1 | (***************************************************************************************************************)
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2 | (****** This is the FeynRules mod-file for the Standard model ******)
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3 | (****** ******)
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4 | (****** Authors: N. Christensen, C. Duhr, B. Fuks ******)
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5 | (****** ******)
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6 | (****** Choose whether Feynman gauge is desired. ******)
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7 | (****** If set to False, unitary gauge is assumed. ****)
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8 | (****** Feynman gauge is especially useful for CalcHEP/CompHEP where the calculation is 10-100 times faster. ***)
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9 | (****** Feynman gauge is not supported in MadGraph and Sherpa. ****)
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10 | (***************************************************************************************************************)
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11 |
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12 | (* ************************** *)
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13 | (* ***** Information ***** *)
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14 | (* ************************** *)
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15 | M$ModelName = "Standard Model";
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16 |
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17 | M$Information = {
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18 | Authors -> {"N. Christensen", "C. Duhr", "B. Fuks"},
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19 | Version -> "1.4.7",
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20 | Date -> "28. 09. 2016",
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21 | Institutions -> {"Michigan State University", "Universite catholique de Louvain (CP3)", "IPHC Strasbourg / University of Strasbourg"},
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22 | Emails -> {"neil@pa.msu.edu", "claude.duhr@uclouvain.be", "benjamin.fuks@cnrs.in2p3.fr"},
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23 | URLs -> "http://feynrules.phys.ucl.ac.be/view/Main/StandardModel"
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24 | };
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25 |
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26 | FeynmanGauge = True;
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27 |
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28 | (* ************************** *)
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29 | (* ***** NLO Variables ****** *)
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30 | (******************************)
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31 |
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32 | FR$LoopSwitches = {{Gf, MW}};
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33 | FR$RmDblExt = { ymb -> MB, ymc -> MC, ymdo -> MD, yme -> Me,
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34 | ymm -> MMU, yms -> MS, ymt -> MT, ymtau -> MTA, ymup -> MU};
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35 |
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36 | (* ************************** *)
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37 | (* ***** Change log ***** *)
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38 | (* ************************** *)
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39 |
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40 | (* 2018 0318: RR: Added CKM angles theta13, theta23, and delta13 *)
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41 | (* v1.4.7: Index issue with the ghost Lagrangian (special thanks to S. Iwamoto *)
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42 | (* v1.4.6: NLO variable added. *)
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43 | (* v1.4.5: Added widths for ghosts. *)
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44 | (* v1.4.4: Changed widths of goldstone bosons to be the same as for the W and Z bosons *)
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45 | (* v1.4.3: Updated conventions for the symmetric structure constants of SU3. *)
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46 | (* v1.4.2: Set FeynmanGauge=True as default again. *)
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47 | (* v1.4: Added SU(2) representation. *)
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48 | (* -> Modification in the field declarations (doublets are added) *)
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49 | (* -> Modification in the Lagrangian (much simpler). *)
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50 | (* v1.3: Added yukawa couplings for all fermions for gauge invariance. *)
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51 | (* Added yukawa couplings for 1st generation fermions to Massless.rst. *)
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52 | (* Updated parameters to PDG 2010. *)
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53 | (* v1.2: Set FeynmanGauge=True as default. *)
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54 | (* Set Gluonic ghosts to be included in both gauges. *)
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55 | (* v1.1: Fixed yukawa couplings in Feynman gauge. *)
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56 | (* Changed yd[n] CKM[n,m] to yd[m] CKM[n,m]. *)
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57 | (* Changed yu[n] Conjugate[CKM[m,n]] to yu[m] Conjugate[CKM[m,n]]. *)
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58 |
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59 | (* ************************** *)
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60 | (* ***** vevs ***** *)
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61 | (* ************************** *)
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62 | M$vevs = { {Phi[2],vev} };
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63 |
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64 | (* ************************** *)
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65 | (* ***** Gauge groups ***** *)
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66 | (* ************************** *)
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67 | M$GaugeGroups = {
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68 | U1Y == {
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69 | Abelian -> True,
