1 | <html> |
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2 | <head> |
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3 | <title>Four-Vectors</title> |
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4 | <link rel="stylesheet" type="text/css" href="pythia.css"/> |
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5 | <link rel="shortcut icon" href="pythia32.gif"/> |
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6 | </head> |
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7 | <body> |
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8 | |
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9 | <h2>Four-Vectors</h2> |
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10 | |
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11 | The <code>Vec4</code> class gives a simple implementation of four-vectors. |
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12 | The member function names are based on the assumption that these |
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13 | represent four-momentum vectors. Thus one can get or set |
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14 | <i>p_x, p_y, p_z</i> and <i>e</i>, but not <i>x, y, z</i> |
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15 | or <i>t</i>. This is only a matter of naming, however; a |
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16 | <code>Vec4</code> can equally well be used to store a space-time |
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17 | four-vector. |
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18 | |
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19 | <p/> |
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20 | The <code>Particle</code> object contains a <code>Vec4 p</code> that |
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21 | stores the particle four-momentum, and another <code>Vec4 vProd</code> |
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22 | for the production vertex. For the latter the input/output method |
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23 | names are adapted to the space-time character rather than the normal |
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24 | energy-momentum one. Thus a user would not normally access the |
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25 | <code>Vec4</code> classes directly, but only via the methods of the |
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26 | <code>Particle</code> class, |
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27 | see <a href="ParticleProperties.html" target="page">Particle Properties</a>. |
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28 | |
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29 | <p/> |
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30 | Nevertheless you are free to use the PYTHIA four-vectors, e.g. as |
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31 | part of some simple analysis code based directly on the PYTHIA output, |
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32 | say to define the four-vector sum of a set of particles. But note that |
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33 | this class was never set up to allow complete generality, only to |
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34 | provide the operations that are of use inside PYTHIA. There is no |
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35 | separate class for three-vectors, since such can easily be represented |
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36 | by four-vectors where the fourth component is not used. |
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37 | |
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38 | <p/> |
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39 | Four-vectors have the expected functionality: they can be created, |
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40 | copied, added, multiplied, rotated, boosted, and manipulated in other |
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41 | ways. Operator overloading is implemented where reasonable. Properties |
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42 | can be read out, not only the components themselves but also for derived |
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43 | quantities such as absolute momentum and direction angles. |
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44 | |
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45 | <h3>Constructors and basic operators</h3> |
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46 | |
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47 | A few methods are available to create or copy a four-vector: |
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48 | |
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49 | <a name="method1"></a> |
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50 | <p/><strong>Vec4::Vec4() </strong> <br/> |
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51 | creates a four-vector with all components set to 0. |
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52 | |
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53 | |
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54 | <a name="method2"></a> |
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55 | <p/><strong>Vec4::Vec4(const Vec4& v) </strong> <br/> |
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56 | creates a four-vector copy of the input four-vector. |
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57 | |
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58 | |
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59 | <a name="method3"></a> |
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60 | <p/><strong>Vec4& Vec4::operator=(const Vec4& v) </strong> <br/> |
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61 | copies the input four-vector. |
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62 | |
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63 | |
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64 | <a name="method4"></a> |
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65 | <p/><strong>Vec4& Vec4::operator=(double value) </strong> <br/> |
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66 | gives a four-vector with all components set to <i>value</i>. |
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67 | |
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68 | |
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69 | <h3>Member methods for input</h3> |
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70 | |
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71 | The values stored in a four-vector can be modified in a few different |
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72 | ways: |
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73 | |
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74 | <a name="method5"></a> |
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75 | <p/><strong>void Vec4::reset() </strong> <br/> |
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76 | sets all components to 0. |
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77 | |
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78 | |
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79 | <a name="method6"></a> |
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80 | <p/><strong>void Vec4::p(double pxIn, double pyIn, double pzIn, double eIn) </strong> <br/> |
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81 | sets all components to their input values. |
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82 | |
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83 | |
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84 | <a name="method7"></a> |
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85 | <p/><strong>void Vec4::p(Vec4 pIn) </strong> <br/> |
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86 | sets all components equal to those of the input four-vector. |
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87 | |
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88 | |
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89 | <a name="method8"></a> |
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90 | <p/><strong>void Vec4::px(double pxIn) </strong> <br/> |
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91 | |
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92 | <strong>void Vec4::py(double pyIn) </strong> <br/> |
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93 | |
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94 | <strong>void Vec4::pz(double pzIn) </strong> <br/> |
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95 | |
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96 | <strong>void Vec4::e(double eIn) </strong> <br/> |
