
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37 | Here comes the list of allowed sets of processes, to combine at will: |
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38 | |
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39 | <flag name="SecondHard:TwoJets" default="off"> |
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40 | Standard QCD <ei>2 -> 2</ei> processes involving gluons and |
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41 | <ei>d, u, s, c, b</ei> quarks. |
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42 | </flag> |
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43 | |
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44 | <flag name="SecondHard:PhotonAndJet" default="off"> |
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45 | A prompt photon recoiling against a quark or gluon jet. |
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46 | |
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47 | <flag name="SecondHard:TwoPhotons" default="off"> |
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48 | Two prompt photons recoiling against each other. |
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49 | |
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50 | <flag name="SecondHard:Charmonium" default="off"> |
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51 | Production of charmonium via colour singlet and colour octet channels. |
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52 | |
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53 | <flag name="SecondHard:Bottomonium" default="off"> |
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54 | Production of bottomonium via colour singlet and colour octet channels. |
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55 | |
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56 | <flag name="SecondHard:SingleGmZ" default="off"> |
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57 | Scattering <ei>q qbar -> gamma^*/Z^0</ei>, with full interference |
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58 | between the <ei>gamma^*</ei> and <ei>Z^0</ei>. |
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59 | </flag> |
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60 | |
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61 | <flag name="SecondHard:SingleW" default="off"> |
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62 | Scattering <ei>q qbar' -> W^+-</ei>. |
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63 | </flag> |
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64 | |
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65 | <flag name="SecondHard:GmZAndJet" default="off"> |
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66 | Scattering <ei>q qbar -> gamma^*/Z^0 g</ei> and |
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67 | <ei>q g -> gamma^*/Z^0 q</ei>. |
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68 | </flag> |
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69 | |
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70 | <flag name="SecondHard:WAndJet" default="off"> |
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71 | Scattering <ei>q qbar' -> W^+- g</ei> and |
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72 | <ei>q g -> W^+- q'</ei>. |
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73 | </flag> |
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74 | |
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75 | <flag name="SecondHard:TopPair" default="off"> |
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76 | Production of a top pair, either via QCD processes or via an |
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77 | intermediate <ei>gamma^*/Z^0</ei> resonance. |
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78 | </flag> |
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79 | |
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80 | <flag name="SecondHard:SingleTop" default="off"> |
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81 | Production of a single top, either via a <ei>t-</ei> or |
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82 | an <ei>s-</ei>channel <ei>W^+-</ei> resonance. |
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83 | </flag> |
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84 | |
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85 | <p/> |
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86 | A further process collection comes with a warning flag: |
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87 | |
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88 | <flag name="SecondHard:TwoBJets" default="off"> |
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89 | The <ei>q qbar -> b bbar</ei> and <ei>g g -> b bbar</ei> processes. |
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90 | These are already included in the <code>TwoJets</code> sample above, |
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91 | so it would be doublecounting to include both, but we assume there |
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92 | may be cases where the <ei>b</ei> subsample will be of special interest. |
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93 | This subsample does not include flavour-excitation or gluon-splitting |
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94 | contributions to the <ei>b</ei> rate, however, so, depending |
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95 | on the topology if interest, it may or may not be a good approximation. |
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96 | </flag> |
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97 | |
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98 | <h3>Cuts and scales</h3> |
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99 | |
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100 | The second hard process obeys exactly the same selection rules for |
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101 | <aloc href="PhaseSpaceCuts">phase space cuts</aloc> and |
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102 | <aloc href="CouplingsAndScales">couplings and scales</aloc> |
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103 | as the first one does. Specifically, a <ei>pTmin</ei> cut for |
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104 | <ei>2 -> 2</ei> processes would apply to the first and the second hard |
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105 | process alike, and ballpark half of the time the second could be |
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106 | generated with a larger <ei>pT</ei> than the first. (Exact numbers |
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107 | depending on the relative shape of the two cross sections.) That is, |
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108 | first and second is only used as an administrative distinction between |
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109 | the two, not as a physics ordering one. |
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110 | |
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111 | <p/> |
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112 | Optionally it is possible to pick the mass and <ei>pT</ei> |
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113 | <aloc href="PhaseSpaceCuts">phase space cuts</aloc> separately for |
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114 | the second hard interaction. The main application presumably would |
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115 | be to allow a second process that is softer than the first, but still |