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70 | CouplingConstant -> g1,
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71 | GaugeBoson -> B,
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72 | Charge -> Y
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73 | },
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74 | SU2L == {
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75 | Abelian -> False,
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76 | CouplingConstant -> gw,
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77 | GaugeBoson -> Wi,
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78 | StructureConstant -> Eps,
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79 | Representations -> {Ta,SU2D},
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80 | Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
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81 | },
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82 | SU3C == {
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83 | Abelian -> False,
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84 | CouplingConstant -> gs,
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85 | GaugeBoson -> G,
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86 | StructureConstant -> f,
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87 | Representations -> {T,Colour},
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88 | SymmetricTensor -> dSUN
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89 | }
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90 | };
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91 |
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92 |
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93 | (* ************************** *)
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94 | (* ***** Indices ***** *)
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95 | (* ************************** *)
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96 |
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97 | IndexRange[Index[SU2W ]] = Unfold[Range[3]];
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98 | IndexRange[Index[SU2D ]] = Unfold[Range[2]];
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99 | IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
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100 | IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
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101 | IndexRange[Index[Generation]] = Range[3];
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102 |
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103 | IndexStyle[SU2W, j];
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104 | IndexStyle[SU2D, k];
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105 | IndexStyle[Gluon, a];
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106 | IndexStyle[Colour, m];
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107 | IndexStyle[Generation, f];
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108 |
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109 |
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110 | (* ************************** *)
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111 | (* *** Interaction orders *** *)
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112 | (* *** (as used by mg5) *** *)
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113 | (* ************************** *)
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114 |
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115 | M$InteractionOrderHierarchy = {
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116 | {QCD, 1},
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117 | {QED, 2}
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118 | };
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119 |
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120 |
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121 | (* ************************** *)
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122 | (* **** Particle classes **** *)
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123 | (* ************************** *)
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124 | M$ClassesDescription = {
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125 |
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126 | (* Gauge bosons: physical vector fields *)
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127 | V[1] == {
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128 | ClassName -> A,
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129 | SelfConjugate -> True,
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130 | Mass -> 0,
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131 | Width -> 0,
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132 | ParticleName -> "a",
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133 | PDG -> 22,
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134 | PropagatorLabel -> "a",
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135 | PropagatorType -> W,
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136 | PropagatorArrow -> None,
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137 | FullName -> "Photon"
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138 | },
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139 | V[2] == {
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140 | ClassName -> Z,
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141 | SelfConjugate -> True,
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142 | Mass -> {MZ, 91.1876},
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143 | Width -> {WZ, 2.4952},
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144 | ParticleName -> "Z",
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145 | PDG -> 23,
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146 | PropagatorLabel -> "Z",
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147 | PropagatorType -> Sine,
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148 | PropagatorArrow -> None,
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149 | FullName -> "Z"
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150 | },
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151 | V[3] == {
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152 | ClassName -> W,
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153 | Unphysical -> True,
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154 | SelfConjugate -> False,
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155 | Mass -> {MWX, Internal},
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156 | Width -> {WWX, Internal},
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157 | ParticleName -> "W+",