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97 | sets the respective component to the input value. |
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98 | |
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99 | |
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100 | <h3>Member methods for output</h3> |
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101 | |
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102 | A number of methods provides output of basic or derived quantities: |
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103 | |
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104 | <a name="method9"></a> |
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105 | <p/><strong>double Vec4::px() </strong> <br/> |
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106 | |
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107 | <strong>double Vec4::py() </strong> <br/> |
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108 | |
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109 | <strong>double Vec4::pz() </strong> <br/> |
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110 | |
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111 | <strong>double Vec4::e() </strong> <br/> |
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112 | gets the respective component. |
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113 | |
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114 | |
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115 | <a name="method10"></a> |
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116 | <p/><strong>double Vec4::mCalc() </strong> <br/> |
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117 | |
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118 | <strong>double Vec4::m2Calc() </strong> <br/> |
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119 | the (squared) mass, calculated from the four-vectors. |
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120 | If <i>m^2 < 0</i> the mass is given with a minus sign, |
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121 | <i>-sqrt(-m^2)</i>. Note the possible loss of precision |
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122 | in the calculation of <i>E^2 - p^2</i>; for particles the |
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123 | correct mass is stored separately to avoid such problems. |
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124 | |
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125 | |
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126 | <a name="method11"></a> |
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127 | <p/><strong>double Vec4::pT() </strong> <br/> |
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128 | |
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129 | <strong>double Vec4::pT2() </strong> <br/> |
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130 | the (squared) transverse momentum. |
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131 | |
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132 | |
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133 | <a name="method12"></a> |
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134 | <p/><strong>double Vec4::pAbs() </strong> <br/> |
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135 | |
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136 | <strong>double Vec4::pAbs2() </strong> <br/> |
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137 | the (squared) absolute momentum. |
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138 | |
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139 | |
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140 | <a name="method13"></a> |
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141 | <p/><strong>double Vec4::eT() </strong> <br/> |
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142 | |
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143 | <strong>double Vec4::eT2() </strong> <br/> |
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144 | the (squared) transverse energy, |
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145 | <i>eT = e * sin(theta) = e * pT / pAbs</i>. |
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146 | |
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147 | |
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148 | <a name="method14"></a> |
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149 | <p/><strong>double Vec4::theta() </strong> <br/> |
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150 | the polar angle, in the range 0 through |
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151 | <i>pi</i>. |
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152 | |
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153 | |
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154 | <a name="method15"></a> |
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155 | <p/><strong>double Vec4::phi() </strong> <br/> |
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156 | the azimuthal angle, in the range <i>-pi</i> through <i>pi</i>. |
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157 | |
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158 | |
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159 | <a name="method16"></a> |
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160 | <p/><strong>double Vec4::thetaXZ() </strong> <br/> |
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161 | the angle in the <i>xz</i> plane, in the range <i>-pi</i> through |
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162 | <i>pi</i>, with 0 along the <i>+z</i> axis. |
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163 | |
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164 | |
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165 | <a name="method17"></a> |
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166 | <p/><strong>double Vec4::pPos() </strong> <br/> |
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167 | |
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168 | <strong>double Vec4::pNeg() </strong> <br/> |
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169 | the combinations <i>E+-p_z</i>. |
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170 | |
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171 | <h3>Friend methods for output</h3> |
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172 | |
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173 | There are also some <code>friend</code> methods that take one, two |
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174 | or three four-vectors as argument. Several of them only use the |
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175 | three-vector part of the four-vector. |
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176 | |
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177 | <a name="method18"></a> |
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178 | <p/><strong>friend ostream& operator<<(ostream&, const Vec4& v) </strong> <br/> |
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179 | writes out the values of the four components of a <code>Vec4</code> and, |
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180 | within brackets, a fifth component being the invariant length of the |
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181 | four-vector, as provided by <code>mCalc()</code> above, and it all |
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182 | ended with a newline. |
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183 | |
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184 | |
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185 | <a name="method19"></a> |
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186 | <p/><strong>friend double m(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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187 | |
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188 | <strong>friend double m2(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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189 | the (squared) invariant mass. |
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190 | |
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191 | |
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192 | <a name="method20"></a> |
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193 | <p/><strong>friend double dot3(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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194 | the three-product. |
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195 | |
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196 | |
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197 | <a name="method21"></a> |