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116 | hard. But one is also free to make the second process harder than the |
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117 | first, if desired. So long as the two <ei>pT</ei> (or mass) ranges |
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118 | overlap the ordering will not be the same in all events, however. |
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119 | |
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120 | <h3>Cross-section calculation</h3> |
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121 | |
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122 | As an introduction, a brief reminder of Poissonian statistics. |
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123 | Assume a stochastic process in time, for now not necessarily a |
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124 | high-energy physics one, where the probability for an event to occur |
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125 | at any given time is independent of what happens at other times. |
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126 | Then the probability for <ei>n</ei> events to occur in a finite |
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127 | time interval is |
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128 | <eq> |
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129 | P_n = <n>^n exp(-<n>) / n! |
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130 | </eq> |
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131 | where <ei><n></ei> is the average number of events. If this |
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132 | number is small we can approximate <ei>exp(-<n>) = 1 </ei>, |
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133 | so that <ei>P_1 = <n></ei> and |
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134 | <ei>P_2 = <n>^2 / 2 = P_1^2 / 2</ei>. |
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135 | |
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136 | <p/> |
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137 | Now further assume that the events actually are of two different |
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138 | kinds <ei>a</ei> and <ei>b</ei>, occuring independently of each |
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139 | other, such that <ei><n> = <n_a> + <n_b></ei>. |
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140 | It then follows that the probability of having one event of type |
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141 | <ei>a</ei> (or <ei>b</ei>) and nothing else is |
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142 | <ei>P_1a = <n_a></ei> (or <ei>P_1b = <n_b></ei>). |
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143 | From |
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144 | <eq> |
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145 | P_2 = (<n_a> + <n_b>)^2 / 2 = (P_1a + P_1b)^2 / 2 = |
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146 | (P_1a^2 + 2 P_1a P_1b + P_1b^2) / 2 |
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147 | </eq> |
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148 | it is easy to read off that the probability to have exactly two |
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149 | events of kind <ei>a</ei> and none of <ei>b</ei> is |
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150 | <ei>P_2aa = P_1a^2 / 2</ei> whereas that of having one <ei>a</ei> |
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151 | and one <ei>b</ei> is <ei>P_2ab = P_1a P_1b</ei>. Note that the |
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152 | former, with two identical events, contains a factor <ei>1/2</ei> |
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153 | while the latter, with two different ones, does not. If viewed |
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154 | in a time-ordered sense, the difference is that the latter can be |
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155 | obtained two ways, either first an <ei>a</ei> and then a <ei>b</ei> |
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156 | or else first a <ei>b</ei> and then an <ei>a</ei>. |
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157 | |
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158 | <p/> |
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159 | To translate this language into cross-sections for high-energy |
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160 | events, we assume that interactions can occur at different <ei>pT</ei> |
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161 | values independently of each other inside inelastic nondiffractive |
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162 | (= "minbias") events. Then the above probabilities translate into |
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163 | <ei>P_n = sigma_n / sigma_ND</ei> where <ei>sigma_ND</ei> is the |
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164 | total nondiffractive cross section. Again we want to assume that |
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165 | <ei>exp(-<n>)</ei> is close to unity, i.e. that the total |
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166 | hard cross section above <ei>pTmin</ei> is much smaller than |
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167 | <ei>sigma_ND</ei>. The hard cross section is dominated by QCD |
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168 | jet production, and a reasonable precaution is to require a |
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169 | <ei>pTmin</ei> of at least 20 GeV at LHC energies. |
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170 | (For <ei>2 -> 1</ei> processes such as |
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171 | <ei>q qbar -> gamma^*/Z^0 (-> f fbar)</ei> one can instead make a |
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172 | similar cut on mass.) Then the generic equation |
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173 | <ei>P_2 = P_1^2 / 2</ei> translates into |
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174 | <ei>sigma_2/sigma_ND = (sigma_1 / sigma_ND)^2 / 2</ei> or |
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175 | <ei>sigma_2 = sigma_1^2 / (2 sigma_ND)</ei>. |
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176 | |
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177 | <p/> |
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178 | Again different processes <ei>a, b, c, ...</ei> contribute, |
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179 | and by the same reasoning we obtain |
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180 | <ei>sigma_2aa = sigma_1a^2 / (2 sigma_ND)</ei>, |
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181 | <ei>sigma_2ab = sigma_1a sigma_1b / sigma_ND</ei>, |
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182 | and so on. |
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183 | |
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184 | <p/> |
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185 | There is one important correction to this picture: all collisions |
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186 | do no occur under equal conditions. Some are more central in impact |
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187 | parameter, others more peripheral. This leads to a further element of |
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188 | variability: central collisions are likely to have more activity |