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158 | AntiParticleName -> "W-",
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159 | QuantumNumbers -> {Q -> 1},
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160 | PDG -> 249,
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161 | PropagatorLabel -> "W",
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162 | PropagatorType -> Sine,
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163 | PropagatorArrow -> Forward,
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164 | FullName -> "W",
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165 | Definitions -> {W[mu_] -> W0[mu] + WT[mu] + WA[mu] + WX[mu]}
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166 | },
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167 | V[240] == {
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168 | ClassName -> W0,
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169 | SelfConjugate -> False,
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170 | Mass -> {MW0, Internal},
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171 | Width -> {WW0, Internal},
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172 | ParticleName -> "W0+",
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173 | AntiParticleName -> "W0-",
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174 | QuantumNumbers -> {Q -> 1},
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175 | PDG -> 240,
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176 | PropagatorLabel -> "W0",
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177 | PropagatorType -> Sine,
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178 | PropagatorArrow -> Forward,
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179 | FullName -> "W0"
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180 | },
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181 | V[241] == {
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182 | ClassName -> WT,
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183 | SelfConjugate -> False,
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184 | Mass -> {MWT, Internal},
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185 | Width -> {WWT, Internal},
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186 | ParticleName -> "WT+",
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187 | AntiParticleName -> "WT-",
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188 | QuantumNumbers -> {Q -> 1},
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189 | PDG -> 241,
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190 | PropagatorLabel -> "WT",
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191 | PropagatorType -> Sine,
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192 | PropagatorArrow -> Forward,
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193 | FullName -> "WT"
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194 | },
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195 | V[242] == {
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196 | ClassName -> WA,
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197 | SelfConjugate -> False,
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198 | Mass -> {MWA, Internal},
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199 | Width -> {WWA, Internal},
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200 | ParticleName -> "WA+",
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201 | AntiParticleName -> "WA-",
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202 | QuantumNumbers -> {Q -> 1},
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203 | PDG -> 242,
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204 | PropagatorLabel -> "WA",
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205 | PropagatorType -> Sine,
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206 | PropagatorArrow -> Forward,
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207 | FullName -> "WA"
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208 | },
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209 | V[243] == {
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210 | ClassName -> WX,
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211 | SelfConjugate -> False,
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212 | Mass -> {MW, Internal},
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213 | Width -> {WW, 2.085},
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214 | ParticleName -> "WX+",
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215 | AntiParticleName -> "WX-",
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216 | QuantumNumbers -> {Q -> 1},
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217 | PDG -> 24,
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218 | PropagatorLabel -> "WX",
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219 | PropagatorType -> Sine,
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220 | PropagatorArrow -> Forward,
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221 | FullName -> "WX"
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222 | },
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223 | V[4] == {
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224 | ClassName -> G,
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225 | SelfConjugate -> True,
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226 | Indices -> {Index[Gluon]},
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227 | Mass -> 0,
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228 | Width -> 0,
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229 | ParticleName -> "g",
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230 | PDG -> 21,
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231 | PropagatorLabel -> "G",
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232 | PropagatorType -> C,
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233 | PropagatorArrow -> None,
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234 | FullName -> "G"
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235 | },
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236 |
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237 | (* Ghosts: related to physical gauge bosons *)
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238 | U[1] == {
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239 | ClassName -> ghA,
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240 | SelfConjugate -> False,
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241 | Ghost -> A,
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242 | QuantumNumbers -> {GhostNumber -> 1},
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243 | Mass -> 0,
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244 | Width -> 0,
