1 |
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2 |
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3 |
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4 |
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5 | (**************** This is the FeynRules model-file for the Hill model **************)
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6 |
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7 | M$ModelName = "HillModel";
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8 |
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9 | M$Information = {Authors -> {"P. Aquino", "C. Duhr"},
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10 | Institutions -> {"Universite catholique de Louvain (CP3)"},
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11 | Emails -> {priscila@fma.if.usp.br, claude.duhr@uclouvain.be},
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12 | Date -> "14. 06. 2009",
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13 | Version -> "1.0",
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14 | References -> "\"The minimal non-minimal Standard Model\", J.J. van der Bij, Phys.Lett.B636:56-59,2006, hep-ph/0603082",
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15 | URLs -> "http://feynrules.phys.ucl.ac.be/view/Main/Hillmodel"};
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16 |
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17 |
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18 | FeynmanGauge=False;
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19 |
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20 | (******* Index definitions ********)
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21 |
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22 | IndexRange[ Index[Generation] ] = Range[3]
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23 |
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24 | IndexRange[ Index[Colour] ] = NoUnfold[Range[3]]
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25 |
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26 | IndexRange[ Index[Gluon] ] = NoUnfold[Range[8]]
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27 |
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28 | IndexRange[ Index[SU2W] ] = Range[3]
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29 |
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30 |
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31 | IndexStyle[Colour, i]
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32 |
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33 | IndexStyle[Generation, f]
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34 |
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35 | IndexStyle[Gluon ,a]
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36 |
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37 |
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38 | (**************** Parameters *************)
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39 |
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40 | M$Parameters = {
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41 |
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42 | (* External parameters *)
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43 |
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44 | \[Alpha]EW == {
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45 | ParameterType -> External,
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46 | BlockName -> SMINPUTS,
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47 | ParameterName -> aEW,
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48 | InteractionOrder -> {QED, 2},
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49 | Value -> 1/132.50698,
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50 | Description -> "Electroweak coupling constant"},
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51 |
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52 | Gf == {
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53 | ParameterType -> External,
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54 | BlockName -> SMINPUTS,
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55 | InteractionOrder -> {QED, 2},
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56 | Value -> 1.16639 * 10^(-5),
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57 | Description -> "Fermi constant"},
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58 |
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59 | \[Alpha]S == {
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60 | ParameterType -> External,
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61 | BlockName -> SMINPUTS,
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62 | ParameterName -> aS,
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63 | InteractionOrder -> {QCD, 2},
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64 | Value -> 0.118,
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65 | Description -> "Strong coupling constant"},
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66 |
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67 | ZM == {
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68 | ParameterType -> External,
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69 | BlockName -> SMINPUTS,
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70 | Value -> 91.188,
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71 | Description -> "Z mass"},
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72 |
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73 | ymc == {
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74 | ParameterType -> External,
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75 | BlockName -> MGYUKAWA,
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76 | Value -> 1.42,