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189 | than the average, peripheral less. Integrated over impact |
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190 | parameter standard cross sections are recovered, but correlations |
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191 | are affected by a "trigger bias" effect: if you select for events |
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192 | with a hard process you favour events at small impact parameter |
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193 | which have above-average activity, and therefore also increased |
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194 | chance for further interactions. (In PYTHIA this is the origin |
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195 | of the "pedestal effect", i.e. that events with a hard interaction |
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196 | have more underlying activity than the level found in minimum-bias |
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197 | events.) When you specify a matter overlap profile in the |
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198 | multiparton-interactions scenario, such an enhancement/depletion factor |
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199 | <ei>f_impact</ei> is chosen event-by-event and can be averaged |
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200 | during the course of the run. As an example, the double Gaussian |
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201 | form used in Tune A gives approximately |
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202 | <ei><f_impact> = 2.5</ei>. The above equations therefore |
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203 | have to be modified to |
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204 | <ei>sigma_2aa = <f_impact> sigma_1a^2 / (2 sigma_ND)</ei>, |
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205 | <ei>sigma_2ab = <f_impact> sigma_1a sigma_1b / sigma_ND</ei>. |
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206 | Experimentalists often instead use the notation |
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207 | <ei>sigma_2ab = sigma_1a sigma_1b / sigma_eff</ei>, |
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208 | from which we see that PYTHIA "predicts" |
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209 | <ei>sigma_eff = sigma_ND / <f_impact></ei>. |
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210 | When the generation of multiparton interactions is switched off it is |
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211 | not possible to calculate <ei><f_impact></ei> and therefore |
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212 | it is set to unity. |
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213 | |
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214 | <p/> |
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215 | When this recipe is to be applied to calculate |
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216 | actual cross sections, it is useful to distinguish three cases, |
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217 | depending on which set of processes are selected to study for |
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218 | the first and second interaction. |
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219 | |
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220 | <p/> |
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221 | (1) The processes <ei>a</ei> for the first interaction and |
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222 | <ei>b</ei> for the second one have no overlap at all. |
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223 | For instance, the first could be <code>TwoJets</code> and the |
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224 | second <code>TwoPhotons</code>. In that case, the two interactions |
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225 | can be selected independently, and cross sections tabulated |
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226 | for each separate subprocess in the two above classes. At the |
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227 | end of the run, the cross sections in <ei>a</ei> should be multiplied |
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228 | by <ei><f_impact> sigma_1b / sigma_ND</ei> to bring them to |
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229 | the correct overall level, and those in <ei>b</ei> by |
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230 | <ei><f_impact> sigma_1a / sigma_ND</ei>. |
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231 | |
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232 | <p/> |
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233 | (2) Exactly the same processes <ei>a</ei> are selected for the |
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234 | first and second interaction. In that case it works as above, |
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235 | with <ei>a = b</ei>, and it is only necessary to multiply by an |
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236 | additional factor <ei>1/2</ei>. A compensating factor of 2 |
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237 | is automatically obtained for picking two different subprocesses, |
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238 | e.g. if <code>TwoJets</code> is selected for both interactions, |
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239 | then the combination of the two subprocesses <ei>q qbar -> g g</ei> |
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240 | and <ei>g g -> g g</ei> can trivially be obtained two ways. |
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241 | |
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242 | <p/> |
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243 | (3) The list of subprocesses partly but not completely overlap. |
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244 | For instance, the first process is allowed to contain <ei>a</ei> |
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245 | or <ei>c</ei> and the second <ei>b</ei> or <ei>c</ei>, where |
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246 | there is no overlap between <ei>a</ei> and <ei>b</ei>. Then, |
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247 | when an independent selection for the first and second interaction |
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248 | both pick one of the subprocesses in <ei>c</ei>, half of those |
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249 | events have to be thrown, and the stored cross section reduced |
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250 | accordingly. Considering the four possible combinations of first |
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251 | and second process, this gives a |
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252 | <eq> |
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253 | sigma'_1 = sigma_1a + sigma_1c * (sigma_2b + sigma_2c/2) / |
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254 | (sigma_2b + sigma_2c) |
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255 | </eq> |
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256 | with the factor <ei>1/2</ei> for the <ei>sigma_1c sigma_2c</ei> term. |
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257 | At the end of the day, this <ei>sigma'_1</ei> should be multiplied |
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258 | by the normalization factor |