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198 | <p/><strong>friend double cross3(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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199 | the cross-product. |
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200 | |
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201 | |
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202 | <a name="method22"></a> |
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203 | <p/><strong>friend double theta(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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204 | |
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205 | <strong>friend double costheta(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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206 | the (cosine) of the opening angle between the vectors, |
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207 | in the range 0 through <i>pi</i>. |
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208 | |
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209 | |
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210 | <a name="method23"></a> |
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211 | <p/><strong>friend double phi(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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212 | |
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213 | <strong>friend double cosphi(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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214 | the (cosine) of the azimuthal angle between the vectors around the |
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215 | <i>z</i> axis, in the range 0 through <i>pi</i>. |
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216 | |
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217 | |
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218 | <a name="method24"></a> |
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219 | <p/><strong>friend double phi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/> |
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220 | |
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221 | <strong>friend double cosphi(const Vec4& v1, const Vec4& v2, const Vec4& v3) </strong> <br/> |
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222 | the (cosine) of the azimuthal angle between the first two vectors |
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223 | around the direction of the third, in the range 0 through <i>pi</i>. |
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224 | |
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225 | |
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226 | <h3>Operations with four-vectors</h3> |
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227 | |
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228 | Of course one should be able to add, subtract and scale four-vectors, |
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229 | and more: |
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230 | |
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231 | <a name="method25"></a> |
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232 | <p/><strong>Vec4 Vec4::operator-() </strong> <br/> |
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233 | return a vector with flipped sign for all components, while leaving |
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234 | the original vector unchanged. |
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235 | |
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236 | |
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237 | <a name="method26"></a> |
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238 | <p/><strong>Vec4& Vec4::operator+=(const Vec4& v) </strong> <br/> |
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239 | add a four-vector to an existing one. |
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240 | |
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241 | |
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242 | <a name="method27"></a> |
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243 | <p/><strong>Vec4& Vec4::operator-=(const Vec4& v) </strong> <br/> |
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244 | subtract a four-vector from an existing one. |
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245 | |
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246 | |
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247 | <a name="method28"></a> |
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248 | <p/><strong>Vec4& Vec4::operator*=(double f) </strong> <br/> |
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249 | multiply all four-vector components by a real number. |
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250 | |
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251 | |
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252 | <a name="method29"></a> |
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253 | <p/><strong>Vec4& Vec4::operator/=(double f) </strong> <br/> |
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254 | divide all four-vector components by a real number. |
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255 | |
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256 | |
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257 | <a name="method30"></a> |
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258 | <p/><strong>friend Vec4 operator+(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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259 | add two four-vectors. |
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260 | |
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261 | |
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262 | <a name="method31"></a> |
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263 | <p/><strong>friend Vec4 operator-(const Vec4& v1, const Vec4& v2) </strong> <br/> |
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264 | subtract two four-vectors. |
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265 | |
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266 | |
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267 | <a name="method32"></a> |
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268 | <p/><strong>friend Vec4 operator*(double f, const Vec4& v) </strong> <br/> |
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269 | |
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270 | <strong>friend Vec4 operator*(const Vec4& v, double f) </strong> <br/> |
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271 | multiply a four-vector by a real number. |
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272 | |
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273 | |
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274 | <a name="method33"></a> |
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275 | <p/><strong>friend Vec4 operator/(const Vec4& v, double f) </strong> <br/> |
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276 | divide a four-vector by a real number. |
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277 | |
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278 | |
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279 | <a name="method34"></a> |
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280 | <p/><strong>friend double operator*(const Vec4& v1, const Vec4 v2) </strong> <br/> |
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281 | four-vector product. |
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282 | |
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283 | |
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284 | <p/> |
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285 | There are also a few related operations that are normal member methods: |
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286 | |
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287 | <a name="method35"></a> |
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288 | <p/><strong>void Vec4::rescale3(double f) </strong> <br/> |
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289 | multiply the three-vector components by <i>f</i>, but keep the |
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290 | fourth component unchanged. |
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291 | |
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292 | |
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293 | <a name="method36"></a> |
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294 | <p/><strong>void Vec4::rescale4(double f) </strong> <br/> |
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295 | multiply all four-vector components by <i>f</i>. |