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77 | OrderBlock -> {4},
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78 | Description -> "Charm Yukawa mass"},
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79 |
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80 | ymb == {
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81 | ParameterType -> External,
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82 | BlockName -> MGYUKAWA,
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83 | Value -> 4.7,
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84 | OrderBlock -> {5},
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85 | Description -> "Bottom Yukawa mass"},
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86 |
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87 | ymt == {
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88 | ParameterType -> External,
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89 | BlockName -> MGYUKAWA,
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90 | Value -> 174.3,
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91 | OrderBlock -> {6},
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92 | Description -> "Top Yukawa mass"},
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93 |
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94 | ymtau == {
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95 | ParameterType -> External,
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96 | BlockName -> MGYUKAWA,
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97 | Value -> 1.777,
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98 | OrderBlock -> {15},
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99 | Description -> "Tau Yukawa mass"},
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100 |
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101 | ymm == {
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102 | Value -> 0.105},
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103 |
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104 | (* Internal Parameters *)
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105 |
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106 | WM == {
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107 | ParameterType -> Internal,
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108 | Value -> Sqrt[ZM^2/2+Sqrt[ZM^4/4-Pi/Sqrt[2]*\[Alpha]EW/Gf*ZM^2]],
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109 | Description -> "W mass"},
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110 |
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111 | sw2 == {
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112 | ParameterType -> Internal,
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113 | Value -> 1-(WM/ZM)^2,
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114 | Description -> "Squared Sin of the Weinberg angle"},
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115 |
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116 | ee == {
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117 | TeX -> e,
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118 | ParameterType -> Internal,
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119 | Value -> Sqrt[4 Pi \[Alpha]EW],
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120 | InteractionOrder -> {QED, 1},
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121 | Description -> "Electric coupling constant"},
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122 |
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123 | cw == {
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124 | TeX -> Subscript[c, w],
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125 | ParameterType -> Internal,
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126 | Value -> Sqrt[1 - sw2],
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127 | Description -> "Cos of the Weinberg angle"},
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128 |
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129 | sw == {
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130 | TeX -> Subscript[s, w],
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131 | ParameterType -> Internal,
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132 | Value -> Sqrt[sw2],
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133 | Description -> "Sin of the Weinberg angle"},
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134 |
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135 | gw == {
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136 | TeX -> Subscript[g, w],
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137 | ParameterType -> Internal,
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138 | Value -> ee / sw,
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139 | InteractionOrder -> {QED, 1},
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140 | Description -> "Weak coupling constant"},
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141 |
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142 | g1 == {
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143 | TeX -> Subscript[g, 1],
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144 | ParameterType -> Internal,
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145 | Value -> ee / cw,
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146 | InteractionOrder -> {QED, 1},
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147 | Description -> "U(1)Y coupling constant"},
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148 |
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149 | gs == {
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150 | TeX -> Subscript[g, s],
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151 | ParameterType -> Internal,