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259 | <eq> |
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260 | f_1norm = <f_impact> (sigma_2b + sigma_2c) / sigma_ND |
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261 | </eq> |
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262 | here without a factor <ei>1/2</ei> (or else it would have been |
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263 | doublecounted). This gives the correct |
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264 | <eq> |
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265 | (sigma_2b + sigma_2c) * sigma'_1 = sigma_1a * sigma_2b |
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266 | + sigma_1a * sigma_2c + sigma_1c * sigma_2b + sigma_1c * sigma_2c/2 |
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267 | </eq> |
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268 | The second interaction can be handled in exact analogy. |
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269 | |
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270 | <p/> |
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271 | For the considerations above it is assumed that the phase space cuts |
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272 | are the same for the two processes. It is possible to set the mass and |
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273 | transverse momentum cuts differently, however. This changes nothing |
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274 | for processes that already are different. For two collisions of the |
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275 | same type it is partly a matter of interpretation what is intended. |
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276 | If we consider the case of the same process in two non-overlapping |
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277 | phase space regions, most likely we want to consider them as |
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278 | separate processes, in the sense that we expect a factor 2 relative |
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279 | to Poissonian statistics from either of the two hardest processes |
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280 | populating either of the two phase space regions. In total we are |
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281 | therefore lead to adopt the same strategy as in case (3) above: |
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282 | only in the overlapping part of the two allowed phase space regions |
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283 | could two processes be identical and thus appear with a 1/2 factor, |
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284 | elsewhere the two processes are never identical and do not |
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285 | include the 1/2 factor. We reiterate, however, that the case of |
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286 | partly but not completely overlapping phase space regions for one and |
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287 | the same process is tricky, and not to be used without prior |
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288 | deliberation. |
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289 | |
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290 | <p/> |
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291 | The listing obtained with the <code>pythia.statistics()</code> |
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292 | already contain these corrections factors, i.e. cross sections |
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293 | are for the occurence of two interactions of the specified kinds. |
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294 | There is not a full tabulation of the matrix of all the possible |
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295 | combinations of a specific first process together with a specific |
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296 | second one (but the information is there for the user to do that, |
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297 | if desired). Instead <code>pythia.statistics()</code> shows this |
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298 | matrix projected onto the set of processes and associated cross |
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299 | sections for the first and the second interaction, respectively. |
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300 | Up to statistical fluctuations, these two sections of the |
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301 | <code>pythia.statistics()</code> listing both add up to the same |
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302 | total cross section for the event sample. |
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303 | |
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304 | <p/> |
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305 | There is a further special feature to be noted for this listing, |
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306 | and that is the difference between the number of "selected" events |
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307 | and the number of "accepted" ones. Here is how that comes about. |
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308 | Originally the first and second process are selected completely |
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309 | independently. The generation (in)efficiency is reflected in the |
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310 | different number of intially tried events for the first and second |
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311 | process, leading to the same number of selected events. While |
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312 | acceptable on their own, the combination of the two processes may |
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313 | be unacceptable, however. It may be that the two processes added |
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314 | together use more energy-momentum than kinematically allowed, or, |
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315 | even if not, are disfavoured when the PYTHIA approach to provide |
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316 | correlated parton densities is applied. Alternatively, referring |
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317 | to case (3) above, it may be because half of the events should |
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318 | be thrown for identical processes. Taken together, it is these |
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319 | effects that reduced the event number from "selected" to "accepted". |
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320 | (A further reduction may occur if a |
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321 | <aloc href="UserHooks">user hook</aloc> rejects some events.) |
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322 | |
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323 | <p/> |
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324 | It is allowed to use external Les Houches Accord input for the |
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325 | hardest process, and then pick an internal one for the second hardest. |
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326 | In this case PYTHIA does not have access to your thinking concerning |
