1 | * Tree-loop interference *
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2 |
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3 | The new lorentz structures present in the higher dimensional operators of the
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4 | SMEFT can mediate processes at tree-level that only arise at loop-level in the
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5 | SM. The most well know example of these are the H^2 X^\mu\nu X_\mu\nu
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6 | operators, with X being a SM gauge field strength tensor, which mediate several
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7 | Higgs production processes via gluon fusion and/or Higgs decays into SM gauge
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8 | bosons. The mixture of tree/loop level SMEFT contributions with loop-level SM
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9 | diagrams makes extracting the EFT dependence of certain processes more involved
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10 | than usual. The main reason for this is the MadGraph5_aMC@NLO (MG5) has a
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11 | special running mode to compute loop-induced (LI) processes that is only
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12 | launched when no tree-level diagrams are found.
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13 |
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14 | From now, we take the gluon-fusion production of Higgs+1 jet as an example,
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15 | with the goal of obtaining the complete lowest-order SMEFT dependence of the
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16 | (differential) cross section.
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17 |
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18 | >> generate p p > h j ...
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19 |
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20 | The operator coefficient cpG induces this process
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21 | at tree-level, while all other SM/SMEFT contributions come from the usual
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22 | diagrams involving a top quark loop. We therefore have 3 types of contributions
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23 | to the amplitude:
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24 |
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25 | a) A(loop)_SM: LI SM
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26 | b) A(loop)_EFT: LI SMEFT (cpd, cpDC, ctG, ctp, cG)
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27 | c) A(tree)_EFT: tree-level SMEFT (cpG)
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28 |
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29 | From which we would like to extract the quadratic polynomial of the SMEFT
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30 | dependence:
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31 |
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32 | 1) SM contributions: |A(loop)_SM|^2
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33 | 2) linear loop EFT contributions: A(loop)_SM*A(loop)_EFT
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34 | 3) linear tree EFT contributions: A(loop)_SM*A(tree)_EFT
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35 | 4) quadratic loop EFT contributions: |A(loop)_EFT|^2
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36 | 5) quadratic tree EFT contributions:|A(tree)_EFT|^2
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37 | 6) quadratic tree-loop EFT contributions: A(loop)_EFT*A(tree)_EFT
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38 |
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39 | Although there is a mixture of tree and loop-level contributions, they are all
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40 | strictly speaking Leading Order (LO), since no contributions exist at a lower
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41 | order for a given operator/the SM. One therefore should not use the full NLO
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42 | capabilities of MG5 as these would generally include further, O(gs^2)
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43 | corrections to the LO numbers. Therefore, a combination of tree-level and LI
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44 | modes is needed to obtain the required numbers.
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45 |
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46 | Pure tree-level (5) and loop-level contributions (1, 2 & 4) can be obtained by
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47 | running the respective LO and LI modes of MG5, as usual.
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48 |
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49 | As previously mentioned, in order to obtain the LI diagrams, MG5 must not find
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50 | any tree-level ones. This can be achieved by restricting out the cpG
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51 | coefficient. In this example, we will assume the existence of two restriction
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52 | cards in the UFO:
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53 |
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54 | restrict_with_cpg.dat
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55 | restrict_without_cpg.dat
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56 |
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57 | with cpG set to non-zero and zero values, respectively.
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58 |
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59 | Loading the first model and generating the default command, requesting some EFT
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60 | contributions (NP=2) will generate only the tree-level cpG diagrams, and can be
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61 | used to compute contribution 5).
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62 |
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63 | >> import model SMEFTatNLO-with_cpg
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64 | >> generate p p > h j NP=2
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65 |
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66 | Loading the second model will launch LI mode and generate the loop diagrams,
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67 | since no tree-level contribution is now possible. This can be used to compute
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68 | contributions 1, 2 & 4 (see different possible commands controlling the NP
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69 | order below).
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70 |
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71 | >> import model SMEFTatNLO-without_cpg
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72 | >> generate p p > h j NP=0 QCD=1 QED=1 [QCD] # 1) SM only
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73 | >> generate p p > h j NP=2 QCD=1 QED=1 [QCD] # 1) + 2) + 4) SM & EFT
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74 | >> generate p p > h j NP^2==2 QCD=1 QED=1 [QCD] # 2) EFT interference only
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75 | >> generate p p > h j NP^2==4 QCD=1 QED=1 [QCD] # 4) EFT squared only
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76 |
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77 | Contributions 3) & 6) however, involve the interference of a tree & loop-level
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78 | diagrams, which is not possible "out of the box" using MG5. The best known
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79 | method to extract these numbers is via the reweighting feature of
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80 | MG5 (https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/Reweight). First, a
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81 | single (somewhat arbitrary) baseline sample is generated and new weights are
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82 | computed for each event corresponding to a matrix-element according to a
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83 | user-specified MG5 generation command. The process is steered by the
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84 | reweight_card.dat file in the Cards/ directory. I this example, we will
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85 | generate a baseline sample with cpG=1 as it is the fastest to generate.