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152 | Value -> Sqrt[4 Pi \[Alpha]S],
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153 | InteractionOrder -> {QCD, 1},
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154 | ParameterName -> G,
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155 | Description -> "Strong coupling constant"},
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156 |
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157 | v == {
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158 | ParameterType -> Internal,
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159 | Value -> 2*MW*sw/ee,
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160 | InteractionOrder -> {QED, -1}},
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161 |
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162 | \[Lambda]0 == {
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163 | TeX -> Subscript[\[Lambda], 0],
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164 | Value -> 0.2,
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165 | InteractionOrder -> {QED, 2},
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166 | ParameterName -> l0},
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167 |
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168 |
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169 | yl == {
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170 | Indices -> {Index[Generation]},
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171 | AllowSummation -> True,
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172 | ParameterType -> Internal,
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173 | ComplexParameter -> False,
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174 | Value -> {yl[1] -> 0, yl[2] -> 0, yl[3] -> -ymtau / v},
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175 | ParameterName -> {yl[1] -> ye, yl[2] -> ym, yl[3] -> ytau},
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176 | InteractionOrder -> {QED, 1},
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177 | Definitions -> {yl[1] -> 0, yl[2] ->0},
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178 | Description -> "Lepton Yukawa coupling"},
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179 |
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180 | yu == {
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181 | Indices -> {Index[Generation]},
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182 | AllowSummation -> True,
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183 | ParameterType -> Internal,
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184 | ComplexParameter -> False,
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185 | Value -> {yu[1] -> 0, yu[2] -> - ymc / v, yu[3] -> -ymt / v},
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186 | ParameterName -> {yu[1] -> yu, yu[2] -> yc, yu[3] -> yt},
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187 | InteractionOrder -> {QED, 1},
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188 | ComplexParameter -> False,
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189 | Definitions -> {yu[1] -> 0},
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190 | Description -> "U-quark Yukawa coupling"},
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191 |
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192 | yd == {
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193 | Indices -> {Index[Generation]},
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194 | AllowSummation -> True,
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195 | ParameterType -> Internal,
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196 | ComplexParameter -> False,
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197 | Value -> {yd[1] -> 0, yd[2] -> 0, yd[3] -> -ymb / v},
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198 | ParameterName -> {yd[1] -> yd, yd[2] -> ys, yd[3] -> yb},
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199 | InteractionOrder -> {QED, 1},
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200 | Definitions -> {yd[1] -> 0, yd[2] -> 0},
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201 | Description -> "D-quark Yukawa coupling"},
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202 |
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203 | cabi == {
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204 | TeX -> Subscript[\[Theta], c],
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205 | ParameterType -> External,
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206 | BlockName -> CKMBLOCK,
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207 | OrderBlock -> {1},
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208 | Value -> 0.488,
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209 | Description -> "Cabibbo angle"},
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210 |
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211 | CKM == {
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212 | Indices -> {Index[Generation], Index[Generation]},
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213 | TensorClass -> CKM,
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214 | Unitary -> True,
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215 | Definitions -> {CKM[3, 3] -> 1,
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216 | CKM[i_, 3] :> 0 /; i != 3,
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217 | CKM[3, i_] :> 0 /; i != 3},
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218 | Value -> {CKM[1,2] -> Sin[cabi],
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219 | CKM[1,1] -> Cos[cabi],
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220 | CKM[2,1] -> -Sin[cabi],
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221 | CKM[2,2] -> Cos[cabi]},