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296 | |
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297 | |
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298 | <a name="method37"></a> |
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299 | <p/><strong>void Vec4::flip3() </strong> <br/> |
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300 | flip the sign of the three-vector components, but keep the |
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301 | fourth component unchanged. |
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302 | |
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303 | |
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304 | <a name="method38"></a> |
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305 | <p/><strong>void Vec4::flip4() </strong> <br/> |
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306 | flip the sign of all four-vector components. |
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307 | |
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308 | |
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309 | <h3>Rotations and boosts</h3> |
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310 | |
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311 | A common task is to rotate or boost four-vectors. In case only one |
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312 | four-vector is affected the operation may be performed directly on it. |
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313 | However, in case many particles are affected, the helper class |
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314 | <code>RotBstMatrix</code> can be used to speed up operations. |
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315 | |
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316 | <a name="method39"></a> |
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317 | <p/><strong>void Vec4::rot(double theta, double phi) </strong> <br/> |
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318 | rotate the three-momentum with the polar angle <i>theta</i> |
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319 | and the azimuthal angle <i>phi</i>. |
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320 | |
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321 | |
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322 | <a name="method40"></a> |
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323 | <p/><strong>void Vec4::rotaxis(double phi, double nx, double ny, double nz) </strong> <br/> |
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324 | rotate the three-momentum with the azimuthal angle <i>phi</i> |
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325 | around the direction defined by the <i>(n_x, n_y, n_z)</i> |
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326 | three-vector. |
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327 | |
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328 | |
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329 | <a name="method41"></a> |
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330 | <p/><strong>void Vec4::rotaxis(double phi, Vec4& n) </strong> <br/> |
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331 | rotate the three-momentum with the azimuthal angle <i>phi</i> |
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332 | around the direction defined by the three-vector part of <i>n</i>. |
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333 | |
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334 | |
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335 | <a name="method42"></a> |
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336 | <p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ) </strong> <br/> |
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337 | boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>. |
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338 | |
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339 | |
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340 | <a name="method43"></a> |
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341 | <p/><strong>void Vec4::bst(double betaX, double betaY, double betaZ,double gamma) </strong> <br/> |
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342 | boost the four-momentum by <i>beta = (beta_x, beta_y, beta_z)</i>, |
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343 | where the <i>gamma = 1/sqrt(1 - beta^2)</i> is also input to allow |
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344 | better precision when <i>beta</i> is close to unity. |
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345 | |
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346 | |
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347 | <a name="method44"></a> |
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348 | <p/><strong>void Vec4::bst(const Vec4& p) </strong> <br/> |
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349 | boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>. |
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350 | |
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351 | |
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352 | <a name="method45"></a> |
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353 | <p/><strong>void Vec4::bst(const Vec4& p, double m) </strong> <br/> |
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354 | boost the four-momentum by <i>beta = (p_x/E, p_y/E, p_z/E)</i>, |
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355 | where the <i>gamma = E/m</i> is also calculated from input to allow |
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356 | better precision when <i>beta</i> is close to unity. |
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357 | |
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358 | |
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359 | <a name="method46"></a> |
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360 | <p/><strong>void Vec4::bstback(const Vec4& p) </strong> <br/> |
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361 | boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>. |
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362 | |
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363 | |
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364 | <a name="method47"></a> |
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365 | <p/><strong>void Vec4::bstback(const Vec4& p, double m) </strong> <br/> |
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366 | boost the four-momentum by <i>beta = (-p_x/E, -p_y/E, -p_z/E)</i>, |
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367 | where the <i>gamma = E/m</i> is also calculated from input to allow |
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368 | better precision when <i>beta</i> is close to unity. |
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369 | |
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370 | |
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371 | <a name="method48"></a> |
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372 | <p/><strong>void Vec4::rotbst(const RotBstMatrix& M) </strong> <br/> |
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373 | perform a combined rotation and boost; see below for a description |
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374 | of the <code>RotBstMatrix</code>. |
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375 | |
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376 | |
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377 | <p/> |
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378 | For a longer sequence of rotations and boosts, and where several |
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379 | <code>Vec4</code> are to be rotated and boosted in the same way, |
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380 | a more efficient approach is to define a <code>RotBstMatrix</code>, |
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381 | which forms a separate auxiliary class. You can build up this |
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382 | 4-by-4 matrix by successive calls to the methods of the class, |
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383 | such that the matrix encodes the full sequence of operations. |
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384 | The order in which you do these calls must agree with the imagined |
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385 | order in which the rotations/boosts should be applied to a |
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386 | four-momentum, since in general the operations do not commute. |