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245 | PropagatorLabel -> "uA",
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246 | PropagatorType -> GhostDash,
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247 | PropagatorArrow -> Forward
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248 | },
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249 | U[2] == {
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250 | ClassName -> ghZ,
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251 | SelfConjugate -> False,
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252 | Ghost -> Z,
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253 | QuantumNumbers -> {GhostNumber -> 1},
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254 | Mass -> {MZ,91.1876},
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255 | Width -> {WZ, 2.4952},
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256 | PropagatorLabel -> "uZ",
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257 | PropagatorType -> GhostDash,
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258 | PropagatorArrow -> Forward
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259 | },
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260 | U[31] == {
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261 | ClassName -> ghWp,
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262 | SelfConjugate -> False,
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263 | Ghost -> W,
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264 | QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
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265 | Mass -> {MW,Internal},
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266 | Width -> {WW, 2.085},
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267 | PropagatorLabel -> "uWp",
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268 | PropagatorType -> GhostDash,
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269 | PropagatorArrow -> Forward
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270 | },
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271 | U[32] == {
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272 | ClassName -> ghWm,
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273 | SelfConjugate -> False,
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274 | Ghost -> Wbar,
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275 | QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
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276 | Mass -> {MW,Internal},
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277 | Width -> {WW, 2.085},
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278 | PropagatorLabel -> "uWm",
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279 | PropagatorType -> GhostDash,
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280 | PropagatorArrow -> Forward
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281 | },
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282 | U[4] == {
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283 | ClassName -> ghG,
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284 | SelfConjugate -> False,
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285 | Indices -> {Index[Gluon]},
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286 | Ghost -> G,
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287 | PDG -> 82,
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288 | QuantumNumbers ->{GhostNumber -> 1},
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289 | Mass -> 0,
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290 | Width -> 0,
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291 | PropagatorLabel -> "uG",
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292 | PropagatorType -> GhostDash,
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293 | PropagatorArrow -> Forward
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294 | },
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295 |
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296 | (* Gauge bosons: unphysical vector fields *)
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297 | V[11] == {
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298 | ClassName -> B,
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299 | Unphysical -> True,
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300 | SelfConjugate -> True,
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301 | Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
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302 | },
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303 | V[12] == {
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304 | ClassName -> Wi,
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305 | Unphysical -> True,
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306 | SelfConjugate -> True,
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307 | Indices -> {Index[SU2W]},
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308 | FlavorIndex -> SU2W,
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309 | Definitions -> { Wi[mu_,1] -> (Wbar[mu]+W[mu])/Sqrt[2], Wi[mu_,2] -> (Wbar[mu]-W[mu])/(I*Sqrt[2]), Wi[mu_,3] -> cw Z[mu] + sw A[mu]}
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310 | },
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311 |
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312 | (* Ghosts: related to unphysical gauge bosons *)
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313 | U[11] == {
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314 | ClassName -> ghB,
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315 | Unphysical -> True,
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316 | SelfConjugate -> False,
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317 | Ghost -> B,
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318 | Definitions -> { ghB -> -sw ghZ + cw ghA}
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319 | },
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320 | U[12] == {
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321 | ClassName -> ghWi,
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322 | Unphysical -> True,
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323 | SelfConjugate -> False,
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324 | Ghost -> Wi,
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325 | Indices -> {Index[SU2W]},
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326 | FlavorIndex -> SU2W,
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327 | Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
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328 | } ,
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329 |