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222 | Description -> "CKM-Matrix"},
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223 |
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224 | f1 == {Value -> 500,
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225 | TeX -> Subscript[f, 1],
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226 | InteractionOrder -> {QED, -1}},
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227 |
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228 | \[Lambda]1 == {Value -> 0.2,
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229 | TeX -> Subscript[\[Lambda], 1],
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230 | InteractionOrder -> {QED, 2},
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231 | ParameterName -> l1},
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232 |
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233 | tha == {Value -> 2.88,
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234 | TeX -> Subscript[\[Theta], a],
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235 | Description -> "Scalar mixing angle"},
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236 |
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237 | ca == {ParameterType -> Internal,
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238 | Value -> Cos[tha],
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239 | TeX -> Subscript[c,a],
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240 | Description -> "Cos of the scalar mixing angle"},
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241 |
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242 | sa == {ParameterType -> Internal,
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243 | Value -> Sin[tha],
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244 | TeX -> Subscript[s,a],
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245 | Description -> "Sin of the scalar mixing angle"}
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246 | }
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247 |
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248 |
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249 | (************** Gauge Groups ******************)
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250 |
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251 | M$GaugeGroups = {
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252 |
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253 | U1Y == {
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254 | Abelian -> True,
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255 | GaugeBoson -> B,
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256 | Charge -> Y,
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257 | CouplingConstant -> ee},
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258 |
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259 | SU2L == {
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260 | Abelian -> False,
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261 | GaugeBoson -> Wi,
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262 | StructureConstant -> ep,
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263 | CouplingConstant -> gw,
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264 | Definitions -> {ep -> Eps}},
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265 |
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266 | SU3C == {
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267 | Abelian -> False,
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268 | GaugeBoson -> G,
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269 | StructureConstant -> f,
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270 | DTerm -> dSUN,
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271 | Representations -> {T, Colour},
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272 | CouplingConstant -> gs}
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273 | }
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274 |
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275 | (********* Particle Classes **********)
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276 |
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277 | M$ClassesDescription = {
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278 |
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279 | (*** Fermions ***)
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280 |
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281 | (* Leptons (neutrino): I_3 = +1/2, Q = 0 *)
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282 | F[1] == {
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283 | ClassName -> vl,
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284 | ClassMembers -> {ve,vm,vt},
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285 | FlavorIndex -> Generation,
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286 | SelfConjugate -> False,
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287 | Indices -> {Index[Generation]},
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288 | Mass -> 0,
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289 | Width -> 0,
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290 | QuantumNumbers -> {LeptonNumber -> 1},
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291 | PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
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292 | PropagatorType -> S,
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293 | PropagatorArrow -> Forward,
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294 | PDG -> {12,14,16},
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295 | FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"} },
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296 |