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327 | the external process, and cannot know whether it overlaps with the |
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328 | internal or not. (External events <ei>q qbar' -> e+ nu_e</ei> could |
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329 | agree with the internal <ei>W</ei> ones, or be a <ei>W'</ei> resonance |
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330 | in a BSM scenario, to give one example.) Therefore the combined cross |
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331 | section is always based on the scenario (1) above. Corrections for |
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332 | correlated parton densities are included also in this case, however. |
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333 | That is, an external event that takes a large fraction of the incoming |
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334 | beam momenta stands a fair chance of being rejected when it has to be |
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335 | combined with another hard process. For this reason the "selected" and |
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336 | "accepted" event numbers are likely to disagree. |
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337 | |
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338 | <p/> |
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339 | In the cross section calculation above, the <ei>sigma'_1</ei> |
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340 | cross sections are based on the number of accepted events, while |
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341 | the <ei>f_1norm</ei> factor is evaluated based on the cross sections |
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342 | for selected events. That way the suppression by correlations |
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343 | between the two processes does not get to be doublecounted. |
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344 | |
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345 | <p/> |
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346 | The <code>pythia.statistics()</code> listing contains two final |
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347 | lines, indicating the summed cross sections <ei>sigma_1sum</ei> and |
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348 | <ei>sigma_2sum</ei> for the first and second set of processes, at |
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349 | the "selected" stage above, plus information on the <ei>sigma_ND</ei> |
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350 | and <ei><f_impact></ei> used. The total cross section |
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351 | generated is related to this by |
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352 | <eq> |
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353 | <f_impact> * (sigma_1sum * sigma_2sum / sigma_ND) * |
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354 | (n_accepted / n_selected) |
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355 | </eq> |
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356 | with an additional factor of <ei>1/2</ei> for case 2 above. |
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357 | |
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358 | <p/> |
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359 | The error quoted for the cross section of a process is a combination |
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360 | in quadrature of the error on this process alone with the error on |
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361 | the normalization factor, including the error on |
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362 | <ei><f_impact></ei>. As always it is a purely statistical one |
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363 | and of course hides considerably bigger systematic uncertainties. |
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364 | |
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365 | <h3>Event information</h3> |
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366 | |
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367 | Normally the <code>process</code> event record only contains the |
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368 | hardest interaction, but in this case also the second hardest |
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369 | is stored there. If both of them are <ei>2 -> 2</ei> ones, the |
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370 | first would be stored in lines 3 - 6 and the second in 7 - 10. |
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371 | For both, status codes 21 - 29 would be used, as for a hardest |
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372 | process. Any resonance decay chains would occur after the two |
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373 | main processes, to allow normal parsing. The beams in 1 and 2 |
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374 | only appear in one copy. This structure is echoed in the |
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375 | full <code>event</code> event record. |
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376 | |
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377 | <p/> |
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378 | Most of the properties accessible by the |
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379 | <code><aloc href="EventInformation">pythia.info</aloc></code> |
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380 | methods refer to the first process, whether that happens to be the |
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381 | hardest or not. The code and <ei>pT</ei> scale of the second process |
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382 | are accessible by the <code>info.codeMPI(1)</code> and |
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383 | <code>info.pTMPI(1)</code>, however. |
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384 | |
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385 | <p/> |
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386 | The <code>sigmaGen()</code> and <code>sigmaErr()</code> methods provide |
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387 | the cross section and its error for the event sample as a whole, |
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388 | combining the information from the two hard processes as described |
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389 | above. In particular, the former should be used to give the |
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390 | weight of the generated event sample. The statitical error estimate |
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391 | is somewhat cruder and gives a larger value than the |
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392 | subprocess-by-subprocess one employed in |
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393 | <code>pythia.statistics()</code>, but this number is |
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394 | anyway less relevant, since systematical errors are likely to dominate. |
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395 | |
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396 | </chapter> |
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397 | |
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398 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |
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