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330 | (* Fermions: physical fields *)
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331 | F[1] == {
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332 | ClassName -> vl,
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333 | ClassMembers -> {ve,vm,vt},
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334 | Indices -> {Index[Generation]},
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335 | FlavorIndex -> Generation,
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336 | SelfConjugate -> False,
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337 | Mass -> 0,
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338 | Width -> 0,
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339 | QuantumNumbers -> {LeptonNumber -> 1},
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340 | PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
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341 | PropagatorType -> S,
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342 | PropagatorArrow -> Forward,
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343 | PDG -> {12,14,16},
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344 | ParticleName -> {"ve","vm","vt"},
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345 | AntiParticleName -> {"ve~","vm~","vt~"},
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346 | FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
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347 | },
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348 | F[2] == {
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349 | ClassName -> l,
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350 | ClassMembers -> {e, mu, ta},
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351 | Indices -> {Index[Generation]},
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352 | FlavorIndex -> Generation,
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353 | SelfConjugate -> False,
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354 | Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
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355 | Width -> 0,
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356 | QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
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357 | PropagatorLabel -> {"l", "e", "mu", "ta"},
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358 | PropagatorType -> Straight,
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359 | PropagatorArrow -> Forward,
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360 | PDG -> {11, 13, 15},
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361 | ParticleName -> {"e-", "mu-", "ta-"},
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362 | AntiParticleName -> {"e+", "mu+", "ta+"},
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363 | FullName -> {"Electron", "Muon", "Tau"}
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364 | },
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365 | F[3] == {
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366 | ClassName -> uq,
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367 | ClassMembers -> {u, c, t},
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368 | Indices -> {Index[Generation], Index[Colour]},
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369 | FlavorIndex -> Generation,
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370 | SelfConjugate -> False,
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371 | Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,173.3}},
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372 | Width -> {0, 0, {WT,1.350}},
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373 | QuantumNumbers -> {Q -> 2/3},
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374 | PropagatorLabel -> {"uq", "u", "c", "t"},
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375 | PropagatorType -> Straight,
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376 | PropagatorArrow -> Forward,
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377 | PDG -> {2, 4, 6},
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378 | ParticleName -> {"u", "c", "t" },
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379 | AntiParticleName -> {"u~", "c~", "t~"},
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380 | FullName -> {"u-quark", "c-quark", "t-quark"}
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381 | },
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382 | F[4] == {
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383 | ClassName -> dq,
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384 | ClassMembers -> {d, s, b},
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385 | Indices -> {Index[Generation], Index[Colour]},
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386 | FlavorIndex -> Generation,
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387 | SelfConjugate -> False,
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388 | Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
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389 | Width -> 0,
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390 | QuantumNumbers -> {Q -> -1/3},
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391 | PropagatorLabel -> {"dq", "d", "s", "b"},
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392 | PropagatorType -> Straight,
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393 | PropagatorArrow -> Forward,
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394 | PDG -> {1,3,5},
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395 | ParticleName -> {"d", "s", "b" },
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396 | AntiParticleName -> {"d~", "s~", "b~"},
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397 | FullName -> {"d-quark", "s-quark", "b-quark"}
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398 | },
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399 |
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400 | (* Fermions: unphysical fields *)
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401 | F[11] == {
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402 | ClassName -> LL,
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403 | Unphysical -> True,
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404 | Indices -> {Index[SU2D], Index[Generation]},
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405 | FlavorIndex -> SU2D,
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406 | SelfConjugate -> False,
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407 | QuantumNumbers -> {Y -> -1/2},
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408 | Definitions -> { LL[sp1_,1,ff_] :> Module[{sp2}, ProjM[sp1,sp2] vl[sp2,ff]], LL[sp1_,2,ff_] :> Module[{sp2}, ProjM[sp1,sp2] l[sp2,ff]] }