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297 | (* Leptons (electron): I_3 = -1/2, Q = -1 *)
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298 | F[2] == {
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299 | ClassName -> l,
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300 | ClassMembers -> {e, m, tt},
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301 | FlavorIndex -> Generation,
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302 | SelfConjugate -> False,
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303 | Indices -> {Index[Generation]},
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304 | Mass -> {Ml, {ME, 0}, {MM, 0}, {MTA, 1.777}},
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305 | Width -> 0,
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306 | QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
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307 | PropagatorLabel -> {"l", "e", "m", "tt"},
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308 | PropagatorType -> Straight,
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309 | ParticleName -> {"e-", "m-", "tt-"},
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310 | AntiParticleName -> {"e+", "m+", "tt+"},
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311 | PropagatorArrow -> Forward,
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312 | PDG -> {11, 13, 15},
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313 | FullName -> {"Electron", "Muon", "Tau"} },
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314 |
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315 | (* Quarks (u): I_3 = +1/2, Q = +2/3 *)
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316 | F[3] == {
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317 | ClassMembers -> {u, c, t},
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318 | ClassName -> uq,
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319 | FlavorIndex -> Generation,
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320 | SelfConjugate -> False,
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321 | Indices -> {Index[Generation], Index[Colour]},
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322 | Mass -> {Mu, {MU, 0}, {MC, 0}, {MT, 174.3}},
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323 | Width -> {0, 0, {WT, 1.50833649}},
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324 | QuantumNumbers -> {Q -> 2/3},
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325 | PropagatorLabel -> {"uq", "u", "c", "t"},
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326 | PropagatorType -> Straight,
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327 | PropagatorArrow -> Forward,
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328 | PDG -> {2, 4, 6},
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329 | FullName -> {"u-quark", "c-quark", "t-quark"}},
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330 |
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331 | (* Quarks (d): I_3 = -1/2, Q = -1/3 *)
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332 | F[4] == {
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333 | ClassMembers -> {d, s, b},
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334 | ClassName -> dq,
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335 | FlavorIndex -> Generation,
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336 | SelfConjugate -> False,
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337 | Indices -> {Index[Generation], Index[Colour]},
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338 | Mass -> {Md, {MD, 0}, {MS, 0}, {MB, 4.7}},
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339 | Width -> 0,
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340 | QuantumNumbers -> {Q -> -1/3},
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341 | PropagatorLabel -> {"dq", "d", "s", "b"},
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342 | PropagatorType -> Straight,
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343 | PropagatorArrow -> Forward,
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344 | PDG -> {1,3,5},
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345 | FullName -> {"d-quark", "s-quark", "b-quark"} },
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346 |
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347 | (*** Gauge bosons ***)
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348 |
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349 | (* Gauge bosons: Q = 0 *)
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350 | V[1] == {
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351 | ClassName -> A,
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352 | SelfConjugate -> True,
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353 | Indices -> {},
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354 | Mass -> 0,
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355 | PropagatorLabel -> "a",
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356 | PropagatorType -> W,
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357 | PropagatorArrow -> None,
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358 | PDG -> 22,
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359 | FullName -> "Photon" },
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360 |
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361 | V[2] == {
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362 | ClassName -> Z,
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363 | SelfConjugate -> True,
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364 | Indices -> {},
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365 | Mass -> {MZ, 91.188},
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366 | Width -> {WZ, 2.44140351},
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367 | PropagatorLabel -> "Z",