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86 | For the baseline sample, we proceed as above & generate some events:
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87 |
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88 | >> import model SMEFTatNLO-with_cpg
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89 | >> generate p p > h j NP=2
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90 | >> output hj_reweight_tutorial
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91 | >> launch hj_reweight
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92 |
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93 | Then depending on the desired contribution, the reweight_card.dat can be
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94 | modified as follows:
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95 |
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96 | Contribution 3), SM loop x cpG interference:
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97 | # specify new matrix element command
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98 | >> change process p p > h j NP=2 QCD=1 QED=1 QCD^2==4 NP^2==2 [virt=QCD] # SM x cphiG
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99 | # launch reweighting
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100 | >> launch --rwgt_name=SMxcpG
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101 | # set parameters
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102 | >> set cpG 1. # set relevant operator to 1
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103 | >> set ctp 0. # set other operators to 0
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104 | >> set ctG 0.
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105 | ...
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106 |
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107 | In the change process command, note the very specific coupling order
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108 | specifications that select the tree-level SMEFT and loop-level SM as well as
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109 | the the [virt = QCD], which forces only the finite part of loops to be
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110 | computed. The events can then be reweighted by launching bin/madevent and
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111 | running the command
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112 |
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113 | >> reweight hj_reweight
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114 |
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115 | Contribution 6), EFT loop x cpG interference:
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116 | >> change process p p > h j NP=2 QCD=1 QED=1 QCD^2==4 NP^2==4 [virt=QCD] # EFT x cphiG including square
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117 |
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118 | # compute one quadratic EFT tree-loop interference contribution
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119 | >> launch --rwgt_name=cpGxctG
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120 | >> set cpG 1. # set first relevant operator to 1
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121 | >> set ctG 1. # set second relevant operator to 1
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122 | >> set ctp 0. # set other operators to 0
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123 | ...
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124 |
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125 | # compute another
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126 | >> launch --rwgt_name=cpGxctp
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127 | >> set cpG 1. # set first relevant operator to 1
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128 | >> set ctp 1. # set second relevant operator to 1
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129 | >> set ctG 0. # set other operators to 0
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130 | ...
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131 |
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132 | This particular matrix element command unavoidably includes quadratic
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133 | contributions from cpG that you donât want. These must be removed a posteriori
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134 | by generating an additional reweighting with cpG = 1 and all others = 0. The
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135 | weight computed here must be subtracted from all other weights in this run to
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136 | to extract the pure interference between cpG and other coefficients.
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137 |
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138 | # compute pure cpG weight to subtract from all others
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139 | >> launch --rwgt_name=cpG_SUB
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140 | >> set cpG 1. # set cpG to 1
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141 | >> set ctp 0. # set all others to zero
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142 | ...
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143 |
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144 | Tips & comments:
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145 | 0) Always specify coupling orders as much as possible & use "display diagrams"
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146 | to verify that MG5 has produced the expected output!
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147 |
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148 | 1) Reweighting has a well known limitation that the phase-space population of
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149 | the event sample is tied to the baseline run. A judicious choice can be made to
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150 | avoid reweighting to parameter points that populate significantly different
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151 | regions, to avoid inducing large MC uncertainties from a large spread of
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152 | weights in the reweighted sample. Using a tree-level cpG baseline, for example,
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153 | may not be ideal for certain processes such as p p > h h as the cross section
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154 | falls to zero around the threshold much faster than the SM prediction.
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155 |
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156 | 2) Each 'change X' command can only be used once in the reweight_card.dat. If
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157 | multiple such commands are needed, they must be split into separate reweighting
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158 | runs.
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159 |
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160 | 3) In principle, reweighting can be used to evaluate all components of the
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161 | quadratic polynomial EFT dependence. In order to obtain the pure loop
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162 | contributions, one should additionally specify:
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163 |
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164 | >> change model restrict_without_cpg
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165 | >> change process p p > h j ...
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166 |
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167 | to swap to the model without cpG and only generate the LI contributions by
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168 | default. The commands for computing contributions 1), 2) & 4) can then be used
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169 | as shown above (bear in mind point 2)).
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170 |
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171 | 4) In principle, this procedure can be combined with jet merging algorithms to
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172 | obtain samples with different jet multiplicities. However, we have not testing
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173 | this feature and therefore cannot provide explicit support.
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174 |
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175 | 5) Other reweight command examples for extracting SM x cpG interferences:
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176 | Higgs + 0j:
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177 | >> change process p p > h NP=2 QCD=0 QED=1 QCD^2==2 NP^2==2 [virt=QCD]
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178 | Higgs + 2j
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179 | >> change process p p > h j j NP=2 QCD=2 QED=1 QCD^2==6 NP^2==2 [virt=QCD]
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180 | di-Higgs
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181 | >> change process p p > h h NP=2 QCD=0 QED=2 QCD^2==2 NP^2==2 [virt=QCD]
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