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409 | },
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410 | F[12] == {
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411 | ClassName -> lR,
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412 | Unphysical -> True,
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413 | Indices -> {Index[Generation]},
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414 | FlavorIndex -> Generation,
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415 | SelfConjugate -> False,
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416 | QuantumNumbers -> {Y -> -1},
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417 | Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
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418 | },
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419 | F[13] == {
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420 | ClassName -> QL,
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421 | Unphysical -> True,
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422 | Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
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423 | FlavorIndex -> SU2D,
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424 | SelfConjugate -> False,
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425 | QuantumNumbers -> {Y -> 1/6},
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426 | Definitions -> {
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427 | QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
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428 | QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
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429 | },
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430 | F[14] == {
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431 | ClassName -> uR,
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432 | Unphysical -> True,
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433 | Indices -> {Index[Generation], Index[Colour]},
|
---|
434 | FlavorIndex -> Generation,
|
---|
435 | SelfConjugate -> False,
|
---|
436 | QuantumNumbers -> {Y -> 2/3},
|
---|
437 | Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
|
---|
438 | },
|
---|
439 | F[15] == {
|
---|
440 | ClassName -> dR,
|
---|
441 | Unphysical -> True,
|
---|
442 | Indices -> {Index[Generation], Index[Colour]},
|
---|
443 | FlavorIndex -> Generation,
|
---|
444 | SelfConjugate -> False,
|
---|
445 | QuantumNumbers -> {Y -> -1/3},
|
---|
446 | Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
|
---|
447 | },
|
---|
448 |
|
---|
449 | (* Higgs: physical scalars *)
|
---|
450 | S[1] == {
|
---|
451 | ClassName -> H,
|
---|
452 | SelfConjugate -> True,
|
---|
453 | Mass -> {MH,125.7},
|
---|
454 | Width -> {WH,0.004170},
|
---|
455 | PropagatorLabel -> "H",
|
---|
456 | PropagatorType -> D,
|
---|
457 | PropagatorArrow -> None,
|
---|
458 | PDG -> 25,
|
---|
459 | ParticleName -> "H",
|
---|
460 | FullName -> "H"
|
---|
461 | },
|
---|
462 |
|
---|
463 | (* Higgs: physical scalars *)
|
---|
464 | S[2] == {
|
---|
465 | ClassName -> G0,
|
---|
466 | SelfConjugate -> True,
|
---|
467 | Goldstone -> Z,
|
---|
468 | Mass -> {MZ, 91.1876},
|
---|
469 | Width -> {WZ, 2.4952},
|
---|
470 | PropagatorLabel -> "Go",
|
---|
471 | PropagatorType -> D,
|
---|
472 | PropagatorArrow -> None,
|
---|
473 | PDG -> 250,
|
---|
474 | ParticleName -> "G0",
|
---|
475 | FullName -> "G0"
|
---|
476 | },
|
---|
477 | S[3] == {
|
---|
478 | ClassName -> GP,
|
---|
479 | SelfConjugate -> False,
|
---|
480 | Goldstone -> W,
|
---|
481 | Mass -> {MW, Internal},
|
---|
482 | QuantumNumbers -> {Q -> 1},
|
---|
483 | Width -> {WW, 2.085},
|
---|
484 | PropagatorLabel -> "GP",
|
---|
485 | PropagatorType -> D,
|
---|
486 | PropagatorArrow -> None,
|
---|
487 | PDG -> 251,
|
---|
488 | ParticleName -> "G+",
|
---|
489 | AntiParticleName -> "G-",
|
---|
490 | FullName -> "GP"
|
---|
491 | },
|
---|
492 |
|
---|
493 | (* Higgs: unphysical scalars *)
|
---|
494 | S[11] == {
|
---|
495 | ClassName -> Phi,
|
---|
496 | Unphysical -> True,
|
---|
497 | Indices -> {Index[SU2D]},
|
---|
498 | FlavorIndex -> SU2D,
|
---|
499 | SelfConjugate -> False,
|
---|
500 | QuantumNumbers -> {Y -> 1/2},
|
---|
501 | Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
|
---|
502 | }
|
---|
503 | };
|
---|
504 |
|
---|
505 |
|
---|
506 | (* ************************** *)
|
---|
507 | (* ***** Gauge ***** *)
|
---|
508 | (* ***** Parameters ***** *)
|
---|
509 | (* ***** (FeynArts) ***** *)
|
---|
510 | (* ************************** *)
|
---|
511 |
|
---|
512 | GaugeXi[ V[1] ] = GaugeXi[A];
|
---|
513 | GaugeXi[ V[2] ] = GaugeXi[Z];
|
---|
514 | GaugeXi[ V[3] ] = GaugeXi[W];
|
---|
515 | GaugeXi[ V[4] ] = GaugeXi[G];
|
---|
516 | GaugeXi[ S[1] ] = 1;
|
---|
517 | GaugeXi[ S[2] ] = GaugeXi[Z];
|
---|
518 | GaugeXi[ S[3] ] = GaugeXi[W];
|
---|
519 | GaugeXi[ U[1] ] = GaugeXi[A];
|
---|
520 | GaugeXi[ U[2] ] = GaugeXi[Z];
|
---|
521 | GaugeXi[ U[31] ] = GaugeXi[W];
|
---|
522 | GaugeXi[ U[32] ] = GaugeXi[W];
|
---|
523 | GaugeXi[ U[4] ] = GaugeXi[G];
|
---|
524 |
|
---|
525 |
|
---|
526 | (* ************************** *)
|
---|
527 | (* ***** Parameters ***** *)
|
---|
528 | (* ************************** *)
|
---|
529 | M$Parameters = {
|
---|
530 |
|
---|
531 | (* External parameters *)
|
---|
532 | aEWM1 == {
|
---|
533 | ParameterType -> External,
|
---|
534 | BlockName -> SMINPUTS,
|
---|
535 | OrderBlock -> 1,
|
---|
536 | Value -> 127.94,
|
---|
537 | InteractionOrder -> {QED,-2},
|
---|
538 | Description -> "Inverse of the EW coupling constant at the Z pole"
|
---|
539 | },
|
---|
540 | Gf == {
|
---|
541 | ParameterType -> External,
|
---|
542 | BlockName -> SMINPUTS,
|
---|
543 | OrderBlock -> 2,
|
---|
544 | Value -> 1.17456*^-5,
|
---|
545 | InteractionOrder -> {QED,2},
|
---|
546 | TeX -> Subscript[G,f],
|
---|
547 | Description -> "Fermi constant"
|
---|
548 | },
|
---|
549 | aS == {
|
---|
550 | ParameterType -> External,
|
---|
551 | BlockName -> SMINPUTS,
|
---|
552 | OrderBlock -> 3,
|
---|
553 | Value -> 0.1184,
|
---|
554 | InteractionOrder -> {QCD,2},
|
---|
555 | TeX -> Subscript[\[Alpha],s],
|
---|
556 | Description -> "Strong coupling constant at the Z pole"
|
---|
557 | },
|
---|
558 | ymdo == {
|
---|
559 | ParameterType -> External,
|
---|
560 | BlockName -> YUKAWA,
|
---|
561 | OrderBlock -> 1,
|
---|
562 | Value -> 5.04*^-3,
|
---|
563 | Description -> "Down Yukawa mass"
|
---|
564 | },
|
---|
565 | ymup == {
|
---|
566 | ParameterType -> External,
|
---|
567 | BlockName -> YUKAWA,
|
---|
568 | OrderBlock -> 2,
|
---|
569 | Value -> 2.55*^-3,
|
---|
570 | Description -> "Up Yukawa mass"
|
---|
571 | },
|
---|
572 | yms == {
|
---|
573 | ParameterType -> External,