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368 | PropagatorType -> Sine,
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369 | PropagatorArrow -> None,
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370 | PDG -> 23,
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371 | FullName -> "Z" },
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372 |
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373 | (* Gauge bosons: Q = -1 *)
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374 | V[3] == {
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375 | ClassName -> W,
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376 | SelfConjugate -> False,
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377 | Indices -> {},
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378 | Mass -> {MW, 80.419},
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379 | Width -> {WW, 2.04759951},
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380 | QuantumNumbers -> {Q -> 1},
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381 | PropagatorLabel -> "W",
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382 | PropagatorType -> Sine,
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383 | PropagatorArrow -> Forward,
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384 | ParticleName ->"W+",
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385 | AntiParticleName ->"W-",
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386 | PDG -> 24,
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387 | FullName -> "W" },
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388 |
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389 | V[4] == {
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390 | ClassName -> G,
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391 | SelfConjugate -> True,
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392 | Indices -> {Index[Gluon]},
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393 | Mass -> 0,
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394 | PropagatorLabel -> {"G"},
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395 | PropagatorType -> C,
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396 | PropagatorArrow -> None,
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397 | PDG -> 21,
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398 | FullName -> "G" },
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399 |
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400 | V[5] == {
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401 | ClassName -> Wi,
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402 | Unphysical -> True,
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403 | Definitions -> {Wi[mu_, 1] -> (W[mu] + Wbar[mu])/Sqrt[2],
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404 | Wi[mu_, 2] -> (Wbar[mu] - W[mu])/Sqrt[2]/I,
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405 | Wi[mu_, 3] -> cw Z[mu] + sw A[mu]},
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406 | SelfConjugate -> True,
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407 | Indices -> {Index[SU2W]},
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408 | FlavorIndex -> SU2W,
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409 | Mass -> 0,
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410 | PDG -> {1,2,3}},
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411 |
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412 | V[6] == {
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413 | ClassName -> B,
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414 | SelfConjugate -> True,
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415 | Definitions -> {B[mu_] -> -sw Z[mu] + cw A[mu]},
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416 | Indices -> {},
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417 | Mass -> 0,
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418 | Unphysical -> True},
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419 |
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420 | (*** Scalars ***)
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421 |
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422 |
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423 | (* physical Higgs: Q = 0 *)
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424 |
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425 | S[1] == {
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426 | ClassName -> h1,
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427 | SelfConjugate -> True,
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428 | Mass -> {Mh1, 78.5},
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429 | Width -> {Wh1, 0.005}},
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430 |
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431 | S[2] == {
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432 | ClassName -> h2,
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433 | SelfConjugate -> True,
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434 | Mass -> {Mh2, 326},
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435 | Width -> {Wh2, 0.005}},
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436 |
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437 | S[3] == {
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438 | ClassName -> H,
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439 | SelfConjugate -> True,
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440 | Unphysical -> True,
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441 | Definitions -> {H -> ca h1- sa h2}},
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442 |
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443 | S[4] == {
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444 | ClassName -> h,