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387 | |
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388 | <a name="method49"></a> |
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389 | <p/><strong>RotBstMatrix::RotBstMatrix() </strong> <br/> |
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390 | creates a diagonal unit matrix, i.e. one that leaves a four-vector |
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391 | unchanged. |
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392 | |
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393 | |
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394 | <a name="method50"></a> |
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395 | <p/><strong>RotBstMatrix::RotBstMatrix(const RotBstMatrix& Min) </strong> <br/> |
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396 | creates a copy of the input matrix. |
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397 | |
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398 | |
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399 | <a name="method51"></a> |
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400 | <p/><strong>RotBstMatrix& RotBstMatrix::operator=(const RotBstMatrix4& Min) </strong> <br/> |
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401 | copies the input matrix. |
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402 | |
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403 | |
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404 | <a name="method52"></a> |
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405 | <p/><strong>void RotBstMatrix::rot(double theta = 0., double phi = 0.) </strong> <br/> |
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406 | rotate by this polar and azimuthal angle. |
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407 | |
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408 | |
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409 | <a name="method53"></a> |
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410 | <p/><strong>void RotBstMatrix::rot(const Vec4& p) </strong> <br/> |
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411 | rotate so that a vector originally along the <i>+z</i> axis becomes |
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412 | parallel with <i>p</i>. More specifically, rotate by <i>-phi</i>, |
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413 | <i>theta</i> and <i>phi</i>, with angles defined by <i>p</i>. |
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414 | |
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415 | |
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416 | <a name="method54"></a> |
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417 | <p/><strong>void RotBstMatrix::bst(double betaX = 0., double betaY = 0., double betaZ = 0.) </strong> <br/> |
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418 | boost by this <i>beta</i> vector. |
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419 | |
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420 | |
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421 | <a name="method55"></a> |
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422 | <p/><strong>void RotBstMatrix::bst(const Vec4&) </strong> <br/> |
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423 | |
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424 | <strong>void RotBstMatrix::bstback(const Vec4&) </strong> <br/> |
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425 | boost with a <i>beta = p/E</i> or <i>beta = -p/E</i>, respectively. |
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426 | |
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427 | |
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428 | <a name="method56"></a> |
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429 | <p/><strong>void RotBstMatrix::bst(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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430 | boost so that <i>p_1</i> is transformed to <i>p_2</i>. It is assumed |
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431 | that the two vectors obey <i>p_1^2 = p_2^2</i>. |
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432 | |
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433 | |
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434 | <a name="method57"></a> |
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435 | <p/><strong>void RotBstMatrix::toCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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436 | boost and rotate to the rest frame of <i>p_1</i> and <i>p_2</i>, |
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437 | with <i>p_1</i> along the <i>+z</i> axis. |
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438 | |
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439 | |
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440 | <a name="method58"></a> |
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441 | <p/><strong>void RotBstMatrix::fromCMframe(const Vec4& p1, const Vec4& p2) </strong> <br/> |
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442 | rotate and boost from the rest frame of <i>p_1</i> and <i>p_2</i>, |
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443 | with <i>p_1</i> along the <i>+z</i> axis, to the actual frame of |
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444 | <i>p_1</i> and <i>p_2</i>, i.e. the inverse of the above. |
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445 | |
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446 | |
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447 | <a name="method59"></a> |
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448 | <p/><strong>void RotBstMatrix::rotbst(const RotBstMatrix& Min); </strong> <br/> |
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449 | combine the current matrix with another one. |
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450 | |
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451 | |
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452 | <a name="method60"></a> |
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453 | <p/><strong>void RotBstMatrix::invert() </strong> <br/> |
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454 | invert the matrix, which corresponds to an opposite sequence and sign |
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455 | of rotations and boosts. |
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456 | |
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457 | |
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458 | <a name="method61"></a> |
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459 | <p/><strong>void RotBstMatrix::reset() </strong> <br/> |
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460 | reset to no rotation/boost; i.e. the default at creation. |
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461 | |
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462 | |
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463 | <a name="method62"></a> |
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464 | <p/><strong>double RotBstMatrix::deviation() </strong> <br/> |
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465 | crude estimate how much a matrix deviates from the unit matrix: |
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466 | the sum of the absolute values of all non-diagonal matrix elements |
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467 | plus the sum of the absolute deviation of the diagonal matrix |
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468 | elements from unity. |
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469 | |
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470 | |
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471 | <a name="method63"></a> |
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472 | <p/><strong>friend ostream& operator<<(ostream&, const RotBstMatrix& M) </strong> <br/> |
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473 | writes out the values of the sixteen components of a |
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474 | <code>RotBstMatrix</code>, on four consecutive lines and |
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475 | ended with a newline. |
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476 | |
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477 | |
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478 | </body> |
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479 | </html> |
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480 | |
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481 | <!-- Copyright (C) 2012 Torbjorn Sjostrand --> |
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