|
---|
574 | BlockName -> YUKAWA,
|
---|
575 | OrderBlock -> 3,
|
---|
576 | Value -> 0.101,
|
---|
577 | Description -> "Strange Yukawa mass"
|
---|
578 | },
|
---|
579 | ymc == {
|
---|
580 | ParameterType -> External,
|
---|
581 | BlockName -> YUKAWA,
|
---|
582 | OrderBlock -> 4,
|
---|
583 | Value -> 1.27,
|
---|
584 | Description -> "Charm Yukawa mass"
|
---|
585 | },
|
---|
586 | ymb == {
|
---|
587 | ParameterType -> External,
|
---|
588 | BlockName -> YUKAWA,
|
---|
589 | OrderBlock -> 5,
|
---|
590 | Value -> 4.7,
|
---|
591 | Description -> "Bottom Yukawa mass"
|
---|
592 | },
|
---|
593 | ymt == {
|
---|
594 | ParameterType -> External,
|
---|
595 | BlockName -> YUKAWA,
|
---|
596 | OrderBlock -> 6,
|
---|
597 | Value -> 173.3,
|
---|
598 | Description -> "Top Yukawa mass"
|
---|
599 | },
|
---|
600 | yme == {
|
---|
601 | ParameterType -> External,
|
---|
602 | BlockName -> YUKAWA,
|
---|
603 | OrderBlock -> 11,
|
---|
604 | Value -> 5.11*^-4,
|
---|
605 | Description -> "Electron Yukawa mass"
|
---|
606 | },
|
---|
607 | ymm == {
|
---|
608 | ParameterType -> External,
|
---|
609 | BlockName -> YUKAWA,
|
---|
610 | OrderBlock -> 13,
|
---|
611 | Value -> 0.10566,
|
---|
612 | Description -> "Muon Yukawa mass"
|
---|
613 | },
|
---|
614 | ymtau == {
|
---|
615 | ParameterType -> External,
|
---|
616 | BlockName -> YUKAWA,
|
---|
617 | OrderBlock -> 15,
|
---|
618 | Value -> 1.777,
|
---|
619 | Description -> "Tau Yukawa mass"
|
---|
620 | },
|
---|
621 | cabi == {
|
---|
622 | ParameterType -> External,
|
---|
623 | BlockName -> CKMBLOCK,
|
---|
624 | OrderBlock -> 1,
|
---|
625 | Value -> 0.227591,
|
---|
626 | TeX -> Subscript[\[Theta], c],
|
---|
627 | Description -> "Cabibbo angle"
|
---|
628 | },
|
---|
629 | th13 == {
|
---|
630 | ParameterType -> External,
|
---|
631 | BlockName -> CKMBLOCK,
|
---|
632 | OrderBlock -> 2,
|
---|
633 | Value -> 0.003508,
|
---|
634 | TeX -> Subscript[\[Theta], 13],
|
---|
635 | Description -> "CKM Theta 13"
|
---|
636 | },
|
---|
637 | th23 == {
|
---|
638 | ParameterType -> External,
|
---|
639 | BlockName -> CKMBLOCK,
|
---|
640 | OrderBlock -> 3,
|
---|
641 | Value -> 0.041539,
|
---|
642 | TeX -> Subscript[\[Theta], 23],
|
---|
643 | Description -> "CKM theta 23"
|
---|
644 | },
|
---|
645 | del13 == {
|
---|
646 | ParameterType -> External,
|
---|
647 | BlockName -> CKMBLOCK,
|
---|
648 | OrderBlock -> 4,
|
---|
649 | Value -> 1.20,
|
---|
650 | TeX -> Subscript[\[Delta], 13],
|
---|
651 | Description -> "CKM delta 13"
|
---|
652 | },
|
---|
653 |
|
---|
654 | (* Internal Parameters *)
|
---|
655 | aEW == {
|
---|
656 | ParameterType -> Internal,
|
---|
657 | Value -> 1/aEWM1,
|
---|
658 | InteractionOrder -> {QED,2},
|
---|
659 | TeX -> Subscript[\[Alpha], EW],
|
---|
660 | Description -> "Electroweak coupling contant"
|
---|
661 | },
|
---|
662 | MW == {
|
---|
663 | ParameterType -> Internal,
|
---|
664 | Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
|
---|
665 | TeX -> Subscript[M,W],
|
---|
666 | Description -> "W mass"
|
---|
667 | },
|
---|
668 | MWX == {
|
---|
669 | ParameterType -> Internal,
|
---|
670 | Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
|
---|
671 | TeX -> Subscript[M,W],
|
---|
672 | Description -> "W mass"
|
---|
673 | },
|
---|
674 | MW0 == {
|
---|
675 | ParameterType -> Internal,
|
---|
676 | Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
|
---|
677 | TeX -> Subscript[M,W],
|
---|
678 | Description -> "W mass"
|
---|
679 | },
|
---|
680 | MWT == {
|
---|
681 | ParameterType -> Internal,
|
---|
682 | Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
|
---|
683 | TeX -> Subscript[M,W],
|
---|
684 | Description -> "W mass"
|
---|
685 | },
|
---|
686 | MWA == {
|
---|
687 | ParameterType -> Internal,
|
---|
688 | Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
|
---|
689 | TeX -> Subscript[M,W],
|
---|
690 | Description -> "W mass"
|
---|
691 | },
|
---|
692 | WWX == {
|
---|
693 | ParameterType -> Internal,
|
---|
694 | Value -> WW,
|
---|
695 | TeX -> Subscript[W,W],
|
---|
696 | Description -> "W width"
|
---|
697 | },
|
---|
698 | WW0 == {
|
---|
699 | ParameterType -> Internal,
|
---|
700 | Value -> WW,
|
---|
701 | TeX -> Subscript[W,W],
|
---|
702 | Description -> "W width"
|
---|
703 | },
|
---|
704 | WWT == {
|
---|
705 | ParameterType -> Internal,
|
---|
706 | Value -> WW,
|
---|
707 | TeX -> Subscript[W,W],
|
---|
708 | Description -> "W width"
|
---|
709 | },
|
---|
710 | WWA == {
|
---|
711 | ParameterType -> Internal,
|
---|
712 | Value -> WW,
|
---|
713 | TeX -> Subscript[W,W],
|
---|
714 | Description -> "W width"
|
---|
715 | },
|
---|
716 | sw2 == {
|
---|
717 | ParameterType -> Internal,
|
---|
718 | Value -> 1-(MW/MZ)^2,
|
---|
719 | Description -> "Squared Sin of the Weinberg angle"
|
---|
720 | },
|
---|
721 | ee == {
|
---|
722 | ParameterType -> Internal,
|
---|
723 | Value -> Sqrt[4 Pi aEW],
|
---|
724 | InteractionOrder -> {QED,1},
|
---|
725 | TeX -> e,
|
---|
726 | Description -> "Electric coupling constant"
|
---|
727 | },
|
---|
728 | cw == {
|
---|
729 | ParameterType -> Internal,
|
---|
730 | Value -> Sqrt[1-sw2],
|
---|
731 | TeX -> Subscript[c,w],
|
---|
732 | Description -> "Cosine of the Weinberg angle"
|
---|
733 | },
|
---|
734 | sw == {
|
---|
735 | ParameterType -> Internal,
|
---|
736 | Value -> Sqrt[sw2],
|
---|
737 | TeX -> Subscript[s,w],
|
---|
738 | Description -> "Sine of the Weinberg angle"
|
---|
739 | },
|
---|
740 | gw == {
|
---|
741 | ParameterType -> Internal,
|
---|
742 | Definitions -> {gw->ee/sw},
|
---|
743 | InteractionOrder -> {QED,1},
|
---|
744 | TeX -> Subscript[g,w],
|
---|
745 | Description -> "Weak coupling constant at the Z pole"
|
---|
746 | },
|
---|
747 | g1 == {
|
---|
748 | ParameterType -> Internal,
|
---|
749 | Definitions -> {g1->ee/cw},
|
---|
750 | InteractionOrder -> {QED,1},
|
---|
751 | TeX -> Subscript[g,1],
|
---|
752 | Description -> "U(1)Y coupling constant at the Z pole"
|
---|
753 | },
|
---|
754 | gs == {
|
---|
755 | ParameterType -> Internal,
|
---|
756 | Value -> Sqrt[4 Pi aS],
|
---|
757 | InteractionOrder -> {QCD,1},
|
---|
758 | TeX -> Subscript[g,s],
|
---|
759 | ParameterName -> G,
|
---|
760 | Description -> "Strong coupling constant at the Z pole"
|
---|
761 | },
|
---|
762 | vev == {
|
---|
763 | ParameterType -> Internal,
|
---|
764 | Value -> 2*MW*sw/ee,
|
---|
765 | InteractionOrder -> {QED,-1},
|
---|
766 | Description -> "Higgs vacuum expectation value"
|
---|
767 | },
|
---|
768 | lam == {
|
---|
769 | ParameterType -> Internal,
|
---|
770 | Value -> MH^2/(2*vev^2),
|
---|
771 | InteractionOrder -> {QED, 2},
|
---|
772 | Description -> "Higgs quartic coupling"
|
---|
773 | },
|
---|
774 | muH == {
|
---|
775 | ParameterType -> Internal,
|
---|
776 | Value -> Sqrt[vev^2 lam],
|
---|
777 | TeX -> \[Mu],
|
---|
778 | Description -> "Coefficient of the quadratic piece of the Higgs potential"