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445 | SelfConjugate -> True,
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446 | Unphysical -> True,
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447 | Definitions -> {h -> sa h1 +ca h2}};
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448 |
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449 | S[5] == {
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450 | ClassName -> phi,
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451 | SelfConjugate -> True,
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452 | Mass -> {Mphi, 120},
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453 | Width -> Wphi,
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454 | PropagatorLabel -> "Phi",
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455 | PropagatorType -> D,
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456 | PropagatorArrow -> None,
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457 | ParticleName ->"phi0",
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458 | PDG -> 250,
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459 | FullName -> "Phi",
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460 | Goldstone -> Z },
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461 |
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462 | S[6] == {
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463 | ClassName -> phi2,
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464 | SelfConjugate -> False,
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465 | Mass -> {Mphi2, 120},
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466 | Width -> Wphi2,
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467 | PropagatorLabel -> "Phi2",
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468 | PropagatorType -> D,
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469 | PropagatorArrow -> None,
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470 | ParticleName ->"phi+",
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471 | AntiParticleName ->"phi-",
|
---|
472 | PDG -> 251,
|
---|
473 | FullName -> "Phi2",
|
---|
474 | Goldstone -> W,
|
---|
475 | QuantumNumbers -> {Q -> 1}},
|
---|
476 |
|
---|
477 |
|
---|
478 | (********* Ghost Fields ****************)(********** Ghosts **********)
|
---|
479 | U[1] == {
|
---|
480 | ClassName -> ghA,
|
---|
481 | SelfConjugate -> False,
|
---|
482 | Indices -> {},
|
---|
483 | Ghost -> A},
|
---|
484 |
|
---|
485 | U[2] == {
|
---|
486 | ClassName -> ghZ,
|
---|
487 | SelfConjugate -> False,
|
---|
488 | Indices -> {},
|
---|
489 | Ghost -> Z},
|
---|
490 |
|
---|
491 | U[31] == {
|
---|
492 | ClassName -> ghWp,
|
---|
493 | SelfConjugate -> False,
|
---|
494 | Indices -> {},
|
---|
495 | Ghost -> W,
|
---|
496 | QuantumNumbers -> {Q-> 1}},
|
---|
497 |
|
---|
498 | U[32] == {
|
---|
499 | ClassName -> ghWm,
|
---|
500 | SelfConjugate -> False,
|
---|
501 | Indices -> {},
|
---|
502 | Ghost -> Wbar,
|
---|
503 | QuantumNumbers -> {Q-> -1}},
|
---|
504 |
|
---|
505 | U[4] == {
|
---|
506 | ClassName -> ghG,
|
---|
507 | SelfConjugate -> False,
|
---|
508 | Indices -> {Index[Gluon]},
|
---|
509 | Ghost -> G},
|
---|
510 |
|
---|
511 | U[5] == {
|
---|
512 | ClassName -> ghWi,
|
---|
513 | Unphysical -> True,
|
---|
514 | Definitions -> {ghWi[1] -> (ghWp + ghWm)/Sqrt[2],
|
---|
515 | ghWi[2] -> (ghWm - ghWp)/Sqrt[2]/I,
|
---|
516 | ghWi[3] -> cw ghZ + sw ghA},
|
---|
517 | SelfConjugate -> False,
|
---|
518 | Indices -> {Index[SU2W]},
|
---|
519 | FlavorIndex -> SU2W},
|
---|
520 |
|
---|
521 | U[6] == {
|
---|
522 | ClassName -> ghB,
|
---|
523 | SelfConjugate -> False,
|
---|
524 | Definitions -> {ghB -> -sw ghZ + cw ghA},
|
---|
525 | Indices -> {},
|
---|
526 | Unphysical -> True}
|
---|
527 |
|
---|
528 |
|
---|
529 | }
|
---|
530 |
|
---|
531 | (*****************************************************************************************)
|
---|
532 |
|
---|
533 | (* SM Lagrangian *)
|
---|
534 |
|
---|
535 | (******************** Gauge F^2 Lagrangian terms*************************)
|
---|
536 | (*Sign convention from Lagrangian in between Eq. (A.9) and Eq. (A.10) of Peskin & Schroeder.*)
|
---|
537 | LGauge = -1/4 (del[Wi[nu, i1], mu] - del[Wi[mu, i1], nu] + gw ep[i1, i2, i3] Wi[mu, i2] Wi[nu, i3])*
|
---|
538 | (del[Wi[nu, i1], mu] - del[Wi[mu, i1], nu] + gw ep[i1, i4, i5] Wi[mu, i4] Wi[nu, i5]) -
|
---|
539 |
|
---|
540 | 1/4 (del[B[nu], mu] - del[B[mu], nu])^2 -
|
---|
541 |
|
---|
542 | 1/4 (del[G[nu, a1], mu] - del[G[mu, a1], nu] + gs f[a1, a2, a3] G[mu, a2] G[nu, a3])*
|
---|
543 | (del[G[nu, a1], mu] - del[G[mu, a1], nu] + gs f[a1, a4, a5] G[mu, a4] G[nu, a5]);
|
---|
544 |
|
---|
545 |
|
---|
546 | (********************* Fermion Lagrangian terms*************************)
|
---|
547 | (*Sign convention from Lagrangian in between Eq. (A.9) and Eq. (A.10) of Peskin & Schroeder.*)
|
---|
548 | LFermions = Module[{Lkin, LQCD, LEWleft, LEWright},
|
---|
549 |
|
---|
550 | Lkin = I uqbar.Ga[mu].del[uq, mu] +
|
---|
551 | I dqbar.Ga[mu].del[dq, mu] +
|
---|
552 | I lbar.Ga[mu].del[l, mu] +
|
---|
553 | I vlbar.Ga[mu].del[vl, mu];
|
---|
554 |
|
---|
555 | LQCD = gs (uqbar.Ga[mu].T[a].uq +
|
---|
556 | dqbar.Ga[mu].T[a].dq)G[mu, a];
|
---|
557 |
|
---|
558 | LBright =
|
---|
559 | -2ee/cw B[mu]/2 lbar.Ga[mu].ProjP.l + (*Y_lR=-2*)
|
---|
560 | 4ee/3/cw B[mu]/2 uqbar.Ga[mu].ProjP.uq - (*Y_uR=4/3*)
|
---|
561 | 2ee/3/cw B[mu]/2 dqbar.Ga[mu].ProjP.dq; (*Y_dR=-2/3*)
|
---|
562 |
|
---|
563 | LBleft =
|
---|
564 | -ee/cw B[mu]/2 vlbar.Ga[mu].ProjM.vl - (*Y_LL=-1*)
|
---|
565 | ee/cw B[mu]/2 lbar.Ga[mu].ProjM.l + (*Y_LL=-1*)
|
---|
566 | ee/3/cw B[mu]/2 uqbar.Ga[mu].ProjM.uq + (*Y_QL=1/3*)
|
---|
567 | ee/3/cw B[mu]/2 dqbar.Ga[mu].ProjM.dq ; (*Y_QL=1/3*)
|
---|
568 |
|
---|
569 | LWleft = ee/sw/2(
|
---|
570 | vlbar.Ga[mu].ProjM.vl Wi[mu, 3] - (*sigma3 = ( 1 0 )*)
|
---|
571 | lbar.Ga[mu].ProjM.l Wi[mu, 3] + (* ( 0 -1 )*)
|
---|
572 |
|
---|
573 | Sqrt[2] vlbar.Ga[mu].ProjM.l W[mu] +
|
---|
574 | Sqrt[2] lbar.Ga[mu].ProjM.vl Wbar[mu]+
|
---|
575 |
|
---|
576 | uqbar.Ga[mu].ProjM.uq Wi[mu, 3] - (*sigma3 = ( 1 0 )*)
|
---|
577 | dqbar.Ga[mu].ProjM.dq Wi[mu, 3] + (* ( 0 -1 )*)
|
---|
578 |
|
---|
579 | Sqrt[2] uqbar.Ga[mu].ProjM.CKM.dq W[mu] +
|
---|
580 | Sqrt[2] dqbar.Ga[mu].ProjM.HC[CKM].uq Wbar[mu]
|
---|
581 | );
|
---|
582 |
|
---|
583 | Lkin + LQCD + LBright + LBleft + LWleft];
|
---|
584 |
|
---|
585 | (******************** Higgs Lagrangian terms****************************)
|
---|
586 | Phi := If[FeynmanGauge, {I phi2, (v + H - I phi)/Sqrt[2]}, {0, (v + H)/Sqrt[2]}];
|
---|
587 | Phibar := If[FeynmanGauge, {-I phi2bar, (v + H + I phi)/Sqrt[2]} ,{0, (v + H)/Sqrt[2]}];
|
---|
588 |
|
---|
589 |
|
---|
590 |
|
---|
591 | LHiggs := Block[{PMVec, WVec, Dc, Dcbar, Vphi},
|
---|
592 |
|
---|
593 | PMVec = Table[PauliSigma[i], {i, 3}];
|
---|
594 | Wvec[mu_] := {Wi[mu, 1], Wi[mu, 2], Wi[mu, 3]};
|
---|
595 |
|
---|
596 | (*Y_phi=1*)
|
---|
597 | Dc[f_, mu_] := I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 (Wvec[mu].PMVec).f;
|
---|
598 | Dcbar[f_, mu_] := -I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 f.(Wvec[mu].PMVec);
|
---|
599 |
|
---|
600 | Vphi[Phi_, Phibar_] := muH^2 Phibar.Phi + \[Lambda]0 (Phibar.Phi)^2;
|
---|
601 |
|
---|
602 | (Dcbar[Phibar, mu]).Dc[Phi, mu] - Vphi[Phi, Phibar]];
|
---|
603 |
|
---|
604 | (*The covariant derivative in terms of physical states is: *)
|
---|
605 | (* ( A + (cw^2-sw^2)/2cwsw Z 1/Sqrt[2]sw W+ ) *)
|
---|
606 | (* D phi = id phi + e ( ) phi *)
|
---|
607 | (* ( 1/Sqrt[2]sw W- -1/2cwsw Z ) *)
|
---|
608 |
|
---|
609 | (*From this we can determine the mixing term. *)
|
---|
610 | (* *)
|
---|
611 | (* L_mix = - MW ( W- dphi+ + W+ dphi- ) - MZ Z dphi0 *)
|
---|
612 | (* This term must be cancelled in the gauge fixing Lagrangian.*)
|
---|
613 |
|
---|
614 |
|
---|
615 |
|
---|
616 | (*************** Yukawa Lagrangian***********************)
|
---|
617 | LYuk := If[FeynmanGauge,
|
---|
618 | Module[{s,r,n,m,i}, -
|
---|
619 | yd[n] CKM[n,m] uqbar[s,n,i].ProjP[s,r].dq[r,m,i] (-I phi2) -
|
---|
620 | yd[n] dqbar[s,n,i].ProjP[s,r].dq[r,n,i] (v+H +I phi)/Sqrt[2] -
|
---|
621 |
|
---|
622 | yu[n] uqbar[s,n,i].ProjP[s,r].uq[r,n,i] (v+H -I phi)/Sqrt[2] + (*This sign from eps matrix*)
|
---|
623 | yu[n] HC[CKM[n,m]] dqbar[s,n,i].ProjP[s,r].uq[r,m,i] ( I phi2bar) -
|
---|
624 |
|
---|
625 | yl[n] vlbar[s,n].ProjP[s,r].l[r,n] (-I phi2) -
|
---|
626 | yl[n] lbar[s,n].ProjP[s,r].l[r,n] (v+H +I phi)/Sqrt[2]
|
---|
627 | ],
|
---|
628 | Module[{s,r,n,m,i}, -
|
---|
629 | yd[n] dqbar[s,n,i].ProjP[s,r].dq[r,n,i] (v+H)/Sqrt[2] -
|
---|
630 |
|
---|
631 | yu[n] uqbar[s,n,i].ProjP[s,r].uq[r,n,i] (v+H)/Sqrt[2]
|
---|
632 | -
|
---|
633 | yl[n] lbar[s,n].ProjP[s,r].l[r,n] (v+H)/Sqrt[2]
|
---|
634 | ]
|
---|
635 | ];
|
---|
636 |
|
---|
637 | LYukawa := LYuk + HC[LYuk]/.HC[v]->v;
|
---|
638 |
|
---|
639 |
|
---|
640 | (************Gauge Fix terms*************************)
|
---|
641 | LGaugeFix := If[FeynmanGauge,
|
---|
642 | Block[{GFG,GFW,GFWbar,GFZ,GFA},
|
---|
643 |
|
---|
644 | GFG[a_] := Module[{mu}, del[G[mu,a],mu] ];
|
---|
645 |
|
---|
646 | GFW := Module[{mu}, del[W[mu],mu] + MW phi2 ];
|
---|
647 | GFWbar := Module[{mu}, del[Wbar[mu],mu] + MW phi2bar ];
|
---|
648 |
|
---|
649 | GFZ := Module[{mu}, del[Z[mu],mu] + MZ phi ];
|
---|
650 |
|
---|
651 | GFA := Module[{mu}, del[A[mu],mu] ];
|
---|
652 |
|
---|
653 |
|
---|
654 | - 1/2*GFG[a]GFG[a] - GFWbar*GFW - 1/2*GFZ^2 - 1/2*GFA^2 ]
|
---|
655 |
|
---|
656 | , 0];
|
---|
657 |
|
---|
658 | (* We can determine the mixing term from this. *)
|
---|
659 | (* *)
|
---|
660 | (* L_mix = MW ( phi+ dW- + phi- dW+ ) + MZ phi0 dZ *)
|
---|
661 | (* This exactly cancels the mixing term from LHiggs. *)
|
---|
662 |
|
---|
663 |
|
---|
664 |
|
---|
665 | (**************Ghost terms**************************)
|
---|
666 | (* Now we need the ghost terms which are of the form: *)
|
---|
667 | (* - g * antighost * d_BRST G *)
|
---|
668 | (* where d_BRST G is BRST transform of the gauge fixing function. *)
|
---|
669 |
|
---|
670 | LGhost := If[FeynmanGauge,
|
---|
671 | Block[{dBRSTG,LGhostG,dBRSTWi,LGhostWi,dBRSTB,LGhostB},
|
---|
672 |
|
---|
673 | (***********First the pure gauge piece.**********************)
|
---|
674 | dBRSTG[mu_,a_] := 1/gs Module[{a2, a3}, del[ghG[a], mu] + gs f[a,a2,a3] G[mu,a2] ghG[a3]];
|
---|
675 | LGhostG := - gs ghGbar[a].del[dBRSTG[mu,a],mu];
|
---|
676 |
|
---|
677 | dBRSTWi[mu_,i_] := sw/ee Module[{i2, i3}, del[ghWi[i], mu] + ee/sw ep[i,i2,i3] Wi[mu,i2] ghWi[i3] ];
|
---|
678 | LGhostWi := - ee/sw ghWibar[a].del[dBRSTWi[mu,a],mu];
|
---|
679 |
|
---|
680 | dBRSTB[mu_] := cw/ee del[ghB, mu];
|
---|
681 | LGhostB := - ee/cw ghBbar.del[dBRSTB[mu],mu];
|
---|
682 |
|
---|
683 | (***********Next the piece from the scalar field.************)
|
---|
684 | LGhostphi := - ee/(2*sw*cw) MW ( I phi2 ( (cw^2-sw^2)ghWpbar.ghZ + 2sw*cw ghWpbar.ghA ) -
|
---|
685 | I phi2bar ( (cw^2-sw^2)ghWmbar.ghZ + 2sw*cw ghWmbar.ghA ) ) -
|
---|
686 | ee/(2*sw) MW ( ( (v+H) - I phi) ghWpbar.ghWp +
|
---|
687 | ( (v+H) + I phi) ghWmbar.ghWm ) -
|
---|
688 | I ee/(2*sw) MZ ( phi2bar ghZbar.ghWp - phi2 ghZbar.ghWm ) -
|
---|
689 | ee/(2*sw*cw) MZ (v+H) ghZbar.ghZ ;
|
---|
690 |
|
---|
691 |
|
---|
692 | (***********Now add the pieces together.********************)
|
---|
693 | LGhostG + LGhostWi + LGhostB + LGhostphi]
|
---|
694 |
|
---|
695 | , 0];
|
---|
696 |
|
---|
697 |
|
---|
698 | (*************** Hill Lagrangian***********************)
|
---|
699 |
|
---|
700 | LHill := 1/2 del[h, mu]^2 - l1 (f1 (h + v^2/2/f1) - HC[Phi].Phi)^2;
|
---|
701 |
|
---|
702 | (*********Total SM Lagrangian*******)
|
---|
703 | Lag := LGauge + LHiggs + LFermions + LYukawa +LGhost + LGaugeFix+ LHill;
|
---|
704 |
|
---|