|
---|
779 | },
|
---|
780 | yl == {
|
---|
781 | ParameterType -> Internal,
|
---|
782 | Indices -> {Index[Generation], Index[Generation]},
|
---|
783 | Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
|
---|
784 | Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
|
---|
785 | InteractionOrder -> {QED, 1},
|
---|
786 | ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
|
---|
787 | TeX -> Superscript[y, l],
|
---|
788 | Description -> "Lepton Yukawa couplings"
|
---|
789 | },
|
---|
790 | yu == {
|
---|
791 | ParameterType -> Internal,
|
---|
792 | Indices -> {Index[Generation], Index[Generation]},
|
---|
793 | Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
|
---|
794 | Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
|
---|
795 | InteractionOrder -> {QED, 1},
|
---|
796 | ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
|
---|
797 | TeX -> Superscript[y, u],
|
---|
798 | Description -> "Up-type Yukawa couplings"
|
---|
799 | },
|
---|
800 | yd == {
|
---|
801 | ParameterType -> Internal,
|
---|
802 | Indices -> {Index[Generation], Index[Generation]},
|
---|
803 | Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
|
---|
804 | Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
|
---|
805 | InteractionOrder -> {QED, 1},
|
---|
806 | ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
|
---|
807 | TeX -> Superscript[y, d],
|
---|
808 | Description -> "Down-type Yukawa couplings"
|
---|
809 | },
|
---|
810 | (* N. B. : only Cabibbo mixing! *)
|
---|
811 | CKM == {
|
---|
812 | ParameterType -> Internal,
|
---|
813 | Indices -> {Index[Generation], Index[Generation]},
|
---|
814 | Unitary -> True,
|
---|
815 | Value -> {CKM[1,1] -> Cos[cabi] Cos[th13], CKM[1,2] -> Sin[cabi] Cos[th13], CKM[1,3] -> Sin[th13] Exp[-I del13],
|
---|
816 | CKM[2,1] -> -Sin[cabi] Cos[th23] - Cos[cabi] Sin[th23] Sin[th13] Exp[I del13], CKM[2,2] -> Cos[cabi] Cos[th23] - Sin[cabi] Sin[th23] Sin[th13] Exp[I del13], CKM[2,3] -> Sin[th23] Cos[th13],
|
---|
817 | CKM[3,1] -> Sin[cabi] Sin[th23] - Cos[cabi] Cos[th23] Sin[th13] Exp[I del13], CKM[3,2] -> -Cos[cabi] Sin[th23] - Sin[cabi] Cos[th23] Sin[th13] Exp[I del13], CKM[3,3] -> Cos[th23] Cos[th13]},
|
---|
818 | TeX -> Superscript[V,CKM],
|
---|
819 | Description -> "CKM-Matrix"}
|
---|
820 | };
|
---|
821 |
|
---|
822 | (* ************************** *)
|
---|
823 | (* ***** Lagrangian ***** *)
|
---|
824 | (* ************************** *)
|
---|
825 |
|
---|
826 | LWPolarMassBase := MW*MW * W0bar[mu] * (WT[mu] + WA[mu] + WX[mu]) \
|
---|
827 | + MW*MW * WTbar[mu] * (W0[mu] + WA[mu] + WX[mu]) \
|
---|
828 | + MW*MW * WAbar[mu] * (W0[mu] + WT[mu] + WX[mu]) \
|
---|
829 | + MW*MW * WXbar[mu] * (W0[mu] + WT[mu] + WA[mu]) ;
|
---|
830 | LPolarMass := LWPolarMassBase;
|
---|
831 |
|
---|
832 |
|
---|
833 | LGauge := Block[{mu,nu,ii,aa},
|
---|
834 | ExpandIndices[-1/4 FS[B,mu,nu] FS[B,mu,nu] - 1/4 FS[Wi,mu,nu,ii] FS[Wi,mu,nu,ii] - 1/4 FS[G,mu,nu,aa] FS[G,mu,nu,aa], FlavorExpand->SU2W]];
|
---|
835 |
|
---|
836 | LFermions := Block[{mu},
|
---|
837 | ExpandIndices[I*(
|
---|
838 | QLbar.Ga[mu].DC[QL, mu] + LLbar.Ga[mu].DC[LL, mu] + uRbar.Ga[mu].DC[uR, mu] + dRbar.Ga[mu].DC[dR, mu] + lRbar.Ga[mu].DC[lR, mu]),
|
---|
839 | FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
|
---|
840 |
|
---|
841 | LHiggsBase := Block[{ii,mu, feynmangaugerules},
|
---|
842 | feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
|
---|
843 |
|
---|
844 | ExpandIndices[DC[Phibar[ii],mu] DC[Phi[ii],mu] + muH^2 Phibar[ii] Phi[ii] - lam Phibar[ii] Phi[ii] Phibar[jj] Phi[jj], FlavorExpand->{SU2D,SU2W}]/.feynmangaugerules
|
---|
845 | ];
|
---|
846 | LHiggs := Simplify[LHiggsBase - LPolarMass];
|
---|
847 |
|
---|
848 |
|
---|
849 | LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
|
---|
850 | feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
|
---|
851 |
|
---|
852 | yuk = ExpandIndices[
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853 | -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
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854 | yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
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855 | yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
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856 | yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
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857 | yuk+HC[yuk]/.feynmangaugerules
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858 | ];
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859 |
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860 | LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
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861 | (* Pure gauge piece *)
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862 | LGh1 = -ghBbar.del[DC[ghB,mu],mu];
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863 | LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
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864 | LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
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865 |
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866 | (* Scalar pieces: see Peskin pages 739-742 *)
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867 | (* phi1 and phi2 are the real degrees of freedom of GP *)
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868 | (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
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869 | gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
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870 | ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
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871 | generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
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872 | doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
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873 | doublet0 = {0, vev/Sqrt[2]};
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874 | Vectorize[{a_, b_}]:= Simplify[{Sqrt[2] Re[Expand[a]], Sqrt[2] Im[Expand[a]], Sqrt[2] Re[Expand[b]], Sqrt[2] Im[Expand[b]]}/.{Im[_]->0, Re[num_]->num}];
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875 | togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
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876 | LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
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877 |
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878 | ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
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879 |
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880 | LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;
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